ML032600982
| ML032600982 | |
| Person / Time | |
|---|---|
| Site: | Davis Besse  |
| Issue date: | 07/15/2002 |
| From: | Bass B, Williams P Oak Ridge |
| To: | Matthew Kirk Office of Nuclear Regulatory Research |
| References | |
| DE-AC05-00OR22725, DOE 1886-N011-9B, FOIA/PA-2003-0018, Job Code Y6533 ORNL/NRC/LTR- | |
| Download: ML032600982 (50) | |
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{{#Wiki_filter:*.",l wi otve Longs computer dated 07118102 08:46am named ORNL 7-1502 Probq~Ifistic.pdr DRAFT NOT FOR ATTRIBUTION 7/15/2002 ORNLRC/LTR-Contract Program or Heavy-Section Steel Technology (HSST) Program Project
Title:
Engineering Technology Division 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: July 2002 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-NOI1-9B NRC JCN NoY6533 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-AC05-000R22725
- Act, FOJ . __ _-_nt _
DRAFT NOT FOR ATrRIBUTION 7/15/2002 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 - July 2002 Date Published - Prepared for the U.S. Nuclear Regulatory Commission Office of Nuclear Regulatory Research Under Interagency Agreement DOE 1886-NO 1-9B
- NRC JCN No. Y6533 OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37831-8063 managed and operated by UT-Battelle, LLC for the U. S. DEPARTMENT OF ENERGY under Contract No. DE-AC05-000R22725
DRAFT NOT FOR ATrRIBUTION 7115/2002 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 nformation, 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.
DRAFT NOT FOR ATTRIBUTION 7/15/2002 Stochastic Failure Model for the Davis-Besse RPV Head P. T. Williams and B. R. Bass Oak Ridge National Laboratory P. 0. Box 2009 Oak Ridge, TN, 37831-8056 Abstract The development of 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 dataobtained 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-elementanalyses of the disk-burst tests, (3) nonlinear, finite-strain, elastic-plasticflnite-element analyses performed for the current study, and (4) a theoreticalcriterion for plastic instability in a circular diaphragm under pressure loading, applied to the disk-burst tests. The final Weibull stochastic model has the scaled form of
'(P,,p xO.308 Pr(RP.P(In)*P)=Fw(PIPrn,)=lCPi(j P 00 ],(P> xO.825) 0 ,(P Px0.825)
A normal distribution is also available for nonzero failure-probability estimates at pressures below the location parameter in the Weibull distribution. Given a computationally predicted burst pressure, Pp, the model gives an estimate of the cumulative probability, Fw, that the true (but unknown) burst pressure
< P. where P is the service pressure under consideration. As an example application, estimates are provided for a bounding calculation of the "as-found" Davis-Besse wastage area.
4
DRAFT NOT FOR ATTRIBUTION 7/15/2002
- 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 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. 5
DRAFT NOT FOR ATTRIBUTION 7/15/2002 Davis Besse Reactor Vessel Head Degradation Head Cutaway View The abovetiwgerm s UewOdis Bas Dao razzle #3 matof vemef head dugadsion behwseun aid nozzle Wi. Ts skesdi was prvdd I Ul NRC by Itt Licanres. Fig. 1. (a) Davis-Besse Nuclear Power Station RPV and (b) sketch of RPV head degradation. 6
DRAFT NOT FOR ATTRIBUTION 7/1S/2002 Typical Pressurized Water Reactor Con to d Oftesft OuMet as, Fig. I (continued) (c) schematic of a typical nuclear power reactor showing the relationship of the CRDM nozzles to the RPV head. 7
DRAFT NOT FOR ATTRIBUTION 7/lS/2002 l Fig. 1. (continued) (d) photographs.of the wastage area with Nozzle 3 removed. 8
DRAFT NOT FOR ATTRIBUTION 7/1S/2002
- 2. Technical Bases The technical bases employed in the construction of the stochastic model are:
(I) 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 [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 ASME PVRC Subcommittee on Effective Utilizationof 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 [21 E eX agE-
- r iu .
I SS 304 A 0.375 0.250 2.625 15 Edge 2 B 0.125 0.125 2.875 6.8 Center 3 C 0.37S 0.125 2.625 7.7 Center 4 A533B A 0.375 0.250 2.625 11 Edge 5 B 0.125 0.125 2.875 5.3 Edge 6 C 0.375 0.125 2.625 6.7 Center 7 ABS-C A 0.375 0.250 2.625 9.8 Edge 8 B 0.125 0.125 2.875 3.75 Edge 9 C 0.375 0.125 2.625 4.94 Edge ' The three materials are representative of reactor core support structures and piping, the reactor pressure vessel, and plant component support structures [2]. 9
DRAFT NOT FOR ATTRIBUTION 7/1S/2002 Table 2. Property Data for Materials In Disk-burst Tests 121 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
- The power-law parameters in Table 2 were fitted for the current study where o = KE" and , E are the effective true stress and effective total true strain, respectively.
Geometry A 0.0~~~~~~~~~ 10.0 Geometry B
.~~~~ . _iI Fig.of2.
th Geometr thre spcmes Cosoh he ikbrtsecmn sdi aldmnin r inces) Imgeo thergtSeRooois-edrd iw f'-ymtysldmdl Fi.2 emti ecitosof the three specimensue ds-us n1](lldmnin r 10
DRAFT NOT FOR ATTRIBUTION 7/15/2002 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-elementmethod 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 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 J 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. 11
ORAFT NOT FOR ATTRIBUTION 7/15/200Z Geometry A
- T~~~~~~~~~~~~~~~~~~~~~~ v:ll _T1 t =0.25 in. r= .375 in. = 1.0in.
r = 0.3 ~ 75 in.~I-- L.-..-.--- I.. - -. ... I __ Geometry B IM** _: R: .m- a a-. - *.a-I I
=0.125 in. r =0.125 in.' 1 in.
I FF--' --- t+----r-4 I I I I I I Geometry C T Zi*=--;E av-4 *P .4,, i*4i ,>-rlNXi -itlil-z riS: v -Rma 1E-=*-:i* }*-
-T ir i- -T-'- -
I I t = 0.125 in. / _ t-.---
=__
--.- i: - ._i- __ i==
4" X= __ _= . r = 0.375 in. 01 i-i i ,- 1.0 111. _'.L_..___ _-i ' 1 . __ __
;1. .t:--
5, - -:
--..C-.-t-j--
_-eS-E-3S. ..--. -:--:E. . 1..--::'=:: HE-= _ ;=_^_ ==--r-t-- :_z _ , 3 in. I 8 5in. 1 I .- Fig. 3. Axisymmetric finite-element meshes used In the analyses of disk-burst tests reported In [21. 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
.3 ic
.3 to E.
0 0.1 0.2 03 0.4 0.5 True Strain -) 06J1112D02.Ki ptw 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 21. (See Table 2). 12
DRAFrS NOT FOR ATTRIBUTION 7/15/2002 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, (,, , ), respectively, is assumed to deform to an axisymmetric shell element with surface length, 8L, deformed thickness, h, radial position, r, and angle q). The nonuniform thickness of the dome reaches its minimum at the pole with polar height H. For a spherical coordinate system with its origin at the center of the dome, the principal strains for the thin-shell (i.e., the strains are assumed constant through the thickness) element are ee=In( r ); en = In ) m= In( h ) (1) A geometric relationship exists between the radius and chord of a circle such that H + a2 2H (2) 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, e9 , strains at any point on the spherical bulge e((z H,a)=e(zjH,a)= n[+ (z )](3) where the geometric parameter z is shown in Fig. 5. Applying the constant volume assumption, i.e.,
+ + Et = 0, produces the following equation for the radial ("thickness") strain e,(z Ha)=-2-#(zjlHa) =In[I+(zHIa2)J4 13
DRAFT NOT FOR ATTRIBUTION 7/15/2002 I to - a Fig. 5. Spherical geometry of deformation assumed In Hill's 151 plastic Instability theory. 14
DRAFT NOT FOR ATTRIBUTION 7/15/2002 The effective strain then becomes E(E~sEO'Eh)-4(E-# - ) + -+*(e ekl) 2=-E^(z ,a) = 2 nl+ 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, o, and hoop, a., membrane stresses and the internal pressure, pi, loading is a7o Afp (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 a pR; (7) and the effective stressU, V(o;r, )' + (a - ,) +(a, - a, 2his a = a; = a=pjR (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 Rp, = 2haP Rdp + pdR =2hd6 + 2Fdh (9) dp= d& + dh dR p, f h R An unstable condition exists at a point of maximum pressure on the surface where dp, = 0. The condition is unstable because any perturbation from this position always involves a reduction in load (pressure), even in a rising stress field. The instability criterion for a deformed bulge of radius R is, therefore, established by the following relation between stress and the deformed geometry for any point on the dome 15
NOT FOR ATTRIBUTION 7/15/2002 DRAFT da dR dh (10) R h or in terms of effective strain d =1 dR (1 1) d -R dE 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 (12) ad 2 4 ) 2 Applying a power-law constitutive form to relate effective stress to effective strain in the plastic region, CY=KE', 9(13) the effective strain at instability is, after a great deal of algebraic manipulation, F =(2n+1) (14) e 1wit where n is the power-law exponent in the constitutive equation, Eq. (13). 16
DRAFT NOT FOR ATTRIBUTION 7/15/2002 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. e,>, = (2n+1) 11 a Calculate the corresponding effective critical stress.
ha,, hr ep
- Calculate the critical thickness.
- Calculate the polar height at the critical condition. H.*=a FxP.. -
H+ 2
- Calculate the corresponding bulge curvature radius. RD = H,' +a 2H,,*fri
- Finally, calculate the predicted burst pressure. bunt = .,&,
r alR., 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 E.* 2(2 - n)(I + 2n) 11-4n (15) (5 17
DRAFT NOT FOR ATTRIBUTION 7/15/2002
- 3. Stochastic Model Development 3.1. Computational and Theoretical Model Results Computational results using the GAPL-3 code were presented in [2]. Converged solutions were obtained for eight of the nine tests. Comparison of experimental and computational centerline deflections showed good agreement for the eight converged cases. In the nonconverged case (ABS-C, geometry C), some difficulty was reported in getting convergence at high pressures. In all cases the experimental data showed a "tailing up" as the pressure approached burst pressure, which the computational model was unable to capture. In general, the prediction of the burst pressure for the eight converged cases showed good agreement with the experimentally-determined burst pressures. Defining a as the ratio of the experimental burst pressure to the computationally-predicted burst pressure, the mean for a was 1.19 with a standard error for the mean of +/-0.0484 and a standard deviation for the sample of 0.137.
The finite-element models using ABAQUS were able to obtain burst pressures for all nine tests, where the predicted burst pressure is defined as the pressure at which a breakdown occurs in the numerical procedure, causing the run to abort. For a nonlinear, finite-strain, static load step, ABAQUS uses automatic sizing of the load increment to maintain numerical stability. The number of iterations needed to find a converged solution for a load increment varies depending on the degree of nonlinearity in the system. If the solution has not converged within 16 iterations or if the solution appears to diverge, ABAQUS abandons the increment and starts again with the increment size set to 25% of its previous value. An attempt is then made at finding a converged solution with this smaller load increment. If the increment still fails to converge, ABAQUS reduces the increment size again. ABAQUS allows a maximum of five cutbacks in an increment before aborting the analysis. Therefore, ABAQUS will attempt a total of 96 iterations with six increments sizes before abandoning the solution. The initial load size for the failing increment was typically already very small due to difficulties in convergence with the previous and final successfully-converged load increment. Equivalent plastic strain contours are shown in Fig. 4 for the geometry A (ABS-C carbon steel) specimen (Test No. 7) at the point of numerical instability. The experimental burst pressure for this specimen was 9.8 ksi, and numerical instability of the solution occurred at approximately 9.05 ksi, for an a = 1.083. Highly localized plastic straining can be observed near the fillet, thus predicting an edge failure for this specimen which did in fact fail at its edge. 18
DRAFr NOT FOR ATTRIBUTION 7/15/2002 pEEQ M* Ot. 7519 El.?a Geometry A ABSC Predicted BP = 9.05 ksi Experimental BP = 9.8 ksi C: amA_%SC A A8AflS mckd 6.2-4 lue un I I 100:014 lt aen DWI t Tn 21 I I Ptlw17jV: Defanu J- 4 2: PEQ VaU Sp T - 0. 9052 0osfarIru4on Sede 9a Ma: *A. OOe,00 Ref. P.C. Riccardella, 'Elasto-Plastic Analysis of Constrained Disk Burst Tests,' ASME Paper No. 72-PVP-12, ASME Pressure Vessels and Piping Conference, New (a) Orleans, LA, September 17-21,1972. rEFQ Of Qt.,75% tO 5 Geometry i A
$ i S 'ii l g R K Predicted BP = 9.05 1ksi 8 ksi
[itEL~ ~~~~~~xeimna 4 BP = . Sust Usk G~ollyi A l CDI: cset ASC GM MAQJStSI d2-4 luJ Stop: Slp-I I naw M, 42: S#~lopI - 0.9052 PtI rw Va: PGQ Delorrud Vr: U Wefaulona~sedefecla: +11.000s40 (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 ATTRIBUTION 7/15/2002 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 +/-O.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. Table 3. Application of Hill's Instability Theory to Nine Disk-burst Tests Test K Ci h~~~~ E~,. 1, H., R.,, a,, h C,jg Pbrst Pbs(e) a (lit) IY ~ W a(in.) (in.) (ksi) (in.) (ksi) (kbsi). 1 162.41 0.27 2.625 0.250 0.561 1.493 3.054 138.84 0.1427 12.98 15 1.156 2 162.41 0.27 2.875 0.125 0.561 1.635 3,345 138.84 0.0714 5.92 6.8 1.148 3 162.41 0.27 2.625 0.125 0.561 1.493 3.054 138.84 0.0714 6.49 7.7 1.187 4 139.41 0.12 2.625 0.250 0.449 1.316 3.276 126.96 0.1596 12.37 11 0.889 5 139.41 0.12 2.875 0.125 0.449 1.441 3.588 126.96 0.0798 5.65 5.3 0.938 6 139.41 0.12 2.625 0.125 0.449 1.316 3.276 126.96 0.0798 6.19 6.7 1.083 7 105.20 0.17 2.625 0.250 0.490 1.383 3.183 92.95 0.1532 8.95 9.8 1.095 8 105.20 0.17 2.875 0.125 0.490 1.514 3.486 92.95 0.0766 4.08 3.75 0.918 9 105.20 0.17 2.625 0.125 0.490 1.383 3.183 92.95 0.0766 4.47 4.94 1.104 A summary of all 26 burst pressure predictions is given in Table 4. Combining the 26 cases into a single sample gives a mean for a of 1.098 with a standard error for the mean of +/-0.0251 and a standard deviation for the sample of 0.1281. Even though Hill's theory is applicable only for center failures, the good agreement between the experiments (including those that failed at the edges) suggests that, for the edge-failure cases, the specimens were also close to a condition of plastic collapse at the center when they failed first at the edge. 20
DRAFT NOT FOR ATTRIBUTION 7/15/2002
- SS304 2 __
~~....
......Geon tB T
1pchwn
. PrO GOOMI try B.I U
u S1 1.5
. . . :..J:: ' ' ' ' ~~.../..'
..... . --:-fK........ ..: ... ..- t -~~~Hr -r 1 ....... . ...-. . ' ' '.'
T . M i U 0.5
~~~~~~~~~.. ... .....
-7~~~~~~~~~~..
K>~~~~~~~~ 0 2 4 6 a 10 12 14 (a) Pressure (ksQl C6212OO2.Ki ptw A533B
~~~~~~~~~~~~~~~~~~~~..... ..... ,-
I I ......... i nilirn .. .. .. .. - ~ in
.....' ' ry ... .. ..
a 1 - .~~~~~~~~~~~~~~.... 4-aT 1l * .... ..14..... I 12k; ........ 4 T.-!l1- a
,~ometzy~ ... .....
'- a - 1 12 1 0 42 6 8 10 t~~~2 14 Lll^l (b) Prnssure (ksl) owirat2ocezc pw ABS-C 0.5 /4 --- 4 - L 1.6- :I.
Bo etIVBl,I , ometYA a 1.2 ------ -t - e. I ..w,,4,,,...... ; I
- 0. *-I
' Ig
/
7-l 41 ,
-Y..-t
[ I ti--t i U 0 2-Ii 4 6 8, 10 . 1- 14i (c) Pressure (ksQ 0W61212002Ksw 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 () ABS-C materials, and 21
ORAFT . NOT FOR AlTRIBUTION 7/1S/2002 ABS-C SS304 Specimen Specimen Failure- _ Failure Geometry C 2 LS304
. .. .. . 4 r4 - -n- - -- - ---
~~~~~~~~~~~~~~~~~~~~~.... ... . .... . ... ^. - S.....
0........ 0
~~~~~~~~~~~~~..... .... ... -----
S. ----....... 1.5 _ ..... ..._'.. AB 1- _B- _;. ...... . A I- ABS-C' 0 0 i. 0
-. . .~~~~~~~~~~~~~~~~~a l r - ,,- #-r-- e-a- ----
0
;:D 0 tD
...... , ..... 3 3 B .. .. . .. .. . ............ ........... .....
0 0 4-
-{ - .~~~~~~~~~~~.. - -- '-. ... .'-'---'-- ..... --- '-= ,
s 0.5
~~~~~~~.... . ........ .... ........ . ... ...... ..... ..... .....
'/
. .t. .. . .. . . . . .. . . . .. . . . . . .. .. . .. ... ..... ..... ..... ..... .....
n 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 7/15/2002 Table 4. Comparison of Experimental Burst Pressures to Three Predictions I 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 Ceater 1.19 6.59 Center 1.17 4 A533B A II 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
DRAFT NOT FOR ATTRIBUTION 7/15/2002 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 statistics for the ratio of experimental burst pressure to predicted burst pressure, a, for the three predictive methods discussed in the previous section. Also shown in the table are the results of combining the three samples into one larger sample of 26 data points. Using the point estimating procedures described in the Appendix, a three-parameter Weibull distribution was fitted to the finite-element model predictions. The results of that fit are shown in Table 6 with a comparison to a normal distribution fit. The Kolomogorov-Smirnoff(K-S) one-sample goodness of fit statistics indicate that both models would be accepted under the null hypothesis that the sample was randomly drawn from the candidate continuous distribution. The normal distribution gives a higher K-S p-value for this sample. A continuous three-parameter Weibull distribution has the following probability density (PDF) and cumulative distribution finctions (CDF) I fw( a,b,c)( ) exp - )] (a>a, (b,cO) 16)
*L i ~~~~~~~~~~~~~~~(1 b~~~~ ( b PrX a=waabc= =-x (- (a >a, (b, c) >0) 24
DRAFT NOT FOR ATTRIBUTION 7/15/2002 Table 5. Descriptive Statistics for the Ratio of Experimental Burst Pressure to Predicted Burst Pressures r1zr1-,Y--11VTt-,f---5rnn=M" i 7W Fr, tesOriptiuve taisticSt RlVOcWMCrdllat1Y7;)kJ1s Ticor ABAQUS CombicciN 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. Weibull Model Parameters and Median Rank Order Statistics for ABAQUS Predictions A533B A 0.89718 2 0.181 A533B B 1.01186 3 0.287 ABS-C A 1.08268 4 0.394 SS 304 B 1.09393 5 0.500 ABS-C C 1.10722 6 0.606 A533B C 1.11041 7 0.713 SS 304 A 1.12876 8 0.819 cQClA r 1Q) 0 n 04 K-S p-value li ~ w e~ bWlb Normal Location 0.848 0.2928 0.4236 Scale 0.232 Shape 2.352 Model Sample Mean 1.0535 1.0549 Variance 0.0087 0.0099 Std. Dev. 0.0930 0.0993 Median 1.0464 1.0939 25
DRAFT NOT FOR ATTRIBUTION 7/15/2002 where a is the location parameter, b is the scale parameter, and c is the shape parameter. The inverse CDF for the Weibull distribution is given by a=a+b[-ln(I-P)]-; O<P<1 (17) The normal distribution function has the form f ( I#ua) p [ i(a _.U)2 2( )]) (18) pr(Xs)=FN(apo)=4=J;1exp[ l(x-p' Combining the three sets into a single sample gives a sample size large enough that additional goodness offit statisticaltests may be carried out. Table 7 shows a ranking of the 26 data points with the median rank order statistic given by i-O.3 (19) n+0.4 Also given in Table 7 are the results of a x2 goodness of fit test comparing the Weibull distribution fitted for the combined data set with a normal distribution. The Z2 statistics were calculated using 5 intervals with 2 degrees of freedom. The resulting p-values shown in the table indicate that both distributions would be accepted by the null hypothesis for the test; however, the Weibull fit has a higher confidence level of 42.9% compared to 7.6% for the normal distribution. The combined dataset was also fitted to two-parameter gamma and log-normal distributions with the results shown in Table 8. Marginal univariate tests using the three empirical distribution function (EDF) statistics of Anderson-Darling, Cramer-von Mises, and Watson were calculated using the GenStat program [9]. For each univariate test, the test statistics are empirical distribution function (EDF) statistics that compare the empirical distribution function of the sample with the theoretical distributions expected under the null hypothesis. These tests provide good power against a wide range of alternatives [O, 11]. The parameters for all four distributions were estimated by GenStat using maximum likelihood estimators. The univariate test results in Table 8 give the Weibull distribution a significance level of approximately 10% (confidence level of 90%) with the other three distributions having acceptable but lower confidence levels; therefore, all four distributions would be accepted by the null hypothesis with the 26
DRAFT NOT FOR ATTRIBUTION 7/15/2002 Weibull distribution indicating a better fit to the data. Figure 6 presents GenStat probability-probability (P-P) plots of the four distributions with order statistics given by (-0.5)n. A qualitative comparison between the plots in Fig. 6 indicates only minor differences between the distributions. Table 7. Weibull Model Parameters and Median Rank Order Statistics for Combined Predictions I Hilrs Theory A533B A 0.8889 0.0265 2 ABAQUS Sohi 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 Ricandella (1972) SS 304 C 1.0405 0.2917 9 ABAQUS Solh. 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 OA053 12 Hilrs 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 Hilrs 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 Ricandella(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 Ricandella (1972) A533B B 1.2619 0.9356 26 Ricarrdella (1972) SS 304 B 1.4167 0.9735 X= Experimental Burst Pressure/Pedicted Burst Pressure W Weisiill PaWeiieters". 2 Significance Test Location 0.825 Distr. X2 p-value DOF Scale 0.308 Nonnal 5.16240 0.07568 2 Shape 2.301 Welbull 1.69430 0.42864 2 Model Sample Mean 1.0975 1.0975 Variance 0.0158 0.0164 Std. Dev. 0.1256 0.1281 Median 1.0876 1.1057 27
DRAFT NOT FOR ATTRIBUTION 7/1S/2002 Table 8. Marginal Univariate Tests for Four Distributions for Combined Sample SuLTh _7 N _ _ _p~Adi~t~ aK?. Normal mean - 1.098 0.44 0.061 0.061 standard deviation - 0.128 Gamma shape - 75.82 0.501 0.073 0.072 scale - 0.01448 Log-Normal log-mean - 0.0867 0.546 0.082 0.08 log-stdv - 0.1153 location - 0.825 Weibull scale - 0.308 0.652 0.103 0.0997 shape - 2.302 Table 9. Critical Values of Marginal EDF Test Statistics Anderson-Darling 0.576 0.656 0.787 0.918 1.092 Cramer-von Mises 0.091 0.104 0.126 0.148 0.178 Watson 0.085 0.096 0.116 0.136 0.163 z 2 2 I I 9 Expected Probability Expected Probability ZI I A41
.0 I
0 75.82 (shape) / 21 / b = 0.308 (scale)
.EAdz b = 0.01448 (scale) e C = 2.302 (shape)
, 0 A 0s 0. 0 0.2 04 0.6 0.8 1 Expected Probabiliy Expected Probability Fig. 6. GenStatP P-P plots of four distributions fitted to combined data set.
28
DRAFT NOT FOR ATTRIBUTION 7/15/2002 For completeness, the continuous two-parameter lognormal distribution is given by f (a A.,I = 7i;=exp [I j, (-r,<a<-o) _______ _______2] (20) Pr(X S a) =Fv(aIA(tpo) exp[ I h(ln(x)I s f (-AI < a <cc) and the two-parameter gamma distribution is described by fr~~b~P~()a [exp bX~) (-->a>-c, b>0) 1 ~~~~~~~(21) Pr(X a)=Fr(a b,c)=P lb(b& [ rabr ( b>o) where b is a scale parameter and c is the shape parameter. Figure 7 compares the probability density functions of the four models to the histogram for the combined data set, and Figs. 8 and 9 show the probability density and cumulative distribution functions for the Weibull and normal models fitted to the n = 9 sample of FEM predictions. The corresponding PDF and CDF plots for the Weibull and normal models fitted to the combined sample of n = 26 are given in Figs. 10 and 11. From these results, it was decided to concentrate on the Weibull and normal distributions. These two distributions will produce bounding estimates on the conditional probability of failure for low pressures relative to the predicted burst pressure with the Weibull distribution being recognized as giving the better fit to the data. 29
DRAFT NOT FOR ATTRIBUTION 7/15/2002 4 I- I-I - aI I I I -I Ii I I C I I I I , I I I I 3.5 3 i 2.5 D &L1.5 0.5 0=1 0.8 0.9 1 1.1 1.2 1.3 1.4 1 .5 Experimental BP/Predicted BP, 07/07/2002.K1 ptw Fig. 7. Histogram of combined data set compared to probability densities of four fitted distributions. 30
DRAFT NOT FOR ATTRIBUTION 7/15/2002 4 4. 0C 3
.0 CZ 0
2 CL f . .God 1 0.6 0.8 1 1.2 1.4 Exp. Burst PressurelPredicted Burst Pressure, 07i1:02,ga p1W Fig. 8. Probability densities for two continuous statistical distributions fitted to the sample of 9 data points for a - experimental burst pressureAFEM predicted burst pressure. 0.8 1 1.2 1.4 Exp. Burst Pressure/Predlcted Burst Pressure, O7W0102.K2 t Fig. 9. Weibull statistical failure model (n = 9) compared to a normal cumulative distribution function, median rank order statistic, and the 90% confidence Interval on the order statistic. ABAQUS FEM solutions were used to develop the models. N.B.: The order statistics and their 90% confidence intervals are shown here for comparative purposes only and were not used In the point-estimate procedures for the parameters of either the Weibull or normal distributions. 31
DRAFT NOT FOR ATTRIBUTION 7/15/2002
- P 2.5 x / e; \ mean = 1.09t Cn Agemedian= 1.088 o2- Welbull Distribution Location 0.825 80 -Scale =0.308 1.5 Shape 2.302 IL~~~~~~~~~~~~~~~~~I 0.5
,n, , , _. e.g, , e , . ... *.e . a, . ...
.6 0.8 1 1.2 1.4 Exp. Burst PressurelPredicted Burst Pressure, ax 07103/02.K1 plt Fig. 10. Probability densities for two continuous statistical distributions fitted to the combined sample of 26 data points for a - experimental burst pressure / predicted burst pressure.
1 , I , , , , ,
- Weibull Distribution - *,
0.8 Location =0.825 /
. Scale =0.308 U. Shape 2.302 .
o - . / Normal Distribution , Iy Mean = 1.0975 IL 0.4 Std Dev =0.1281 J E 0.2 95/ . p t n +0.4 O f . l ;.* I l I 0.6 0.8 1 1.2 1.4 Exp. Burst PressurelPredicted Burst Pressure, a 07tf3/02.K2 ptw Fig. 11.WeibuU statistical failure model (n - 26) compared to a normal cumulative distribution function, median rank order statistic, and the 90% confidence Interval on the order statistic. Models developed with combined sample. N.B.: The order statistics and their 90% confidence Intervals are shown here for comparative purposes only and were not used in the point-estimate procedures for the parameters of either the Welbull or Normal distributions. 32
DRAFT NOT FOR ATrRXBUTIdt 7/15/2002 The recommended Weibull stochastic model of failure has the following form 2.302(a-0.825)lD2 [ (a-0.825 1 fw,(a) = 0.308 ~.0.308 ) XL~ 0.308 (ja> 0.825) 0 ,(a 5 0.825) (22) a xp _(a-~~~~~0.825 .~.O Pr(Xf-a)= a jfw()d~c={ [ ~le I [ r0x[(J
.2)3l](a > .8825) 308 0 ,(a5 0.825) where a is the ratio of the true (but unknown) burst pressure to the calculated burst pressure. If nonzero estimates are required for values of a <0.825, then the following normal distribution may be used f =~~ I.
f(a)= 0.1281'2x expF -(a-1.0975 P-2 1(_-952 0.1281 ] <aa (23) Pr(X 5 a) = I Lexp 1(S-1.0975)\ d{, ( <a < 0.1281 2 l, 2 0.1281 These models may also be considered as statistical estimates of the expected predictive accuracy of computational methods applied to burst pressure calculations for constrained diaphragms. 33
DRAFT NOT FOR ATTRIBUTION 7/15/2002
- 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. 12. An adjusted stress-strain curve (see Fig. 13) 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 CauseAnalysis 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. 14 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. [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. 15). For the predicted burst pressure of 6.65 ksi, the normal and Weibull statistical failure models can be scaled to provided estimates of cumulative probability of failure as a function of internal service pressure for the specific condition of the wastage area simulated by the finite-element analysis. Examples of the scaled Weibull model are shown in Figs. 16 and 17 for normalized internal pressure and direct internal pressure, respectively. The scaled Weibull model has the form (PIP)= 2 (302 P230 ) exp -( P(P>Ppx0 B-P3 ) 825) PPxO.308 P,. xO.308 P, BxO.308 P
\2P302 (24)
Pr
- 5 P IP,) I P 825 ,(P> Prnp x0. 825)
,p x 0.308 0 ,(P P, xO.825) where Pap is the predicted burst pressure (either normalized by the operating pressure or in dimensional form), and PPB(Q,.e) is the unknown true burst pressure. The scaled normal distribution is A (PI PP) = I P,p XO.I2872 ex13 2
( Pp X 12975 1? x 0. 281 )2] (25) Pr(PP(.) PIP=P)= 12 exp P XO Jd 34
DRAFT NOT FOR ATTRIBUTION 7/15/2002 Table 10. Wastage-Area-Footprint Geometry Data g>E ark a,s a>S^>Ai4',t4-iEfl¶iflU*
$1,nWtp,' ,,.sa Mo~n~IiI1M As-Found Footprint 35.36 30.36 16.4122 4.1194 98.89 99.33 -117.16 75.26 197.41 <0.9004,-0A35> c0A351,0.9004>
Adjusted Footprint 025 in. 40.06 31.78 16.4301 4.1255 129.02 11031.31 -141.35 99.00 245.71 <0.3943, 4.4476> <0.4476,0.3943> for Bounding Calculation Footprint centroid is In global coordinates. Global coordinate system ha 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 1.
"As Found" Footprint Area = 35.36 In' Perimeter 30.36 In.
35
DRAFT NOT FOR ATTRIBUTION 7/15/2002 Table 10 (continued) Details of Wastage Area Footprint Before Adjustment for Bounding Calculation (Figure taken from Fig. 13 ref. 1121)
- 117IAwi oh tc1 Lewis iii IIrno C'y.I
. X f itch..Sw1 4-i 1,ritizdI I. t Pt'~n J I
. I . _ .,
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) 36
DRAFT NOT FOR ATTRIBUTION 7/15/2002 (a) Submodel of Wastage Area Nozzles 3, 11, 15, and 16 Cladding SS30B Weld (0.297 In. thick) Base Material with Wastage Area (b) v Fig. 12. Finite-element global and submodels of the Davis-Besse bead 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. 37
DRAFT NOT FOR ATTRIBUTION 7/1S/2002 I ooo AX oog - \A ' ~K i I / ,~~~~ 0 0 0 a ," A 000 0 00 0 0 Global Model of Davis-Besse 0 00 0 0 a 00 RPV Head and Closure Flange
~~~~1b o~:o --- tl 00 0 0 0 0 00 o o 4<
"/
/ws m R11 l, ... .... . .
SECTCRA . _ _ _ _ . _ _ !ft i V - Oak Ridge National Laboratny l __-] RIAf C .* _ _- Davis-Besse RPV Head INowsse __tCCW~ nvaC and Closure Flange ___ __ _C
.f~~ _tmtmw _
KM_ Ct^>l _ IW 0o~104-m il~ll^
- INA-MM o
!l 7;W-n - T- -,,- - _ 0_ . _ i,
_tss... (c) - _" 002O _ __ _.__ Arg Fig 12 (continued) (c) geometry of RPV head and closure flange used In global model, 38
DRAFT NOT FOR ATTRIBUTION 7/15/2002
+4k N' N Six h e A-")
t9
©
© I I
© 1 © %I I
I
+ 0 I 4-I I
I/ I
© '9 I I/
I I L9 3 ALDIMENSK3NS IN INES Computational Sdences Oak Ridge National Laboratory and Engineering Dvision Modeling and Simulon Caop Heavy-Section Steel Technology Program UT-Battelle, LLC OY' P. T. WAU4 SIZE TITLE Dait$eSo RPVHead OWNO. l REV A, l Foapoht;al WastW Ames I CNK BY. SCALE IDATE 21 APRIL 2D2 SHEET (d) Fig 12 (continued) (d) relative location of submodel within full RPV head, 39
DRAFT NOT FOR ATTRIBUTION 7/15/2002 C P~~~ 0-0
+ 5--
P. -R3' I _ _ _ .--. _- Coiputatonal Sdkmces Oak Ridge National Laboratory end Engneerlng Division Modefing and Simul1dan Grop Heavy-Section Steel Technology Program UT-Battelle, LLC DRtet P tWMfms 1 SIE TE TVtc1r9 of W" Au t o NO. __A ft B*iAuy f SUMctsI CHK 11Mt YSCIF DATE 42t2 SHET 1-.' I I Fig 12 (continued) (e) geometry of submodel relative to Nozzles 3, 11, 15, and 16. 40
DRAFT NOT FOR ATTRIBUTION 7/1S/2002
. . . . t , . I . I 1. I , i -, I -, i 80 t Framatome 1 14.992 £,¢ _.
t Cunt _ ,-. ~ MW - lt = 69.65 ksi
,.V 60 , c4,, = 61.64 ksi A W jfS s Adjusted SS38Curve
______________________ for Bounding Calculation 40 gU t For both SS308 curves 0 = 94.3 59 £,0194l 25 tZ CT.0B¢Zkslr ... 30.96 I = I-kr uniform elongation - 11.15% SS 308
- _ SS 308 ladiusted
-* - A8W-101
~-AW-102 M
20 At 600 F -,- A8W-103
; =s 25,5;71 ksi -- A8W-104 v = 0295 -. - A8W-105 A8W-106 A8W d at 5 F 0 0.05 0.1 0.15 0.2 True Strain (-) 06/1012002.K1 ptw Fig. 13. 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.
41
NOT FOR ATTRIBUTION 7/15/o20 DRAFT Adjuikd' FrI& Nd.4 .1. I' A N-Caiid of
- rint
~~a~~~.i1Mkrm OS P-WiL Fig. 14. Geometry of adjusted wastage area footprint. Lower figure is a PhotoworksD-rendered image of the submodel with the adjusted "as-found" footprint.
42
DRAFT NOT FOR ATTRIBUTION 7/15/2002 1 020. I I0.15 Ad ^.n (a) UD on 1 d00 D.AD 0.0 1.5 L. 2.0 4* S0 ' 0 (b) m Fig. 15. Effective plastic-strain histories at two high-strain locations in the wastage area: (a) near the center and (b) near Nozzle 3. 43
DRAFT NOT FOR ATrRIBUTION 7/15/2002 Internal Pressure I Operating Pressure 2 2.5.i : .,:! ' 3 3.5 4 4.5 I ..... .... . . . . ,. .
- ! ! ! 1 Weibull Distnbution ...... -j -1'0 ...:--.i --I-.
Location = 2.534 ...... L. . . . . . . .. . . .:- --: . . : .- I-0 Scale -0.946 ....... . .I- , . . : . ..I . I 'I ' ' 0.8 Shape 2.302
. I . _ I
- Z : ::::.
4-0 ~~~~~~.. .... ...._........ ..... ....... - : : - Normal Predicted i; 0,6 Z'
.0 Operating =2.165ksi Burst = 6.65 ksla 3
0 . Pressure Pressure :7f9E it 0.4II ,M"- - --11---,%-.11---'- I Un,. 4-E 0.2 Weibull Distribution C.) .. I . Location 0.825
. Scale =0308 El U 0.6 0.8
. ....U......L.
1 1.2 Shape= 2.3021 1.4 Exp. Burst Pressure/Predicted Burst Pressure, a 07103102.K5 ptw Fig. 16. Application of the failure statistical criterion produces a cumulative probability of failure (based on a Welbull distribution) curve for the Bounding Case condition. Cumulative probability of failure as a function of normalized Internal pressure. Internal Pressure ksl) 4 4.8 5.6 6A 7.2 8 8.8 9.6 1 I , , .. j I 4 I I I i , J II I I _, Weibul Distribution :
.. Location = 5.486 . .. 7.i S.. Scale
_Shape =2.302 6.291 I . . . . 0.8
.-'-1*--fl-- - '
V
. .. . ......... _ ..z .
-.0 l-0.6
. Pi Predicted
.0 Burst =6.65 ksia
.0 0.~PressureI E
.2
' ' ! ... 1 --! -- 1 ' ' - -' ' - .......
E 0.2 . -- - -i Weibull Distribution E0 -- - Location = 0.825 Scale =0308 Shape =2.302 0.6 0.8 1 1.2 1.4 Exp. Burst PressurelPredicted Burst Pressure, 07103/02.K4 ptw Fig. 17. Application of the failure statistical criterion produces a cumulative probability of failure (based on a Welbull distribution) curve for the Bounding Case condition. Cumulative probability of failure as a function of Internal pressure. 44
DRAFT- NOT FOR ATTRIBUTION 7/1S/2002 As discussed above, the bounding calculation predicted a burst pressure of 6.65 ksi which has a cumulative probability of failure of 0.244. For pressures below 5.486 ksi (at the position of the location parameter), the Weibull model predicts a zero probability of failure. The model based on a normal distribution estimates a cumulative probability of failure of 8.43xlO-10 at the operating pressure of 2.165 ksi and 8.89 x109 at 2.5 ksi. See Table 11 for additional estimates. Table 11. Estimated Cumulative Probability of Failures for the Bounding Calculation 2.155 7.84E-10 0 2.165 8.43E-10 0 2.200 1.09E-09 0 2.225 1.30E-09 0 2.250 1.55E-09 0 2.275 1.86E-09 0 2.300 2.22E-09 0 2.325 2.65E-09 0 2.350 3.1SE-09 0 2.375 3.76E-09 0 2.400 4.47E-09 0 2.425 5.32E-09 0 2.450 6.32E-09 0 2.475 7.50E-09 0 2.500 8.89E-09 0 45
DRAFT NOT FOR ATTRIBUTION 7/15/2002
- 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) experimentaldata 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-elementanalyses of the disk-burst tests also reported in [2] (GAPL-3 discrete-element code[3]), (7) nonlinear, finite-strain, elastic-plasticfinite-element analysesperformed for the current study (ABAQUS finite-element code[4]) of the nine disk-burst test specimens reported in [2], and (8) a theoretical criterionfor plastic instability in a circular diaphragm under pressure loading, due to Hill [5] (as cited in [6]), applied to the disk-burst tests. The final Weibull model has the scaled form of e _(jP-' xO.825 I >IP .85 Pr(fP(lTh) P)= F(PP)I [ P xO.308P) x0825) (26) 0 ,(P P xO.825) Given a computationally predicted burst pressure, Pp, the model gives an estimate of the cumulative probability that the true but unknown burst pressure P,, ) P. where P is the service pressure under consideration. A normal distribution is also available for nonzero failure-probability estimates at pressures below the location parameter in the Weibull distribution. 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 0.244. For pressures below 5.486 ksi (at the position of the location parameter), the Weibull model predicts a zero probability of failure. The model based on a normal distribution estimates a cumulative probability of failure of 8.43 xl 0-1 at the operating pressure of 2.165 ksi and 8.89 x109 at 2.5 ksi. 46
DRAFT NOT FOR ATTRIBUTION 7/15/2002 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 ComputerProgramfor 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, EngineeringSolid Mechanics, FundamentalsandApplications, 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-PVP49, ASME Pressure Vessels andPipingConference, May 1971.
- 8. J. Cbakrabarty and J. M. Alexander, "Hydrostatic Bulging of Circular Diaphragms," J. Strain Anal.
5(3), (1970) 155-161.
- 9. GenStatcR'Release 6.1 Reference Manual, VSN International, Oxford, UK, 2002.
- 10. M. A. Stephens, EDF Statistics for Goodness of Fit and Some Comparisons," Journal of the American StatisticalAssociation 69 (1974), 730-737.
- 11. J. A. Aitchison, The StatisticalAnalysisof CompositionalData, Chapman and Hall, London, 1986.
- 12. S. A. Loeblein, Root Cause Analysis Report, Significant Degradation of Reactor Pressure Vessel Head,CR 2002-0891, Davis-Besse Power Station, April 15,2002.
47
Appendix A - Statistical Point-Estimation Techniques for Weibull Distributions The three parameters for the Weibull distributions of a (=experimental burst pressure/predicted burst pressure) were calculated using a combination of two point-estimation procedures, Maximum Likelihood and the Method of Moments. The parameters to estimate are the location parameter, a, of the random variate, the scale parameter, b, of the random variate, and the shape parameter, c. Maximum likelihood estimators for the shape parameter c' and the scale parameter b can be derived from the likelihood function, L, for the Weibull distribution. The Weibull density is given by wmala,b,c) = Cyc- exp yc), for b (A.1) (y=(a-a)/b,a>a,b,c>O) and the corresponding likelihood function is the joint density (see Ref.[A1]) (given the location parameter, a) L(bclaa)= jjc aQ)-al1 E)( -a C- (A.2) The maximum likelihood (L) estimators for the scale, b', and shape parameters, c', are defined as the unique values of (b',c') that maximize the joint probability that the N members of the sample set all come from the same parent population. The AL estimators are, therefore, calculated by finding the stationary point of Eq. (A.2). Upon taking the logarithm of Eq. A.2), the derivatives with respect to the individual parameters (b',c') are set to zero. The resulting ML estimator for the shape parameter, c', is found by solving iteratively for c' in the following nonlinear equation N a(ln(L(c')) f(ap0_a)C ln(a)-a) 1 _______ -a i X n')-o (A.3) Upon obtaining a solution for c', the Ml estimator for the scale parameter, b', follows directly from 48
Nibs(l)) = ' (A.4) For the ML point estimators for (b',c'), the location parameter, a, was assumed given. The Method of Moments (MM) can now be applied to provide a point estimate for the location parameter, a . In the Method of Moments, the sample moments are used as estimators for the population moments. The MM point estimator for the scale parameter, b , is (given the shape parameter, c), b = 2 /[F(1 +2/c)-r 2 (l +I/c)] (A.5) where m2 is the second moment of the sample about the sample mean and r is Euler's gamma function. The MM estimator for the location parameter, a* , follows from a =n4-b5 F(l+l~c) (A.6) where ml is the I" crude moment of the sample (the sample mean) and the sample moments are defined by N ml = aI) j~ N
)
N (Iml - (A.7) in2 = From Ref. [B.2], a moment estimator for the shape parameter, c, also exists 4.104683-1.148513~f-+ 0.44326(4) 2 -0.O53025(0)3 (.8)
+ 1..139547 where is the sample skewness. However, for sample sizes as small as 20, there will be a high level of uncertainty in the (a*,b',c') estimates derived from c* (Ref. A.2]).
49
The three parameters for the Weibull distribution of a were estimated through the following iterative sequence:
- 1. For the discrete set (a(E), i = 1,N), calculate the sample moments, (m,m 2 ) from Eqs. (A.7).
- 2. Select a trial value for the location parameter, a.nal where atial < min(a(;),i = 1,2,...N).
- 3. Calculate ML estimates for (c',b) from Eqs. (A-3)-(A.4) by letting a = aia,.
- 4. Calculate MM estimates for (a*,b) from Eqs. (A.5)-(A.6) by letting c = c' as determined in Step 3.
- 5. Calculate a relative deviation between the trial a,a1 and the MM estimate of a*from Step 4 by a = 0tria - a (A.9) atfial
- 6. Given tokrance as a pre-selected convergence tolerance, if 8 > etolernce then select a new trial location parameter, aia,, and repeat Steps 3-6 until convergence, defined as o6 S Ctolerance -
Upon convergence, there will be two triplets (atr,,ab,c') and (a*,b*, c) where in general atjdal a* and b'
- b although b' was typically close to b in this study. The triplet (a ,b' c) was taken as the converged estimate for the parameters of the Weibull distribution for a.
References Al. A. Ghosh, "A FORTRAN Program for Fitting Weibull Distribution and Generating Samples," Computers & Geosciences 25, (1999) 729-738. A2. K. 0. Bowman and P. T. Williams, Technical Basisfor StatisticalModels of Extended K,, and K&1 FractureToughness Databasesfor RPV Steels, ORNLINRC/LTR-99/27, Oak Ridge National Laboratory, Oak Ridge, TN, February 2000. 50}}