Basic probabilistic considerations on safety of prestressed concrete pressure vessels

NUCLEAR ENGINEERING AND DESIGN 29 (1974) 338-345. © NORTH-HOLLAND PUBLISHING COMPANY

BASIC

PROBABILISTIC

CONSIDERATIONS

ON

PRESSURE

SAFETY

OF

PRESTRESSED

CONCRETE

VESSELS*

M. S H I N O Z U K A Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 1002 7, USA and L.C. S H A O Structural Engineering Branch, Directorate of Licensing, US Atomic Energy Commission, Washington, DC, USA Received 10 September 1973

The purpose of the present study is to lay the foundation for the probabilistic safety analysis pertinent to prestressed concrete pressure vessels of nuclear generating stations. We place a major emphasis on the consistency with which various aspects of structural and statistical significance are analyzed and then combined into a probabilistic formulation amenable to practical risk assessment and assurance. In particular (1) prestressed concrete pressure vessels subjected to loads of various temporal nature are considered, (2) the probabilistic algebra used in risk assessment is basically that of the second moment-first order approximation, and (3) the method thus established is applied to reliability analysis of prestressed concrete reactor vessels (PCRVs) to estimate the probability level of ultimate failure. The result of the present study, indicating the probability level of the vessel reliability depends entirely on the assumptions concerning statistical aspects of loading conditions and resisting capacities of the vessel and on the approximations in relation to stress and structural analysis and ultimate failure conditions. Therefore, the resulting probability values are illustrative and comparative rather than factual at this time. The authors are solely responsible for the content of this paper which does not represent the official view of the USAEC.

1. Introduction The procedure o f probabilistic safety analysis pertinent to prestressed concrete pressure vessels o f nuclear generating stations is described. A t t e n t i o n is focused on the consistency with which various aspects o f structural and statistical significance are analyzed and then combined into a probabilistic formulation amenable to practical- risk assessment and assurance. In particular, this paper considers the following points: (1) Prestressed concrete pressure vessels subjected to loads of various temporal nature are considered. A m e t h o d o f estimation o f reliability levels o f these vessels is described. * Invited paper H1/4* presented at the Second International Conference on Structural Mechanics in Reactor Technology, Berlin, Germany, 10-14 September, 1973. This work was partially supported by National Science Foundation Grant, GK-24925.

(2) The probabilistic algebra used in this estimation is basically that o f the second m o m e n t - f i r s t order approximation. Accordingly, the degree o f uncertainty in any variable involved is expressed either in terms o f standard deviation or coefficient o f variation, with the source o f the uncertainty consisting o f m o d e l i n g error, intrinsic randomness and statistical variability in its mean value. (3) The m e t h o d thus established is applied to reliability analysis o f prestressed concrete reactor vessels (PCRVs) to estimate the probability level o f loss o f structural integrity categorized as ultimate failure. The ultimate structural failure as investigated here consists o f a n u m b e r o f possible modes o f failure such as tension failure in vertical direction through horizontal crosssections o f the vessel cylinder under unpressurized or pressurized crack conditions, tension failure in circumferential direction with unpressurized or pressurized

M. Shinozuka, L.C Shao, Safety of PCR Vs cracks opening along generator of the vessel cylinder, and crack initiation in bulk concrete within the head structure. The major purpose of this study is to establish a consistent and logical probabilistic method of reliability assessment for PCRVs. The result of this study, indicating the probability level of the vessel reliability, depends entirely on the assumptions concerning statistical aspects of loading conditions and resisting capacities of the vessel and on the approximations in relation to stress and structural analysis and ultimate failure conditions. Therefore the resulting probability values are more illustrative and comparative than factual.

339

ability theory in which only the knowledge of mean values and variances of R and S are needed. Secondly, we shall use the first-order approximation to estimate these mean values and variances in terms of mean values and variances of design and other variables Xo, XI . . . . . Yo, Y1. . . . . of which R and S are functions. In particular, let X o denote the modeling error associated with Ro(X 1, X 2. . . . ) representing a theoreticalexperimental model for R. Then, the resisting capacity R may be written as [1] R = XoRo(Xl, X 2. . . . ).

(3)

Similarly, the load S may be written as S = YoSo(YI, Y 2 , . . . ) .

2. Basic reliability analysis The classical reliability analysis is based on the assumption that we know the probability distributions FR(r ) and Fs(s) of the resisting capacity R of the structure and load S applied to it. If the probability pf of failure is defined as the probability P(R <~ S) that R will be less than or equal to S, then it is a straightforward exercise of the probability theory to show that pf =P(R ~
(1 - Fs(r)}fR(r ) dr,

(1)

--oo

where fR(r) and fs(s) are the density functions of R and S. Under the assumption of normality of both R and S, eq. (1) becomes pf = ~b(--(R - ff)/(o 2 + o~) l/z)

(2)

in which ~(')indicates the standardized normal distribution. In reality, however, both FR(r ) and Fs(s ) have to be estimated on the basis of currently existing technological knowledge and available statistical data. There is in fact a general consensus that it is totally impractical to seek the exact information on the probability distributions of R and S. Furthermore, R and S are in general functions of a number of design variables of random characteristics. This also makes it impractical to pursue either analytically or numerically the exact expressions of FR(r) and Fs(s ). Such consensus and observation will be reflected in the following analysis. First, we shall perform a reliability analysis on the basis of the second moment reli-

(4)

In addition to the modeling errors X o and Yo, we consider the variabilities in Xi and Yi (i = l, 2, 3 . . . . ) as the source of uncertainty in R and S. These variabilities consist of (1) modeling errors associated with individual variables, (2) their intrinsic randomness, and (3) statistical uncertainties of their mean values. The estimate of their variances should include all these factors. For brevity, we shall dispense with detailed discussion on these variabilities and refer the reader to ref. [1]. I f R and S are given in eqs (3) and (4), it follows from the first-order approximation and the assumed independence that R = XoRo(X1, X 2. . . . ) = XoRo(XI, X2 . . . . ),

(5)

S = YoSo(Y1, Y2 . . . . ) = YoSo(Y1, Y2 . . . . ),

(6)

x-'~[ORo ~2

varRo=~,|~-~ varXi, varS - w [ O S ° ' ~ 2 o-+~yi) var Yi,

(i= 1, 2 . . . . )

(7)

(i= 1,2 . . . . )

(8)

where R is the mean value of R, var Ro is the variance of Ro, etc. It follows again from the first-order approximation that V2 : II2o + V2 o'

Vs2 -- V]% + V~o ,

(9)

where VR is the coefficient of variation of R, etc. Let 13 denote the safety index defined by t3 : (R - S)/(o~ + 05) 1/2,

(10)

then, 3 measures the distance between/~ and S i n terms of the standard deviation of the random vari-

340

M. Shinozuka, L.C. Shao, Safety o f PCR Vs

able Z = R - S. Since the probability of failure is p f = P ( R - S <~ 0), we can show that pf = P(~ ~< -13) in which ~ is the standardized random variable of Z; = ( Z - Z ) / o z . In fact, ifR andS are both normal, pf =dp(-~) as shown in eq. (2). Hence, the safety index is directly related to the probability of failure through the distribution function of ~. In this study, we use the safety index 13as a measure of safety. Obviously, the larger the value of/3, the safer the structure. It is often convenient and sometimes necessary to convert a 13 value into a corresponding probability value, although we must understand that such a value indicates at best the order of magnitude of the probability. If such a necessity arises in what follows, we use eq. (2).

3. Classification of loads The loads that are to be considered for the safety analysis of prestressed concrete reactor vessels may be classified into the following three classes according to their temporal characteristics. P e r m a n e n t load

Dw: dead load; To: operating temperature load. S u s t a i n e d load

Ft:

prestress load at time t(0 ~< t < oo); Po: operating cavity pressure load; L: live load; Dc: equipment load for refueling, etc. during regular shutdown period. S h o r t - t e r m load

Feq: earthquake load; Ta: accident-incident temperature load; Pm: maximum cavity pressure load; Pa: unpressurized crack pressure load; Pe: pressurized crack pressure load. Although those loading conditions that will contribute significantly to the ultimate failure (as defined in tiffs study) of the vessels are all listed above, obviously there are other sources of loading that may have to be included in the list for completeness. However, these sources of loading are disregarded because the probabilistic treatment of the problem will not be affected by the inclusion of these sources of loading. The operating temperature load is considered as a permanent load because it exhibits little change through

the transition from the operating period to the regular shutdown period and back to the operating period. The sustained load is defined as the load that generally maintains a constant value over either the operating period (say, 11 months) or the regular shutdown period (say, 1 month). In what follows, L g denotes the kth period of the plant operation and L~ the period for the plant shutdown immediately following Lk. The prestress load varies with time. In approximation, however, it is considered as constant during each stretch of Lk or L~ with its intensity gradually decreasing as k increases. Hence, the prestress load is a sustained load. The operating cavity pressure load, the live load and the equipment load are all sustained loads. However, the first two are active only during the plant operation while the last is to be considered only for the period of regular plant shutdown. The short-term loads are those which can occur from time to time and last for a short period of time. The earthquake load and the extremely high cavity pressure load caused by a combination of serious accidents are typical examples. The latter may represent a possible but extremely rare accident involving among others the malfunction of safety valves. Note that such an accident may follow a severe earthquake possibly with an increased (conditional) probability. The accident-incident temperature load and the maximum cavity pressure load are assumed to result from minor accidents involving cooling pumps, cooling pipes, etc. We further assume that the consequences of these accidents can be corrected and remedied within a short period of time and therefore these loads are disregarded in the analysis. The probabilistic characteristics of permanent and sustained loads can usually be described in terms of the probability distribution functions of their magnitude or intensity. For short-term loads, however, the probability laws of their occurrence in time are needed in addition. In this study, we assume that the law of occurrence is that of the Poisson arrival since the nature of its occurrence is rare and/or accidental.

4. Reliability analysis for ultimate failure For a systematic reliability analysis of prestressed concrete pressure vessels (PCRVs) for ultimate failure, we must consider combinations of a number of events in

M. Shinozuka, L.C. Shad, Safety o f PCR Vs

which some or all of the loads investigated in the preceding section will occur. For this purpose, let E12(k) be the event that permanent and sustained loads will be acting during Lk. An assumption is then made concerning the loading condition that may produce the unpressurized and pressurized crack loads, Pa and PcWe assume that such a condition arises either as a result of an extremely severe earthquake or as a consequence of the event Eac representing combinations of serious accidents independent of earthquakes. An earthquake will result in the damage and/or malfunction of a variety of degrees in the vessel system depending on its intensity, say, peak ground acceleration A. Therefore, we shall classify the earthquake into four classes I, II, III andlV: 0 < A ~ a 3 for IV. The use of the modified Mercalli intensity scale also appears possible without altering the following analysis significantly. The engineering significance of these classes are as follows. Earthquakes of class I will produce only minor dynamic effects and will not interfere with normal plant operations. Earthquakes of class II will have some dynamic effect which, however, will not result in any structural damage. Earthquakes of class III may cause local structural yielding. Finally, earthquakes classified as class IV are so severe that they may result in those serious accidents leading to Pa or Pc. If we parallel the standard terminology, earthquakes of classes I, II, III and IV will probably belong to normal, severe environmental, extreme environmental and failure load categories, respectively. Let ctk denote the maximum among the intensity values of the earthquakes which will occur in Lk and let E*(k) denote the event that a k ~< as and let Ee(k ) denote the event that as < ctk, where E*(k) includes the event of non-occurrence ot eartiaquakes in Lk. We assume that the return period of earthquake is Te and that the probability distribution function of A is FA(a) with FA(O) = 0. Then, it can be shown that the probability distribution function of ak is

F,~a) = exp [--Xk{1 -- FA(a)}],

(11)

where kk = Lk/Te. Furthermore, let Ea(k ) denote the occurrence of Eat (with return period Ta) at least once in Lk. We can now construct the following three mutually exclusive combinations of events that can occur in Lk: E(1) = El2 N E* ¢3 E*, E(2 ) = El2 N E e N E* and

341

E(3) = El2 N E* N E a where the asterisk indicates nonoccurrence of event and "k~ is suppressed for simplicity. Also, the joint event Ee N E a is assumed to be the null event because its probability is small and because the earthquake and the event Eac cannot occur simultaneously. It then follows that 3

~. P{E0.))= 1,

(12)

j=l

where P{E(2)} = P(Ee) = 1 -- F,~(a3), P{E(3)} = P(Ea) = 1 - e -u ,

(13)

with IX = # k

Lk/Ta.

=

(14)

Writing Fu for the event of ultimate failure, the probability Pk that the ultimate failure of the vessel will occur during Lk is given by 3

3

pk = ~. P{Fu n E q ) } = Z P{FuIEu)}P(Eo)}. /=l

/=l

/

~

@'°/g'/"

\ \

~.

) duct

t

~

/

k Secondary

/

/K,~Circumferential Tendons Typical Vertical / Tendon

t

20' Head

Wall

30'

I

Fig. 1. Hypothetical prestressed concrete reactor vessel.

M. Shinozuka, L.C Shao, Safety o f PCR Vs

342

Since E0) will not result in Pa or Pc, we have P{Ful EO) } = 0. Therefore, 3

Pk = ~ P{FulE(j)}P{Eo')}"

(15)

j=2

In this study, we consider a hypothetical multicavity vessel as shown in fig. 1. It is a thick-walled cylindrical concrete structure and has a main cylindrical cavity containing the reactor core. Equally spaced four secondary cylindrical cavities surround the main cavity. We assume that the vessel is constructed of high-strength concrete reinforced with high-strength deformed bars and prestressed by post-tensioning systems. In the vertical direction, a linear prestressing system is applied to develop concentric vertical prestress around the core and secondary cavities. In the circumferential direction, the prestressing is developed uniformly from the bottom to the top through the tendons applied around the outer surface of the vessel. For this hypothetical vessel, consider the modes of ultimate failure as listed in table 1. These modes of failure are all caused by excessive pressure loads Pa or Pc- The event Fu of ultimate failure can now be written

mode 4, unpressurized and pressurized crack conditions are considered. The pressurized crack condition develops as the cavity liners or liner joints break and the pressurized coolant leaks into the concrete cracks in the wall so that one or more of the crack surfaces defined in table 1 are pressurized. Therefore, it is assumed that the probability of having pressurized crack conditions is much smaller than that of having unpressurized crack conditions. We assume that the conditional probabilities for the unpressurized and pressurized conditions to occur given E(2) are respectively ra2 and rb2('~ ra2 ). Also, let ra3 and rb3(~ra3 ) denote respectively the conditional probabilities for the unpressurized and pressurized crack conditions to occur given E(3 ). Then, P{FulEU)} = rale* + rbiP~,

(/' = 2, 3)

(16)

where Pa* and Pg are conditional probabilities of ultimate failure respectively under unpressurized and pressurized conditions. The tendon stress Oia and oib for mode i respectively under unpressurized and pressurized crack conditions after the yielding of liners and rebars can be written as (see table 2 for definition of symbols):

as O'la =

4

fu=UFui,

-- A L c O L c Y -- AL1OL1Y -- W } A T ,

i=1

where Fui is the ultimate failure in mode i. In all these cases except for mode 4, the concrete in the wall cracks as the cavity pressure builds up and provides no resisting capacity as the ultimate failure occurs. For each mode of failure except again for Table 1. Modes of ultimate failure. Mode

Pressurized crack

Description

l la lb 2a 2 2b

no yes no yes

axial direction (horizontal cracks) circumferential direction (cracks through main cavity

1

3 4

3a 3b

no yes no

{7rp(R2c + 4R~) + pAD -- ABOI3y

and concrete) circumferential direction (cracks through main cavity, generator cavity and concrete) crack initiation in concrete within heads

olb=Ola + O q p ( n ( R 2 - R 2 - 4 R ~ ) Oia = (P(Rc + 27R1)

(17)

AD)/AT,

(18)

tl3Ouy - tLCOLCY

-- 7 t L l O L 1 Y } / t T , (rib = Oia + aiP(Ro - Rc - 27R1)/tT,

(19) (20)

where i = 2 and 3, 3' = 0 f o r i = 2 a n d 3 ' = 1 fori=3. As to the vessel head, we shall consider the crack initiation in concrete to be the ultimate failure. On the basis of ref. [2] and using symbols defined in table 2, the tensile stress in concrete at the critical location is written as J'i = ½{(P/3 - OOTtT/Ro) 2 + (3rp)2/(2h)2} 1/2 - ½(P/3 + OOTtT/Ro).

(21)

For the safety analysis, we compare stresses Oia and Oib with corresponding yield strength O~v of axial and o~v of circumferential tendons and ft with concrete tensile strength 3'~ by computing the (3 values as given in eq. (10). The mean values and variances of o~¥,

M. Shinozuka, L.C. Shao, Safety o f PCR Vs

343

Table 2. Description of variables. Geometrical variables Variable

Mean value

Coefficient of variation

Definition

Rc R1 Ro Rp r h AT AB ALC ALl AD tT tB tLC tL1

216 in. 90 in. 600 in. 492 in. 186 in. 240 in. 2500 in. 2 4 0 0 0 in. 2 1020 in. 2 1134 in. 2 27648 in. 2 3.4 in. 1.5 in. 0.75 in. 1.0 in.

0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01

see fig. 1 see fig. 1 see fig. 1 see fig. 1 distance between critical location and vessel axis thickness of head total cross-sectional area of vertical tendons total cross-sectional area o f vertical rebars horizontal cross-sectional area of main cavity liner horizontal cross-sectional area o f secondary cavity liners total projected area of ducts on horizontal plane average cross-sectional area of circumferential tendons per unit height of vessel average cross-sectional area o f circumferential rebars per unit height of vessel thickness o f main cavity liner thickness o f secondary cavity liner Analytical variables

t~1 ~2 ~3

0,783 0,50 0,735

0.2 0.2 0.2

coefficients such that siP = pressure acting on pressurized cracks

Material properties Variable

Mean value (psi)

Coefficient o f variation

Strength at * 2.0 percentile (psi)

Definition

oBy aLC Y --- OLl Y o~y o~y f~ OoT

94 41 241 251

384 529 622 690 761

0.1 0.1 0.1 0.1 0.1 0

75 33 192 200

yield strength of rebars yield strength o f cavity liners (main & secondary) yield strength of axial tendons yield strength o f circumferential tendons tensile strength o f concrete prestress in circumferential tendons

Ec EB ET EL

5 29 28 30

106 106 106 106

0 0 0 0

Young's Young's Young's Young's

x x x ×

000 000 000 000 605 164 500

modulus modulus modulus modulus

of of of of

concrete rebars tendons cavity liners (main & secondary)

Other variable W

43 8 2 8 500 #

0

1/2 o f vessel weight

* The probability o f a specimen to exhibit the strength less than this value is 0.02.

M. Shinozuka, L.C Shao, Safety of PCR Vs

344 ~r

OTV and ft' are evaluated from available data while those of Oia, oib a n d f t are by means of eqs (6) and (8) assuming that there is no modelling error. Under the assumption that the simultaneous occurrence of these failure modes will not affect the resisting capacities of the individual modes, we can show that

ld2

163 "6 :>., 1(5 •

has a lower bound Pl/being the largest among P{FuilEO. )} (i = 1 ~ 4) and an upper bound

c3

xQ_

4

Pu/= E

ld s

P{Fu,IEo)}.

1(56

i=1

From eq. (16), we have

P{FuiIE(j)} = raiP~ + rbiP~b,

(22)

in which P ~ and Prb are respectively the conditional probabilities of ultimate failure in mode ia and ib and P~a = P~b for mode 4 because of its independence on pressurized crack conditions. Since quite often one o f the failure modes dominates; Pl/ and Pui are frequently of the same order of magnitude. Once P{FuIEq)} is estimated, the probability pg is evaluated from eq. (15). The probability of failure for the entire service life of the vessel is then evaluated in approximation as N

Pf = ~. Pk, k=l

where N is the number o f operating periods in the service life.

5. Numerical example The/3 values associated with individual failure modes are computed following the procedure described in the preceding section. The mean values and coefficients o f variation o f various variables are listed in table 2. As to the probabilistic nature o f earthquake, we assume that the peak acceleration has the Weibull distribution

FA(a) = 1 -- exp {-- [(a -- ao)/C] B},

a >1a o.

(23)

An estimation for return periods for various values of

1(57 1(58

0

1000

2000

3000

4000

5000

6000

mean cavity pressure ~ (psi)

Fig. 2. Probability of ultimate failure as a function of mean pressure.

peak accelerations for San Francisco Bay Area is fitted to eq. (23) with Te = 8 yr, a o = 0.05 g (g = acceleration due to gravity), B = 0.85 and C = 0.045 g. In this study, however, we consider a plant at a location o f less seismic activity with Te = 20 yr, ao = 0.05 g, B = 0.85 and C = 0.02 g. Under the further assumption that a 3 = 1.0g, we obtain P(Ee)= P(E(2)) = 1 x 10 -13. The excessive pressure build-up p resulting from Pa or Pc is treated as a random parameter: the mean value /~indicates our estimate for such pressure build-up. The coefficient of variation Vp reflects the degree o f belief in this estimate. Fig. 2 shows the probability o f ultimate failure based on the 13value plotted as a function of/S with Vp = 0.10. The probabilities associated with curves ia and fb represent respectively/~za and P~b (i = 1, 2, 3) while curve 4 corresponds to P~a = e~bFor/~ = 1000 psi, reading probability values from fig. 2 and using eq. (22) under the assumption o f ra2 = 10 -2 and rb2 = 10 -4, we obtainP{FulE(2)} = 2 x 10 -s and therefore P(FulE(2))P{Eo)) = 2 x 10 -is. The probability of ultimate failure o f the vessel resulting from earthquake-induced accidents for the service life o f

M. Shinozuka, L.C. Shao, Safety of PCR Vs 40 yr is then 8 x 10 -17. Such a level o f probability can be regarded as representing impossibility. If it is assumed that ra3 = ra2 and rb3 = rb2, then we obtain P{FuIE(3)) = 2 x 10 -s. Since P(E(3)) = 1 - exp (Lk/Ta), the return period Ta o f Eac is a significant quantity. Purely for the sake of illustration, consider Ta --- 10 000 years. Even for such a large return period, P(E(3)) = 10 -4 and P(FulE(a)}P(E(a)} = 2 x 10 -'9. Then, tile probability of ultimate failure for the service life resulting from earthquake-independent accidents is a commanding 8 x 10 -8. Other combinations o f various parameter values obviously result in different probability values. A future study should be undertaken to determine

345

realistic values for the parameters involved, particularly in the earthquake-independent accidents Eac.

References [1] A.H.-S. Ang and C.A. Cornell, Reliability bases of structural safety and design, In" Modern Concepts of Structural Safety and Design, Meeting Preprint 2023, ASCE National Structural Engineering Meeting, San Francisco, Apr. (1973). [2] S.L. Paul et al., Strength and behavior of prestressed concrete vessels for nuclear reactors, Vol. 1, Structural Research Series 346, University of Illinois, Urbana, Illinois, July (1969).