ML031040235

ORNL/NRC/LTR-, Stochastic Failure Model for Davis-Besse Rvp Head

ML031040235
Person / Time
Site:	Davis Besse
Issue date:	08/31/2002
From:	Bass B, Williams P Oak Ridge
To:	Office of Nuclear Reactor Regulation
References
DE-AC05-00OR22725, FOIA/PA-2003-0018, JCN Y6533 ORNL/NRC/LTR-
Download: ML031040235 (44)

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DRAFT NOT FOR ATTRIBUTION 8/04/2002 ORNL/NRC/LTR-Contract Program or Project

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Heavy-Section Steel Technology (HSST) Program Stochastic Failure Model for the Davis-Besse RPV Head Letter Report P. T. Williams B. R. Bass August 2002 Responsible NRC Individual and NRC Office or Division 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 I 886-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 Inform;' an i Cins rScord was de!eted

?c.

ac-:.,

o Freedom of Information Act, ee.uioD.s 7 FOIA-

-OO3-OOi

DRAFT NOT FOR ATTRIBUTION 0tJ04/02 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-ACO5-00OR22725 2

DRAFT NOT FOR ATTRIBUTION 0t/04/02 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 UT43attelle, 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 Imnpied, or assumes any legal lability 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 tade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imnply Its endorserent.

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.

3

DRAFT NOT FOR ATTRIBUTION 08/04/02 Stochastic Failure Model for the Davis-Besse RPV Head P. T. Williams and B. R. Bass Oak Ridge National Laboratory P. O. Box 2009 Oak Ridge, TN, 37831-8056 Abstract The development of a stochastic model is described in this report in which the uncertainty associated with predictions of burst pressure for circular diaphragms using computational or analytical methods is estimated. It is postulated that the trends seen in predicting the burst pressure with nine experimental disk-burst tests (using materials, geometries, and pressure loadings 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 model will provide an estimate of the cumulative probability that the true burst pressure will be less than any given service pressure.

The stochastic model was 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.

The resulting Log-Laplace model has the scaled form of

(

-JAS4 Pr(^f S9SP) = Fl(SP IPop = 2 1l.1057xPp, (SP115x 2(l.1057x

(SPZ

l.I057xP,,)

Given a computationally-predicted burst pressure, P., the model gives an estimate of the cumulative probability, FL^ that the true (but unknown) burst pressure P,,,,,,) is less than a specified service pressure, SP.

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 (1158. The stochastic model estimates a cumulative probability of failure of 4.14x1077 at the operating pressure of 2.165 ksi and 2.15x106 at the set-point pressure of 2.5 ksi.

4

DRAFT NOT FOR ATTRIBUTION 09/04/02

1. Introduction 1.1. Objective This report presents a stochastic model 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 model provides an estimate 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 a statistical model of 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 model; Section 3 presents the details of the stochastic model; Section 4 demonstrates an application of the 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.

S

DRAFTY NOT FOR ATTRIBUTION 08/04/02 Davis Besse Reactor~esel Head Degradaion Head Cutaway View czxy No uIozie gi. Nedo h Wu prtood onrs onP by #W Liceme.

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

6

DRAFT NOT FOR ATTRIBUTION 08/04/02 Typical Pressurized Water Reactor Corol Rod Drive Mchanbs Rea"o

%Vnet Head A" dota&U Imap)

FILg. 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 ALTRIBUTION 08/04/02 Fig. 1. (continued) (d) photographs of the wastage area with Nozzle 3 removed.

8

DRAFT NOT FOR ATTRIBUION 08/04/02

2. Technical Bases The technical bases employed in the construction of the stochastic model 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 [2J (GAPL-3 discrete-element code[3]),

(3) nonlinear, finite-strain, elastic-plastic finite-element analyses performed for the current study (ABAQUS finite-element code[4]) 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 [2] 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 I

SS 304 A

0.375 0.250 2.625 15 Edge 2

B 0.125 0.125 2.375 6.3 Center 3

C 0.375 0.125 2.625 7.7 Center 4

A533B A

0.375 0.250 2.625 11 Edge S

B 0.125 0.125 2.375 5.3 Edge 6

C 0.375 0.125 2.625 6.7 Center 7

ABS-C A

0.375 0.250 2.625 9.3 Edge s

B 0.125 0.125 2.875 3.75 Edge 9

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

9

DRAFT NOT FOR ATTRIUTION 0e/04/02 Table 2. Property Data for Materials in Disk-burst Tests 121 SS304 34 84 054 34.07 129.36 0432 162.41 0.27 A-533B 74 96 0.17 74.15 112.32 0.157 139.41 0.12 ABS-C 39 64 0.31 39.08 83.84 0.270 105.20 0.17 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.

Geometry A Geometry B

,U r

.125R Fig. 2. Geometric descriptions of the three disk-burst specimens used In Ill (all dimensions are Inches). Images on the right are Photoworks-rendered views of VS-symmetry solid models of the three specimens.

10

NOT FOR ATTIBTIOB N0 08/04/02 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-hinge 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., incorn-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.2 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 CAX8M). 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.

11

DRAFT NOT FOR ATTRIBUTION 08/04/02 I

I I

I J

Gcoinctry A 0.25 in.

r= 0.375 in.

I Gcometry B

,!n.

r.

.0....

II-0. 125 in.

r =0. 125 in.'

/

1.0 in.

1.0 in.

I

/

I 1 1 Geometry C I

1=0.125 in.

/

r 0.375 in.

1 ~

~~~~

3in.

j^~-~~~

T 1.0 in.

I 5 in.

I I

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

Quadratic 8-node axisymmetric (CAX8R) elements with reduced integration were used In a nonlinear finite-strain elastic-plastic analysis of the three disk-burst geometries with three materials.

140 12D.

ED t

E 60

_30S^^00_

40 0

0.1 02 0.3 0.4 0.5 Tme Strain (-1 OW11 r"2V1O P Fig. 4. True stress vs true strain curves of the three materials used in the disk-burst tests compared to SS308 at 600 "F. 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).

12

DRAFT NOT FOR ATTRIBUTION 08/04/02 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, (ro,Or,,h), respectively, is assumed to deform to an axisymmetric shell element with surface length, 8L, deformed thickness, h, radial position, r, and angle

. 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) clement are o =In -)

e

=In

= (i;6 EA =In ((1)

A geometric relationship exists between the radius and chord of a circle such that RH +a 2 (2

2H a) 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, E,, strains at any point on the spherical bulge e,(zjH,a)=e,(zjH,a)= In1+ (z H )])

where he geometric parameter z is shown in Fig. 5. Applying the constant volume assumption, ie.,

CO + e, + EA = 0, produces the following equation for the radial ("thickness") strain Eh(z IH,a) = -2e,(z IHa) = In [ (

2 (4) 13

DRAFT NOT FOR ATTRIEUTION 08/04/02 FIg. 5. Spherical geometry of deformation assumed In mill's 151 plastic Instability theory.

14

DRY~

DNo FOR ATRIBUTION 08/04/02 The effective strain then becomes E.e -E. Y^=gE-e

+ (E,_E,)I +(E, _E,)2 =-E,,(zjH,a)=2ln 1+( 2 )]

(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, at,. and hoop, a,, membrane stresses and the internal pressure, pi, loading is Pi

+(6)

For a spherical dome, R =

R, and a state of equibiaxial stress is assumed to prevail near the pole of the dome with the principal stresses being or,

,C

= R; a =° (7) 2h' and the effective stress,

=4(-as)

+(C.-

,)' + (a-a, is a=a, =,, = p0 (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 Bpi =2ha' Rdp,+ pdR =2hda+26dh (9) dpf d=d dh dR pi 67 hR 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 08/04/02 d6' dR dh

=

a~

R h

or in terms of effective strain (10)

(11)

I da lI dR F-F=1+

RdE 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 I dU 3

1( 2 E) 0 di' 2 4 E t 2

(12)

Applying a power-law constitutive form to relate effective stress to effective strain in the plastic region,

?F=Ki, 9(13) the effective strain at instability is, after a great deal of algebraic manipulation,

!,,, = 4 (2n +1) 11 (14) where n is the power-law exponent in the constitutive equation, Eq. (13).

16

DRAFT NOT FOR ATRXBUTION 08/04/02 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:

• Calculate the effective critical strain.

• Calculate the corresponding effective critical stress.

Calculate the critical thickness.

eF,, = 4l (2n+l) 11 C,,,, = K E h.e, = So exp(-E.,,,,)

Calculate the polar height at the critical condition.

Calculate the corresponding bulge curvature radius.

Finally, calculate the predicted burst pressure.

H.*=a E.I HS

=

exp( 2)-l

=H2 + a2 2H.,

R,2h PR.,

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

- nX + 2n) 11-4n (15) 17

DRAFT NOT FOR ATTRIBUTION 08/04/02

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-Ccarbon 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 staining can be observed near the fillet, thus predicting an edge failure for this specimen which did in fact fail at its edge.

I8

DRAFT NOT FOR ATTRIBUTION 08/04/02 INE2 lw OII.

lsq

.47. 17.- I 74..I Geometry A ABSC Predicted BP = 9.05 ksi Experimental BP = 9.8 ksi 2

&KIs Oak bOPut, VA CM: ^_AISCUc ABAQAIS ozd6.2.-4 Sue Juni It 51:00.:4 ImElrn DWI Iv Tos m32 uamna 42: Stp TIem -

0.9062

" Iteis V a U

a mEE c d Qt Ref. P. C. Riccardella, Elasto-Plastic Analysis of Constrained Disk Burst Tests.'

ASME Paper No. 72-PVP-I2, ASME Pressure Vessels end Piping Conference, New Orleans, LA, September 17-21, 1972.

(a)

GeometryA ABSC Predicted BP = 9.05 ksi Experimental BP = 9.8 ksi aaSO 0 St baM ?SY A I

0m: Cwel.AIIIC C~

MAIWIISI,

-yd&.g.4 Tue j, I

oww 42: S$1 h0 im -a 0.902 PINImty Vas PESO Dof of V. : U OK aout Ian Seeda Sew-

.s Mr.m (b)

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

DRAFT NOT FOR A1TRZBUTION 08/04/02 Figure 5 compares the predicted centerline deflection load histories with the experimentally-observed deflections at failure (estimated from Figs. 3 and 4 in [2]). 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. (15),

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 Ea, H,* R.*

CF.*

he P,,

PI,,W) a Cm.)

(n.)

Orsi)

(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 3345 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 i0.0251 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 ATTRIBUTION 08/04/02 0

1I i

i0 E

t (a)

SS304 2

Geometry B.

• 1 00 2

4 6

I 10 12 14 Pressure (ksI MW122002. Vtw 1.5

-A533B 1,,.*- --

/-

Iti

tt ;

.G............

/.

1 I

t l...

E

/..

......... i.....,.....

0.!-

4 1

4.........

0 2 4

a 10 12 14 ne A

III (b)

Pressure (ksQ Irw1120020 -w ABS.C i

II

.1-- -'

Om A

Y..............

_ I',

,.1 I.

1 0.1

^

i--d

.. A..

't ' . -'-'.......................

I...:

1 L 1..

... L..

-0 2

4 6

a 10 12 14 (C)

Pressure (ksQ)

SWt2J0mpw 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 08/04/02 ABS-C Specimen Failure SS304 Specimen Failure Geometry C 2

on0 M

0 C:

U:

C CD 0

-4a i!

CD 0

1.5 0

0.5

-~...i

~.

.S34

........1.

..... t...............................

M S:

Aallu C.

^i

-A533Be

........33B--...........

t.;.

/..................

.............. {.j

R;..............................

A^.................................. I...........

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 08/04/02 Table 4. Comparison of Experimental Burst Pressures to Three Predictions I

SS 304 A

is MEe 12.3 Edge 12 12.98 Center 1.16 13.29 Edge 1.13 2

B 6.8 Cee 4.3 Edg 1.42 5.92 Center 1.15 6.22 Edge 1.09 3

C 7.7 Center 7.4 Center 1.04 6.49 C

1.19 6.59 Center 1.17 4

A5333 A

it Edge 9.1 Edge 1.12 12.37 Center 0.89 12.26 Edge 0.90 5

B 5.3 Edg 4.2 Edge 1.26 5.65 Center 0.94 5.24 Edge 1.01 6

C 6.7 Cene

6.

Center 0.99 6.19 Center 1.08 6.03 Ed 1.11 7

ABS-C A

9.8 Edge 8

Edge 1.23 8.95 Center 1.10 9.05 Edge 1.08 I

B 3.75 Edge 3

Edge 1.25 4.03 Center 0.92 4.19 Edge 0.S9 9

C 4.94 Edp I

4.47 Center 1.10 4.46 Edge/Center 1.11 23

DRAFT NOT FOR ATrRIBUTION 08/04/02 3.2. Development of Stochastic Model of Failure The development of a stochastic model is described in this section in which the uncertainty associated with predictions of burst pressure for circular diaphragms using computational or analytical methods is 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 model will provide an estimate 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 feasible and based on the best current knowledge, the actual geometry of the wastage area footprint.

Table 5 summarizes some descriptive statisticsifor 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 a stochastic model with at treated as a random variate. Combining the three sets into a single sample produced a sample size large enough for the application of the computer program Expert File [9]. Also given in Table 6 is a ranking of the 26 data points where the median rank order statistic is i-0.3 p-- )

II (16)

The Expert Fite [9] computer program was used to develop a stochastic model of the sample data presented in Table 6. Using a combination of heuristic criteria and Goodness of Fit statistics, twenty six nonnegative 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 (MM), and Quantile Estimates. Table 8 compares three Goodness of Fit statistics (Anderson-Darling, X, and Kolmogorov-Smirnoff (K-S)) for the top five distributions. None of these distributions were rejected by the tests. Figure 6 shows a density/histogram overplot of the top three distributions.

24

DRAFrb NOT FOR ATTRIBUTION 08/04/02 Table 5. Descriptive Statistics for the Ratio of Experimental Burst Pressure to Predicted Burst Pressures

.......zr..

a k'1¶ a,,-..f n-

~

lf*nns-f,,na.n~rtr...et~.t2.f

.wuA.Aam;.'

c AA Sample Size Mean Standard Error Median Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Confidence Level(95.0O%)

8 1.1902 0.0484 1.2223 0.1368 0.0187

-0.0506 0.0007 0.4314 0.9853 1A167 0.1144 9

1.0576 0.0374 1.0953 0.1123 0.0126

-1.4799

-0.5892 0.2979 0.8889 1.1868 0.0863 9

1.0549 0.0331 1.0939 0.0993 0.0099

-0.4349

-0.9683 0.2739 0.8943 1.1682 0.0764 26 1.0975 0.0251 1.1057 0.1281 0.0164 0.2593 0.1714 0.5277 0.8889 1.4167 0.0517 Table 6. Combined Sample Used in Development of Stochastic Model 00 31 I

Hills hy A533B 2

ABAQUS Soln.

ABS-C 3

ABAQUIS Soln.

A533B 4

Hil's Theoy ABS-C 5

Iiilrs They A533B 6

Riaurrdella (1972)

A533B 7

ABAQIS Sdn.

A533B 8

Ricala (1972)

SS 304 9

ABAQI.E San.

ABS-C 10 Hill's Thbay A533B 11 ABAQLS Soln.

SS 304 12 Hill's Theary ABS-C 13 HilrsThey ABS-C 14 ABAQT.S Sdn.

ABS-C 15 ABAQUS Soiln.

A533B 16 Ricardella (1972)

A533B 17 ABAQLlS Socn.

SS 304 18 Hif's hory SS 304 19 HllrsTheory SS304 20 ABAJQUS Soln.

SS 304 21 iil'sfheory SS304 22 Riarrdeda (1972)

SS 304 23 Riorrdella (1972)

ABS-C 24 Rikandela (1972)

ABS-C 25 Ricardella (1972)

A533B 26 Ricammrdla (1972)

SS 304 A

0.8889 B

0.8943 A

0.8972 B

0.9180 B

0.9382 C

0.9853 B

1.0119 C

1.0405 A

1.0827 C

1.0829 B

1.0939 A

1.0953 C

1.1042 C

1.1072 C

1.1104 A

1.1224 A

1.1288 B

1.1479 A

1.1560 C

1.1682 C

1.1868 A

1.2195 A

1.220 B

1.2500 B

1.2619 B

1A167 0.0265 0.0644 0.1023 0.1402 0.1780 0.2159 0.2538 0.2917 0.3295 0.3674 0.4053 0.4432 0.4811 0.5189 0.5568 0.5947 0.6326 0.6705 0.7083 0.7462 0.7841 0.8220 0.8598 0.8977 0.9356 0.9735 ca = ExperixmM Burst Pressurd~redicted Bus Pressure 25

DRAFT NOT FOR ATrRIBUTION 08/04/02 Table 7. Non-negative Bounded Continuous Distributions Investigated -

Ranke odness of Fit I -Lag-Laplace Locatian Deiault 0

Scale ML estimate 1.1057 Shape MLestimate 11.45441 2-Beua Lower endpoint MOM estimntae 0.61449 Upper endpoint MOM estimate 1.73866 Shape oI MOM estimate 7.95564 Shape #2 MOM estimate 11.33552 3-Gamma Locatien Defult 0

Scale ML esimate 0.01444 Shape ML estimate 76.01293

4. Eflang Location Defiult 0

Scale ML estimate 0.01444 Shape ML estimate 76

5. Log-Loistic Location Deliaut 0

Scale MLestimate 1.095S6 Shtape ML estimate 15.21867 6 -Wetbull Locatien Delmult 0

Scale MLestimate 1.15333 Shape ML estimate 9.03948 7.-Lognormal Location Default 0

Scale MLestimate 0.08641 Shape ML wtimate 0.11516

• RaIndom Walk Location Default 0

Scale ML estiae 0.92335 Shape MLestitate 69.1St 9.11sent Gausn Location DeAu 0

Scale ML estimate 1.09747 Shape MLestimate 3-23451

10. n Pason Tpe V Lacation Default 0

Scale ML estimate SIA2582 Shape MLudimate 75.1346 II. I-swrted Weibull Loatoln Default 0

Scale ML estmate l.02327 Shape MLortimatc S38335 12-Weibul(E)

Location Quandle estimate 0.33S4 Scale MLestimate 0.21562 Shape MLe te 1.15S68 13 -ayeiggh(E)

Location Quantleestimate 0.33w Scale ML estimate 0.24352 14-Ekng(E)

Location Qun1ti etimate 0.3tU Scale MLutiemte 0.2062 Shape NL tlmate I

15-G3amaE Location uantileestinate

.38334 Scale ULimate 0.21819 Shape MLW&mte 095616 16 -. aponncial(E)

Location ML eim-e 0.8Q39 Scale ML imate 0Q2057 17 -rearacn Type V(E)

Location Qtlemtinate 0.33834 Scale Dsth I

Shape4l MLstimate 1.00117 Shpe #2 MLestimate 5.43892 isL-ogneM=aE)

Location Quantle etimate 0.38834 Scale MLtimat 4.17414 Shape ML estimate 1.6365 19 -Rando WdkE)

Location Quanic estimate 0.33334 Scale ML temate 699.32509 Shape MLetmate 4.32644 2

Pardo(E)

Location MLestimate UQS9 Shape ML etimate 4.976 21.atSquam Location Qissa estimate 0.Q3S4 4C

l.

tate 0.72313 22 -Wald Location Deftult a

Shape ML Climate 48.03951 23-Raleigh Location DW t 0

Scale MLostimate 1.10463 24-Exponential Location Ddauh 0

Scale ML stimate 1.09747 25-WWdd Lcation Quntile estimate QSSSU Shape ML simte 1.43E-03

26. lInerse GaussianE) Location Qatie timate 0.89E.04 Scale MLestimate 0e20E Shape ML estimate 1.44E-03 26

DRAFT NOT FOR AfTRIBUTION 08/04/02 Table 8. Goodness of Fit Statistics for Top Five Stochastic Models 1

Log-Laplace 0.44952 2.15385 0.59218 2

Beta 0.44697 4.92308 0.81037 3

Gamma 0.46050 3.53846 0.81894 4

Erlang 0.46050 3.53846 0.81897 5

Log-Logistic 0.46271 2.15385 0.74682 Density/Histogram Overplot 0.3C Cem e:

o 0.2 2 @

% 0.11 Co 0.10I 6.05 1.31 Interval Midpoint for a Fig. 6. Overplot of probability densities with histogram for top three fitted stochastic models.

27

DRAFT NOT FOR ATTRIBUTION 08/04/02 The general three-parameter Log-Laplace continuous distribution has the following probability density function,fi,, and cumulative distribution function, Fu, I r (Y-oA

a<x<b fxp(xlasb~c)= 2b(

b fora)

ax c x-a)

x2tb Pr(X < x) = FP (x I a,b,c)=j for a and the percentile function (inverse cumulative distribution function) is a+bexp {In(2P)

P O.5 QJp(P lab,c) =x=

x-(n[2(1-P)]1 a+bexp l ;P>0.5 b,c) > O (17) 20, (b,c)>0 for (O<P<I)

(18) where a is the location parameter, b is the scale parameter, and c is the shape parameter.

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

28

DRAFT NOT FOR ATTRIBUTION 08/04/02 Probability-Probability Plot 1.01 1 -

1ii*

0.7 0.21

[

/

Samiple Who PA0 Ofg

.tmyI.

El.L~q.La~g&-aw" d-wspc".uu al-w1 denp.q.-aS1Ol O3S-ts~w~psa4AflTW Fig. 7. Probability-probability plot comparing top three fitted distributions.

A 1

,2 if 0

co

.02 E

0 0.8 0.6 OA 0.2 S....

.y a

,~.

.L.

Gc :lr! :

9@@~@fi*twog-Lpae t

e 0.13 9

_R Nl __

s.

,.., I

1. ... ')u

_.w n

0.8 1

1.2 1.4 1.6 Experimental BPlPredicted BP, a 08104/2002.K2 ptw Fig. 8. Log-Laplace statistical failure model (n - 26) compared to a beta and gamma cumulative distribution functions.

29

DRAFr NOT FOR ATTRIBUTION 08/04/02 From the Expert Fito [9] analysis, the optimal (Log-Laplace) stochastic model of failure has the following form 10.4544 1 5.17971 a

fLp(a 10,1.1057,11A5441)= 5 (1.710-)

7 9710

-1.1057)

0<a<1.1057

a21.1057 (19) 1 a

4 2 1.1057)

,-I a

-1'45 2 1.1057)

0<a<1.1057 441

a21.1057 where a is the ratio of the true (but unknown) burst pressure to the calculated burst pressure. The percentile function is given by QL,(P10,1.1057,11.A5441)=a= I e05 In(2p)

1P0.5

1.

1057exp {l[2( -P)]. ;P>0.5 for (O<P<1)

(20)

This stochastic model will be used to provide statistical estimates of the expected predictive accuracy of computational methods applied to burst pressure calculations for constrained diaphragms.

30

DRAFT NOr FOR ATTRIBUTION 08/04/02

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 [10]. As an estimate of the uncertainty in the current wastage area measurements, the footprint was extended by approximately 0.25 inches (see Table 9 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).

For the predicted burst pressure of 6.65 ksi, the Log-Laplace statistical failure model can be scaled to provided 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. An example of the scaled Log-Laplace model is shown in Fig. 13. The scaled Log-Laplace model has the following form 5

.17971(

Sp 10.45441 S.791057 P r.24S 5.1797.1057xPI_

O<SP<1.1057xP,.

SP21.1057xP,,

(21) i(1.057xPJ,)

0<SP<1.1057xPp PrP,()gSP) =FL.p(SP I PP) =I SP IllAM41 where, SP, is the service pressure under consideration, Psp is the predicted burst, and Ppf(Q,.,) is the unknown true burst pressure. The scaled percentile function is Qp(PI0Jl.I057xpb, 11.45441) =SP= I

°.057xPpexp {f ln(2P)1}

P < 0.5 P-ln[2(1-P)]

for (OlP<1)(22) 1.1057xP,,exp 11.45441

P>O35 31

DRAFT NOT FOR ATTRIBUTION 08/04/02 Table 9. Wastage-Area-Footprint Geometry Data

~~~Jm MUNI-r > 1 As-Found Footpqnt I

3536 3036 16A122 4.1194 98J9 969933

-117.16 75.26 197.41 cO.904.O,43S>

cO.4351.0.9004>

AdjustedFootprint 0.25i.

40.06 31.78 16.4301 4.1255 129.02 11031J1

-14135 99.O0 245.71

<0.943.4.4476>

<0.4476.0.94P3 f1r Bounding Calculation Fooqprint enraid is in global eordinates.

Global coordinat aysteq his its zas aligned with die vetical centerline ofahe vessel.

The x-y plane ofthe global coordinate systeau is a boz aal phsoe widt hie x-s dalong tbelebtween die enerlines oNozzsle 3 nd I.

,'Adjusted" Footprint a - 25 In.

Area - 40.06 In' Perimeter-31.78 In.

Centroid of I /Footprint EM E~,

"As Found" Footprint /

Area - 35.36 In' Perimeter - 3036 in.

r,=ra + a x;= -r, cos(4,)

Y, = r, sin(C,)

32

D5 A,1r DNOT FOR ATTRIBUTION 08/04/02 Table 9 (continued) Details of Wastage Area Footprint Before Adjustment for Bounding Calculation (Figure taken from Fig. 13 ref. 1101)

Amn ofA1 to critm 1hg,Vi Pans -1 lCycle t

2 fIPOAM D ', ns i.,Ii.

I 0

-0.639

-1.895 1

-0.334

-2.280 2

0.000

-2.235 3

0.500

-2A92 4

1.000

-2.522 5

1.500

-2.482 6

2.000

-2.581 7

2.500

-2.730 8

3.000

-2.769 9

3.500

-2.759 10 4.000

-2.789 11 4.500

-2.819 12 5.000

-2.819 13 5.500

-2.759 14 6.000

-2.700 15 6.500

-2.621 16 7.000

-2.512 17 7.500

-2.364 18 8.000

-2.216 19 8.500

-2.087 20 9.000

-1.712 21 9.135

-1.000 22 9.000

-0.555 23 8.500 0.137 24 25

• 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 8.000 0.334 7.500 0.483 7.000 0.582 6.500 0.829 6.000 1.046 5.500 1.303 5.000 1.778 4.500 2.460 4.000 3.023 3.500 3.300 3.000 3.221 2.500 3.250 2.000 3.300 1.500 3.349 1.000 3.240 0.500 3.122 0.000 3.000

-0.210 2.578

-0.364 2.000

-0.242 1.985 Origin of local coordinate system located at centerline of Nozzle 3. (inches) 33

DRAFr NOT FOR ATTRIBUTON 0S/04/02 (a)

Submodel of Wastage Area 16,935 elements 52,887 nodes Nozzles 3, 11, 15, and 16 Base Material with Wastage Area pWw W4WZ2 (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.

34

DRAFT NOT FOR ATTRIBUTION 08/04/02 I_. _-.;

I-...-...

_L Global Model of Davis-Besse RPV Head and Closure Flange L4 SECTMoNA (c) l Oak Endge Hatonal Laboaty

___~~

S_ B O

G usl_

r OM S

Davis-Besse RPV Head and Cosmre Flange in_

1_--

i Fig 9. (continued) (c) geometry of RPV head and closure flange used In global model, 35

DRAFT NOT FOR ATTRIBUTION 08/04/02

© 0

3 N

4

©

© kI I

I I

I I 4-ALLOW.NSIOdS sJ ICKs I

I I

I

/

I

/

/

/

L, a

Snes Oak Ridge National Laboratory "nd Enineeg Drsion Moweging VW nSIa aGm Heavy-Secdon Steel Technology Program UT-Battelle, LUC ORaY PLr tU seWGNO.

s I R Ca( II.

SCALE AI 21=2 APIL (d)

Fig 9. (continued) (d) relative location of submodel within full RPV head, 36

DRAFT NOT FOR ATTRIBUTION 08/04/02 Fig 9. (continued) (e) geometry of submodel relative to Nozzles 3, 11, 15, and 16.

37

DRAFT NOT FOR ATTRIBUTION 0/

- T' -' j..-..

t - -

' ' 't' '

1

' -J.- -- I D8/04/02 80 V Framatome a

= 114.992 C0-21 sce'nR gsV true 61z" 600 OF 7

X a,

()

2 601--

2,,

= 69.65 ksi

~c-,,

61.64 ksi Adjusted SS308 Curve for Bounding Calculation DScurves q,, = 94.359 ELX9 40 For both SS3(

uniformelongat 3(

uniform elongati

).96 ksi on= 1.15%

SS 308 a ~

q AnlR itj i

20 0j0

-At 600 OF E - 25,571 ksl v- 0.295 A8W-101 A8W-102 A8W-103

-0 A8W-104 A8W-105 ABW-106 ASW data at 550 IF LA I

I I

I I

I I

I.

0.05 0.1 True Strain (-)

0.15 0.2 06/10/2002.K1 ptw Flg. 10. Adjusted SS308 stress vs. strain curve used In the bounding-case calculations compared to curves from a range of ASW 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.

38

DRAFT NOT FOR ATrRIBVTION 08/04/02 Fig. 11. Geometry of adjusted wastage area footprint. Lower figure Is a Photoworks-rendered Image of the submodel with the adjusted "as-found' footprint.

39

DRAFT NOT FOR ATTRIBUTION 08/04/02 12.

IIo (a)

• AO add-A

'm 120 l

1 b) fs e

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

40

DRAFT NOT FOR ATTRIBUTION 08/04/02 Internal Pressure (ksi) 5 6

7 8

9 10 11

............ ;-- r -,:-:-...

0.9 _

Predicted Burst i--.f...........

a 0.8 Pressure=.

6.65 ksi LL0.7

..7........

L0.6

..3 Log-Laplace

.0 0.6 5..

2

....... Median = 1.1057 Mean

= 1.1142 Variance = 0.01959 0.3..

j.

St0.13998-0.2 0.1.

....1 0

0.8 1

1.2 1.4 1.6 Experimental BP/Predicted BR, a0O8/04/2002.K3 Ptw 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.

41

DRAFT NOT FOR ATTRIBUTION 08/04/02 As discussed above, the bounding calculation predicted a burst pressure of 6.65 ksi which has a cumulative probability of failure of 0.158. The stochastic model estimates a cumulative probability of failure of 4.14x10-7 at the operating pressure of 2.165 ksi and 2.15x10 4 at the set-point pressure of 2.5 ksi. See Table 10 for additional estimates.

Table 10. Estimated Cumulative Probability of Failures for the Bounding Calculation 2.155 3.92E-07 2.165 4.14E-07 2.175 4.36E-07 2.185 4.60E-07 2.195 4.84E-07 2.205 5.10E-07 2.215 5.37E-07 2.225 5.66E-07 2.235 5.96E-07 2.245 6.27E-07 2.255 6.60E-07 2.265 6.94E-07 2.275 7.30E-07 2.285 7.67E-07 2.295 8.07E-07 2.305 8.48E-07 2.315 8.91E-07 2.325 9.36E-07 2.335 9.83E-07 2.345 1.031E-06 2.355 L.08E-06 2.365 1.14E-06 2375 1.19E406 2.385 1.25E-06 2.395 1.31E-06 2.405 1.38E-06 2.415 1.45E-06 2.425 1.52E-06 2.435 1.59E-06 2.445 1.67E-06 2.455 1.75E-06 2.465 1.83E-06 2.475 1.92E-06 2.485 2.01E-06 2.495 2.10E-06 2.500 2.15E-06 42

ADA CT NOT FOR ATTRIBUTION 08/04/02

5. Summary and Conclusions A stochastic model 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 has been developed from the following technical bases:

(5) 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, (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-plasticflnite-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.

The resulting Log-Laplace model has the scaled form of 2 l.lO57xP,, )

0<SP<1.1057xpr PrLPpp, S SP]=F,,(SPIP,,)=

I1 SP 4

(23)

1.

1SP2l.057xP,,

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

i.e., Pr[P,,(-) S SP].

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. The stochastic model estimates a cumulative probability of failure of 4.14xl0 7 at the operating pressure of 2.165 ksi and 2.15x04V at the set-point pressure of 2.5 ksi.

43

DRAFT NOT FOR ATTRIBUrION 08/04/02 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. P. C. Riccardella, "Elasto-Plastic Analysis of Constrained Disk Burst Tests," Paper No. 72-PVP-12, presented at the ASME Pressure Vessels and Piping Conference, September 17-21, 1972, New Orleans, LA.

3.

A. L. Thurman, GAPL-3-A Computer Program for the Inelastic Large Deflection Stress Analysis of a Thin Plate or Axially Symmetric Shell with Pressure Loading and Deflection Restraints, WAPD-TM-791, Bettis Atomic Power Laboratory, Pittsburgh, PA, June 1969.

4. ABAQUS/Standard User's Manual, v. 6.2, Hibbit, Karlsson, and Sorensen, Inc., Pawtucket, RI, 2001.

5. R. Hill, "A Theory of the Plastic Bulging of a Metal Diaphragm by Lateral Pressure," Philos. Mag.

(Ser. 7) 41, (1950) 1133.

6. A. R. Ragab and S. E. Bayoumi, Engineering Solid Mechanics, Fundamentals and Applications, CRC Press LLC, Boca Raton, FL, 1999.

7. W. E. Cooper, E. H. Kotteamp, and G. A. Spiering, "Experimental Effort on Bursting of Constrained Disks as Related to the Effective Utilization of Yield Strength," Paper No. 71-PVP-49, ASME Pressure Vessels and Piping Conference, May 1971.

B. J. Chakrabarty and J. M. Alexander, "Hydrostatic Bulging of Circular Diaphragms," J. Strain Anal.

5(3), (1970) 155-161.

9. A. M. Law, Expert Fite User's Guide, Averill M. Law & Associates, Tuscon, Arizona, May 2002.

10. S. A. Looeheicn, Root Cause Analysi Report, Significant Degradation of Reactor Pressure Vessel Head, CR 2002-089 1, Davis-Besse Power Station, April 15, 2002.