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Reliability of nuclear structures
Nuclear Engineering and Design 114 (1989) 235-245 North-Holland, Amsterdam
RELIABILITY
OF NUCLEAR
STRUCTURES
235
*
G.I. SCHUELLER Institut fur Mechanik, Universitiit Innsbruck, A-6020 Innsbruck, Austria Received 22 July 1988
The reliability of nuclear structures relates directly to the reliability of mechanical and electronic systems. Nuclear structures may be viewed as part of operational system as well as part of standby safety system. Except for proof loading, most of the load types, seen by nuclear structures, such as containments, primary piping, etc. during their design life, are random in time and space. Structural resistances, governed mainly by the respective material properties show also random characteristics. Methods of structural reliability, as discussed here, provide the means to process rationally the available information. Hence, they are an indispensable tool within the frame work of probabilistic risk analyses (PRA's). The probabilistic analysis is exemplified by determining the reliability of a containment structure sited in an area with strong seismicity.
1. Introduction
W i t h i n the first generic risk study of nuclear power p l a n t s the aspect of the reliability estimation of the structures involved, such as the c o n t a i n m e n t , p r i m a r y piping, etc, received considerable less a t t e n t i o n t h a n the reliability analysis of the mechanical as well as electronic systems [1]. In this context it is interesting to note that the failure probabilities of the p r i m a r y piping (for the large a n d the small L O C A respectively) were estim a t e d from sample statistics which also included d a t a from n o n nuclear power plants. The failure probabilities of the c o n t a i n m e n t due to L O C A for example were d e t e r m i n e d b y i n t r o d u c i n g n o r m a l l y distributed " e r r o r b o u n d s " on the estimated expected m a x i m u m pressure. F o r e a r t h q u a k e analysis the investigators did not leave deterministic g r o u n d s at all. This procedure was quite surprising since d u r i n g S M i R T ' 7 3 a n u m b e r of presentations were given which suggest m u c h more sophisticated ways to calculate structural reliabilities, i.e. failure probabilities of pipings [2], pressure vessels [3] a n d c o n t a i n m e n t s [4], etc. Subsequently, these m e t h o d s were developed further (see e.g. [5,6,7]) a n d then utilized in risk studies [8], [9]. A r o u n d this time the U S N R C initiated the S S M R P - mainly carried out b y L L N L -
which c o n c e n t r a t e d on p l a n t specific studies (see e.g. [10]). This effort was then c o m p l e m e n t e d by the Load C o m b i n a t i o n Program, carried out by B N L (see e.g. [11]). Both in Europe a n d the U S quite sophisticated reliability studies o n c o n t a i n m e n t a n d pipings followed a m o n g m a n y others, see e.g. [12-15]. In this context it should be mentioned, that Jaeger [16] stressed already quite at an early stage the i m p o r t a n c e of structural reliability studies in context with reactor safety a n d initiated a series of S M i R T Post Conference Seminars on this subject [17-19]. As within a r a t h e r short period of time, the interest in the subject grew considerably, the Scientific C o m m i t t e e of S M i R T decided to devote a new Division ( M / M 1 ) to this subject [20-22]. N a t u rally, this brief historical review can be b y n o m e a n s exhaustive. Limited space does not allow to list a n d discuss all activities which c o n t r i b u t e d to the developm e n t of the field. Needless to say that the T h r e e Mile Island incident as well as the Tschernobyl accident underlined the significance of such studies.
2. Reliability analysis 2.1. Concept
* Principal Division Lecture, Division M, 9th International Conference on Structural Mechanics in Reactor Technology (SMiRT-9), Lausanne, Switzerland, Aug. 17-21 1987. 0029-5493/89/$03.50
The basic concept of classical reliability analysis which will be outlined briefly in the following (for a
© 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.
G.L SchuFller / Reliability of nuclear structures
236
more detailed t r e a t m e n t see e.g. [23]) - recognized the fact, that most variables which are involved in determining the load processes as well as the structural resistance, are of r a n d o m characteristics. The relation between these variables might be defined for example by a limit state surface such as g(X) = g(X,, X2.... X,)=0
(1)
where X i represent the r a n d o m variables defining structural resistances a n d loads respectively. The limit state is reached when the condition g ( X ) < 0 is met, which is associated with the notion of failure. Therefore, failure must always be seen in context with the limit state. It can be defined by the analyst or the designer for example as the exceedance event of a particular deflection (serviceability limit sate), the f o r m a t i o n of hinges, or the f o r m a t i o n of a chain of hinges describing a collapse mechanism. D e p e n d i n g on the particular problem, these limit states can b e considered either individually or combined. The probability that a particular limit sate will be reached, i.e. the failure probability may be expressed as:
p f = S D / x ( i l ) dll,
(2)
w h e r e .fx('q) is the j o i n t d e n s i t y f u n c t i o n of X1, X 2 , . . . , X~ and Df is the d o m a i n of failure in the
n-dimensional space. A schematic sketch for two variables is shown in fig. 1. The variables involved in eq. (2) m a y certainly be correlated. C o m p u t a t i o n a l m e t h o d s to solve eq. (2) are described in detail e.g. in [23], a brief discussion is given in the following section. T h e failure probability as defined by eq. (2) is obviously a time invariant expression. M a n y cases, however, such as the earthquake analysis, impact analysis, etc., are certainly to be treated as a time variant problem. This includes o n one h a n d the occurrence p r o b a b i l i t y of the load event and on the other h a n d the conditional distribution with respect to time a n d space. This m a y be exemplified by referring to the e a r t h q u a k e load case. T h e earthquake event occurrence m a y b e m o d e l e d by so called stochastic processes. In particular, the Poisson, or modified Poisson processes, are utilized for this purpose. The event itself is also a (small scale) time d e p e n d e n t , n o n s t a t i o n a r y process with r a n d o m l y distributed amplitudes, frequencies, lengths, etc., a n d a spacial distribution. These e a r t h q u a k e strong m o t i o n records m a y be described conveniently by so called evolutionary spectra. In addition to the consideration of the time variance of the load process, the structural resistance m a y also be considered as a time d e p e n d e n t process. The effects which c o n t r i b u t e to this decrease in structural resistance m a y be either due to corrosion, crack propagation,
5"1, tx(~)
~
~
~ , ' ~
i,\<; ",D"*3t ', t : < " / .
':.:2)'/:(..
~tx)<0
",
S<~, . ,d ~ <
~
glgl=O
Fig. 1. Failure probability - Schematic sketch (two dimensional case).
237
G.L Schu~ller / Refiability of nuclear structures
aging, etc., or a combination of it. For this case the reliability may be expressed as
TT(t)=exp{-- fothT(t' )
dt'},
x2
K2~ D" ~/
f(x)
~.~l,I
(3)
where
hT(t ) =fT(t)/(1 -- F T ( t ) )
(4)
is called the hazard rate and (5b).
fT(t)
mx 1
is defined by eq. X1D.X2D ¢~ntir~t~ ~ tl~ ~sign Ix~nt
2.2. Computational methods
where N is the number of simulations and sampling distribution, and 1
(5a)
hv(Xi)
the
for X e failure domain, otherwise.
In addition, the statistical error of this estimate can be calculated without difficulties [26]. For time variant structural resistance, the above equation reads
~=~
/x(X,(t))
fT= -N1 i= l[g(Xi(t))
Fig. 2. Computational procedure to calculate failure probabilities (two dimensional case).
can then be represented either in the time or in the frequency range, respectively.
2.3.2. External events
1 N e l = N i=~=ll[g(xi) _<0] hr(Xi),fx(X')
0
--g(X__l=O. t = t 3
Xl
Various methods to solve eq. (2) both computationally accurate and efficient have been suggested so far. For a critical discussion of these methods it is referred to e.g. [24], [25]. One particular method, which satisfies these criteria sufficiently well is based on importance sampling procedures. The failure probability can then be computed by the following expression
l[g(Xi)<-O]=
~"
(5b)
where g(Xi(t)) changes with each time interval or load cycle (see fig. 2). As pointed out in section 2.4 below, this change may be modeled by linear or nonlinear fracture mechanical considerations. Both the time variant and invariant cases respectively are included in a computer software package, called ISPUD [26].
2.3. Load analysis 2.3.1. General To model, i.e. predict rare (extreme) events, generally Poisson type processes are utilized. In most cases the parameters i.e. expected frequencies of occurrence of intemal or external man made or natural hazards can be estimated from observations. The load event itself
Nuclear power plant structures must survive external events, such as man made or natural hazards (for example air plane impact, external pressure wave or earthquakes) with a very high probability. The target level may be specified as design requirements. Both types of hazards are considered to be site dependent. While in the case of aircraft impact occurrence on NPP's a statistical analysis of airplane crashes is instrumental [27] a rather sophisticated model building, which is based on geophysical considerations including aspects of wave propagation, it to be applied in the earthquake load case. More details on the latter type of load analysis will be discussed below.
2.3.3. Internal events Except for test loading, the occurrence of internal load events are either due to operation or malfunction of the systems, i.e. the operational and the accidental loading conditions respectively. An important example of the second type is the LOCA (loss of coolant accident) which occurs when the primary piping falls with a break of a certain size (100 cm2 and 1000 cm2 for small and large LOCA respectively is generally postulated). The expected occurrence rates of internal accidental load events are estimated either directly from observations or they have to be calculated by utilizing either fault and event trees to predict the (non observed) occurrence rate of a larger events from a sequence of (observed) smaller events, or by utilizing load spectra along with fracture mechanical methods.
238
G.I. Schu~ller / Reliability of nuclear structures
2.4. Structural resistance The stochastic nature of the structural resistance is mainly due to the random characteristics of the material properties. This applies to the time invariant yield or compressive concrete strength as well as to the time variant corrosion and crack propagation under cyclic loading which reduce the structural resistance. In this context it should be pointed out that the quality of construction, which may be expressed in terms of initial crack distributions, also plays a very important role. Structural reliability considerations in context with quality control provide also a quantitative basis for the decision whether to utilize high strength (brittle) or low strength (ductile) types of steel [28].
2.5. Structural response analysis The structures of NPP's to be analyzed within a reliability analysis are generally discretized by utilizing finite elements. This applies also to the soil on which the entire structure is based. Depending on the properties of the loading, either l i n e a r / n o n l i n e a r static or dynamic structural analyses have to be carried out. In the linear range, the stochastic dynamic response analysis is performed most conveniently in the frequency range, i.e. by applying power spectral analysis. For nonlinear dynamic analysis, time domain calculations for a number of samples are still required. At the present stage, the application of equivalent linearization techniques are still confined to structures with a small number of DOF's. Many site conditions also require nonlinear soil-structure interaction analysis.
CONTROl. A/~I) S&eltTV ~ S (SZNSOP.S.L~CI
PIPES. W~J. pUMPS, HatJIGRIIS
ETC. CONTAINMENT. PP.I~SlIRg LARGI~ PIpZS.LeTC.
Fig. 3. Interaction between systems. Moreover, as mentioned above, in case of a LOCA the load depends on the size of the pipe leak or break. The failure probabilities of PP as well as containments under large and small LOCA's are already discussed in some detail in [15], [30], [13], [31]. Hence, as an additional example, in the following, the reliability analysis of a containment structure located in a seismic active area is shown below. Note, that the outlined procedure should only serve as a guideline w.r. to how such a procedure may be developed.
3.2. Containment structure under seismic loading
3. Application to nuclear structures
3.2.1. General In seismic inactive areas, such as the F.R. Germany, some simplified procedures with respect to the seismic risk analysis are acceptable (see e.g. [13], [31]). However, in seismic active zones, more sophisticated procedures are required. Needless to say that there is a strong interrelationship between seismic risk and structural analysis respectively. This statement implies, that simplifying assumptions made in the seismic risk analysis part may significantly affect the structural part. Hence the level of sophistication of both analyses should be in balance.
3.1. Interaction between structural and systems reliability
3.2.2. Seismic risk model
Among the various structures and components of a NPP, the RPV, the containment and the PP structures received most attention from structural reliability analysts. When calculating the reliability of nuclear structures, it is most important to realize, that the structure under consideration is part of the entire system (fig. 3). Functionally, a structure may be considered from two points of view, namely, as a part of the operational system and as a part of standby safety system [29]. In fig. 4 it is shown clearly, that loads, to which the PP may be exposed to can be determined from the operational parameters as well as from the function - - or malfunction - of the safety system.
The seismic risk model has to be based on a time variant reliability function, i.e. by (see e.g. [32]) LT(t ) = fRLvln(t)fR(r)
dr
(6)
where L x ( t ) defines the probability that the structure survives the seismic hazard within a time interval (0, t) and f n ( t ) the P D F of the structural resistance. L x l n defines the survival probability conditioned on a particular realization of the structural resistance r = R: LTin(t )=
Y'. p ~ ( 1 - - F T i n ) k, k=O
(7)
G.I. Schuie'ller / Reliability of nuclear structures SAFETY SYSTEM
[1 I [ LOADTRI~E [ OPERATIONAL[ [
I TRANSIENT
I) ~
I IL°SS Or
OPERATIONAL
I
LEA.'(
LOADS IS l ^ecmer~r
~ II II
LOAD~
P
I
I I,.
~
1
r I
-
_"l'
~ l
239
&CCIDENT~:~LASSES
Sl
K1
K2
YE,S
NO lI ~ ; I ' I
1
~rlRll ~'~1"II NAL CHARACTE RIb'TI C S i
(KAFKA 19821
Fig. 4. Structure as part of an operational system [29].
where k is the number of the occurring load events and Fw(t ) the time variant failure probability, i.e. the complement to the survival probabihty
is a measure of its variance. Its relation to the PSD (Power Spectral Density) GA(~o) is a RMS =
FT( t) = I -- L T ( t ).
fo°°GA (~o) doo
(10)
(8)
AS mentioned above, for modeling the earthquake event occurrence the simple Poisson process proves to be applicable, hence Pk may be modeled by
where ~0 defines the circular frequency (2qrf[Hz]). With this information at hand, the conditional failure probability, as utilized in eq. (7) can be defined:
F.rl ~ = /Apflk.AfA( a ) d a , ( v t ) k e -°t, pk(k)
=
k!
k=0,1,
oo,
(11)
(9)
....
where u is the mean rate of event occurrence to be estimated from observed data. It is well known that this process is based on the assumption, that the occurrence in adjacent time intervals are independent. This, however, does not reflect physical reality (see e.g. [33]). Nevertheless, keeping in mind the present knowledge of the seismic process, which is still quite insufficient, this model may still considered to be appropriate. Note, that the conditional probability of failure F-rl R is defined as the probability that the structural response exceeds the realization of the resistance during an earthquake occurrence. The earthquake event itself, which is described by strong motion records, and hence the structural response, may be defined by a continuous stochastic process. Depending on the type of limit state condition to be analyzed, general random vibration theory may be applied. This implies the assumptions that the structural behavior is linear and the response is a weakly stationary (Gaussian) process. Both assumptions do not apply exactly. However, for simplifying reasons they are introduced, nevertheless. The earthquake process, generally defined it terms of ground acceleration a is characterized by its R M S (Root Mean Square) value, which
where fA(a) is the P D F of the earthquake acceleration at a particular site. Naturally this distribution, depends strongly on the knowledge on seismic, tectonic and geological parameters of the ~articular region, i.e. the site under investigation. Due to lack of sufficient information, the (maximum) ground accelerations are generally determined by empirical attenuation laws of the following type a = b I e b2M A -b3,
(12)
where b 1, b2, b 3 are constants to be determined by regression analysis based on observed events, M is the Richter magnitude and zl the hypocentral distance. In most cases detailed information on the tectonic process, i.e. fault mechanisms the likely fault depth (seismic zone) is scarce and consequently the distribution of future faults can only be described with considerable uncertainties. Hence, it is most advantageous to assume either a homogeneous seismic region - or if information permits - several subregions Z. N o w the probability of the acceleration to exceed a certain value at a particular site may be defined as [34]:
P [ A > a ] = ~_,P[A > a I Z = z ] p z ( z
).
(13)
Z
The parameters of this equation are defined in the
240
G.L Schugller / Reliabifity of nuclear structures
Appendix A. The shape of the suitable power spectrum PSD of the ground acceleration could be determined accurately if the necessary data, i.e. time histories were available. Needless to say that this is generally not the case. Therefore, an artificial - usually the Kanai-Tajimi - spectrum is utilized for this purpose:
GA(°~) G°[1-(W/Wg)] + 4~g(w/wg) 22
2
(14)
2'
where wg and ~s denote the central frequency and the band width respectively. On one hand these parameters are adjusted to the geological information available and on the other hand they should be adjusted to the structural eigenfrequencies. The latter ensures a conservative approach. The third parameter, Go , which scales the spectrum, may be related to the RMS-value of the acceleration (see eq. (10)). As attenuation laws for RMS-values are generally not available the establishment of a relation between R M S - and the maximum value of the acceleration is to be sought. Theoretical considerations obviously do not provide such bounds, however, in practice the maximum acceleration is generally bounded by area x ---- 2 - 3 aRM s (see e.g. [35]). 3.2.3. Structural refiability analysis This analysis refers to the assumption made in the section above as well as to the relations defined by eq. (6) to eq. (9). Since linear structural properties are utilized, the assumptions of stationarity and normality of the seismic input, also holds for the structural response process. Hence, the P D F of the load effect consisting of the value of the dead load and the standard deviation of the earthquake loading - is fully determined. The latter may be obtained by integrating the PSD of the response, G s ( ~ ), i.e. by
,-, Applying modal analysis, the standard deviation may be expressed by the modal variances Os2, and covariances o2, j , respectively:
OS =
E t=l
2 2 2 + ~iLi°s,ii
~, ~b,~bjLiLjo~,i j
,
(16)
i=1 J=l
where ~b, is the modal shape of the displacement and load effect respectively and L, is the load participation
factor (as defined by eq. (18) below) - which, according to eq. (15) may be obtained by the auto- and cross-spectral densities G s , , ( w ) and Gs, jj(w ). The cross spectral density for the modal shapes i and j is =
OA
(17)
where /4i is the transfer function of the i-th mode, H ~ the complex conjugate and GA the PSD of the ground acceleration as defined by eq. (14). The load participation factors are defined by
,TM = *TM,--. '
(18)
where g~i is the eigenvector of the mode shape and M the mass matrix. To illustrate the analysis a numerical example is carried out in the Appendix B.
4. Conclusions It has been shown that reliability analyses of nuclear structures are an indispensable tool within the context of PRA of NPP's. The probabilistic analysis processes the maximum on input information on structural loads and resistance available and hence also provides additional information over traditional, deterministic analysis in terms of failure probabilities and reliabilities, respectively. Only todays computational possibilities allow such an increase in processing information. This paper also attempts to give an overview over the developments of structural reliability applied to nuclear structures. A m o n g various examples, for which it is referred to the literature, one particular aspect, i.e. the reliability analysis of a containment structure under earthquake loading is discussed in some detail in order to exemplify the procedure. Finally it is shown that the analysis allows a rational treatment of all uncertainties involved.
Appendix A. P D F of ground acceleration
Following for example [34] the probability that the acceleration A exceeds certain value at a particular site may be defined by P [ A > a] = ~ _ , P [ A > a [ Z = z] p z ( z ) , z
(13)
G.L Schu#ller / Reliability of nuclearstructures
Appendix B. Numerical example of reliability analysis of a containment under seismic excitation
where Z defines a particular subregion
P[A > a l Z l = f~fyP[A > al Z, Rlf,~(x, y ) dx dy, x, y ~ Z
(A1)
and
P[A>alZ,
241
B.1. General remarks The theoretical developments as outlined in section 3.2 is now exemplified by applying it to a particular case study [34], [36]. For this purpose the analysis is based on data and considerations as shown below.
A ] = P [ b , e b ~ M A - b ' > a l Z, A] B. 2. Evaluation of the probability distribution of the maximum horizontal ground acceleration (A2)
In eq. (13) the total probability from all subregions is established, where p z ( z ) is the probability that an event will occur in a particular subregion Z. It is well known that the occurrence rate of earthquakes may be determined by log N = a - bm,
(A3)
where N is the number of earthquakes (per year and km 2) with a magnitude equal or greater than m; a and b are constants to be taken from earthquake catalogs. The P D F of the magnitude is
N(m,) -N(m) F ~ ( m ) = N(mi ) _ N ( m u )
,
(A4)
where m~ and m I are - depending on the seismic region - the geophysically possible largest and smallest magnitude respectively. Furthermore the probability of occurrence of earthquakes in a particular zone may be expressed by
Within the contact of a geological study a model for the tectonic process was established which resulted in the estimation of a seismic zone (Fig. B.1). The depth of the source varies from about 10-30 km to about 100 km, the attenuation law for the m a x i m u m horizontal ground acceleration could be deduced from seismic intensity maps which were based on seven earthquake occurrences with a range of magnitudes from 5.8-7.25, assuming source depths of 10 and 20 km respectively. Fig. B.2 shows the attenuation law of the region. Based on a geological study the upper limits of magnitudes are determined, and consequently four different seismic regions proposed with maximum magnitudes 5.5, 6.5, 7.5 and 8.0 respectively. Their orientation depends on the respective thrust mechanism. Furthermore, two different frequency distributions of the earthquake occurrence are given as shown in fig. B.3. The whole seismic region may be subdivided into eight subregions as shown in fig. B.4.. One can observe unrealistic discontinuities between the seismic activities of such regions. It is quite natural to enlarge the more active subregions (see dashed
z
N ( z ) f~fyh(X, y ) dx dy pz(z)=
, Y',N(z) f f h ( x , "x"y Z
x, y ~ Z ,
0
y) dx dy
I
100
50
D I S T A N C E IN K I L O M E T E R S 150
tU
(A5)
o
50 MOHO
where h(x, y) is the frequency of earthquake occurrence at the location (x, y). The P D F of an earthquake occurrence at a certain distance A is
~(x, y)=
h(x, y)
fh(x,
y ) d x dy
200
(A6)
il~
100 _a
Discontinuity
----•
MOHO
liiiiiil~i~
Oceanic Basement
Data
Fig. B.1. Seismic mechanism.
250 eP
242
G.L SchuUIler / Reliability of nuclear structures Y
a [cm/s2l
i I I
1200
I i
I |
Based on Geological Study 800
II
/ 8.0
i
1
I1
I
~85 I
6.5
o
I /f-
L.-t=._
II
I SiTE
_~./-~ I l~.sff"
t
I
IOO.O
i
;
200.0
[kml
~
115M • I
"%~ I
Fig. B.2. Attenuation law for the m a x i m u m ground acceleration in magnitudes (Source depth h = 10 km),
YL
85
er~n~
lines in fig. B.3.). For this case the P D F ' s of maximum ground acceleration are shown in fig. B.5.
20cm/s z
--~min ~ 50~m/s2
~-1~-" --,~_ __ ~
* -120
I
B,3. Determination of the PSD of the horizontal ground acceleration As there is no advanced geological information available for the site, the utilization of a sophisticated power spectrum is not feasable. Measurements of microtremors indicate a predominant period of 0.6 seconds. Based on the soil profile (see fig. B6) which provides information to a depth of 35 m, but with no details on bedrock formation, one can estimate the upper limit of
I
the lowest resonance frequency to be about 3 Hz. This is valid for a S-wave - vertical incident - by assuming a layer over a half space, where the layer thickness is 35 m and the shear velocity 400 m / s . It is quite obvious that under these circumstances the PSD of the horizontal ground acceleration can not be estimated i.e. fitted with
log fA(a)
32
_.I,~I!__L~_I
Fig. B.4. Seismic provinces according to the attenuation law (of fig. B.2) and a lower bound of m a x i m u m acceleration of 20 c m / s 2.
I i--I I 11~-~-I~-~IJ,I , ~ _
-10
! 1
,~
Fig. B.3. Frequency distribution (h(x, y)) of earthquake occurrences.
100
300
500
a [ c m / s 2]
Fig. B.5. Conditional density function of the m a x i m u m ground acceleration for the seismic region (with reference to one event).
243
G.I. Schu@ller / Reliability of nuclear structures
r--I --i I
I
I
r-7 i I IL_J
I
I
I
lO
I
I
I
20
30
35
Depth
---
[m]
Fig. B.6. Typical soil profile of the site.
high confidence. In order to obtain at least a conservative approach, it is suggested to take into account the dynamical properties of the N P P structure. These properties may be described by its eigenfrequencies in context with the participation factors as shown in fig. B.7. The parameters of the Kanai spectrum are chosen such that the first two eigenfrequencies of the structures which reveal the largest participation factors are enveloped by the peak of the power spectrum. This leads to a Kanai frequency of 2 Hz which, incidentally, is in agreement with the information provided by the geological study. Furthermore, in order to concentrate on one hand the seismic energy around the first two eigenfrequencies of the structure and on the other hand not to select a too peaked spectrum - which in fact would not reflect the geological data at all - the Kanai damping value ~g is chosen to be 0.5.
ture under consideration. This is mainly due to the transparent relationship between load and load effects (see eq. (15)-(18)). Load participation factors are a first but fundamental information on the dynamical sensitivity of the structure. Hence a conservative approach to define the loading function is feasible or, - for cases where a definite description of the loading function is available - the structure may be optimized for minim u m load effects. In the present study (see also fig. B.7) a conservative load spectrum is chosen. The dynamic properties of the sample structure - a containment are listed in table B1. The dominance of the first mode shapes of the loading, which may be concluded from the modal variances, is a direct consequence of the choice of the load spectrum.
B. 4. Structural analysis
A cross section located in the transition between the spherical dome and the cylindrical shell of the reinforced concrete shielding is analyzed as an example (see No. 1 of fig. B.8). Due to the additional design against aircraft impact a disproportional large wall thickness of
The modal analysis is most advantageously applied to assess the dynamical behavior of the particular struc-
B.5. Failure probability of the structure
G(f)
2,0
/
Kanai - Spectrum
Table B1 Dynamic properties of NPP structures under earthquake excitation Mode
Natural frequency [Hz]
Damping [%]
Load participation factor
Modal variances 10 -6 [m2]
1 2 3 4 5
1.06 1.94 5.03 5.73 8.78
15 15 7.5 2.4 7.1
331 435 181 3 66
30.334 8.860 63 51 5
1, o
o,o
~o
4.0
6,o
8,0
f IHz]
Fig. B.7. Comparison between load participation factors (relative values) and load spectrum.
244
G.L Schu~ller / Reliability of nuclear structures log PflE,A
-10
-11
-
I 100
I 300
1
I,
a [ c m / s 21
500
Fig. B.9b. Conditional failure probabilities in the vertical cross section No. 1.
Fig. B.8. Sample structure containment.
Table B2 Failure probabilities TT(t ) of cross section for a design life of 40 years Horizontal Vertical Earthquake occurrence rate v events per year
< 1.0 × 10-10 8.6 × 10-6
2.0 m is required. The b e n d i n g reinforcement, however, is relatively small, which leads to the fact that the failure of the b e n d i n g r e i n f o r c e m e n t is the expected failure mechanism. Hence, the r a n d o m variable of significance is the axial force of the reinforcement, which is modeled b y a n o r m a l distribution with a coefficient of variation of 0.055 a n d a m e a n value of 602 N / m m 2. The c o n d i t i o n a l failure probability, given the earthquake event E is s h o w n in fig. B.9, for a vertical a n d horizontal cross section respectively. F o r a design life of 40 years the failure p r o b a b i l i t y F ( t ) = 1 - L ( t ) is shown in table B2. N o t e that the failure probabilities of this critical cross section are well below those values which are generally accepted in codified design.
Acknowledgement Part of this overview p a p e r is b a s e d o n a j o i n t research with S i e m e n s - K W U , Offenbach. Special t h a n k s are due to N.J. Krutzik for his support a n d c o n t i n u o u s interest in the work. R.J. Scherer's (University of Karlsruhe) help, particularly in carrying out the numerical example, is also deeply appreciated.
28.7
Iog PfiE,A
References
-15 ~ -20 -
-25 -
-30-
-35 I
i
I
100
300
500
w,
a [cm/s2l
Fig. B.9a. Conditional failure probabilities of the horizontal cross section No. 1.
[1] Reactor Safety Study, Wash. 1400, US NRC, Washington DC (1975). [2] A.M. Freudenthal, Reliability of reactor components and systems subject to fatigue and creep, Nucl. Engrg. Des. 28, No. 2 (Sept. 1974) 169-217. [3] P.E. Becher and A. Pedersen, Application of statistical linear elastic fracture mechanics to pressure vessel reliability analysis, 2nd Int. Conf. on Struct. Mech. in React. Techn., Berlin, 10-14 Sept., Paper M 6 / 4 (1973). [4] G.I. Schu~ller, On the structural reliabihty of reactor safety containments, Nucl. Engrg. Des. 27 (1974) 426-433.
G.L Schu~ller / Reliability of nuclear structures [5] A.B. Lidiard and M. Williams, A simplified analysis of pressure vessel reliability, J. Br. Nucl. Energy Soc. 16, No. 3 (July, 1977) 207-223. [6] G.I. SchuSller, A concept for the reliability assessment of the containment of a PWR, Proc. Int. Conf. Nucl. Systems Rel. Engr. and Risk Assessment, eds. J.B. Fussell and G.R. Burdick, SIAM, Philadelphia (1977) pp. 443 -464. [7] P. Kafka and G.I. Schu~ller, Probabilistische Zuverl~issigkeitsbeurteilung von Sicherheitsbehiiltern am Beispiel des Druckwasserreaktors, 2. GRS-Fachgespr~tch, GRS-Bericht No. 13, K~ln (1979) pp. 39-51. [8] W. Marshall (Chm.,), An assessment of the integrity of PWR pressure vessels, UKAEA Dept., H.M. Stationery Office, London, UK (Oct. 1976). [9] Deutsche Risikostudie-Kernkraftwerke, Verlag TI~IVRheinland, K61n (1979). [10] P.D. Smith, Seismic safety margins research program Program plan, Revision II, UCID-17824, Rev. II, Lawrence Livermore Laboratory, Livermore, Calif. (Aug. 1978). [11] H. Hwang, P. Brown, M. Reich and M. Shinozuka, Reliability assessment of reinforced concrete containment structures, NUREC/CR-3227 (1982). [12] W.A. von Riesemann, T.E. Blejwas, A.W. Dennis and R.L. Woodfin, NRC containment safety margins program for LWR's, Nucl. Engrg. Des. (1982). [13] G.I. Schu~ller, P. Kafka, R. WeUein, F.H. Wittmann, N.F. Zorn, A. Berning, T. Boulhadour, M. Mihashi and J. Bauer, Zuverl~issigkeitsbeurteilung f'flr den SicherheitseinschluB am Beispiel des Druckwasserreaktors, BMFT-Report 150 201-C, Mi~nchen (1981). [14] D.O. Harris, E.Y. Lim, D.D. Dedhia, H.H. Woo and C.K. Chou, Fracture mechanics models developed for piping reliability assessment in light water reactors - Piping reliability project, NUREG/CR-2301, UCRL-15490, Wash. D.C. (1982). [15] R. Sch~ifer, G.I. Schu~ller and P. Kafka, Probabilistische Untersuchung des Rissfortschrittverlialtens von Reaktorkomponenten, Bericht TF-1605, Industrieanlagen-Betriebsges, mbH., Ottobrunn (MSxz 1984). [16] T.A. Jaeger, Safety and reliability analysis of nuclear reactors, Proc. Int. Conf. Struct. Safety and Reliability (ed. A.M. Freudenthal) Pergamon Press, pp. 323-334 (1972). [17] F.J. Witt (Ed.), Proc. 1st Int. Sem. on Struct. Rel. of Mech. Comp. and Subass. of Nuclear Power Plants, Nucl. Engr. Des. 50(2) (1978) 143-215. [18] J.R. Benjamin, G.I. SchuSller and F.J. Witt (Ed.), Proc. 2nd Int. Sem. on Struct. Rel. of Mech. Comp. and Subass. of Nuclear Power Plants, Nucl. Engr. Des. Special Vol. 60, No. 1 (Sept. 1980). [19] G.I. SchuSller, J.R. Benjamin, D. Costes and F.J. Witt (Ed.), Proc. third Int. Sem. on strnct. Rel. of Mech. Comp. and Subass. of Nuclear Power Plants, Nucl. Engr. Des. Spec. Vol. 71(3) (1982) 263-446. [20] A.H.-S. Ang and G.I. Schu~ller (Div. Coord.), Reliability and risk analysis of nuclear power plants, Vol. M, SMiRT 7 (North-Holland Publ. Co., Amsterdam, 1983).
245
[21] A.H.-S. Ang and G.I. SchuSller (Div. Coord.), Structural Reliability, Vol. M1, SMiRT 8 (North-Holland Publ. Co., Amsterdam, 1985). [22] A.H.-S. Ang and G. Apostolakis and P. Kafka (Div. Coord.), Structural Reliability - Probabilistic Safety Assessment, Vol.; M, SMiRT 9, Ed. F.H. Wittmann (Balkema Publ. Co., Rotterdam, 1987). [23] G.I. SchuSller and M. Shinozuka (Editors), Stochastic Methods in Structural Dynamics (Martinus Nijhoff Publ., Dordrecht, The Netherlands, 1987). [24] G.I. SchuSller and R. Stix, A critical appraisal of methods to determine failure probabilities, Structural Safety 4 (1987) 293-309. [25] G.I. SchuSller, C.G. Bucher, U. Bourgund and W. Ouypornprasert, On efficient computational schemes to calculate structural failure probabilities, in: Lecture Notes in Engrg. 31: Stochastic Structural Mechanics, Eds. Y.K. Lin, G.I. Schu~ller (Springer Verlag, Berlin, Heidelberg, New York, 1987) pp. 388-410. [26] U. Bourgund and C.G. Bucher, Importance sampling procedures using design point (ISPUD) - A user's manual, Report No. 8-86, Innsbruck (1986). [27] L. Siitterlin, Zur Auslegung kerntechnischer Anlagen gegen Einwirkungen yon auBen. Teilaspekt, Betrachtungen iiber das Risiko bei Flugzeugabsturz auf ein Kernkraftwerk (Zwischenbericht), IRS-W-12 (1975). [28] G.F. Oswald and G.I. SchuSller, On the reliability of deteriorating structures, Proc. 4th Int. Conf. on Appl. of Statistics and Prob. in Soil & Struct. Engrg. (ICASP-4), Ed. G. Augnsti et al. (Pitagora Editrice, Bologna, 1983) pp. 597-608. [29] P. Kafka, Interface between structural and system analysis, Nucl. Engrg. Des. 71 (1982) 355-357. [30] G.I. Schu~ller, N.C. Hampl and H.J. Pradlwarter, Fragility of primary piping, Nucl. Engrg. Des. 110 (1988) 221-227. [31] G.I. SchuSller, Impact of probability risk assessment on containment, Nucl. Engr. Des. (1983) 203-217. [32] G.I. SchuSller, Einf'flhrung in die Sicherheit und Zuverl~issigkeit von Tragwerken (W. Ernst und Sohn Verlag, Berlin, 1981). [33] A.S. Kiremidjian and T. Anagnos, Stochastic slip - Predictable model for earthquake occurrences, Bull. Seism. Soc. of America 72, No. 2 (April 1984) 739-755. [34] R.J. Scherer, N.J. Krutzik, G.I. SchuSller and S. Freyst~itter, PRA of NPP structures under earthquake conditions - Seismic risk analysis, Proc. 8th Int. Conf. Struct. Mech. React. Techn., Vol. M1 (North Holl. Publ. Comp., Amsterdam, 1985). [35] T.C. Hanks and R.K. McGuire, The character of high frequency strong ground motion, Bull. Seism. Soc. of America 71, pp. 2071-2095. [36] N.J. Krutzik, G.I. SchuSller , R.J. Scherer, PRA of NPP structures under earthquake conditions - Structural reliability analysis, Proc. 8th Int. Conf. Struct. Mech. React. Techn., Vol. M1 (North Holl. Publ. Comp., Amsterdam, 1985).
RELIABILITY
OF NUCLEAR
STRUCTURES
235
*
G.I. SCHUELLER Institut fur Mechanik, Universitiit Innsbruck, A-6020 Innsbruck, Austria Received 22 July 1988
The reliability of nuclear structures relates directly to the reliability of mechanical and electronic systems. Nuclear structures may be viewed as part of operational system as well as part of standby safety system. Except for proof loading, most of the load types, seen by nuclear structures, such as containments, primary piping, etc. during their design life, are random in time and space. Structural resistances, governed mainly by the respective material properties show also random characteristics. Methods of structural reliability, as discussed here, provide the means to process rationally the available information. Hence, they are an indispensable tool within the frame work of probabilistic risk analyses (PRA's). The probabilistic analysis is exemplified by determining the reliability of a containment structure sited in an area with strong seismicity.
1. Introduction
W i t h i n the first generic risk study of nuclear power p l a n t s the aspect of the reliability estimation of the structures involved, such as the c o n t a i n m e n t , p r i m a r y piping, etc, received considerable less a t t e n t i o n t h a n the reliability analysis of the mechanical as well as electronic systems [1]. In this context it is interesting to note that the failure probabilities of the p r i m a r y piping (for the large a n d the small L O C A respectively) were estim a t e d from sample statistics which also included d a t a from n o n nuclear power plants. The failure probabilities of the c o n t a i n m e n t due to L O C A for example were d e t e r m i n e d b y i n t r o d u c i n g n o r m a l l y distributed " e r r o r b o u n d s " on the estimated expected m a x i m u m pressure. F o r e a r t h q u a k e analysis the investigators did not leave deterministic g r o u n d s at all. This procedure was quite surprising since d u r i n g S M i R T ' 7 3 a n u m b e r of presentations were given which suggest m u c h more sophisticated ways to calculate structural reliabilities, i.e. failure probabilities of pipings [2], pressure vessels [3] a n d c o n t a i n m e n t s [4], etc. Subsequently, these m e t h o d s were developed further (see e.g. [5,6,7]) a n d then utilized in risk studies [8], [9]. A r o u n d this time the U S N R C initiated the S S M R P - mainly carried out b y L L N L -
which c o n c e n t r a t e d on p l a n t specific studies (see e.g. [10]). This effort was then c o m p l e m e n t e d by the Load C o m b i n a t i o n Program, carried out by B N L (see e.g. [11]). Both in Europe a n d the U S quite sophisticated reliability studies o n c o n t a i n m e n t a n d pipings followed a m o n g m a n y others, see e.g. [12-15]. In this context it should be mentioned, that Jaeger [16] stressed already quite at an early stage the i m p o r t a n c e of structural reliability studies in context with reactor safety a n d initiated a series of S M i R T Post Conference Seminars on this subject [17-19]. As within a r a t h e r short period of time, the interest in the subject grew considerably, the Scientific C o m m i t t e e of S M i R T decided to devote a new Division ( M / M 1 ) to this subject [20-22]. N a t u rally, this brief historical review can be b y n o m e a n s exhaustive. Limited space does not allow to list a n d discuss all activities which c o n t r i b u t e d to the developm e n t of the field. Needless to say that the T h r e e Mile Island incident as well as the Tschernobyl accident underlined the significance of such studies.
2. Reliability analysis 2.1. Concept
* Principal Division Lecture, Division M, 9th International Conference on Structural Mechanics in Reactor Technology (SMiRT-9), Lausanne, Switzerland, Aug. 17-21 1987. 0029-5493/89/$03.50
The basic concept of classical reliability analysis which will be outlined briefly in the following (for a
© 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.
G.L SchuFller / Reliability of nuclear structures
236
more detailed t r e a t m e n t see e.g. [23]) - recognized the fact, that most variables which are involved in determining the load processes as well as the structural resistance, are of r a n d o m characteristics. The relation between these variables might be defined for example by a limit state surface such as g(X) = g(X,, X2.... X,)=0
(1)
where X i represent the r a n d o m variables defining structural resistances a n d loads respectively. The limit state is reached when the condition g ( X ) < 0 is met, which is associated with the notion of failure. Therefore, failure must always be seen in context with the limit state. It can be defined by the analyst or the designer for example as the exceedance event of a particular deflection (serviceability limit sate), the f o r m a t i o n of hinges, or the f o r m a t i o n of a chain of hinges describing a collapse mechanism. D e p e n d i n g on the particular problem, these limit states can b e considered either individually or combined. The probability that a particular limit sate will be reached, i.e. the failure probability may be expressed as:
p f = S D / x ( i l ) dll,
(2)
w h e r e .fx('q) is the j o i n t d e n s i t y f u n c t i o n of X1, X 2 , . . . , X~ and Df is the d o m a i n of failure in the
n-dimensional space. A schematic sketch for two variables is shown in fig. 1. The variables involved in eq. (2) m a y certainly be correlated. C o m p u t a t i o n a l m e t h o d s to solve eq. (2) are described in detail e.g. in [23], a brief discussion is given in the following section. T h e failure probability as defined by eq. (2) is obviously a time invariant expression. M a n y cases, however, such as the earthquake analysis, impact analysis, etc., are certainly to be treated as a time variant problem. This includes o n one h a n d the occurrence p r o b a b i l i t y of the load event and on the other h a n d the conditional distribution with respect to time a n d space. This m a y be exemplified by referring to the e a r t h q u a k e load case. T h e earthquake event occurrence m a y b e m o d e l e d by so called stochastic processes. In particular, the Poisson, or modified Poisson processes, are utilized for this purpose. The event itself is also a (small scale) time d e p e n d e n t , n o n s t a t i o n a r y process with r a n d o m l y distributed amplitudes, frequencies, lengths, etc., a n d a spacial distribution. These e a r t h q u a k e strong m o t i o n records m a y be described conveniently by so called evolutionary spectra. In addition to the consideration of the time variance of the load process, the structural resistance m a y also be considered as a time d e p e n d e n t process. The effects which c o n t r i b u t e to this decrease in structural resistance m a y be either due to corrosion, crack propagation,
5"1, tx(~)
~
~
~ , ' ~
i,\<; ",D"*3t ', t : < " / .
':.:2)'/:(..
~tx)<0
",
S<~, . ,d ~ <
~
glgl=O
Fig. 1. Failure probability - Schematic sketch (two dimensional case).
237
G.L Schu~ller / Refiability of nuclear structures
aging, etc., or a combination of it. For this case the reliability may be expressed as
TT(t)=exp{-- fothT(t' )
dt'},
x2
K2~ D" ~/
f(x)
~.~l,I
(3)
where
hT(t ) =fT(t)/(1 -- F T ( t ) )
(4)
is called the hazard rate and (5b).
fT(t)
mx 1
is defined by eq. X1D.X2D ¢~ntir~t~ ~ tl~ ~sign Ix~nt
2.2. Computational methods
where N is the number of simulations and sampling distribution, and 1
(5a)
hv(Xi)
the
for X e failure domain, otherwise.
In addition, the statistical error of this estimate can be calculated without difficulties [26]. For time variant structural resistance, the above equation reads
~=~
/x(X,(t))
fT= -N1 i= l[g(Xi(t))
Fig. 2. Computational procedure to calculate failure probabilities (two dimensional case).
can then be represented either in the time or in the frequency range, respectively.
2.3.2. External events
1 N e l = N i=~=ll[g(xi) _<0] hr(Xi),fx(X')
0
--g(X__l=O. t = t 3
Xl
Various methods to solve eq. (2) both computationally accurate and efficient have been suggested so far. For a critical discussion of these methods it is referred to e.g. [24], [25]. One particular method, which satisfies these criteria sufficiently well is based on importance sampling procedures. The failure probability can then be computed by the following expression
l[g(Xi)<-O]=
~"
(5b)
where g(Xi(t)) changes with each time interval or load cycle (see fig. 2). As pointed out in section 2.4 below, this change may be modeled by linear or nonlinear fracture mechanical considerations. Both the time variant and invariant cases respectively are included in a computer software package, called ISPUD [26].
2.3. Load analysis 2.3.1. General To model, i.e. predict rare (extreme) events, generally Poisson type processes are utilized. In most cases the parameters i.e. expected frequencies of occurrence of intemal or external man made or natural hazards can be estimated from observations. The load event itself
Nuclear power plant structures must survive external events, such as man made or natural hazards (for example air plane impact, external pressure wave or earthquakes) with a very high probability. The target level may be specified as design requirements. Both types of hazards are considered to be site dependent. While in the case of aircraft impact occurrence on NPP's a statistical analysis of airplane crashes is instrumental [27] a rather sophisticated model building, which is based on geophysical considerations including aspects of wave propagation, it to be applied in the earthquake load case. More details on the latter type of load analysis will be discussed below.
2.3.3. Internal events Except for test loading, the occurrence of internal load events are either due to operation or malfunction of the systems, i.e. the operational and the accidental loading conditions respectively. An important example of the second type is the LOCA (loss of coolant accident) which occurs when the primary piping falls with a break of a certain size (100 cm2 and 1000 cm2 for small and large LOCA respectively is generally postulated). The expected occurrence rates of internal accidental load events are estimated either directly from observations or they have to be calculated by utilizing either fault and event trees to predict the (non observed) occurrence rate of a larger events from a sequence of (observed) smaller events, or by utilizing load spectra along with fracture mechanical methods.
238
G.I. Schu~ller / Reliability of nuclear structures
2.4. Structural resistance The stochastic nature of the structural resistance is mainly due to the random characteristics of the material properties. This applies to the time invariant yield or compressive concrete strength as well as to the time variant corrosion and crack propagation under cyclic loading which reduce the structural resistance. In this context it should be pointed out that the quality of construction, which may be expressed in terms of initial crack distributions, also plays a very important role. Structural reliability considerations in context with quality control provide also a quantitative basis for the decision whether to utilize high strength (brittle) or low strength (ductile) types of steel [28].
2.5. Structural response analysis The structures of NPP's to be analyzed within a reliability analysis are generally discretized by utilizing finite elements. This applies also to the soil on which the entire structure is based. Depending on the properties of the loading, either l i n e a r / n o n l i n e a r static or dynamic structural analyses have to be carried out. In the linear range, the stochastic dynamic response analysis is performed most conveniently in the frequency range, i.e. by applying power spectral analysis. For nonlinear dynamic analysis, time domain calculations for a number of samples are still required. At the present stage, the application of equivalent linearization techniques are still confined to structures with a small number of DOF's. Many site conditions also require nonlinear soil-structure interaction analysis.
CONTROl. A/~I) S&eltTV ~ S (SZNSOP.S.L~CI
PIPES. W~J. pUMPS, HatJIGRIIS
ETC. CONTAINMENT. PP.I~SlIRg LARGI~ PIpZS.LeTC.
Fig. 3. Interaction between systems. Moreover, as mentioned above, in case of a LOCA the load depends on the size of the pipe leak or break. The failure probabilities of PP as well as containments under large and small LOCA's are already discussed in some detail in [15], [30], [13], [31]. Hence, as an additional example, in the following, the reliability analysis of a containment structure located in a seismic active area is shown below. Note, that the outlined procedure should only serve as a guideline w.r. to how such a procedure may be developed.
3.2. Containment structure under seismic loading
3. Application to nuclear structures
3.2.1. General In seismic inactive areas, such as the F.R. Germany, some simplified procedures with respect to the seismic risk analysis are acceptable (see e.g. [13], [31]). However, in seismic active zones, more sophisticated procedures are required. Needless to say that there is a strong interrelationship between seismic risk and structural analysis respectively. This statement implies, that simplifying assumptions made in the seismic risk analysis part may significantly affect the structural part. Hence the level of sophistication of both analyses should be in balance.
3.1. Interaction between structural and systems reliability
3.2.2. Seismic risk model
Among the various structures and components of a NPP, the RPV, the containment and the PP structures received most attention from structural reliability analysts. When calculating the reliability of nuclear structures, it is most important to realize, that the structure under consideration is part of the entire system (fig. 3). Functionally, a structure may be considered from two points of view, namely, as a part of the operational system and as a part of standby safety system [29]. In fig. 4 it is shown clearly, that loads, to which the PP may be exposed to can be determined from the operational parameters as well as from the function - - or malfunction - of the safety system.
The seismic risk model has to be based on a time variant reliability function, i.e. by (see e.g. [32]) LT(t ) = fRLvln(t)fR(r)
dr
(6)
where L x ( t ) defines the probability that the structure survives the seismic hazard within a time interval (0, t) and f n ( t ) the P D F of the structural resistance. L x l n defines the survival probability conditioned on a particular realization of the structural resistance r = R: LTin(t )=
Y'. p ~ ( 1 - - F T i n ) k, k=O
(7)
G.I. Schuie'ller / Reliability of nuclear structures SAFETY SYSTEM
[1 I [ LOADTRI~E [ OPERATIONAL[ [
I TRANSIENT
I) ~
I IL°SS Or
OPERATIONAL
I
LEA.'(
LOADS IS l ^ecmer~r
~ II II
LOAD~
P
I
I I,.
~
1
r I
-
_"l'
~ l
239
&CCIDENT~:~LASSES
Sl
K1
K2
YE,S
NO lI ~ ; I ' I
1
~rlRll ~'~1"II NAL CHARACTE RIb'TI C S i
(KAFKA 19821
Fig. 4. Structure as part of an operational system [29].
where k is the number of the occurring load events and Fw(t ) the time variant failure probability, i.e. the complement to the survival probabihty
is a measure of its variance. Its relation to the PSD (Power Spectral Density) GA(~o) is a RMS =
FT( t) = I -- L T ( t ).
fo°°GA (~o) doo
(10)
(8)
AS mentioned above, for modeling the earthquake event occurrence the simple Poisson process proves to be applicable, hence Pk may be modeled by
where ~0 defines the circular frequency (2qrf[Hz]). With this information at hand, the conditional failure probability, as utilized in eq. (7) can be defined:
F.rl ~ = /Apflk.AfA( a ) d a , ( v t ) k e -°t, pk(k)
=
k!
k=0,1,
oo,
(11)
(9)
....
where u is the mean rate of event occurrence to be estimated from observed data. It is well known that this process is based on the assumption, that the occurrence in adjacent time intervals are independent. This, however, does not reflect physical reality (see e.g. [33]). Nevertheless, keeping in mind the present knowledge of the seismic process, which is still quite insufficient, this model may still considered to be appropriate. Note, that the conditional probability of failure F-rl R is defined as the probability that the structural response exceeds the realization of the resistance during an earthquake occurrence. The earthquake event itself, which is described by strong motion records, and hence the structural response, may be defined by a continuous stochastic process. Depending on the type of limit state condition to be analyzed, general random vibration theory may be applied. This implies the assumptions that the structural behavior is linear and the response is a weakly stationary (Gaussian) process. Both assumptions do not apply exactly. However, for simplifying reasons they are introduced, nevertheless. The earthquake process, generally defined it terms of ground acceleration a is characterized by its R M S (Root Mean Square) value, which
where fA(a) is the P D F of the earthquake acceleration at a particular site. Naturally this distribution, depends strongly on the knowledge on seismic, tectonic and geological parameters of the ~articular region, i.e. the site under investigation. Due to lack of sufficient information, the (maximum) ground accelerations are generally determined by empirical attenuation laws of the following type a = b I e b2M A -b3,
(12)
where b 1, b2, b 3 are constants to be determined by regression analysis based on observed events, M is the Richter magnitude and zl the hypocentral distance. In most cases detailed information on the tectonic process, i.e. fault mechanisms the likely fault depth (seismic zone) is scarce and consequently the distribution of future faults can only be described with considerable uncertainties. Hence, it is most advantageous to assume either a homogeneous seismic region - or if information permits - several subregions Z. N o w the probability of the acceleration to exceed a certain value at a particular site may be defined as [34]:
P [ A > a ] = ~_,P[A > a I Z = z ] p z ( z
).
(13)
Z
The parameters of this equation are defined in the
240
G.L Schugller / Reliabifity of nuclear structures
Appendix A. The shape of the suitable power spectrum PSD of the ground acceleration could be determined accurately if the necessary data, i.e. time histories were available. Needless to say that this is generally not the case. Therefore, an artificial - usually the Kanai-Tajimi - spectrum is utilized for this purpose:
GA(°~) G°[1-(W/Wg)] + 4~g(w/wg) 22
2
(14)
2'
where wg and ~s denote the central frequency and the band width respectively. On one hand these parameters are adjusted to the geological information available and on the other hand they should be adjusted to the structural eigenfrequencies. The latter ensures a conservative approach. The third parameter, Go , which scales the spectrum, may be related to the RMS-value of the acceleration (see eq. (10)). As attenuation laws for RMS-values are generally not available the establishment of a relation between R M S - and the maximum value of the acceleration is to be sought. Theoretical considerations obviously do not provide such bounds, however, in practice the maximum acceleration is generally bounded by area x ---- 2 - 3 aRM s (see e.g. [35]). 3.2.3. Structural refiability analysis This analysis refers to the assumption made in the section above as well as to the relations defined by eq. (6) to eq. (9). Since linear structural properties are utilized, the assumptions of stationarity and normality of the seismic input, also holds for the structural response process. Hence, the P D F of the load effect consisting of the value of the dead load and the standard deviation of the earthquake loading - is fully determined. The latter may be obtained by integrating the PSD of the response, G s ( ~ ), i.e. by
,-, Applying modal analysis, the standard deviation may be expressed by the modal variances Os2, and covariances o2, j , respectively:
OS =
E t=l
2 2 2 + ~iLi°s,ii
~, ~b,~bjLiLjo~,i j
,
(16)
i=1 J=l
where ~b, is the modal shape of the displacement and load effect respectively and L, is the load participation
factor (as defined by eq. (18) below) - which, according to eq. (15) may be obtained by the auto- and cross-spectral densities G s , , ( w ) and Gs, jj(w ). The cross spectral density for the modal shapes i and j is =
OA
(17)
where /4i is the transfer function of the i-th mode, H ~ the complex conjugate and GA the PSD of the ground acceleration as defined by eq. (14). The load participation factors are defined by
,TM = *TM,--. '
(18)
where g~i is the eigenvector of the mode shape and M the mass matrix. To illustrate the analysis a numerical example is carried out in the Appendix B.
4. Conclusions It has been shown that reliability analyses of nuclear structures are an indispensable tool within the context of PRA of NPP's. The probabilistic analysis processes the maximum on input information on structural loads and resistance available and hence also provides additional information over traditional, deterministic analysis in terms of failure probabilities and reliabilities, respectively. Only todays computational possibilities allow such an increase in processing information. This paper also attempts to give an overview over the developments of structural reliability applied to nuclear structures. A m o n g various examples, for which it is referred to the literature, one particular aspect, i.e. the reliability analysis of a containment structure under earthquake loading is discussed in some detail in order to exemplify the procedure. Finally it is shown that the analysis allows a rational treatment of all uncertainties involved.
Appendix A. P D F of ground acceleration
Following for example [34] the probability that the acceleration A exceeds certain value at a particular site may be defined by P [ A > a] = ~ _ , P [ A > a [ Z = z] p z ( z ) , z
(13)
G.L Schu#ller / Reliability of nuclearstructures
Appendix B. Numerical example of reliability analysis of a containment under seismic excitation
where Z defines a particular subregion
P[A > a l Z l = f~fyP[A > al Z, Rlf,~(x, y ) dx dy, x, y ~ Z
(A1)
and
P[A>alZ,
241
B.1. General remarks The theoretical developments as outlined in section 3.2 is now exemplified by applying it to a particular case study [34], [36]. For this purpose the analysis is based on data and considerations as shown below.
A ] = P [ b , e b ~ M A - b ' > a l Z, A] B. 2. Evaluation of the probability distribution of the maximum horizontal ground acceleration (A2)
In eq. (13) the total probability from all subregions is established, where p z ( z ) is the probability that an event will occur in a particular subregion Z. It is well known that the occurrence rate of earthquakes may be determined by log N = a - bm,
(A3)
where N is the number of earthquakes (per year and km 2) with a magnitude equal or greater than m; a and b are constants to be taken from earthquake catalogs. The P D F of the magnitude is
N(m,) -N(m) F ~ ( m ) = N(mi ) _ N ( m u )
,
(A4)
where m~ and m I are - depending on the seismic region - the geophysically possible largest and smallest magnitude respectively. Furthermore the probability of occurrence of earthquakes in a particular zone may be expressed by
Within the contact of a geological study a model for the tectonic process was established which resulted in the estimation of a seismic zone (Fig. B.1). The depth of the source varies from about 10-30 km to about 100 km, the attenuation law for the m a x i m u m horizontal ground acceleration could be deduced from seismic intensity maps which were based on seven earthquake occurrences with a range of magnitudes from 5.8-7.25, assuming source depths of 10 and 20 km respectively. Fig. B.2 shows the attenuation law of the region. Based on a geological study the upper limits of magnitudes are determined, and consequently four different seismic regions proposed with maximum magnitudes 5.5, 6.5, 7.5 and 8.0 respectively. Their orientation depends on the respective thrust mechanism. Furthermore, two different frequency distributions of the earthquake occurrence are given as shown in fig. B.3. The whole seismic region may be subdivided into eight subregions as shown in fig. B.4.. One can observe unrealistic discontinuities between the seismic activities of such regions. It is quite natural to enlarge the more active subregions (see dashed
z
N ( z ) f~fyh(X, y ) dx dy pz(z)=
, Y',N(z) f f h ( x , "x"y Z
x, y ~ Z ,
0
y) dx dy
I
100
50
D I S T A N C E IN K I L O M E T E R S 150
tU
(A5)
o
50 MOHO
where h(x, y) is the frequency of earthquake occurrence at the location (x, y). The P D F of an earthquake occurrence at a certain distance A is
~(x, y)=
h(x, y)
fh(x,
y ) d x dy
200
(A6)
il~
100 _a
Discontinuity
----•
MOHO
liiiiiil~i~
Oceanic Basement
Data
Fig. B.1. Seismic mechanism.
250 eP
242
G.L SchuUIler / Reliability of nuclear structures Y
a [cm/s2l
i I I
1200
I i
I |
Based on Geological Study 800
II
/ 8.0
i
1
I1
I
~85 I
6.5
o
I /f-
L.-t=._
II
I SiTE
_~./-~ I l~.sff"
t
I
IOO.O
i
;
200.0
[kml
~
115M • I
"%~ I
Fig. B.2. Attenuation law for the m a x i m u m ground acceleration in magnitudes (Source depth h = 10 km),
YL
85
er~n~
lines in fig. B.3.). For this case the P D F ' s of maximum ground acceleration are shown in fig. B.5.
20cm/s z
--~min ~ 50~m/s2
~-1~-" --,~_ __ ~
* -120
I
B,3. Determination of the PSD of the horizontal ground acceleration As there is no advanced geological information available for the site, the utilization of a sophisticated power spectrum is not feasable. Measurements of microtremors indicate a predominant period of 0.6 seconds. Based on the soil profile (see fig. B6) which provides information to a depth of 35 m, but with no details on bedrock formation, one can estimate the upper limit of
I
the lowest resonance frequency to be about 3 Hz. This is valid for a S-wave - vertical incident - by assuming a layer over a half space, where the layer thickness is 35 m and the shear velocity 400 m / s . It is quite obvious that under these circumstances the PSD of the horizontal ground acceleration can not be estimated i.e. fitted with
log fA(a)
32
_.I,~I!__L~_I
Fig. B.4. Seismic provinces according to the attenuation law (of fig. B.2) and a lower bound of m a x i m u m acceleration of 20 c m / s 2.
I i--I I 11~-~-I~-~IJ,I , ~ _
-10
! 1
,~
Fig. B.3. Frequency distribution (h(x, y)) of earthquake occurrences.
100
300
500
a [ c m / s 2]
Fig. B.5. Conditional density function of the m a x i m u m ground acceleration for the seismic region (with reference to one event).
243
G.I. Schu@ller / Reliability of nuclear structures
r--I --i I
I
I
r-7 i I IL_J
I
I
I
lO
I
I
I
20
30
35
Depth
---
[m]
Fig. B.6. Typical soil profile of the site.
high confidence. In order to obtain at least a conservative approach, it is suggested to take into account the dynamical properties of the N P P structure. These properties may be described by its eigenfrequencies in context with the participation factors as shown in fig. B.7. The parameters of the Kanai spectrum are chosen such that the first two eigenfrequencies of the structures which reveal the largest participation factors are enveloped by the peak of the power spectrum. This leads to a Kanai frequency of 2 Hz which, incidentally, is in agreement with the information provided by the geological study. Furthermore, in order to concentrate on one hand the seismic energy around the first two eigenfrequencies of the structure and on the other hand not to select a too peaked spectrum - which in fact would not reflect the geological data at all - the Kanai damping value ~g is chosen to be 0.5.
ture under consideration. This is mainly due to the transparent relationship between load and load effects (see eq. (15)-(18)). Load participation factors are a first but fundamental information on the dynamical sensitivity of the structure. Hence a conservative approach to define the loading function is feasible or, - for cases where a definite description of the loading function is available - the structure may be optimized for minim u m load effects. In the present study (see also fig. B.7) a conservative load spectrum is chosen. The dynamic properties of the sample structure - a containment are listed in table B1. The dominance of the first mode shapes of the loading, which may be concluded from the modal variances, is a direct consequence of the choice of the load spectrum.
B. 4. Structural analysis
A cross section located in the transition between the spherical dome and the cylindrical shell of the reinforced concrete shielding is analyzed as an example (see No. 1 of fig. B.8). Due to the additional design against aircraft impact a disproportional large wall thickness of
The modal analysis is most advantageously applied to assess the dynamical behavior of the particular struc-
B.5. Failure probability of the structure
G(f)
2,0
/
Kanai - Spectrum
Table B1 Dynamic properties of NPP structures under earthquake excitation Mode
Natural frequency [Hz]
Damping [%]
Load participation factor
Modal variances 10 -6 [m2]
1 2 3 4 5
1.06 1.94 5.03 5.73 8.78
15 15 7.5 2.4 7.1
331 435 181 3 66
30.334 8.860 63 51 5
1, o
o,o
~o
4.0
6,o
8,0
f IHz]
Fig. B.7. Comparison between load participation factors (relative values) and load spectrum.
244
G.L Schu~ller / Reliability of nuclear structures log PflE,A
-10
-11
-
I 100
I 300
1
I,
a [ c m / s 21
500
Fig. B.9b. Conditional failure probabilities in the vertical cross section No. 1.
Fig. B.8. Sample structure containment.
Table B2 Failure probabilities TT(t ) of cross section for a design life of 40 years Horizontal Vertical Earthquake occurrence rate v events per year
< 1.0 × 10-10 8.6 × 10-6
2.0 m is required. The b e n d i n g reinforcement, however, is relatively small, which leads to the fact that the failure of the b e n d i n g r e i n f o r c e m e n t is the expected failure mechanism. Hence, the r a n d o m variable of significance is the axial force of the reinforcement, which is modeled b y a n o r m a l distribution with a coefficient of variation of 0.055 a n d a m e a n value of 602 N / m m 2. The c o n d i t i o n a l failure probability, given the earthquake event E is s h o w n in fig. B.9, for a vertical a n d horizontal cross section respectively. F o r a design life of 40 years the failure p r o b a b i l i t y F ( t ) = 1 - L ( t ) is shown in table B2. N o t e that the failure probabilities of this critical cross section are well below those values which are generally accepted in codified design.
Acknowledgement Part of this overview p a p e r is b a s e d o n a j o i n t research with S i e m e n s - K W U , Offenbach. Special t h a n k s are due to N.J. Krutzik for his support a n d c o n t i n u o u s interest in the work. R.J. Scherer's (University of Karlsruhe) help, particularly in carrying out the numerical example, is also deeply appreciated.
28.7
Iog PfiE,A
References
-15 ~ -20 -
-25 -
-30-
-35 I
i
I
100
300
500
w,
a [cm/s2l
Fig. B.9a. Conditional failure probabilities of the horizontal cross section No. 1.
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245
[21] A.H.-S. Ang and G.I. SchuSller (Div. Coord.), Structural Reliability, Vol. M1, SMiRT 8 (North-Holland Publ. Co., Amsterdam, 1985). [22] A.H.-S. Ang and G. Apostolakis and P. Kafka (Div. Coord.), Structural Reliability - Probabilistic Safety Assessment, Vol.; M, SMiRT 9, Ed. F.H. Wittmann (Balkema Publ. Co., Rotterdam, 1987). [23] G.I. SchuSller and M. Shinozuka (Editors), Stochastic Methods in Structural Dynamics (Martinus Nijhoff Publ., Dordrecht, The Netherlands, 1987). [24] G.I. SchuSller and R. Stix, A critical appraisal of methods to determine failure probabilities, Structural Safety 4 (1987) 293-309. [25] G.I. SchuSller, C.G. Bucher, U. Bourgund and W. Ouypornprasert, On efficient computational schemes to calculate structural failure probabilities, in: Lecture Notes in Engrg. 31: Stochastic Structural Mechanics, Eds. Y.K. Lin, G.I. Schu~ller (Springer Verlag, Berlin, Heidelberg, New York, 1987) pp. 388-410. [26] U. Bourgund and C.G. Bucher, Importance sampling procedures using design point (ISPUD) - A user's manual, Report No. 8-86, Innsbruck (1986). [27] L. Siitterlin, Zur Auslegung kerntechnischer Anlagen gegen Einwirkungen yon auBen. Teilaspekt, Betrachtungen iiber das Risiko bei Flugzeugabsturz auf ein Kernkraftwerk (Zwischenbericht), IRS-W-12 (1975). [28] G.F. Oswald and G.I. SchuSller, On the reliability of deteriorating structures, Proc. 4th Int. Conf. on Appl. of Statistics and Prob. in Soil & Struct. Engrg. (ICASP-4), Ed. G. Augnsti et al. (Pitagora Editrice, Bologna, 1983) pp. 597-608. [29] P. Kafka, Interface between structural and system analysis, Nucl. Engrg. Des. 71 (1982) 355-357. [30] G.I. Schu~ller, N.C. Hampl and H.J. Pradlwarter, Fragility of primary piping, Nucl. Engrg. Des. 110 (1988) 221-227. [31] G.I. SchuSller, Impact of probability risk assessment on containment, Nucl. Engr. Des. (1983) 203-217. [32] G.I. SchuSller, Einf'flhrung in die Sicherheit und Zuverl~issigkeit von Tragwerken (W. Ernst und Sohn Verlag, Berlin, 1981). [33] A.S. Kiremidjian and T. Anagnos, Stochastic slip - Predictable model for earthquake occurrences, Bull. Seism. Soc. of America 72, No. 2 (April 1984) 739-755. [34] R.J. Scherer, N.J. Krutzik, G.I. SchuSller and S. Freyst~itter, PRA of NPP structures under earthquake conditions - Seismic risk analysis, Proc. 8th Int. Conf. Struct. Mech. React. Techn., Vol. M1 (North Holl. Publ. Comp., Amsterdam, 1985). [35] T.C. Hanks and R.K. McGuire, The character of high frequency strong ground motion, Bull. Seism. Soc. of America 71, pp. 2071-2095. [36] N.J. Krutzik, G.I. SchuSller , R.J. Scherer, PRA of NPP structures under earthquake conditions - Structural reliability analysis, Proc. 8th Int. Conf. Struct. Mech. React. Techn., Vol. M1 (North Holl. Publ. Comp., Amsterdam, 1985).