ML20064N737

Reliability Analysis of Containment Strength.Sequoyah and McGuire Ice Condenser Containments

ML20064N737
Person / Time
Site:	Mcguire, Sequoyah, McGuire, 05000000
Issue date:	08/31/1982
From:	Bluhm D, Fanous F, Greimaann L, Greimann L, Ketalaar D, Ketelaar D, Sabri A, Woldetinsae A IOWA STATE UNIV., AMES, IA
To:	Office of Nuclear Reactor Regulation
References
CON-FIN-A-4131 IS-4753, NUREG-CR-1891, NUDOCS 8209100222
Download: ML20064N737 (268)

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NUREG/CR-1891 IS-4753 l

l Reliability Analysis of Containment Strength Sequoyah and McGuire Ice Condenser Containments Prepared by L. Greimann, F. Fanous, A. Sabri, D. Ketelaar, A. Wolde-Tinsae, D. Bluhm Amee : aboratory towa State University Pr pared for U.S. Nuclear Regulatory Commission I

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. DISCLAIMER This book was prepared as an account of work sponsomd by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or as-sumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, pro-duct, or process disclosed, or represents that its use would not infringe privately owned rights. Refemnce herein to any specific commercial product, process , or service by trade name , trade-mark, manufacturer, or otherwise, does not necessarily con-stitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not neces-sarily state or reflect those of the United States Government or any agency thereof.

Available from GP0 Sales Program Division of Technical Information and Documen: Control U. S. Nuclear Regulatory Commission Washington, D. C. 20555 Printed copy price: $7.50 and National Technical Information Service Springfield, Virginia 22161

NUREG/CR-1891 IS-4753

_ _ = _ _ -. ___

Reliability Analysis of Containment Strength Sequoyah and McGuire Ice Condenser Containments Manuscript Completed: April 1982 Date Published: August 1982 Prepared by L. Greimann, F. Fanous, A. Sabri, D. Ketelaar, A. Wolde-Tinsae, D. Bluhm Ames Laboratory towa State University Ames, IA 50011 Prepared for Division of Engineering Office of Nuclear Reactor Regulation U.S. Nuclear Regulatory Commission Washington, D.C. 20566 NRC FIN A4131

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ABSTRACT The Sequoyah and McGuire ice condenser containment vessels were designed to withstand pressures in the range of 12 to 15 psi. Since pressures of the order of 28 psi were recorded during the Three Mile Island incident, a need exists to more accurately define the strength of these vessels. A best estimate and uncertainty assessment of the strength of the containments was performed by applying the second mo-ment reliability method. Material and geometric properties were sup-plied by the plant owners. A uniform static internal pressure was as-sumed. Gross deformation was taken as the failure criterion. Both ap-proximate and finite element analyses were performed on the axisymmet-ric containment structure and the penetrations. The predicted strength for the Sequoyah vessel is 60 psi with a standard deviation of 8 psi.

For McGuire, the mean and standard deviation are 84 psi and 12 psi, re-spectively. In an Addendum, results by others are summarized and com-pared and a preliminary dynamic analysis is presented.

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TABLE OF CONTENTS Page ABSTRACT iii I. INTRODUCTION 1 1.1 Background 1 1.2 Objective and Scope 1 1.3 Containment Description 1 II. UNCERTAINTY ANALYSIS 3 2.1 Probablistic Safety Analysis 3 2.2 Second Moment Method - One Failure Mode 5 2.2.1 Invariant Second Moment Method 6 2.2.2 Non-normal Distributions 11 2.3 Multiple Failure Modes 14 III. PARAMETER STATISTICS 16 3.1 Material Parameters 16 3.1.1 Yield and Ultimate Stress 16 3.1.2 Approximate Fracture Stress 16 3.2 Geometric Parameters 22 3.3 Resistance Modeling Error - General 23 3.4 Load Modeling Error 26 IV. FAILURE CRITERIA 27 4.1 Containment Shell 28 4.2 Attached Pioing Equipment 29 4.3 Combined Failure Criterion 30 V. APPROXIMATE STRUCTURE ANALYSIS 34 5.1 Stiffened Axisymmetric Shell 34 5.1.1 Failure Criteria 34 5.1.2 Modeling Error 36 5.1.3 Application to Containnent Vessels 37 5.1.4 Sensitivity Analysis 38 5.2 Penetration Intersections 39 5.2.1 Failure Criteria 39 5.2.2 Modeling Error 42 5.2.3 Application to Containment Vessels 43 5.2.4 Sensitivity Analysis 44 5.3 Anchor Bolts 44 5.4 Combined Failure 46 v

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VI. FINITE ELEMENT ANALYSIS 47 6.1 ANSYS Finite Element Program 47 6.2 Failure Criteria 48 6.3 Stiffened Axisymmetric Shell 48 6.3.1 Finite Element Modeling Guidelines 48 6.3.2 Modeling Error 51 6.3.3 Applcations 53 6.3.3.1 Finite Element Model 53 6.3.3.2 Results 54 6.3.4 Uncertainty Analysis 55 6.4 Penetration Analysis 56 6.4.1 Modeling Guidelines 56 6.4.2 Experimental vs ANSYS Results 57 6.4.3 Application 58 6.4.3.1 Finite Element Model 58 6.4.3.2 Results 60 6.4.4 Uncertainty Analysis 60 6.5 Combined Failure Modes 61 VII.

SUMMARY

OF RESULTS 63 7.1 Summary 63 7.2 Conclusions 63 7.3 Recommendations 64 VIII. LIST OF REFERENCES 66 IX. APPENDIX 71 X. TABLES 72 XI. FIGURES 92 XII. ADDENDUM 122 12.1 Static Pressure 122 12.1.1 Ames Laboratory (January 1980) 122 12.1.2 R&D Associates 123 12.1.3 TVA 124 12.1.4 NRC Research 124 12.1.5 Franklin Research Institute 125 12.1.6 Offshore Power Systems 126 12.1.7 Ames Laboratory (September 1980) 127 12.1.8 Summary 127 12.2 Dynamic Pressure 128 12.2.1 Introduction 128 12.2.2 Preliminary Finite Element Analysis 129 12.2.3 Approximate Dynamic Analysis 131 12.2.4 Summary 133 12.3 List of References for Addendum 139 vi

I. INTRODUCTION

1.1 Background

The purpose of the containment vessel in a nuclear power plant is to prevent the spreading of radioactivity from an event with a low probability of occurrence which may release radioactivity within the vessel. The probability that the containment will not leak radioactiv-ity must be acceptably low. The design pressure for ice condenser con-tainments is typically in the range of 12 to 15 psi. Because the peak pressure recorded during the Three Mile Island (TMI) explosion incident was 28 psi, there is a need to more accurately define the probability of leakage of these containments.

1.2 Objective and Scope The objective of this work is to review the structural design as-pects of the containments at the Sequoyah and McGuire nuclear power plants, as related to the TMI incident pressure pul se.

This objective is accomplished by making a best estimate and un-certainty assessment of the pressure strength of the steel containments at Sequoyah and McGuire. The scope of this assessment was limited to manageable proportions by using:

. Second moment reliability theory (Sec. 2)

. Uniform static internal pressures (Sec. 3.4)

. Gross deformation as the failure criterion (Sec. 4).

Semi-empi rical approximate methods are used to analyze the stiffened axisymmetyric shell and each penetration (Sec. 5). A more refined non-linear finite element analysis is applied to analyze the axisymmetric stiffened shell and, also, to analyze one penetration in each conyain-ment (Sec. 6).

1.3 Containment Description The steel containment vessels for both the Sequoyah and McGuire nuclear power plants are basically cylindrical shells with hemispheri-cal shell tops. Both vessels have ring (circumferential) and stringer (longitudinal) stiffeners. Figs.1.1 and 1.2 show the geometry of the axisymmetric vessels (Unit 1 for each pl ant) . At elevations where

4 2

thicknesses vary circumferentially, the minimum thickness is shown.

j Material properties will be discussed in Sec. 3. There are a total of 193 and 258 penetrations in the Sequoyah and McGuire vessels, respec-tively. Details of these penetrations are not included in this report, but were shown on drawings provided by the plant owners.

I i

l i

i i

i

, II. UNCERTAINTY ANALYSIS 2.1 Probabilist Safety Analysis Failure of a nuclear power plant containment vessel is considered to occur when radioactivity is released. The probability of such a failure, F, must be acceptably low. If the events, E, which may n

cause the failure event are independent, the probability of containment failure, P f, can be written approximately (for small probabilities of failure) as [1,2]*

Pf=[P(F/E)P(E}n n (2-II where P(E n ) is the probability of occurrence of event n and P(F/En )

is the conditional probability of failure given event En. Typically, E

n represents severe events with a small probability of occurrence, such as a major earthquake, a direct tornado hit or a loss of coolant accident. It is impractical to design for certain events with a very low probability of occurrence. For these events (core meltdown, geo-logic faul t directly under facili ty), P(F/En ) is one.

For the current task, the study is limited to the failure proba-bility for one event - the explosion situation identified in the TMI incident:

PfTMI = P(F/ETMI) P(ETMI) (2-2)

The study is not concerned with the determination of the probability of occurrence of this event, P(ETMI), e.g., human error, equipment malfunction. It is not the purpose of this work to judge the adequacy of the containment designs against the TMI type incident. The work intends only to present information useful in that judgment. The final

• Numbers in brackets refer to entries in the List of References.

[ Vertical bars in the right margin indicate changes (expanded remarks and/or corrections) made in this report in response to reviewers' comnents to initial draft of November 1980.]

Judgment can be made only by bringing this and other factors to bear, e.g., the probability of the event itself and the probability of fail-ure that society is willing to accept. It is, however, aimed at deter-mining the conditional probability of failure P(F/ETMI), that is, the probability of failure given the TMI explosion incident. This con-ditional probability will herein be referred to as pf = P(F/ETMI) (2-3)

The reader will note that the actual probability of failure for the TMI incident is given by Eq. 2-2, i.e. , PfTMI = pf P(ETMII -

The failure criteria for a structure can be written as G(xj) < 0 (2-4) where G is the failure function, which may include several failure modes and xj are structural parameters such as material properties, geometry, loads and modeling error. The parameters are considered as uncorrelated random variables. Failure does not occur if G(xj) is greater than zero. The probability of failure pf can be written as o

pf = P(G < 0) = f f(G)dG (2-5) in which f(G) is the probability density function of G. Various other forms of this general relationship are given in the literature [3,4,5],

e.g.,

pf = P(xj in D) = f f(x )d j xj = f dF(x j) (2-6) 3 D where f(xj) is the joint probability density function of xj, F(xj) is the joint probability distribution function and D is the failure domai n where G(xj) is less than zero.

In some cases, the xj parameters can be separated into two groups: x, r resistance parameters, and x, q load parameters. The failure function can then be written as [1,2,3,4,5,6,7,8,9, many others]

G(xj) = R(xc ) - Q(xq ) < 0 (2-7) i where R(x r ) is the resistance function and Q(x q ) is the load func-tion. In this form, the probability of failure can be written as Pf = P(R-Q<0) = ff f r(R) f (Q) q dR dQ (2-8) where fr (R) and f q(Q) are the probability density functions for R and Q, and D is the failure region where R-Q<0 or R<Q. Al ternatively, Q

Pg= f fq (Q) f f r(R) dR dQ = f f (Q)F q (Q) r dQ (2-9)

It is generally recognized, even in the most recent literature

[2,3,4,5], that the evaluation of the structural safety by the direct evaluation of the above integrals is impractical. First, th6 required probability distribution functions are seldom, if ever, known, and se-condly, the evaluation of the multiple integrals is practically not l

feasible. In this regard, several investigators have attempted to de-velop approximate methods to evaluate the failure probability. Since

the current task is not research oriented, a summary of these methods will not be made here. Refs. 3, 4 and 5 present a good review of the state of the art. The simplest generally-accepted approach is called the second moment method.

2.2 Second Moment Method - One Failure Mode The second moment method for probability statements has been advo-cated for some time [1,2,3,4,5,7,8,9,10,11,12]. The method, as out-lined below, follows the developments in these references, principally

l Ref. 4.' The second moment method has several advantages, the biggest of which is its simplicity. Uncertainty is expressed in terms of the first and second moment of the random variables, xj . Hence, exact information about the probability density function is not needed - only the first and second moments. The method uses a linearized form of the failure criterion which allows separation of the load and resistance functions as illustrated in Eq. 2-7.

2.2.1 Invariant Second Moment Method For the second moment method, the failure function is again writ-ten as in Eq. 2-4. The uncertainty of the variables xi is expressed by their mean pj (first moment of probability density distribution) and their standard deviation ej (square root of second moment). The mean and standard deviation of G are obtained by linearizing G with the first two terms of a Taylor series pg = G(xj) (2-10) aG(xj )

2 = j;nl og ax i "i

~

Most early approaches performed the linearization at the mean of xj, i .e. , xj equal pj ; however, several investigators have shown that a better approximation is obtained if the linearization is performed at the design (or Rackwitz) point on the failure surface, G(xj) = 0 (2-12)

In this way, the Taylor series expansion of G(xj) takes place in the upper tail of the load distribution and the lower tail of the resis-tance distribution [3,4,10,133, which is the design point or most like-ly region of failure. Linearization of the failure surface at the de-I sign point has the additional advantage that the method now becomes invariant under a change in formulation of the failure criteria, e.g.,

changing a load variable to a resistance variable.

l l

The second moment method does have limitations which should be pointed out. First, if more exact information is available on the pa-rameter distributions, it cannot be logically included. Second, a linear approximation to the failure surface may not be acceptable [3].

A safety index is defined for the second moment method as (2-13) 6 = "G It is generally assumed that the distribution of G can be approximated as normal in the vicinity of the design point [4] so that the probabil-ity of failure can be found as by Eq. 2-5 as pf = P(G<o) = 4(-8) (2-14) where e is the standard normal integral.

A more general formulation of the second moment method, equivalent to above, is as a nonlinear minimization problem, [4,10,13,14]

2 [uj-xj\

minimize s = I I I (2-15)

( i /

constraint G(xj) = 0 (2-16)

This formulation can be shown to be, by the Lagrange multiplier method, equivalent to solving the equations [4,14]

xj = uj - 8 aj oj (2-17)

G(xj) = 0 (2-18) 2 -1/2 BG(xj ) lag (xj) I where q = ax "i E ax "i

! All three formulations (Eqs. 2-13, 2-15 and 2-17) are equivalent.

A simple example will illustrate the significance of the design point minimization of the failure function. Suppose the failure func-tion is taken as the nonlinear function

G(x) = 1 -

x

[=0 (2-19)

If linearization is performed at the mean I

G =G x,g h (x p) (2-20)

=1 b+2--3( x- ) (2-21) 2 9 p where p is the mean of x. The mean and standard deviation of G, are given, using Eqs. 2-10 and 2-11, as p

g =1- (2-22)

" p og = (2-23)

" p where o is the standard deviation of x. The safety index, with lineri-zation about the mean, is 2

=

s = p( p -1) (2-24) p Zo i

On the other hand, if linearization is performed about the design point xd on G(xd) equal zero or i

xd = 1 1 (2-25)

the linearized G has the form Gd=G + (x=1) (2-26) x=1 x=1

=0 2 (x-1) (2-27)

The mean and standard deviation of Gd becomes p

g =2(p- 1) (2-28) og d

and the safety index is s=d

=

(2-30) o p(p + 1) 8" The reader will note that sd is almost always less than s p since G( p) is almost always greater than zero, i.e., p is almost always greater than one. Also, the value of s would p change if G were form-ulated as G(x) = x2 - 1 = 0 (2-31) whereas sd would not. Thus, as stated previously, sd is invariant under a coordinate transformation.

If the failure function G is linearized about the design point, careful interpretation of p and o is necessary.

Gd Gd Wi th reference to Eqs. 2-28 and 2-29 above,. pg and Gd are not actually the mean and standard deviation of the f(G) but are first order approximations of the mean and standard deviation of f(G).

They can also be considered to be the mean and standard deviation of a normal distribution 4(G) which is a first approximation to f(G) in the vicinity of the design point.[4]

Returning to the formulation of Eq. 2-15, the minimization problem lends itself to a graphical interpretation. If the basic variables are transformed to variables, yj, with zero mean and unit standard devia-tion xg-u g y9 = (2-32)

,i the second moment method can be stated as minimize 62= I yf (2-23) constraint G(yj) = 0 (2-34)

The equation for 6 is seen to represent a hypersphere and the minimiza-tion process determines the minimum distance between the origin of yj and the failure surface G(yj) equals zero. This is illustrated in Fig. 2-1 for a two-parameter failure surface. The f(G) and its approx-imating normal function 4(G) are also indicated.

For the special case of a linear failure surface, the derivatives of G are constant. Thus, the distinction between the mean point and design point linearization of the failure function becomes unnecessary.

For example, suppose the failure function G is linear in R and Q G=R-Q (2-35) where R and Q are normally distributed. Eqs. 2-10, 2-11 and 2-13 give

[6,7,8,9]

6= (2-36) 2 2

as the safety index. The formulation of Eq. 2-15 with the Lagrange multiplier method will yield identical results. The probability of failure, by Eq. 2-14, is py = P(R-Q<0) = P(R<Q) = e(-8) (2-37) 2.2.2 Non-normal Distributions If the distribution of the variables is non-normal, their distri-bution can be incorporated by transforming to standard normal variables

[4,14]. For example, if zj is a lognormally distributed structural parameter, the transformation zj=e (2-38) where xj is normally distributed, is employed. In many cases, the lognormal distribution assumption is appropriate because it eliminates problems associated with negative values of the parameters. For exam-ple, a lognormal distribution assumption for the material yield strength correctly states that the probability of a negative yield strength is zero whereas a normal distribution assumption would provide a finite (but small) probability of negative values. With the trans-formatien of Eq. 2-38, the mean and variance for the normally distrib-uted xj is given as px = fn (pz/ Yz + II (2-39) 2 o = In(Vz + 1) where V z equal to oz / Uz is the coefficient of variation of z. If Vz is small with respect to one (say Y zless than 0.3)

J 2 2 14 16 =V 2 in(V +1) = V 7V +7V (2-40) 2 (V + 1)_1/2 =1 12 34 =1 7V+gV so that px = An ( uz )

(2-41) o x =V z are approximate values of the mean and standard deviation of the trans-formed variables.

For the simple case of G in Eq. 2-35, if R and Q are assumed to be lognormally distributed, the transformation r = in R (2-42) q = in Q will transform the parameters to the normally distributed r and q.

Thus, the failure criterion becomes G(r,q) = er _ eq = 0 (2-43) where again, G is taken to be normally distributed in the vicinity of l

the design point. Application of the Lagrange multiplier method to Eq.

l 2-15 with G from Eq. 2-43 gives I 9 6= (2-44)

/2 2 Mr+ q l

or, in terms of the original variables R and Q [6],

tn  % _ An  %

2 2

g. YR+1 VQ+1 2 2 1/2

[tn(VR + II

• A"IYQ + II3 where VR*9M /

(2-46)

Vn = qn/ q are coefficients of variation of R and Q, respectively. For small V R and Vg , the approximations of Eq. 2-40 apply and Eq. 2-45 becomes approximately [8,9]

8= (2-47)

VR+YQ Since G is taken to be normally distributed in the vicinity of the design point, the probability of failure is pr = P(R-Q<0) = P(R<Q) = 4(-8) (2-48)

Several investigators [6,8,9,10,11'l c'ioose to write (for lognormal R and Q)

G'(R,Q) = An R-in Q (2-49) instead of Eq. 2-35. Again, making the transformation- of Eq. 2-42 and applying Eq. 2-15 gives results identical to Eq. 2-45. The failure probability for normally distributed G' is understood as pf = P(G'<0) = P(in R - AnQ<0) = 4(-8) (2-50)

which is, of course, the same as Eq. 2-48. The fact that the formula-tions of G in Eqs. 2-35 and 2-49 give identical results (Eq. 2-45) in-dicates that the safety index of Eq. 2-15 is invariant under at least ,

this particular coordinate transformation (Eq. 2-42). In general, it is approximately true.

Coordinate transformations for other non-normal distributions are given in Ref. 4. Only lognormal distribution assumptions will be used in this work because of the ease with which they Lre handled. However, they obviously represent distribution functions of a specific analyti-cal form and, as such, thei r application is limited.

2.3 Multiple Failure Modes -

If a structure can fail by more than one failure mode, estimates of the probability of failure of the structural system can be obtained.

In this case, the safe region of the structure is defined by the inter-section of the safe regions of the individual failure criteria Gm IXi)=0 (2-51) where m denotes the failure mode number. This is illustrated in Fig.

2-2 for three failure modes. For each individual failure mode, the minimum 6 can be obtained by the minimization method in Eq. 2-15. Let sm be the minimum S for each failure mode m. If, as before, each G, is assumed to be normally distributed in the vicinity of the de-sign point, the failure probability for each failure mode is 1

Pfm = P(Gm <0) = 4(- %) (2-14) l 1

l

,- e

Bounds on the probability of failure for the structural system are given by [4,10,14,15,16,17,18]

max. pfm = P(Gm < 0) = 4(-em ) (2-52) or e(-min g) < pf < r 4 (-g) where the sum is taken over all failure modes and max pf, denotes the maximum failure probability of all modes. If one mode predomi-nates, the bounds become very close.

Lind [14] suggests that it is of ten appropriate to derive a safety index for the structural system which includes all failure modes since this may be more convenient in practical design than the probability

~

statements. He defines the generalized safety index 6 for the mul-tiple failure mode case as the inverse of the normal error function of the reliability i = e'I(1-pf) (2-53)

In terms of the probability bounds of Eq. 2-52 e -I(1-r pg) < i < e-I(1 - max pf,)

or e -l[1 - te (- g)] < i < mins, (2-54) as bounds on the generalized safety index.

III. PARAMETER STATISTICS The formulation in Sec. 2, second-moment reliability theory, re-quires the first and second moment of the structural parameters (xg in Eq. 2-4) to approximate the first and second moment of the failure function. In this chapter, these parameter statistics (mean and stan-dard deviation) will be discussed for the Sequoyah and McGuire nuclear power plant containments.

3.1 Material Parameters 3.1.1 Yield and Ultimate Stress The material for both containment vessels is A516 Grade 60 steel with a specified minimum yield of 32 ksi and a specified ultimate of between 60 and 80 ksi. The mean values of the yield and ultimate strength were furnished by TVA (for Sequoyah) and Duke Power (for McGuire) and are listed in Table 3-1. The standard deviation for these properties was supplied by Duke and assumed to apply to Sequoyah. Al-suming that the properties of this steel are similar to those for typi-cal structural steels, e.g., A-36 and A-441, the mean and standard de-viation for the other pertinent structural parameters in Table 3-1 [19, pg 1467, and 20, pg 1440] can be used. The distribution type (normal or log-normal) is also indicated. The anchor bolts are SA320-L43 with a minimum specified yield strength of 105 ksi. The mean and standard dev'iation as given in Table 3-1 are assumed to be similar to A-490 bolts [20, pg 1433].

l 3.1.2 Approximate Fracture Stress

An additional material property which will be of interest in pre-l dicting containment leakage (see Sec. 4) is the fracture stress for the s teel . In Sec. 4, it will be shown that the probability of brittle fracture is much smaller than the probability of gross containmer.t deformation. To demonstrate this, an approximate (conservative) value of fracture stress is required. The authors acknowledge that the

n!

following discussion is rather tentative but adequate to serve the put-} ..s s..

poses of Sec. 4. _

Among other items, the fractcre stress is dependent upon the frac-ture toughnest of the steel which is frequently characterized by .1 critical stress intensity factor, K IC. The determination of KIC for a typical ductile steel over a wide temperature range is practical- , f ly impossible by direct methods. A typical variation of K ig theo-retically requires that no plasticity exists at the crack tip [21 (Chap. 3)]. This condition is sufficiently satisfied if plane straf o{

conditions exist at the crack tip, or test specimens are sufficientif e thick to prevent signf ficant through thickness straining. ASTM Specid fications [22] have translated this into the requirement that J

y=h( ) < 0.4 (3-1)

Y .

where B is the specimen width. As y increases significantly beyond, j this limit, increasing amounts of through thickness training occur and j [

a condition of planc stress is approached. AdmittedVy;,. the above plane ,e strain conditions do not exist in the contaf,nment vessel ; however, /

plane strain conditions do represent a lower bound case. '- '

With reference to Fig. 3-1, it is possible to satisfy the criter- ,.'

l ion of Eq. 3-1 for ductile steels onlysat low temperatures - near and i "

below the NDT (nil ductility transition temperature). At these -low ,

e temperatures, the material is sufficiently brittle (low KICI 'that a practical specimen size, B, can be used. At higher temperatures, the ,

behavior becomes increasingly inelastic and B becomes exessivel'y large. - , 4 ' y ,_

In this region, K IC cannot be dethrmined directly. In fact, ,even .

to define a K IC I" ' this, region is questionable. (KIC in the' upper shelf region can be ' approximated by' correlation with other test l methods, e.g., Charpy V-Notch tests (21 (Chap. 6)].) In practical tenns, this means that at low temperatures (relative to NDT) fracture

( by crack propagation is 'possible whereas at temperatures significantly

^

above the NDT, gross yielding of the section will most likely occur even in the presence of a large crack.

/ r I

s Other factors shocid be mentioned which affect the use of K IC

,; ,,r ' da ta . In a typical real structure, thicknesses are significantly less

, than B from .Eq. 3.1. Thus, plane strain conditions at the crack tip may not be achieved. In this regard, aK c is often defined as the critical sness intensity for plane stress. This value depends upon, among other paraneters, the material thickness. (The critical stress intensity factor for plane strain, KlC, is the lower bound of Kc '

i.e., the value of K c for large thicknesses.) However, in complex structures with intersecting plates (stiffeners, penetrations) and sig-7-

nificant welds in complex geometries and residual stresses, the dis-tinction between plane stress and plane strain cannot be made. In this study, it will be assumed that plane strain conditions of maximum re-

<' straint exist near the crack and, hence, that KIC is applicable.

Loading rate also affects values of KIC. At fast loading rates, dynamic KIC values are defi ne:! as kid, the critical stress intensity factor for impact loading and plane strain conditions.

A common technique for obtaining kid curves from KIC data is to shift the curve in Fig. 3-1 to the right by 160*F for mild ductile steels [21 (pg.129)]. This shift effectively accounts for the reduc-tion in critical stress intensity for impact loadings. For this work, the lower values of K Id will be used instead of KIC for two

~

reasons: (1) the loading is actually dynamic (explosive), and (2) the ASME code adopts this approach through the use of K IR curves

[21,23]. (K IR is the lower bound of KIC and kid I

, Unfortunately, K Id data for the containment vessel steel,

' A-516 Gr. 60, ic not available. However, the ASME code recommendations y for KIR[23] are approximately applicable for steel s with yield strengths below 50 ksi. As illustrated in Ref. 21 (Fig. 15.1), there is considerable scatter in the experimental data used to develop l~

K IR. Additionally, KIR is a function of the operating tempera-ture relative to the NDT, as in Fig. 3-1. The NDT of the A516 steel is j -

assumed to be -30 F. From Fig.15.1 of Ref. 21, the mean of the test I

data for KIR can be represented approximately by (kip, in., and F i units)

I

II+ N

+26.8) (3-2) kid " *o (1.2e which is a shifted form of the ASME equation for KIR. The quantity xoaccounts for scatter in the data (see Sec. 3.3). Approximately, Uxo"1 (3-3) oxo = 0.17 For these containment vessels, the temperature range is approximately 0*F to 100 F. The following statistics of T are used:

q = 50*F (3-4)

T = 25'F These temperature statistics are conservative and give a higher than likely probability that the containment will reach a low temperature.

Under accident conditions, the temperatures will, most likely, be ele-vated. With the above statistics for xo and T and with Eq. 3-2, the statistics on kid can be found as pK u = 105 ksi in (3-5) gg = 31 ksi in These values are similar to other reported values [24,253 Also, the required Charpy V-notch impact value for this material is 15 ft-lb at -

30"F [26]. This corresponds approximately to a kid of 47 ksi in at -30*F [21 (Eq. 6.2)] which compares to 54 ksi in given by Eq. 3-2.

Again, these are conservative stati stics.

Linear elastic fracture mechanics states that a partially through surface crack will propagate through the thickness when

Kt= Kid (3-6)

K i is the stress intensity factor for a partially through surface crack, typically written as Kg = Mf 8 (3-7) where f is the stress remote from the crack or the stress which would exist in the vicinity of the crack if the crack were not present. (The symbol f is used here for stress since o has been reserved for standard deviation.) The quantity a represents the crack dimension (crack depth for a partially through crack). The factor M accounts for the differ-ent types of stress in the vicinity of the crack (extensional, bend-ing), the shape of the crack (semi-elliptical , semi-circular, straight), and the local geometry of the structure (penetration, weld detail ) . By their very nature, the crack size a and magnification f actor M are random quantities, dependent upon crack shape, local structural geometry, crack location, quality of the material and welds, and inspection and repair techniques.

It is beyond the scope of this work to review possible forms of M except to list a very limited number of references [21,27,28,29,30,31].

Typically, M ranges between about one and two. For this study, the following (conservative) statistics will be assumed for M uM = 1.5 (3-8) g = 0.15 M will be assumed to be lognormally distributed. These estimates are subjective.

Crack size statistics are very difficult to define without a very careful inspection of the vessel, including welds. Such an inspection is probaMy not feasible at this point. ( As will be shown in Sec. 4, t

I

the probability of failure by fracture is very small. Thus, precise determination of fracture properties does not appear necessary.) The vessels were inspected by die penetrant techniques. The maximum allow-able crack size is about 3/16 in. [32]. It will be assumed that, be-fore inspection, one crack in ten is greater than 3/16 in. deep. If the inspection is 95 percent effective in detecting these large cracks

[33] and the detected cracks have been repaired, approximately one crack in 200 will be greater than 3/16 in, after inspection. This con-dition, in conjunction with an assumed mean crack size of 1/16 in.

gives the following statistics for the crack size a pa = 0.0625 in.

(3-9) ca = 0.052 in, if the crack size is lognarmally distributed. Fracture at only one point (a hypothetical crack) is considered here. The probability of fracture at this point will be compared to the probability of gross yielding at this same point in Sec. 4.3.

The fracture criterion of Eq. 3-6 can be formulated as f=F c (3-10) where F c , the fracture stress, is defined as [34]

Id F

c

= (3-11)

M [a Statistics for the quantities KIC, M, and a have been approximated i n Eqs. 3-5, 3-8 and 3-9, respectively. Since these quantities are assumed to be lognormally distributed, Fc will also be lognormal dis-tributed. The statistics of F care listed in Table 3-1. (Note that the large standard deviation of F c is caused by the large variations in crack size and kid I

These values of F are quite conservative and serve only as a first approximation for applying fracture mechanics principles. A better ap-proximation can be obtained by collecting actual data for the material fracture properties and crack sizes, shapes and locations. However, as will be discussed in Section 4.3, the values listed here serve the in-tended purpose, i.e., they demonstrate that the probability of fracture is quite low. The final results are quite insensitive to the particu-lar value of the fracture stress.

3.2 Geometric Parameters No as-built measurements of the containment vessels exist. How-ever, fabrication and erection tolerances were established and (presum-ably) met during the construction process. Tolerances on plate thick-ness and size are given in Ref. 35. Tolerances on erection dimensions were supplied by TVA for Sequoyah and by Duke Power for McGuire during site visits. The nominal values and tolerances are listed in Table 3-

2. The mean value of the geometric parameter is taken as the average of the maximum and minimum limits of that parameter as specified by the tolerance extremes. The standard deviation is taken as one-third of the allowable tolerance from the mean [6 (pg 112)]. This is equivalent to assuming that 99.73 percent of the as-built dimensions fall within the prescribed tolerances. The mean and standard deviations of the geometric parameters are listed in Table 3-2. They are assumed to be normally distributed.

The values listed in Table 3-2 imply that quantities such as the thickness are random but uniform throughout. This is, of course, not true. Thicknesses have a spatial variation in actuality and could be idealized as random processes that are functions of the spatial vari-ables. However, the random process approach would be an analytical over-sophistication and would make the following work intractable. The following work (Secs. 5.1.4 and 5.2.4) demonstrates that the results are insensitive to the statistics of the geometeric parameters because the coefficient of variation of these quantities is relatively small.

l The assumption selection (uniform vs random process) is, therefore, l

1

I immaterial since either choice will not significantly influence the answer.

3.3 Resistance Modeling Error - General In the practical case, the failure function G (Eq. 2-4) is not precisely known. At best, a theoretical model is available to predict the failure of a real structure. Variability is introduced into both the prediction of the resistance of the structure and the applied load.

In this section, va ri abili ty in the resistance prediction is considered.

Imperfections in the resistance model result from various sources.

First, several assumptions are typically involved in the formulation of a model. Though these assumptions may be relaxed for more sophistica-ted models, there remains some uncertainty as to their effect. To a limited extent, the uncertainty of the model can be quantified by com-paring the theoretical results to results from highly idealized experi-mental models. However, uncertainty also exists in this experimental Wo rk .

Another source of uncertainty of the resistance model is in its application to a real structure. The real structure typically has many details which cannot be accurately modeled analytically or experimen-tally. Hence, another set of assumptions is introduced into the analy-sis of the real structure. Typically, application of the prediction model to the real structure involves more uncertainty than in its ap-plication to experimental structures.

The resistance modeling error is considered as an additional structural parameter, x o xg

=ht (3-12) where R is the actua* in-service resistance and R t is the predicted or theoretical resistance. By the extended reliability formulation

[8,9], the modeling error is taken as

xo = a6 (3-13) where 6 represents the basic variability of the theoretical resistance model with respect to experimental results and a represents the varia-bility between experimental results and in-service conditions. Thus, a accounts for imperfections in the experimental modeling of real struc-tures, e.g., boundary conditions, welds, residual stresses. (The ex-tended reliability formulation was not introduced into the other param-eter variabilities because these variabilities are presumably based upon a large number of samples of in-service conditions.) The random variable a and 6 are taken to be lognormally distributed so that. xo is al so lognormally distributed.

The mean and standard deviation of a cannot be quantified ration-ally but remain a judgment of the engineer. Actual values can only be determined by testing real, as-built containments which is, of course, prohibitively expensive. In lieu of this, the approach herein will be to adopt the typical values suggested by other investigators [9,36,37].

Hence, the mean of A is taken equal to one. This implies that the mean of test data fits the mean of in-service behavior. Typical values of the coefficient of variation of a are 0.05 [9], 0.02, 0.05 and 0.07

[36], and 0.05 [37]. A value of 0.05 will be used in this work.

The values of the mean and standard deviation of 6 will be deter-mined as the various analytical methods are presented in the following chapters. The general procedure will be to tabulate values of the in 6 where 6 is the ratio of the experimental resistance of a model to the theoretical resistance. The mean and variance of this tabulation are calculated by usual means as m and s2 With 95 percent confidence, one can say (if the error is normally distributed) that s

0.025 m- <

En 6 < m + (3-14) n n 2 2 (n-1) s 2 2

< ,in 6 < (n-1)s 2

(3-15)

X 0.025 X 0.975 l

where n is the number of specimens, t o.025 is the value of t such that the area under the Student -t distribution to the right is 0.025, and x2 0 025 x20*9/5 are the val ue of x 2 for n-1 degrees of freedom such that the area under the chi-square distribution to the i left and right is 0.025 and 0.975, respectively [38]. For this work,  ;

conservative values will be used, i.e.,

l p =m-(3-16) in 6 n

2 a

2 , (n-1)s (3-17) in 6 2 D.975 and, by Eq. 2-40, p6"*

(3-18) 2 2 V6 * **P I "in 6I -I where 6V is the coefficient of variation of 6.

In summary, the 'mean and coefficient of variation of the resis-tance modeling error (lognormally distributed) will be taken as 90 =96 (3-19) 2 V*=V o + (0.05) 2 (3-20)

Details for the various analysis methods are presented in the following chapters.

3.4 Load Modeling Error The exact nature of the pressure loading experienced at TMI is poorly defined. Information supplied to the authors indicate only that the peak recorded pressure was 28 psi and resulted from an explosion incident. The spatial variation and time history of the pulse were not available to us. As a first approximation, the load is assumed to be a uniform, static internal pressure with up = 28 psi (3-21) o p

=0 (3-22)

Most likely, the peak pressure was recorded at an interior point of the containment, so that the pressure could be reduced as it radiated to the shell walls. Also, the pulse length may be short so that the dy-namic effect of the pressure could be more or less than the static ef-fect depending upon the ratio of the pulse length to the local deforma-tion mode period. In lieu of more specific information and to obtain a first approximation within the project time constraints, the above uni-form static pressure is used. In view of the uncertainties of the load information, it is, of course, not consistent to take op equal zero.

However, with regard to Eqs. 2-2 and 2-3, this report is aimed at de-termining a conditional probability; that is, the probability of fail-ure given that the applied pressure is 28 psi. A zero value of op is, thus, appropriate.

f

- w

IV. FAILURE CRITERIA The purpose of the containment vessel is to prevent the spreading of radioactivity resulting from e.n event of low probability (in this case, the TMI explosion incident). Thus, the containment vessel is intended to be a leakproof barrier against release of radioactivity.

Failure of the containment is, therefore, considered to occur when radioactivity is released or, in other words, when leakage of the con-tainment occurs. Leakage can occur in at least two areas associated with the containment:

- Leakage of the containment vessel itself either in the shell wall or the peneteration intersections;

- Leakage of piping and/or other equipment passing through or attached to the containment shell at locations remote from the containment shell.

Leakage of the containment vessel itself will occur when a crack or defect propagates completely through the wall of the containment shell or penetration intersection. In this study, the forces required to propagate the crack are provided by the internal pressure.

Prior to leakage of the vessel wall or penetration intersection, it is quite possible that leakage could be induced into the attached piping by gross deformation of the vessel wall. Thus, if the vessel wall does not leak, the vessel will continue to expand under increasing internal pressure until the piping or some other attachment is so grossly deformed that it leaks. These failures are considered to oc-cur at locations removed from the shell itsel f. Additionally, gross deformation of the vessel may induce other types of undesirable behav-ior such a failure of the equipment in the annular space between the containment and shield building walls, failure of instrumentation and control devices passing through the shell wall, and failure of the an-chor bolts and leakage barrier at the containment base. Expansion bel-lows between the attached equipment and the shield building are pro-vided in each plant. However, these allowances for expansion are limited to the elastic deformation of the containment.

A schematic representation of the pressure-displacement curve for a pressure vessel is shown in Fig. 4-1. In this figure, 61 represents the displacmeent at which leakage (fracture) occurs in the vessel wall or penetration intersection. Leakage of the piping and/or failure of attached equipment at a point removed from the shell is identified as occurring at 6 2. In all likelihood, 62 will be significantly smaller than 6i because of the inherent ductility of the large diameter, thin-shell containment vessel . Quantification of these leakage failure is discussed below.

4.1 Containment Shell The characterization of the fracture of ductile steel structures is still very much in the development stage [21]. No generally recog-nized technique has established itself among structural analysts. How-ever, two limi ting cases are reasonably well agreed upon.

First, fracture of an initially uncracked structure, e.g., tension specimen, will occur when the stresses reach the ultimate tensile strength of the material, Fu (see Table 3-1).

f=F u I4-II Here f is the applied tensile stress. (Actually, f should be interpre-ted as the tensile stress on the net area (gross area minus crack area) but for crack areas much smaller than the gross area, f can be inter-preted as the tensile stress on the uncracked area.) This limit im-plies that imperfections in the real structure are smaller than or equal to the imperfections in a machined tensile specimen.

At the other limit, failure of a perfectly elastic structure with an initial crack will occur when the stress intensity Ky reaches the limiting value of the critical stress intensity K Id. By Sec.

3.1.2, this is equivalent to the failure condition f=F c (4-2) where F c is the material fracture strength (see Table 3-1).

Most real cases fall somewhere between these limits depending upon the relative values of the fracture strength and the ultimate tensile strength. Application of the ultimate tensile strength (UTS) limit implies that small cracks and imperfections do not exist or, if they do exist, do not affect the material strength. Application of the linear elastic fracture mechanics (LEFM) implies that no local yielding of the structure occurs, even at the crack tip. Both these implications are obviously incorrect. Ultimate tensile strength is affected by imper-fections and local yielding does occur in structures before fracture.

Many techniques have been proposed to account for plasticity at the crack tip (elastic plastic fracture mechanics, EPFM) - the missing link in the above two limits [21 (Chap.16),34,39,40]. The difference in the theoretical approaches is usually much less than the scatter in the experimental results of fracture tests. In this regard, at least for the current state of the art, a two parameter fracture interaction curved [34] of the form 2 2 G i = 1 - F(b) - (b)

F

<0 (4-3) c u is appropriate. This function is used herein as the failure function for fracture of the containment vessel wall.

4.2 Attached Piping and Equipment Leakage of piping and failure of other equipment passing through or attached to the containment shell wall is probably even more diffi-cult to predict than fracture of the shell itsel f. However, it can reasonably be assumed that failure of these pieces will not occur if deflection of the containment shell is kept within reasonable limits.

To accurately define this limit, it would be necessary to examine indi-vidually each component associated with the containment. Because of the large number of such components and their indeterminacy, this is not considered practically feasible.

One approach would be to select some (fairly arbitrary) deflection limit for the shell below which failure of associated components would not occur -- say a few inches. The approach which will be taken here is to assume failure of associated components will occur when "exc es-sive" plastic deformations of the shell occur or when deflections in-crease " rapidly " for "small" increases in load. In real structures with strain hardening materials and large displacement effects, a plas-tic pressure would correspond with this large plastic deformation.

Numerous definitions of the plastic pressure have been adopted in the literature [24,41]. Each definition is associated with varying degrees of plastic deformation. The half linear slope method will be adopted here, primarily because it is recommended in the ASME Code

[24,41,42]. In this method, the plastic pressure is defined as the pressure at the intersection of the pressure-displacement curve with a straight line having a slope equal to one-half the initial slope (pc in Fig. 4-2) . In other words, the plastic pressure is equal to the pressure which produces deformation twice that of the elastic deforma-tion at the same pressure.

In summary, leakage or failure of the piping and attached equip-ment at points remote from the containment shell will be taken to be governed oy the failure function G2

• Pc - p < 0 (4-4) where p is the applied pressure and pc is the plastic pressure as defined by the hal f linear slope method.

4.3 Combined Failure Criterion As alluded to in Sec. 3.1 and Fig. 4.1, it is reasonable to expect that leakage of the piping and attachments (general yielding of the containment governed by 2G ) will occur before fracture (leakage gov-erned by G ) of the containment shell itself. In terms of probabili-g ties of failure, one would expect that the probability of leakage of the shell, P(G <0,){ Event A), is much less than the probability of g

leakage of the piping and attachments, P(G2 <0,){ Event B), where G and g

G 2 are the failure functions in Eqs. 4-3 and 4-4, respectively. This expectation is verified in the following paragraphs.

Now, by the addition and multiplication formulas of probability, the probability of failure can be written as l

P(A B) = P(A) + P(B) - P(B) . P(A B) (4-5) in which P(A) = P(G <0),

i P(B) = P(G 2<0) and P( A B) is a conditional probability that can be referred to as the probability of shell leakage (A) given that attachment leakage (B) has occurred. Now G 2 less than zero (Event B) implies that the plastic pressure pc of the containment has been reached (see Eq. 4-4). At this condition, the stresses in the containment are greater than or equal to Fy in the region of failure, or P(B) - P(f > Fy ) (4-6)

In a similar manner, the fracture criterion of Eq. 4-3 can be approxi-mately written as l

l l P(A) - P(f > Ff) (4-7) in which i

l Fc Fu F =

f Xg(F2 + F2)

I where X o has been introduced to represent the modeling error of this f ailure criterion. Now, the conditional probability in Eq. 4-5 can be l written l

l l

P( AlB) = P(f > Ff lf > Fy) (4-8)

Since the stresses cannot be greater than F y because of the elastic-perfectly plastic assumption, one has P(AlB)=P(f>Flf=F)=P(Fy>F) f y f (4-9)

If the modeling error is taken to have a mean of one and a standard deviation of 0.08 [39] and the statistics of F, y Fu and F c are as given in Table 3-1 for McGuire (Sequoyah is similar), Eqs. 2-15, 2-16 and 2-14 can be applied to find P(AlB)=0.020 (4-10)

The conditional possibility P(Bl A) is also of interest, hence P(Bl A) = P(f > Fy lf > Ff )

By a Venn diagram P(BlA)> P(f > Fy o Ff > Fy lf > Ff)

Introducing the multiplication rule for the probability of the inter-section gives P(B A)= P(f y> F Ff>FOf>F)*

y f P(Ff>Fy f>F)f Now the first term on the right is one because of the conditions im-posed on f. The second term is simply P(F f >F)y since the two events are independent. Hence

P(BjA) > P(F f>F)=1 y - P(Fy>F)f and, by Eqs. 4-9 and 4-10 P(Bl A) > 0.98 (4-11)

Thus, by the multiplication rule, P(A) = P(B) - < P(B) (4-12)

A Eq. 4-5 now becomes P(ADB) < 0.9996 P(B) - P(B) (4-13) as the approximate probability of failure. Eq. 4-12, indeed, shows that the probability of fracture (A) before yield (B) is very small,*

or, the probability of containment leakage is negligible with respect to the probability of attachment leakage. In other words, there is a high probability that the containment vessel has the required ductility to reach the collapse pressure before brittle fracture. This probabil-ity of fracture can be neglected in probability calculations. Hence, the resul ts are insensitive to the fracture stress.

In summary, since the probability of containment leakage is negli-gible, it will be sufficiently accurate for the purposes of this pro-ject to consider only leakage of attachments. Thus, the failure cri-terion to be used is G = pc - p < 0 (4-14) where, again, pc is the plastic pressure as defined in Section 4.2.

• There are actually many cracks in the structure, each representing a potential failure mode and, thus, the discussion of Sec. 2.3 applies

[433 However, we are here comparing the probability of fracture at a typical " poi nt " to the probability of yielding at the same typical

" poi nt ". Only one crack is assumed to occur at this " poi n t " .

V. APPR0XIMATE STRUCTURAL ANALYSIS The Sequoyah and McGuire containment vessels are analyzed by ap-proximate methods in this section. In addition to providing approxi-mate resul ts , these methods provide useful guidelines for the more exact finite element analyses described in the following section.

5.1 Stiffened Axisymmetric Shell 5.1.1 Failure Criteria As discussed in Sec. 4, failure of the containment vessel is as-sumed to occur when large increases in deflections occur for relatively small increases in pressure, or when the pressure reaches the plastic pressure, pc (Eq. 4-11). An approximation to the plastic pressure, pc, is given by limit analysis theory [24], which defines a limit pressure, po, at which a plastic mechanism is formed. Classical limit analysis assumes rigid - perfectly plastic materials and small deflections, i.e., strain hardening and large displacement effects are neglected. For the approximate analysis discussed herein, the limit pressure is reached and a limit mechanism is formed. ( A more appropri-ate definition of the plastic pressure will be used in connection with the finite element analysis in Sec. 6. With the finite element analy-sis, the theoretical pressure-displacement curve will be available and pc can be obtained by the half linear slope method. For the simple model discussed here, the pressure-displacement curve would be elastic

- perfectly plastic and the half linear slope collapse pressure corres-ponds to the limit pressure.)

For the approximate analysis, a limit mechanism is assumed to occur when the entire structural system, including stiffeners, yields.

The stresses in the shell are assumed to be uniformly distributed at the limit mechanism. For equilibrium in the circumferential and axial directions of a cylindrical shell, respectively,

.- . = _ - .. .-

pr s t=fo tsi+f3 A g (5-1)

[s2*f t xs2 + f2 A2 where f x,fe = axial and circumferential shell stresses r = radius

t = shell thickness p = pressure s ,s = ring and stringer spacings 3 2 A .A2 = ring and stringer areas g

f ,f2 = ring and stringer stresses 3

(The symbol f is used here for stress since o has been reserved for standard deviation.)

As mentioned above, when the stresses are at yield, a limit mech-anism forms. For the stiffeners, this means

) f i=Fy (5-2) l f2=Fy For the biaxial stress state in the shell, yielding is governed by the l von Mises yield criterion 2

fx+f2_ffxe=Fy 9 (5-3)

The assumption that all stresses are at yield implies that sufficient

, ductility is present in the shell wall to permit a redistribution of load from the shell wall into the stiffeners. This model also assumes that the stringers and rings are totally effective at the limit

. mechanism.

k i

Introducing the yield criteria, Eqs. 5-2 and 5-3, into Eq. 5-1 4

)

gives the theoretical limit pressure pot for the stiffened cylinder as 2

Pot r 4 - (2a2 - ai) y = ai + 3 (5-4) y where ai = At /ts i a2 = A2/ts 2 Similar reasoning shows the limit pressure for a stiffened sphere to be Potr /

y=at+ a2 + g4-3(at-a2)2 (5-5) y The random value of the limit pressure can be written as (see Eq. 3-12)

Po = Xo Pot (5-6) where x o is the modeling error relating theory to the actual struc-ture (see Sec. 3.3). Following Eq. 4-11, the failure function becomes G = po - p (5-7) in which p is the applied pressure. The random structural parameters (x i in Eq. 2-4) are x o, F y, r, t, A i, s i, A 2, s 2, and p.

5.1.2 Modeling Error Before proceeding to the application of this anaylsis to the j actual containment vessels, it is appropriate to discuss the statistics of the modeling error, x.

o Unfortunately, no plastic strength

i experimental results for sti f fened cylinders with internal pressure could be found. The values of po and on are, thus, highly subjec-tive. In the extreme case of unstiffened uniform cylinders of infinite length, the values of 96 and o6 (see Sec. 3.3) would be one and zero, respectively. (In fact, this experimental situation is frequent-ly used to obtain Fy and/or to verify the von Mises yield criterion.

lience the only variability in tha theoretical versus actual would be represented by the variability in F .) y For this analysis, it will be assumed that most of the experimental results fall within i 30 percent of the thoeretical value. (This is admittedly a crude approximation but one we are forced to accept by the lack of experimental data.) If this limit is taken as two standard deviations, the values of 96 and a6 are 1.00 and 0.15. By Eqs. 3-19 and 3-20, the statistics of x o arc po = 1.0 (5-8) oo = 0.16 where xo is taken to be lognormally distributed.

5.1.3 Application to Containment Vessels The geometry of the two containment vessels is illustrated in l

Figs. 1-1 and 1-2. The stringers were neglected in the McGuire vessel (A2 equals zero) because they are not continuous, i.e., there is a 1/2" gap between each stringer and the ring webs. The statistics of the structural parameters are listed in Sec. 3 and Eq. 5-8. The failure criterion of Eq. 5-7 was used in conjunction with the second moment l

method of Eq. 2-15 to predict the safety index s at each ring eleva-

tion. The minimization procedure of the advanced first order second moment method in Eq. 2-15 is carried out using an iterative numerical I procedure based on Eqs. 2-17 and 2-18. Bounds on system failure proba-bility and safety index (Eqs. 2-52 and 2-54) were also calculated. The results are tabulated in Table 5-1. (Note that the safety index and

failure probabilities are associated with an applied pressure of 28 psi

- see Sec. 3.4. ) Minimum mean limit pressures, which were calculated near Elev. 788 ft. for Sequoyah and Elev. 736 ft. through Elev. 816 ft. for McGuire (Eq. 5-6), are also listed. The safety index, failure probability, and mean limit pressure at each ring elevation are listed in the Appendix.

5.1.4 Sensitivity Analysis One benefit to be derived from the approximate analysis (besides an approximate safety index) is a sensitivity study. Intuitively, one would expect that, if the coefficient of variation of a particular ran-dom parameter xj is relatively small, the solution for the safety in-dex would be insensitive to this parameter. This is evident when the Taylor series expansion of Eq. 2-11 is examined. In particular, the coefficient of variation of the geometric parameters (Table 3-2) are significantly smaller than the coefficient of variation of the material properties (Table 3-1) and the modeling errors (Sec. 3.3). One would, therefore, expect the reliability of the containment vessels to be relatively insensitive to variations in the geometric parameters. This expectation is verified below.

For this approximate analysis, the failure function is given in Eqs. 5-6 and 5-7. If the randomness of the geometeric parameters is neglected (coefficient of variation is small), the formulation for 8 in Eq. 2-15 can conveniently be solved in closed form. Since the yield strength (F y), modeling errors (x o), and applied pressure (p) are taken to be lognormally distributed, this solution is a=

• A (5-9)

/V2 +y2 + y2 gx o F p where P" "p / "p g

and Vxo , V Fy and V p are the coefficient of variation of the

(

modeling error, yield strength, and applied pressure, respectively.

Note that po is now lognormally distributed since it is the product of lognormally distributed parameters. Also, note the similarity with Eq. 2-47.

The application of Eq. 5-9 to the two nuclear containment vessels results in the safety indices in Table 5-2. Comparison of these re-sults with those of Table 5-1 shows that, indeed, the safety index and probability of failure is insensitive to the geometric variables. This observation will be very useful in the finite element analysis.

The minimum mean limit pressure is also listed in Table 5-2.

Since po is lognormally di stributed, the standard deviation of po can be found from Eqs. 5-7 and 5-9 (or, by first order methods from Eq.

5-6) as u 2 -1/2 in p 2 opo =g o 2

-V p

(5-10)

_7 .

where upo is taken as the minimum mean limit pressure and 3 is the generalized safety index of Eq. 2-54. Eq. 5-10 defines an upper and lower bound to opo because of the upper and lower bound to 3 in Eq. 2-54. These bounds on opo are also listed in Table 5-2. Note that V p is zero in view of the discussion in Sec.

3.4. l 5.2 Penetration Intersections As a continuation of the approximate structural analysis, the Sequoyah and McGuire penetrations are analyzed using the method out-lined in this section.

5.2.1 Failure Criteria As presented in Sect 4.2, leakage is assumed to occur in the pene-trations at points remote from the containment when excessive plastic deformations take place. The limit pressure is approximately the pres-sure at which this occurs. For a penetrated vessel, the theoretical

limit pressure Pot is assumed to occer when a yield mechanism is formed [24]. To calculate this pressure, the following two equations

[44,45] are used:

Cylinder-Cylinder Intersection

[162(1) +228(1)(A)+210]K+155 P

=* p (5-11) co 108K +[228(N) + 228] K + 152 D ,

Cylinder-Sphere Intersection Pot = Ps Pso (5-12a) in which ps is found from 1 {[1 + (1) ] (1-p s)) = P s [1-(I) ] I $ [ 1(1 Ip)]

s (5-12b)

D T D D DT d Dt and where t = penetration wall thickness T = vessel wall thickness d = penetration diameter D = vessel diameter K = d/D gD/T Pco = 2 Fy T/D i Pso =4FyT/D Note that pco and pso are the limit pressures of an unstiffened cylinder and sphere, respectively, according to the maximum principal stress failure theory. As indicated by Eqs. 5-4 and 5-5, these limit

! pressures are conservative because the effects of stiffeners are neg-lected. Eq. 5-11 gives the theoretical limit pressure of a cylinder-nozzle intersection. Eq. 5-12b is solved by Newton's method to find i

ps. The theoretical limit pressure for a sphere-nozzle intersection is then determined using Eq. 5-12a.

Both Eqs. 5-11 and 5-12 are developed for penetrations which are flush with the inside surface of the vessel. Thus, they neglect any strengthening effect of the penetration wall which intrudes into the vessel -- as is the case for the containment vessels. The effect of the intrusion is investigated in Ref. 45 for nozzles in spherical shells. An expression for the limit pressure is obtained which resem-bles, in certain respects, Eq. 5-12.

If the intrusion extends significantly into the vessel, the limit mechanism [45 (Mech. No. 6)] includes the formation of a positive and negative plastic hinge in both the interior and exterior portion of the penetration (Fig. 5-la). Eqs. 5-11 and 5-12 presented herein are based on a mechanism with plastic hinges in only the exterior portion (Fig.

5-lb). The effect of the intrusion can be approximately included by using an equivalent penetration thickness in Eqs. 5-11 and 5-12. If only membrane plastic work is involved, the mechanism in Fig. 5-12b would give the same limit pressure as that in Fig. 5-la if an equiva-lent thickness of 2t is used. If, at the other extreme, only bending plastic work is involved, the equivalent thickness would be g2 t. For the present case, both membrane and bending plastic work are involved and the equivalent thickness is somewhere between these two limits. As an approximation, the geometric mean, 23/t or 1.68t, will be used here.

The random limit pressure is written as po = Xo pot (5-13) in which x o is the modeling error relating theory to the actual structure (see Sec. 3.3) . Following Eq. 4-11, the failure function becomes G = po - p (5-14)

in which p is the applied pressure. The random structural parameters are xo , Fy , d, D, t, T and p.

Tne complex geometry and associated difficulty in fabrication of the penetration intersections are a potential cause of significant un-detected cracks. Hence, the probability of fracture relative to the probability of ductile flow is higher than at other locations in the shell. However, the discussion in Sec. 4.3, especially Eq. 4-12, is still appropriate, in the authors' opinion.

5.2.2 Modeling Error As mentioned previously, the modeling error xo is defined as a factor relating theory to the actual structure. Ref. 44 gives some tabulated experimental results of the limit pressure for cylinder-cylinder and sphere-cylinder intersections. Only experimental data for intersection failures (Mode 3, Sec. 5.2.1) were used to find the model-ing error. A total of 11 experimental results were used for the cylinder-cylinder intersection and 12 for the sphere-cylinder intersec-tion. Failure of the vessel or penetration wall (modes 1 and 2) is not considered in this section.

The ratio, 6 , between the experimental and the theoretical limit pressures can be calulated for the chosen experiments as

• • (5-15) 6=

Pot Assuming that 6 is lognormally distributed, the mean and the standard deviation of in 6 for the experimental data are Cylinder Sphere (5-16) m in 6 0.0323 0.248 sin 6 0.0835 0.111 Following Sec. 3.3, conservative values for these statistics are found for the 95 percent confidence interval as

43-Cylinder Sphere (5-17)

Un6 t -0.017 0.185

'in 6 0.147 0.188 Using Eq. 2-39, the mean and standard deviation of the lognormally distributed 6 becomes Cylinder Sphere (5-18) 96 0.994 1.23 p6 0.147 0.232 Referring to Sec. 3.3 (Eqs. 3-19 and 3-20) the statistics for the modeling error xocan be found as Cylinder Sphere (5-19) p o

0.99 1 23 oo 0.16 0.24 5.2.3 Application to Containment Vessels The structural parameters for each penetration were obtained from drawings of the containment vessels. The statistics for these param-eters are listed in Tables 3-2 and 3-2. Statistics for thicknesses not listed in Table 3-2 were interpolated. Tolerances for the penetration diameters were taken as i one percent.

The failure criterion of Eq. 5-14 was used in conjunction with the second moment method of Eq. 2-15 to predict the safety index for each penetration. Bounds on the system failure probability and safety index were also calculated. The results of this analysis are listed in Table 5-3. The minimum mean limit pressures were found to occur at the pene-trations at Elev. 767 ft. , Az. 266 for Sequoyah and Elev. 758.75 ft.,

Az. 20 for McGuire. There are several penetrations identical to these

at other locations in the containment shells. Complete results for each individual penetration are listed in the Appendix.

5.2.4 Sensitivity Analysis As in the discussion of Sec. 5.1.5, the results are expected to be relatively insensitive to the geometric parameters. In the case of the penetrations, this can be verified by using the closed form solution for the safety index in Eq. 5-9. The results for bounds to the safety index and failure probability using Eq. 5-9 are given in Table 5-4.

Again, by compdrison with Table 5-3, the reliability is seen to be in-sensitive to the geometric ' parameters. Minimum mean limit press'ure and bounds to the limit pressure standard deviation (Eq. 5-10) are also

' listed in Table 5-4.

5.3 Anchor Bolts

'- The Sequoyah and McGcire containment vessels are held down by a number of high strength bolts distributed at about 4* around the con-tainment base. olts anchor the containment vessel to the con-

, crete foundation. st,ontainment weight also acts downward to prevent uplift by internal pressure but this force is relatively small and is neglected here.) The a .chor bolts will yield when the internal pres-sure reaches the limit pressure. Summing vertical forces at the 1(mit pressure when all the bol ts have yiel ded gives

( ,

pot nr =nAb Fy 't '(5-20) where i n = number of bolts A

b = bolt cross-sectional area i Fy = bolt yield strength 1 r = containment radius

' pot = tneoretical limit pressure at bolt yielding.

-( .

The random value of the limit pressure can be expressed as (see Eq.

3-12) po = Xo pot (5-21) in which x o is the modeling error relating theory to the actual structure. For this analysis, it will be assumed that most of the experimental results fall within 30 percent of the theoretical value.

The modeling error statistics are then the same as in Eq. 5-8, i.e.,

po = 1.00 (5-22) oo = 0.16 (5-23)

Following Eq. 4-11, the failure function becomes G = po - p (5-24) in which p is the applied pressure. The random structural parameters (xj in Eq. 2-4) are x,o A' b

f, y r, and p. Ti<e , statistics of these structural parameters are listed in Tables j-1 and '3-2 and Eq.

5-23. The second moment reliability method summari ed in Sec. 2-2 (Eq.

2-15 and 2-16) is applied. The resulting limit pressure, safety in-dices, and failure probabilities are tabulated in Table 5-5. (Note that the safety index and the failure probability are associated with l

an applied pressure of 28 psi.)

! Following the discussion of Sec. 5.1.4, the results are expected to be relatively insensitive to the geometric parameters. In the case of the anchor bolts, this can be verified using the closed form solu-tion fcr the safety index given in Eq. 5-9. The safety index, failure probability and the limit pressure standard deviation are listed in Table 5-6. Comparing these results with those given in Table 5-5, one can see that the anchor bolt reliability is insensitive to the geo-metric parameters.

5.4 Combined Failure Modes As discussed in Secs. 5.1, 5.2 and 5.3, each failure mode was studied individually. A sensitivity study demonstrated that the safety index was insensitive to the geometric parameter statistics and could adequately be represented by Eq. 5-9. In this section all the failure modes (shell, penetration intersection and anchor bolts) are combined to predict the bounds for the safety index and failure probability of the complete structure. Thus, the results of Table 5-2 (shell failure modes), Table 5-4 (penetration intersection failure modes) and Table 5-6 (anchor bolt failure mode) are combined using Eqs. 5-9 and 2-54.

Bounds on the structure failure probability and the safety index are evaluated and listed in Table 5-7. The minimun mean limit pressure and I

bounds on its standard deviation (Eq. 5-10) are also listed.

As shown in Tables 5-2 and 5-3, the minimum mean limit pressure for both containments is controlled by the stiffened shell failure modes. Thus, the strength of the peneterated shell is larger than the strength of the unpenetrated shell . (This result is not unexpected and, in fact, provides the basis for the ASME area replacement rule.)

However, in the case of McGuire, the strength of te controlling pene-tration is only slightly larger than the unpenetrated shell. The stif-fened shell and the controlling penetration will be analyzed in Sec. 6.

In both containments, the anchor bolt limit pressure is relatively high and has no effect on the failure probability.

A word of caution - only the failure modes discussed above were analyzed. They were selected as being the most likely modes. However, many other modes are possible, e.g., welds, expansion bellows, person-nel and equipment hatches, seals, and foundation liners. To examine all potential failure modes would require a much more extensive effort.

The assumption has been made here that, as shown for the penetration and anchor bolt modes, the strength in all these other modes is greater than the strength in the shell failure modes.

<p ,

m . _ __ _

VI. FINITE ELEMENT ANALYSIS The approximate analyses in the previous section provide useful information in that they indicate critical failure modes which deserve further analysis. Finite element analysis methods are applied in this section to perform a more refined analysis of the stiffened axisymmet-ric shell and certain controlling penetrations.

6.1 ANSYS Finite Element Program i

ANSYS [47] is a large-scale, general purpose computer program for the solution of several classes of engineering analysis problems. The l

program has the capability of analyzing two and three dimensional i

structures, piping systems, two dimensional axisymmetric solids, three dimensional solids and nonlinear problems. The ANSYS program is also capable of solving static, dynamic and heat transfer problems. The l ANSYS program has the capability of generating and plotting the struc-tural input data for the finite element models. Plotting routines are 4

also available for plotting the distorted geometry.

The ANSYS program has two options available to include geometric nonlineari ty. The first is called large displacement analysis and is accomplished by updating nodal coordinates to formulate the element 1 stiffness matrix [48,49]. The second option is called stress stiffen-ing and is accomplished by adding the geometric stif fness matrix l [48,49] to the usual linear element stiffness matrix. The stress stif-fening matrix depends upon element stresses obtained from the previous l iteration. The stress stiffening solution represents a first approxi-l mation to large displacement effects.

l The ANSYS program provides a plastic material capability with I several options for material nonlinearity. The option employed here is

called classical bilinear kinematic hardening. An elastic perfectly l plastic material property is used (no strain hardening).

To accomplish the nonlinear (material and geometry) solution, an

, iterative approach is used within the ANSYS program. The procedure is to increase the applied load by small increments, called load steps, and allow the program to iterate until it converges to a final

solution. The solution is said to be converged if the ratio of the change in the plastic strain, ac p, to the elastic strain, c, e re-ferred to as the plastic convergence ratio, is less than a specified value. The smaller the load step, the fewer the required number of iterations. In the program, a value of 0.01 is used for this ratio unless otherwise specified.

6.2 Failure Criteria The nonlinear fini te element analysis predicts the theoretical pressure-displacement curve. As discussed in Sec. 4, failure is as-sumed to occur when the displacements become "large", or the pressure reaches a plastic pressure value as calculated by the half linear slope method [24,41,42]. (See Fig. 4-2.) The theoretical plastic pressure pct will be used in calculating the structural safety index and probability of failure in conjunction with the sensitivity analysis discussed previously. The random plastic pressure pc is written as pc " Xo Pct (6-1) in which x o is the modeling error relating theory to the actual structure (see Sec. 6.3.2).

6.3 Stiffened Axisymmetric Shell 6.3.1 Finite Element Modeling Guidelines The containment vessel wall (axisymmetric shell) can be modeled by axisymmetric solid elements. The ANSYS program provides two different four-sided isoparametric elements of this type. One has four corner nodes (STIF42) and the other has four corner nodes and four midside nodes (STIF82) as shown in Fig. 6-1.

To study the accuracy of both STIF42 and STIF82 and the various options in ANSYS, a smooth closed end clyindrical shell was analyzed.

The following options were employed:

Plastic Convergence Geometric Nonlinearity Element Ratio Option STIF42 0.03 Stress stiffening (SS)

STIF42 0.03 Large displacement (LD)

STIF82 0.03 SS STIF82 0.05 SS STIF82 0.10 SS The cylinder was modeled as shown in Fig. 6-2. For STIF42, an element length (height) of /rt/4 was used, in which r and t represent the shell radius and thickness, respectively. Twice this length was used with STIF82. One element was used through the shell thicknesses.

Fig. 6-3 shows the radial deformation of the cylinder at an inter-nal pressure of 35 psi. At this pressure, the structure is in the elastic region. The results using STIF82 and STIF42 without either stress stiffening or large displacements are very close to those found from classical shell theory [50]. Also, STIF82 with the stress stif-fening option gives results which are close to those calculated using STIF42 with either the stress stiffening or large displacement options.

STIF82 with an element length of about grt/2 will be used in this work.

Fig. 6-4 shows the pressure-displacement curves for the different element types and the different plastic convergence ratios. The re-

! sults of the theoretical plastic pressure using the half linear slope method are given in Table 6-1. The percentage difference in the theo-retical plastic pressures (with respect to pct for a convergence ratio of 0.03) is listed in Table 6-1. Computer CPU time is also tabu-lated. Fig. 6-4 and Table 6-1 indicate that there is no significant l difference between the results obtained from STIF42 and STIF82. Fur-ther, stress stiffening adequately accounts for large displacement l effects in the range of interest at a large savings in computer time.

, Hence stress stiffening will be used in this work.

Table 6-1 illustrates that a higher plastic convergence ratio results in shorter CPU times and a larger error in the theoretical l

, plastic pressure. Comparing the CPU time used and the difference in pct, it was decided to use a plastic convergence ratio of 0.1. The error in the theoretical plastic pressure, pct. will be accounted for, somewhet, in the modeling error x o (Eq. 6-1).

The Sequoyah and McGuire containment vessels are reinforced by ring and vertical stiffeners. The vessel in Fig. 6-5 was used to in-vestigate possible finite element models for the stiffeners. The cyl-inder was modeled by the isoparametric elements mentioned above. The vessel, without stringers, was analyzed using two different ring mo-dels. Initially, five isoparametric axisymmetric elements were used to idealize the web and two elements for the flange of the ring. Next, one element was used in the web and one in the flange. Both idealiza-tions gave approximately the same predicted theoretical plastic pres-sure (within three percent). Due to the noticeable saving in computer time, one-element idealizations of the web and flange will be used.

The stringers obviously introduce a non-axisymmetric character to the problem, i.e., the stringers cause the displacements to vary cir-cumferentially. In principle then, the structural behavior is three-dimensional. However, a three-dimensional idealization of the entire containment vessels is beyond a reasonable scope for this project. To retain the axisymmetric idealization, a stringer is idealized as a beam with properties uniformly distributed around the circumference. The basic assumption, then, is that the circumferential variation of dis-placement is negligible.

The circumferential variation of displacement for the Sequoyah containment has been independently studied by two investigators

[51,52]. Their work is unpublished and summarized here.* Both authors considered the behavior of a typical panel of the Sequoyah containment bounded by two rings and two stringers. At a point midway between the rings, Ref. 51 found that the ratio of the displacement at the stringer to the displacement midway between the stringers (center of panel) is

• Copies of [51,52] are attached in the Addendum.

about 1.08. This ratio is maintained up into the nonlinear range. In Ref. 52, a linear analysis of a slightly different-sized panel showed the ratio to be 1.20.

A two-dimensional beam element (STIF23) was used to model the vertiical stiffeners (stringers). This beam element has stress stif-fening and nonlinear capabilities. Since no hoop effects are present in the stringers, STIF23 can be used with axisymmetrical elements.

Properties are input on a per-radian basis.

The linear constraint equation option in ANSYS was used to ideal-ize the ring and stringer connection to the shell wall. This option relates the displacements of selected nodal points through a specified equation. The cylinder / stringer connection idealization is shown in Fig. 6-6 together with the associated linear constraint equations. In effect, a rigid link connects the stringer node to the shell node.

The cross-section of the ring was assumed to remain rigid and to translate and rotate with the stringer node. Fig. 6-7 shows the ring web connection to the cylinder and to the stringer with the associated constraint equations. Node 4 was constrained to Node 1 only in the vertical direction to prevent any artificial stiffening of the web, e.g., to permit through thickness straining. The ring stiffener flange was also constrained to the stringer node as shown in Fig. 6-8.

Fig. 6-9 illustrates the pressure-di splacement curves for the vessel in Fig. 6-5 with and without stiffeners. Using the half linear slope method, there is about a 23 percent increase in the theoretical plastic pressure when the ring and vertical stiffeners are added.

6.3.2 Modeling Error l

The modeling error, x, o is defined as a factor relating theory

! to the actual structure (see Sec. 3.3. and Eq. 6-1). Ref. 53 provides some experimental results in the form of pressure-displacement curves for closed-end smooth cylinders. (Unfortunately, no experimental re-suits for stiffened cylinders were found.) A total of six such cylin-( ders (Cylinders 31, 32, 33, 34, 35 and 36) were analyzed using STIF82 with a plasticity convergence ratio of 0.1. The theoretical pressure-displacement relationship was found and the theoretical plastic l

i I

pressure was evaluated as discussed in Sec. 6.2. The experimental plastic pressures were obtained from the experimental pressure-displacement curves by the hal f linear slope method.

The ratio, 6, between the experimental and theoretical plastic pressures was calculated as in Eq. 5-15. Assuming that 6 is lognorm-ally distributed, the eean and the standard deviation of in 6 is found as min 6 = 0.153 (6-2) sin 6 = 0.24 Following Sec. 3.3, conservative values for these statistics are found

~

from the 95 percent confidence interval as pin 6 = 0.123 (6-3)

"in d = 0.059 Using Eq. 2.39, the mean and standard deviation of the lognormally distributed 6 becomes p6 = 1.13 (6-4) o6 = 0.067 which represents the variability between theoretical and experimental results on smooth cylinders. The experimental models appear to have a plastic pressure about 13 percent larger than that predicted by theory.

This increase is not considered reliable and will be conservatively neglected here. The error in the ring and vertical stiffener models is assumed to be incorporated into the factor A. Referring to Sec. 3.3 (Eqs. 3-19 and 3-20) the statistics for the modeling error x o are taken to be

l 1

j po = 1.00 (6-5) oo = 0.083 Admittedly, these values are somewhat subjective. As discussed in Sec.

3.3, more confidence can be developed in these statistics only by anal-yzing more experimental results and, especially, by testing more full-scale containments. This is beyond the scope of the present study.

6.3.3 Applications The ANSYS finite element computer program was used to analyze the Sequoyah and McGuire containment vessels shown in Figs. 1-1 and 1-2, respectively. The guidelines outlined in Sec. 6.3.1 were used to model these vessels. The mean dimensions given in Table 3-2 were employed in the finite element analysis.

6.3.3.1 Finite Element Model For the Sequoyah containment vessel, the stringers are welded to the containment wall and the ring stiffeners. The linear constraint equations discussed in Sec. 6.3.1 were used to model this connection.

Two hundred fifty STIF82 elements and 123 STIF23 beam elements were used to idealize the containment vessel. The total number of nodes was 1414. At the time this analysis was performed, the mean material yield strength had not been furnished by the Sequoyah owners. A mean yield strength of 35.2 ksi was used in the ANSYS analysis. The actual mean value was provided later (see Table 3-1). A uniform internal pressure of 35 psi was initially applied to the Sequoyah vessel model and incre-mented by 5 psi. At a pressure of 50 psi convergence to the specified plastic ratio did not occur within 20 iterations. About 7-1/2 hours of computer time on a PRIME 400 minicomputer were used for this analysis.

In the McGuire containment vessel, there is a 1/2" gap between the stringer and the ring webs. Linear constraint equations similar to Fige 6.6 were used at each end of each stringer to model this gap. Two hundred thirty-eight STIF82 elements and 130 STIF23 elements were used

to model the containment. The number of nodes was 1365. The uniform internal pressure for the McGuire containment was started at 55 psi with increments of 5 psi up to 80 psi. The pressure increment was then changed to 1 psi to ensure convergence within each load step. The solution did not converge to the specified plastic ratio after ten iterations at a pressure of 85 psi. The run time on the PRIME 400 was approximately 7-1/2 hours.

6.3.3.2 Results The results of the finite element analysis are summarized in Figs.

6-10 through 6-15. Figs. 6-10 and 6-11 are plots of the applied pres-sure versus the maximum radial displacement for the Sequoyah and McGuire containment vessels. This displacement occurs at about Elev.

783' for Sequoyah and about Elev. 751' for McGuire. For the Sequoyah vessel, plastic deformation starts at a pressure of about 40 psi and increases rapidly at a pressure of 45 psi. For the McGuire containment vessel, plastic defonnation starts at about 70 psi and increases rapid-ly at 84 psi . Since convergence did not occur at the last load step, the radial displacement is taken to be very large. This is represented by the nearly horizontal line in Figs. 6-10 and 6-11. Using the half linear slope method, the theoretical plastic pressures, pct, are 45 psi and 84 psi for Sequoyah and McGuire, respectively. As mentioned above, 35.2 psi was used as the mean yield strength of the Sequoyah containment. The corrected theoretical plastic pressure is obtained by multiplying 45 psi by 47.2/35.2 to obtain 60 psi as the predicted plas-tic pressure for Sequoyah.

Figs. 6-12 and 6-13 show the applied pressure versus the maximum effective von Mises strain. For Sequoyah, this maximum strain occurred

! at about Elev. 783' and at about Elev. 751' for McGuire. These plots indicate the strains at the predicted plastic pressures are not exces-sive - of the order of two times the elastic strain. The ductile steel used in these containments will almost certainly be able to tolerate these strains without fracture [21, pg 529]. In other words, the ves-sels will almost certainly reach the plastic pressure before leakage.

This observation reinforces a similar conclusion in Sec. 4.3.

l

Fig. 6-14 shows the deflected shape of the Sequoyah containment vessel near the plastic pressure. The maximum displacement occurs at the smallest shell thickness. The deflection between the ring stiffen-ers is slightly more than at the ring locations. The deflected shape of the McGuire containment vessel near the plastic pressure is shown in Fig. 6-15. Since this vessel has a more uniform thickness, the radial displacement is almost uniform.

An examination of the stress results for both of these analyses indicates, that, as expected, the shell yields first about midway be-tween stiffeners. At this point the rings are below their yield value.

However, as the pressure is increased, the ductile shell continues to strain with no change in stress. Forces are, thereby, redistributed to the rings. Eventually, the rings themselves reach their yield stress and, for all practical purposes, a collapse mechanism is formed. In this mechanism, the shell and rings are completely yielded. This cor-responds to the last (non-converged) load step of the nonlinear solu-tion. At this point, the stringers are also almost at total yield in tension in the vicinity of the maximum displacement. (The stringer axial loads are 0.96 and 0.89 of the stringer tensile yield strengths for Sequoyah and McGuire, respectively.) This behavior tends to con-finn the mechanism assumptions in Sec. 5.1.1.

6.3.4 Uncertainty Analysis In Sec. 5.1 it was concluded that the safety index is insensitive to the geometric structural parameters and could adequately be predic-ted using Eq. 5-9. The theoretical plastic pressure, pct, obtained in Sec. 6.3.3, is used in conjunction with Eqs. 5-9 and 5-10 to perform the uncertainty analysis for the Sequoyah and McGuire containment

vessel s. This leads to the standard deviation of the plastic pressure as*

/2 2 o

p

=p p gV +Y p (6-6)

The statistics for material yield strength are given in Table 3-1 and for the modeling error in Eq. 6-5 The mean and the standard deviation of the plastic pressure for the axisymmetric shell failure mode become 60 psi and 5.9 psi, respectively, for Sequoyah and 84 psi an 8.3 psi, respectively, for McGuire.

6.4 Penetration Analysis The controlling penetration for each containment vessel (see Sec.

5) was analyzed using the finite element method through the ANSYS pro-gram.

6.4.1 Modeling Guidelines The element used for this analysis was STIF48, the only ANSYS shell element possessing both stress stiffening and plasticity charac-teristics. STIF48 is a three node triangular element with six degrees of freedom per node. Following the suggestion of the ANSY documenta-tion, a curved shell pressure loading option was utilized which elimi-nates equivalent nodal moments that cause fictitious bending stresses in the element. Two linear trial runs, with and without this option, proved this to be true.

• Note that the collapse pressure is, to the first order, proportional to F. Dimensional analysis principles would give p and Fy /E as the only dimensionless products involving force difn/ Fen 5 ions.

The term, Fy /E, is, typically, considered to be a second order effect so that P c 15 proportional to Fy when other parameters are held constant.

l

In an effort to minimize CPU time without sacrificing accuracy, an attempt was made to develop some modeling guidelines. Parameters felt to warrant study included element size, aspect ratio, mesh characteris-tics, and model size. An extensive study was not possible, but by linearly analyzing two models - a smooth pressurized cylinder and a pressurized cylinder with a penetration - and comparing their behavior with theoretical and experimental results, some very general guidelines developed.

Element length at the vessel-penetration intersection is control-led not by rt/2, but rather by central angle size of the penetration.

An arc length of 10 degrees with respect to the penetration is an ap-proximate maximum value. Maximum element size in the penetration re-mote from the intersection is 30 degrees of arc length. Maximun ele-ment size in the vessel remote from the intersection is 20 degrees of arc length with respect to the vessel. Element aspect ratios should approach 1:1 in critical areas of the intersection, but 1:3 is satis-factory in areas remote from the intersection.

Characteristics of the mesh involve two concepts. First, and most obvious, the mesh should have smooth transitions from fine areas around the intersection to coarse areas remote from the intersection. Second, the mesh, wherever possible, should be generated with three sets of parallel lines instead of four [48, pg 245]. The latter mesh gives rise to unequal equivalent nodal forces.

Finally, trial runs indicated that the model of the penetration should extend at least one penetration diameter, d, away from the in-tersection; similarly, the vessel model should also extend at least a distance, d, away from the intersection.

6.4.2 Experimental vs ANSYS Results To facilitate some sort of comparison of ANSYS results to predic-ted results, a nonlinear analysis of a pressure vessel with a penetra-tion was performed using the aforementioned guidelines. The results were compared to an experimental model [54] of the same dimensions (d =

3.762 ", t = 0.125 ", D = 5.789 ", T = 0.187 ") . The disp 1acement of the point on the vessel-penetration intersection and in the symmetry plane

perpendicular to the vessel axis was plotted versus the internal pres-sure. Both the experimental and ANSYS curves are shown in Fig. 6-16.

Lack of time and money precluded additional model runs, therefore preventing a more complete error analysis. However, the similarity of the two curves in Fig. 6-16 does indicate certain factors. Fi rst, STIF48 appears to give fairly reliable results. Second, the modeling guidelines which were applied appear to be satisfactory. The percent error in the theoretical plastic pressure, pc t, for this particular case is 10 percent. Last, an approximation to the CPU time for a non-linear run was obtained which directly influenced later modeling decisions.

6.4.3 Application 6.4.3.1 Finite Element Model The finite element model of the controlling Sequoyah penetration (Elev. 767', Az. 266') was developed with certain considerations. This penetration has a diameter of 24 in, and a thickness of 0.375 in. The containment vessel has a thickness of 0.625 in. in the vicinity of the penetration but is reinforced locally to 1.5 in. To minimize CPU time and still obtain accurate results, as small a portion of the structure as possible was chosen. Since an identical penetration is located 5 degrees (centerline to centerline) away from the modeled penetration, a i plane of symmetery was assumed at half that arc length, 2.5 degrees.

Also, to minimize the stiffening effect of the circumferential rings and to obtain a conservative result for other similar penetrations, the penetration was assumed to lie midway between two rings. The model, thus, had quarter symmetry. The penetration was madeled with a 24 in.

intrusion and a 24 in. protrusion, or one diameter away from the inter-section. The model consists of 253 elements and 183 nodes. STIF48 was used for all elements, including the rings. The mesh is shown in Fig.

6-17. All four edge-planes of the vessel, along with the longitudinal symmetry plane of the penetration, were modeled with symmetry boundary conditions. The transverse edges of the penetration were assumed to be free edges.

A uniform pressure was applied in increments to the internal ves-sel face and the external face of the penetration intrusion. Axial forces were applied to both the penetration intrusion and the vessel to account for pressure end loading. The initial load, 20 psi, was used to establish an initial slope for the pressure / displacement curve. An estimate of the yield pressure was obtained by extrapolating the ele-ment stresses to yield. This estimate was 35 psi, which was the second load, or load step. The succeeding load steps were determined by con-sidering the pressure-displacement curve and rate of convergence of the previous load step. Load steps of 5 psi were used successfully. In the interest of saving the time and expense of unnecessary computer processing, the loading das continued only until a minimum acceptable pressure was achieved. For Sequoyah, the plastic pressure of the un-penetrated shell was determined to be 60 psi (see Sec. 6.3). There-fore, the penetration analysis was concluded upon reaching 65 psi suc-cessfully. The computer run time was about 9 hours1.041667e-4 days <br />0.0025 hours <br />1.488095e-5 weeks <br />3.4245e-6 months <br />.

The controlling McGuire penetration (Elev. 758.75', Az. 20') was modeled with similar considerations. The penetration is a Schedule 60,12.75 in, diameter pipe. The containment shell thickness is 1.0 in.

There is no local reinforcement. The penetration is one of several identical penetrations arranged in two parallel rows between two cir-cumferential rings. Again, a symmetry plane was assumed midway between the penetrations (2.5 centerline to centerline) or 1.25' away from the controlling penetration centerline. To conservatively minimize the stiffening effects of the rings, the two rows of penetrations were as-sumed to lie at one-third points between the rings. A horizontal plane of symmetry was then assumed midway between the two horizontal rows of l penetrations. The penetration was modeled with a 12 in. intrusion and a 12 in. protrusion, or one diameter away from the intersection. The McGuire model consists of 455 elements, all using STIF48, and 315 nodes. The finite element mesh is shown in Fig. 6-18. All four edge-

! planes of the vessel, along with the longitudinal synmetry plane of the

[ penetration, were modeled with symmetry boundary conditions. The remaining transverse edges of the penetration were assumed free edges.

I i

A uniform pressure was applied in increments to the internal ves-sel face and the external face of the penetration intrusion. Axial forces were applied to both the penetration intrusion and the vessel to account for pressure end loading. Using loading criteria similar to the Sequoayh analysis, the initial load was taken to be 10 psi with an estimated initial yield pressure of 32 psi. Again, varying load steps were applied until a pressure of 88 psi was successfully achieved, well above the controlling theoretical plastic pressure of 84 psi for the axisymmetric shell as determined from the containment analysis (Sec.

6.3). The approximate run time was 22 hours2.546296e-4 days <br />0.00611 hours <br />3.637566e-5 weeks <br />8.371e-6 months <br />.

6.4.3.2 Resul ts The pressure versus displacement curves are shown in Fig. 6-19 (Sequoyah) and Fig. 6-20 (McGuire). In each case, the displacement refers to the displacement of the end of the penetration protrusion measured radially from the axis of the vessel [24]. These curves show that the theoretical plastic pressure for the controlling penetrations for Sequoyah and McGuire are greater than 65 psi and 88 psi, respec-tively.

Plots of the applied pressure versus the maximum effective von Mises strain are shown in Figs. 6-21 and 6-22. In both cases, the strains of the above pressures are not more than six times the yield strain. As discussed in Sec. 6.3.3.2, the ductile steel of this con-tainment will almost certainly tolerate these strains without fracture

[21, pg 529].

6.4.4 Uncertainty Analysis As mentioned above, it did not seem reasonable to spend the com-puter time and funds which would have been required to continue the pressure-displacement plots of Figs. 6-19 and 6-20 up to the theoreti-cal plastic pressure. An uncertainty assessment of the penetration f ailure modes can, therefore, not be performed. However, the mean plastic pressures for the controlling penetrations are above 65 psi and 88 psi for Sequoyah and McGuire, respectively. These plastic pressures are significantly above those calculated for the unpenetrated shell in

Sec. 6.3 - 60 psi and 84 psi, respectively. The approximate results show that the standard deviation of the plastic pressure of the pene-tration analysis (Table 5-4) is not significantly different from the standard deviation of the plastic pressure of the unpenetrated shell (Table 5-2) . In terms of the uncertainty analysis of the entire con-tainment, therefore, one can reasonably neglect the probability of failure of the penetrations with respect to the shell. Or, as stated in Sec. 5.4, the strength of the penetrated shell is greater than the strength of the unpenetrated shell .

6.5 Combined Failure Modes As summarized in the previous portions of Sec. 6, the finite ele-ment was used to analyze two failure modes (axisymmetric stiffened shell and one penetration intersection) for each of the two contain-ments. These modes were indicated by the approximate methods of Sec. 5 to be the most likely to occur. A finite element analysis of all the possible failure mo. des would be well beyond the scope of this project.

And yet, failure by these unanalyzed modes is certainly possible. To approximate the effect of the unanalyzed failure modes on the safety index and failure probability, the following approach will be adopted.

The system mean failure pressure will be taken to be equal to the mini-mum mean plastic pressure for the analyzed failure modes: 60 psi and 84 psi for Sequoyah and McGuire, respectively (Sec. 6.3.4). In both cases, the unpenetrated axisymmetric shell has been shown to control, i.e., to have the minimum mean plastic pressure (Sec. 6.4.4). The coefficient of variation of the plastic pressure will be taken as [37]

2 2 2 V = V +V F + V, (6-7) y where Vm represents the effect of the unanalyzed failure modes. The approach suggested in Eq. 6-7 implies that the effect of the unanalyzed failure modes can be expressed in the form of a multiplying factor similar to the modeling error. The authors realized that this is, at best, a very approximate approach and that it has pitfalls. However,

it is expedient. It is not economically feasible to analyze all fail-ure modes by finite element. In fact, a primary motivation for the development of approximate methods is to be able to analyze many fail-ure modes. The finite element method is here visualized as a means of refining the approximate results. For this reason, no 8 values or failure probabilities are calculated for the finite element analysis. l The coefficient of variation of the modeling error and material yield strength have been discussed previously in conjunction with Eq. 6-6.

The difference between the upper and lower bounds to the plastic pres-sure standard deviation is due to the additional failure modes (see Eq.

2-54 and 5-10) . An appropriate value of Vmwhich approximately ac-counts for all the shell, penetration and bolt failure modes is 2 2 V, = Y -V p (6-8) p where V Pcu and V Pct are the upper and lower bound coeffici-ents of variation for the plastic pressure. For the approximate re-suits in Table 5-7, V m is 0.037 and 0.099 for Sequoyah and McGuire, respectively. For this work, Vm will be taken as [37]

Vm= 0.10 (6-9)

With this value of V m and Eq. 6-7, the final predicted mean and stan-dard deviation of the plastic limit pressure are listed in Table 6-2.

VII.

SUMMARY

OF RESULTS 7.1 Summary The Sequoyah and McGuire ice condenser containment vessels were designed to withstand pressures in the range of 12 to 15 psi. Because the peak recorded pressure at TMI was 28 psi, there is a need to nore accurately define the strength of these vessels. A second moment relf-ability method was applied to perform a best estimate and uncertainty analysis of the containment strengths. Material property statistics and geometric tolerances were furnished by the plant owners. The load-ing was assumed to be a uniform static internal pressure. Gross defor-mation (displacement twice the elastic value at the same load) was taken to be the failure criteria. Fracture of the vessel wall was shown to have a low probability of occurrence. The final results are insensitive to the ultimate strength and fracture stress and to the randomness of the geometric parameters. However, the calculated mean and standard deviation of the pressure strength are dependent upon the randomness of the yield stress and modeling error.

The complete vessel and all the penetrations were analyzed by approximate methods based on classical limit anaylsis theory. For Sequoyah, the mean limit pressure was found to be 59 psi with a stan-dard deviation of 10 psi. For McGuire, the mean was 77 psi and the standard deviation was 15 psi.

An axisymmetric finite analysis was performed of each vessel using the ANSYS computer program with nonlinear material and geometric options. Also, the controlling penetration for each vessel, as deter-mined by the approximate analysis, was analyzed using ANSYS. The resulting mean and standard deviation of the plastic pressure for Sequoyah were 60 psi and 8 psi, respectively. For McGuire, the mean and standard deviations were 84 psi and 12 psi, respectively.

7.2 Conclusions For the failure modes investigated, the mean plastic static pres-sure strength of the Sequoyah containment vessel is 60 psi with a

a standard devatiation of 8 psi. The corresponding pressure for McGuire is 84 psi with a standard deviation of 12 psi.

7.3 Recommendations There are at least four areas where the scope of work presented herein was limited. These limitations could be removed by:

Dynamic analyses. The TMI pressure pulse is potentially dynamic, i.e., a pulse length of the order of the structural period. (No exact information has been furnished.) In this regard, dynamic analyses should be performed using realistic pressure-time-space relationships.

Additional failiure modes. As nentioned in Sec. 5.3 and 6.5, only a

limited set of failure modes was examined in this study. A more comprehensive program could be undertaken to examine other modes.

Approximate analyses. The approximate analyses appear to give rea-sonable results for these cases at a large savings in time. The limits of these approximations should be defined.

Distribution assumptions. The assumption of normal or lognormal distributions for input parameters introduce errors of various amounts in the reliability estimates. The error should be quanti-fied.

- Fracture. The fracture properties of the material (stress intensity factors, crack shapes and size; temperature) should be investigated more thoroughly to establish more certainly the tentative results of

Sec. 4.3.

I - Experimental results. Perhaps the biggest shortconing of the analy-sis reported herein is the lack of correlation with experimental data. No experimental data could be found for stiffened cylinders

under internal pressure. Hence, the modeling error which was used was quite subjective. Although extensive experimental data exist for shell penetrations, very little exists for large r/t values.

Also, there was not enough time or money to analyze the existing data with the finite element program. Future work should be devoted to: (1) finding (or obtaining) experimental results for stiffened cylinders with internal pressure and correlating these results with

a finite element analysis, and (2) correlating existing penetration experimental results with a finite element analysis.

l VIII. LIST OF REFERENCES

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2. Kiureghian, A.D., ' Reliability Analysis Under Stoichastic Load,"

Journal of Structural Division, ASCE, Vol.106, ST2, Feb.1980, pp. 411-429.

3. Fiessler, B., Neuman, H-J and Rackwitz, R., Tuadratic Limit 4

States in Structural Reliability," Journal of Engineering Mechan-ics, ASCE, Vol. 105, EM4, Aug. 1979, pp. 661-676.

4. Rackwitz, R., " Practical Probabilistic Approach to Design," Tech-nical University of Munich, May 1976.

5. Veneziano, D., "New Index of Reliability," Journal of Engineering Mechanics, ASCE, Vol .105, EM2, April 1979, pp. 277-296.

6. Kapur, K.C. and Lamberson, L.R., Reliability in Engineering De-sign, John Wiley & Sons, New York,1977.

7. Freudenthal, A.M., Garrelts, J.M., Shinozuka, M., "The Analysis of Structural Safety," Journal of Structural Division, ASCE, Vol.

90, STI, Feb. 1966, pp. 267-325.

8. Ang, A.H.-S., ' structural Risk Analysis and Reliability Based Design," Journal of Structural Division, ASCE, Vol. 99, ST9, Sept.

1973, pp. 1891-1910.

9. Ang, A. H-S. and Cornell, C. A. , ' Reliability Bases of Structural Safety and Design," Journal of Structural Division, ASCE, Vol.

100, ST9, Sept. 1974, pp. 1755-1769.

10. Hasofer, A.M. and Lind, N.C., " Exact and Invariant Second-Moment Code Format," Journal of Engineering Mechanics, ASCE, Vol.100, EMI, Feb. 1974, pp. 111-121.

11. Ravindra, M.K. and Galambus, T.V., ' toad and Resistance Factor Design for Steel," Journal of Structural Division, ASCE, Vol.

104, ST6, Sept. 1978, pp. 1337-1353.

12. Ravindra, M.K. and Singh, A.K., ' Reliability Assessment of ASME Code Equations for Nuclear Components," Reliability Engineering in Pressure Yessels and Piping, Second National Conference on Pres-sure Vessels and Piping, ASME, San Francisco, June 1975, pp. 29-36,

13. Lind, N.C., " Optimal Reliability Analysis by Fast Convolution,"

Journal of Engineering Mechanics, ASCE, Vol.105, EM3, June 1979, pp. 447-452.

14. Lind, N.C., "Formul ation of Probabilistic Design," Journal of Engineering Mechanics, ASCE, Vol.103, EM2, April 1977, pp. 273-284.

15. Moses, F., " Reliability of Structural Systems," Journal of Struc-tural Division, ASCE, Vol . 100, ST9, Sept. 1974, pp. 1813-1820.

16. Moses, F. and Stevenson, J.D. , " Reliability-Based Structural De-sign," Journal of Structural Division, ASCE, Vol . 96, ST2, Feb.

1970, pp. 221-244.

17. Cornell , C. A. , " Bounds on the Reliability of Structural Systems,"

Journal of Structural Division, ASCE, Vol. 93, ST1, Feb.1967, pp.

171-200.

18. Ang, A. H-S. and Ma , H-F. , 't)n the Reliability of Framed Struc-tures," Proceedings of Specialty Conference on Pobabilistic Me-chanics and Structural Reliability, ASCE, Tucson, Jan. 10-12, 1979, pp. 106-111.

19. Galambos, T.V. and Ravindra, M.K., " Properties of Steel for Use in LRFD," Journal of Structural Division, ASCE, Vol.104, ST9, Sept.

1978, pp. 1459-1468.

20. Fisher, J.W. et al . , " Load and Resistance Factor Design Criteria for Connections," Journal of Structural Division, ASCE, Vol.104, ST9, Sept. 1978, pp. 1427-1441.

21. Rol fe, S.T. and Barsom, J.M., Fracture and Fatigue Control in Structures--Applications of Fracture Mechanics, Prentice Hall, i 1977.

! 22. " Standard Method of Test for Plane-Strain Fracture Toughness of I

Metallic Materials," ASTM Designation E399-74, Part 10, ASTM An-nual Standards.

l l

)

4

23. ASME Boiler and Pressure Vessel Code,Section III, Division 1 -

Subsection NA, Appendix G, Article G-2000.

24. Gerdeen, J.C., "A Critical Evaluation of Plastic Behavior Data and a Unified Definition of Plastic Loads for Pressure Vessel Compon-ents," draf t of final report to Task Group on Characterization of Plastic Behavior of Structures, Pressure Vessel and Research Com-mittees, Welding Research Council, April 9. 1979.

25. Oldfiel d, W. , et al . , " Statistically Defined Reference Toughness Curves," Third International Conference on Pressure Yessel Tech-nology, Tokyo, ASME, April 1977, pp. 703-715.

26. ASME Boiler and Pressure Vessel Code, Sec. III, Table I-1.1,1971.

27. Schmi tt, W., Keim, E., Wellein, R. and Bartholome, G. , 'tinear Elastic Stress Intensity Factors for Cracks in Nuclear Pressure Vessel Nozzles Under Pressure and Temperature Loading," Interna-tional Journal of Pressure Vessels and Piping, Vol. 8,1980, pp.

41-68.

28. Cesari, F., ' Evaluation of Stress Intensity Factors for Internally Pressurized Cylinders with Surface Flaws," International Journal of Pressure Vessels and Piping, Vol. 7, 1979, pp. 199-227.

29. Merkle, J.G. , "A Review of Some Existing Stress Intensity Factor j

Solutions for Part-Through Surface Cracks," Oak Ridge National Laboratory, Report ORNL-TM-3983, Jan. 1973.

30. ASME Boiler and Pressure Vessel Code, Sec. XI, 1974 ed. , App. A.

31. Newman, J.C., "A Review and Assessment of the Stress-Intensity Factors for Surface Cracks," Part-Through Crack Fatigue Life Pre-dictions, ASTM STP 687, 1979, pp. 16-42.

32. ASME Boiler and Pressure Vessel Code, Sec. III, Par. NB535,1974.

33. Packman, P.F. et al . , ' Reliability of Defect Detection in Welded Structures," Reliability Engineering in Pressure Vessels and Pip-ing, Second International Conference on Pressure Vessels and Pip-ing, ASME, June 1975, pp. 15-28.

34. Larsson, H. and Bernard, J. , Tracture of Longitudinally Cracked Ductile Tubes," International Journal of Pressure Vessels and Piping, Vol. 6, 1978, pp. 223-243.

35. " Standard Specifications for General P,equirements for Steel Plates for Pressure Vessels," ASTM Standards,1979, Spec. A20, Table 1 and Table 6.

36. Ravindra, M.K., Lind, N.C. and Siu, W., " Illustration of Reliabil-ity-Based Design," Journal of Structural Division, ASCE, Vol.100, ST9, Sept. 1974, pp. 1789-1811.

37. Morales, W.J . , Duke, J.M. and Mazumdar, M., " Reliability of Slightly Oval Cylindrical Shells Against Elastic-Plastic Col-lapse," Reliability Engineering in Pressure Vessels and Piping, Second National Congress on Pressure Vessels and Piping, ASME, San Francisco, June 1975, pp. 37-49.

38. Miller, I. and Freund, J.E., Probability and Statistics for Engi-neers, Prentice-Hall, Englewood Cl i f fs, 1977.

39. Harrison, R.P., Darlaston, B.J .L. and Townley, C.H.A., Tailure Assessment of Pressure Vessels Under Yielding Conditions," Third International Conference on Pressure Vessel Technology, Tokyo, ASME, April 1977, pp. 661-668.

40. Adams , N.J . I . , "An Analysis and Prediction of Failure in Tubes,"

Third International Conference on Pressure Vessel Technology, Tokyo, ASME, April 1977, pp. 685-694.

41. Hayakawa, T., Yushida, T. and Mii, T., 'tollapse Pressure for the Small End of a Cone-Cylinder Junction Based on Elastic-Plastic Analysis," Third International Conference on Pressure Vessel Tech-nology, Tokyo, ASME, April 1977, pp. 149-156.

42. ASME Boiler and Pressure Vessel Code, Soc. III, Div.1, Par. II-1430, 1975, Winter Addenda.

43. Okamara, H. , Watnabe, K. and Naito, Y. , "Some Crack Problems in

Structural Reliability Analysis," Reliability Approach in Struc-tural Engineering. Japan-U.S. Joint Seminar on Reliability Ap-proach in Structural Engineering, May 1974, Maruzen, Tokyo,1975, pp. 243-257,

44. Rodabaugh, t.C. and Cloud, R.L., " Assessment of the Plastic Strength of Pressure Vessel Nozzles," Journal of Engineering for Industry, ASME, November 1968, pp. 636-643.

f

45. Gill, S.S., 'The Limit Pressure for a Flush Cylindrical Nozzle in a Spherical Pressure Vessel," International Journal of Mechanical Science, Vol. 6, 1964, pp. 105-115.

46. Dinno, K.S. and Gill, S.S. , " Limit Pressure for a Protruding Cyl-indrical Nozzle in a Spherical Pressure Vessel," Journal of Mech-anical Engineering Science, Vol. 7, No. 3, 1965, pp. 259-270.

47. ANSYS, Engineering Analysis System, User's Manual and Theoretical Mant-1,, Swanson Analysis Systems, Inc., Houston, Pa.

48. Zienkiewicz, 0.C., The Finite Element Method, 3rd ed., McGraw-Hill Bock Company, Inc., New York, 1977.

49. Gallagher, R.H., Finite Element Analysis Fundamentals, Englewood Cliffs, New Jersey, Prentice Hall, Inc., 1975.

50. Roark, R.J. and Young, W.C., Formulas for Stress and Strain, 5th ed., McGraw-Hill Book Company, Inc., New York, 1975.

51. Orr, Richard, oral presentation at Advisory Committee on Reactor Safety meeting, Washington, D.C., Sept. 2, 1980.

52. Zudans, Z. , oral presesntation at Advisory Committee on Reactor Safety meeting, Washington, D.C., Sept. 2,1980, and letter to Dr.

R. Savio, U.S. Nuclear Regulatory Commission, Aug. 29, 1980.

53. Augusti, G. and d' Agostino, S., " Experiments on the Plastic Behav-ior of Short Steel Cylindrical Shells Subject to Internal Pres-sure," First International Conference on Pressure Vessel Technol-ogy, ASME, 1970, pp. 363-375.

54. Ellyin, F. , " Experimental Investigation of Limit Loads of Nozzles in Cylindrical Vessel , " WRC Bulletin 219, Sept. 1976.

.I

IX. APPENDIX The limit pressure, safety index and failure probability (p=28 psi) from the approximate analysis of each failure mode are listed in this Appendix. The results represent the application of second moment reliability theory in cojuunction with the minimization principle of Eq. 2-15 and the statistics of Table 3-1 arsd 3-2. A computer library subroutine was used to perform the minimization. The results of the approximate axisymmetric shell analysis (Sec. 5.1.3) for Sequoyah and McGuire are presented in Tables 9-1 and 9-2 respectively. The approxi-mate anchor bolt analysis results (Sec. 5.3) are listed in Tables 9-5 and 9-6. All the results are summarized in Sec. 5. Run times and costs are quite small. For example, the results in Table 9-3 (193 failure modes) were obtained for a cost of $6.97 on an Advanced System 6 computer.

_m- .-ea.. --.i aa%* . ___-._a-i=_ eA.,..a--w .--A--- m_e*A - . O at A 4.. . 2_. a ,--a+w.. .a.4.--.L.m -Le_m. .a,h_Jde 6 4A 4 4_ _m .... _, _aa _m .__-

0 1 s 72- l t.

4 s

4

.k 4 \' % s 1

I i

f 10.0 TABLES 4

N

\

l I

Y s

i t

l l

1

.._ - _... _,-, - --_--,.... _ . - - - - - . .- - . - - , . , _ _ , . - ~ -- . ___-_ -, __.~.--. --.....

_. -. _. . ~ . ._ .. -. . . - _ . . - . - - .--

)

i 1 l 1

, TABLE 3-1. Mean and Standard Deviation of Material Properties

\ \

?

l

) Property Mean Standard Deviation i . .

j Modulus of elasticity (normal) 29,000 kai 1740 kai 4 Poisson's Ratio (normal) 0.3 0.009 f Yield Stress (lognormal)-Sequoyah 47.2 ksi 2.50 kai

. Yield Stress (lognormal)-McGuire 46.9 kai 2.50 kai

! Ultimate (lognormal)-Sequoyah 66.2 kai 1.80 ksi J

! Ultimate (lognormal)-McGuire 67.0 kai 1.80 kai Fracture Stress (lognormal) 197 kai 102 ksi l

Bolt Yield Stress (lognormal) 105 kai 2.50 ksi. i 2

TABLE 3-2. Mean and Standard Deviation of Geometric Properties (Normally Distributed)

Nominal Standard i Property (in) Tolerance (in) Mean (in) Deviation 4

(in)

I Thickness 1 1/4" -0.010,+0.063 1.277 0.012

) 1 3/16 -0.010,+0.059 1.212 0.012 1

i 1 1/16 -0.010,+0.053 1.084 0.011 1 -0.010,+0.050 1.020 0.010 j 3/4 -0. 010,+0. 045 0.768 0.0092 4 11/16 -0.010,+0.041 0.703 0.0095 5/8 -0.010,+0.038 0.639 0.0080 1/2 -0.010,+0.035 0.513 0.0075 7/16 -0.010,+0.035 0.450 0.0075

.! Length & Width 1,b i 1/4" tb 0.083 Stiffener Spacing s 1 0.005 s s 0.00167s

! Radius (Sequoyah) 690 1 3.45" 690 1.15 (McGuire) 690 i 1.5" 690 0.50 Anchor Bolt Diameter (upset) j (Sequoyah) 2.58 i .01 2.58 .0033 j (McGuire) 3.75 i .015 3.75 .005 i

j i,

1 i

i I

TABLE 5-1. Reliability Results with Approximate Stiffened Shell Analysis Sequoyah McGuire Minimum Mean Limit Pressure 59.1 psi 77.5 psi Safety Index (p = 28 psi)

Upper Bound 4.5 6.1 Lower Bound 4.4 5.7 Failure Probability (p = 28 psi)

Upper Bound 5.0 (10-6) 5.6 (10-9)

Lower Bound 3.9 (10-6) 5.4 (10-10)

TABLE 5-2. Reliability Results with Approximate Stiffened Shell Analysis-Sensitivity Study Sequoyah McGuire Minimum Mean Limit Pressure 59.1 psi 77.5 psi limit Pressure Standard Deviation Upper Bound 10.0 psi 13.8 psi Lower Bound 9.8 psi 12.9 psi Safety Index (p = 28 psi) i Upper Bound 4.5 6.1 Lower Bound 4.4 5.7 Failure Probability (p = 28 psi) i Upper Bound 4.7 (10-6) 5.2 (10-9)

Lower Bound 3.7 (10-6) 5.1 (10-10) i l

.-. u. - .- - . - .-

1 TABLE 5-3. Reliability Results for Approximate Penetration Analysis Sequoyah McGuire Minimum Mean Limit Pressure 67.2 psi 78.5 psi Safety Index (p = 28 psi)

Upper Bound 5.1 6.1 Lower Bound 4.9 5.2 Failure Probability (p = 28 psi)

Upper Bound 6.1 (10-7) 6.9 (10-8)

Lower Bound 1.4 (10- ) 7.3 (10-10) i i TABLE 5-4. Reliability Results for Approximate Penetration Analysis--

Sensitivity Study (Eq. 5-9)

Sequoyah McGuire Minimum Mean Limit Pressure 67.2 psi 78.5 psi Limit Pressure Standard Deviation i

Upper Bound 12.0 psi 15.3 psi Lower Bound 11.4 psi 13.4 psi Safety Index (p = 28 psi)

Upper Bound 5.1 6.1 i Lower Bound 4.9 5.3 Failure Probability (p = 28 psi)

Upper Bound 5.8 (10- ) 6.6 (10- )

Lower Bound 1.3 (10- ) 7.0 (10-10) i

+

J 4

- - - - - . - , - - - . . _ ,m_m. _ _ _ , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _

TABLE 5-5. Reliability Results for Anchor Bolts Sequoyah McGuire Minimum Mean Limit Pressure 66 psi 141 psi Safety Index (p = 28 psi) 5.4 10.1 Failure Probability (p = 28 psi) 4.0 (10-8) 23 (10-24)

TABLE 5-6. Reliability Results for Anchor Bolts--Sensitivity Study (Eq. 5-9)

Sequoyah McGuire Minimum Mean Limit Pressure 66 psi 141 psi Limit Pressure Standard Deviation 10.5 psi 22.5 psi Safety Index (p = 28 psi) 5.4 10.1 Failure Probability (p = 28 psi) 4 (10-8) 2.3 (10-24) 1 TABLE 5-7. Reliability Results for Approximate Structural Analysis Sequoyah McGuire 59.1 psi 77.5 psi Minimum Mean Limit Pressure Limit Pressure Standard Deviation 10.0 psi 15.0 psi Upper Bound 9.8 psi 13.0 psi kwer Bound Safety Index (p = 28 psi)

Upper Bound 4.5 6.1 Lower Bound 4.4 5.3 Failure Probability (p = 28 psi)

Upper Bound 5.5 (10-6) 7.1 (10-8) f Lower Bound 3.7 (10-6) 7.0 (10-10)

I l

TABLE 6-1. Plastic Pressure for Cylinder in Fig. 6-2 With Different ANSYS Options STIF NO. 42 42 82 82 82 Option S.S* L.D S.S* S.S* S.S*

Plastic Convergence Ratio 0.03 0.03 0.03 0.05 0-1 p (psi) 71 71 71 73 79

% Difference --- ---

3 11 cpu (sec) 1327 2338 1170 765 415 a) S.S = Stress Stiffening Option b) L.D = Large Displacement Option TABLE 6-2. Plastic Pressure Results Sequoyah McGuire Minimum Mean 60 psi 84 psi Standard Deviation 8 psi 12 psi

l TABLE 9-1 SE00UY AH CONTAINMENT VESSEL f SHELL LIMIT PRESSURE H1NG MEAN LIMIT SAFETY PROBASILITY ELEV. PRESSURE INDEX OF FAILURE 680.80 122.67 8.85 0.4423E-18 691 20 113 25 8.37 0.2782E-16 701.60 107.20 8.04 0.4352E-15 713.50 99.38 7.59 0.1592E-13 721 00 95.4o 7.35 0.9908E-13 730.30 75.41 5.93 0.1472E-08 740.60 69 61 5 45 0.2449E-07 750.10 65.1J 5.06 0.2135E-06 759.60 65 20 5.06 0.2062E-06 769.10 6S.79 5.12 0 1551E-06 778.60 63.60 4.91 0.4472E-06 788.10 59.08 4.47 0.3889E-05 791.40 83.45 6.52 0.3494E-10 796.00 83.86 6.56 0.26o7E-10 799 80 85.02 6.64 0.1575E-10 803.80 83.72 6.55 0.2884E-10 809.50 81.96 6 42 0 6697E-10 815.40 81.69 6.40 0.7597E-10 821.40 70.18 5.49 0.1978E-07

SUMMARY

MINIMUM FAILudE LIMIT PRESSURE = 59.077 SAF E TY 4.42< INDEX < 4.47 (BETA)

PROBABILITY 0.38d9E-05< OF < 0.4957E-05 FAILURE 9

l TABLE 9-2 MCGUIRE CONTAthMENT VESSEL SHELL LIMIT PRESSURE RING MEAN LIMIT SAFETY PROSABILITY ELEV. PRESSURE INDEX OF FAILURE 727.83 82.39 6.46 0.5151E-10 736.42 77.94 6.13 0.4427E-09 746.42 77.52 6.10 0.5418E-09 756.92 77.52 6.10 0 5418E-09 766.42 77.52 6.10 0.5418E-09 776.42 77.52 6.10 0.5418E-09 786 42 77.52 6.10 0.5418E-09 796 42 77.52 6.10 0.5418E-04 806.42 77.52 6.10 0 5418E-09 816.42 77.52 6.10 0.5418E-09 826.42 78.08 6.14 0 4129E-09 635.42 78.68 6.19 0.3091E-09 845.00 95.57 7.34 0.1047E-12

SUMMARY

MINIMUM F AILURE LIMIT PRESSURE = 77.520 SAFETY 5.71< INDEX < 6.10 l

(BETA) l PRO 8ABILI TY 0.5418E-09< OF < 0.5551E-08 FAILURE l

TABLE 9-3 SEQUOY AH CONT AINMENT VESSEL PENETR ATION LIMIT PRESSURE ELEV. AZ1M. MEAN LIMIT SAFETY PROBABILITY PRESSURE INDEX OF FAILURE 699.71 0 0 0 96.93 7 29 0.1539E-12 822.75 0 0 0 86.44 5.56 0.1343E-07 714.00 4 18 46 81.28 6.26 0 1979E-09 697.00 7 0 0 105.99 7.82 0.2728E-14 708.00 7 38 44 93.46 7 08 0.7380E-12 697.00 8 30 0 105.99 7.82 0 2728E-14 711.42 9 30 0 101.29 7 55 0.2184E-13 716.50 9 30 0 101.29 7.55 0.2184E-13 711 50 17 0 0 111.34 8.11 0 2564 E-15 715.50 17 0 0 124.04 8.75 0.1108E-17 715.50 21 0 0 111.84 8 13 0.2069E-15 711.50 21 0 0 111.84 8.13 0 2069E-15 711 50 25 0 0 111 84 8 13 0.2069E-15 715.50 25 0 0 111.84 8.13 0.2069E-15 711 50 29 0 0 111 84 8.13 0 2069E-15 715.50 29 0 0 111.84 8 13 0.2069E-15 712.00 38 30 0 96.93 7.29 0 1539E-12 710 75 56 30 0 124.04 8.75 0 1108E-17 705.50 57 0 0 103 18 7 66 0.9436E-14 696 30 62 0 0 146.89 9.73 0 1100E-21 i 714.00 65 0 0 105.36 7.78 0 3637E-14 1 715.00 65 0 0 105.36 7.78 0.3637E-14 697.00 78 30 0 111.84 8 13 0 2069E-15 717.00 90 0 0 104.03 7.71 0.6508E-14 715.50 93 0 0 111 84 8.13 0.2069E-15 715 50 97 0 0 111.84 8 13 0.2069E-15 715.50 101 0 0 111.84 8.13 0.2069E-15 715.00 104 0 0 105.52 7.79 0.3374E-14 714.00 104 0 0 105 52 7.79 0.3374E-14 715.50 114 0 0 111.84 8.13 0.2069E-15 716.67 116 0 0 103 18 7.66 0.9436E-14 711.50 116 30 0 111 84 8.13 0.2069E-15 i 715.50 118 30 0 111 84 8.13 0.2069E-15 l

822 75 120 0 0 86.44 5.56 0.1343E-07 712 50 120 30 0 111 34 8 11 0 2564E-15 711 50 151 0 0 111.84 8.13 0.2069E-15 715.50 151 0 0 111 84 8.13 0.2069E-15 711.50 155 0 0 111.84 8 13 0.2069E-15 715.50 155 0 0 111 84 8 13 0.2069E-15

TABLE 9-3 (CONTINUED) 711.50 159 0 0 111 84 8.13 0.2069E-15 1s5.50 159 0 0 111.84 8.13 0.2069E-15 715.50 163 0 0 124.04 8.75 0.1108E-17  ;

711.50 163 0 0 111.84 8.13 0 2069E-15 )

697.00 171 30 0 105 99 7.82 0.2728E-14 ,

708.00 172 21 16 93.46 7.08 0. 73 8 0 E- 12 )

697.00 173 0 0 105.99 7 82 0.2728E-14 714.00 175 41 14 81.28 6.26 0.1979E-09  !

714.00 184 18 46 81.28 6.26 0 1979E-09 697 00 187 0 0 105.99 7.82 0. 27 28 E- 14 {

708.00 187 38 44 93.46 7.08 0.7380E-12 697.00 189 30 0 105.99 7.82 0.2728E-14 715.50 197 0 0 124.04 8 75 0.1108E-17 711.50 197 0 0 111.84 8.13 0.2069E-15 711.50 201 0 0 111 84 8 13 0.2069E-15 715.50 201 0 0 111.84 8.13 0.2069E-15 711.50 205 0 0 111.84 8 13 0.2069E-15 715.50 205 0 0 111.84 8 13 0.2069E-15 711.50 209 0 0 111 84 8 13 0.2069E-15 715.50 209 0 0 111.84 8.13 0.2069E-15 688.50 209 0 0 105 99 7 82 0 2728E- 14 685.50 209 0 0 104.03 7.71 0.6508E-14 688.50 218 0 0 96 93 7.29 0.1539E-12 688.50 222 0 0 96.93 7.29 0.1539E-12 717.00 236 0 0 105.02 7 76 0.4210E-14 705.00 236 0 0 105.02 7.76 0.4210E-14 704.00 236 0 0 105.36 7.78 0.3637E-14 697.00 236 30 0 111.84 8.13 0.2069E-15 714.00 236 30 0 96.93 7.29 0.1539E-12 717.00 237 0 0 105.02 7.76 0.4210E-14 705.00 237 0 0 105 62 7.79 0.3238E-14 716.50 240 0 0 124.04 8.75 0.1108E-17 710.00 240 0 0 111.84 8.13 0.2069E-15 697.00 243 30 0 124.04 8.75 0.1108E-17 716.00 243 30 0 124.04 8.75 0.1108E-17 748.50 248 30 0 111 84 8 13 0.2069E-15 752.50 248 30 0 111.84 8 13 0.2069E-15 822 75 245 0 0 86.44 5.56 0.1343E-07 756.50 248 30 0 124.04 8.75 0 1108E-17 I

773.50 248 30 0 111.84 8.13 0. 206 9 E-15 767.00 248 30 0 124.04 8.75 0.1108E-17 l 707.50 249 0 0 111.84 8.13 0.2069E-15 728.33 252 30 0 96.93 7.29 0.1539E-12 l l 737.30 255 0 0 146.89 9.73 0.1100E-21 l 775.92 261 0 0 67.21 5.14 0.1365E-06 748.50 262 0 0 103.18 7.66 0.9436E-14 688.23 261 51 10 67 21 5 14 0.1365E-06

. = - - -

TABLE 9-3 (CONTINUED) 748.50 265 0 0 103 18 7.66 0.9436E-14 767.00 266 0 0 67.21 5.14 0 1365E-06 748.50 268 0 0 103.18 7.66 0 9436 E-14 710.00 277 30 0 104.03 7 71 0 6510E-14 706.00 277 30 0 93.46 7.08 0.7380E-12 697.50 277 30 0 102 27 7 61 0.1412E-13 700 00 278 0 0 100 26 7.49 0.3467E-13 709.00 278 30 0 104.03 7.71 0.6510E-14 719.50 280 0 0 96.93 7.29 0 1539E-12 697.50 280 0 0 104.00 7.70 0.6566E-14 706.50 280 30 0 105.69 7.80 0.3133E-14 700.00 281 30 0 101 29 7 55 0.2184E-13 709.00 281 30 0 94.63 7.15 0.4360E-12 704.50 281 30 0 99 18 7.43 0.5616E-13 706.00 283 0 0 105.02 7.76 0.4210E-14 697.50 282 30 0 101 29 7 55 0.2184E-13 695.00 282 30 0 103.84 7.70. 0.7064E-14 741 63 285 0 0 274.55 13.40 0.3066E-40 705.50 286 30 0 104.36 7.72 0.5634E-14 697.00 286 30 0 102 44 7.61 0.1321E-13 700.50 287 0 0 105.62 7.79 0.3238E-14 695 00 287 0 0 101 29 7.55 0.2184E-13 698.00 287 30 0 105.02 7.76 0.4210E-14 709.00 287 30 0 96 93 7.29 0.1539E-12 700.50 288 0 0 105.02 7.76 0.4210E-14 699 50 288 0 0 105 34 7.78 0.3660E-14 697.00 288 30 0 104.36 7.72 0 5634E-14 775.92 289 0 0 96 93 7.29 0.1539E-12 785.00 289 0 0 103.55 7.68 0 8062E-14 705.50 298 30 0 98 07 7.36 0.9242E-13

. 700.50 289 0 0 105.02 7.76 0.4210E-14

! 712.50 290 0 0 105 02 7.76 0.4210E-14 786 00 290 0 0 102.63 7.63 0.1212E-13 718.00 290 0 0 105 02 7.76 0.4210E-14 718.00 291 0 0 105.02 7.76 0.4210E-14 l 700.50 291 0 0 105 02 7.76 0.4210E-14 709.00 291 30 0 96.93 7.29 0 1539E-12 697 50 291 30 0 101 29 7.55 0.2184E-13 718.00 292 0 0 104.36 7.72 0 5634E-14 700.50 292 0 0 103.84 7.69 0.7110E-14 704.00 292 0 0 104.36 7.72 0.5634E-14 705 50 292 0 0 104.36 7.72 0. 5634 E-14 700.50 293 0 0 103 84 7.69 0 7110E-14 726.25 293 0 0 96.93 7.29 0.1539E-12

, 718.00 293 0 0 104.36 7.72 0.5634E-14 785 50 293 0 0 97 06 7.30 0 1462E-12 I 705.50 293 30 0 104.36 7.72 0 . 56 34 E-14 l

I

TABLE 9-3 ( CONTI NUED )

704.00 293 30 0 104.36 7.72 0 5634E-14 700.50 294 0 0 103.84 7.69 0.7110E-14 718.00 294 0 0 104 36 7.72 0.5634E-14 710.00 294 0 0 104.03 7.71 0.6508E-14 696.50 294 0 0 104.36 7.72 0. 56 34 E-14 707.25 294 45 0 99.18 7.43 0.5616E-13 785 50 296 30 0 97.06 7.30 0.1462E-12 705.50 298 30 0 98.07 7.36 0.9242E-13 711 50 299 30 0 106 20 7.83 0.2496E-14 697.50 299 30 0 104.03 7.71 0 6508E-14 707 50 299 30 0 104.36 7.72 0. 5634 E- 14 699.50 299 30 0 104.36 7.72 0.5634E-14 j 727 25 299 30 0 104.36 7.72 0. 5634 E- 14 725.25 299 30 0 104.36 7 72 0. 563 4 E- 14 752.01 300 0 0 99.18 7.43 0.5616E-13 688.00 300 0 0 105.54 7.79 0.3356E-14 748.50 300 0 0 99 18 7.43 0 . 5616 E- 13 737.00 300 0 0 104.36 7.72 0.5634E-14 735.00 300 0 0 104 36 7.72 0. 5634 E- 14 711.50 300 30 0 106.20 7.83 0.2496E-14 694.96 300 30 0 93.46 7.08 0.7380E-12 714.00 301 0 0 96.93 7.29 0.1539E-12 704.50 301 C 0 99 18 7.43 0.5616E-13 710.00 301 0 0 100.26 7.49 0.346TE-13 700.50 301 0 0 104 36 7.72 0 . 563 4 E- 14 698.50 303 0 0 104.36 7.72 0.5634E-14 687.00 301 0 0 105 99 7.82 0.2728E-14 770.50 301 15 0 105.41 7.78 0 3543E-14 771.50 301 15 0 105.41 7.78 0.3543E-14 772.50 301 15 0 105.41 7.78 0 3543E-14 773.50 301 15 0 105 41 7.78 0.3543E-14 774.50 301 15 0 105.41 7.78 0.3543E-14 775.50 301 15 0 105 41 7.78 0.3543E-14 776.50 301 15 0 105.41 7.78 0 3543E-14 777 50 301 15 0 105 41 7.78 0.3543E-14 725.25 301 30 0 104.36 7.72 0 5634E-14 i 727 25 301 30 0 104.36 7.72 0. 563 4 E- 14 l

697 50 301 30 0 104.36 7.72 0.5634E-14 832.33 301 30 0 82.09 5.31 0 . 55 6 8 E-0 7 830.33 304 0 0 82.09 5 31 0 5568E-07 l 832.33 306 0 0 84.69 5.46 0.2372E-07 698.50 306 0 0 104.36 7.72 0. 5634 E- 14 700.00 306 0 0 104.36 7.72 0 5634E- 14 696.00 306 30 0 124.04 8.75 0.1108E-17 700.00 307 30 0 104.36 7.72 0 5634E-14 698.50 307 30 0 104.36 7.72 0 . 56 3 4 E- 14 830.33 308 0 0 84 69 5.46 0.2372E-07

~84~

TA8LE 9-3 (CONTINUED) 711.50 331 0 0 111 84 8.13 0 . 20 69 E- 15 711.50 335 0 0 111.84 8.13 0.2069E-15 711.50 339 0 0 111 84 8.13 0.2069E-15 711.50 343 0 0 111.84 8.13 0.2069E-15 715.50 343 0 0 124.04 8.75 0.1108E-17 711.50 346 30 0 104.36 7.72 0. 5634 E- 14 713.50 346 31 0 104.36 7 72 0. 56 34 E- 14 716.50 350 30 0 101.29 7.55 0.2184E-13 711.42 350 30 0 101 29 7.55 0 . 218 4 E- 13 697.00 351 30 0 105.99 7 82 0.2728E-14 708.00 352 21 16 93 46 7 08 0.7380E-12 697.00 353 0 0 105.99 7.82 0.2728E-14 714.00 355 41 15 81 28 6 26 0.1979E-09 SUM M ARY MINUMIM F AILURE LIMIT PRESSURE = 67.208 SAFETY 4.85< INDEX < 5.14 (8 ETA)

PROBASILITY 0.1365E-06< OF < 0.6095E-06 FAILURE l

l l

l TADLE 9-4 MCGUIRE CO NT A I NME N T VESSEL PENETRATION LIMIT PRESSURE ELEV. AZIM. MEAN LIMIT SAFETY PROSABILITY PRESSURE INDEX OF FAILURE 715.75 5 0 0 149 31 9.82 0.4720E-22 760.50 5 58 43 113.15 8.20 0.1218E-15 753.00 14 10 0 117.84 8 43 0.1658E-16 764.00 16 0 0 118.27 8.46 0.1378E-16 754.75 17 30 0 78.46 6.05 0. 73 0a E- 0 9 758.75 20 0 0 78.46 6.05 0.730SE-09 758.75 22 30 0 78.46 6 05 0. 73 0 8 E-0 9 754.75 22 30 0 78.46 6.05 0.7308E-09 758.75 25 0 0 78.46 6.05 0.7308E-09 754.75 25 0 0 78.46 6 05 0.7308E-09 754.75 27 30 0 78.46 6.05 0.7308E-09 758.75 27 30 0 78.46 6.05 0.7308E-09 758.75 30 0 0 78.46 6.05 0. 73 0 8 E- 0 9 754.75 30 0 0 78.46 6.05 0.7308E-09 754.75 36 0 0 78.46 6.05 0. 73 0 8 E- 09 758.75 36 0 0 78 46 6.05 0.7308E-09 762.75 36 0 0 78.46 6 05 0. 73 08 E-09 752 17 57 30 0 78.46 6.05 0. 73 0 8E-09 755.65 57 30 0 78.4o 6.05 0.7308E-09 670.58 57 30 0 78.46 6 05 0. 73 0 8 E-09 755 65 60 0 0 78.46 6.05 0. 73 0 8 E-0 9 741 00 62 0 0 137.59 9.34 0.4602E-20 755.65 62 30 0 78.46 6.05 0. 73 0 8 E- 0 9 760.75 62 30 0 78.46 6.05 0 . 73 08 E- 0 9 755 65 65 0 0 78.46 6.05 0.7308E-09 760.75 65 0 0 78.46 6.05 0.7308E-09 755.65 68 0 0 78.46 6 05 0. 7308E-09 760.75 68 0 0 78 46 6.05 0. 73 0 S E-09 752.17 68 0 0 78.46 6.05 0.7308E-09 752.17 74 0 0 78.46 6 05 0.7308E-09 755.46 74 0 0 78.46 6.05 0. 73 0 S E- 0 9 761.58 75 59 48 78.46 6.05 0.7308E-09 755.46 77 30 0 78.46 6.05 0.7308E-09 752 16 77 30 0 78.46 6.05 0. 73 08 E-0 9 755.46 80 0 0 78.46 6.05 0.7306E-09 752.16 80 0 0 78.46 6 05 0. 73 0 8 E- 0 9 755.46 82 30 0 78.46 6.05 0.7308E-09 752.16 82 30 0 78.46 6.05 0.7308E-09 755.46 85 0 0 78.46 6.05 0.7308E-09

l 1

TABLE 9-4 ( CONTI NUED )

752.16 85 0 0 78.46 6.05 0.7308E-09 761.50 90 59 48 78.46 6.05 0.7308E-09 748.67 91 30 0 78.46 6.05 0. 73 08 E- 0 9 755.65 91 30 0 78.46 6.05 0.7308E-09 748.67 94 0 0 78.46 6.05 0. 73 08 E- 09 755.66 94 0 0 78.46 6.05 0.7308E-09 748.67 96 30 0 78.46 6.0S 0.7308E-09 755.65 96 30 0 78.46 6.05 0.7308E-09 748.67 99 0 0 78.46 6.05 0.7308E-09 755.65 102 0 0 78.46 6.05 0.7308E-09 755.65 99 0 0 78.46 6.05 0.7308E-09 748.67 102 0 0 78.46 6 05 0. 73 08 E-09 761.58 102 0 0 78.46 6.05 0.7308E-09 758.67 110 0 0 78.46 6.05 0.7308E-09 762.25 110 0 0 78.40 6 05 0. 73 0 d E- 0 9 730.25 111 0 0 149.31 9.82 0.4720E-22 734 00 111 0 0 149.31 9.82 J.4720E-22 758.67 112 30 0 78 46 6.05 0.7308E-09 762.25 112 30 0 78.46 6 05 0.7308E-09 734.00 115 0 0 149.31 9 82 0.4720E-22 758 67 115 0 0 78.46 6 05 0.7308E-09 762.25 115 0 0 78.46 6 05 0.7308E-09 768 67 117 30 0 78.46 6 05 0.7308E-09 762.25 117 30 0 78.46 6.05 0.7308E-09 734 00 119 0 0 149.31 9.82 0.4720E-22 734.00 120 0 0 78 46 6.05 0.7308E-09 762.25 120 0 0 78.46 6.05 0.7308E-09 734.00 123 0 0 149.31 9 82 0.4720E-22 734.00 129 0 0 149.31 9.82 0.4720E-22 734.00 134 0 0 149.31 9.82 0.4720E-22 734.00 138 0 0 149.31 9.82 0 4720E-22 750.75 141 30 0 78.46 6.05 0.7308E-09 754.75 141 30 0 78.46 6.05 0.7308E-09 758 75 141 30 0 76.46 6.05 0.7308E-09 762.75 141 30 0 78.4o 6.05 0.7308E-09 750.75 144 0 0 78.46 6.05 0.7308E-09 754.75 144 0 0 78 46 6.05 0.7308E-09 750.75 144 0 0 78.46 6.05 3.7308E-09 762.75 144 0 0 78.46 6.05 0.7308E-09 730.25 145 0 0 149.31 9.82 0.4720E-22 734.00 145 0 0 149.31 9 82 0.4720E-22 730.25 149 0 0 149.31 9 82 0.4720E-22 734.00 151 0 0 150.73 9.87 0.2777E-22 754.75 152 30 0 78.46 6.05 0.7308E-09 758.75 152 30 0 78 46 6.05 0. 73 0 d E- 0 9 739.00 153 0 0 97.20 7.31 0.1383E-12 744 17 153 0 0 97.20 7.31 0.1383E-12

TABLE 9-4 ( CO N T I NUED )

730.25 154 0 0 149 71 9.83 0.4044E-22 754.75 155 0 0 78.46 6 05 0.7308E-09 738.75 155 0 0 78.46 6 05 0.7308E-09 734.00 156 0 0 150.73 9.88 0.2635E-22 758.75 157 30 0 78 46 6 05 0. 72 62 E-09 762 75 157 30 0 73.46 6 05 0.7308E-09 739.75 157 30 0 117.59 8.42 0 1822E-16 744.17 157 30 0 97.20 7.31 0.1383E-12 750.75 160 0 0 78.46 6.05 0.7308E-09 754.75 100 0 0 78.46 6 05 0.7308E-09 758.75 160 0 0 78.46 6 05 0.7309E-09 762.75 160 0 0 78.46 6 05 0.7308E-09 730 25 162 0 0 149.71 9.83 0.4044E-22 734.00 to2 0 0 150.73 9.88 0.2635E-22 758.75 162 30 0 78.46 6 05 0.7308E-09 761.50 164 0 0 118.27 8.46 0.1378E-16 730.25 to6 0 0 149.31 9.82 0.4720E-22 734 00 166 0 0 149.31 9.82 0.4720E-g2 739.00 166 0 0 97.20 7.31 0.1383E-12 744.17 166 0 0 117 59 8 42 0.1822E-16 753.00 166 0 0 117.84 8.43 0.1658E-16 744.17 171 30 0 97.20 7.31 0.1383E-12 730 25 172 0 0 149.36 9.82 0 4624E-22 734.00 172 0 0 149.36 9.82 0 . 46 2 4 E-2 2 744.17 172 0 0 117.59 8 42 0 18 22 E- 16 ,

760.50 174 1 7 113.15 8.20 0.1218E-15 726.00 175 0 0 97.20 7.31 0.1383E-12 751.75 175 0 0 149.31 9.82 v.4720E-22 744 17 175 0 0 118.27 8.46 0.1378E-16 734.00 177 0 0 150.73 9.88 0.2635E-22 730 25 177 0 0 149.31 9 82 0.4720E-22 726.00 184 0 0 193.75 11.34 0.4339E-29 744.50 184 0 0 97.20 7.31 0.1383E-12 740 35 134 0 0 118.27 8.46 0.1378E-16 731.75 184 30 0 149.31 9 82 0.4720E-22 760.50 183 58 9 113.15 8 20 0 1218E-15 i 730 25 185 0 0 149.31 9.82 0.4720E-22 1

734.00 185 0 0 149.31 9.82 0.4720E-22 739.75 188 0 0 97.20 7.31 0 1383E-12 730.25 189 0 0 149.31 9.81 0.4978E-22 734 00 189 0 'O 149.31 9.81 0.4978E-22 744 17 189 0 0 97.20 7.31 0.1383E-12 730.25 194 0 0 149.31 9.82 0.4720E-22 739.75 194 0 0 144.31 9.82 0 4720E-22 l

734.00 194 0 0 150.73 9.88 0.2635E-22 744.17 194 0 0 117.59 8.42 0.1822E-16 753.00 0 0 0 117.84 8.43 0.1658E-16

TABLE 9-4 ( CON T I N UE D )

761.50 196 0 0 118.27 8.46 0.1378E-16 730.25 201 0 0 149.31 9 82 0.4720E-22 734.00 202 0 0 149.31 9.82 0.4720E-22 755.00 203 0 0 149.31 9.82 0.4720E-22 761 50 203 0 0 149.31 9.82 0.4720E-22 730.25 205 0 0 149.31 9 82 0.4720E-22 734.00 206 0 0 117.59 8.42 0.1822E-16 755.00 207 0 0 97.32 7.31 0.1314E-12 739.75 208 0 0 149.31 9.82 0.4720E-22 l 744.17 208 0 0 149.31 9 82 0.4720E-22 I 761.50 209 0 0 117.59 8.42 0.1822E-16 730.25 210 0 0 149.31 9.82 0 4720E-22 734.00 210 0 0 150.73 9 88 0.2635E-22 755.00 211 0 0 149.31 9.82 0.4725E-22 762.75 215 0 0 149.31 9 82 0.4720E-22 751.50 216 30 0 117.59 8.42 0.1822E-16 730.25 217 0 0 149.31 9 82 0.4720E-22 734.00 219 0 0 150.73 9.88 0.2635E-22 761.50 220 0 0 117.59 So42 0.1822E-16 730 25 221 0 0 149.31 9.82 0.4720E-22 752.00 221 0 0 78.46 6.05 0. 73 08E-0 9 730.25 225 0 0 149.31 9.82 0.4720E-22 734.00 225 0 0 149.31 9.82 0.4720E-22 730.25 229 0 0 149.31 9.82 0.4720E-22 734 00 231 0 0 150.73 9.88 0.2635E-22 730 75 236 0 0 149 31 9.82 0.4720E-22 734 00 236 0 0 149.31 9.82 0.4720E-22 739.75 236 0 0 97.20 7.31 0.138 3 E - 12 744 17 236 0 0 97.20 7.31 0 13d3E-12 761.50 23o O O 97.20 7.31 0.1383E-12 755 00 237 0 0 97.20 7.31 0.1383E-12 751.50 239 0 0 117.59 8 42 0.1822E-16 730 25 240 0 0 149.31 9 82 0.4720E-22 734.00 240 0 0 149.31 9.82 0.4720E-22 739.75 240 0 0 97.20 7.31 0.1383E-12 744.17 240 0 0 117.59 8 42 0.1822E-16 760.50 242 0 0 113.80 8.23 0.9629E-16 739.75 244 0 0 97.20 7.31 0.1383E-12 744 17 244 0 0 97.20 7.31 0.1383E-12 730.25 248 0 0 149.J1 9.82 0.4720E-22 734.00 249 0 0 149.31 9.82 0 4720E-22 739.75 248 0 0 97.20 7.31 0.1384E-12 744.17 248 0 0 97.20 7.31 0.1384E-12 769.50 249 0 0 168.78 10 52 0.3502E-25 799.75 249 0 0 230.26 12.33 0.3260E-34 809.75 249 0 0 230.26 12.33 0.3260E-34 760.50 250 0 0 117.59 8.42 0 1822E-16

TABLE 9-4 (CONTINUEO) 814.25 250 30 0 168.78 10.52 0.3502E-25 818.25 250 30 0 168.78 10.52 0.3502E-25 824.25 250 30 0 168.78 10.52 0.3502E-25 730.25 252 0 0 149.31 9 82 0.4720E-22 734 00 252 0 0 149.31 9.82 0.4720E-22 739.75 252 0 0 97.20 7.31 0.1383E-12 744 17 252 0 0 97.20 7.31 0.1383E-12 769.50 252 0 0 90 99 6.91 0.2347E-11 810.50 254 0 0 90 99 6.91 0 2347E-11 814.50 254 0 0 90.99 6.91 0.2347E-11 818.67 254 0 0 90.99 6.91 0.2347E-11 822.50 254 0 0 90.99 6 91 0.2347E-11 800.50 254 0 0 90.99 6 91 0.2347E-11 804.50 254 0 0 90.99 6 91 0.2347E-11 769.50 254 30 0 90.99 6.91 0.2347E-11 782.19 255 0 0 137.59 9 34 0.4602E-20 769.50 257 0 0 90.99 6 91 0 2347E-11 819.75 258 0 0 168.78 10.52 0.3502E-25 824.25 258 0 0 170.04 10 57 0.2055E-25 799.75 258 30 0 170.04 10.57 0.2055E-25 804.75 258 30 0 163.78 10.52 0.3502E-25 809.75 259 0 0 170.04 10.57 0.2055E-25 814.25 259 0 0 168.78 10.52 0.3502E-25 769.50 259 30 0 90.99 6.91 0.2347E-11 733.50 261 51 44 165 62 10.43 0.9304E-25 812.00 265 0 0 173.64 10.69 0.5563E-26 819.75 2o5 0 0 170.04 10.57 0.2055E-25 824 25 265 0 0 170.04 10.57 0.2055E-25 750.75 278 30 0 78.46 6 05 0.7308E-09 754.75 278 30 0 78.46 6 05 0.7308E-09 750.75 281 0 0 94 67 7.15 0 4281E-12 754.75 281 0 0 78.4o 6.05 0.7308E-09 760.00 281 0 0 168.78 10 52 0 3502E-25 769.42 281 0 0 90.99 6.91 0.2347E-11 l 769.42 283 30 0 90.99 6 91 0.2347E-11 l 769.42 286 0 0 90.99 6.91 0.2347E-11 1

787.25 288 0 0 248.17 12.80 0.7907E-37

( 769.42 288 30 0 90 99 6.91 0.2347E-11 754.75 291 0 0 78.46 6.05 0.7308E-09 760.00 291 O O 168.78 10.52 0. 350 2 E-2 5 769.42 291 0 0 90.99 6.91 0. 234 7E- 11 822.42 291 30 0 230.26 12.33 0.3260E-34 754.75 293 30 0 78.46 6.05 0.7308E-09 769.42 293 30 0 90.99 6.91 0 2347E-11 818.67 295 0 0 90.99 6.91 0.2347E-11 822.42 295 0 0 90.99 6.91 0 2347E-11 754 75 296 0 0 78.46 6.05 0.7308E-09

TABLE 9-4 (CONTINUED) 760.00 296 0 0 170.04 10.57 0.2055E-25 818.67 297 30 0 90.99 6.91 0.2347E-11 822.42 297 3J 0 90.99 6.91 0.2347E-11 754.75 298 30 0 94 67 7.15 0.4410E-12 818.67 300 0 0 90 99 6.91 0.2498E-11 822.42 300 0 0 90.99 6.91 0.2498E-11 754.75 301 0 0 94.67 7.15 0 4281E-12 760.00 301 0 0 168.78 10.52 0.3502E-25 810.50 301 0 0 230.26 12.33 0.3260E-34 761.58 307 0 0 230 26 12.33 0.3260E-34 754.75 323 0 0 78.46 6.05 0.7308E-09 758.75 323 0 0 78.46 6.05 0.7308E-09 754.75 332 30 0 78.46 6.05 0.7308E-09 758.75 332 30 0 78.46 6.05 0.7308E-09 754.75 335 0 0 78.46 6 05 0.7308E-09 758.75 335 0 0 78.46 6 05 0.7308E-09 75*.75 337 30 0 78.46 6.05 0.7308E-09 761.58 339 0 0 118.27 8.46 0.1378E-16 754.75 340 0 0 78.46 6.05 0.7308E-09 754.75 342 30 0 78.46 6.05 0.7308E-09 762.75 345 0 0 78.46 6.05 0.7308E-09 758.75 345 0 0 78 46 6 05 0.7308E-09 751.75 355 30 0 149.31 9.82 0.4720E-22 760.50 356 0 51 113.15 8.20 0.1218E-15 753.00 0 0 0 117.84 8.43 0.1658E-16 872.75 211 0 0 123.28 7.32 0.1270E-12 872.75 218 0 0 123 28 7.32 0.1270E-12 872.75 225 0 0 123.28 7.32 0.1270E-12 872.75 232 0 0 123.28 7.32 0.1270E-12 872.75 240 0 0 123.28 7.32 0.1270E-12 872.75 247 0 0 123.28 7.32 0 1270E-12

SUMMARY

MINUMIM F AILURE LIMI T DRESSURE= 78 459 5AFETY 5.27< INDEX < 6.05 (BETA)

PROBABILITY 0.7309E-09< OF < 0.6875E-07 FAILURE

TABLE 9-5 SEQUOYAH CONTAINMENT VESSEL ANCHOR BOLTS LIMIT PRESSURE ANCHOR BOLT LIMIT PRE SSURE = 66.061 S AF ET Y INDEX = 5.369 PROGABLITY OF FAILURE = 0.3960E-07 TABLE 9-6 MCGUIRE CONTAINMENT VESSEL ANCHOR BOLTS L IMIT PRESSURE ANCH3R BOLT LIMIT PRESSURE = 141.112 S AF ET Y INDEX = 10.116 PROBABLITY OF FAILURE = 0.2343E-23 l

I 11.0 FIGURES i

i

t*

I

,1 = 9 /16"

,1

• I / 2 "

/

/

/ 8tS* 4 {"

• / BO9'6i"

, @k y eor 9i" 799'9{"

f 796' 0"

/ 791' 6" 79;. 4 f

• 788' j" 782'0" E 3 778' E }"

772'6" 3 769'I" 763*d' i

3 759'6l*

~

753'6" 3

750'II" 744'0"

-# ;3 74Q'6I" 734'6" _ .

3 7 30' 3l"

~

725'0" _ .

, 72i' 6)"

7 [

715' 3)" -

713' 6" 3

-c 1

E 705'7" _ .

h 3 701' 6l" I

695'IOf" ._ .

( 69t* 2i" l

686'2" l  :

679'9I" i . __

o _60 D I*

L 690" g WELD LOCATION siirF LOCATim Figure 1-1. Sequoyah Containment Vessel Geometry

l

,s

\, \\l\

• ,6 WELD LOCATION STIFE LOCATION 835'9" 835' 9" e o .c 826'5" ,y 826'5" 817' 5" -, 816'. 5" 807'5" 8__0_6' 5" o

797'5" .

y 796' 5" 787'5" ., y 786'5" 777'5" 776_' 5" I _ m .. c O W U U 767'5" 766'5" y

157'5" -, y 756' ll" 747' 5" -' O 746' 5" j =

690.75,,

7 37 ' 5"_ _ ..,y 736'5" 728'4" y 727'10"

$>, 722' 9"  !

690"  ;

u 691" 1

Figure 1-2. McGuire Containment Vessel Geometry

O a

3y2 Eq. 2-33 1 S=Eyf co.

1. 3 g]N

= G<0) h

[

• gg Eq. 2 -34 ~ cp (-g)

G ( yi,y2 ) = 0 SAFE REGIOD DESIGN POINT

[ FAILURE REGION (a) Failure Function in Design Parameter Space (b) Probability Density of Failure Function Figure 2-1. Graphical Representation of Safety Index

l u Y2 u G2(Y >ya)=0 i

I G3(y,,Y2)= 0

,/ B3

/ SAFE REGION

/

L~' ~~ q z y,

\ 42

\ gl l7, (y, Y2) =0

\ G ,

Figure 2 '6 - Safe Region for Multiple Failure Modes

Kc" I

y=0.' ~s ELASTIC-PLASTIC s FAILURE s

\

ELASTIC FRACTURE 1 _

NDT 'T Figure 3-1. Effect of Temperature on Critical Stress Intensity Factor

)

..-. . - - . - - = . - -

1 1

d pi _________________

l l

l p _ . _ _ _ _ _ _ _ _

l I

I 1

l l l 1 I I

I I

I i

' I

8 82 81 Figure 4-1. Pressure-Displacement Curve for Pressure Vessel

P a Pc

/

/

/

/

/

/

/

/ tan 6=

/ g l/2 tanh

+

=

82 8 Figure 4-2. Definition of IIalf-El P#

Plastic Pressure

---~_

-100-i i I i I

I i

I t s

- ,u I

,> l (o ) Protruding Penetration (b) Flush Penetration Figure 5-1. Limit Mechanism for Cylindrical Penetration in Spherical

-101-L. K L. O .K Pi ,

<>N I -

J I  : :J M

STIF 42 STIF 82 Figure 6-1. Axisymmetric Shell Element M kr ,

k ; k-

< - , o <,

i < -, < +

< -, i , , <,

< ,---< , < + ,  :

g

-O W

'I,690"  ;

' ~ '

."m

_a ~

-, c "'

{ f690"_'

' ~

o

- o , ,

0.7 5"-<~ - O.75"- p < -e-< ,

l <H g 4,

i' It STIF 42 STIF 82 Figure 6-2. Finite Element Model for Closed End Cylinder

-102-

+ THEORY c STIF 42 W/SS W/L.D.

n SS 60 2 STIF 42 and 82 50 _ W/0 SS or LD.

vi h.1 I

o Z

40 -

v Z

o 30 -

l F-4 0

9 20 -

46 -

IO -

Ju 0 I I I I I IA -

O.I 0.2 0.3 0.4 0.5 0.6 0.7 DISPLACEMENT (INCHES)

Figure 6-3. Radial Deformation vs Location (at p = 35 psi)

-103-

+ STIF 42 W/ SS, L.D.

STIF 82 W/ SS (0.03) e STIF 82 (0.05) 10 0

( '

90 -

C en 80 -

Q-70 -

LL) 60 -

/

x 50 o

,/

$ 40 -

,/

L1.1 /

m 30 -

/

a. ,/

20 -

,/

lO ,/

O i i i 1.0 2D 3.0 DISPLACEMENT (INCHES)

Figure 6-4. Pressure-Displacement Using Different Element Type and Convergence Criteria

-104-l l SHELL WALL '

q mwnswwj RING I

l (690" OE'r -

STRINGER l

%%S%\hWG i

1 Figure 6-5. Vessel with Ring and Vertical Stiffeners

-105-

. q_

b .

STIF 82 _

SHELL ELEMENT STIF 23

=

STRINGER RIGID ELEMENT REGION y O 8

@ 3

,-U CONSTRAINT EQUATIONS U=U 2 i V2= Vi -b6 i Figure 6-6. Cylinder-Stringer Connection Idealization

-106-STIF 82, WEB ELEMENT STRINGER NODE b c (FIG. 6-6V)\ = -

SHELL A  :\ FLANGE NODE NODE 1 ' i [(FIG. 6-8)

(FIG. 6-6) o g

~

v

/

U CONSTRAINT EQUATIONS i

U=Ui 3 + 06 V3= V i - b6 i V4 = V i + c6, ,

l Figure 6-7. Ring Web Connection Idealization

-107-l WEB NODE (FIG. 6-7 ) @ ,

/2 STIF 82 b g, , FLANGE ELEMENT STRINGER NODE h/2 / jV (FIG. 6-6 ) < ,

_ c J j-u CONSTRAINT EQUATIONS Us = U i + h/2 6 i Vs=V+c6i i U7 = U i - h/2 6 i Figure 6-8. Ring Flange Connection Idealization

-108-100 -

w/ STIFFENERS f

80 -

C 60 -

/ .

w s f ,/ \w/o STIFFENERS m 40 - //

m //

d //

//

[ #

20 si

//

/

OP 1 I I I I I I I 2 3 4 5 6 7 DISPLACEMENT (INCHES)

Figure 6-9. Pressure-Displacement for the Vessel in Fig. 6-5

-109-60 -

O 8

^

/

/

~

/

W /

T /

a 30 -

D /

D /

W i /

SO '

0 /

l ,/

lO -

/i

/Il

/

o i i  !!

1.0 2.0 li DISPLACEMENT (INCHES) i Figurc 6-10. Pressure-Maximum Displacement Curve, Sequoyah Containment (Fy = 35.2 ksi) v----m- , ~- _ - - , - , . , - - - , , , ,

e

-110-i 10 0 -

l C 80 -

/ jj w /

k /

v /

w 60 T

/

s' D

D /

/ ,

D /

w 40 -

le2 /

T --' /

k l /

p 20 -

.Il

/ 7_!

/

O ' ll i i 1.0 2.0 I DISPLACEMENT (INCHES)

Figure 6-11. Pressure-Maximum Displacement Curve - McGuire f

Containment

-111-60 50 -

o a 40 -

m Q.

w 30 -

x a

m e

20 -

Q.

10 -

0 i i  !!

ll 0.0 01 0.002 STRAIN Figure 6-12. Pressure-Maximum Effective Membrane Strain Curve - Sequoyah l

Containment Vessel l l

i 1

? l l

l i l

)

I

-112-lO0 -

O  !! o

[ 80 ll w

1 a 60 m

m w

T -

a_ @

20 -

O i i i i !!

O.001 0.002 0.003 ii STRAIN Figure 6-13. Pressure-Maximum Effective Membrane Strain Curve -

McGuire Containment Vessel

-113-

\

9 9

9 9

W 9

9

\ 9 9

9

\ *

=

m I

I

\ 9 l \

m

///////

Figure 6-14. Deflected Shape of Sequoyah Containment Vessel Near Plastic Pressure

i

-114-f.

i

)

i i

\

\ \

i

\

\ .

\

\

t \

s 4

a.

4 1 4

4 1

4 i

4 i I 4

}

i i I 4

4 -

t 4

[ i 4 I 1 L 1

1 1 e h

Figure 6-15. Deflected Shape of McGuire Containment Vessel I Near Plastic Pressure i

i i

' . - _ . . . . _ - . . . _ _ . _ , _ _ _ _ _ . _ _ _ - - _ - - , _ - . _ _ - - , _ _ ~ . _ , . _ - . , - - - - _ _ , - . . . _ - _ - - - _ - . _ _ . . . . _ --

-115-1200 EXPERIMENTAL 1000 -

- ANSYS (O ',

D- s' /

800 -

,/ , '

U '

,/

x i's D //

w 600 -

' /,

. ii

& i,

/'

400 -

b

,,o*

/

/,

200 - '/

/, '

/

0 I I I I I l.

0 0D04 0.008 0.012 0.016 0.020 0.024 DISPLACEMENT (INCHES)

Figure 6-16. Pressure-Displacement for Experimental Model l

l l

i

-116-N0ZZLE RING STIFFENER N/NZ

/ k Ak N

< AE 24 M7Nnz:b

> \Y\

CONTAINMENT g VESSEL So Figure 6-17. Ring Stiffener, Penetration and Containment Vessel Mesh Layout (Sequoyah vessel)

-117-e NA y NOZZLE RING  :

STIFFENERS nn  ? 3 Qa 7 CONTAINMENT gXxi VESSEL N

Figure 6-18. Ring Stiffener, Penetration and Containment Vessel Mesh Layout (McGuire Vessel)

-118-10 0 m

- gQ -

m CL

/

w x 60 -

D s' W /

/

D '

W 40 -

T /

Q. /

/

/

20 -

y' 0 ' I I O l.0 2.0 DISPLACEMENT (INCHES)

Figure 6-19. Pressure-Displacement of Penetra61on End -

Sequovah Vessel

-119

• 10 0 -

/

0 -

m 80 -

1 -

/

W T 60 -

/

/

a '

D 7 D /

W /

m 40 /

s' 0-

/

/

/

20 -

/

/

/

/

O I l O 1.0 2.0 DISPLACEMENT (INCHES)

I Figure 6-20. Pressure-Displacement of Penetration End -

McGuire Vessel l

l

.~ _ _ _ - _ _ . _ -_ - - - - _ _ _ _ _ _ _ _ _ _ _ _ _ _

-120-

! j 10 0 C

W 80 -

a.

v 9

LLI tr 60 -

a M

M W

Eg ___

20 -

O I I I I I I l 0 0.0 01 0.002 0.003 0.004 0.005 0.006 0.007 STRAIN Figure 6-21. Pressure - Maximum Strain - Sequoyah Penetration

{

,, ,n,, , , - , - - - , , , - - , . - - - - - . - - - . - - - - , - - - , . , . - , - - - . . - - - - - , -. _ , , _ --. --- ,. , -- -

--.-,.n,,

-121-10 0 c 80 -

(n CL v

g 60 -

a (O

(n w 40 -

ct Q.

20 -

0 I I i 1 0 0.002 0.004 0.006 0.000 0.010 STRAIN Figure 6-22. Prcssure - Maximum Strain - McGuire Penetration i

-122-12.0 ADDENDUM

SUMMARY

AND CRITIQUE OF INDEPENDENT ANALYSES OF SEQU0VAH CONTAINMENT On September 2,1980, a meeting of the Advisory Committee on Reac-tor Safeguards (ACRS) subconnittee on Structural Engineering was held in Washington, D.C. At that meeting, the results of several indepen-dent analyses of the Sequoyah containment vessel were presented. A summary of these results and a critique by this author are presented herein. All of these analyses examined the strength of the vessel under uniform static internal pressure. The results of a f ;st order approximation to the dynamic strength obtained by this author are also presented in this Addendum.

12.1 Static Pressure 12.1.1 Ames Laboratory (January 1980) '

On January 16, 1980, Anes Laboratory was requested to make a first order approximation to the static strength of the Sequoyah containment.

In this analysis, the total ring and stringer areas were "sneared" to form an equivalent shell [A]*. Stresses in the equivalent shell were assumed to be uniform at f (12-1)

= [ex-f g= PI (12-2) eo

• References for the Addendun are listed at the end. Copies of the ref-erences are attached

-123-where et o = t(1 + at) t ex = t( 1 + a2I are the equivalent thicknesses and the other terms are defined in Sec.

5.5.1. The von Mises yield criterion is applied to the biaxial stress state in the shell. Since the rings are under axial stress, the limit strength was obtained when Eq. 12-1 is set equal to the yield stress, Fy (taken to be 32 ksi in this work). The ASME area replacement rule was assumed to be satisfied so that penetrations did not control. The static pressure was reported to be 36 psi (+30, -10 percent) . Burst pressure predictions were not considered reliable because of the limit-ed ductility of the stiffened shell.

Comments: The assumption of uniform stress in the equivalent

" smeared" shell at the maximum pressure is not consistent with a limit mechanism. Since the shell and stiffeners are in biaxial and uniaxial stress states, respectively, stresses will not be the same in the stif-feners and shell at the limit load (nor in the linear elastic range).

Local bending effects are not included nor are they demonstrated to be unimportant, i.e. , the limits of " smearing" are not defined. However, the report, submitted four working days after it was requested, did serve its intended purpose.

12.1.2 R&D Associates The above results by Ames Laboratory were presented to the Commis-sioners of the Nuclear Regulatory Commission. Commissioner Victor Gilinsky requested R&D Associates to critique the Ames Laboratory (Jan-uary 1980) analysis. Their work employed a linearly elastic analysis to shcw that the stringers are only about 40 percent effective and the rings are totally ineffective [B]. Locally high bending stresses were shown to exist near the rings and stringers but they were shown not to ,

, affect the vessel strength. A burst analysis is not appropriate for this vessel since other features, such as holddown bolts, will fail first. The predicted strength, based on an F y of 32 ksi and the"

= _ _ - _ _ _ _

9

-124-l von flises yield criterion, was 27 psi. A nore detailed finite eleaent analy:;is and an experinental panel test were recomended.

Comments: This work represents a reasona5le approxiniation to the

! linear behavior of the stiffened shell. In essence, the analysis cal-culates the strength of an unstiffened shell of infinite length. Lo-i cally high bending stresses are, indeed, not important insofar as the linit state is concerned. However, the results certainly are a lower bound to the limit pressure. If the shell has any ductility capacity j whatsoever (which it nost certainly does), stress redistribution will

! occur between the stif feners and shell and pressures beyond 27 psi will be obtained.

12.1.3 TVA The analysis by TVA [C] considered several failure nodes: anchor-l age, penetrations (bellows and valves), personnel locks /equipnent 1 hatch, seals and shell pl a te. The shell plate was shown to control.

The shell was conservatively analyzed as an unstiffened cylinder with a naterial yield stress of 45.7 ksi (lowest value of actual mill tests).

f The von tiises failure criterion gave a pressure strength of 38.2 psi.

Connents: The TVA analysis is conservative and gives essentially ,

the sane resul ts as the R&D Associates analysis. Sinilar coments

apply.

12.1.4 NRC Research HRC Research personnel subnitted a critique of the Anes Laboratory j (January 1980) and the R&D Associates analyses [D]. This nenorandun states that the stiffeners should be expected to add some strength to f

! the shell. An independent analysis was performed in which the string-

! ers were idealized as Seans spanning between ring stiffeners. Pressure applied to the inside shell surface was assumed to be resisted by cir-cunferential tension in the shell plus bending of the stringer. Local shell bending effects were ignored. The maximun pressure was assuned l

I to occur when a plastic bean nechanisn formed in the stringer and the

]

shell yielded in tension. For an assumed Fy of 32 ksi and the von itises yield cri terion , a predicted strength was given as 34 psi.

I

~.__ _._ _ _ . _ . _ _ _ _ _ _ _ _ _ _ _ . - - _ _ - _ _ . - _ _ _ _ _

-125-4 Comments: This analysis represents ar interesting approach to in-corporating stringer bending effects. It gives results similar to the 1 Ames Laboratory (January 1980) results. However, the results are con-servative in that transverse neridinal shear forces are neglected in the shell free body diagram. The net result is that no force is assuned to be transnitted from the shell directly to the rings. The effect of j the ring on the ci rcumferential shell stresses is, consequently, neglected. Additionally, at the limit load, the stringer is predon-inately an axial force member, rather than a bending nenber (See Sec.

6.3.3.2). Al so, radial deflection of the rings at the ends of the

stringers may not permit the development of the full stringer bean mechanisn hypothesized here.

i 12.1.5 Franklin Research Institute Zenons Zudans of the Franklin Research Institute reported on his critique of the Anes Laboratory (January 1980) and R&D Associates work

[E]. He concluded that both analyses are incorrect in the manner in

! which the rings and stringers are treated. Of the Anes Laboratory

) work, for example, he stated that the " calculations of limit pressure

. . . are ner.ningless. Accordingly, the conclusion, that the ring will yield first (at 35.7 psi) is not realistic. " Mr. Zudans proposed a ,

model of his own which includes four separate analyses:

(1) Axisynnetric, ring stiffened shell (5/8 in.) without string-ers. This elastic analysis demonstrated that the rings have no effect on the elastic stresses in the shell nidway between rings if no stringers are present.

(2) 'Axisynnetric, ring sti f fened shell (5/8 in.) with sneared stringers. The hoop stresses in the shell with the stringers were shown to be alnost uniform in the clastic range and significantly below those of Analysis (1).

(3) Curved shell panel (5/8 in.) with one ring and one stringer.

The elastic panel analysis denonstrated that the "snearing" I technique used in Analysis (2) above is valid. Thus, the bending and nenbrane sti f fnesses of the stringer can be f sneared ci rcunf erenti ally .

4 1

-126-(4) Axisymmetric, ring stif fened shell (1/2 in.) with smeared stringers. (Sane as Analysis (2) except 1/2 in, shell). The pressure at which the average hoop stress midway between rings reached the yield stress was predicted to be 30.3 psi.

An Fy at 32 ksi with the maximun shear stress criterion was used. j Conments: This analysis confirms at least two aspects of the original Anes Laboratory work: (a) stringers can be "sneared" if their axial and bending stiffnesses are included (Analysis (3) above), and (b) the hoop stresses in the shell with the stringers are almost uni-form between rings (Analyses (2), (3), and (4) above). Thus, although the author strongly objects to the Ames Laboratory assumptions, he tends to confirm them. Beyond initial yielding, i.e., with large dis-placement and force redistribution effects, and near the limit load,

! the Ames Laboratory assumptions become even more realistic. At this stress level, stringer bending strength and stiffness are negligible and only axial effects need to be incorporated into the smearing pro-cess. The above direct quote from the Franklin Research Institute report indicates that they may not have understood that the analysis by Anes Laboratory was intended to be an approximate limit analysis. Cer-tainly, the ring will not yield first but, as confirmed by the current report (Sec. 6.3.3.2) , the limit strength is certainly controlled by the rings.

The results of this analysis were very useful in confirming the stringer snearing process used in the finite element analysis reported. ,

in the text of this report (Sec. 6.3.1).

i l

12.1.6 Offshore Power Systems Offshore Power Systems performed a nonlinear analysis of a typical 1/2 in, curved panel bounded by two stringers and two rings (a geometry very similar to Analysis (3) by Franklin Research Institute) [F]. The nonlinear finite element analysis was performed using ANSYS and an Fy of 45 ksi. These results demonstrated that the circunferential varia-tion of displacement in the panel is negligible - even in the nonlinear range. Additionally, the ring and shell stress vary little circumfer-

-127-l entially. No maximum pressure strength was predicted from the finite elenent results. An analysis with " smeared" rings, independent of the finite element analysis, gives predicted strength of 50.8 psi and 56.8 psi for 9' 6 " and 6' 6 " ring spacings, respectively.

Comments: As with the Franklin Research Institute results, the finite element results confirm the stringer in the original Ames analy-sis. The Offshore Power Systems work is also referred to in the text of thi report as verification of the smeared stringer technique used in the nonlinear finite element analysis (Sec. 6.3.1). The predicted strengths from the " smeared" analysis are based on essentially the same assumptions as the original Ames work.

12.1.7 Ames Laboratory (September 1980) -

The latest complete Ames Laboratory analysis is summarized in the text of this report. Preliminary results were presented at the ACRS meeting. The approximate analysis reported herein is a revised version s  ;,_

of the January 1980 work. A complete mechanism was assuned to forn -

with stiffeners and shell at yield (see Sec. 5.1)K The approximate .y limit pressure is 59 psi. An axisynmetric, nonlineir finite element analysis of the complete containment was also perforned (see Sec. 6.3). y The plastic pressure was shown to be 60 psi. An ry of 47.2 ksi (riean (;,

~

value) was used in each of these analyses. , ,.

Comments: The finite element model employed here appears to give + ' i the most complete analysis of the nidainment. The stringer smearing ( '

process is confirmed by the Franklin .Research Institute and Offshorex -

analyses. <

Power Systems' ,~_

. w, -

12.1.8 Summary '

~

To provide sone basis for a comparison' of the variouhapproachei ,y summarized above, it is useful to: (1) convert each result tu .an Fy ,-[

of 47.2 ksi by a direct ratio, (2) use ,the von Mises yi' eld.-criterion s x

(multiply the result of Franklin Research Inititute by 2//3), .and (3) ~ ^ ,

classify resul ts. This process gives: x A

-128-4 R & D Associates 40 psi first yield w/o stiffeners TVA 40 psi first yield w/o stiffeners NRC Research 50 psi stringer beam mechanism Franklin Research 51 psi nembrane yield Ames Laboratory (Jan.1980) 53 psi yield w/ smeared stiffeners Offshore Power Systems 53 psi yield w/sneared stiffer.ers Anes Laboratory (Sept.1980) 59 psi limit mechanism Ames Laboratory (Sept.1980) 60 psi finite element In spite of the apparent larga variation in the predictions for the Sequoyah strength, there is, in essence, really only one practical

. question to be answered: How much ductility capacity does the existing containment have? If the ductility is such that the membrane strains must remain below yield, then a pressure based on first yield is appro-priate. First yield will probably occur between 40 psi (if stiffeners are neglected) and 50 psi (if stiffeners are included). Figs. 6-10 and

^

6-12 of the current report also confirm this. If, on the other hand, the containment vessel has a ductility capacity of at least two, force redistribution will be permitted to occur and the stiffening will be-come more effective. The strength of the vessel could then be taken as j 60-psi. As documented in this report (Chap. 4 and Fig. 6-12), a duc-tility capacity of two is certainly probable (see also [21, pg. 529])

so that a 60 psi strength is reasonable. The statistical distribution

' of the ductility capacity of the containnent vessel should be deter-

~

mined to quantitatively define this probabili ty.

_12.2 Dynamic Pressure

~12 2 1 Introduction As mentioned el sewhere in this report, the explosion incident i

u identified at_.TMI may have produced dynanic pressures which varied with I time. In particular, if a hydrogen explosion occurs within the rela-i.tvely confined volume of a lower compartment, significant dynanic pressure could develop. A prelininary estimate of the dynamic pressure i capacity of the- Sequoyah containment is presented in this section. An 4

Y

.-y-n , - ~ - - - - , - - . - , - , - . , - - - - . - - , . , - - r ,, ,. .

---r-- - -- . , - - - - - . - ,,

-129-explosion in the upper compartment or in the ice condenser conpartnent is assumed to be relatively unconfined and of little significance. Dy-namic pressures were considered in only the lower compartnents (Elev.

693' to 719.5') listed in Table 12-1. No information was furnished to us regarding the actual time and spatial variation of the potential ex-plosive pressures for Sequoyah. The analyses sunmarized below are in-tended to be very preliminary. In this regard, several simplifying as-sumptions have been made. More effort should be devoted to this work; see Reconnendations in Sec. 7.3.

12.2.2 Preliminary Finite Element Analysis On July 3,1980, Anes Laboratory was requested to nake a prelini-nary calculation of the strength of the Sequoyah containment vessel subject to a dynamic pressure [G (copy of report is attached)]. A dynanic pressure was assumed to act in a lower compartment over an arc length of 60 degrees. The pressure in the compartment was assumed to vary from a naximun pressure, yp , at time zero to a zero pressure at 0.030 sec. The pi;lse nagnitude and length are quite arbitrary and would be dependent upor. compartment size, explosion characteristics and venting properties. (An initial pul se representing the detonation phase of the explosion was also included, but its momentun was shown to be relatively insignificant.)

Since the pressure loading is not axisymmetric, the response will not be axisymmetric. The non-axisymnetric response was assuned to be dominated by the rings. A typical ring (Elev. 713.5') with an effec-tive shell width was idealized by STIF23 nonlinear bean finite elements

( see Sec. 6.3.1) . Elastic springs, tangential to the ring, were used to model the resistance of the shell below the ring. An Fy of 39 ksi was used for the dynamic analysis. A dynamic transient solution was obtained using ANSYS with naterial and geometric nonlinearities. Three separate analyses were performed with three separate maximum dynanic pressures, i.e., for py equal to 10, 50 and 100 psi.

The resulting maximun strains and displacements are summarized on page 12 of the attached Ref. G. They can be summarized in a non-dimensional form as

1

-130-Ductility Demand 1000 py/F y ut ux j 0.26 0.4 0.3 1.28 4.8 2.2 2.56 24.9 11.7 where the maximum dynanic pressure has been non-dimensionalized with respect to the material yield strength, the strain ductility demand is l

, maximum strain c yield strain (12-3)

I i and the displacement ductility demand is maxinun displacenent j "6 " yield displacement (12-4)

The non-dinensionalization of the maximum pressure is convenient for extending the results to other material yield strengths. The yield strain is 39/29000 or 1345 micro in./in. The yield displacement is, quite arbitrarily, taken as the clastic displacement at the ASME half-linear-slope pressure (Sec. 4.2). From page 10 of Ref. G, the yield t displacement for an F y of 32 ksi is 35 psi /(20 psi /in.) or 1.75 in.

For an Fy of 39 ksi the yield displacement is 2.13 in. which was used

to calculate the above u6*

The predicted strength of the vessel is dependent upon the ductil-ity capacity of the vessel -- as in the static case (Sec.12.1.8). In Ref. G, a maximun dynamic pressure was conservatively predicted, based j upon a strain ductility capacity of two. However, a ductility capacity l of two for displacement seems more consistent with the ASME definition i of the static plastic pressure by the half-linear-slope method. For a l di splacement ductili ty capacity of two, a 1000 p y/Fy ratio of 1.2 l

l i

i j

- - - . - . - - - , - - - , . . - - - - - - ,. . . - - - - , - - ~ - - , , - - - - ,

-131-

{ is interpolated from the above table. Using the actual mean material yield stress of 47.2 ksi gives a predicted dynamic pressure strength of 57 psi.

12.2.3 Approximate Dynamic Analysis l A simple approximate analytical model for estimating the strength of the Sequoyah steel containment vessel under a dynamic pressure act- '

ing in a lower compartment over some arc length is presented in this section.

The transient response of the ring which was obtained in Sec.

{ 12.2.2 indicated that nost of the energy absorbed is predominately due 1

to membrane action. A nonlinear static analysis of the ring was also performed in Ref. G. The results showed that, as the limit load is approached, the cross sections in the vicinity of a equal zero (Fig.

12-1) went into pure plastic tension. What may be terned complex hin-i ges formed in the vicinity of ao (actually, slightly beyond g) g on either side of the centerline shown in Fig.12-1. The energy absorbe i at each end of the complex hinges is the summation of the work done by

the reduced plastic moment and the tensile force at the section. The results obtained in Sec.12.2.2 suggest that a simpler analytical model

~

may provide a first approximation to the strength of the containment vessel. A section of the containment vessel will be modeled by a sin-gle degree-of-f reedon systen.

To obtain a simple analytical model, it is assumed that, at the limit load, a typical ring section with an effective shell width col-lapses by the formation of plastic hinges on either side of the center-line at a oand a fully plasticized section in pure axial tension at a

{ equal zero. The reactive stresses are assumed to be shearing stresses as shown in Fig. 12-1. The deformation of the mechanism under constant pressure during plastic collapse is shown in Fig.12-2. The arc curva-ture is assumed to remain constant during collapse. Secondary effects like the influence of large deformations on the limit load are neglect '

ed. The work done by the external loads is assumed to be absorbed by a

4

., -132-I the two plastic hinges (assuning no reduction in the plastic moment, M) p and by axial extensi_on at a equal zero. The contribution of the shearing stresses to the strain energy is neglected.

i From Fig. 12-2, for a virtual displacement, 0, the axial exten-j sion, 6, is 2G o 6 = 40 Rg sin 7 (12-5) t where R o is the radius of the containment wall. The external work

< done by the distributed load acting on arc ABD may be taken as, 2 2G o external work = 4 0 Rg (sin 7)ps gp 4

where po is the limit pressure and s r is the ring spacing. Equat-ing internal work with external work, we obtain 2 2G o 2M p e+P6=40Rg y (sin 7)ps g p (12-6) where M p is the plastic moment and Py is the axial yield load of the effective ring section.

Substitution for 6 in Eq.12-6, gives 2 2a o 2 2o g 2Mp + P 4R g sin 7 = 4R g (sin 7)ps gr (12-7)

The limit load of the ring considered in Sec.12.2.2 will now be estimated using the simple analytical model described above and the results will be compared with those obtained from the nonlinear (mate-rial and geometric) finite element analysis of Sec.12.2.2. Using the same ' geometric and material quantities as in Sec.12.2.2 [G, pg 3] and 4

a steel yield stress, F,y equal 32 ksi, the static limit pressure estimated by Eq. 12-7 is 30 psi as compared with 35 psi from Sec.

12.2.2 [G, pg 10].

e e

.cw. -- , - , --- . , , r-, - - _

_ _ - , - - ~ . --,-_-,-.- , y. ,-----,,,,p3 -.--w - , , , - -y- --,,, m.

-133-The nonlinear finite element analysis of Sec.12.2.2, using the actual mean material yield stress of 47.2 ksi and a displacement duc-tility capacity of two, predicted a dynamic pressure strength of 57 psi. For a yield stress of 47.2 ksi, Eq. 12-7 predicts a static limit pressure, po, of 44 psi. Assuming td /T is still 0.75 as in Ref. G, and the displacement ductility capacity is two as in Sec.12.2.2, Fig.

12-3 gives py /po of 0.88 for the one degree-of-freedon model with a linearly decaying pressure. Therefore, the maximum dynamic pressure

! predicted by this model is 50 psi. Again, the one degree-of-freedom model yields a more conservative maximum dynamic pressure. This may partly be attributed to the omission, in the one degree-of-freedon mod-el, of the strain energy due to shearing stresses. In the finite ele-ment study the shearing stresses were approximated by linear springs.

A hydrogen explosion in a lower compartment may be assumed to im-pose a ;ynamic pressure consisting of a detonation phase followed by a venting phase. As stated in Sec.12.2.2, the impulse may be idealized to consist of only the venting phase. For the dynamic analyses to be performed in the remainder of this section, the venting time is assumed to be infinite (conservative). Thus, the hydrogen explosion in a lower compartment has been idealized as a suddenly applied constant pressure (Fig. 12-4) acting on the arc of the containment vessel subtended by the compartment in question. With this approximation and a displace-ment ductility capacity of two, Fig. 12-4 gives p /py g as 1.33 where po is found from Eq. 12-7. Data and the estimated maximum dynamic pressures, p, y for the lower level compartments are summarized in Table 12-1. A ring spacing of 120 inches and a yield stress of 47.2 ksi were used in computing the approximate maximum dynamic pressures.

The minimum value is 33 psi.

12.2.4 Summary Preliminary estimates of the dynamic pressure capacity of the lower compartment region in the Sequoyah containnent were made. A dynamic, transient finite element analysis of a typical ring with a linearly decaying dynamic pressure was performed. For a displacement f

e I

e T

e 1

-134-ductili ty capacity of two, the predicted dynamic press _re strength is 57 psi. An approximate, one degree-of-freedom analysis of a typical ring was also conducted. For this analysis, the dynamic pressure was assumed to be suddenly applied and constant with time. The predicted strength is 33 psi for a given displacement ductility capacity of two.

Both of these analyses must be considered quite approximate and, in the authors' opinion, conservative. In particular, the results obtained from the one degree-of-freedom analyses can be considered very conser-vative since the venting times were assumed to be infinite. More so-phisticated analysis techniques with fewer assumptions shoul d be applied.

Table 12-1. Estinated Maxinum Dynamic Pressures (Lower Level Compartments)

Azimuth Shell Limit Max. Dyn.

Compartment (degrees) Arc thickness Pressure Pressure (approx.) (degrees) (in.) (psi) (psi)

Accumulator Room 270-326 56 1 1/2 73 55 Fan Room 326- 34 68 1 1/2 73 55 Accumulator Room 34- 54 20 1 1/2 87 65 Instrument Room 54-126 72 1 1/16 44 33 Accumulator Room 126-146 20 1 1/16 47 43 Fan Roon 146-214 68 1 1/16 44 33 Accumulator Room 214-270 56 1 1/16 44 33

l

-135-E l FULLY PLASTIClZED SECTION r IN PURE AXIAL TENSION  !

L j PLASTIC HINGE Ass\' \$t g 1 fl [s / j'/ -

C PLASTIC HINGE s l Po r /

/

\ ,

s s

l

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. CONTAINMENT WALL N l s,l,/

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REACTIVE SHEARING STRESSES Fig. 12-1. Collapse Mechanism for Ring Section with Effective Shell Width I

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,j io 4o Fig. 12-3. Maximum response of elasto-plastic one-degree systems (undamped) due to triangular load pulses with zero rise time. (U.S. Army Corps of Engineers: " Design of Structures to Resist the Effects of Atomic Weapons,"

Manual EM 1110-345-415, 1957.)

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-138-01 10 to 4C

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!""(,""" ii,i,'*"'" i 1.0 10 40 01 eg/r Fig. 12-4. Maximum response of elasto-plastic one-degree systems (undamped) due to rectangular load pulses. (U.S. Army Corps of Engineers: " Design of Structures to Resist the Effects of Atomic Weapons," Manual EM 1110-345-415, 1957.)

t

-139-1 i

12.3 LIST OF REFERENCES FOR ADDENDUM Copies of References attached following this page.

A. Greimann, L.F. (Ames Lab.), Letter report to Dr. F.P. Schauer, NRC, Washington, DC, January 22, 1980.

B. Hubbard, H.W. (R&D Associates), Letter report to Dr. R.L. Tedesco, 8913 Wooden Bridge Road, Potomac, Washington 20851, July 25,1980 (presented orally at ACRS meeting, September 2,1980, by F. Parry).

C. TVA, oral presentation at ACRS meeting, September 2, 1980.

D. Bagchi, G. (NRC Research), Memorandum to F.P. Schauer, NRC, Washington, DC, August 17, 1980.

E. Zudans, Z. (Franklin Research Institute), letter report to Dr. R.

Savio, NRC, Washington, DC, August 29, 1980.

F. Orr, R. (Offshore Power Systems), oral presentation at ACRS meeting, September 2, 1980.

G. Creimann, L.F., letter report to Dr. F.P. Schauer, NRC, Washington, DC, July 18, 1980 and supplement of July 30, 1980.

-140-l REFERENCE A (

Greimann, L.F. (Ames Lab.), letter report to Dr. F.P. Schauer, NRC, Washington, DC, January 22, 1980.

i i

-141-

,y lowa State University Ames. lowa 50011 HIIIes 1.l

-e "'

inhornfor Energy 6 Mineral ResourcesyResearch Institute January 22, 1980 Dr. F. P. Schauer, 2 v % Chief Olvision of Systems Safety Office of hoclear Reactor Regulation Nuclear Regulatory Comnission Washington, DC 20555

SUBJECT:

AMES LABORATORY TECHNICAL ASSISTAhCE TO THE DIVISION OF SYSTEM SAFETY, NUCLEAR REACTOR REGULATION " REVIEW OF NUCLEAR PLANTS STRUCTURAL DESIGh" (FIN NG. A-4131). PRELIM-l NARY CALCULATION OF ULTIMATE STRENGTH OF SEQUOYAH AND MCGulRE CONTAINHENT VESSELS

Dear Dr. Schauer:

As you requested in our telephone conversation of January 16, 1980, I have performed a preliminary calculation of the ultimate strengths of the Sequoyah and McGuire Containment Vessels. The following assumptions and limitations apply to these calculations:

(1) Uniform static internal pressure loading.

(2) Shell stiffeners are " smeared" for stress c41culation.

(3) Von Mises failure criterlon applies.

(4) Penetrations do not control; i.e., ASME area-replacement rule is satisfied.

(5) Ultimate strength is defined as the pressure at which stresses in the equivalent " smeared" shell reach the minimum specified yleid stress. (Burst pressures are not considered reliable at this time because of the potentially limited ductility of the vessel.)

Coples of the calculations are enclosed. In sunnery, the preliminary calculated ultimate strengths are 36 psi for the Sequoyah and 47 ps!

for the McGuire containment vessel. In r.y Judgment, the actual ulti-mate strengths are prabably between -10 percent and +30 percent of these values. The actual value may be less because Assumption (1) under estimates the shell stresses, although the shell does not control.

The actual value may be greater because Assumption (1) overestimates the ring stresses, which do control; Assumption (5) Is conservative; and the actual material yleid strength is probably greater than the minimum speelfled.

-142-Dr. F. P. Schauer January 22, 1980 As per the statement of work on the subject project, we intend to continue to refine the above estimates of the ultimate strength and the associated uncertaintles. If you have any questions, please c,ntact me.

$1ncerely, 1

Lowell F. Greimann Pr$Je:t Engineer Enclosure cc: Director, Division of System Safety (Attn: B. L. Grenier) w/ encl.

Selwyr D. Bluhm, Head, Praject Engineering w/ enc 1.

-143-4 befaof3b Ontaintnent Ulhenpre Swnph pr. L. C Greimann V l~arwc

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- __., ..e REFERENCE B  % ~

Hubbard, H.W. (R&D Associates), letter report to Dr. R.L.

• Tedesco, 8913 Wooden Bridge Road', Potomac, Washington 20851, July 25, 1980 (presented orally at ACRS meeting, September 2, ., #-

1980, by F. Parry) - 5 g res0 . j .\ ~s?* k w s

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                             ,     25 July 1980
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                   %                                                                                                                                                                               I
  ;,            3                  D'r. ' Rober t. .L. Tedesco

( 8913 1:ooden' Bridge Road

  !                                Potomac, niryl.ind 20854
 .                             .                                               1   -

Dear Dr.Tedesco:

i The enclo' sed document is a critique of the Ames .malysi~s i of the Sequoyah. containment structure. The critique

                              was performed by R & D Associates at the request of i                                 Commissioner' Victor Gilinsky, who asked that a copy be i, ,. ,-                       ,,

supplied to you on its completion. Very truly yours, f 4 4 L

e. Ila rn.on W. 11ubl,s.i rd t-HWH/dl

Enclosure:

                                         "!'equoyah Containment Analysi n , " .lul y 1980, 2                                                                                          (1 cy).

y s, E i,

 !                                                                            +

i i 1 me 4: r..w.i n v. .u . t. .: 'a DE L RE Y. C AL D CRN8 A ".'. '"

• 18 . t ' 'At . 1h

                                                                                                                                                                     ."/.'**1'
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                                            -157-i SEQUOYAH CONTAINMENT ANALYSIS

1. IS.'P.ODUCTID:;

l This letter report is in response to a rer;uest from the U.S. .;uclear Regulatory Commission to review a.nd critique the ultimate strength analyses of the Secuoyah containment. The description of the contain::.ent vessel and the analysis for review were provided in the NRC Informaticn Report dated 22 April 19 5 0,., Re f . SF.CY-80-107A. The tasks reauested in the work statement were as follows:

1. To what extent are the assumptions in the analyses conservative?

2. To what extent is the calculated ultimate strength i conservative?

3. ';ha

                    .-    t are the uncertainti3s in the analyses, methods, l                  and models?

i 4. To what extent is there assurance of no gross leak-age from the vessel at stresses abcVe the design stress and yield stress?

5. How would the analyses and results be altered if the stresses are caused by ignitien ' detonation of 300-600 kg of hydrogan distributed unifor:nly and nonuniformly in the containment?

i 6. To what extent can distributed ignition sources mitigate the effects of hydrogen?

• This report will cover the first four tasks of the work state-ment. A report on the hydrogen problem, tasks 5 and 6, will be issued separately. A preliminary briefing of the analyses conducted by RDA was given to Ccmmissioner Gilinsky and Dr.

J. Austin at RPA o n I S t h .1uly 198 0.

                                    -158-

2. U.% KOROC;D - SdOCOYAli CO:4 TAI:iMENT VESSEL DESIGN The containment vessel for Sequoyah is a low-leakage, free-standing steel structure consisting of a cylindrical wall, a hemispherical dome, and a bottom liner plate encased in con-crete. Figure 1 shows the outline and configuration of the containment vessel.

The structure consists of side walls measuring 113 feet S-5/8 inches in height from the liner on the base to the spring line of the dome and has an inside diameter of 115 feet. The bottom liner plate is 1/4 inch thick, the cylinder varies from 1-3/8 inch thickness at the bottom to 1/2 inch thick at the spring line and the dome varies from 7/16 inch thickness at the spring line to 15/16 inch thickness at the apex. The containment vessel is provided with both circumferen-tial and vertical stiffeners on the exterior of the shell. These stiffeners are required to satisfy design requirements for expansion and contraction, seismic forces, and pressure transient loads. The circumferential stiffeners were installed , on approximately 20-foot centers during erection to insure stability and aligr. ment of the shell. Vertical stiffeners are spaced at 4 degrees and other locally stiffened areas are pro-vided for penetration, etc., as required The design of the containment vessel was to the require-i ments of the AS *.E code, Section III, Subsection B. The Code , e includes cases 1177-5, 1290-1, 1330-1, 1413, 1431, and the Winter 1968 Addenda. The following pressures and temperatures were used in the design of the vessel: Overpressure test (1) 13.5 psig Maximum internal pressure (2) 12.0 psig at 200*F Design internal pressure (2) 10. 8 psig at 220 *F

                                                                       -159-Figure 1.         Sequoyah Containment Vessel i

15/16" PL i 9/16"PL\ C s %

                                                           . _ _ _ _ _ _ _ . . _ _ _                           N N
                                                                                                                    \
                                                                                                                                     \

1/2" PL g

                                                                                                                                          \

EL 815.3 EL 809.5 . 4

                                                                                                            +                                \

EL 803.7 7/16" PL EL 799.8 t { EL 796.0 El 791.5 EL 788.0 )3 e _ _ . _ _ _ _ _ _ l

                                ), ,, p                                                                                                           l

EL 778.5 c

5/8" PL EL 769.0 c: 5/S" PL l

EL 759.5 c 5/8" PL l EL 750.1 c: Sfga pt t l EL 700.5 c: 11/16" PL l FL 730.3 r= : 3/4" PL j ' I EL 7:1.5 9 1" PL l FL 713.5 c= 1 1/6" PL  !  ! I I EL 70'.5 c= 1 3/16" PL j tt 69i.i a i i/4 n. I I El 679.78 / Ib" I'I l

              !.      ....u-                                                         ,                                                   . . . . . . . .
            /                                                                        i 9 1.2" x 1/" PL C. 4' CENTERS

                                            -160-Leakage rate test pressure                                      T2.0 psig Design external pressure                                           0.5 psig l

Lowest service metal temperature 30 F Operating ambient temperature 120* F Operating internal temperature 120* F (1) 1.25 times design internal pressure as required by ASME Code, UG-100 (b) . (2) See Paragraph N-1312(2) of Section III of the ASME Code which states that the " design internal pressure" of the vessel may differ from the " maximum containment pressure" but in no case shall the design internal pressure ha locc than 90 percent of the maximum con-tainment internal pressure. The steel plate used is to ASME specifications SA-516 grade 60 with a yield stress of 32,000 psi, an ultimate stress of 60.000 psi and a Young's Modulus, E, of 28 x 10 6 psi at 70*F. For the above code, the maximum shear stress criterion yields an equivalent maximum membrane principal stress, in the hoop direction, given by: hoop s tress = PR- = allowable stress, where P = 10.8 psi R = 690 in. (the given allowable stress in the 1977 version or the code is 16,500 psi (i.e., approximately 1/2 the yield stress)). Hence, 10.8 x 690 . t= 16,500 = 0.452 2n. Thus, the minimum plate thickness of 12 inch s.itisfies the basic code requirements. Originally the vessel was designed with enly seven ring stiffeners and local vertical stiffeners .it penetration i

i I

                                 -161-Tegions. 04 tailed buckling analysis and .    ..ic excitation analysis showed, however, that additional rings and vertical stiffeners would be required and the final configuration of Figure 1 resulted. It should be noted that tha longitudinal, or meridional, stresses in a cylindrical membrane are only half of the hoop stress and hence do not contribute to the maximum shear criterion of the ASME Code. Further the dome stresses are all of the same type    (" meridional" as opposed to " hoop")

and hence with the plate thicknesses used the dcme membrane stresses are much less than the critical cylindrical stresses. 3 THE ANALYSIS OF A SHELL NITH RING AND STRINGER STIFFENERS The application of rings and stiffeners to a nembrane structure is well known in aircraft structural anlaysis and must be trea ted with caution since local bending stresses can be induced. It was noted that the analysis provided in the reference document SECY-80-107A used a " smearing" technique whereby the rings and longitudinal stiffeners (or " stringers") are smeared out over the membrane thickness thereby increasing the effective thickness of the membrane and hence its pres-sure capability. It is well known, however, in aircraft structural analysis that in general this cannot be done. The problem is succinctly described in the following extract from " Analysis and Design of Flight Vehicle Structures," E. F. Bruhn, Purdue University, Tri-State Offset Company, 1965. (Library of Congress Card 964-7896). Because of functional requirements over and abcva t hose of a simple pressure vessel, the pressurized cabin shell of an airplane has a number of stress analysis problems peculiar to its configuration. Several of the more general of these will be considered here. To stabilize the shell wall in transmitting heavy tail loads through the fuselage, loi.f +udinal stringers are added. These same stringers will also help to carry the meridional

                                     -162-pressure loads. The skin and stringers must, of co arse, have equal strains in the longitudinal directions but, because the skin is in. a two-dimensional state of stress, they connot         !

have equal longitudinal stresses: hence the following a nalys is . Let the meridional (longitudinal) stresses in the skin and stringers be S and S g, respectively. S will be the 3 t tangential (hoop) stress in the skin. We have t" If N is the total number of stringers, each of cross sectional area A g, then equilibrium longitudinally requires P m R~ = 2n RtSM + ~^L L. The condition of equal longitudinal strain in the skin and stringers yields E c=S =S g -pS t where u is Poisson's ratio (= .27 for steel). Solving these three equations one finds PR t 't g _ PR (1 + 2ua) _ PR (1 + 0.54 .d M 2t (1 + a) 2t (1 + a) g 2 _ 111 (1 - 2 u) , PR 0.46 L 2t (1 + a) 2t (1 + a) where a = NA L/2n Rt is the ratio of total stringer a: ca to skin area. A little study will shcw that t(1 + a) i:: a sert

                                         -163-or " effective shell wall thickness":      it is the result of taking all the cross sectional area (skin plus stringers) and distributing it uniformly around the perimeter. On this basis, the results arc a little disappointing:        the stringers are carrying only 40% of the stress one might expect if the net longitudinal load (P n R ) were distributed evenly over the entire cross sectional area (2 n Rt (1 + a)). Thus the meridional skin stresses are reduced by the factor (1 + .6 a)/

i (1 + a) from what they would be without the strinjers. 1 -

]

Because of the necessity for transmitting various concen-t trated loads from within the cabin and from the wings and tail to the main shell and because it is also necessary to provide some lateral restraint which will stabilize the stringers and skin against an overall instability failure, the pressurized fuselage of an airplane contains a considerable number of rings and frames distributed along the length of the shell. These rings are seldom, if ever, spaced closely enough such that they can be considered ef fective in carrying a part of the hoop stressas (in the way the stringers were effective in carrying part of the meridional stress). Rather, they act more like widely spaced restraining bands having the effect shown exaggerated in Figure 2. Figure 2. Restraining rings along a pressurized tank. The action is r'epresentative of a fuselage with widely spaced rings inside.

                                    --  - ~    _. - _-

k 1 t i U.v.-.v.v.-/ It is obvious that the rings in this case will produce secondary bending stresses in the skin and hence may have a detrimental effect on the simple membrane stress system.

                                                                         -164-equally harmful are the tensile '_o.o.:mjs def eloped in the TiVets joining the skin and rings.                                                      (End,of Extract)

4. STRINGER EFFECTIVENESS 1

Following the method of Section 3 above and Figure 3 illustrates the application of the longitudinal stiffeners to the Sequoyah vessel. In calculating the meridional stresses an " effective" pressure is used, which is the internal pressure of the container less that pressure which is needed to support the structural weight above the section under consideration. Thus, at the c'ritical 1/2 inch plate section (top of the cylinder) a dome weight of about 550,000 lb has to be supported and this is equivalent to an internal pressure of about 0.37 psi, and the internal pressure has to exceed this value before a meridional tension stress can be achieved. At the base the < equivalent pressure to offset the overall weight of the con-ta iner (about 2. 3 million Ib) is 1.54 psi. It is seen from Figure 3 that the stringers are stressed to only about 40% of the amount of the meridional stress in the membrane. Of the total longitudinal load the membrane carries 935 and the stringers only 74. It is therefore clearly incorrect to assume that the stringer cross sectional area can be " smeared" out fully over the membrane - the smearing tech-nique can be used but by using about 400 of the stringer cross sectional area. 5 RING STIFFENER EFFECT The analysis of thin walled cylinders with ring Etiffeners is treated in detail in " Beams on Elastic Foundation" by M.

    !!ctenyi (University of Michigan Press 1946) pages 83-84.

Figure 4 shows the results of this analysis applied to the cylindrical section of the Sequoyah vessel. It is seen that the ring stiffeners have to be spaced very much closer than S0 inches to have any appreciable reduction on the membrane

Figure 3. Stringar Effectiveness e PEr "AliALYSIS MD DESIGN Of FLIGili STRIflGER VElllCLE STRUCTURES" BRlillN t (SKIN THICKNESS) I-- # e __

                                                                                                          }

# TOTAL LOAD

( i z \ l = aR V l h -'k , A i i TOTAL STRiflGER C.S. AREA = A, STRESS = S( H = 690" t = 1/2" SKifi STRESS (MEMBRAtlE): St (1100P) = PR/t, Sg (MERID10flAl.) = 7 P = 12 Psi (EFFECTIEf) i LO!!GITIJDillAL EQUILIBRill!1: mR P = 2rRtSg + ASg A = 90 x 4.75 = 42 7. in' EQillL L0iiGIT11DillAL STRAlti i. = - - (STRillGER) l Sg - pS t ( r,lH) (is = 0.3, P0iS50:15 RATIO) ! _ _" E PR/2L = 821D l 501.il f l0H: Sg = PR 1 + 0.6a 2 't , l' t o . Sg = 7E35 A l

t =l-(-[;i,-

                                                                         *

• 27 RT St = 2767 NOTE: SKIf4 CARRIES 16.8 MILLION LB (931) STRINGERS 1.2 MILLION LB (75) 0F TOTAL LOAD OF 18 MILLION LB.

i Pigure 4. Ring Sti2fener Effect L __ . 5..___ J i i j NL' N s . -

                                     '        i                   t                         t 2R           l'                  p                         p I                   l                         l
                                         -  - - j~ ~ . s _ _             -

_j~.

                                                                                      ~ "BEllD 7.ftr-                     ,,1100P D = 6'/s"                                                                    .
                                                                                                  .;B EtiD

t. = 5/8", 1/2"

                                                                                                                                                  -1.82 ASYMr.

i - M m ! 27.5"(1/2) I 30.7" (5/8) d /

                                  ~

1.0 1.00 ASYMP. WR/tT -- - - ll00P REF. IIETEliVI (l:EAl% O!! El.ASTIC f0VilDAT10ll5)

                                                              /

f s, . 20 40 60 80 100 S(IN)

                                    -167-hoop stress. Further local bending stresses at the attachment to the ring which are greater than the unmodified hoop stress are generated when the ring spacing is in excess of 27.5" to 20.7" (respectively for 1/2" and S/8" plate) .      Since the l

actual design ring spacings are at 10 ft two conclusions may l be drawn:

a. Membrane hoop stresses in a considerable region between the ring stiffeners is for practical purposes not in-fluenced by the ring stiffeners.

b.- A local bending stress at the ring attachment to the shell is induced and this stress is some 80% higher then the simple mambrane hoop stress. Thus, the critical region for hoop stress will be the 1/2 inch plate midway between the two rings. (This occurs between rings at elevations 778.5 and 788.0 shown in Figure 1). This section has the upper 2/3 of 1/2 inch plate and the lower 1/3 of 5/8 inch plate, and hence the mid-section area of criticality is in the 1/2 inch plate). In this case the critical internal pressure may be calculated as follows: yield stress (= 32,000 psi) = PR/t (R = 690 in., t = 1/2 in.) giving P = 23.2 psi This corresponds to the Boiler Code Max Shear Stress criterion for yield. If ultimate strength is used then tl}is pressure would be scaled up in the ratio of ultimate to yield stresses (60,000 to 32,000 psi) giving a value of 47.'5 psi. The corresponding longitudinal stress would be half the hoop stress in a simple unstiffened cylinder. As shown in Section 4, the membrane longitudinal stress is reduced by a factor of 0.S7 due to the presence of the stringers. An alternative method to the minimum shear stress method of the Boiler Code is to use Von Mises criteria which determines the critical

                                     -168-slfess as a function of botn the hoop stress le I and the               l longitudinal or meridional stress (.j) .          This is given by:

1 1 c

• N +# -

crit M t Mt In this case og = 0.5 x 0.870 = 0.435 a t Hence c . = 0.858a t crit Hence, for the von Mises criteria the critical pressures corresponding to yield and ultinate stresses are respectively 26.8 and 50.3 psi. 6 ALTERNATIVE PANEL ANALYSES An alternative approach, in order to determine local stress regions induced by the rings and stringers, is to consider the cylinder to be a number of rectangular panels framed by ring sectors and stringer sections as shown in Figure 5. Thus, . the cylinder is composed of a number of panels approximately 4 ft by 10 ft as shown with thicknesses varying from 1/2 in. to 1 3/8 in. A comparison of the bending stiffness of the panel and the rings and stringers is shcwn in Figure 5. The cross sectional moment of inertia about the bending axis is a measure of the stiffness of a beam. In the case of a panel bending as a beam there is an additional term due to a Poisson's Ratio (a) contribution. This is, however, only a 10% effect 2 (proportional to 1 - p , and L = 0.27) and is neglected in calculating the moment of inertia of the panel. From Figure 5 it is seen that in bending about the XX axis the stringers are over twenty ::mes as stiff as the skin, even though the skin is curved acrcss the bending axis thereby in-creasing its effective moment of inertia by some 50%. For bending about the longitudinal axis YY the relative stiffness '

1 Figure 5. Panel Arrangement e tiOT 70 SCALE t Y/ o = 0.4 2"

                                                                                                 #            ./ L = 0.629 p g gir,                                                                          t    -

r -- ( 18a'x 1 1/4a) ;ST _. _ z__ /fd}-/ / j li f -

                                                                                                                 =U x/"
                  /                                                             I STRIi;GER     = 35.73 Ir:4 I

( )( R)

                   ,               (91/2' x 1/2")

S.5.C_][,X_, ,X, i PAffEL 5'

                        .           ('N 4 ' x 10 '

t = 1/2" - 1 3/8") , t = . r,? !, -

             /         N
                                                                                                ,.['                           _ _ .

N/x Ig _. i 120" = L_j 4 E Iggp;g = 607.5 IN I SKlil

• 2 44 Ill

           /   [

Y iiOTE: 1-u TEGLECTED (10". EFFECT) SEC't1 y-y

                                                        -170-i is even higher (about 250 to 1).                          The analysis of Figure 5 i       were carried out for a 5/8 inch thick skin. The relative stiffness will be even higher for a 1/2 inch thick skin since 3

the skin moment of inertia involves a t term. It is clear from these considerations that an analysis of I the skin as a panel held rigidly at the boundaries should be made (i.e., encastre edges). The legitimacy of this encastr'e l assumption is strengthened when one considers that adjacent panels help in keeping the ring and stringer edges from twisting. For example, symmetry in the cross section across a stringer i in the XX direction ensures that the stringer cannot twist for panel bending in about the YY axis. Two flat plate analysis have been carried out following the methods of " Formulas For Stress and Strain" - R. J. Roark, 5th Edition McGraw-Hill Book Co. (1975), pages 392 and 408.

a. Simple flat plate analysis This is presented in Figure 6. For an encastr'e edged plate Table Sa on page 392 of the reference volume gives a value for the maximum bending stress at A & B (the midpoints of the long 2 2 sides) as a = 0.5 Pb /t . For the plate under consideration l this gives initial stresses for yielding at a pressure of 6.94

psi. At this pressure the inner plate fibers at A & B will ,

I just begin to yield in tension, and the outer plate fibers in these locations will be compressed to a stress of 32,000 psi. At a value of about 1.5 ti: es this pressure (or 10.4 psi) ' yielding will occur through the entire plate section at A & B. (This is known as a " plastic hinge"). Ultimate yielding stresses of the surface fibers at A & B will be reached at a pressure of 13.0 psi. The table, re ferenced above, shows that the stress at the midpoint of the pla te (C in Figure 6) is half that occurring at A & B, and is in the opposite sense (i.e., tensile on the

outside, compression on the inside). However the plate is not c- . .- . - - _ _ _ _ .. - . _ , - _ - . . - - - - - _ . - . ._ . ---

I Figure 6. Plat Plate Analysis e REi: H0 ARK "FORi4ULAS FOR STRESS Afil) STRAlii" vAiit .N ,' MAX. n. IAT A, B) [b=48" I , b = 48"

                             =       -                               , .0 5. .P.b. 2.

(t=1/2") t i FOR s = 32,000 PSI o = 60,000 PSI P = 6.94 PSI P = 13.0 PSI CErlif R s p' = 0.5 9,4 ,

                 =

o C Z B FOR o = 32,000 PSI o = 60,000 psi

                  ,,       .,A    .C      o P = 13.9 PSI    P = 26.0 ft0TE:    FULL PLASTIC llINGE DEVELOPS AT A, B
               ...                                                AT 1.5 x 6.9 = 10.4 PSI

                                  -172-a truly " fla t" plate and the inalysis of Sechion 4 is mo re appropriate to the center of the plate Which is mainly subject to the hoop tension. There would undoubtedly be some coupl.-

co:sination of bending stresnes due to the ring and stringer constraints coupled with the hoop and meridional membrane streuses. A careful analysis with a finite element code would be required to resolve this point and this is beyond the scope of this review.

b. Large deflection plate analysis (" quilting" ef fect)

The analysis of (a) assumes a flat plate and makes no allow-ance for the finite deflections of the plate. The formula of page 408 of the referenced work makes allowance for the plate deflection. These results are summarized in Figure 7. Again

 .axicum stresses occur at the midpoints of the long sides.

The resulting stress is a combina tion of bending and membrane stnesses. Yielding (at 32,000 psi stress) of the inner fibers at A & B begins at an internal pressure of 7.8 psi. Only 6 1/2% of the total stress is due to the membrane contribution.

c. Comments on the maximum stress loading at A & B The onset of yield could occur at the inner fibers at the mid-points of the long edges of the half inch plate sections at an internal pressure of 7.8 psi, assuming the more realistic

" quilting" analysis. However, this is at local points only and full plastic hinging would not occur until about ll.7, psi. Even then local stress relief might well occur and for a "one-shot" prcssurization it is not clear whether thTs would result in leakage. It would be a serious problem if many cycles of pressurination were encountered when cracking due to "LCF" (low cycle fatigue) might well eccur. More serious, however, is the pure membrane stress induced in the 1/2 inch skin at 26.8 psi. This is a ransion ever the whole cross section of the panel and wculd occur over several inches of the vertical panel centerline.

1 i Figure 7. Large Dt?flection Plat Pla te Analys i:, i (" Quilting Ef fect") l

. -- - - 18 1 43". _ . .-.s e REF: ROARE "F04MOLAS FOR STRESS AND STRAlif' SLti Lilliioli, 4

plait ilif f.rt;E',5 = L 4 b t/2 [ _ t/2 STRESS FIELD--COMBINED TENSION AhD DENDIhG a = 120" - - o0= DIAMAGM (MEMBRANO STRESS AT A AND D l o = TOTAL STRESS BEllDING AND DIAPitRAGM AT A AilD D i Y;AX = OUT OF PLAfiC PLATC DEFLECTI0il AT C y 4 4 Y/t = Fj (Pb /Et ) \ I DEFLECTI0il AtiD 2 4 4 ogb /Et2 = F2 (Pb /Et ) STRESS COEFFICIErlTS v E = 28 x 106 PSI 2 4 4 ob /Et 2=F 3 tPb /Et ) J t = 1/2" t = S/8" I o PSI 32,000 60,000 32,000 60,000 "D PSI 2100 5000 1500 4100  ! 1 ( P PSI 7.8 15.9 11.8 23.2 j j v 33x itis .24 .40 .20 .35 1 ' I l

I i

                                ~174-

7. liOLD DO'WN BOLT STRESSES i

Figure 8 depicts the tension stress in the hold down bolts as the internal pressure is increased. The bolts are pre-stressed to a level of 25,000 psi and this bolt tension is not increased until the internal pressure overcomes the container weight as well as the preload tension. This occurs at an internal pressure of 17.3 psi. Increasing pressure will produce bolt yield stress at 64.5 psi and the ultimate bolt stress of 125,000 psi would be reached at an internal pressure of 77.1 psi. 'The latter, however, could not realistically be achieved since gross leakage would occur as soon as the bolts yield.

8.

SUMMARY

OF STRESS ANALYSES, CONCLUSIONS AND RECOMMENDATIONS-- Figure 9 summarizes the stress analyses described above to-gether with the AMES " smeared" shell/ stiffener analyses of SECY-80-107A. The RDA analysis leads to the following conclusions.

a. The AMES analysis is optimistic.

1. The ring stiffeners are not amenable to the smearing technique--the spacing is such that the hoop stress in the mid-region between the rings is essentially unaffected.

2. The stringers are only partially amenable to smearing--the stringers only carry 40% of the Toad that would be expected with " equal" area effective-ness between membrane and stringers.

3. Having " smeared out" the rings and stringers they cannot be put back in to carry load. This leads to the rather surprising case of one of the rings being the " weak" element in the system.

4. The ultimate burst analysis is clearly incorrect--

the hold down bolts would yield first.

Lg T A L I O D D

                                       /  I      I
                                 "     D  S      S 8 A      P      P 5 O
                                   . L    0      0 2 E       0      0 R

S P 0 0, i 0 C T 5 5 8 L B 0 2 O L 1 1 s B K e 0

                                           =      =

s 8 3 1 y s U e 1 1 o ' r t e

• e S

t l o D 0 i6 n w o ) D I S ) P I d S l 3 P l o 7 ( I 1 P ( L . T A 8 l i i l G ) L0 R e I E I S 4 E T r W ) P l u I l I g T S 1 i N P . F E M 7 5 7 N ( I 4 A 6 o T ( N E O o T C A D M

           +     L   I E   T                                                 0 D     I   L                                              I2 A     Y   U O                                                  0 L     T   T E     L   L R     O   O P     B   B D
             -     -    -                                 A S O P A     B   C                                    L K E 5 R

P 2

         -     ~

0 2 0 0 0 0 0 - 1 8 6 4 2 3gghN>e

Figure 9. Sequoyah Containtment Vessel - Summary of S tresses r------------------- - - - - - - - - - - - --- -- ----- --- g g ggg g EEiiiOD , ;f STEESSES YlELD STRESS ULI. STRESS , PSI IW i y AMES "5MEARED" } RlilG STIFFEilER PURE MEE3RAtlE 1 35.G 1 G6.7 SilELL/STIFFEllER ANALYSIS 2 5/8" SKIN (VON MISES) 2 38 G 2 72.Il  : 2 RDA SilELL/STlFFENER PURE MEMBRAilE ' 1/2" SKIN 23.2/ 2G.8 ANALYSIS (BOILER CODE-MAX. 43.5/50.'3 SilEAR STRESS / VON MISES) 1

                                    -       . . . .                   .     .              _.          . _ . _. ._.. ... _ . .. _ _.__      _ _ . . . ._    _i 3

PDA FLAT Pl. ATE 1/2" skill PURE BEilDitlG 1 A) G.9 1 13.0 i A!!T\ LYSIS 1 MIDDLE OF LONG- (BOILER CODE-MAX SilEAi< B)10.4  ; , (EilCASTRE' EDGES) STRESS) EDGE 2 A)13.9 2 2G.0 ,E i 2 CENTER OF A) YlELD AT MAX FIBER B)20.85 i PLATE B) FULL PLASTIC lilNGE q RDA LARGE DEFLECTION 1/2" SKIN COMBINED BENDING AND g) 7,3 i FLAT PLATE AtlALYSIS MIDDLE OF LONG- TENSION (BOILER CODE- 15.9  : (ENCASTRE' EDGES) EDGE MAX SHEAR STRESS) B) 11.7 - A) YlELD AT MAX FIBER  ! B) FULL PLASTIC HINGE MATERIAL ASME SA 516 GRADE 60 o YlELD STRESS 32,000 PSI o ULTIMATE STRESS 60,000 PSI o E 28 X 106 PSI a 70'F

i l l -177-

b. 7he above four conclusions answer the first three tasks of the work statement. A preliminary answer to the fourth task--the question of leakage above the design point is given by the following summary of the panel / membrane analysis.

Recommendations are also presented to refine these answers.

1. Onset of local yielding could occur at about 8 psi, but this is not considered a problem since local yielding could lead to stress relief. Full plas-tic hinging would not theoretically occur until 12' psi. This could Icad to local cracking for a repeated pressurizing case (low cycle fatigue) but may not be important for a "one-shot" loading.

2. Gross membrane yielding could occur at about 27 psi.

This corresponds to the ASME code value of 23 psi limit loading. It is interesting to note that an elastic-platic analysis carried out by Sandia gives a nominal failure pressure of 27 + 3 psi. It appears from this simplified analysis that the progres-sion of events with increasing pressure, begins with pure bending resistance and small local clastic fiber deformations , and progresses through combined bending and tensile resistance (quilting) with larger elastic deformations. Eventually local zones of plastic yielding will culminate in a state such that the final resistance mode is pure membrane tension in the skin material alone. This final state will only occur if the skin material is sufficiently ductile to avoid local ruptg,re by tearing or cracking with the internal bending resistance nullified by yielding. Furthermore this final state will be reached independently of the properties of the stiffeners e

        " Report On Systems Analysis Task, Reactor Safety Study Methodology ApplicationsProgram, Sequoyah #1 Power Plant,"

Draft Report 1978, Asselin, Carlson, Gramond, Hickman, Fedele, Cybulskis and Wooton.

                                                                     -178-
            '   .-   n o to in:iai tely stif f) so long is the spacing raf the ring st i f feners is greater than about 60 inches for the 1/2 inch plate. The final state would then be pure membrane resistance with an equivalent longitudinal thickness which includes the partial effect of longitudinal stif feners and wi th hoop thickness equal to the unmodified plate thickness.

The resulting limit load pressure about 27 psi is thus probably a reasonable estimate of failure onset. The structure may fail locally below this value but will probably not survive much above thrs value whatever the properties of stiffeners as currently spaced. Eased on these analyses and conclusions it is recommended that further analyses and experimental verification be carried out:

a. A detailed finite element code analysis should be carried out to clarify the location, extent and profile of stress concentrations.

b. A full scale excastre panel should be pressurized to failure including a full strain gage and stress coat instrumentation. This would not be difficult or expensive since the panel size is only 10 f t by 4 ft, and the severity and effect of the local stresa concentrations could be readily evaluated. The pres-surination should be carried out in two stages.

1. Up to 13.5 psi and back to :ero (to simulate

, the containment acceptance pressure test +. The panel should then be examined carefully for local de f o rma tion s, etc. These weuld likely be shown up by stress coat or crack detection methods.

2. Pressurization to failure with full instrumentation readino at selected pressure increments.

i

r. _ _ . . _ . . . _. . , _ _ _ . _ _ _ _ . _ _ _ . _ - _ _ _ _ , _ _ _ . _ _ _ - . _ _ . .__ -

i

                                 -179-                         ,

REFERENCE C TVA, oral presentation at ACRS meeting, September 2, 1980

                                  -180-PRESENTATION OUTLINE I. OVERVIEW OF PRESENTATION

11. DESIGN OF CONTAINMENT VESSELS ORIGINAL DESIGN -

REVISED DESIGN - DESIGN / CONSTRUCTION PROCEDURES DESCRIPTION OF CONTAINMENT 111. CONTAINMENT CAPACITIES - CRITICAL SECTIONS EVALUATION OF THE CRITICAL SECTION R ESU LTS IV. CONCLUSIONS & RECOMMENDATIONS -

481-E/ 6/53 El 809.5 El00?7 E/ 799.0 == El 79S 0 ab El 7S/5 ab El 786.0 = El 778.5 = El 769.0 = El 759.5 = EQUIPMENT NATCH El 750./ = t 0 740.5 = 1 l El 730.3 a: El 72/5 m PERSC//NEL El 7/3.5 = LOCKS El 70/.5 = El 69/./ d-E/ 679.70 Steel Contain:nent Vessel

                          ~182-l DETERMINATION OF CRITICAL SECTIONS ANCHORAGE PEN ETR ATIONS (PIPING)

BELLOWS VALVES PERSONNEL LOCKS / EQUIPMENT HATCH SEALS SHE LL PLATE

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                                          -190-l EVALUATION OF CONTAINMENT VESSEL BETWEEN                l STIFFENERS AT ELEVATIONS 778'-6" AND 778' MATE RI AL -

PLATE- SA 516 OR 60 STI F F ENE RS - SA 516 &R 60 WELD- E7018 MATERI AL PROPERTIES - STRESSES YlELD TENSlLE SHELL SPECIFIED CODE MINIMUM 32 KSI 60 KSI PLATE & < LOWEST TEST VALUE 45.7 KSI 65.'.KSI STI F F EN E RS 47.2 KSI 66.2 KSI i MEAN TEST VALUE WELD SPECIFIED CODE MINIMUM 60 KSI 72 KSI

                                 -191-3_A\

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               -193-METHODS OF ANALYSES FINITE ELEMENT SHELL MODEL PANEL MEMBRANE FAILURE CRITERI A MAXIMUM SHEAR STRESS VON MISES

                                                       ~~

CO,\ ~~A .\ V E \ 3 HESS AE PS'G) SPECIFIED CODE MIN. LOWEST ACTUAL CRITICAL CRITICAL SECTIONS PRESSURE MAX SHEAR VON MISES M,'AX SHEAR VON MISEE SHELL PLATE utTiuATE 43.5 50.2 '47. - 5 YlELD 23.2 26.8 33.1 38.2 PENETRATIONS WELDED SPARE YlEl.D 83.0

                                                                  ~

SOLTED HEAD YlELD 1355.0 BE LLOWS YlELD 10 0.9 , 2 ELECTRICALS ULTIMATE 10 0.0 VACUUM RELIEF VALVES a m ATE 47.8 PERSONNEL LOCK YlELD 3l.l 'EOUIPMENT HATCH ULTIMATE 73.0

                                                                               =

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                                   -195-l REFERENCE D Bagchi, G. (NRC Research), Memorandum to F.P. Schauer, NRC, Washington, DC, August 17, 1980.

                                                         -196-
    , , n e r, ,                                UNITED ST Af f s

.; 'g NUCLE AR REGUL ATORY COMMisslot.

                    '                        WA$HINGTON. O C 20555 q \M./l
  .                                          AUG 8 71980 I;Ett0RAllDUM FOR:      F. P. Schauer, Chief Civil Engineering Branch FR0fi:                  Goutam Dagchi, Chief Structural Engineering Research Branch

SUBJECT:

AliALYSIS OF SEQUOYAH C0flTAINf1EllT CAPACITY I reviewed two separate analyses of the Sequoyah Containment Structure referenced below: (1) Analysis by Ames as a part of SECY-80-107A (2) Critique of SECY-80-107A by R&D Associates I feel that the critique in Reference 2 above treated the effects of meridional stiffeners independently of the ring stiffeners and vice versa. The network of stiffeners should serve as another strength ele-ment to provide resistance to the shell membrane beyond its first yield capacity. I tried to take an independent look at this and developed a force equilibrium nodel to utilize both the ring and meridional stiffeners. Enclosed is a copy of calculations for your review. I hope you will find thea useful for your safety evaluation. liy conclusions are that the containment strength is 34 psi at gross yield and is governed by the thinnest section. This value is closer to that calculated in Reference (1), 35.5 psi than the 27 psi estimated in Reference (2). It is my opinion that tne ultimate capacity of the containment is around 45 to 50 psi internal pressure, considerably higher than the strength at gross yield.

                                                       ]          g       m Goutam Bagchi, Chief Structural Engineering Research Branch Division of Reactor Safety Reseech

Enclosure:

Calculations cc: T. E. Murley, RES L. C. Shao, RES D. G. Eisenhut, NRR J. P. Knight, NRR C. N. Kelber, RES J. F. Costello, RES C. P. Siess, ACRS M. Bender, ACRS

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                                              -198-

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                                 -204-REFERENCE E Zudans, Z. (Franklin Research Institute), 1ctter report to Dr. R. Savio, NRC, Washington, DC, August 29, 1980

205- -A

.00. Franklin Research Center A Division of The Franklin Institute August 29, 1980 Dr. R. Savio Staff Engineer Advisory Committee on Reactor Safeguards U.S. Nuclear Regulatory Commission Washington, D.C. 20555

References:

1) Dr. L. Greimann, Ames Laboratory: Ultimate Strength Characteristics of the Sequoyah and McGuize Containments, January 21, 1980.

2) R&D Associates Report: "Sequoyah Containment Analysis,"

July 25, 1980. Subj ect: Review of Sequoyah Containment Structural Analyses by Ames and R&D Associates and an Independent Analyses of a Portions of Sequoyah Containment.

Dear Dr. Savio:

As per your instructions I reviewed the analyses and conclusions reached in subject referenced Reports. Detailed findings of this review are given in the Enclosures 3 and 4. In summary neither of the two reports support their conclusions with argtsnenst of adequate rigor. Ames conclusion (for 5/8" thick section) that plastic limit load is reached at p = 35.6 psi is derived from an erroneous assumption which results in ring yielding at this prensure! Similarly, R&D Associates conclusion that full membrane hoop stress will develop in a considerable region of 1/2" section between the rings at p = 23.2 psi is based on neglecting the effect of the stringers. To offer a rational estimate of the containment strength, I performed four (4) independent analyses using a reduced model of the Sequoyah containment. The first three (3) analyses addressed the 5/8" thick section (considered critical by Ames.). The conclusion based on these three (3) analyses, indicates that: 5/8" thick shell section full yielding vill occur at 34.3 psi, and that essentially an entire panel _(between the stringers and stiffeners) vill yield at 38.6 pai. These are conservative numbers, non-linearities will stiffen the structure and higher pressures will likely be required to produce gross distortion. The Benjamin Franklin Parkway Philadelphia, Pa.19103 (215)448-1000 TWX 710 6701889

                                                              -206-Dr. R. Savio                                                            Augu st 29, 1980 ACRS 4

The fourth analysis addressed the area of 1/2" thick shell (considered critical by R&D Associates) . The conclusion based on the fourth analysis l indicates that the first yielding of 1/2" thick section will occur at the I point where 1/2" thick shell joins the 5/8" thick shell at 30.3 psi, and that essentially the entire panel between elevations 778' 0-5/8" and 691' 0-1/2" will yield at 34.7 psi. Details of four (4) analyses are given in the Enclosures 1 through 5. Briefly, these analyses were performed in the following manner. First analyses modelled a portion of the containment building between Elev. 730' to 769' with circumferential ring stiffeners, but peglected the meridional stiffeners (artineergi). The reason for this analysis was to demonstrate the response of a shell without meridional stiffeners. Results of this analysis confirmed the fact that given the spacing of the rings as per Sequoyah design in this area, full hoop stresses would develop in the region couple feet away from the ring and cause membrane yielding of 0.625 in. thick shell at about 28 psi. The stresses for this case are shown in Figure 2, Enclosure 1. It is also clear that plastic hinges would develop in the shell at a considerable lower pressure. The second analysis was performed (for the same area as the first analysis) with added stiffeners in the model. While the rings were modelled exactly within the linear elastic theory, the stringers were smeared out to represent their axial stiffness and meridional bending stittness in an average manner. Pertinent details of this analysis are given in Enclosure 1. Figure 3 Enclosure 1 shows the hoop and axial stress distribution for this case. .It is noted that the hoop stress is much more uniform than in the previous case, Figure 2, Enclosure 1, and that the average axial stress varies along the meridian. The largest hoop stress in this case predicts total cross section plasticity at 36 psi (see Page 6, Enclosure 1) . The results further indicate that the entire shell section between the elevation 740' to 759' would yield with the internal pressure loading in the range from 36 to 38 psi. The third analysis was performed to prove the validity of the method used for " smearing out" meridional stif fener. This was a finite element analysis of a portion of the containment between El. 744' and 755'. Here, symmetry boundary conditions were imposed on all sides of the model and axial loads applied at one end of the model such that the end remained flat. Details of this analysis are given in Enclosure 2. This analysis confirmed the shell of revolution analysis results as obtained with rings and smeared out stiffeners. It further showed that hoop stresses generated in the midspan between the rings and stringers are slightly higher than those produced by shell of revolution analysis. However, the basic finding that the stringers are significant in reducing the hoop stress remained. The results of this analysis indicate that a gross shell yielding at this location will occur over essentially the entire span of this model between rings and stiffeners in the range of the internal pressure 34.3 to 38.6 psi.

l l

                                                      -207-l l           Dr. R. Savio                                               August 29, 1980 ACRS Another interesting result is that essentially full axial stress is developed in the shell at the rings since the stringers at that location contribute very little to axial stiffness due to significant bending strest developed in the stringers. Figure 5 of Enclosure 2 shows this case.

Fourth analysis was performed for the region between elevations 778' to 791' by using shell of revolution method with rings and smeared out meridional stiffeners. Pertinent details of this analysis are given in Enclosure 3. Figure 7, Enclosure 5 shows hoop and axial stress distribution for this case. The largest hoop stress in this case predicts total cross section plasticity at 31.9 psi (see Page 5, Enclosure 5). The results further indicate that the entire section between the ring at elevations 778' 0-5/8" and 791' 0-1/2" will yield at 34.7 psi. There was no finite element analysis done for this region, however, the same amount of stress change (from shell to finite element) can be anticipated here as was found in comparing the second and third analysis. Accordingly, a reasonable plastic limit load in this region is 30.3 pai. Because of the structural discontinuities, the local stress at the shell surf ace vary significantly from the average stress upon which gross plasticity pressure was derived. However, formation of plastic hinges locally is not a significant contributor to failure for a one time loading. To define a pressure at which the structure would reach its ultimate capacity, it is first necessary to select the mode of failure of concern. If the leakage is the concern, R&D Associates computed value of p = 64.5 psi (producing hold down bolt yielding) is a reasonable value. Other design details around the penetrations and at discontinuities, however, may produce leakage path at a pressure lower than 64.5 psi. The ultimate structural capacity of the Sequoyah is strongly dependent on as built condition of the specific details and requires nonlinear inelastic analysis under consideration of strain hardening and strain rate effects. Generally, however, dynamic structural capability exceeds the static capability, in particular, if the loading is of impulsive type with short load application time as compared to the length of the lower natural period of the containment. l Also previous analyses of pressure vessel closures indicate that the collapse 1 pressure (defined collapse pressure in ASME Code is equal to the load which ! produces deformation twice that of the elastic deformation at the same load) usually occurs at pressures in excess of the pressure to produce first plasticity in a cross section.

                                                           -208-Dr. R. Savio                                                                       August 29, 1980 ACRS Accordingly, the computed internal pressure for plastic limit at p = 30.3 psi can be considered a reasonable lower limit.

Very truly yours, enons Zudans ces enior Vice President, Engineering encls. t

          , _ -               . - - . . - . -, ~ . -               .,          -                 . .-- -
                                                                                                                ,.-.-e ,

                                         -209-Enclosure 1 SHELL OF REVOLUTION ANALYSIS FOR EL. 730' TO 769' REGION Shell of revolution model representing portion of containment between A-A  and B-B, Figure 1 was made. It was assumed that the meridional rotation was zero at Sections A-A and B-B to simulate the fact that, due to approximately uniform distance between the rings, such rotation would be zero in a full containment model. This is deemed to be a good assumption. At A-A it was further assumed that the shell was fixed axially and free to expand radially.

At B-B, axial force per unit length of meridian, equal to the end closure pressure loading of 20 psi was applied, and free radial expansion allowed. The rings were modelled as circular plates, hence represented exactly within the linear theory of shells. The meridional stiffeners (stringers) were included in one analysis, such that their contribution to meridional bending and axial stiffness is correctly represented. For another analysis these meridional stiffeners were ignored. Analysis with rings and meridional stiffeners produced stresses shown in Figure 3. Analysis with rings, but without meridional stiffeners produced stresses shown in Figure 2. As anticipated, if stiffeners are neglected, full membrane stress is developed some distance away from the rings. This is due to the f act that distance between the rings is in the order of 5 to 7 times the characteristic length of the cylinder, i.e., edge effects at the ring do not propagate in the shell. Full yield would develop at locations between the rings at a pressure

                                     ,32,000t For locations shown in Figure 3 P " 22 851 x 32,000 2 28 pai

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                                        -220-Enclosure 3 Comments to:

Dr. L. Creimann, Ames Lab. Analysis:

 "Ultimatimate Strength Characteristics of the Sequoyah and McGuire Containments," January 21, 1980.

1. Assumption of " smeared out rings" by use of an equivalent cir-circumferential thickness t

0 " * + Ar/8r is not conservative since the distance between the rings range from 4 to 7 characteristics length of the cylinder. All edge effects (at the ring) will decay completely towards midspan between the rings and full hoop , stress will develop as if the rings were not there. This is confirmed in Figure 2 of the Shell of Revolution Analysis, Page 3, Enclosure 1.

2. Assumption of 'bmeared ouf meridional stiffeners (stringers) is a common practice of the industry when shell of revolution type of the analysis is used. However, a correct " smearing out" method considers the bending stiffness of the stringer in addition to the axial stiffness.

Since the bending stiffness of stringers is significant as compared to the shell bending stiffness, stringers transfer loads to rings and if properly spaced.can reduce the hoop stress in the shell considerably. Typical " smearing out method" normally used in shell of revolution analysis is shown in Pages 5 to 7 of Attachment 1, Ames method not acceptable.

3. Assumptions used in Ames report will not yield a reliable burst pressure. Burst pressure depends strongly on load history, fracture characteristics of welds, local geometry (as built), materials hardening parameters, all of which can be considered (approximately) by use of a large deformation, inelastic analysis or by testing.

i l l l

                                         -221-

4. Calculations of limit pressure for stringers and ring stiffeners as described in the report are meaningless. Accordingly, the conclusion, that the ring stiffeners will yield first (at 35.57 psi) is not realistic.

Hoop stresses in ring stiffeners are considerably lower than those in the shell (see Attachments 1 and 2).

5. Penetration replacement area calculation is not in compliance with the ASME Code. Paragraph NB-3334.2 (Page 73- 1980 edition) requires that the reinforcing normal to the shell vall shall be within 0.5 /121.5(3) = 9.5 in.

of the wall and 75% of the reinforcing must be located at most (NB-3334.1) 0.5 /690(0.75) = 11.37 in. from the finished diameter of the equipment hatch. Using these limits we find that the available reinforcing is only 2 2 x (9.5x3) = 57.00 in 2(1.5-0.75) 11.37 - 17.01 74.01 in < 135 (75% of 180) Accordingly, the conclusion reached in the report is not substantiated.

                                            -222-Enclosure 4 Comments to:

R&D Associates Report: "Sequoyah Containment Analysis," 25 July 1980. R&D report presents an interesting discussion of stringer and skin interaction (Page 6). The results would be exact if: 1) full hoop stress a =.p R/t can be developed and if one deals with an infinite cylinder without king stiffeners. As shown in Enclosure 2, rings and stringers will cause significant reduction of hoop stresses and, due to bending of stringers, axial stress S in the stringer will vary considerably over the span between the rings. Also, the presence of stringers reduces the - hoop stress in the shell at midspan between the rings. Accordingly, the results of the Page 6 analysis, while correct for conditions assumed, are not realistic for determination of the plastic limit load. Alternate panel analysis, Pages 12 to 17, is based on the assumption of rigid boundary and flat panel. These assumptions are totally inappropriate here, because we deal with curved shell and all boundaries are able to deflect. Accordingly, conclusion based on this analysis are not applicable to subject containment. Applicable ring, stiffener and shell interaction is presented in Enclosure 2. Bolt yield pressure of 64.5 psi, found by R&D is acceptable to this reviewer. i t l

                                                     -223-Enclosure 5 SHELL REVOLUTION ANALYSIS FOR EL. 778' TO 798' Shell of revolution model representing the portion of the containment, i              Figure 6, was used for the analysis. All basic assumptions used in the second analysis, Enclosure 1, were applied here as well, including the method of " smearing-out" the meridional stiffeners.

Qualitative results for this analysis are similar to those for the

  ;           second analysis. The stress distribution in the shell is shown in Figure 7.

The highest hoop membrane stress of 20,039 psi is found at the junction of 5/8" and 1/2" shell sections. Its increase over that found in the second analysis is due to decrease in shell thickness and also due to the reduction of the ring size. This hoop stress would cause yield at p = 20 039 (32,000) = 31.9 psi This pressure exceeds that predicted by R&D Associates and it reflects the positive influence offered by the meridional stiffeners and rings in reducing the hoop stress from 27,600 psi to 20,039 psi. The membrane hoop stress in the span between elevations 778' 0-5/8" to 788' 0-5/8" varies between 20,039 psi and 17,390 psi. This entire section will yield in the range of pressures 31.9 psi to 36.8 psi. There was no finite element analysis performed for this model. Based on the findings when comparing the second and third analysis, it is reasonable to assume that the lowest pressure required to yield the junction of 5/8" and 1/2" sections will be I

                                               = 30.3 psi (31.9)f6.1 The model, Figure 6, extends to the spring line between cylinder and the sphere. It is anticipated that the discontinuity effects of the sphere      l l

will not affect adversely the critical section identified in this analysis, , in particular since the transition section is strongly reinforced with rings. l

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                      %                             Y 7                                   .

Figure 6. Shell of Revolution Model for Sequoyah Containment, 1/2" Section 1 1

s s s s ss s e se e r er r t rt t S tS S S p l a 0 l! rot 1 t 'q

                                                                  ! lI f ,'                                  5 oo        f       x                                               2 oH          x   A H         A                                                              d e            e                                                        e ed          e   d                                                          z di        d     i y

is i s l st s t a nu n u n I0 I O A

                                                ~              -

7 n s 0 o I ,I

                                            '                                                                0      i t
                                          \                 ' _i'                                            2        c s                           _                                                 e s                                                                             S
                                    \                               ~                                               l l

_s -

                                                                   -                                                h e
                                ,.                                                                                  S S 2
            ,Il               i                                          f   -

0 E 5 H C

                                                                                                                    /

1 t j 1 n s s' . N e m I n

                                                                                 ._                                 i x                                                                      ..                                t a

n _ ~ o

         ~                                                                                                          C

_ h IJ

         -                 l      I
                                                                               ?
                                                                                ~
                                                                                 ~N              _-

0 0 1 a o y u

                -                                                                              ~

q e

                  -                                                                           ~                     S n

i

                      -                                                                  ,'                           s e

s

                       -                                                                ~                             s e
                                                                                         .                     0      r g                                                                    .,

5 t S

                                                                                             '                       7 e

r u g i _ - F

                         -                                                                     2                0

_ _ - _ __ _ __~ _ _~ ~_ - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 5 0 5 0 5 2 0 7 5 2 0 7 I 2 2 I 1 I p y W'

                               -226-REFERENCE F Orr, R. (Offshore Power Systems), oral presentation at ACRS meeting, September 2, 1980.

                                               -227-RING
                                       #                       (16" X 1 1/4~)   .

R=690" (B) (A)

                                  \s STRINGER 7     IS 1/2" X 1/2")

I PANEL g - g o, (C) y 7 (4' x & -gn t=1/2') (D) 1 l ll I ( ) LOCATION I i I l I N

               !                l i                i e               . i
                .               n    l:.

i i i f

                                      '                  ISOMETRIC VIEri OF 4HE IDEALIzEc pgngt

1 i 1 -228-i (B) (A)

                          \

l l l i J i ( ) LOCATION I , .I

                        !                 (C)

;i

h

                        !     l!     ISOMETRIC VIEW OF

, 1 A QUARTER PANE. . I

                         !    ii (D)'Q

                                         -229-B-B 561 563 565 567 569 551 N/N/          j       SSS L

9.5" 101 103 105 107 109 g RING A-A 401 Z=57.0_101 103 105 107 109 551 251 *01 91 ~ 91 99 541'/\ 531

           /       81            81
                                     /N/N      :           89

"i\/ \ N/ '"ax/

                                      /N,\ /.L_      4 g;;,me 501
     \ /\.

4  : 51 51 N/N/ 59

     / \/                             / -\/\

481

     \ \

1 31 STRINGER 31

                                       \             '

39

     /\'/

471 s  : 21 21

                                      / -\ /'N  ;           29
       \/ \                           N/ 'NJ/

451 1 Z=0.0 - 301 151 1 3 5 7 9 i 9.5 _ e'= 0- e=2= A FINITE ELEMENT MCCEL OF THE QUARTER PANEL

                                          -230-l STRESSES DUE TO 12 PSI INTERNAL PRESSURE LOAD STRESS    VON MEMBER             LOCATION                FIBER        INTENSITY MISES (ksi)   (ksi) 1/2" PANEL PLATE    CENTRAL REGION      OUTER SURFACE     18.3    15.9 0F THE PANEL        MID SURFACE       14.0    12.1 (C)                  INNER SURFACE      9.6    8.4 NEAR MID SPAN        OUTERSURFACI     10.5     9.3 0F THE RING         MID SURFACE       12.0    10.5 (B)                  INNER SURFACE    13.6    12.8 NEAR MID SPAN        OUTER SURFACE      8.6    7.5 0F THE STRINGER      MID SURFACE       14.1   12.2 (D)                  INNER SURFACE    20.2    17.6

                              -231-STRESSES DUE TO 12 PSI INTERNAL PRESSURE LOAD STRESS    VON '

MEM EP. LOCATION FIBER INTENSITY MISES (ksi) (ksi) RING MID SPAN NEAR INNER 9.2 9.2 (B) EDGE NEAR OUTER 9.5 9.4 EDGE RING-STRINGER NEAR INNER 9.3 9.3 JUNCTION EDGE (A) NEAR OUTER 9.5 9.1 EDGE

                                                   -232-STRESSES DUE TO 12 PSI INTERNAL PRESSURE LOAD STRESS          VON MEMBER                LOCATION                FIBER              INTENSITY       MISES (ksi)         (ksi)

STRINGER MID SPAN NEAR INNER 3.1 3.0 (0) EDGE NEAR OUTER 6.9 6.8 EDGE RING-STRINGER NEAR INNER 6.5 6.0 JUNCTION EDGE (A) NEAR OUTER 4.9 4.5 EDGE

                                         -233-l l     60     -

l 50 -

 ,,   40     -

U o_ ( A) t(B) o 3 30 - (D)

                             .C) d

c. 20 -

d E

  #                                                YlELD 6TREM6rTH 3   10       .                                  = 45 kSI A         t            t         a 0         0.5          1.0       1.5         z.o       2.5 RADIAL    DISPLACEMENT (N)

PREssu RE - Dis PLACE NCMT CUkVE5 BASED ON GEOMETRIC. T HATERI AL NON- LINEAR AN ALYSIS

234-l STRESSES DUE TO 46 PSI INTERNAL PRESSURE LOAD MEMBER LOCATION FIBER HOOP VON STRESS NISES (ksi) (ksi) RING MID SPAN NEAR INNTER 39.4 39.5 EDGE (B) NEAR OUTER 37.0 37.0 EDGE l l J. .__ ..

  !                                             !      NEAR INNER          !   38.2           38.2 RING-STRINGER JUNCTION         .I          EDGE           i j                            (A)             i l       NEAR OUTER         l   38.9           38.9 l                                            i           EDGE

{ LOCATION FIBER LONGI- VON MEMBER TUDINAL MISES i STRESS (ksi) i (ksi) l STRINGER  !!!D SPAN NEAR INNER 11.5 12.0 EDGE I (D) I NEAR OUTER 22.0 21.8 l EDGE l RING-STRINGER NEAR INNER 18.1 22.9 JUNCTION EDGE (A) i NEAR OUTER i EDGE -10.2 10.2 ,, -c~ -- ,, - + - . - - - , - - , - , , ,

l 1

                                                            -235-C O N YAIN?1CNT          YlELD        PRES $ UKE e

SASE CASE

0. 5 INCH PLd TC N0 ol fNH L YttLD = 32, CCC p, = 2 3 , 2. pu

 ._FAcr e s2_. .(. ~, ')

Acrt,dt Y!s t D F, = 46

                                                                                                           /. 4. /

32. FAcron {FY 2 Vt.n o'IsgS ?tELO Fz = 1. lT [tt c s.' H 1_ S T I FFEN 1N

• 16 " x I '4 Hou P o vER 9 ' f " LEN G.r H -

fy = l +~ IE X I.W = j. '$q ll4 r 0.5 I!. s a % u c.c. y oca l ' (, " Len 5, u - Fy = 1 + li ': ;. *1G' -  :. 5 I

                                                                                             ~7S < 0. 6 F In sTL E LEML :47             fn:.LL CM,'AdsL1-y                  =        p, x       F,   x, F     x F3
                                          =     23,7. x        f. 41 x f..'5 x 1, 35"                   m:

Sc. 5 ,.,3 , _ S E L Ll :.1rt 4__ -. VM U L AT S Ms.s;5 U.v E C APA 61Ls rY n , x F, x FL x Fu

                                        =     2 3.~1      .< l.41 x      1. IF 4  l. fl ff. $ p; ;,

                                          -236-REFERENCE G Greimann, L.F. (Ames Lab.), letter report to Dr. F.P. Schauer, NRC, Washington, DC, July 18, 1980 and supplement of July 30, 1980.

                                                                        -237-SUPPLEMENT T0:     PRELIMINARY CALCULATIONS, ULTIMATE STRENGTH FOR HYDROGEN EXPLOSION, SEQUOYAH CONTAINMENT VESSEL (submitted 7/18/80)

T0: F. P. Schauer, Chief Structural Engineering Branch Division of System Safety Office of Nuclear Reactor Regulation Nuclear Regulatory Comission Washington, DC 20555 BY: Lowell Greimann Ames Laboratory Iowa State University Ames, IA 50011 DATE: July 30, 1980 Idealized Pressure Loading The dynamic pressures associated with detonation are not significant for two reasons (see pg. 6). The detonation time is much smaller than the period of motion of the structure t d T , 0.0000025 0.040 = 6.25(10-5) The impulse associated with detonation I , is negligible with d respect to the impulse from the venting pressure, yl , Id 1.375(10-5)p p= 0.015 p = 9.2(10 ) y v or I d is about 0.1% of ly. Ductility Capacity - Lower Bound A lower bound on the ductility capacity of a pressure vessel is U C This value is a design recomendation by the Welding Institute in the United Kingdom and is based upon experimental evidence.* Also, the ASME Rolfe & Barsom, Fracture and Fatigue Control in St uctures Prentice Hall, 1977, pp. 527-530. i j

                                        -238-Boller and Pressure Vessel Code ** previously defined the collapse pressure of a vessel as the pressure at which the displacement is two times the displacement at first yielding. Note that both of these values      I are design reconinendations. As such, they incorporate some (unspecified) factor of safety--probably between 2 and 3.      With this conservative value )

of ductility capacity, the ultimate venting pressure would be (from page 14) about 31 psi. ASME Boiler and Pressure Vessel Code, Section I'I, Division 1, par.1430, Appendix II, 1974.

                                        -239-PRELIMINARY CALCULATIONS ULTIMATE STRENGTH FOR HYDR 0 GEN EXPLOSION                -

SEQUOYAH CONTAINMENT VESSEL To: F. P. Schauer, Chief Strucutral Engineering Branch Division of System Safety Office of Nuclear Reactor Regulation Nuclear Regulatory Commission Washington, DC 20555 -

                                                                                         =

By: Lowell Greimann Ames Laboratory Iowa State University, Ames, Iowa 50011 July 18,1980 CONTENTS Sumary Typical Compartment 1 . Assumptions 1.1 Ring Idealization 2 - Pressure Loading 6 Finite Element Idealization 7 Static Solution. 9

• Dynamic Solution 11 Failure Criteria 15 Ultimate Strength 20 o

a f

                                                               .a m.-_. _ _ _ _

                                            -240-

SUMMARY

This report sumarizes the calculations performed to predict the ultimate strength of the containment vessel for the Sequoyah Nuclear Power Plant subject l to an explosion in a lower compartment. The analysis is intended to be a first approximation which can be refined when time pennits. Assumptions

         . Behavior is controlled by the containment rings.       The shell wall acts as a membrane transmitting forces to the rings. Thus, the analysis considers a typical ring in the vicinity of a iower compart-ment (pg. 1.1).

f

         . The shell below a typical ring remains elastic (pg. 4).

i

         . The steel was taken to have a static yield strength of 32 ksi and a L

dynamic strength of 39 ksi.

          . The dynamic loads from the explosion can be represented by (1)Animpulseof 1.37 (10-5) pyk-sec/in 2at time zero which approximate 5the detonation phase.

(2) A dynamic pressure which tecreases linearly from yp at time zero to zero pressure at 0.030 sec. to approximate the venting phase, where py is the maximum value of the venting 0 pressure (pg. 6). The pressure was applied over a 60 arc of the vessel. Ductility limits on the vessel control the allowable strains which may be permitted (pg. 15).

                                         -261-Analysis A typical ring in the lower compartment region was idealized by a number of beam type finite elements with nonlinear material and geometric capability (pg.7). The stiffness of the shell below the ring was approximated by linear springs (pg. 7). Time dependent forces M applied to the ideal-ization (pg. 8). This idealization was analyzed by the ANSYS computer program.

Results Static Solution: The static solution was obtained by incrementing the pressure from 0 to l 50 psi. The approximate static plastic pressure is 34 psi (pg.10). Dynamic Solution: A nonlinear transient solution was obtained for the idealization dis-cussed above. Three dynamic analyses were performed with a maximum venting pressure, py , of 10, 50 and 100 psi, respectively. The following results were obtained: py 9 6 0 max 10 psi 0.4 0.6 in. d 50 psi 4.8 4.6 in. 100 psi 24.9 24.9 in. where 90 is the ductility demand (maximum strain / yield strain) and 6 ,x is the maximum displacement (pg.12). Ductility Capacity: The ductility capacity (allowable strain / yield strain) of the ring was taken as n e

                                     =5 Ultimate Load:

Ultimate load is defined as the maximum value of the venting pressure at which the ductility demand is equal to the ductility capacity. The ultimate pressure is found as 51 psi (pg. 20).

e. - toms-,-m- a m m.4 A 4-w A = m.-J,mw - -,,

n _ . -AM.am-a , ,.agm A.--x-> w -R k _

                                                                                                                                                                               -en-am,-u waLa           -a --sm-.ha                   .A+K&       A  d       a.an I                                                                                                                      262-3 gpiOS; w-                                          $7;p/S?M                                      - Si9voYAH                                            NPP i

l Lowst ccMPns'7Mi,v.75 \ N Tyf!l 00'"P"N'N co,Qof d,\) Wu Ce perw c W,il  ;

                                                                                                                                                                         /
                                                                                                                                                            /
                                                                                                                                                            \

d

• l ll s , ,

w,bx 4

                                                                                       \M v                                                                                                       e ,e p 1

i f . i I

i .

4

                                                                                                                                                                                       ~

\

4

                                                                                                                                                                                                 ~                    ,  -
                                                                                      =-                                         ~~

y-***,o***'" - " ff ///j,.p / 1 I

   ._.,.-_..,--,-------,_m.                  --r-.     .--,-- _ . , . . . ...--                 __           -
                                                                                                                        ..m-,y-.                  -,w-       y.. - .         -                    ,              .,.-,m.   .- - m,
                                                                                                                                                                                                                                   .-       - . , --   wym--    -

             - ~ . . _ .        _ _ _ _     _     __    _    .   .__         __  _
                                            -243-The following analysis is considered as a first approximation to the strength of the Sequoyah containment vessel subject to a hydrogen explosion in a lower compartment. Time constraints require that several simplifying assumptions be made to make the problem tractable.

Primary Assumption Behavior is basically non-symmetric since the loading is non-symmetric. Non-symmetric behavior of a circular cylinder is controlled primarily by its bending stiffness. In this case, the predominate bending stiffness is provided by the rings. Therefore, the non-symmetric behavic-r will be controlled by the rings. Assumption: The behavior of the containment vessel is controlled by the rings in the lower compartment region. The shell wall acts only as a membrane which transmits the dynamic pressures to the rings. (This assumption would not be as valid if the disturbance were axi-symmetric, for which case the effect of the ring extends only on the order of /rt (a few feet) along the shell. However, for non-symmetric disturbances, the effect of the rings extends a couple diameters (many ring spacings). Hence, for this non-symmetric loading case, the rings will be quite effective in prov.iding stiffness to the shell between stiffeners.) (Note: Deformations of the shell wall between rings will, of course, be somewhat larger than deformations of the rings. However, the deformation between rings need only be sufficiently large to carry the pressure to the rings by membrane action.) This assumption can be relaxed by a more sophisticated analysis which is beyond the present time constraint. Other Assumptions

       . Shell below a typical ring is elastic (p. 229).
       . Venting pressure decreases linearly to zero at 0.030 sec (p. 231).
       . Ductility capacity of the ring is limited (p.

                                          -244                       ;

e Conshc Rinq se \o s e e n p ar fr u s-f-Iclealigdh - xi h-x 4 A N-

• N

                                                            ./                      .
                                                                                      \
                                                                                        \

f  ; I i s g(t) 1 t/so' -- ha

                                                                                               \

M m,~s, a<ua 4- spoy/ex) k ,s nnj ays~ 4,,

                                                                 ,an- y       s/ sun S-          nhj spdcIn9 ya) =              dy n~ic y-nxe Sny       Case         Se&n is
                  /       _
                          /
                           /

f J

                           /
                           /
                                /. i --     <         l     - h(
             '         ,  N                                               fa <//cAu           /~ d/

t of sua c _ m u

                                         -245-h       hyci! /S9               //
                                          //

vg' p yrs' s dk;>rk (~

                          }- =      l//

h d, = it. f " i,=l%" S =- l 2 0 " b=J .,5 a si 2f  :- f 9 5 (AMC i,9. t. 2 W

                   <OaAx p .- s ! y . p ~ 4 1 8 = W72 id *
                             ~                  //                           //

y= 3 22 + 2= 693 7

                            .Z~ = Z +'?? 3          si #

2 =- 2sv.1 A' (p/uk .:c<a o \ ma o/vAu ) Ma4"s/ Rope < des ) j' = Efcoo J ri [l /S4 Ec) = .92 Isc (4/r. A*. 4 A,. bds, M Gas, f ,/ynom*k = Ef E3 i '\ /fSf, y, 6

                                  =     ?.!44 (/d      )xsel~+            (irm k-(kr th Pch            :         _ kreef
                                    = l. 4 9 2 (10~'.)   e see V f/A Reasonable approach is by numerical methods (FE).

E.A. Witner, et al., "Large Dynamic Deformations of Beams, Rings, Plates and Shells," AIAA Jour., v. 1, No. 8, Aug. 1963, pp 1848-1857. Hodge, "The Influence of Blast Characteristics on the Final Deformation of Circular Cylindrical Shells," J. of App. Mech., v.23, n.4, Dec. 1956, p. 617. Owens & Symonds, " Plastic Deformations of a Free Ring Under Concentrated Dynamic Loading," J. of App. Mech., v.22, n.4, Dec. 1955, pp 523-529. I

                                     -246-Spring Stiffness, k (lower portion of shell)

Neglect forces in shell above ring. Neglect inertial forces in shell below ring. Assume lower shell remains elastic. l Lower Shell W l'  % Ct

                                               , ti l

I

                                            ~

j///, /, <u \

                      -             {

g= 6So" From deep beam theory, containment vessel acts as deep beam, cantilevered from base, under action of asymmetric dynamic load. 4

htKik(i - -

V 3EI AG _T = tr # + ( <n kse shell) A=MR v = skey skp En k = 2. E = 2. 6 G

                               -                  2
            -    O                c 25G[b . t1    I         O    2N AG        L         \es           -

As h = 23 Mobii.) , 6 : ilsoo km

                                       -287 f = 1.799 (10-5) h/k or V=ka k=f=55600k/in. stiffness Select k (equivalent springs) such that above stiffness is obtained.

h 7tr l Sp(t h ') a= AS q= ba/I~,4 b g , s7,,,3 4,,p n f i

               \l =    4        0 5:49 Ro9 e
                                          -          k "e

f 1 Y 4 d b k St/ 6 d& d s e,< us y~) = h s.# + E V= 4 A AR I r* P

                                          )y s#o do = 4 J_. ti R. L 43
                      = 17 k.R. h at      _4 = K =                 25. r,5 k/in/in 1T R
   @-              A g =. A A si^o =            A__\__/ siAo- =

Y sine-

            $=                                         K.           FR au% -          sku       %       lis6 h h d L hms f VG h L5)

                                         -248-PRESSURE LOADING Simulated Dynamic Pressures f(f) n
        ' fv                                                                             i
                                                                                         \

f- t -

                                              ---                                ; 7 4 2.5 m                                                             30we qw                                                   - - - . .

MionaN&o venmn3 Let py = venting pressure. Assume detonation pressure = 10 py . (as per material furnished by NRC) Assume detaonation time interval = 2.5 usec. Assume venting time interval = 30 msec. Note: Detonation pressure & time not very important since impulse associated with detonation is very small relative to venting impuse; also, detonation time is very small with reference to structural period. IDEALIZED DYNAMIC PRESSURE f d)A tmp\se = x.s7s 6o~')s e-sec 4,c kwssac, y - p- Gt=o

                                                            *^%

N x So mpe

                                             -249-Finite Element Idealization Ring
                               , .e 6D '#
                                          @4    ,

83 i4 '

                ,                                           Si g                                                      4 o                                             .          e 17                                           3S              3
      !!y p                                   - x Elements are two-dimensional beam elements (STIF23 in ANSYS) with plastic deformation capability and stress stiffening capability (1st order approx.

to large disp.). A = 74.72 in , I = 2479.5 in', Z = 258.4 in 4 Material E = 29000 ksi, p = 1.492 (10-6) ksec /in F = 32 ksi (static), F = 39 ksi (dynamic). y No strain hardening. Support Springs (P OS u to 5 o 8 y s @ h if g 2 h s

                      @                                                       f 8

0, 7 @ 4'e 2 w.i s a y 3 cm,r ,4 ,uesss) 4, = 4 (MR= rs.ss(%h131) SioG v/,;, 6 g 2.s2.

                                       -250-Loads
                                                            . .'\
                                                                     'x,                         (

i N i s 1

                 ,/                                                                              .

e

                                                                                       .+        ;

f/C / . 3 i i p- / ,.y i l 1 F /' 2 l '

                                                     '~

W . F,l,., =*;} I fG)= f'&) S & C) = f(:') /20 't131)

                           =-    19500         y C+)

w/w p Cf) tr1 ries .i.s on page 233

                                    .' note . p M)       in  <si      mih i
       /nilu /     Cet une c       (L-     a'ys:m,,,c. ame ry.ci,)

l

                 /mhi/        .D/rp      =o i ki-% '     W/a,t} =         !"l'"l'C N>n/ Me.,s W s's/ Mau          =f A([/?3.1)                     nodi 2 f3 liyde       =
                                                 !.M5ito'0 p S(2 4931)                    n.a :fi
                  /mds;                    !.31s(if') pr (! o)             ,4,9             ,.

y, se,4 = ~ l.112 Clo '.) 79 7% f,.- nd4/ wn dry ,p-e.s s *1c A4d' /n/A s / Ve,b n 9 i r pada//y n ey/ifiJ .h l

                                                -251-Solution Idealization on page 234 run in ANSYS with loads on page 235.                <

Static Static Solution p increased from 10 to 50 psi in increments of 5 psi F = 32 ksi (yield strength) y Convergence Criteria Acp < 0.01. C e (Not satisfied above 30 psi, but not rerun because static solution not that useful.) Deflection at Node 1 vs Pressure (see following page) Deflected Shape (right 1/2 of model) ( w

                             \.

4 4

                                             \
                                                \
                                                       ..    :        i e        %
                                                        -g g     4        4
                                                                            , = ase o           to"      20

f J l L

                                                                                                                                 -.                                                                                                              i l

1 50 - 20.0 psi /in

                                                                                                                              ' ' y -" ~                                                                                                         '
                                                                                                                   /p ..

4o --I 8 .

.                      t

) i G

          -            !.                                                      N     Plastic Pressure                                                                                                                                            '

[ (by ASME method) 30 - Y .

                                                                                                                                                                                                                                   ~

1

• t I

9 Deflection vs Pressure 2D Static Solution

10 psi /in 1

i to-- 1 s f

                                  /

i

                          ,                 . . s, .    .                         .s .
                                                                                        .g..---.              ; . _ .  .._g_.--....-4-.----.                 ,

_ 4 t. 8- io is in 16 Deflection at Node 1 (in.) t

                                         -253-Dynema     So/urtm Dyosrric     doa./.s De/an a.s'M       f Vendnj          -

p,.2.33+23S~ p, = w muxi<< w ym r%) p.s.sw c

                    =     /0 ,,       so ,        too        p.s e

(/Ar" l ay/:e-nie =e4 7 u )

    /r: 4 ya 4n       s h l o ,v a / , 4 t AriSFS         srwaua/ Suyejes/s                    s & = 7/~g o usAw        7' ir p. <. : o' s/                   mM+

7~ ~

                                  ?f K% () @ '~s96e [Co$,{ W31$5)                    _-
                                                 /)                  / rad W
                                  =     87/        Md
                                                    ~

M = /.492(to '$79 v 7zg 4139X3) t - a

                                             /                   v.s tua~<. ie 3o s/ src (o.1s~ s        mas,       .a  so' or c.

is </Ae4a .)

                                   = o.o go S           x s<cg:,

Tz s.o 4.s n 21/foo4*E-9?/ M% d ' 4! -- s. o o /4 m 3o e/Se At= 0 Oo/ s-c.-c. (% lo ol's or - w re-ru /f /w p . ,So d wh

• 7~ 4 0 0 4 pr )

Mahid Ys/c' 579er 5 fy = 8 7 rs e' (A e ccw J A, dipA .s+om n 4,

                                                -256-Results Disp. at Node 1 vs Time (see following page 240) only results for p = 50 psi plotted Deflected Shape
                                                    '               =

So fse'

                             ,                      '..          t= s.or7 u<

r-

                                               .. y               .

3

                                                                                ~~
                                                    '_' ,             . . -                            l 1      i                       _

i- ! O j g* Dis t

                                 . . _ _ _ _ _              ...1 Maximum Strain Output was searched for maximum strain in all elements, c max Ductility requirement is c                 /c       =   p where c is yield strain D            y (1345 u in/in)

Only membrane strains used (mid-surface). See later sections for use of this strain in defining failure, p c p 6 v max (membrane) D max 10 psi 504 y in/in 0.4 0.6" 50 psi 6487 - 4.8 4.6" 100 psi 33520 u 24.9 24.9" (See following page for plot.)

                                                                                                 -255-

i. .

e . 9

                                                                                , ,                              s  04b &c l
                .    '                                                                                                                        .        I h-                                                                                                                        9 l

i

                  . 7                     .                                                                                                      !     .' .

! d  : ,

                                          ;                      -                   i                                                                 .
                  . \ 3,    .             ',                     !                   l                                               s           .
               ... .                  - t.                  .        .
                    >                                            :                    .                                   1 t                  '                      i I                                                                                                            $    .

pa de psi I} L.

                                        .t                                                 -

e o O jo 3e .yo & p fa 7, UNf Sft) 5 4 -+ . . 1 I po . -.-. - - . . . . , . g, . . . . . _ _ .. _ .. .. me 60 - - v

                                                                                                                       ~

so- - ~'- y, .

                                                                                                     -. MMou       h, NMa*/           W
                                                                                                                            ., v< Ay &we y                                      y               =         y g

o 5 to is to 15

                                                                                             #/                          6 3

l

I i i

!                                     -256-kJ/ err   fribrik Esdw !<          duch/lX,             A+ <sI             o'      de/./        -A>
            ,prews/ A-dup e uA%:*<

(e, s4w // c5. .. O 6 w< i %s. ') ,

   .                       S,-                        /

(i)l 6 i s +-a

                                .,t =

7 )-d s +-si ,>/maha/ f, = u/A,,,, 4 <. >: A l - M xsi Gss-d w< ' i rej binftf 5lGfki he 'hr /? friar'A:

                        ' n s p. b r a i . /. 6 .r -J a J c <s : :.i . .'                         '

f' Sn o // fr/r/' j eo e))sfs *r-f ^l$ff'fj ' */ '. .- x.c = m (

                              .t, .

r r #1r re 4 ra d, d c 'w

                                    =

i.n 4 ta

i. n .4 ch 4 p -4Aj /4<~rA sr 3. f f( = fllJ// i 5'/M.f S

a. =- cri e /- sirc.

Nl#/ S. 7" }?7 /k f v. l'$. Sk T'Y" , hdi& A <l k9[C (% Mi/ or; K&vcAr,'s , Pre <% A4 //, /177 virw d= fjg " 12 % a s +ss) e 6, i' (epva6/ /ksAi s/re.cs

t 4<. dy E

                                                         = . s<.

G n = +,4 a

                                                                  =

4-1/$

                        -257-t 34est /e hsihj kc/v b 5xg =        Cft bi.e /

d y<> w <,

                                   /0ae'i9 Gyrsxien *}<           cvr ve            r$/      of   cla h.
              & if-SHF                cur v<

N. W2 *ASHr B,he q' R V Ce n& , See-k 2zt' Div. / - suAsee 44 Auf Ap,~.4 k 6, At k rle - re w . J't A / #/ , fi./S/ f ,pg 4?7 Ezd  % s.ons (rnss)

                        /. 2 C                               + 24 6   ksc'jf 7" = dwp a ador                 NOT ffsjm 80'f)

Gs= 10.58 mc G

   -biln<     h        4ffM x, -       x ,<

ots x< 5 =/ x.u g, /* /2 k5 5c 1

                --         /<                                               (2')

5

                 &            f; = k c 6 s&s = _ k.n
                                                                /.It R 45;S ^                  .
                                 = -                    -
                                                           = 2/S.Z ZS t
                                    /.12 {fT(f)

E

                                                                                                             -258-ff.G)                                      ~ /di,'w e           w.M       no       crat.,

S=/ r [ / era w.H p/,iu ,s/,4,,, en444,, if .rt) - s/ cen! Ap ( so Mrup4 / die /w yitMinj .c u p/ss 4i a ou ) i,. = / IIro/ cese /rAn /j hss cra:/.? ' s ;)//.r.A c' 2 rt ' . /n /Nw a 7434i p/r c yra t c k ove As. tes  :. .s e pm fe ,,; ,4,,o esta cwve K, = S,. A s<-e. [ 5", Je4 3 F.P. hLian. , 8. J. l. L%,/e sh, c M. 4. 77-}

                                                                                                "S /s          Aw.ss. ~.t aJ .%sm M. s/,

Wh L44 y t'% J.d~," J& ,/n l. t%$ sn ft/ P<Ani/<jp , sysg e , /r77. B / p( N ,Lars.ssn .J. En ned "Ra ck sh ' i /vdni res e M ./ A c4M ?'.<.4.i ,' < ln - d V N S72h f , Ml 4,/f7f g ?23- 293 I s< ass ,wA l= fy (S/res3es </ yi</d .) dA S~

                                                                           $                                                                          x,.
                                                                                                     /,..- a /;                       .a 9,n                                                                       6F = c w & fn- -
                                                                                                                                                    '

• h" d M=c d"Nbh '"/"'h

                                                                                                   =      5. 0 /
                                                 %        /rbfbfJ sfrash                                 the b $            Cfn lnted                       Ch 4/i/Ask d                                                     i     oM            S Ance.1                  %                 yie/e/

skein be4< /c rsye oece rr. \ i

                                                   -259-Other approaches to ductility limits crack opening displacement (1, p. 530 & 532].

C + o .2.S

                          = & =- f +0

• C S F 2sf L d

y wM E, wika/ cop \ Sc d64ha] * 4 " l'A'ISA hr) (fiy n.1, N. h, t c.//, /4.17 )

                                          $, (f,,n ;         e o.z ~                        (a.ooe,;,.)

a;y.it.iz,nn,16.a) 0 008 - +o.: S = lS 4 x=' - 2 w (o.m/ws)(,f D sqn Pec ~<ne- c4 dia - /A eb. des sw fun.bnwi,) f S.

                          =     2.                        D /s / ,, /=ij /4 /t a-a .e
                     .a., =-  0 ** 4       = 2'. 7 7                       shr /, dwL g ss3

c. od/3f 5

                          =   1
                                       +o.2 6                            c: / # /                Ses
                    .u.'                                                             '

i 1TS (3 4 2.

                                                                            ,t ~-

6- % soo - ri(<-0w$)/p5.s X<,, - l.2 hr Surfac<. (ero ck.

                             &        $~    to O
                               .u., =    4/16 Summary Ductility capacity, p ,cf vessel is probably between 5 and 15 (dependent upon actual material properties and defects - in material and welds). Pene-trations, with the resulting additional welding and complex geometry, will limit ductility. Defonnations of the shell between rings (assumption, p. 228) will limit ductility. The ductility capacity of the ring will be selected as p

c

                                                     =5   .

t I

                                      -260-U/ Ind l'            Wh/ntj/k                                                     i
                    ~

U/AM *Ie S&< np// Jr :dr N ,. A 4<. M l m m- vi & s/ -/4 w,-Sny y .u m

  ,4       ~sJ            /,~ n,,-          w,r       o.:       acca-            ,

s =

                                          %)~x. vgl, /nr<y c                     -
               ,                M*                            *
                       /

fT D ) fj' A & A, ;l f-> .,: As Dur4A h t"cy u

                             -uc       -      5                 , pg :W           i y/e4,n,;/e           A /4     9
                           <g      -            <c c

449 ' S X , k i<- j.)? . / 4 , in scien x /.7ibc sl'<ve.n'; f r t'.S Svt f. . b! I i l " "'/ ( A/ok' , // Ad c " "' I' 5'"

                                                            )     $            '

46 7St ) ,

[,RC, r oau 335 U.S. NUCLEAR REGULATORY COMMISSION UR[GICR-18N BIBLIOGRAPHIC DATA SHEET 15-4753 4 TITLE AND SUBTlTLE lade volume r No., o!appecperant) 2. (Leave bimk) Reliability Analysis of Containment Strength 3. RECIPIENT'S ACCESSION NO. Sequoyah and McGuire Ice Condenser Containments

7. AUTHOR (S) 5. DATE REPORT COMPLE TED L. Greinann, F. Fanous, A. Sabri, D. Ketelaar, ooN,o j u AR A. Wolde-Tinsae, D. Bluhm April 1982 9 PE RF ORMING OHGANIZATION NAME AND MAILING ADDRESS (inclu* Ira Codel DATE REPORT ISSUED

] l " ^" Ames Laboratory lowa State University " hst

                                                                                          , ,t,,,, ,,,,,

1982 Ames, Iowa 50011

8. (Leove Nanki

12. SPONSOHING ORGANIZATION NAME AND MAILING ADDRE SS (/nclude I,p Codel Office of Nuclear Reactor Regulation Nuclear Regulatory Commission ,,',,y,g~

Washington, D.C. 20555 A4131 13 TYPE OF REPORT PE RIOD cove RE D //nclustre defes/ Technical Report September 1, 1979 to September 30, 1980 15 SUPPLEMEN TARY NOTES 14 (tenve etek; 16 ABSTH ACT 200 words or lessl The Sequoyah and McGuire ice condenser containment vessels were designed to withstand pressures in the range of 12 to 15 psi. Since pressures of the order of 28 psi were recorded during the Three Mile Island incident, a need exists to more accurately define the strength of these vessels. A best estimate and uncertainty assessment of the strength of the containments was performed by applying the second moment reliability method. Material and geometric properties were supplied by the plant owners. A uniform static internal pressure was assumed. Gross deformation was taken as the failure criterion. Both approximate and finite element analyses were per-formed on the axisymmetric containment structure and the penetrations. The predicted strength for the Sequoyah vessel is 60 psi with a standard deviation of 8 psi. For McGuire, the mean and standard deviation are 84 psi and 12 psi, respectively. In an results by others are surmiarized and compared and a preliminary dynamic Addendym, analysis is presenfed. 17 KEY WORDS AND DOCUME NT ANALYSIS 17a DESCRIPTORS ( 17b. IDE N TIFIE RS ~ OPE N E NDE D TERYS 18 AV AIL ABIL4TY STATEMENT 19 SECURITY L ASS (Th,s reporrt 21 NO OF P AGES Unclassified Unlimited 20 gCgggsgra.s,,e 22 PaiCE N RC F ORM 33S tis ,n i

UNITED STATES ,, ,, ,, NUCLEA2 CEGULATORY COMMISSION Postau a esispaeo WASHINGTON. D.C. 20566 ,$5*"l, 3,, c re nea m. Su OFFICIAL BUSINESS PiNALTY FOR PRIVATE USE. sYO 1 ANFO 120SS5078dIt US NRc I l 0L. ADM DIV CFPUBLICATIGNS MvT OK PO L i cy & pDR NUREG COPY LA 212 DC 20555 WASHINGTON

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