ML20091D206
| ML20091D206 | |
| Person / Time | |
|---|---|
| Site: | Indian Point  |
| Issue date: | 05/31/1984 |
| From: | Chang M, Hwang H, Kawakami J, Reich M BROOKHAVEN NATIONAL LABORATORY |
| To: | NRC OFFICE OF NUCLEAR REGULATORY RESEARCH (RES) |
| References | |
| CON-FIN-A-3226 BNL-NUREG-51740, NUREG-CR-3641, NUDOCS 8405310082 | |
| Download: ML20091D206 (51) | |
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j i iEG/CR 3641
[. NUAEG 51740 i
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RELIABILITY ASSESSMENT l
OF INDIAN POINT UNIT 3 CONTAINMENT STRUCTURE l
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J. Kawakami. H. Hwang, M.T. Chang, and M. Reich t
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Date Published - January 1984 l
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STRUCTURAL ANALYSIS OlVISION i
DEPARTMENT OF NUCLEAR ENERGY, BROOKHAVEN NATIONAL LABORATORY UPTON. LONG ISLAND. NEW YORK 11973 l
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NUREG/CR 3NI BNL NUREG 51740 AN.RD i
RELIABILITY ASSESSMENT OF INDIAN POINT UNIT 3 CONTAINMENT STRUCTURE i
1 J. Kawakami, H. Hwang, M.T. Chang, and M. Reich 1
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Manuscript Completed October 1983 Date Pubilshed January 1984 i
STRUCTURAL ANALYSIS OlVISION I
DEPARTMENT OF NUCLEAR ENERGY BROOKHAVEN NAT10NAL LABORATORY I
UPTON, LONG ISLAND, NEW YORK 11973 1
l PREPARED FOR UNITED STATES NUCLEAR REGULATORY COMMISSION WA8HINGTON, D.C. 20665 CONTRACT NO. DE-AC02 76CH00016 FIN A 3226
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I Abstract In the current design criteria the load cabinations specified for design of concrete containment structures are in the deterministic formats. However, by applying the probability-based reliability method developed by BNL to the con-crete containment structures designed according to the criteria, it is possi-l ble to evaluate the reliabiltty levels implied in the cu/ rent design criteria.
For this purpose, the reliability analysis is applied to the Indian Point Unit-No. 3 contatronent.
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The details of the contatronent structure such as the geometries and the rebar arrangements, etc., are taken from the working drawings and the final safety j
analysis reports. Three kinds of loads are considered in the reliability I
analysis. They are, dead load (D), accidental pressure due to a large LOCA (P), and earthquake ground acceleration (E). Reliability analysts of the containment subjected to all combinations of loads is performed. The results are presented in this report.
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Table of Contents Page i
Abstract...............................................................
til List of Tables.........................................................
vit ListofFigures........................................................
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A c k n ow l ed g e me n t........................................................ ix l
1.
I N TR OD UC T I O N.......................................................
1 2.
CO N TA I NiE N T DE 5C R I P T I O N............................................
l 2.1 Ge um 0 t ry......................................................
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- 2. 2 Design Luads..................................................
2 2.3 R e b a r A r r a ng em e n t s............................................
3 1
l 3.
CO N T A l hM E Ki H00E L L t dG..............................................
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4.
MA T E R I AL PR O PE R T I E S................................................
5 5.
PR UD A B I L I S T IC HO DE LS F O R L 0AD S.....................................
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l 5.1 D e ad L o ad.....................................................
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5.2 Ac c i den t a l Pr e s s ur e...........................................
6 5.3 E a rt hqua k e Gr ound Acc el e ra tion................................
7 6.
FINITE ELEME NT ANAL YS IS OF THE C0NTAt hME NT.........................
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6.1 $ t a t i c An al y s i s............................................... 10 i
6.2 Dynamic Analysis..............................................
10 7.
L l H I T S TAT E FOR T H E C0 N T A l hME N T.................................... 10 1
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l TableofContents(Cont'd)
D U.
R EL I A B I L I TY ANAL ) 515 R E SU L T 5....................................... 11 8.1 De ad Load and Accidental Pres sure (D +P )....................... 12 8.2 Dead Load and Earthquake Gr oun d Accel e ra t i on (0 +E )..................................... 12 U.3 Dead Load. Earthquake Ground Acceleration a nd Ac cid en t al Pr es sur e (D + E +P )............................... 13 8.4 Ove r al l L imi t 5ta t e Probab i li ty............................... 14 9.
R E SULTS F OR D IF FER E NT L IMi t S TAT E.................................. 14
- 10. CO N" L U0 t hG R EM A RK S................................................. 15 References.............................................................
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4 VI.
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Ll5T OF TABLES 1
Iable M
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1 Cy l i nde r R e i n f o r c um e n t..........................................
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D omo R e i n fo r c e re n t...............................................
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L o ad P a r am e t e rs..................................................
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S t r e s s e s D ue t o De a d L o ad........................................
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5 5 t r e: se s Due t o Un i t Pr e s s ur e....................................
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Na t ur a l F r eq ue n c i e s..............................................
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Coordinates of Limit state Surface (Elerent 289 - 312 Loc a l X-0 t r ect i on )...........................
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Condi tional Limi t 5 tate Prob 4bility p(D +P),,,,,,,,,,,,,,,,,,,,,,,
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9 Condi t ional Limi t State Probabili ty p(D+E ).......................
26 10 Condi ti onal Limi t Sta te Probabli t ty p(0 +E *P ).....................
21 Lifetime Limit State Probabilities (BasedonLinearStress01stributton)............................
28 12 Lt fetime Limit State Probabilities r
i (Ba sed on Nonli near stres s 01 str ibut t on)..........................
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LIST OF FIGURES Ftgure Page 1
Cros s Sect ion of Conta lment (Sect ion A-A).......................
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Cros s Sect ion of Conta i nnen t (Sect ion B-B ).......................
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Sk e tc h of Cont a l men t st ru c t ur e..................................
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Tire Histories of Design Pressure and Tertperature................
33 Typical Rebar Arrangement for Cf i t ndrical Wal l...................
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He r i dio n al R eba r Arra ngeren t (Doce )..............................
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Three-Dimens ional Model f o r Conta lmen t..........................
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S ide Vi ew o f Con ta i ncent Model...................................
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T op V i ew o f C o n t a t m e n t Mo de l....................................
ad 10 C r os s Se ct io n o f Con t a i nmen t Model...............................
39 11 Three Dimensional $ ketch of Mode Shapes (Modes 1 and 2)..........
40 12 Three-Dimensional 56 etch of Mode Shapes (Modes 15 and 16)........
41 13 Limit St att Surf ace (Elements 289 - J12, Local X Direction)......
42 14 Condtttonal Limit State Probability Distributtons................
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ACKNOWLEDEME NTS The authors wish to exprest their appreciation to Mr. H. Ashar of the Nuclear Regulatory Commission for his advice and support during various phases of this study. The authors also wish to thank the Power Authority of the State of New York and United Engineers and Contractors for providing the drawings and test data.
Thanks are due to Mr. D. $umannakate for preparing the figures and to Ms.
Diana Votruba for the typing of this manuscript.
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t 1.
INTRODUCTION Concrete contalment structures in the United States are currently designed j
I according to the A5M codel and other supplementary requirements such as
$tandard Review Plan ($RP)2, etc. The load cabinations specified in these criteria are in the deterministic format and the reliablitty levels implied in l
the load cabinations are not stated explicitly. For tha safety evaluation of the nuclear structures, however. It is important to know these reliability levels.
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l The Structural Analysis Olvision of Brookhaven National Laboratory (BNL) has j
been developing a probability based reliability analysts mthodology for nu.
l clear structures, particularly for concrete contalment structures.3*6 An j
tmpurtant feature of this methodology is the incorporation of finite elemnt
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analysts and rende vibration theory. By utillaing this method, it is pos-I sible to evalutte the safety of nuclear structures under various static and dynetc loads f 1 terms of limit state probability.
By applying the reliability analysts method to the concrete contatraent struc=
l tures designed according to the criterta antioned above, it is possible to l
evaluate the reltability levels implied in the current design criteria. For j
this purpose, the reltability analysis is applied to the Indian Point Unit No.
l J contatraent structure. The results of the reltability analysts are pre-sented in this report.
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2.
CONTAINPENT DC$CRIPf a0N 2.1 Geometry The Indian Point unit No. 3 nuclear power plant employs a pressurtied water I
reactor nuclear stem supply system furninnet by Westinghouse Electric Cor.
poration. The containment structure completely encloses the entire reactor i
and reactor coolant synta and ensures that essentially no leakage of radioac.
tive meterials to the environannt would result even if gross fatture of the reactor cootent systm were to occur. The structure also provides biological shielding for normal and accident situations.
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1 Two typical cross-sections of the containment are shown in Figs. I and 2. The reactor containment structure consists of a vertical right cylinder with a hemispherical dame on the top. The cylinder-dome system is built on a base-nut with thickness 9'-0".
Thus, the cylinder-dome system is considered to be fixed at the base in the present analysis for simplicity. The concrete con-tainrent structure is illustrated in Fig. 3.
The thickness of the dome is equal to 3'-6", whereas the thickness of the cylindrical wall is 4'-6".
The inside radius of the dome and the cylinder is equal to 67'-6".
The height of the cylindrical wall is 148'-0" and the total height of the containment is 219'-0". These dimensions are also shown in Fig. 3.
2.2 Design Loads The containment structure is subjected to various static and dynamic loads during its lifetime. In this study, only three types of loads are taken into consideration. They are: dead load, accidental internal pressure and earth-quake ground acceleration. From reviewing the drawings and the Final Safety Analysis Report (FSAR)7, it is found that there is no live load acting on the containnent structure.
The dead loads arise mainly from the weights of the dome and the cylinder.
3 The weight density of the reinforced concrete is taken to be 150 lb/ft,
The accidental internal pressure is assumed to be caused by a large Loss of Coolant Accident (LOCA). The time history of the accidental pressure is shcwn in Fig. 4, which is taken from FSAR. The accidental pressure is considered as a quasi-static load and is uni formly v.istributed on the containment wall. The design value of the pressure is 47 psi.
From FSAR, the design value of the ground acceleration for the Operational Ba-sis Earthquake (OBE) is determined to be 0.1 g applied horizontally and 0.05 g-applied vertically. Additionally, the ground acceleration for the Design Ba-sis Earthquake (DBE) is determined to be 0.15 g horizontally and 0.10 g verti-cal ly.
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2.3 Rebar Arrangements The contaiment wall is reinforced with hoop, meridional and diagonal rebars.
A typical rebar arrangement for the cylindrical wall is shown in Fig. 5.
The hoop and meridional rebars are divided into two groups and each group is placed close to the wall surface. The diagonal rebars are only in one group, which is placed close to the outer surface.
The details of the rebar arrangements for the cylindrical portion of the con-tainment are described below. Each group of the hoop reinforcement consists of two layers of No.,18 rebars with 14 inch spacing and remains constant throughout the cylindrical portion of the containment. There are two kinds of the meridional rebars: primary and secondary.- Each group of the primary meridional rebars contains one layer of No.18 rebar with 12 inch spacing.
The primary rebars remain constant in the cylindrical portion and extend into the dome of the containment. The secondary meridional rebars are placed at the bottom third of the cylinder, i.e., from the basemat to the elevation of 54'-10" (zero elevation at base is used.in this report). The amount of sec-ondary rebars is varied at different elevations as summarized in Table 1.
It is noted that there is one out of six No.18 secondary rebars continued from the bottom to the top. For simplicity, it is replaced by smaller size rebars in Table 1.
In addition some secondary bars are bent at the lower portion of s
the contairment, these bends are neglected.
For the dome portion of the contaiment, each group of the hoop reinforcements consists of one. layer of No.14 rebars with 8 inch spacing for vertical angle from zero (spring If ne) to 55*.
However, spacing of 8-1/4 inch is used near the outside face from zero to 9.5*.
For vertical angle from 55*. to 90' (top),
one layer of No.14 rebars with 9 inch spacing is used.
The meridional rebar arrangements in the dome portion is sketched in Fig. 6.
The primary meridional reinforcement of the cylinder extends into the dome along the meridians. The distance between _ meridional bars along the parallel decreases as one moves up' from the springline, and therefore, the quantity of....
~ ~ ~ steel per unit length along the meridian increases above the design value.
When this later steel quantity becomes twice the design value, i.e., at 60" from the springline, each pair of meridional bars are combined into one bar, by the help of appropriate transition splices. The reinforcenent quantity is halved again at 75*, 83* and 86* from the springline. Thus, the quantity of meridional steel per unit length along the parallel is kept between the design value and twice the design value, to avoid reinforcement congestion and waste of material. A summary of dome reinforcements is shown in Table 2.
A layer of diagonal rebars at +45* and -45* with the vertical is placed near the outer surface to take the in-plane seismic shear forces as shown in Fig.
5.
These seismic diagonal bars extend into the bottom third of the dome. The amount of reinforcing is varied at different elevations and is shown in Tables 1 and 2.
3.0 CONTAINMENT MODELLING In order to utilize the finite element analysis results in computing the limit state probabilities, the containnent modelling should be made in such a way that the local coordinates of the elements nave the same directions as those of the rebars. This is very easy to achieve in this study, since all the rebars are in the hoop ano meridional directions.
The finite element utilized in the analysis is the shell element as described in the SAPV conputer code. A three-dimensional finite element model as shown in Fig. 7 is used for the structural analysis of the containment. The side and top views of the containnent are shown in Figs. 8 and 9, respectively.
Also, a detailed cross-sectional view of the containment is shown in Fig.10.
As can be seen from this figure, the containnent is divided into 23 layers.
Except at the top of the dome, each layer has 24 elements such that the nodal points are taken every 15* in the circumferential direction. This discretiza-tion requires a total of 553 nodes and 540 elements.
l The boundaries of the elements are made such that it matches the change of the rei nforcement s.
Hence, the amount of reinforcenents in most of the elenents is the same as shown in Tables 1 and 2.
However, the meridional rebars in the done portion are varied in different elevations, an average value of the meri-dional reinforcement in these elements is used. It is noted that the diagonal rebars, which provide in-plane seismic shear resistance, are not included in the present analysis.
A welded steel liner with a minimum thickness of 1/4-inch is attached to the inside face of the concrete shell to insure a high degree of leak-tightness.
The liner is disregarded as a load carrying structural component in the analy-sis. Furthermore, such other complications as penetrations, personal lock and equipment hatches are not included in the study.
4.0 MATERIAL PROPERTIES In order to perform a reliability analysis on a containment structure, it is necessary to determine the actual material properties. In the present seidy, the mean values of the material properties are used in the analysis. The variation of material properties will be included in the sensitivity studies in the future. The properties for the concrete and rebars are summarized as follows:
A) Concrete The minimum compressive strength of concrete at 28 days used for the Indian Pont Urdt No. 3 ccetairenent is 3000 psi. The weight density of the concrete 3
is taken to be 150 lb/ft. Young's modulus and Poisson's ratio are 3.1 x 106 psi and 0.2, respectively. For the 28 day compressive strength f, a statistical analysis of the available data was carried out at BNL to c
detennine its statistical characteristics. The mean value and standard deviation are estimated to be 4896 osi and 627 psi, respectively. -
B) Reinforcing Bars As can be seen from Tables 1 and 2, No.18 rebars are the main reinforcement used in the containment structure. Hence, the statistics for No.18 rebars is used to represent all other types of rebars. Young's modulus and Poisson's ratio are taken to be 29.0 x 106 ps! and 0.3, respectively. A statistical analysis was carrid out for the No.18 rebars, the mean and standard devia-tion of the yield strength fy are estimated to be 71.8 and 5.18 ksi, respec-tively.
5.
PROBABILISTIC MODELS FOR LOADS Various static and dynamic loads act on the contaiment structure during its li fetime. These loads may be caused by normal operating, environmental and accidental conditions. Since the loads intrinsically involve random and other uncertainties, an appropriate probabilistic model for each load must be estab-lished.
5.1 Dead Load As mentioned in Section 2.2, the dead load primarily arises from the weights of the containment wall. It is noted that there are some uncertainties as to the actual magnitude of.the dead load.8 For the purpose of the this analy-sis, however, dead load is assumed to be deterministic and is equal to the~
design value, which is cmputed based on the weight density of reinforced 3
concrete as 150 lb/ft,
- 5. 2 Accidental Pressure The accidental pressure is considered as a quasi-static load and it is uni-
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fonnly distributed on the containment wall. The accidental pressure is idealized as a rectangular pulse and will occur in accordance with the Poissen law during the coataiment lifc. Under these assumptions, three parameters are required to model the internal pressure: the occurrence rate Ap (per year), the mean duration udP (in seconds) and the intensity P..
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For the Indian Point Unit No. 3 containment, the mean duration is taken to be 1200 seconds. This value is obt31ned from the approximation of the time his-tory shown in Fig. 4.
According to the Indian Point Probabilistic Safety 9
Study, the mean occurrence rate for a large LOCA is 2.16 x 10-3/yr, which is used in the analysis.
The intensity of the accidental pressure is treated as a Gaussian random var-iable. Unfortunately, there is no actual data to determine its statistics, i.e., the mean value and the standard deviation. Nevertheless, the consensus survey of the nuclear structural loads, which was carried out by BNL, indi-cates that the ratio of the mean value to the design value is 0.89 and the coefficient of variation is 0.12.10 Since the design value of the acci-dental pressure for tM Indian Point Unit No. 3 containment is 47 psi, the mean value, P, and the standard deviation, ap, are computed as follows:
P = 47 x 0.89 = 41.83 psi p = 0.12 x 41.83 = 5.02 pst o
5.3 Earthquake Ground Acceleration The eartnquake ground acceleration is assumed to act only along the global x direction. It is further assumed that the ground acceleration can be idealized as a segment of finite duration-of 'a stationary Gaussian process with mean zero and a Kanai-Tajimi spectrum. The Kanai-Tajimi spectrum has the following expression:
1 + 4c (,f,9)2 2
U 99"(w)
S
=S (1) 1 - (w/w )2 2 + 4g (w/w )2 g
g where the parameter So represents the intensity of the earthquake and w g and cg are the dominant ground frequency and the critical damping, respec-t i vely.- The values of og and cg depend on the soil conditions of the con-taiment site. For the Indian Point Unit No. 3 Power Plant, 'the soil condi-tion is determined as rock.7 For such a soil condition, Reference 11 recom- ~-
mends that the values of wg and cg in Eq.- 1 are equal to 8x rad /sec and -
0.6, respectively. The mean duration I'dE of the earthquake acceleration is assumed to be 15 seconds in this study. The peak ground acceleration A,
1 given an earthquake, is assumed to be A1=pog g where pg is the peak factor which is assumed to be 3.0 in this study. The standard deviation of the ground acceleration, o, is conputed by integrating the Kanai-Tajimi
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g spectral density function with respect to e, Explictly written, the standard deviation o is g
g"kwg (2
+ 2c )
(2) o g
The peak ground acceleration A, given an earthquake, can be rewritten as 1
A1 = a (,
(3) g where
=pfu(2c + 2c )
(4) a g
g g
g If the earthquake occurs in accordance with the Poisson law at a rate AE per year, it is easy to show that the probability distribution F (a) of the A
annual peak ground acceleration A is related to the probability distribution FA(a) of A in the following fashion.
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F (a) = exp {-A El - FA (a)D A
E 1
or (5)
FA(a)'=1+
fn F (a)
A g
Therefore, if a indicates the minimum peak ground acceleration for any o
A1(a ) = 0 and hence, ground shaking.to be considered an earthquake, F o
Assuming that F (a) is of the extreme distribution.
AE = - in F (a ).
A o A
of Type - 11, i.e.,
F (a) = exp[-(a/u)-Q]
(6)
A where a and n are two parameters to be detennined. By least square fitting to 9
the harzard curve given in the PRA study, we find a = 3.14 and u = 0.0135.
From Eqs. 5 and 6, we obtain:
FA (a) = 1 - (a/a p a)ao W
o y
E = 1.64 x 10-2/ year for Under these conditions, one finds that A ao = 0.05 g.
Combining Eqs. 3 and 7, and writing Z for
, we further obtain the probability distribution and density functions of Z in the forms, respectively, 1
F (z) = 1 - (ag /a ) a Z
z o (8) for z 2 a /ag o
gz/a )-( +1) f (Z) * "( g/a )(o 2
o o
The infonnation about the maximum earthquake ground acceleration, amaxe which represents the largest earthquake possible to occur at a particular site, is needed in order to detennine the limit state probability.
In this study, amax is chosen to be equal to 0.71 g.
The parameters of the loading conditions described in this section are summarized in Table 3.
6.
FINITE ELEMENT ANALYSIS OF THE CONTAIPMENT When a reinforced concrete containment is subjected to static and dynamic-loads, its cross-section will usually produce cracks, the extent of. which
~ depends on the load history. Because of the complexity of the various load combinations, however, it is difficult tc predict a priori the crack patterns for all conceivable combinations of loadings. Wilile a linear elastic analysis cannot take into account the temporal variations of the structural stiffness which result from such a dependence on load-history,-it will nevertheless, in most instances yield approximately correct stress resultants for the various i
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sections of the structure. This is especially the case if the section materi-al properties are adjusted to reflect the conc.ete cracking. To account the cracking effect, the stiffness of the element is taken to be one half of that i
of the uncracked section in this study.
ti.1 Static Analysis As mentioned in Section 5, dead load and accidental pressure are considered to be static loads acting on the containment. Using the finite element model de-scribed in Section 3, a static analysis of the containment due to dead load alone was performed, and results are shown in Table 4.
Similarily, the anal-ysis of the containnent due to unit accidental pressure alone is also carried out and the results are shown in Table 5.
These results are going to be used in the reliability analysis.
6.2 Dynamic Analysis For dynamic analysis of structures, modal analysis is employed. Hence, the dynamic characteristics of the structures are represented by the natural fre-quencies and associated mode shapes. Using the model described in Section 3, the first twenty (20) ratural frequencies are evaluated ar.d are sho'wn in Ta-ble 6.
The mode shapes for the first and second pairs of bending modes (modes 1,2,15,16) are shown in Figs.11 and 12.
It is important to choose the significantly participating modes for the reliability analysis. In this study, only the first and second pairs of bending modes are included in the analysis.
7.
LIMIT STATE FOR THE CONTAlf#1ENT For the reinforced concrete containment, if an onset of the structural failure is of interest, the limit state may be defined as follows. - The structural re-sponse is considered to have reached the limit state if the rebars begin to yield (in tension or compression) and/or if the crushing strength of the con-crete is reached at the cross-section's extreme fiber during the service life of the containment.
The analytical expressions for the limit state introduced previously are as follows:
(9) fs 1 fy fc 1 0.85 fc (10) is the steel yield strength, where fs is the stress in the rebars and fy fc is the conpressive concrete stress at the extreme fibers and fc is the concrete compressive strength.
Based on, (a) the above definition of the limit state, (b) the assumption of a linear stress-strain relationship, and (c) the conventional theory of rein-forced concrete, which asserts that concrete cannot take any tension, the limit state surface in tenns of the membrane stress t and bending moment m/ unit length can be established for a specific cross-section at the finite element boundaries.12 A typical limit state surface for a element is shown in Fig.13, and the coordinates are listed in Table 7.
Point 'a' represent a limit state under pure (uniform) compression and point 'g' a limit state under pure (uni form) tension.
In the same figure, lines I (aE and li3'), lines II (approximated by isi and c'e'), lines III ('ei and e'f') and lines IV (fg and f'g) indicate those parts of the limit stete surface in which the
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limit states are reached in concrete crushing with cross-sections remaining uncracked (lines I), in concrete crushing with partially cracked cross-sec-tions (lines II), in yielding of rebars in tension with partially cracked cross-sections (lines III) and in yielding of rebars in tension with totally.
cracked cross-sections (lines IV).
8.
RELIABILITY ANALYSIS RESULTS l
.The reliability analysis methodology used in this study is summarized in Ref.
3.
Based on this reliability analysis method,'the structural model, loading conditions and the limit state described in the preceding section, the re-liability analysis for the Indian. Point Unit No. 3 containment structure was
'made.
The results are presented in this section.
( t W
W
8.1 Dead Load and Accidental Pressure (D+P) fue load characteristics of the dead load und the accidental pressure due to a large LOCA are described in Sections 5.1 and 5.2, respectively. A reliability analysis is carried out to estimate the probability that the limit state will be reached under the simultaneous action of this accidental pressure and the dead load during the forty year lifetime of the contaiment.
The conditional limit state probability for each element is tabulated in Table 8 and plotted in Fig.14. From the table, it can be seen that the critical olements are the elements denoted as 289 to 312, which are located in the first layer of the dome section just above the spring line. Since the struc-ture and the loads are both axisymmetrical in this case, all the elements located at the same level have the same limit state probability. The limit i
state is reached as the hoop rebars in the critical elements begin to yield.
When these f.onditional probabilities are multiplied by the expected number tap = 8.64 x 10-2 of such simultaneous occurrences during the contaiment 11fe of forty years, the unconditional limit state probabilities P (D+P) f under the simultaneous action of D and P is obtained, and is equal to 6.79 x 10-6, 8.2 Dead Load and Earthquake Ground Acceleration (D+E)
An idealization of the earthquake ground acceleration is presented in.Section 5.3.
For the conbination of the dead load and earthquake ground acceleration, the conditional limit state probability P(D+E) for the containment are shown in Table 9 and plotted in Fig.14. From the tabic, it can seen that the cri-tical eierents are elements 6, 7,18 and 19. These elements are located in-the lowest finite element layer and immediately adjacent to the global x axis.
In this case, the limit state.is rcached as the meridional reinforcing bars in the critical elements begin to yield. The locations of the critical elements and the manner in which the limit state is reached are obviously consistent with the structural and loading symmetry with respect to the x-axis under this particular load combination. The lower and upper bounds of the condi-tional limit state probability P(D+E) are found to be 0.92' x 10-7 and 1.64 x 10-7, respectively. N1tiplying these bounds by the expected number tie = 6.56 x 10-1 of simultaneous occurrences of D and E, the lower and upper bounds of the unconditional limit state probability P (D+E) under this f
load combination during the containment 11fe of forty years are obtained as 0.60 x 10-7 and 1.07 x 10-7, respectively. Since these two bounds are in a relatively narrow range within the same order of magnitude, any value be-tween these two bounds may be used as a reasonable approximation for the limit state probability.
- 8. 3 Dead Load, Earthquake Ground Acceleration and Accidental Pressure (D+E+P)
The probabilistic characteristics of the loads indicated in the sub-section title above are described in Section 5.
Under the combination of these loads, the conditional limit state probability are listed in Table 10 and also plot-ted in Fig. 14.
The critical elements are found to be elements 294, 295, 306 and 307, which are located immediately adjacent to the global x-axis (when projected onto a horizontal plane) at the first layer above the springline.
This is the same level at which the critical elements are found under the D+P load conbination (Section 8.1).
The manner in which the limit state is reached is also the same as for the D+P combination (i.e., yielding of the hoop rebars). The lower and upper bounds of the conditional limit state probability P(D+E+P) under this load combination are 4.89 x 10-4 and 5.80 x 10-4, respectively. N1tiplying these bounds by TAD $E+P
= 5.46 x 10-8 provides the lower and upper bounds of the unconditional probability under this load combination during the containnent life of forty years; the lower bound = 2.67 x 10-11 and the upper bound = 3.17 x 10-11 Comparing the limit state probabilities under the load combination D+P with those under the conbination D+E+P, it is observed that (1) the mode in which the limit state is reached (yielding of the hoop rebars) in the critical elements is the same and (2) the critical elements under the current load combination D+E+P comprise the four elements which are most stressed by the additional earthquake load aniong those critical elements under the load com-bination D+P.
These observations suggest that the accidental pressure P.is a I N-
dminant factor ir controlling the conditional limit state probability. The substantial reduction in the values of the lower and upper bounds of the un-conditional probability P/D+E+P), as compared with P/D+P), is primarily attributable to the fact that TD+E+P = 5.46 x 10-8 is much smaller than T AD+P = 8.64 x 10-2, 8.4 Overall Limit State Probability The limit state probabilities evaluated in the preceding section are those at the critical elements within the containment under various load combinations.
While the limit state probability of the contaiment as a whole, or the system limit state probability, under a certain load combination is always larger than that of the critical elements, the author's experience in structural re-liability analysis suggests that the difference between the system limit state probability and the limit state probability of the critical elements is toler-able for the type of load-structure system under consideration. There fore, for the sake of analytical simplicity and camputational economy, the present study approximates the containment limit state probability under each load cabination by the critical element limit state probability as evaluated in Sections 8.1 to 8.3.
The limit state probabilities, conditional and uncon-ditional, under various load cabinations are si.mmarized in Table 11. Unfier the assumption that the containment will not fail under dead load alone, the overall containment limit state probability Pf is then obtained as the sum of the limit state probabilities under all these (mutually exclusive) load c m binations. Hence, the containment limit state probability is 6.85 x 10-6
- 6.90 x 10-6 for its lifetime of forty years.
9.
RESULTS FOR DIFFERENT LIMIT STATE The results presented in the previous section (i.e., Section 8) are corre-sponding to the onset nf the structure failure. If more substantial failure of structures is of interest, the limit state representing this condition can be defined based on the reinforced concrete ultimate strength theory. Essen-tially, the limit state is reached when a maximum compressive strain at the extrane fiber of the cross-section is equal to 0.003, while the yielding of rebars is permitted. Under this definition of the limit' state, a limit state surface for each element may be constructed. Utilizing this limit state sur-face, the reliability analysis of Indian Point Unit 3 containment undr the loading described in Section 5, is also carried out. The results are sum-marized in Table 12. For most load combinations, the limit state probabili-ties in Table 12 is about one order of magnitude less than those shown in Table 11, while the critical elements remain the same.
- 10. CONCLUDING REMARKS The reliability analysis method developed by BNL is applied to the Indian Point Unit No. 3 contalment structure under dead load, accidental pressure and earthquake ground acceleration. The results are presented in this report.
This is the first attmpt to carry out the reliability analysis for the exist-ing containment structures in order to evaluate the reliability levels implied in the design criteria. It is noted that the estimated reliability levels are affected by the judgements made in the design process and the assumptions made in the reliability evaluation. In order to reasonably assess the reliability levels implied in the design criteria, it is necessary to continue the efforts by carrying out the reliability analysis for other existing contaiment struc-tures designed according to these criteria.
i 1
l,
l REFERENCES 1.
ACI-ASME Joint Technical Committee, " Code for Concrete Reactor Vessels and Containments, ASME Boiler and Pressure Vessel Code Section 111 -
Divi sion 2",1980.
2.
USNRC, Standard Review Plan Section 3.8.1, NUREG-0800, Rev.1,1981.
3.
Hwang, H., Shir.ozuka, M., Brown, P. and Reich, M., " Reliability Assess-ment of Reinforced Concrete Containment Structures", BNL-NUREG-51661, MREG/CR-3227, February,1983.
4.
Kako, T., Shinozuka, M., Hwang, H. and Reich, M., " FEM-Based Random Vibration Analysis of Nuclear Structures Under Seismic Loading",
SMiRT-7 Conference Paper K 7/2, Chicago, IL, August 22-26, 1983.'
5.
Shinozuka, M., Kako, T., Hwang, H. and Reich, M., " Development of a Reliability Analysis Method for Category I Structures" SMiRT-7 Con-ference Paper M 5/3, Chicago, IL, August 22-26, 1983.
6.
Shinozuka, M., Kako, T., Hwang. H., Brown, P. and Reich, M., " Estimation of Structural Reliability Under Combined Loads", SMiRT-7 Conference Paper M 2/3, Chicago, IL, August 22-26, 1983.
7.
Power Authority of the State of New York, " Indian Point Nuclear Power Plant, Unit 3, Final Safety Analysis Report", Docket 50286.
8.
Hwang, H., Wang, P.C. and Reich, M., "Probabilistic Models for Opera-tional and Accidental Loads on Seismic Category-1 Structures",
BNL-NUREG-51682 NUREG/CR-3342, June 1983.
9.
Power Authority of the State of New York, " Indian Point Probabilistic Safety Study", Docket 50247, March 1982.
i REFERENCES (Cont'd)
- 10. Hwang, H., Wang, P.C., Shoaman, M. and Reich, M., "A Consensus Estima-tion Study of Nuclear Power Plant Structural Loads", BNL-NUREG-51678, NUREG/CR-3315, May 1983.
- 11. Ellingwood, B., Satts, M.E., " Characterization of Earthquake Forces for Probability-Based Design of Nuclear Structures", BNL-NUREG-51587, NUR EG/CR-2945, Sept.1982.
- 12. Chang, M., Brown, P., Kako, T., Hwang, H., " Structural Modeli ng and Limit State Identification for Reliability Analysis of RC Containment Structures", SMIRT-7 Conference Paper M 3/2, August 22-26, 1983.
J 4
4 J -, -,
m.
.. _ _,. ~. - _ _. _. _., _ _. _. _. _. _.. _ _. _ _....... -.... _. _... _
Table 1 Cylinder Reinforcement.
i 4c Elevation Hoop Heridional Diagonal Primary Secondary
[
O to 25'-0"
' 2#18 914fn 1#18 012in 1#18 012in 1#18 0 30in L
~ 25 '-0" to 45 '-5" 1 #11 0 12in -
'45 '-5" to 50 '-3" 2#11'0 361n l'
~.m 50 '-3" to 54 ' -10" -
1#11 0 361n.
54 '-10" to 110 '-6" 110 '-6" to ' 148 '-0" 1#18 9 60in + 1#14 0 60in
.I t
t y
p, a-_,
Table 2 Dome Reinforcement.
2, Angle From Spring-Line (Degrees)
Hoop Meridional Diadonal 0*-9.5*
' 9 8-1/4" outside 1# 18 9 0.796*
1#18+1#14 9 4' (horiz. dist.)
1#14 '
, 9 8.0" inside 9.5'-18.5*
1#14 9 8.0 in J.
18.5*-35' 1#11 9 2* (borz. di st.)
3 5*-5 5* -
55'-60*
1#14 9 9.0 in 60*-75*
1#18 9 1.593*-
75*-83*
1#18 9 3.136' T
83*-86*
1#18 9 6.37' 86*-90*
1#18 9 12.74*
l l
l Table 3 Load Paraneters.
l t
[
l Load Load Paraneters l
l Dead Load (D)
- Deterministic and time invariant i
l Accidental Pressure
- 0ccurrence rate Ap = 2.16 x 10-3/ year i
due to a LOCA (P) 4tean duration udp = 1200 seconds i
- P = Gaussian with mean value F = 41.83 psi and standard deviation op = 5.02 psi Earthquake Lead (E)
- stationary randon process (a segment of 15 seconds) with a Kanai-Tajimi spectrum 1+4C[,f,a)2 2
o sgg(w) = s 1-(w/w)2]2+4g(,j,g)2 U
g where wg = 8e rad /sec and Cg = 0.6
- Distribution function of Z = (
l l
l F (z) = 1 - (agz/a ) a l
Z g
where g = P /wwg[l/(2Cg) + 2C ]
o g
g l
t l
with Pg = 3.0, a = 0.05g and a = 3.14; n
amax = 0.71g I
- 0ccurrence rate )E = 1.64 x 10-2/ year
- Mean duration pdE = 15 seconds !
Table 4 Stresses Due to Dead Load.
Local X-Direction Local Y-Direction j
Element Axial Stress Moment Axial Stress Moment i
Number Sxx(psi)
Mxx(Pound-in/in)
Syy (psi)
Myy (pound-in/in) i.
529 - 540
- 35.4
- 0.144 x 104
- 35.3
- 0.163 x 104 505 - 528
- 34.8
- 0.144 x 104
- 35.6
- 0.132 x 104
]
481 - 504
- 33.4
- 0.137 x 104
- 35.8
- 0.134 x 104 l
457 - 480
- 30.7
- 0.132 x 104
- 36.4
- 0.131 x 104 433 - 456
- 26.4
- 0.126 x 104
- 37.4
- 0.129 x 104 j
409 - 432
- 19.0
- 0.123 x 104
- 39.2
- 0.141 x 104 385 - 408
- 4.09
- 0.112 x 104
- 42.6
- 0.135 x 104 361 - 384 17.7
- 0.252 x 103 49.2 0.120 x 104 r
337 - 360 30.5 0.614 x 103
- 57.6 0.356 x 104 313 - 336 31.9 0.107 x 104
- 56.5 0.531 x 104 4
289 - 312 27.9 0.150 x 10.
- ol.6 0.724 x 104 265 - 288 23.4 0.649 x 103
- 59.2 0.324 x 104 241 - 264 14.7
- 0.678 x 103
- 66.0
- 0.339 x 104 a
217 - 240 5.67
- 0.102 x 104
- 75.4
- 0.510 x 104 l
193 - 216 0.53
- 0.465 x 103
. gg,4
- 0.232 x 104 i
169 - 192
- 0.65
- 0.139 x 102 111.8
- 0.694 x 102 145 - 168 0.19
- 0.683 x 102
- 140.5
- 0.341 x 103 121 - 144 1.55 0.378 x 102-158.7 0.169 x 103 97 - 120
- 1.07 0.826 x 103
- 174.3 0.413 x 104
{
73 - 96
- 11.2 0.133 x 104
- 18 9.9 0.664 x 104 49 - 7?
- 22.9 0.656 x 103
- 197.8 0.328 x 104 25 - 48
- 32.4
- 0.114 x 104
- 203.0
- 0.568 x 104 1-24
- 39.6
- 0.450 x 104
- 208.2
- 0.225 x 105 NOTE:
1.
Local X direction is the same as hoop direction.
l 2.
Local Y direction is the same as meridional ' direction. i
Table 5 Stresses Due to Unit Pressure.
r r
Local X-Direction Local Y-Direction Element Axial Stress Moment Axial Stress Moment Number Sxx(psi)
Mxx (pound-in/in)
Syy (psi)
Myy (pound-in/in) 529 - 540 9.74 0.201 x 101 9.73 0.540 x 102 505 - 528 9.72 0.574 x 101 9.73 0.262 x 102 481 - 504 9.72
- 0.560 x 101 9.72
- 0.112 x 102 457 - 480 9.74
- 0.538 x 101 9.72
- 0.472 x 101 433 - 456 9.75
- 0.272 x 101 9.72
- 0.236 x 101 409 - 432 9.76 0.241 x 101 9.72 0.138 x 102 385 - 408 9.72 0.172 x 101 9.69 0.533 361 - 384 9.91
- 0.277 x 102 9.68
- 0.111 x 103 337 - 360 10.6
- 0.'320 x 102 9.78
- 0.111 x 103 313 - 336 11.0
- 0. 712 x 102 8.61
- 0.291 x 103 289 - 312 11.8
- 0.139 x 103 8.65
- 0.642 x 103 265 - 288 12.4
- 0.439 x 102 7.68
- 0.219 x 103 241 - 264 13.7 0.106 x 103 7.68 0.529 x 103 217 - 240 14.8 0.134 x 103 7.68 0.670 x 103 193 - 216 15.4 0.651 x 102 7.68
'0.320 x 103 169 - 192 15.4
- 0.150 x 101 7.68
- 0.748 x 101 145 - 168 15.5
- 0.258 x 102 7.68
- 0.129 x-103 121 - 144 15.9 0.156 x 102 7.68 0.781 x 102 97 - 120 15.0 0.277 x 103 7.68 0.138 x 104 73 - 96 11.5 0.439 x 103
.7.68 0.219 x 104 49 - 72 7.61 0.179 x 103 7.68 0.897 x'103 25 - 48 4.50
- 0.454 x 103 7.68
- 0.227 x 104 1-24 2.26
- 0.162 x 104 7.68
- 0.809 x 104
Table 6 Natural Frequencies.
1 Mode Nunter Frequencies (cycles /sec) 1 2.869 2
2.869 3
4.186 4
4.186 5
4.587 6
4.587 l-7 5.501 8
5.501 9
6.001 10 8.028 11 8.028 12 8.115 13 8.115 14 8.147 15 8.215 16 8.215 3
17 8.389 18 8.389 19 9.310 20 9.310 NOTE: 1) Based on E = Ec/2, Ec = uncracked concrete stif fness.
- 2) 1st pair of bending modes: Mode 1 and Mode 2.
- 3) 2nd pair of bending modes: Mode 15 and Mode 16.
i.
.. _ _ _ _ _ _. _... -. _ = - _ -. - _ - _ _ _ _ _
a i
i Ta'ble 7 Coordinates of Limit State Surface l
(Element 289 - 312 Local X-Direction).
Point Coordinate i
j t (psi) m (Ib-in/in) a
- 4.56 x 103 1.13 x 104
?
l c
- 2.28 x 103 8.84 x 105 i
c'
- 2. :7 x 103
- 8.73 x 105 e
- 2.21 x 102 8.47 x 105 I
e'
- 2.10 x 102
- 8.71 x 105
,i f
5.14 x 102 1.96 x 105 i.
]
f' 5.29 x 102
- 2.21 x 105 1
I g
8.29 x 102
- 2.36 x 104 i
f i
NOTE: See Figure 13 for reference.
i f
I i
4 l
i l
i Table 8 Conditional Limit State Probability p(D+P),
l j-Local Critical Element No.
Di rection P(D +P )
Log 10 j
529 - 540 (531)
X 4.02 x 10-13
- 12.4 505-528(510)
X 3.59 x 10-13
- 12.4 481 - 504 (426)
X 5.36 x 10-13
- 12.3 457 - 480 (462)
X 9.08 x 10-13
- 12.0 433-456(432)
X 1.98 x 10-12 11,7 409 - 432 (414)
X 5.46 x 10-12 11.3 385 - 408 (390)
X 1.83 x 10-17
- 16.7
]
361 - 384 (316)
X 2.27 x 10-17
- 16.6 337-360(342)
X 5.87 x 10-13
- 12.2 t
I 313 - 336 (318)
X 5.54 x 10-6
- 5.16 289 - 312 (294)
X 7.86 x 10 "
- 4.10 289, 290.. 312
{
265 - 288 (270)
X 7.40 x 10-34
- 33.1 241 - 264 (246)
X 5.32 x 10-2R
- 24.3 217 - 240 (222)
X 3..'5 x 10-19
- 18.5 193 - 216 (198)
X 5.31 x 10-17
- 16.3 l
169 - 192 (174)
X 3.75 x 10-17
- 16.4
.f 145 - 168 (150)
X 8.20 x 10-17
- 16.1 l
121 - 144 (126)
X 4.12 x 10-15 14,4 j
97-120(102)
X 7.68 x 10-18 17, g f
73 - F6 (78)
X 5.12 x 10-41
- 40.3 l
49 - '/2 (54)
X 1.45 x 10-141
- 100 25 - 48 (30)
Y 3.97 x 10-193
- 100 l
1-24(6)
Y 1.77 x 10-39
- 38.8 4
l Table 9 Conditional Limit State Probability p(D+E),
local Critical Eltment No.
Direction p(D+E)
Logg(D+E)
Elements 529 - 540 (531)'
0
- 100 505 - 528 (510) 0
- 100 481 - 504 (486) 0
- 100 457 - 480 (462) 0
- 100 433 - 456 (432) 0
- 100 409 - 432 (414) 0
- 100 385 - 408 (390)
X 2.88 x 10-76
- 75.5 361 - 384 (316)
X 9.89 x 10-23
- 42.0 337 - 360 (342)
X 8.39 x 10-34
- 33.1 313 - 336 (318)
X 1.48 x 10-27
- 26.8 289-312(294)
X 2.55 x 10-32
- 31.6 265 - 288 (270)
Y (1.39 - 1.39) x 10-77
- 76.9 241 - 264 (246)
Y (J.82 - 3.83) x 10-50 49,4 217 - 240 (222)
Y (2.94 - 2.99) x 10-35
-(34.5-34.5) 193-216(198)
Y (3.69 - 3.86) x 10-15
. (24.4 - 24.4) 169 - 192 (174)
Y (5.51 - 6.58) x 10-15
-(14.3-14.2) 145 - 168 (150)
Y (1.37-1.96)x10-9
- (8.86 - 8.71) 121-144(126)
Y (2.43 - 4.54) x 10-9
-(8.46-8.34)97-120(102)
Y (1.41 - 1.73) x 10-9
- (8.85 - 8.76) 73 - 96(78)
Y (3.69-3.89)x10-12
-(11.4-11.4) 49 - 72(54)
Y (1.46 - 1.68) x 10-10
. (9,84 9.76) 25 - 48 (30)
Y (3.48 - 4.76) x 10-9
- (8.46 - 8.32) 1-24 (6)
Y (0.917 - 1.64) x 10-7
- (7.04 - 6.79) 6, 7, 18, 19 Table 10 Conditional Limit State Probability p(D+E+P),
Local Critical Element No.
Direction P (D +E +P )
tog p (D +E +P )
Elements 529 - 540 (531)
X (4.23 - 4.65) x 10-13
- 12.4 1
505 - 528 (510)
X (4.97 - 5.44).t 10-13
- 12.3 481 - 504 (486)
X (0.992 - 1.09)x 10-12
- 12.0 457 - 480 (462)
X (2.72 - 3.00) x 10-12
-11.6 - -11.6 433 - 456 (432)
X (1.64 - 1.86) x 10-11
-10.8 - -10.7 409 - 432 (414)
X (3.21 - 3.81) x 10-10
- 9,49 - -9.42 385 - 408 (390)
X (0.863 - 1.01) x 10-11
-1).1 - -11. 0 361 - 384 (J16)
X (2.53 - 2.63) x 10-9
- 8.60 - -8,58 337 - 360 (342)
X (3.11 - 3.32) x 10-7
- 6.51 - -6.48 313-336(318)
X (1.38 - 1.63) x 10-4
- 3.86 - -3.79 289 - 312 (294)
X (4.09 - 5.80) x 10-4
- 3.31 - -3.24 294,295,306,307 265-288(270)
Y (9.85 - 9.93) x 10-21
- 20.0 241 - 264 (246)
Y (1.24 - 1."4) x 10-14
- 13.9 217 - 240 (222)
Y (7.34 - 8.55) x 10-12
-11,1 - -11.1 193 - 216 (198)
Y (3.72 - 4.47) x 10-10
- 9.43 - -9.35 169 - 192 (174)
Y (1.61 - 2.20) x 10-7
- 6.79 - -6.66 l
145 - 168 (150)
Y (1.25 - 2.00) x 10-5
- 4.90 - -4.70 121-144(126)
Y (0.828 - 1.21) x 10-5
- 5.08 - -4.92 97 - 120 (102)
Y (5.44 - 7.48) x 10-6
- 5.26 - -5.13 73 - 96(78)
Y (4.85 - 5.76) x 10-8
- 7.31 - -7.24-49 - 72(54)
Y (1.41 - 1.75) x 10-7
- 6.85 - -6.76 25 - 48(30)
Y (1.40-1.75)x10-7
- 6.85 - -6.76 1-24(6)
Y (0.852 - 1.08) x 10-7
- 7.07 - -6.96 l
l l
l Table 11 Lifetime Limit State Probabilities.
l (Based on Linear Stress Distribution) i Expected Unconditional Load Number of Conditional Limit Limit State Critical Combination Occurrences State Probabilities Probabilities Element D+P 8.64 x 10-2 7.86 x 10-5 6.79 x 10-6 289,290...,312 t
0+E 6.56 x 10-1 (0.92 - 1.64)x10-7 (0.60-1.07)x10-7 6,7,18,19 D +E +P 5.46 x 10-8 (4.83 - 5.80)x10-4 (2.67-3.17)x10-Il 294,295,306,307 (6.85 - 6.90)x10-6 Ove ral l r
NOTE: Assuming the containment will not fail under dead load alone.
l I
Ii l
l l
l :
2 l
Table 12 Lifetime Limit State Probabilities.
(Based on Nonlinear Stress Distribution) i Expected i
l Load Number of Conditional Limit Unconditional-Critical Combination Occurrences State Probabilities Probabilities Element j
j-0+i 8.64 x 10-2 3,99 x 10-7 3.46 x 10-8 289,290.... 312 i
i l
0 +E -
6.56 x 10-1 1.02 x 10-8 6.72 x 10-9 6,7,18,19 i4.n'
,0 +E +P 5.46 x 10'*8 (1.57-1.68)x10-5 (8.55-9.14)x10-13 294.295,306,307 e,
l l
Overall 4.13 x 10-8 l
0
,c l
NOTE: Assuming the containment w111 not fail under dead load alone.
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