ML042660219

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ORNL Report, ORNL/NRC/LTR-02/10 Stochastic Failure Model of the Davis-Besse RPV Head.
ML042660219
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Site: Davis Besse Cleveland Electric icon.png
Issue date: 09/01/2004
From: Bass B, Williams P
Oak Ridge
To: Matthew Kirk
Division of Engineering Technology
References
DE-AC05-00OR22725, DOE 1886-N653-3Y, Job Code Y6533 ORNL/NRC/LTR-02/10
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ORNL/NRC/LTR-02/10 Contract Program or Heavy-Section Steel Technology (HSST) Program Project

Title:

Subject of this Document: Stochastic Failure Model for the Davis-Besse RPV Head Type of Document: Letter Report Authors: P. T. Williams B. R. Bass Date of Document: September 2004 Responsible NRC Individual M. T. Kirk and NRC Office or Division 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

ORNL/NRC/LTR-02/10 Stochastic Failure Model for the Davis-Besse RPV Head P. T. Williams B. R. Bass Oak Ridge National Laboratory Oak Ridge, Tennessee Manuscript Completed - September 2002 Date Published - September 2004 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-AC05-00OR22725 2

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 liability 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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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 set of six 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 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 the pressure at numerical instability, PNI , for a specific configuration of the wastage area, the scaled stochastic models provide estimates of the failure pressure with a specific associated probability.

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-plastic finite-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-six 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. Due to the small sample size (n = 26) used in the stochastic model development, no definitive claim can be made that one distribution is significantly superior to the other five; however, the Log-Laplace is shown to have the highest ranking given the available data, and it produces the highest failure probabilities when extrapolating to service pressures well below the range of the data, e.g., to the nominal operating pressure or safety-valve set-point pressure. It is, therefore, recommended that the Log-Laplace stochastic model be applied in future studies for the Davis-Besse wastage-area problem.

As an example application, estimates are provided for a bounding calculation of the as-found Davis-Besse wastage area. The bounding calculation predicted a PNI value of 6.65 ksi. From the Log-Laplace stochastic model, the corresponding median failure pressure is 7.35 ksi. The Log-Laplace model also estimates a cumulative probability of failure of 4.14 x 107 at the operating pressure of 2.165 ksi and 2.15 x 106 at the safety-valve set-point pressure of 2.5 ksi. Using all six distributions, the average probability of failure is 6.91 x 108 at 2.165 ksi, 3.60 x 107 at 2.5 ksi, and 0.2155 at 6.65 ksi.

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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 licensees 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 Commissions (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 proposed candidate 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.

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

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Fig. 1 (continued) (c) schematic of a typical nuclear power reactor showing the relationship of the CRDM nozzles to the RPV head.

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Fig. 1. (continued) (d) photographs of the wastage area with Nozzle 3 removed.

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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[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] with extensions by Chakrabarty and Alexander [6] (as cited in [7]), 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 [8]. This test program employed a range of materials and specimen geometries that were relevant to components in a nuclear power plant steam supply system1. 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 [2]

Effective Experim ental Results T est M aterial G eom etry Fillet D iaphragm D iaphragm Burst Location of N um ber R adius T hickness Radius Pressure Failure (in.) (in.) (in.) (ksi) 1 SS 304 A 0.375 0.250 2.625 15 Edge 2 B 0.125 0.125 2.875 6.8 C enter 3 C 0.375 0.125 2.625 7.7 C enter 4 A 533B 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 C enter 7 A BS-C A 0.375 0.250 2.625 9.8 Edge 8 B 0.125 0.125 2.875 3.75 Edge 9 C 0.375 0.125 2.625 4.94 Edge 1

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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Table 2. Property Data for Materials in Disk-burst Tests [2]

Yield Strength Ultimate Strain at True Stress True Ultimate Log Strain Power Law Fit*

Material 0.2% offset Strength Ultimate 0.2% offset Stress at Ultimate K n (ksi) (ksi) (-) (ksi) (ksi) (-) (ksi) (-)

SS304 34 84 0.54 34.07 129.36 0.432 162.41 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 n

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

Fig. 2. Geometric descriptions of the three disk-burst specimens used in [1] (all dimensions are inches). Images on the right are Photoworks-rendered views of 1/2-symmetry solid models of the three specimens.

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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., incom-pressible with a Poissons ratio, = 0.5), for linear-elastic deformation = 0.3; (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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Fig. 3. Axisymmetric finite-element meshes used in the analyses of disk-burst tests reported in [2].

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.

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 [2]. (See Table 2).

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2.3. Theory - Hills Plastic Instability Theory A plastic instability theory due to Hill [5] for a pressurized circular diaphragm constrained at the edges is presented in [7]. 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, (r0 , r0 , h0 ) , respectively) is assumed to deform to an axisymmetric shell element with surface length, L , 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) element are r L h

= ln  ; = ln  ; h = ln (1) r0 r0 h0 A geometric relationship exists between the radius and chord of a circle such that H 2 + a2 R= (2) 2H where a is the effective radius of the undeformed diaphragm. Using Eqs. (1) - (2) and the geometry shown in Fig. 5, ref. [7] derives the following relations for the meridional, , and hoop, , strains at any point on the spherical bulge zH

( z l H , a) = ( z l H , a) = ln 1 + (3) a 2

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

+ + h = 0 , produces the following equation for the radial (thickness) strain 2

1 h ( z l H , a ) = 2 ( z l H , a ) = ln (4) 1 + ( zH / a )

2 13

Fig. 5. Spherical geometry of deformation assumed in Hills [5] plastic instability theory.

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The effective strain then becomes z H 2

( ) + ( h ) + ( h ) = h ( z l H , a) = 2ln 1 + 2 2 2

( , , h )

2 (5) 3 a 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, , and hoop, , membrane stresses and the internal pressure, pi , loading is pi

+ = (6)

R R 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 pi R

= =  ; r = 0 (7) 2h 1

( ) + ( r ) + ( r ) , is 2 2 and the effective stress , =

2 2

pi R

= = = (8) 2h 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 Rpi = 2h Rdpi + pi dR = 2hd + 2 dh (9) dpi d dh dR

= +

pi h R An unstable condition exists at a point of maximum pressure on the surface where dpi = 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

d dR dh

= (10)

R h or in terms of effective strain 1 d 1 dR

=1+ (11) d R d 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 1 d 3 1 2

= 1+

d 2 4 (12) 2 Applying a power-law constitutive form to relate effective stress to effective strain in the plastic region,

=Kn , (13) the effective strain at instability is, after a great deal of algebraic manipulation, 4

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

An alternative instability criterion was developed by Chakrabarty and Alexander [6] which was based on a Tresca yield surface. The critical effective strain was found to be 2(2 n)(1 + 2n) crit = (15) 11 4n 16

For a given material and diaphragm geometry ( n, a, h0 ), the pressure at the instability condition (i.e., the burst pressure) can be determined by the following procedure:

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  • Calculate the effective critical strain. crit = ( 2n + 1) 11
  • Calculate the corresponding effective critical stress. crit = K n
  • Calculate the critical thickness. hcrit = h0 exp ( crit )
  • Calculate the polar height at the critical condition. H crit = a exp crit 1 2

2 H crit + a2

  • Calculate the corresponding bulge curvature radius. Rcrit =

2 H crit 2hcrit crit

  • Finally, calculate the predicted burst pressure. pburst =

Rcrit 17

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 as the ratio of the experimental burst pressure to the computationally-predicted pressure at numerical instability, the mean for 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 pressure at numerical instability, PNI , 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 = 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.

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(a)

(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

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 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 Hills failure criterion are presented in Table 3. The mean for 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 [6], Eq. (15),

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

Table 3. Application of Hills Instability Theory to Nine Disk-burst Tests Test K n a h0 crit H crit R crit crit h crit P NI P burst(exp)

(ksi) (in.) (in.) (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 PNI values is given in Table 4. Combining the 26 cases into a single sample gives a mean for 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 Hills 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.

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(a)

(b)

(c)

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

(d)

Fig. 5. (continued) (d) ABAQUS FEM vertical deflection histories at the center of Geometry C, all three materials compared to specimen failure.

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Table 4. Comparison of Experimental Burst Pressures to Three Predictions Experimental Riccardella's ASME Paper Hill's Plastic Instability Theory ABAQUS Solutions Test Material Geometry Burst Location Pressure at Location Exp. BP Pressure at Location Exp. BP Pressure at Location Exp. BP Number Pressure (BP) of Failure Instability (P NI ) of Failure P NI Instability (P NI ) of Failure P NI Instability (P NI ) of Failure P NI (ksi) (ksi) (ksi) (ksi) 1 SS 304 A 15 Edge 12.3 Edge 1.22 12.98 Center 1.16 13.29 Edge 1.13 2 B 6.8 Center 4.8 Edge 1.42 5.92 Center 1.15 6.22 Edge 1.09 3 C 7.7 Center 7.4 Center 1.04 6.49 Center 1.19 6.59 Center 1.17 4 A533B A 11 Edge 9.8 Edge 1.12 12.37 Center 0.89 12.26 Edge 0.90 5 B 5.3 Edge 4.2 Edge 1.26 5.65 Center 0.94 5.24 Edge 1.01 6 C 6.7 Center 6.8 Center 0.99 6.19 Center 1.08 6.03 Edge 1.11 7 ABS-C A 9.8 Edge 8 Edge 1.23 8.95 Center 1.10 9.05 Edge 1.08 8 B 3.75 Edge 3 Edge 1.25 4.08 Center 0.92 4.19 Edge 0.89 9 C 4.94 Edge 4.47 Center 1.10 4.46 Edge/Center 1.11 23

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 the ratios of experimentally-observed failure pressures in the nine disk-burst tests in [2] to calculated PNI values will be representative of the predictive accuracy of computational estimates of the burst pressure in the Davis-Besse wastage-area problem. Given a calculated PNI 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, specifically providing a failure pressure with its associated probability. 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. Accurate material properties are also an important input to the analysis.

Table 5 summarizes some descriptive statistics for the ratio, , of experimental burst pressure to the calculated pressure at numerical instability 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 treated as a random variate. Combining the three sets into a single sample produced a sample size large enough to make a reasonably 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 i 0.3 P( i ) = (16) n + 0.4 The Expert Fit© [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 (MM),

and Quantile Estimates. Table 8 compares three Goodness of Fit statistics (Anderson-Darling, 2 , 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 Fit© computer program.

The remaining twenty 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 absolute rating by the 24

Expert Fit© software. Figure 6 shows a density/histogram overplot of the six candidate continuous distributions.

Table 5. Descriptive Statistics for the Ratio of Experimental Burst Pressure to Predicted Burst Pressures Descriptive Statistics Riccardella (1972) Hill's Theory ABAQUS Combined 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 1.4167 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 Rank Method Material Geometry

  • Order Statistic 1 Hill's Theory A533B A 0.8889 0.0265 2 ABAQUS Soln. ABS-C B 0.8943 0.0644 3 ABAQUS Soln. 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 Ricarrdella (1972) A533B C 0.9853 0.2159 7 ABAQUS Soln. A533B B 1.0119 0.2538 8 Ricarrdella (1972) SS 304 C 1.0405 0.2917 9 ABAQUS Soln. ABS-C A 1.0827 0.3295 10 Hill's Theory A533B C 1.0829 0.3674 11 ABAQUS Soln. 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 ABAQUS Soln. ABS-C C 1.1072 0.5189 15 ABAQUS Soln. A533B C 1.1104 0.5568 16 Ricarrdella (1972) A533B A 1.1224 0.5947 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 Ricarrdella (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

= Experimental Burst Pressure/Prssure at Numerical Instability 25

Table 7. Continuous Distributions Investigated - Ranked by Goodness of Fit Model Parameters Point Estimator Parameter Values 1 - Log-Laplace Location Default 0 Scale ML estimate 1.1057 Shape ML estimate 11.45441 2 - Beta Lower endpoint MOM estimate 0.61449 Upper endpoint MOM estimate 1.78866 Shape #1 MOM estimate 7.95564 Shape #2 MOM estimate 11.38552 3 - Gamma Location Default 0 Scale ML estimate 0.01444 Shape ML estimate 76.01293 4 - Log-Logistic Location Default 0 Scale ML estimate 1.09586 Shape ML estimate 15.21867 5 - Normal Mean ML estimate 1.09747 Standard Dev. ML estimate 0.12811 6 - Weibull Location Default 0 Scale ML estimate 1.15383 Shape ML estimate 9.03948 7 - Lognormal Location Default 0 Scale ML estimate 0.08641 Shape ML estimate 0.11516 8 - Random Walk Location Default 0 Scale ML estimate 0.92335 Shape ML estimate 69.18788 9 - Inverse Gaussian Location Default 0 Scale ML estimate 1.09747 Shape ML estimate 82.23451 10 - Pearson Type V Location Default 0 Scale ML estimate 81.42582 Shape ML estimate 75.1846 11 - Inverted Weibull Location Default 0 Scale ML estimate 1.02827 Shape ML estimate 8.88835 12 - Weibull(E) Location Quantile estimate 0.88884 Scale ML estimate 0.21562 Shape ML estimate 1.15868 13 - Rayleigh(E) Location Quantile estimate 0.88884 Scale ML estimate 0.24352 14 - Erlang(E) Location Quantile estimate 0.88884 Scale ML estimate 0.20862 Shape ML estimate 1 15 - Gamma(E) Location Quantile estimate 0.88884 Scale ML estimate 0.21819 Shape ML estimate 0.95616 16 - Exponential(E) Location ML estimate 0.8889 Scale ML estimate 0.20857 17 - Pearson Type VI(E) Location Quantile estimate 0.88884 Scale Default 1 Shape #1 ML estimate 1.00117 26

Model Parameters Point Estimator Parameter Values 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 8.89E-04 Scale ML estimate 0.20862 Shape ML estimate 1.44E-03 27

Table 8. Continuous Distributions That Passed All Goodness of Fit Tests Rank Model Relative Score Rating Anderson-Darling 2 Statistic K-S 1 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 Fig. 6. Overplot of probability densities with histogram for fitted stochastic models.

28

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 function, fLP, and cumulative distribution function, FLP, c x a c 1 for a < x < b 2b b f LP ( x l a, b, c ) = c 1 for a 0, (b, c) > 0 c xa 2b b for x b (17) 1 x a c for a < x < b 2 b Pr( X x ) = FLP ( x l a, b, c) = c for a 0, (b, c) > 0 1 xa 1 2 b for x b 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, fBe, and cumulative distribution function, FBe, x a 1 1 x a 2 1 1

b a b a f Be ( x l a, b, 1 , 2 ) = for a < x < b (b a ) B(1 , 2 )

0 otherwise (18) x f ( l a, b,1 , 2 ) d FBe ( x l a , b, 1 , 2 ) = a Be for a < x < b 0 otherwise 1

where B (1 , 2 ) = u1 1 (1 u )2 1 du , a is the lower endpoint, b is the upper endpoint, 1 is the first 0

shape parameter, and 2 is the second shape parameter.

29

Gamma Distribution The Gamma distribution has the following probability density function, fGa, and cumulative distribution function, FGa,

( x ) 1 x exp for x >

f Ga ( x l , , ) = ( )

0 otherwise (19) x FGa ( x l , , ) =

f ( l , , ) d Ga for x >

0 otherwise where is the shape parameter, is the scale parameter, is the location parameter, and

( x ) = exp( u )u x 1du .

0 Normal Distribution The Normal distribution has the following probability density function, fN, and cumulative distribution function, FN, 1 ( x µ )2 f N ( x l µ , ) = exp for all real numbers x 2 2 2 2

(20) z FN ( x l µ , ) = ( z ) =

f N ( l 0,1) d for z = ( x µ ) /

where µ is the mean (location parameter) and is the standard deviation (scale parameter).

30

Random Walk Distribution The Random Walk distribution has the following probability density function, fRW, and cumulative distribution function, FRW 1/ 2 1 ( x ) 2 exp 2 for x >

f RW ( x l , , ) = 2 ( x ) 2 ( x )

0 otherwise (21) 1 1 ( x )

(x )

FRW ( x l , , ) = 2 1 exp + 1 ( x ) for x >

(x )

0 otherwise where is the shape parameter, is the scale parameter, is the location parameter, and is defined in Eq. (20).

Inverse Gaussian Distribution The Inverse Gaussian distribution has the following probability density function, fIG, and cumulative distribution function, FIG 1/ 2

( x )2 2 ( x )3 exp for x >

f IG ( x l , , ) = 2 ( x )

2 0 otherwise (22) x 2 x 1 + exp + 1 for x >

FIG ( x l , , ) = x x 0 otherwise where is the shape parameter, is the scale parameter, is the location parameter, and 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

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

Fig. 8. Log-Laplace statistical failure model (n = 26) compared to a beta and gamma cumulative distribution functions.

32

As a specific example from the Expert Fit© [9] analysis, the Log-Laplace stochastic model of failure has the following form 10.45441 5.17971  ; 0 < < 1.1057 1.1057 f LP ( l 0,1.1057,11.45441) = 12.45441 5.17971  ; 1.1057 1.1057 (23) 1 11.45441

0 < < 1.1057 2 1.1057 Pr( X ) = FLP ( l 0,1.1057,11.45441) = 11.45441 1

1 2 1.1057  ; 1.1057 where is the ratio of the true (but unknown) burst pressure to the calculated pressure at numerical instability, PNI. The percentile function is given by QLP ( p l 0,1.1057,11.45441) =

ln(2 p )

exp ln (1.1057 ) + 11.45441  ; p 0.5 (24) p = for ( 0 < p < 1) exp ln (1.1057 ) ln 2 (1 p )  ; p > 0.5 11.45441 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., 0.8889 x PNI SP 1.4167 x PNI , where, SP, is a service pressure, and PNI is the calculated pressure at numerical instability 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, but the relative ranking of these distributions may be sensitive to sample size. Due to the small sample size (n = 26) used in the stochastic model development, no definitive claim can be made that one distribution is significantly superior to the other five; however, the Log-Laplace is shown to have the highest ranking given the available data, and it produces the highest failure probabilities when extrapolating to service pressures well below the range of the data, e.g., to the nominal operating pressure or safety-valve set-point pressure.

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 33

(2) the combined data set (n = 26). The two stochastic models were then scaled by the calculated PNI of 6.65 ksi for the as-found condition. 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.

Table 9. Sensitivity of Cumulative Probability of Failure to Sample Size:

As-Found Condition (see Sect. 4)

Internal Normal Distribution Pressure ABAQUS Combined (ksi) n =9 n =26 6.65 0.2902 0.2233 2.155 1.04E-12 7.81E-10 2.165 1.17E-12 8.40E-10 2.200 1.53E-13 1.08E-09 2.225 2.02E-13 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 2.400 1.39E-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

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 [10, 11] for the cladding material. The geometry of the wastage area footprint was taken from Fig. 13 in the Root Cause Analysis Report [12]. 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 based on ultrasonic testing (UT) measurements on a 1/2 inch grid as depicted in Fig. 14 of ref. [12]) 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) was attained. Decreasing the cladding thickness from 0.24 inches to 0.1825 inches (the minimum design allowable) resulted in a calculated pressure at numerical instability of 5.18 ksi. 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.

As an example, the Log-Laplace statistical failure model 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 10.45441 5.17971  ; 0 < SP < 1.1057 x PNI 1.1057 x PNI f LP ( SP l PNI ) = 12.45441 SP 5.17971 1.1057 x P  ; SP 1.1057 x PNI NI (25) 1 SP 11.45441

0 < SP < 1.1057 x PNI 2 1.1057 x PNI Pr( BP( true ) SP ) = FLP ( SP l PNI ) = 11.45441 1 SP 1 2 1.1057 x P  ; SP 1.1057 x PNI NI where, SP, is the service pressure under consideration, PNI is the calculated pressure at numerical instability, and BP(true) is the unknown true burst pressure. To calculate as estimated burst (failure) pressure, BPp , with probability, p, the scaled percentile function is applied 35

QLP ( p l 0,1.1057 x PNI ,11.45441) =

(26) ln(2 p )

exp ln (1.1057 x PNI ) + 11.45441  ; p 0.5 BPp = for ( 0 < p < 1) exp ln (1.1057 x P ) ln 2 (1 p )  ; p > 0.5 NI 11.45441 Table 10. Wastage-Area-Footprint Geometry Data Centroid of Wastage Moments of Interia Eigenvalue Extraction for Prinicipal Moments and Directions Description Scaling Factor Area Perimeter Area Footprint About the Centroid Principal Moments Principal Directions xc yc I xx I yy I xy I1 I2 I1 I2 (in2) (in.) (in.) (in.) (in4) (in4) (in4) (in4) (in4) < nx, ny > < nx, ny >

As-Found Footprint 1 35.36 30.36 16.4122 -0.1194 98.89 9699.33 -117.16 75.26 197.41 <0.9004, -0.4351> <0.4351, 0.9004>

Adjusted Footprint 0.25 in. 40.06 31.78 16.4301 -0.1255 129.02 11031.81 -141.35 99.00 245.71 <0.8943, -0.4476> <0.4476, 0.8943>

for Bounding Calculation 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 x-axis along the line between the centerlines of Nozzles 3 and 11.

36

Table 10 (continued) Details of Wastage Area Footprint Before Adjustment for Bounding Calculation (Figure taken from Fig. 13 ref. [12])

Point x* y* Point x* y*

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

(a)

(b)

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

(c)

Fig 9. (continued) (c) geometry of RPV head and closure flange used in global model (B&W proprietary dimensions have been blacked out),

39

(d)

Fig 9. (continued) (d) relative location of submodel within full RPV head, 40

(e)

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

41

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

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

0.35 Center (Clad/base) 0.30 Center (mid clad)

Center (Inner Surface)

Average Thru Thickness 0.25 Effective Plastic Strain 0.20 0.15 clad thickness = 0.24 in. (constant) bounding case 0.10 0.05 P.T. Williams 6/10/2002 0.00 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Pressure (ksi)

(a) 1.80 1.60 Edge (Clad/base)

Edge (mid clad)

Edge (Inner Surface) 1.40 Average Thru Thickness 1.20 Effective Plastic Strain 1.00 0.80 clad thickness = 0.24 in. (constant) bounding case 0.60 0.40 0.20 P.T. Williams 6/10/2002 0.00 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Pressure (ksi)

(b)

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

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

As discussed above, the as-found bounding calculation predicted a PNI value of 6.65 ksi which has a cumulative probability of failure of 0.158 for the Log-Laplace model. Additional predicted burst pressure percentiles can be calculated including from Eq. (26):

1% - BP0.01 = 5.226 ksi 5% - BP0.05 = 6.015 ksi 50% - BP0.50 = 7.353 ksi 95% - BP0.95 = 8.990 ksi 99% - BP0.99 = 10.346 ksi The Log-Laplace stochastic model also estimates a cumulative probability of failure of 4.14 x 107 at the operating pressure of 2.165 ksi and 2.15 x 106 at the safety-valve 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 6.91 x 108 at 2.165 ksi, 3.60 x 107 at 2.5 ksi, and 0.2155 at 6.65 ksi.

Note in Table 11, that as the internal pressure decreases from PNI down to a nominal operating pressure, the variability in the failure probability estimates increases significantly. The standard deviation of the six estimates, when normalized by the sample mean, increases from 0.13 at 6.65 ksi to 2.44 at 2.165 ksi. The average values in Table 11 are dominated (at the lower tail) by the Log-Laplace distribution. For this reason, we recommend adopting the Log-Laplace model for future studies as the most appropriate distribution based on the available data. Note also that the Log-Laplace model produces the highest failure probabilities of the six candidates when extrapolating down into the lower tail of the distribution.

Table 11. Estimated Cumulative Probability of Failures for the Bounding Calculation Failure Probability at Internal Pressure Distribution Parameters Point Estimator Paramter Values Relative Score 2.165 ksi 2.5 ksi 6.65 ksi Log-Laplace Location Default 0 98 4.14E-07 2.15E-06 0.1582 Scale ML estimate 1.1057 Shape ML estimate 11.45441 Beta Lower endpoint MOM estimate 0.61449 93 0 0 0.2340 Upper endpoint MOM estimate 1.78866 Shape #1 MOM estimate 7.95564 Shape #2 MOM estimate 11.38552 Gamma Location Default 0 89 8.17E-19 1.50E-15 0.2236 Scale ML estimate 0.01444 Shape ML estimate 76.01293 Normal Mean ML estimate 1.09747 83.33 8.44E-10 8.90E-09 0.2234 Standard Dev. ML estimate 0.12811 Random Walk Location Default 0 75 0 0 0.2269 Scale ML estimate 0.92335 Shape ML estimate 69.18788 Inverse Gaussian Location Default 0 71 4.01E-29 1.79E-22 0.2269 Scale ML estimate 1.09747 Shape ML estimate 82.23451 Average= 6.91E-08 3.60E-07 0.2155 StdDev= 1.69E-07 8.77E-07 0.0283 StdDev/Average= 2.44 2.44 0.13 46

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:

(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[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 [7]), applied to the disk-burst tests.

Among the twenty-six 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. 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 1 SP 11.45441

0 < SP < 1.1057 x PNI 2 1.1057 x PNI Pr PBP ( true ) SP = FLP ( SP l PNI ) = 11.45441 (27) 1 SP 1 2 1.1057 x P  ; SP 1.1057 x PNI NI Given a computationally-determined pressure at numerical instability, PNI, and service pressure, SP, the model gives an estimate of the cumulative probability of nonexceedance of the true but unknown burst pressure, PBP ( true ) , i.e., Pr PBP ( true ) SP .

Due to the small sample size (n = 26) used in the stochastic model development, no definitive claim can be made that one distribution is significantly superior to the other five; however, the Log-Laplace is shown to have the highest ranking given the available data, and it produces the highest failure probabilities when extrapolating to service pressures well below the range of the data, e.g., to the nominal 47

operating pressure or safety-valve set-point pressure. The Log-Laplace stochastic model is, therefore, the recommended candidate for future applications to the Davis-Besse wastage-area problem.

As an example application, estimates are provided for a bounding calculation of the as-found Davis-Besse wastage area. The bounding calculation predicted a PNI value of 6.65 ksi. From the Log-Laplace stochastic model, the corresponding median failure pressure is 7.35 ksi. Taking the average of the estimates from all six distributions produces a probability of failure of 6.91 x 108 at 2.165 ksi, 3.60 x 107 at 2.5 ksi, and 0.2155 at 6.65 ksi.

These results for the as-found Davis-Besse wastage area can be considered bounding due to the following factors:

(a) The modeled wastage-area footprint is slightly larger than the exposed-cladding area reported in [12].

(b) The minimum cladding thickness of 0.24 inches reported in [12] was used in this analysis.

(c) A lower-bound stress-strain curve was constructed from the available tensile data for the cladding material.

(d) The reinforcing effect of the J-groove weld was not included in the simulation.

48

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

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

7. A. R. Ragab and S. E. Bayoumi, Engineering Solid Mechanics, Fundamentals and Applications, CRC Press LLC, Boca Raton, FL, 1999.
8. 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.
9. A. M. Law, Expert Fit© Users Guide, Averill M. Law & Associates, Tuscon, Arizona, May 2002.
10. Personal communication with Computational Mechanics Corporation of Columbus, Columbus, OH, March 2002, Data from PIFRAC database.
11. Nuclear Systems Materials Handbook, Vol. 1, Design Data, Section 1A for 308/308L weld, Table II.
12. S. A. Loehlein, Root Cause Analysis Report, Significant Degradation of Reactor Pressure Vessel Head, CR 2002-0891, Davis-Besse Power Station, April 15, 2002.

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