ML032600983

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Stochastic Failure Model for Davis Besse RPV Head
ML032600983
Person / Time
Site: Davis Besse 
Issue date: 08/14/2002
From: Bass B, Williams P
Oak Ridge
To: Matthew Kirk
Division of Engineering Technology
References
DE-AC05-00OR22725, DOE 1886-N653-3Y, FOIA/PA-2003-0018, Job Code Y6533 ORNL/NRC/LTR-
Download: ML032600983 (48)
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i' -ie Torn Steve Long's computer dated 08/16/02 12:01pm named

.s r ORNL Failure Criterion_08_14_02.pdf DRAFT NOT FOR ATTRIBUTION 8/14/2002 ORNL/NRC/LTR-Contract Program or Project

Title:

Subject of this Document:

Type of Document:

Authors:

Heavy-Section Steel Technology (HSST) Program Stochastic Failure Model for the Davis-Besse RPV Head Letter Report P. T. Williams B. R. Bass Date of Document:

Responsible NRC Individual and NRC Office or Division August 2002 M. T. Kirk Division of Engineering Technology Office of Nuclear Regulatory Research Prepared for the U. S. Nuclear Regulatory Commission Washington, D.C. 20555-0001 Under Interagency Agreement DOE 1886-N653-3Y NRC JCN No. Y6533 OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37831-8056 managed and operated by UT-Battelle, LLC for the U. S. DEPARTMENT OF ENERGY under Contract No. DE-AC05-00OR22725 Is ~ ~ ~

I A~ct, eA~e;mPio---------

FOIA-I II/

DRAFT NOT FOR ATTRIBUTION 08/14/02 ORNLINRCILTR-Stochastic Failure Model for the Davis-Besse RPV Head P. T. Williams B. R. Bass Oak Ridge National Laboratory Oak Ridge, Tennessee Manuscript Completed - August 2002 Date Published -

Prepared for the U.S. Nuclear Regulatory Commission Office of Nuclear Regulatory Research Under Interagency Agreement DOE 1886-N653-3Y NRC JCN No. Y6533 OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37831-8063 managed and operated by UT-Battelle, LLC for the U. S. DEPARTMENT OF ENERGY under Contract No. DE-AC05-000R22725 2

DRAFT NOT FOR ATTRIBUON 08/14/02 CAUTION This document has not been given final patent clearance and Is for Internal use only. If this document Is to be given public release, It must be cleared through the site Technical Information Office, which will see that the proper patent and technical information reviews are completed In accordance with the policies of Oak Ridge National Laboratory and UT-Battelle, LLC.

This report was prepared as an account of work sponsored 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 assumes any legal liabflity or responsibility for the accuracy, completeness, or usefulness of any Information, apparatus, product, or process disclosed, or represents that Its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute 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 necessarily state or reflect those of the United States government or any agency thereof.

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DRAFT NOT FOR ATTRIBUTION 08114/02 Stochastic Failure Model for the Davis-Besse RPV Head P. T. Williams and B. R. Bass Oak Ridge National Laboratory P. 0. Box 2009 Oak Ridge, TN, 37831-8056 Abstract The development of several stochastic models is described in this report in which the uncertainties associated with predictions of burst pressure for circular diaphragms using computational or analytical methods are estimated. It is postulated that the trends seen in predicting the burst pressure with nine experimental disk-burst tests (using materials, geometries, and pressure badings relevant to the Davis-Besse analysis) will be representative of the computational predictions of the burst pressure in the Davis-Besse wastage area problem. Given a computational prediction of burst pressure for a specific configuration of the wastage area, the scaled stochastic models provide an estimate of the cumulative probability that the true burst pressure will be less than any given service pressure.

The stochastic models were developed from the following technical bases:

(1) experimental data obtained during disk-burst tests with loadings, geometries, and materials relevant to the Davis-Besse pressure loading, wastage-area footprint, and cladding, (2) nonlinear, large-deformation, elastic-plastic discrete-element analyses of the disk-burst tests, (3) nonlinear, finite-strain, elastic-plasticflnite-element analyses performed for the current study, and (4) a theoretical criterion for plastic instability in a circular diaphragm under pressure loading, applied to the disk-burst tests.

Among the twenty-seven continuous distributions investigated, six passed all of the heuristic and Goodness of Fit tests applied in the analysis. The six distributions, ranked in relative order, are: (1) Log-Laplace, (2) Beta, (3) Gamma, (4) Normal, (5) Random Walk, and (6) Inverse Gaussian (Wald).

As an example application, estimates are provided for a bounding calculation of the "as-found" Davis-Besse wastage area. The bounding calculation predicted a burst pressure of 6.65 ksi which has a cumulative probability of failure of 0.158 using the Log-Laplace model. The Log-Laplace model also estimates a cumulative probability of failure of 4.14x1 0-7 at the operating pressure of 2.165 ksi and 2.15x10 4 at the set-point pressure of 2.5 ksi. Using all six distributions, the average probability of failure is 6.91x10-" at 2.165 ksi, 3.60x10-7 at 2.5 ksi, and 0.2155 at 6.65 ksi.

4

DRAFT NOT FOR ATTRIBUTION 08/W4/2

1. Introduction 1.1. Objective This report presents stochastic models of failure for the stainless steel cladding in the wastage area of the Davis-Besse Nuclear Power Station reactor pressure vessel (RPV) head. For a given internal pressure, the statistical models provide estimates of the cumulative probability (probability of nonexceedance) that the exposed cladding will have failed at a lower pressure. The failure mode addressed by this model is incipient tensile plastic instability (i.e., plastic collapse) of the cladding.

1.2. Background The following was taken from ref. [1].

On February 16, 2002, the Davis-Besse facility began a refueling outage that included inspection of the vessel head penetration (VHP) nozzles, which focused on the inspection of control rod drive mechanism (CRDM) nozzles, in accordance with the licensee's commitments to NRC Bulletin 2001-01, "Circumferential Cracking of Reactor Pressure Vessel Head Penetration Nozzles," which was issued on August 3, 2001. These inspections identified axial indications in three CRDM nozzles, which had resulted in pressure boundary leakage. Specifically, these indications were identified in CRDM nozzles 1, 2, and 3, which are located near the center of the RPV head.... Upon completing the boric acid removal on March 7, 2002, the licensee conducted a visual examination of the area, which identified a large cavity in the RPV head on the downhill side of CRDM nozzle 3. Followup characterization by the ultrasonic testing indicated wastage of the low alloy steel RPV head material adjacent to the nozzle. The wastage area was found to extend approximately 5 inches downhill on the RPV head from the penetration for CRDM nozzle 3, with a width of approximately 4 to 5 inches at its widest part.

See Fig. 1. for a photograph of the Davis-Besse RPV, a schematic of a typical nuclear power reactor, and a sketch and photographs of the wastage area.

1.3. Scope In support of the investigation by the United States Nuclear Regulatory Commission's (NRC) Office of Nuclear Regulatory Research, the Heavy-Section Steel Technology Program at Oak Ridge National Laboratory has developed statistical models for a specific failure mode for the exposed stainless steel cladding in the cavity of the Davis-Besse RPV head. Section 2 reviews the technical bases employed in the development of the models; Section 3 presents the details of the stochastic models; Section 4 demonstrates an application of the top-ranked Log-Laplace model to the results of a bounding calculation for the "as found" condition of the wastage area; and Section 5 provides a summary and conclusions.

5

DRAFT NOT FOR ATTRIBUTION 08/14/02 Davis Besse Reactor Vessel Head Degradallotn Head Cutaway View TM above Ngwe shows the ODuais Sesmabr vessel head ddatgoi# hot Beeii nazt #3 end nozzle Ott, W skoeth was pwovad lo fte NRC by Me LUnsee.

Fig. 1. (a) Davis-Besse Nuclear Power Station RPV and (b) sketch of RPV head degradation.

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DRAFT NOT FOR ATTRIBUTION 08/14/02 Typical Pressurized Water Reactor Cam~trm Rod ID&*o Mocilnis cOnreurnt mooit Nozze, kInktWozze Fig. 1 (continued) (c) schematic of a typical nuclear power reactor showing the relationship of the CRDM nozzles to the RPV head.

7

DRAFT NOT FOR ATTMRBUTION 08/14/02 Fig. 1. (continued) (d) photographs of the wastage area with Nozzle 3 removed.

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DRAFT NOT FOR ATTRIBUTION 08/14/02

2. Technical Bases The technical bases employed in the construction of the stochastic models are:

(1) experimental data obtained during disk-burst tests reported by Riccardella [2] with loadings, geometries, and materials relevant to the Davis-Besse pressure loading, wastage-area footprint, and

cladding, (2) nonlinear, large-deformation, elastic-plastic discrete-element analyses of the disk-burst tests also reported in [2] (GAPL-3 discrete-element code[3]),

(3) nonlinear, finite-strain, elastic-plastic finite-element analyses performed for the current study (ABAQUS finite-element code[4D of the nine disk-burst test specimens reported in [2], and (4) a theoretical criterion for plastic instability in a circular diaphragm under pressure loading, due to Hill [5] (as cited in (6]), applied to the disk-burst tests.

2.1. Experimental - Disk-Burst Tests In the early 1970s, constrained disk-burst tests were carried out under the sponsorship of the PVRC Subcommittee on Effective Utilization of Yield Strength [7]. This test program employed a range of materials and specimen geometries that were relevant to components in a nuclear power plant steam supply system'. The geometries of the three test specimens analyzed in 12] are shown in Fig. 2, the test matrix is shown in Table 1, and the properties of the three materials are presented in Table 2. The nine disk-burst tests produced three center failures and six edge failures over a range of burst pressures from 3.75 to 15 ksi as shown in Table 1.

Table 1. Test Matrix for Disk-burst Tests 121 We.

g ffi g ~~~~~~~U~rffffe 1

SS 304 A

0.375 0.250 2.625 Is Edge 2

B 0.125 0.125 2.875 6.8 Center 3

C 0.375 0.125 2.625 7.7 Center 4

A533B A

0.375 0.250 2.625 11 Edge 5

B 0.125 0.125 2.875 5.3 Edge 6

C 0.375 0.125 2.625 6.7 Center 7

ABS-C A

0.375 0.250 2.625 9.8 Edge I

B 0.125 0.125 2.875 3.75 Edge 9

C 0.375 0.125 2.625 4.94 Edge The three materials are representative of reactor core support structures and piping, the-reactor pressure vessel, and plant component support structures [2].

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DRAFT NOT FOR ATTRIBUTION 08/14/O2 Table 2. Property Data for Materials In Disk-burst Tests 121

  • The power-law parameters in Table 2 were fitted for the current study where a = KE and a, e are the effective true stress and effective total true strain, respectively.

L__

6.0io.e Fig. 2. Geometric descriptions of the three disk-burst specimens used In IIJ (all dimensions are inches). Images on the right are PhotoworksZ-rendered views of %-symmetry solid models of the three specimens.

10

DRAFT NOT FOR AlTRIBUTION 08/14/02 2.2. Computational - Axisymmetric Discrete-Element and Finite-Element Models The results of a computational study were presented in [2] in which the nine tests were simulated using the GAPL-3 computer code [3]. GAPL-3 applied the discrete-element method using a two-layered system of elements: one layer for the strain-displacement field and a second layer for the stress field to perform an elasto-plastic large-deformation analysis of stresses, strains, loads, and displacements of thin plates or axisymmetric shells with pressure loading. At each incremental load step, the code iterated to resolve both geometric and material nonlinearities, thus establishing a condition of static equilibrium. The GAPL-3 code did not account for the reduction in thickness of the diaphragm with increasing load, and, therefore, was unable to demonstrate the "tailing up" of the experimental center-deflection histories. As discussed in

[2], the thin-shell approximation of the GAPL-3 code is not strictly valid in the fillet region. The GAPL-3 model did include a plastic-binge type of strain redistribution, but the strain concentration effect due to the fillet radius was not accounted for, since the predicted strain distribution in the cross-section of the fillet was linear by assumption. These approximations in the analysis were driven by the limitations of the computer resources available at the time of the study in 1972.

The current study reanalyzed all nine disk-burst tests using the ABAQUS [4] finite-element code. With current computing power, many of the simplifying assumptions required in 1972 could be removed to provide a more detailed analysis. The fundamental assumptions made in the current study are:

(1) the material is assumed to be homogenous and isotropic before and throughout plastic defor-mation; (2) the material is assumed to be free of pre-existing defects; (3) the volume of the material undergoing plastic deformation is assumed to be constant (i.e., incom-pressible with a Poisson's ratio of 0.5);

(4) the hydrostatic component of the stress tensor has no effect on yielding; and (5) the plastic deformation follows incremental J2 flow theory (Mises yield criterion) with its associated flow rule (Levy-Mises) and isotropic strain hardening.

The finite-element meshes shown in Fig. 3 were developed using 8-node quadratic, axisymmetric, solid elements with reduced integration (ABAQUS element type CAX8R). The material property data given in Table 2 were used to fit power-law constitutive models for the plastic region of the three materials (see Fig. 4). The analyses applied a nonlinear finite-strain procedure with an incrementally increasing pressure load applied from zero up to the load at which numerical instabilities caused ABAQUS to abort the execution.

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DRAFT NOT FOR ATRlBUTION 08/14/02 GeornetryA

-'T:--F -1 t

t 0.25 in.

j j

in.

r= 0.375 in.

Geometry B

'*fl1~~~~~~~~~in 1 ~~~~~10 n

r=O.125m~~~

§: :.Iff:.-

,--t-..

I Geometry C Frr~~~~~~~~~.r 1= 0.125 in.

I r0.7 1417 4

.0min t

~~~~3 in.

i

'I5 in.

I~~~~~~~~~~~~~~S I

Fig. 3. Axisymmetric finite-element meshes used In the analyses of disk-burst tests reported in 12].

Quadratic 8-node axisymmetric (CAXSR) elements with reduced integration were used In a nonlinear finite-strain elastic-plastic analysis of the three disk-burst geometries with three materials.

140-:---................

120 -

i330 A733B

, 100 p

800

/

S 0

5 60 Y

a30B t 600 F 40 0

O.1 0.2 0.3 0.4 0.5 True Strain (-)

0611f002 KI OW Fig. 4. True stress vs true strain curves of the three materials used in the disk-burst tests compared to SS308 at 600 IF. These three test material curves were developed using a power-law strain-hardening model fitted to yield and ultimate strength/strain data for each material given in 121. (See Table 2).

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DRAFT NOT FOR ATTRIBUTION 08/14/02 2.3. Theory - Hill's Plastic Instability Theory A plastic instability theory due to Hill [5] for a pressurized circular diaphragm constrained at the edges is presented in [6]. Figure 5 shows the geometry of the diaphragm, both undeformed and deformed, along with the nomenclature used in the development of the theory.

The geometry of deformation is assumed to be a spherical dome or bulge of radius, R. The undeformed ring element (defined by its position, width, and thickness, (rejrok), respectively) is assumed to deform to an axisymmetric shell element with surface length, AL, deformed thickness, h, radial position, r, and angle q. The nonuniform thickness of the dome reaches its minimum at the pole with polar height H. For a spherical coordinate system with its origin at the center of the dome, the principal strains for the thin-shell (i.e., the strains are assumed constant through the thickness) element are In I( r- ); E# =In ( n, );

k=I (

)

(l)

A geometric relationship exists between the radius and chord of a circle such that R =H 2 +a2 (2) 2H where a is the effective radius of the undeformed diaphragm. Using Eqs. (1) - (2) and the geometry shown in Fig. 5, ref. [6] derives the following relations for the meridional, e,, and hoop, es, strains at any point on the spherical bulge

.e,(z I H, a) = *v.(zI H, a) - In [I+(

2 i](3) where the geometric parameter z is shown in Fig. 5. Applying the constant volume assumption, i.e.,

Em + e, + Em = 0, produces the following equation for the radial ("thickness') strain eq.

lI H,a) = -2e,(z l H,a) = Inl+(Hla)]

(4) 13

DRAFT NOT FOR ATTRIBUTION 08/14/02 h

Fig. 5. Spherical geometry of deformation assumed in Hill's 151 plastic instability theory.

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DRAFT NOT FOR ATTRIBUTION 08/14/02 The effective strain then becomes E(Ets~f e)

J3 (ev-

)' + (l-Ej, )' + (88-Eh )

-(z I H, a) =2 In 1 + (a2

)]

(5)

The maximum radial strain, therefore, occurs at the pole of the spherical bulge. Applying the thin-walled assumption (which is not made in the computational finite-element model) for an axisymmetric shell element, the equilibrium relation between the meridional, a,, and hoop, ae, membrane stresses and the internal pressure,pi, loading is Oo + d =A (6)

R?,

R9 h

For a spherical dome, R. = R = R, and a state of equibiaxial stress is assumed to prevail near the pole of the dome with the principal stresses being at=e=p2';R (7) pjRo~=; 0. =O0 CO 2h 7

and the effective stress,

=Ua(#T)

+(,

°,)

(

a)

,is

=

=

pJaR=

(8)

To establish an instability criterion, a surface can be constructed in pressure, effective stress, and deformation/strain space by expressing Eq. (8) as a total differential of the form Rp, = 2h0 Rdpi + pdR = 2hdr + 2Udh (9) dp, dr + dh dR p,

a h

R An unstable condition exists at a point of maximum pressure on the surface where dp, = 0. The condition is unstable because any perturbation from this position always involves a reduction in load (pressure),

even in a rising stress field. The instability criterion for a deformed bulge of radius R is, therefore, established by the following relation between stress and the deformed geometry for any point on the dome 15

DRAFT NOT FOR ATTRIBUTION 08/14/02 d

=dR dh a

R h

(10) or in ternms of effective strain I ddF=1 IdR U dF R dF (11)

If the instability condition is attained, it will first occur at the point of maximum effective strain at the top of the dome (at z = H) such that Eq. (11) can be stated as Id7 3 1 (2Vf i) 2f-dE =2-4t-lt+j-)

(12)

Applying a power-law constitutive form to relate effective stress to effective strain in the plastic region, (13) the effective strain at instability is, after a great deal of algebraic manipulation, 6011 = 4 (2n + 1)

(14) where n is the power-law exponent in the constitutive equation, Eq. (13).

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DRAFT NOT FOR ATTRIBUTION 08/14/02 For a given material and diaphragm geometry ( n, a, ho), the pressure at the instability condition (i.e., the burst pressure) can be determined by the following procedure:

4 Calculate the effective critical strain.

E

-(2n + 1)

Calculate the corresponding effective critical stress.

&,,;, =K E' Calculate the critical thickness.

h,= ho exp(-,,j,)

Calculate the polar height at the critical condition.

Hw = a Sexp(d

) 1 Calculate the corresponding bulge curvature radius.

R,=, H.,, + a' 2H, Finally, calculate the predicted burst pressure.

V=

RH, An alternative instability criterion was developed by Chakrabarty[8] which was based on a Tresca yield surface. The critical effective strain was found to be 2(2-n)(l + 2n)

(15) aF.

11-4n 1) 17

DRAFT NOT FOR ATTRIBUTION 08/14/2

3. Stochastic Model Development 3.1. Computational and Theoretical Model Results Computational results using the GAPL-3 code were presented in [2]. Converged solutions were obtained for eight of the nine tests. Comparison of experimental and computational centerline deflections showed good agreement for the eight converged cases. In the nonconverged case (ABS-C, geometry C), some difficulty was reported in getting convergence at high pressures. In all cases the experimental data showed a "tailing up" as the pressure approached burst pressure, which the computational model was unable to capture. In general, the prediction of the burst pressure for the eight converged cases showed good agreement with the experimentally-determined burst pressures. Defining a as the ratio of the experimental burst pressure to the computationally-predicted burst pressure, the mean for a was 1.19 with a standard error for the mean of +/-0.0484 and a standard deviation for the sample of 0.137.

The finite-element models using ABAQUS were able to obtain burst pressures for all nine tests, where the predicted burst pressure is defined as the pressure at which a breakdown occurs in the numerical procedure, causing the run to abort. For a nonlinear, finite-strain, static load step, ABAQUS uses automatic sizing of the load increment to maintain numerical stability. The number of iterations needed to find a converged solution for a load increment varies depending on the degree of nonlinearity in the system. If the solution has not converged within 16 iterations or if the solution appears to diverge, ABAQUS abandons the increment and starts again with the increment size set to 25% of its previous value. An attempt is then made at finding a converged solution with this smaller load increment. If the increment still fails to converge, ABAQUS reduces the increment size again. ABAQUS allows a maximum of five cutbacks in an increment before aborting the analysis. Therefore, ABAQUS will attempt a total of 96 iterations with six increments sizes before abandoning the solution. The initial load size for the failing increment was typically already very small due to difficulties in convergence with the previous and final successfully-converged load increment.

Equivalent plastic strain contours are shown in Fig. 4 for the geometry A (ABS-C carbon steel) specimen (Test No. 7) at the point of numerical instability. The experimental burst pressure for this specimen was 9.8 ksi, and numerical instability of the solution occurred at approximately 9.05 ksi, for an a- = 1.083.

Highly localized plastic straining can be observed near the fillet, thus predicting an edge failure for this specimen which did in fact fail at its edge.

18

DRAF-r NOT FOR ATTRIBUTION 08/14J02 FEEQ CM, ceas ASOC MA8AS3laud3d 6.2-4 lue Jur L iStw.$

StP-I I no grnti 42: Ste SiP

-1P 0.9052 prinw Var PEWQ Oglarud V:

U Dlrrgttan Scale Flact: 44.M0900r Geometry A ABSC Predicted BP = 9.05 ksi Experimental BP = 9.8 ksi IIt 11:00:01 Euxtarn M4Ilg tine 200 (a)

(b)

Ref. P. C. Riccardella, 'Elasto-Plastic Analysis of Constrained Disk Burst Tests,'

ASME Paper No. 72-PVP.12, ASME Pressure Vessels and Piping Conference, New Orleans, LA, September 17-21,1972.

PEEQ S

X ^&

Geometry A l

s'5vf~g l s g

, ll LPredicted BP = 9.051

[t11t~~~~~~AAR, Exeimna BP 9.4 1l Il' m

I ls cnli V

C 1A:

MASC Oda MAQIf St Otd d 6.2-4 ~us L i~f ta WW I

nawanoM 42: Slo tm 0.9032 1

rud Va: U

-s DlrfI0a Sce*

fact ar:

1

  • 00 ks!

.8 ks!

Fig. 4. Equivalent plastic strain contours for the Geometry A (ABS-C carbon steel) specimen at the point of numerical Instability. Highly localized plastic straining provides a precondition for plastic collapse at the edge of the specimen. (ABAQUS analysis results) 19

DRAFr NOT FOR ATTRIBUTION 08/W4/2 Figure 5 compares the predicted centerline deflection load histories with the experimentally-observed deflections at failure (estimated from Figs. 3 and 4 in [2D. The "tailing up" of the experimental deflection curves near the point of failure is predicted by the model, indicating that the computational simulations are capturing the final localized "necking" of the diaphragm. For the nine ABAQUS predictions, the mean for a was 1.055 with a standard error for the mean of +/-0.0331 and a standard deviation for the sample of 0.0993.

The results of applying Hill's failure criterion are presented in Table 3. The mean for a was 1.058 with a standard error for the mean of +/-0.0374 and a standard deviation for the sample of 0.1123. The calculations were repeated using the theoretical critical strain of Chakrabarty and Alexander [8], Eq. (I5),

with the resulting burst pressures being essentially identical to those given in Table 3.

Table 3. Application of Hill's Instability Theory to Nine Disk-burst Tests pi est He,,,

R q, c,,,

h j, P&,,,

P bumt&V) a (ksi)

(iii.}~- i(Ifl~)

(in.)

(in.)

(ksi)

(in.)

(ksi)

(ksi) 1 162.41 0.27 2.625 0.250 0.561 1.493 3.054 138.84 0.1427 12.98 15 1.156 2

162.41 0.27 2.875 0.125 0.561 1.635 3.345 138.84 0.0714 5.92 6.8 1.148 3

162.41 0.27 2.625 0.125 0.561 1.493 3.054 138.84 0.0714 6.49 7.7 1.187 4

139.41 0.12 2.625 0.250 0.449 1.316 3.276 126.96 0.1596 12.37 11 0.889 5 139.41 0.12 2.875 0.125 0.449 1.441 3.588 126.96 0.0798 5.65 5.3 0.938 6

139.41 0.12 2.625 0.125 0.449 1.316 3.276 126.96 0.0798 6.19 6.7 1.083 7

105.20 0.17 2.625 0.250 0.490 1.383 3.183 92.95 0.1532 8.95 9.8 1.095 8

105.20 0.17 2.875 0.125 0.490 1.514 3.486 92.95 0.0766 4.08 3.75 0.918 9

105.20 0.17 2.625 0.125 0.490 1.383 3.183 92.95 0.0766 4.47 4.94 1.104 A summary of all 26 burst pressure predictions is given in Table 4. Combining the 26 cases into a single sample gives a mean for a of 1.098 with a standard error for the mean of +/-0.0251 and a standard deviation for the sample of 0.1281. Even though Hill's theory is applicable only for center failures, the good agreement between the experiments (including those that failed at the edges) suggests that, for the edge-failure cases, the specimens were also close to a condition of plastic collapse at the center when they failed first at the edge.

20

DRAFT NOT FOR ATTRI8UTION 08/14/02 SS304 IA I

I I.

I P

1L I

4 I a L...

r 0*1 I

(a)

(b) i.,.... I - :

1,

. Geomr try E~~~~.

7 I

-' i '" '

am!

1--and Failure try A

/ - ~ ~ ~ ~ ~ ~ ~......

0 2

4 4

a 10 12 14 Pressure e

(ks l) 061212I002.Ki p1w 1.5 A.33B

....... 4 i w ria 0 8 K - - i - -~~~~~~~~~~~.

Me I"h 0.2 - -----

-0 2

4 a

a 10 12 14 (C)

Pressure (ksl)

W01220102.K3 Prw Fig. 5. Comparison of experimental centerline vertical deflections at failure to ABAQUS FEM vertical deflection histories at the center of the Geometry A and B specimens for (a) SS 304, (b) A533-B, and (c) ABS-C materials, and 21

DRAFT NOT FOR ATTRIBUTION 08/14/02 ABS-C Specimen Failure_

SS304 Specimen Failure Geometry C 2

U, 0

C, 0

0 00 0

In.

0 4..

0 S

1.5 0

0.5 ABS~~

1 A533.

A.33B.-.

!.~~......

~...

  • ~

~

~~~~~~........

(

5~~~~~~.

I..

I............

0 5 t~.

0 0

2 4

6 8

10 12 14 (d)

Pressure (ksi) 06/12/2002.K4 ptw Fig. 5. (continued) (d) ABAQUS FEM vertical deflection histories at the center of Geometry C, all three materials compared to specimen failure.

22

DRAFT NOT FOR ATTRIBUTION 08/14/2 Table 4. Comparison of Experimental Burst Pressures to Three Predictions 23

DRAFT NOT FOR ATTRIBUTION 08/14/02 3.2. Development of Stochastic Model of Failure The development of several stochastic models is described in this section in which the uncertainties associated with predictions of burst pressure for circular diaphragms using computational or analytical methods are estimated. It is postulated that the trends observed in estimating the burst pressure with the nine disk-burst tests in [2] will be representative of the predictive accuracy of computational estimates of the burst pressure in the Davis-Besse wastage-area problem. Given a prediction of burst pressure for a specific configuration of the wastage area, the scaled stochastic models will provide estimates of the cumulative probability that the true burst pressure will be less than a given service pressure. This postulated linkage of the test specimens to the Davis-Besse problem is obviously an approximation, since the wastage area footprints are not identical to the circular diaphragms used in the tests. The appropriateness of this linkage is in part, therefore, dependent on the ability of the finite-element models to capture, as accurately as is feasib le and based on the best current knowledge, the actual geometry of the wastage area footprint.

Table 5 summarizes some descriptive statistics for the ratio of experimental burst pressure to predicted burst pressure, a, for the three predictive methods discussed in the previous section. Also shown in the table are the results of combining the three samples into one larger sample of 26 data points. This combined sample was used to develop the stochastic models with a treated as a random variate.

Combining the three sets into a single sample produced a sample size large enough to make a thorough statistical analysis of a range of continuous distributions feasible. Also given in Table 6 is a ranking of the 26 data points where the median rank order statistic is P(

1)= i03 (16) n+0.4 The Expert Fit0 (9] computer program was used to develop several stochastic models of the sample data presented in Table 6. Using a combination of heuristic criteria and Goodness of Fit statistics, twenty six continuous distributions were tested with the results shown in ranked order in Table 7. The point-estimation procedures noted in Table 7 include Maximum Likelihood (ML), Method of Moments (MAO, and Quantile Estimates. Table 8 compares three Goodness of Fit statistics (Anderson-Darling, X2, and Kolmogorov-Smirnoff (K-S)) for the top six distributions. None of these distributions were rejected by the Goodness of Fit tests, and all received an absolute rating of Good by the Expert Fito computer program.

The remaining nineteen distributions investigated were either rejected by one or more of the Goodness of Fit tests at some significance level and/or received a less than Good heuristic rating by the Expert Rio software. Figure 6 shows a density/histogram overplot of the six continuous distributions.

24

DRAFT NOT FOR ATTRIBUTION 08/14/02 Table 5. Descriptive Statistics for the Ratio of Experimental Burst Pressure to Predicted Burst Pressures Sample Size 8

9 9

26 Mean 1.1902 1.0576 1.0549 1.0975 Standard Error 0.0484 0.0374 0.0331 0.0251 Median 1.2223 1.0953 1.0939 1.1057 Standard Deviation 0.1368 0.1123 0.0993 0.1281 Sample Variance 0.0187 0.0126 0.0099 0.0164 Kurtosis

-0.0506

-1.4799

-0.4349 0.2593 Skewness 0.0007

-0.5892

-0.9683 0.1714 Range 0.4314 0.2979 0.2739 0.5277 Minimum 0.9853 0.8889 0.8943 0.8889 Maximum 1A167 1.1868 1.1682 1.4167 Confidence Level(95.0%)

0.1144 0.0863 0.0764 0.0517 Table 6. Combined Sample Used In Development of Stochastic Model 1

Hill's Theory A533B A

0.8889 0.0265 2

ABAQUS SoIIL ABS-C B

0.8943 0.0644 3

ABAQUS Sodn.

A533B A

0.8972 0.1023 4

Hill's Theory ABS-C B

0.9180 0.1402 5

Hill's Theory A533B B

0.9382 0.1780 6

Ricarrdelia (1972)

A533B C

0.9853 0.2159 7

ABAQUS Soln.

A533B B

1.0119 0.2538 8

Ricarrdefla (1972)

SS 304 C

1.0405 0.2917 9

ABAQUS Soin.

ABS-C A

1.0827 0.3295 10 Hill's Theory A533B C

1.0829 03674 11 ABAQUS Soba SS 304 B

1.0939 0.4053 12 Hill's Theory ABS-C A

1.0953 0.4432 13 Hill's Theory ABS-C C

1.1042 0.4811 14 ABAQUSSoln ABS-C C

1.1072 0.5189 15 ABAQUS Soin A533B C

1.1104 0.5568 16 Ricarrdella (1972)

A533B A

1.1224 05947 17 ABAQUS Soln.

SS 304 A

1.1288 0.6326 18 Hill's Theory SS 304 B

1.1479 0.6705 19 Hill's Theory SS 304 A

1.1560 0.7083 20 ABAQUS Soln.

SS 304 C

1.1682 0.7462 21 Hill's Theory SS 304 C

1.1868 0.7841 22 Ricarrdella (1972)

SS 304 A

1.2195 0.8220 23 Ricarrdedla (1972)

ABS-C A

1.2250 0.8598 24 Ricarrdella (1972)

ABS-C B

1.2500 0.8977 25 Ricarrdella (1972)

A533B B

1.2619 0.9356 26 Ricarrdella (1972)

SS 304 B

1.4167 0.9735 a = Experimental Burst Pressure/Predicted Burst Pressure 25

DRAFT NOT FOR ATTRIMUTION 08/14/02 Table 7. Continuous Distributions Investigated - Ranked by Goodness of Fit I - Log-Laplace Location Default 0

2 - Beta 3 - Gamma 4 -Log-Logistic 5 - Nomnal 6 - Weibull 7 - Lognormal 8 - Random Walk 9 - Inverse Gaussian 10 - Pearson Type V 11 - Inverted Weibull 12 - Weibull(E) 13 - Rayleigh(E) 14-Erlang(E) 15 - Gamma(E) 16 - Exponential(E) 17 - Pearson Type VI(E)

Scale Shape Lower endpoint Upper endpoint Shape #1 Shape #2 Location Scale Shape Location Scale Shape Mean Standard Dev.

Location Scale Shape Location Scale Shape Location Scale Shape Location Scale Shape Location Scale Shape Location Scale Shape Location Scale Shape Location Scale Location Scale Shape Location Scale Shape Location Scale Location Scale Shape #1 ML estimate ML estimate MOM estimate MOM estimate MOM estimate MOM estimate Default ML estimate ML estimate Default ML estimate ML estimate ML estimate ML estimate Default ML estimate ML estimate Default ML estimate ML estimate Default ML estimate ML estimate Default ML estimate ML estimate Default ML estimate ML estimate Default ML estimate ML estimate Quantile estimate ML estimate ML estimate Quantile estimate ML estimate Quantile estimate ML estimate ML estimate Quantile estimate ML estimate ML estimate ML estimate ML estimate Quantile estimate Default ML estimate 1.1057 11.45441 0.61449 1.78866 7.95564 1138552 0

0.01444 76.01293 0

1.09586 15.21867 1.09747 0.12811 0

1.15383 9.03948 0

0.08641 0.11516 (I

0.92335 69.18788 0

1.09747 82.23451 0

81A2582 75.1846 0

1.02827 8.88835 0.88884 0.21562 1.15868 0.88884 0.24352 0.88884 0.20862 1

0.88884 0.21819 0.95616 0.8889 0.20857 0.88884 l

1.00117 26

DRAFT NOT FOR ATRIBUTION 08/14/02 Shape #2 ML estimate 5.43892 18 - Lognormal(E)

Location Quantile estimate 0.88884 Scale ML estimate

-2.17414 Shape ML estimate 1.86865 19 - Random Walk(E)

Location Quantile estimate 0.88884 Scale ML estimate 699.32509 Shape ML estimate 4.82644 20 - Pareto(E)

Location ML estimate 0.8889 Shape ML estimate 4.8976 21 - Chi-Square Location Quantile estimate 0.88884 d.f.

ML estimate 0.72313 22 - Wald Location Default 0

Shape ML estimate 48.03951 23 - Rayleigh Location Default 0

Scale ML estimate 1.10463 24-Exponential Location Default 0

Scale ML estimate 1.09747 25 - Wald(E)

Location Quantile estimate 0.88884 Shape ML estimate 1.43E-03 26 - Inverse Gaussian(E)

Location Quantile estimate 8.89EM04 Scale ML estimate 0.20862 Shape ML estimate 1.44E-03 27

DRAFT NOT FOR ATTRIBUTION 08/14/2 Table 8. Continuous Distributions That Passed All Goodness of Fit Tests I

Log-Laplace 98 Good 0.44952 2.15385 0.59218 2

Beta 93 Good 0.44697 4.92308 0.81037 3

Gamma 89 Good 0.46050 3.53846 0.81894 4

Normal 83 Good 0.39325 1.23077 0.74664 5

Random Walk 75 Good 0.50448 3.53846 0.85840 6

Inverse Gaussian 71 Good 0.50514 3.53846 0.85891 DensItylllistogram Oveiplot 0.tj I

r-et£i 0.7 OS 0.s PA4 0.2 0.1 0.0 II' 7N.

Interval Midpoint MIieMg 3t~~M1.

I-.Lo:.L*if U3 1-5FNA r-1W..

104,W 06aw In I3B-am I, -M Gus 03D'a"n Fig. 6. Overplot of probability densities with histogram for fitted stochastic models.

28

DRAFT NOT FOR ATTRIBUTIO8 08/14/02 The six distributions in Table 8 have the following analytical forms:

Log-Laplace Distribution The Log-Laplace distribution has the highest relative ranking among the twenty-six distributions investigated. The general three-parameter Log-Laplace continuous distribution has the following probability density functionfLp, and cumulative distribution function, FLP, c (x-a f fLp(x Ia, b,c) =2b b

)

c (x -a 2b( b for a <x O for x Ž b (17)

}(x~a}

for a<xcb Pr(X S x) = FLp(xI a, b c)=

1=

1x-a f

I-I(x-J"for x~:b fora>O, (b,c)>O where a is the location parameter, b is the scale parameter, and c is the shape parameter.

Beta Distribution The Beta distribution has the following probability density function, f8. and cumulative distribution function, Fe,

{(a)[(x-a) f.,(xja,b^c,a 2 =

(b-a)b

-a 1,

fora<xcb l(b-a)B (,a,,

2) 0 otherwise (18) zf fo. (4 I a, bo;, %) d4 F,(xlaba,,a,)=..

0 for a <x y otherwise where a is the shape parameter, P is the scale parameter, r is the location parameter, and R(x) = fexp(-u)u'du.

Normal Distribution The Normal distribution has the following probability density function, fN, and cumulative distribution function, FN, fN (x U CT) expt -(X-)2 for all real numnbersx N 1

~~~~~~~~~~~~~~(20 FN(X IJ1,)= ¢?(Z) = lN({,J)d forz=(x-p)/a where p is the mean (location parameter) and a is the standard deviation (scale parameter).

30

DRAFT NOT FOR ATTRIBUTION 08/14/02 Random Walk Distribution The Random Walk distribution has the following probability density function,fR-, and cumulative distribution function, FRW A (laj3y)=(2r~a7 expl 21 T ]l forx>7 0

otherwise (21)

(D f7(x-I)

Ia(x y)} -

FRW(XlaaP)=

2a I,

1 expp T

-[P(x-Y)+lJfr 0

otherwise where a is the shape parameter, P3 is the scale parameter, y is the location parameter, and 4) is defined in Eq. (20).

Inverse Gaussian (Wald) Distribution The Inverse Gaussian (also known as the Wald distribution) distribution has the following probability density function,fiG, and cumulative distribution function, FIG fIG(xja, PY)=I [27r(x-r)y j exp[2p2(X,)] forx> y 0

otherwise (22) b(Dx -'y -) 1 T_+exp[2a ]P[-(D Y +1 1 forx >y

,(x Ia,, ly)P

_Y P3 P~-y lxL Y

J 0

otherwise where a is the shape parameter, /3 is the scale parameter, y is the location parameter, and 4) is defined in Eq. (20).

Figures 7 and 8 compare the probabilities and the cumulative distribution functions, respectively, of the top-three ranked models.

31

DRAFT NOT FOR ATTRIBUTION 08/14/02 I

._ I LL S.-0

.0 g.0 E

0 Probablfity-Probab]ty PMot 0.2 4-

j- -----

I--

0.5 oJ_

I =. _ _ _ l i C.2

~~__

_t_

W___ I ~-

t-i..............

.0 e

.1 0.2

  • J 4A
0.

IA

0.

t O'

0.9 1.0 Fig. 7. Probability-probability plot comparing top three fitted distributions.

4 II 0.8 0.6 0.4 0.2

=

,,r

-T

.j Gamma.

./

I

.Log-Laplace vtar.........

Median =1.1057 Mean

=1.1142 Variance =0.01959 StdDe 01399 8

w.

I ft-s U --

0.8 1

1.2 1.4 1.6 Experimental BP/Predicted BP, a 0o80412002.K2 p1w Fig. 8. Log-Laplace statistical failure model (n - 26) compared to a beta and gamma cumulative distribution functions.

32

DRAFT NOT FOR ATTRIBUTION 08/14/02 As a specific example from the Expert FRt [9] analysis, the Log-Laplace stochastic model of failure has the following form 5.1797 1 (

a 1

O<a<1.1057 fp(a 10, 1. 105 7,11.4544 1) =

1.1057j

~~S441 ~~~~(23)

Pr(X

  • a) =

I~

(

0,1. 1 057,11.4544 1) j2 J1;0<a01.05 2 1.1057 J a

(21.1057 where a is the rafio of the true (but unknown) burst pressure to the calculated burst pressure. The percentile function is given by I 1.1057exp Tin-24) l}

P 50.5 Q~p(P 10, 1. 105 7,11.4544 1) = a fln[2 ( P)]

for ( < P <1)

(24) l1.1057expl P[,,,

>.5 The stochastic models in Table 8 can be used to provide statistical estimates of the expected predictive accuracy of computational methods applied to burst pressure calculations for service pressures within the range of the data used to develop the model, i.e., 0.8889xP,p 5 SP 5 1.4167 xPsp, where, SP, is a service pressure, and Pep is the predicted burst pressure for the condition under investigation. Extrapolating significantly beyond the range of the data becomes somewhat problematic due to the small sample size of twenty-six data points. All six models in Table 8 are plausible candidates to describe the population from which the sample in Table 6 was drawn, and the relative ranking of these distributions may be sensitive to sample size.

Table 9 provides an example of the sensitivity of the fitting process to the sample size for the case of the "as-found" cavity condition (to be discussed in the next section). Normal distributions were fitted to two samples from the predictions of the disk-burst tests: (1) the ABAQUS finite-element results (n = 9) and (2) the combined data set (n = 26). The two stochastic models were then scaled by the failure pressure for the "as-found" condition (failure pressure = 6.65 ksi). Extrapolating beyond the range of the data for the "as-found" case study produces approximately three orders-of-magnitude difference in estimated failure probability at the operating pressure of 2.165 ksi. This difference in estimated failure probability decreases as the service pressure increases towards the range of data used to develop the models.

33

DRAFT NOT FOR ATTRIBUTION 08/14/02 Table 9. Sensitivity of Cumulative Probability of Failure to Sample Size:

'As-Found" Condition (see Sect. 4) 6.65 0.2902 0.2233 2.155 1.04E-12 7.81E-10 2.165 1.17E-12 8.40E-l0 2.200 1.53E-13 1.08E-09 2.225 2.02E-1 3 1.30E-09 2.250 2.68E-13 1.55E-09 2.275 3.53E-13 1.85E-09 2.300 4.66E-13 2.21E-09 2.325 6.13E-13 2.64E-09 2.350 8.05E-13 3.14E-09 2.375 1.06E-12 3.75E-09 2A00 139E-12 4.46E-09 2.425 1.81E-12 5.30E-09 2.450 2.37E-12 6.30E-09 2.475 3.09E-12 7.48E-09 2.500 4.03E-12 8.87E-09 34

DRAFT NOT FOR ATTRiU ioN0/

08/14/02

4. Application of Stochastic Model to Bounding Calculation A bounding calculation was carried out for the "as-found" condition of the wastage area in the Davis-Besse head. The finite-element model used in the analysis is shown in Fig. 9. An adjusted stress-strain curve (see Fig. 10) was constructed to lower-bound the available data for the cladding material. The geometry of the wastage area footprint was taken from Fig. 13 in the Root Cause Analysis Report [1O]. As an estimate of the uncertainty in the current wastage area measurements, the footprint was extended by approximately 0.25 inches (see Table 10 and Fig. 11 for a geometric description of the adjusted footprint).

A uniform cladding thickness of 0.24 inches (the minimum cladding thickness value shown in Fig. 14 of ref. [10]) was assumed in the model. The finite-element model was then loaded with increasing pressure until the point of numerical instability at an internal pressure of 6.65 ksi (see Fig. 12). In the following, an example is provided of how the statistical distributions in Table 8 can be scaled and applied to the analysis of failure of the cladding in the wastage area.

For the predicted burst pressure of 6.65 ksi, the Log-Laplace statistical failure model as an example can be scaled to provide estimates of cumulative probability of failure (or probability of nonexceedance) as a function of internal service pressure for the specific condition of the wastage area simulated by the finite-element analysis. The scaled Log-Laplace model (see Fig. 13) has the following form SP 10.4S44 I5.1797 I_

5.17971w SP

1. 0 5 7 xP,,)
O<SP<1.1057xPvp
SP 21.1057 xP,g (25)

P r(FP,,

pf*)<SP) = F~,,(SP IP

~11A5441 I

SP A

2 1.1057xPp I I SUxP~u J11 SP Ž1.1057 xP,,,

where, SP, is the service pressure under consideration, P5p is the predicted burst, and PPBQt, ) is the unknown true burst pressure. The scaled percentile function is 1.1057xPpexp ln(2P) 1

PS 0.5 111.45441 QL, (P 10J. 1057 x PPl 1.4544 1) = SP =

'pI_1n[2(I-P)]J 1.1057xP

P>0.5

,spem 1 11.45441 f

for (0 < P< 1)(26) 35

DRAFT NOT FOR ATTRIBUTION 08/14/02 Table 10. Wastage-Area-Footprint Geometry Data Footprint centroid Is in global coordinates.

Global coordinate system has its z-axis aligned with the vertical centerline of the vessel.

The x-y plane of the global coordinate system is a horizontal plane with the xs-axis along the line between the centerlines of Nozzles 3 and I1.

KAdjusted~ Footprint

/

a - 025 I.

Area - 40.06 in" Perimeter 31.78 In.

"As Found" Footprint /

Area - 35.36 int Perimeter = 30.36 in.

.r1;P ra +/- a 0 =-r, cos(41)

) r, shin(§)

36

DRAFT NOT FOR ATTRIBUTION 08/14/2 Table 10 (continued) Details of Wastage Area Footprint Before Adjustment for Bounding Calculation (Figure taken from Fig. 13 ref. [101) i_~~

A h

e

--~-to1-. ~

hirtr.u tit4k Mvf 1t

_infli

-L.

0

-0.639

-1.895 24 8.000 0.334 1

-0.334

-2.280 25 7.500 0.483 2

0.000

-2.235 26 7.000 0.582 3

0.500

-2.492 27 6.500 0.829 4

1.000

-2.522 28 6.000 1.046 5

1.500

-2.482 29 5.500 1.303 6

2.000

-2.581 30 5.000 1.778 7

2.500

-2.730 31 4.500 2.460 8

3.000

-2.769 32 4.000 3.023 9

3.500

-2.759 33 3.500 3.300 10 4.000

-2.789 34 3.000 3.221 11 4.500

-2.819 35 2.500 3.250 12 5.000

-2.819 36 2.000 3.300 13 5.500

-2.759 37 1.500 3.349 14 6.000

-2.700 38 1.000 3.240 15 6.500

-2.621 39 0.500 3.122 16 7.000

-2.512 40 0.000 3.000 17 7.500

-2.364 41

-0.210 2.578 18 8.000

-2.216 42

-0.364 2.000 19 8.500

-2.087 43

-0.242 1.985 20 9.000

-1.712 21 9.135

-1.000 22 9.000

-0.555 23 8.500 0.137 Origin of local coordinate system located at centerline of Nozzle 3. (inches) 37

DRAFT NOT FOR ATTRIBUTION 08/14/02 (a)

Submodel of Wastage Area CladdlngN SS308 Weld (0.24 In. thick) 15, and 16 (b) v Fig. 9. Finite-element global and submodels of the Davis-Besse head and wastage area. The displacements at the vertical side boundaries of the submodel are driven by the global model. Both models are exposed to the same internal pressure loading.

38

DRAFT NOT FOR ATTRIBUTION 08/1402 i

i,

_ i_,

I....

... i l --

S 1

-. i i

I pfil-

.. -,I I

1 1T!

.i Global Model of Davis-Besse RPV Head and Closure Flange I

i ic Ix 41*7~~~~~~~~~~~~~~~~~~1

-'4-ZWI 4' I

,wqnwetG,,,www...

a

_~~~fl

..~~~~~~l~l~

~.A h.U$

....Lt......

tK..

~~

~

.-... a a,. -. _.... o....

..r.. v..

w~~as~aio 4

1 1--

_._,L r t I f A b l :-

bi-UCL"hflC4I4 4,,'

~-

>_.Oavis esse RPV I ad '

fr and Cosure Flange Yi tC.- O S ZS4,)C,

IX L

i. X Fig 9. (continued) (c) geometry of RPV head and closure flange used In global model, 39

DRAFT NOT FOR ATTRIBUTION 08/W4/2

\\\\

N~ \\

N N

I I

I I

I I

I

/

I

/

I

/

J I

I I

I I

/

I

+

0 0

© 4-

© AU. DIMENS IONS V WHES Corputional Sciences Oak Ridge National Laboratory and Engineerng Division Modeling and Slmulation Group Heavy-Section Steel Technology Program UT-Battelle, LLC A

Fbotprfrsot Walngo Aren

, BHsY.

PT.lhpI ITITLE I

V H-ol 1

NO.

RI

" 8 If.,

I W.ALE IO~ATIE 21 APRM.L2002 1 OHEE IA.'1 1up I Fig 9. (continued) (d) relative location of submodel within full RPV head, 40

DRAFT NOT FOR ATTRIBUrION 08/14/02 I

Fig 9. (continued) (e) geometry of submodel relative to Nozzles 3, 11, 15, and 16.

41

DRAFT NOT FOR ATTRIBUTION 08/14/02 SS308 Curve be n

600 °F\\ '

_lt = 69.65 ksi y 60

<v>=1.,

= 61.64 ksi Un z {, * ;

t Adjusted SS308 Curve 12 for Bounding Calculation For both SS308 curves l

94.359 S 4

~, 40 cre m

2 Gf,,/

CS 2-)e 30.96 ksi 1-uniform elongation = 11.15%

SS 308 SS 308 ladiustec'

_e-ABW-101 ABW-102 20 At600°F ABW-103 E =25571Jlsi A8W-104 v =0Q295 A8W-105 A8W-1D6 A8W data at550 °F 0

I 0

0.05 0.1 0.15 0.2 True Strain (")

06/10/2002.K1 ptw Fig. 10. Adjusted SS308 stress vs. strain curve used in the bounding-case calculations compared to curves from a range of A8W heats. Strain hardening In the adjusted curve was reduced to lower-bound all of the data. The offset yield strength and strain at ultimate strength were retained from the unadjusted SS308 curve received from Framatome.

42

DRAFT NOT FOR ATTRIBUTION 08/14/02 A*.

rad idpor:

.!rug' Fig. 11. Geometry of adjusted wastage area footprint. Lower figure Is a Photoworks-rendered Image of the submodel with the adjusted "as-found" footprint.

43

DRAFT NOT FOR ATTRIBUTION 08/14/02 1.t II 01 (a) pvst40s I1.00I, jIGM.

0.0 1.0 Lf *

.0 4.0 1.0 1.0 7.0 (b)

Mu,, pi Fig. 12. Effective plastic-strain histories at two high-strain locations In the wastage area: (a) near the center and (b) near Nozzle 3.

44

DRAFT NOT FOR ATTRIBUTION 08/14/02 Internal Pressure (ksi) 5 6

7 8

9 10 11 0.9,

Predicted Burst..

s 0.8 Pressure=

0~~~~~~~~~~~~~~

6.6 5ksi

. l..

X,.

r r rtr 2:,

0_6 Log-Laplace 2

AlMedian 1.1057

_0R.4 E.~....

VarMeianc 0.01957 75 0.3 j

St e

=0.1399 E

.I.............

0.8 1

1.2 1.4 1.6 Experimental BP/Predicted BP, a

.,b04.2002..

p.w Fig. 13. Application of the failure statistical criterion produces a cumulative probability of failure (based on a Log-Laplace distribution) curve for the Bounding Case condition. Cumulative probability of failure as a function of internal pressure.

45

DRAFr NOT FOR ATTRIBUTION 08/14/2 As discussed above, the bounding calculation predicted a burst pressure of 6.65 ksi which has a cumulative probability of failure of 0.158 for the Log-Laplace model. The Log-Laplace stochastic model also estimates a cumulative probability of failure of 4.14x10-7 at the operating pressure of 2.165 ksi and 2.15xl106 at the set-point pressure of 2.5 ksi. See Table 11 for additional estimates from all six models.

For the six distributions in Tables 8 and 11, the average probability of failure is 6.91x1O-' at 2.165 lsi, 3.60x IO' at 2.5 ksi, and 0.2155 at 6.65 ksi.

Table 11. Estimated Cumulative Probability of Failures for the Bounding Calculation Log-Laplace Location Default 0

98 4.14E-07 2.15E-06 0.1582 Scale ML estimate 1.1057 Shape ML estimate I11A5441 Beta Lower eandpoint MOM estimate 0.61449 93 0

0 02340 Upperendpoint MOM estimate 1.78866 Shapefl MOM estimate 7.95564 Shave#2 MOM estimate 1138552 Gamma Location Default 0

89 8.17E-19 1.50E-15 02236 Scale ML estimate 0.01444 Shave ML estimate 76.01293 Nonnal Mean ML estimate 1.09747 83.33 8.44E-10 8.90E-09 02234 Standard Dev.

ML estimate 0.12811 Random Walk Location Defauft 0

75 0

0 0.2269 Scale ML estimate 0.92335 Shave ML estimate 69.18788 Inverse Gaussian Location Default 0

71 4.01 E-29 1.79E-22 0.2269 Scale ML estimate 1.09747 Shape ML estimate 82.23451 Averaee-6.91E.08 3.60E107 0.2155 Standard Deviation-1.69E-07 8.77E.07 0.0283 46

DRAFT NOT FOR ATTRIBUTION 08/14/02

5. Summary and Conclusions Six stochastic models of the probability of failure associated with a computational prediction of the plastic collapse of the exposed cladding in the wastage area of the Davis-Besse RPV head have been developed from the following technical bases:

(5). experimental data obtained during dik-burst tests reported by Riccardella [2] with loadings, geometries, and materials relevant to the Davis-Besse pressure loading, wastage-area footprint, and cladding, (6) nonlinear, large-deformation, elastic-plastic discrete-element analyses of the disk-burst tests also reported in [2] (GAPL-3 discrete-element code[3]),

(7) nonlinear, finite-strain, elastic-plasticfinite-element analyses performed for the current study (ABAQUS finite-element code[4]) of the nine disk-burst test specimens reported in [2], and (8) a theoretical criterion for plastic instability in a circular diaphragm under pressure loading, due to Hill [5] (as cited in [6]), applied to the disk-burst tests.

Among the twenty-seven continuous distributions investigated, six passed all of the heuristic and Goodness of Fit tests applied in the analysis. The six distributions ranked in relative order are: (1) Log-Laplace, (2) Beta, (3) Gamma, (4) Normal, (5) Random Walk, and (6) Inverse Gaussian (Wald). As an example of how the stochastic models may be applied to the Davis-Besse wastage area problem, the top-ranked Log-Laplace model has the scaled form of 11.45441 Pr[P,,.( sSP] =F,(SP IPp up (27)

I)

(1 5 I

SP 1.A1057xP4p Given a computationally predicted burst pressure, PBP, and service pressure, SP, the model gives an estimate of the cumulative probability of nonexceedance of the true but unknown burst pressure, PBPtr..)

i.e., Pr [PaP>,,) 5SP].

As an example application, estimates are provided for a bounding calculation of the "as-found" Davis-Besse wastage area. The bounding calculation predicted a burst pressure of 6.65 ksi which has a cumulative probability of failure of 0.158 by the Log-Laplace model. The Log-Laplace stochastic model also estimates a cumulative probability of failure of 4.14x1 0I at the operating pressure of 2.165 ksi and 47

DRAFT NOT FOR ATTRIBUTION 08/14/02 2.15 x 106 at the set-point pressure of 2.5 ksi. For all six distributions, the average probability of failure is 6.91x1 O at 2.165 ksi, 3.60x10-7 at 2.5 ksi, and 0.2155 at 6.65 ksi.

References

1. Recent Experience with Degradation of Reactor Pressure Vessel Head, NRC Information Notice 2002-11, United States Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation, Washington, DC, March 12, 2002.
2.

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