...

ORNL2004-FracturePressureVessels.pdf

by user

on
Category: Documents
15

views

Report

Comments

Transcript

ORNL2004-FracturePressureVessels.pdf
NUREG/CR-6854
ORNL/TM-2004/244
Fracture Analysis of
Vessels – Oak Ridge FAVOR,
v04.1, Computer Code: Theory
and Implementation of
Algorithms, Methods, and
Correlations
Oak Ridge National Laboratory
U.S. Nuclear Regulatory Commission
Office of Nuclear Regulatory Research
Washington, DC 20555-0001
AVAILABILITY OF REFERENCE MATERIALS
IN NRC PUBLICATIONS
NRC Reference Material
Non-NRC Reference Material
As of November 1999, you may electronically access
NUREG-series publications and other NRC records at
NRC’s Public Electronic Reading Room at
http://www.nrc.gov/reading-rm.html.
Publicly released records include, to name a few,
NUREG-series publications; Federal Register notices;
applicant, licensee, and vendor documents and
correspondence; NRC correspondence and internal
memoranda; bulletins and information notices;
inspection and investigative reports; licensee event
reports; and Commission papers and their attachments.
Documents available from public and special technical
libraries include all open literature items, such as
books, journal articles, and transactions, Federal
Register notices, Federal and State legislation, and
congressional reports. Such documents as theses,
dissertations, foreign reports and translations, and
non-NRC conference proceedings may be purchased
from their sponsoring organization.
NRC publications in the NUREG series, NRC
regulations, and Title 10, Energy, in the Code of
Federal Regulations may also be purchased from one
of these two sources:
1. The Superintendent of Documents
U.S. Government Printing Office
Mail Stop SSOP
Washington, DC 20402–0001
Internet: bookstore.gpo.gov
Telephone: 202-512-1800
Fax: 202-512-2250
2. The National Technical Information Service
Springfield, VA 22161–0002
www.ntis.gov
1–800–553–6847 or, locally, 703–605–6000
A single copy of each NRC draft report for comment is
available free, to the extent of supply, upon written
request as follows:
Address: Office of the Chief Information Officer,
Reproduction and Distribution
Services Section
U.S. Nuclear Regulatory Commission
Washington, DC 20555-0001
E-mail:
[email protected]
Facsimile: 301–415–2289
Some publications in the NUREG series that are
posted at NRC’s Web site address
http://www.nrc.gov/reading-rm/doc-collections/nuregs
are updated periodically and may differ from the last
printed version. Although references to material found
on a Web site bear the date the material was accessed,
the material available on the date cited may
subsequently be removed from the site.
Copies of industry codes and standards used in a
substantive manner in the NRC regulatory process are
maintained at—
The NRC Technical Library
Two White Flint North
11545 Rockville Pike
Rockville, MD 20852–2738
These standards are available in the library for
reference use by the public. Codes and standards are
usually copyrighted and may be purchased from the
originating organization or, if they are American
National Standards, from—
American National Standards Institute
11 West 42nd Street
New York, NY 10036–8002
www.ansi.org
212–642–4900
Legally binding regulatory requirements
are stated only in laws; NRC regulations;
licenses, including technical
specifications; or orders, not in
NUREG-series publications. The views
expressed in contractor-prepared
publications in this series are not
necessarily those of the NRC.
The NUREG series comprises
(1) technical and administrative reports
and books prepared by the staff
(NUREG–XXXX) or agency contractors
(NUREG/CR–XXXX), (2) proceedings of
conferences (NUREG/CP–XXXX),
(3) reports resulting from international
agreements (NUREG/IA–XXXX),
(4) brochures (NUREG/BR–XXXX), and
(5) compilations of legal decisions and
orders of the Commission and Atomic
and Safety Licensing Boards and of
Directors’ decisions under Section 2.206
of NRC’s regulations (NUREG–0750).
NUREG/CR-6854
ORNL/TM-2004/244
Fracture Analysis of
Vessels – Oak Ridge FAVOR
v04.1, Computer Code: Theory
and Implementation of
Algorithms, Methods, and
Correlations
Manuscript Completed: September 2004
Date Published: August 2007
Prepared by
P.T. Williams, T.L. Dickson, and S. Yin
Oak Ridge National Laboratory
Managed by UT-Battelle, LLC
Oak Ridge, TN 37831-8063
E. Focht, NRC Project Manager
Prepared for
Division of Fuel, Engineering and Radiological Research
Office of Nuclear Regulatory Research
U.S. Nuclear Regulatory Commission
Washington, DC 20555-0001
NRC Job Code Y6533
ii
Abstract
The current regulations to insure that nuclear reactor pressure vessels (RPVs) maintain their
structural integrity when subjected to transients such as pressurized thermal shock (PTS) events
were derived from computational models developed in the early-to-mid 1980s. Since that time,
advancements and refinements in relevant technologies that impact RPV integrity assessment
have led to an effort by the NRC to re-evaluate its PTS regulations. Updated computational
methodologies have been developed through interactions between experts in the relevant disciplines of thermal hydraulics, probabilistic risk assessment, materials embrittlement, fracture
mechanics, and inspection (flaw characterization). Contributors to the development of these
methodologies include the NRC staff, their contractors, and representatives from the nuclear
industry. These updated methodologies have been integrated into the Fracture Analysis of
Vessels – Oak Ridge (FAVOR, v04.1) computer code developed for the NRC by the Heavy
Section Steel Technology (HSST) program at Oak Ridge National Laboratory (ORNL). The
FAVOR, v04.1, code represents the baseline NRC-selected applications tool for re-assessing the
current PTS regulations. Intended to document the technical bases for the assumptions,
algorithms, methods, and correlations employed in the development of the FAVOR, v04.1, code,
this report is one of a series of software quality assurance documentation deliverables being
prepared according to the guidance provided in IEEE Std. 730.1-1995, IEEE Guide for Software
Quality Assurance Planning. Additional documents in this series include (1) FAVOR, v01.1,
Computer Code: Software Requirements Specification, (2) FAVOR, v01.1, Computer Code:
Software Design Description, and (3) FAVOR, v04.1, Computer Code: User’s Guide.
iii
iv
Foreword
The reactor pressure vessel is exposed to neutron radiation during normal operation. Over time,
the vessel steel becomes progressively more brittle in the region adjacent to the core. If a vessel
had a preexisting flaw of critical size and certain severe system transients occurred, this flaw
could propagate rapidly through the vessel, resulting in a through-wall crack. The severe
transients of concern, known as pressurized thermal shock (PTS), are characterized by rapid
cooling (i.e., thermal shock) of the internal reactor pressure vessel surface that may be combined
with repressurization. The simultaneous occurrence of critical-size flaws, embrittled vessel, and a
severe PTS transient is a very low probability event. The current study shows that U.S.
pressurized-water reactors do not approach the levels of embrittlement to make them susceptible
to PTS failure, even during extended operation well beyond the original 40-year design life.
Advancements in our understanding and knowledge of materials behavior, our ability to
realistically model plant systems and operational characteristics, and our ability to better evaluate
PTS transients to estimate loads on vessel walls have shown that earlier analyses, performed
some 20 years ago as part of the development of the PTS rule, were overly conservative, based on
the tools available at the time. Consistent with the NRC’s Strategic Plan to use best-estimate
analyses combined with uncertainty assessments to resolve safety-related issues, the NRC’s
Office of Nuclear Regulatory Research undertook a project in 1999 to develop a technical basis to
support a risk-informed revision of the existing PTS Rule, set forth in Title 10, Section 50.61, of
the Code of Federal Regulations (10 CFR 50.61).
Two central features of the current research approach were a focus on the use of realistic input
values and models and an explicit treatment of uncertainties (using currently available uncertainty
analysis tools and techniques). This approach improved significantly upon that employed in the
past to establish the existing 10 CFR 50.61 embrittlement limits. The previous approach included
unquantified conservatisms in many aspects of the analysis, and uncertainties were treated
implicitly by incorporating them into the models.
This report is one of a series of 21 reports that provide the technical basis that the staff will
consider in a potential revision of 10 CFR 50.61. The risk from PTS was determined from the
integrated results of the Fifth Version of the Reactor Excursion Leak Analysis Program
(RELAP5) thermal-hydraulic analyses, fracture mechanics analyses, and probabilistic risk
assessment. This report is the theory manual for the probabilistic fracture mechanics code
Fracture Analysis of Vessels, Oak Ridge (FAVOR). The FAVOR code is used to assess
structural integrity of pressurized-water reactor pressure vessels during postulated pressurized
thermal shock transients.
_____________________________
Brian W. Sheron, Director
Office of Nuclear Regulatory Research
U.S. Nuclear Regulatory Commission
v
vi
Contents
Page
Abstract ......................................................................................................................................... iii
Foreword ........................................................................................................................................ v
Contents........................................................................................................................................ vii
List of Figures ............................................................................................................................... ix
List of Tables................................................................................................................................ xii
Executive Summary ................................................................................................................... xiii
Abbreviations............................................................................................................................... xv
Acknowledgments...................................................................................................................... xvii
1.
Introduction ....................................................................................................................... 1
2.
Pressurized Thermal Shock Events ................................................................................. 5
2.1
2.2
2.3
Historical Review........................................................................................................... 5
Current NRC Regulatory Approach to PTS............................................................... 6
Contributions of Large-Scale Experiments to the Technical Basis for PTS
Assessment...................................................................................................................... 7
3 . Structure and Organization of the FAVOR Code............................................................... 10
3.1
FAVOR – Computational Modules and Data Streams ............................................ 10
3.2
FAVOR Load Module (FAVLoad) ............................................................................ 11
3.2.1
Thermal-Hydraulic Transient Definitions ........................................................ 11
3.2.2
Required Vessel Geometry and Thermo-Elastic Property Data ..................... 12
3.2.3
Deterministic Analyses ........................................................................................ 12
3.2.4
Flaw Categories Used in FAVOR....................................................................... 13
3.3
FAVOR PFM Module (FAVPFM)............................................................................. 14
3.3.1
FAVPFM Flowchart............................................................................................ 16
3.3.2
Beltline Configurations and Region Discretization .......................................... 18
3.3.3
Treatment of the Fusion-Line Along Welds...................................................... 20
3.3.4
Warm Prestressing .............................................................................................. 22
3.3.5
Probability Distributions .................................................................................... 25
3.3.6
Truncation Protocol ............................................................................................ 30
3.3.7
Conditional Probability of Initiation (CPI) ....................................................... 30
3.3.8
Post-Initiation Flaw Geometries and Orientations........................................... 35
3.3.9
Conditional Probability of Failure (CPF) by Through-Wall Cracking.......... 37
3.3.10 Multiple Flaws ..................................................................................................... 38
3.3.11 Ductile-Tearing Models in FAVOR ................................................................... 38
3.3.11.1
Ductile-Tearing Model No. 1 (implemented in FAVOR, v04.1) .............. 43
3.3.11.2
Ductile-Tearing Model No. 2 (implemented in FAVOR, v03.1) .............. 45
3.3.12 Initiation-Growth-Arrest (IGA) Submodel ....................................................... 50
3.3.13 Ductile-Tearing Submodel .................................................................................. 61
vii
3.3.14 Ductile Tearing as an Initiating Event............................................................... 64
3.4
FAVOR Post Module – FAVPost ............................................................................... 65
4 . Probabilistic Fracture Mechanics ......................................................................................... 67
4.1
Deterministic Analyses ................................................................................................ 67
4.1.1
Thermal Analysis................................................................................................. 68
4.1.2
Stress Analysis ..................................................................................................... 71
4.1.3
Linear-Elastic Fracture Mechanics (LEFM) .................................................... 74
4.1.3.1 Mode I Stress-Intensity Factors ..................................................................... 74
4.1.3.2 Inner Surface-Breaking Flaw Models –Semi-Elliptic and Infinite Length 74
4.1.3.3 Embedded Flaw Model ................................................................................... 81
4.1.3.4 Inclusion of Residual Stresses in Welds......................................................... 84
4.1.3.5 Inclusion of Crack-Face Pressure Loading for Surface-Breaking Flaws... 86
4.2
Sampled LEFM Material and Correlative Properties ............................................. 87
4.2.1
Reference Nil-Ductility Transition Temperature, RTPTS, at EOL Fluence..... 87
4.2.2
Radiation Embrittlement .................................................................................... 88
4.2.3
Fast-Neutron Fluence Attenuation and Sampling Distribution ...................... 92
4.2.4
ORNL 99/27 KIc and KIa Databases .................................................................... 92
4.2.5
Index Temperature RTNDT – Uncertainty Classification and Quantification 98
4.2.6
Index Temperature RTArrest – Uncertainty Classification and Quantification
103
4.2.7
Plane-Strain Static Cleavage Initiation Toughness – KIc ............................... 107
4.2.8
Plane-Strain Crack Arrest Toughness – KIa .................................................... 109
4.2.9
Material Chemistry –Sampling Protocols ....................................................... 112
4.3
NRC RVID2 Database .............................................................................................. 117
4.4
Discrete Flaw Density and Size Distributions ......................................................... 117
4.5
Summary of Sampling Distributions and Protocols ............................................... 121
5 . Summary and Conclusions .................................................................................................. 128
References .................................................................................................................................. 129
Appendix A – Background and Antecedents of FAVOR, v04.1............................................ A-1
Appendix B – Stress-Intensity Factor Influence Coefficients................................................ B-1
Appendix C – Listings of KIc And KIa Extended Databases ................................................... C-1
Appendix D – Summary of RVID2 Data for Use in FAVOR Calculations.......................... D-1
Appendix E – Statistical Point-Estimation Techniques for Weibull Distributions ............. E-1
Appendix F – Development of Stochastic Models for ΔRTepistemic and ΔRTarrest .....................F-1
viii
List of Figures
Figure
Page
1. The beltline region of the reactor pressure vessel wall extends from approximately one
foot above the active reactor core to one foot below the core ...............................................2
2. FAVOR data streams flow through three modules: (1) FAVLoad, (2) FAVPFM, and (3)
FAVPost. ..................................................................................................................................10
3. The FAVOR load generator module FAVLoad performs deterministic analyses for a
range of thermal-hydraulic transients...................................................................................11
4. Flaw models in FAVOR include infinite-length surface breaking flaws, finite-length semielliptic surface flaws (with aspect ratios L / a = 2, 6, and 10), and fully elliptic embedded
flaws. All flaw models can be oriented in either the axial or circumferential directions..13
5. The FAVPFM module takes output from FAVLoad and user-supplied data on flaw
distributions and embrittlement of the RPV beltline and generates PFMI and PFMF
arrays........................................................................................................................................15
6. Flow chart for improved PFM model implemented in FAVPFM showing the four
primary nested loops – (1) RPV Trial Loop, (2) Flaw Loop, (3) Transient Loop, and
(4) Time Loop . .........................................................................................................................17
7. Fabrication configurations of PWR beltline shells (adapted from [3]): (a) rolled-plate
construction with axial and circumferential welds and (b) ring-forging construction with
circumferential welds only......................................................................................................19
8. FAVOR uses a discretization of the RPV beltline region to resolve the variation in
radiation damage in terms of plate, axial weld, and circumferential weld major regions
which are further discretized into multiple subregions. ......................................................21
9. Example of warm prestressing: (a) loading history with pressure applied to the inner
surface and the temperature at the crack tip, (b) load path for a flaw showing two WPS
regions. (cpi is the instantaneous conditional probability of initiation). ............................24
10. Example probability density functions for (a) normal and logistic and (b) uniform,
Weibull, and lognormal continuous distributions. ...............................................................29
11. Interaction of the applied KI time history and the Weibull KIc statistical model for an
example flaw. ...........................................................................................................................32
12. The parameter cpi(τ)(i,j,k) is the instantaneous conditional probability of initiation
(cleavage fracture) obtained from the Weibull KIc cumulative distribution function.
CPI(i,j,k) is the maximum value of cpi(τ)(i,j,k). (Note: i = transient index, j = RPV trial index,
and k = flaw index) ..................................................................................................................34
13. Δcpi(τn)(i,j,k) is the increase in cpi(τn)(i,j,k) that occurs during each discrete time step. When
the maximum value of cpi(τn)(i,j,k) is reached, negative values of Δcpi(τn)(i,j,k) are set to
zero. (Note: i = transient index, j = RPV trial index, and k = flaw index) ..........................34
14. At the time of initiation, the three categories of flaws are transformed into infinitelength flaws: (a) Category 1 semi-elliptic surface breaking circumferential flaws become
360 degree circumferential flaws, (b) and (c) Category 2 and 3 embedded flaws become
inifinite-length axial or 360 degree circumferential flaws at the same depth. Category 1
flaws are only oriented in the circumferential direction......................................................36
15. Given a JR curve in power-law model form and current flow stress, σ f , the initiation
toughness, JIc , and local tearing modulus, TR, are uniquely defined). ...............................42
16. Flowchart for PFM model – the Initiation-Growth-Arrest (IGA) submodel can be
viewed as a Monte Carlo model nested within the larger PFM Monte Carlo model. For a
ix
given flaw, the IGA submodel is called after the CPI for the current transient has been
calculated. Note: ++ notation indicates increment index by 1, e.g., i++ means i=i+1......51
17. (a) Flow chart for Initiation-Growth-Arrest Model – The IGA Propagation submodel is
only called for flaws with increasing CPIs. The weld-layering scheme is also shown for
Initiation-Growth-Arrest Model. No through-wall resampling is carried out for plates or
forgings.....................................................................................................................................52
17 (continued) (b) IGA Propagation submodel to test for Stable Arrest (no failure) and
Vessel Failure...........................................................................................................................53
17 (continued) (c) Unstable-Ductile-Tearing submodel to test for either stable tearing to a
new flaw position, a*, or unstable ductile tearing that fails the vessel. ..............................54
18. An example Category 2 flaw (a) initiates, (b) expands into an infinite-length flaw,
(c) advances to new weld layer and resamples chemistry content to calculate new RTNDT,
(d) continues growth until either failure by net-section plastic collapse of remaining
ligament or stable crack arrest. The potential for arrest and subsequent re-initiation is
also modeled.............................................................................................................................55
19. IGA mesh used to estimate dJapplied / da using second order central finite-difference ratio.64
20. The FAVOR post-processor FAVPost combines the distributions of conditional
probabilities of initiation and failure calculated by FAVPFM with initiating frequency
distributions for all of the transients under study to create distributions of frequencies of
RPV fracture and failure........................................................................................................66
21. Isoparametric mapping from parameter space to axisymmetric 1 Euclidean space using
3-node quadratic basis functions. ..........................................................................................69
22. One-dimensional axisymmetric finite-element model used in FAVOR to calculate both
temperature and stress histories through the wall of an RPV. ...........................................70
23. Influence coefficients, K*, have been calculated for finite semi-elliptical flaws with aspect
ratios L / a = 2, 6, and 10 for Ri / t = 10..................................................................................77
24. Crack-surface loading cases for determining finite 3D flaw influence coefficients:
(a) uniform unit load, (b) linear load, (c) quadratic load, and (d) cubic load....................77
25. Influence coefficients have been computed for both infinite axial and 360 degree
circumferential flaws. .............................................................................................................79
26. Superposition allows the use of an equivalent problem to compute the stress intensity
factor.........................................................................................................................................80
27. Influence coefficients, K*, represent stress intensity factor per unit load applied to the
crack face. ................................................................................................................................80
28. Geometry and nomenclature used in embedded flaw model..............................................82
29. Resolution of computed nonlinear stress profile into the linear superposition of effective
membrane and bending stresses. ...........................................................................................82
30. Weld residual stress through-thickness distribution developed for use in RPV integrity
analyses.....................................................................................................................................85
31. Relationship between the change in the fracture-toughness index temperature
( ΔT0 ≈ ΔRTNDT ) change in the 30 ft-lbf CVN transition temperature ( ΔT30 ) for welds and
plates/forgings produced by irradiation. The difference in the best-fit slopes is
statistically significant.............................................................................................................91
32. ORNL 99/27 KIc database including modified ASME KIc curve that served as a lowerbounding reference curve in the development of a new transition index temperature. ...94
33. KIa databases (a) ORNL 99/27 KIa database and (b) Extended KIa database.....................95
34. Cumulative distribution function of the observed difference in RTNDT
0 and To
(with a size of 1T) using data in the ORNL 99/27 database.
.................99
35. Cumulative distribution function of the difference (designated as ΔRTepistemic ) between
RTNDT0 and a new lower-bounding reference index designated RTLB. ................................99
36. The ΔRTLB for HSST Plate 02. The lower-bounding transition reference temperature,
RTLB , was developed from 18 materials in the ORNL 99/27 database, where for each
material RTLB = RTNDT 0 − ΔRTLB ............................................................................................100
x
37. Comparison of cumulative distribution functions developed for RTNDT0-T0 and
RTNDT0-RTLB . ..........................................................................................................................102
38. Lognormal distribution of ΔRTARREST = TK Ia − T0 as a function of T0 ..................................105
39. Lognormal probability densities for ΔRTArrest as function of T0.........................................105
40. Proposed adjustment to RTLB arises from observed offset between Δ RTLB CDF and
RTNDT – T0 CDF at median (P = 0.5). ....................................................................................106
41. Weibull statistical distribution for plane-strain cleavage initiation fracture toughness,
KIc, with prescribed validity bounds. The ORNL 99/27 KIc database was used in the
construction of the model. .....................................................................................................108
42. Lognormal statistical distribution for plane-strain crack arrest fracture toughness, KIa,
constructed using the (a) Model 1: ORNL 99/27 KIa database normalized by the arrest
reference temperature, RTArrest and (b) Model 2: Extended KIa database normalized by
the arrest reference temperature, RTArrest. ...........................................................................110
43. Weld fusion area definitions for (a) axial-weld subregion elements and
(b) circumferential subregion elements................................................................................119
43. (continued) (c) Plate subregion element. .............................................................................120
A1. Depiction of the development history of the FAVOR code...............................................A-3
F1. Comparison of the initial Weibull model, W0, for ΔRTepistemic with the ODR model:
(a) probability density functions and (b) cumulative distribution functions. ..................F-18
F2. Comparison of ODR Weibull model, WODR, for ΔRTepistemic with the models for
Case 1 (W1) and Case 2 (W2): (a) probability density functions and (b) cumulative
distribution functions............................................................................................................F-19
F3. Comparison of initial model in FAVOR, W0, with Case 2, W2. ........................................F-20
F4. Comparison of ODR model, WODR, with Case 2, W2. ........................................................F-21
F5. Data used to develop the lognormal statistical model for ΔRTarrest as a function of T0 F -22
F6. Model developed from ODR analysis of log-transformed data. ......................................F-23
F7. Variance of ODR model compared to final model. ...........................................................F-24
F8. Comparison of ODR model with final model. ...................................................................F-25
xi
List of Tables
Table
Page
1. Large-Scale PTS Experiments and Performing Organizations .............................................9
2. Illustration of Computational Procedure to Determine CPI and CPF for a Postulated
Flaw (Warm Prestress Not Included)....................................................................................33
3. Applied Flaw Orientations by Major Region.........................................................................35
4. Sources for Ductile-Tearing Data ...........................................................................................39
5. Chemical Composition of Materials Used in the Ductile-Tearing Model Development ....39
6. Summary of Ductile-Tearing Data Used in the Ductile-Tearing Model Development ......39
7. Summary of ORNL 99/27 KIc Extended Database.................................................................96
8. Summary of KIa Extended Database .......................................................................................96
9. Chemistry and Heat Treatment of Principal Materials: ORNL 99/27 Database ..............97
10. Materials Used from the ORNL 99/27 KIc Extended Database ........................................100
11. Values of Lower-Bounding Reference Temperature ........................................................101
12. ORNL 99/27 KIa Database – Reference-Transition Temperatures ..................................105
B1. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical Surface
Flaws: R / t = 10 and a / t = 0.01 ...........................................................................................B-2
B2. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical Surface
Flaws: R / t = 10 and a / t = 0.0184 .......................................................................................B-3
B3. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical Surface
Flaws: R / t = 10 and a / t = 0.05 ...........................................................................................B-4
B4. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical Surface
Flaws: R / t = 10 and a / t = 0.075 .........................................................................................B-5
B5. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical Surface
Flaws: R / t = 10 and a / t = 0.1 .............................................................................................B-6
B6. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical Surface
Flaws: R / t = 10 and a / t = 0.2 .............................................................................................B-7
B7. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical Surface
Flaws: R / t = 10 and a / t = 0.3 .............................................................................................B-8
B8. Influence Coefficients for Inside Axial Semi-elliptical Surface Flaws:
R / t = 10 and a / t= 0.5 ..........................................................................................................B-9
B9. Influence Coefficients for Inside Circumferential Semi-elliptical Surface Flaws:
R / t =10 and a / t = 0.5.........................................................................................................B-10
B10. Influence Coefficients for Inside Axial Infinite-Length Surface Flaws, R / t = 10 ......B-11
B11. Influence Coefficients for Inside Circumferential 360 Degree Surface Flaws, R / t = 10
............................................................................................................................................B-13
C1. Static Initiation Toughness KIc Extended Database...........................................................C-2
C2. Crack Arrest Toughness KIa ORNL 99/27 Database .........................................................C-8
C3. Crack Arrest Toughness KIa Extended KIa Database – Large Specimen Data..............C-11
F1. ΔRTepistemic Ranked Data with Order-Statistic Estimates of P ..........................................F-2
F2. ODRPACK Results of ODR Analysis of ΔRTepistemic Model Equation ..............................F-6
F2. ODRPACK Results of ODR Analysis of ΔRTepistemic Model Equation (continued)..........F-7
F3. Data Used in the Development of the ΔRTarrest Model ....................................................F-12
F4. ODRPACK Results of ODR Analysis of ΔRTarrest Model Equation ...............................F-13
F4. ODRPACK Results of ODR Analysis of ΔRTarrest Model Equation (continued)...........F-14
F4. ODRPACK Results of ODR Analysis of ΔRTarrest Model Equation (continued)...........F-15
xii
Executive Summary
This report is one of a series of reports that summarize the results of a 5-year project conducted
by the U.S. Nuclear Regulatory Commission’s (NRC) Office of Nuclear Regulatory Research.
This study sought to develop a technical basis to support revision of Title 10, Section 50.61, of
the Code of Federal Regulations (10 CFR 50.61), which is known as the pressurized thermal
shock (PTS) rule and the associated PTS screening criteria in a manner consistent with current
NRC guidelines on risk-informed regulation. The figure below illustrates how this report fits into
the overall project documentation.
Summary Report – NUREG-1806
Results
Models, Validation, & Procedures
PFM
• Procedures, Uncertainty, & Experimental
Validation: EricksonKirk, M.T., et al.,
“Probabilistic Fracture Mechanics:
Models, Parameters, and Uncertainty
Treatment Used in FAVOR Version 04.1,”
NUREG-1807.
• FAVOR
• Theory Manual: Williams, P.T., et al.,
“Fracture Analysis of Vessels – Oak
Ridge, FAVOR v04.1, Computer Code:
Theory and Implementation of
Algorithms, Methods, and
Correlations,” NUREG/CR-6854.
• User’s Manual: Dickson, T.L., et al.,
“Fracture Analysis of Vessels – Oak
Ridge, FAVOR v04.1, Computer Code:
User’s Guide,” NUREG/CR-6855.
• V&V Report: Malik, S.N.M., “FAVOR
Code Versions 2.4 and 3.1
Verification and Validation Summary
Report,” NUREG-1795.
• Flaw Distribution: Simonen, F.A., et al.,
“A Generalized Procedure for Generating
Flaw-Related Inputs for the FAVOR
Code,” NUREG/CR-6817, Rev. 1.
• Baseline: Dickson, T.L. and Yin, S.,
“Electronic Archival of the Results of
Pressurized Thermal Shock Analyses for
Beaver Valley, Oconee, and Palisades
Reactor Pressure Vessels Generated with
the 04.1 version of FAVOR,”
ORNL/NRC/LTR-04/18.
• Sensitivity Studies: EricksonKirk, M.T.,
et al., “Sensitivity Studies of the
Probabilistic Fracture Mechanics Model
Used in FAVOR,” NUREG-1808.
TH
• TH Model: Bessette, D., “Thermal
Hydraulic Analysis of Pressurized
Thermal Shock,” NUREG/1809.
• RELAP Procedures & Experimental
Validation: Fletcher, C.D., et al.,
“RELAP5/MOD3.2.2 Gamma Assessment
for Pressurized Thermal Shock
Applications,” NUREG/CR-6857.
PRA
• Procedures & Uncertainty: Whitehead, D.W.
and Kolaczkowski, A.M., “PRA Procedures
and Uncertainty for PTS Analysis,”
NUREG/CR-6859.
• Uncertainty Analysis Methodology: Siu, N.,
“Uncertainty Analysis and Pressurized
Thermal Shock, An Opinion,” USNRC, 1999.
• Experimental Benchmarks: Reyes, J.N.,
et. al., Final Report for the OSU APEX-CE
Integral Test Facility, NUREG/CR-6856.
• Experimental Benchmarks: Reyes, J.N.,
Scaling Analysis for the OSU APEX-CE
Integral Test Facility, NUREG/CR-6731
• Uncertainty: Chang, Y.H., et all.,
“Thermal Hydraulic Uncertainty Analysis
in Pressurized Thermal Shock Risk
Assessment,” University of Maryland.
This Report
• Baseline: Arcieri, W.C., “RELAP5
Thermal Hydraulic Analysis to Support
PTS Evaluations for the Oconee-1,
Beaver Valley-1, and Palisades Nuclear
Power Plants,” NUREG/CR-6858.
• Beaver: Whitehead, D.W., et al., “Beaver
Valley PTS PRA”
• Sensitivity Studies: Arcieri, W.C., et al.,
“RELAP5/MOD3.2.2Gamma Results for
Palisades 1D Downcomer Sensitivity
Study”
• Palisades: Whitehead, D.W., et al.,
“Palisades PTS PRA”
• Consistency Check: Junge, M., “PTS
Consistency Effort”
• Oconee: Kolaczkowski, A.M., et al., “Oconee
PTS PRA”
• External Events: Kolaczkowski, A.M., et al.,
“Estimate of External Events Contribution
to Pressurized Thermal Shock Risk”
• Generalization: Whitehead, D.W., et al.,
“Generalization of Plant-Specific PTS Risk
Results to Additional Plants”
The Fracture Analysis of Vessels – Oak Ridge (FAVOR, v04.1) computer program has been
developed to perform a risk-informed probabilistic analysis of the structural integrity of a nuclear
reactor pressure vessel (RPV) when subjected to an overcooling event. The focus of this analysis
is the beltline region of the RPV wall. Overcooling events, where the temperature of the coolant
in contact with the inner surface of the RPV wall rapidly decreases with time, produce temporally
dependent temperature gradients that induce biaxial stress states varying in magnitude through the
vessel wall. Near the inner surface and through most of the wall thickness, the stresses are tensile,
thus generating Mode I opening driving forces that can act on possible surface-breaking or
xiii
embedded flaws. If the internal pressure of the coolant is sufficiently high, then the combined
thermal plus mechanical loading results in a transient condition known as a pressurized-thermal
shock (PTS) event.
In 1999 ORNL, working in cooperation with the NRC staff and with other NRC contractors,
illustrated that the application of fracture-related technology developed since the derivation of the
current pressurized-thermal-shock (PTS) regulations (established in the early-mid 1980s) had the
potential for providing a technical basis for a re-evaluation of the current PTS regulations.
Motivated by these findings, the U.S. Nuclear Regulatory Commission (NRC) began the PTS Reevaluation Project to establish a technical basis rule within the framework established by modern
probabilistic risk assessment techniques and advances in the technologies associated with the
physics of PTS events. An updated computational methodology has been developed through
research and interactions among experts in the relevant disciplines of thermal-hydraulics,
probabilistic risk assessment (PRA), materials embrittlement, probabilistic fracture mechanics
(PFM), and inspection (flaw characterization). Major differences between this methodology and
that used to establish the technical basis for the current version of the PTS rule include the
following:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
The ability to incorporate new detailed flaw-characterization distributions from NRC
research (with Pacific Northwest National Laboratory, PNNL),
the ability to incorporate detailed neutron fluence regions – detailed fluence maps from
Brookhaven National Laboratory, BNL,
the ability to incorporate warm-prestressing effects into the analysis,
the ability to include temperature-dependencies in the thermo-elastic properties of base
and cladding,
the ability to include crack-face pressure loading for surface-breaking flaws,
a new ductile-fracture model simulating stable and unstable ductile tearing,
a new embrittlement correlation,
the ability to include multiple transients in one execution of FAVOR,
input from the Reactor Vessel Integrity Database, Revision 2, (RVID2) of relevant RPV
material properties,
fracture-toughness models based on extended databases and improved statistical
distributions,
removal of the implicit conservatism in the RTNDT transition temperature,
a variable failure criterion, i.e., how far must a flaw propagate into the RPV wall for the
vessel simulation to be considered as “failed” ?
semi-elliptic surface-breaking and embedded-flaw models,
through-wall weld residual stresses, and an
improved PFM methodology that incorporates modern PRA procedures for the
classification and propagation of input uncertainties and the characterization of output
uncertainties as statistical distributions.
This updated methodology has been implemented in the Fracture Analysis of Vessels – Oak
Ridge (FAVOR, v04.1) computer code developed for the NRC by the Heavy Section Steel
Technology (HSST) program at Oak Ridge National Laboratory (ORNL). The FAVOR, v04.1,
code represents the baseline NRC-selected applications tool for re-assessing the current PTS
regulations. This report documents the technical bases for the assumptions, algorithms, methods,
and correlations employed in the development of the FAVOR code.
xiv
Abbreviations
ASME
ASTM
BNL
CCA
C(T)
CDF
CPI
CPF
CRP
CVN
DTE
EFPY
EPFM
EPRI
EOL
FAVOR
FEM
HAZ
HSST
IPTS
LEFM
LOCA
NESC
NIST
NRC
ORNL
PDF
PFM
PNNL
PRA
PTS
PWHT
PWR
RCW
RG1.99
RG1.154
RPV
RVID
SIFIC
SMD
10CFR50.61
TMI
T-E
American Society of Mechanical Engineers
American Society for Testing and Materials
Brookhaven National Laboratory
compact crack-arrest test specimen
compact tension fracture-toughness test specimen
cumulative distribution function
conditional probability of initiation
conditional probability of failure (as indicated by through-wall cracking)
copper-rich precipitate
Charpy V-Notch test specimen
differential-thermal expansion
effective full-power years
elastic-plastic fracture mechanics
Electric Power Research Institute
end-of-licensing
Fracture Analysis of Vessels – Oak Ridge
finite-element method
heat-affected zone
Heavy Section Steel Technology Program
Integrated Pressurized Thermal Shock Program
linear-elastic fracture mechanics
loss-of-coolant accident
Network for Evaluating Structural Components
National Institute for Standards and Technology
United States Nuclear Regulatory Commission
Oak Ridge National Laboratory
probability density function
probabilistic fracture mechanics
Pacific Northwest National Laboratory
Probabilistic Risk Assessment
pressurized thermal shock
post-weld heat treatment
pressurized water reactor
recirculating cooling water
NRC Regulatory Guide 1.99, Revision 2, Ref. [12]
NRC Regulatory Guide 1.154, Ref. [11]
reactor pressure vessel
Reactor Vessel Integrity Database, Version 2, Ref. [129]
stress-intensity influence coefficients
stable matrix defect
Title 10 of the Code of Federal Regulations, Part 50, Section 50.61, Ref. [10]
Three-Mile-Island nuclear reactor
thermo-elastic
xv
T-H
UMD
WOL
WPS
thermal-hydraulic
unstable matrix defect
wedge-open loading test specimen for fracture toughness
warm prestressing
xvi
Acknowledgments
The development of the new methodologies and models incorporated into FAVOR, v04.1, has
been the result of a long and fruitful collaboration with many colleagues. The contributions of the
NRC staff including Dr. L. Abramson, D. Bessette, Dr. N. Chokshi, Dr. E. Hackett, D. Jackson,
W. Jones, D. Kalinousky, Dr. M. Kirk, Dr. S. Malik, M. Mayfield, T. Santos, Dr. N. Siu, and R.
Woods are gratefully acknowledged. The new approaches to conditional probability of initiation
and failure and the treatment of multiple flaws were developed in collaboration with Professors
M. Modarres, A. Mosleh, and Dr. F. Li of the University of Maryland Center for Technology
Risk Studies. The new flaw-characterization distributions were developed by D. Jackson of the
NRC and Drs. F. Simonen, S. Doctor, and G. Schuster at Pacific Northwest National Laboratory,
and the new detailed fluence maps were developed by W. Jones and T. Santos of the NRC and
Dr. J. Carew of Brookhaven National Laboratory. Dr. K. Bowman of the Computer Science and
Mathematics Division at Oak Ridge National Laboratory (ORNL) developed the statistical
procedures that were applied in the development of the Weibull fracture-toughness model for
FAVOR. Drs. M. Sokolov and S. K. Iskander of the Metals and Ceramics Division at ORNL
carried out the survey of fracture-toughness data that produced the ORNL 99/27 extended
fracture-toughness database. Dr. B. R. Bass, head of the Heavy Section Steel Technology
Program at ORNL, provided the survey of fracture-arrest data from the Large-Specimen
experiments carried out in the 1980s. Drs. E. Eason and J. Wright of Modeling and Computing
Services, Boulder, Colorado and Prof. G. R. Odette of the University of California at Santa
Barbara developed the new irradiation-shift model implemented in FAVOR, v04.1. In addition to
developing the ductile-tearing model implemented in this version of FAVOR, Dr. M. Kirk of the
NRC led a Working Group in the development of the new fracture-toughness models in FAVOR.
Other members of this Working Group included, in addition to the authors, Dr. R. K. Nanstad and
J. G. Merkle of the Metals and Ceramics Division at ORNL, Professor Modarres and Dr. F. Li of
the University of Maryland Center for Technology Risk Studies, Dr. M. Natishan of PEAI, and
Dr. B. R. Bass. J. G. Merkle with Dr. Nanstad developed the lower-bounding reference
temperature approach that was adopted in the uncertainty analysis of the reference-nil-ductility
transition temperature. Several conversations with Prof. R. Dodds of the University of Illinois,
Prof. K. Wallin of VTT, Finland, and Dr. C. Faidy of Electricité de France were most helpful in
the course of this effort. There were also contributions from many members of the nuclear
industry.
xvii
xviii
1. Introduction
The Fracture Analysis of Vessels – Oak Ridge (FAVOR, v04.1) computer program has been
developed to perform a risk-informed probabilistic analysis of the structural integrity of a nuclear
reactor pressure vessel (RPV) when subjected to an overcooling event. The focus of this analysis is
the beltline region of the RPV wall as shown in Fig. 1. Overcooling events, where the temperature of
the coolant in contact with the inner surface of the RPV wall rapidly decreases with time, produce
temporally dependent temperature gradients that induce biaxial stress states varying in magnitude
through the vessel wall. Near the inner surface and through most of the wall thickness, the stresses are
tensile, thus generating Mode I opening driving forces that can act on possible surface-breaking or
embedded flaws. If the internal pressure of the coolant is sufficiently high, then the combined thermal
plus mechanical loading results in a transient condition known as a pressurized-thermal shock (PTS)
event.
In 1999, Dickson et al. [1] illustrated that the application of fracture-related technology developed
since the derivation of the current pressurized-thermal-shock (PTS) regulations (established in the
early-mid 1980s) had the potential for providing a technical basis for a re-evaluation of the current
PTS regulations. Based on these results, the U.S. Nuclear Regulatory Commission (NRC) began the
PTS Re-evaluation Project to establish a technical basis rule within the framework established by
modern probabilistic risk assessment techniques and advances in the technologies associated with the
physics of PTS events. An updated computational methodology has been developed over the last four
years through research and interactions among experts in the relevant disciplines of thermalhydraulics, probabilistic risk assessment (PRA), materials embrittlement, probabilistic fracture
mechanics (PFM), and inspection (flaw characterization). This updated methodology has been
implemented in the Fracture Analysis of Vessels – Oak Ridge (FAVOR, v04.1) computer code
developed for the NRC by the Heavy Section Steel Technology (HSST) program at Oak Ridge
National Laboratory (ORNL). The FAVOR, v04.1, code represents the baseline NRC-selected
applications tool for re-assessing the current PTS regulations. This report is intended to document the
technical bases for the assumptions, algorithms, methods, and correlations employed in the
development of the FAVOR code.
1
Fig. 1. The beltline region of the reactor pressure vessel wall extends from approximately one
foot above the active reactor core to one foot below the core (adapted from [2]).
2
This baseline release of the new FAVOR (version-control code v04.1) implements the results of the
preparatory phase of the PTS Re-evaluation Project in an improved PFM model for calculating the
conditional probability of crack initiation (by plane-strain cleavage initiation) and the conditional
probability of vessel failure (by through-wall cracking). Although the analysis of PTS has been the
primary motivation in the development of FAVOR, it should also be noted that the problem class for
which FAVOR is applicable encompasses a broad range of events that include normal operational
transients (such as start-up and shut-down) as well as additional upset conditions beyond PTS.
Essentially any event in which the reactor pressure vessel (RPV) wall is exposed to time-varying
thermal-hydraulic boundary conditions could be an appropriate candidate for a FAVOR analysis of
the vessel’s structural integrity.
In support of the PTS Re-evaluation Project, the following advanced technologies and new
capabilities have been incorporated into FAVOR, v04.1:
•
the ability to incorporate new detailed flaw-characterization distributions from NRC
research (with Pacific Northwest National Laboratory, PNNL),
•
the ability to incorporate detailed neutron fluence regions – detailed fluence maps from
Brookhaven National Laboratory, BNL,
•
the ability to incorporate warm-prestressing effects into the analysis,
•
the ability to include temperature-dependencies in the thermo-elastic properties of base and
cladding,
•
the ability to include crack-face pressure loading for surface-breaking flaws,
•
a new ductile-fracture model simulating stable and unstable ductile tearing,
•
a new embrittlement correlation,
•
the ability to include multiple transients in one execution of FAVOR,
•
input from the Reactor Vessel Integrity Database, Revision 2, (RVID2) of relevant RPV
material properties,
•
fracture-toughness models based on extended databases and improved statistical
distributions,
•
a variable failure criterion, i.e., how far must a flaw propagate into the RPV wall for the
vessel simulation to be considered as “failed” ?
•
semi-elliptic surface-breaking and embedded-flaw models,
•
through-wall weld residual stresses, and an
•
improved PFM methodology that incorporates modern PRA procedures for the
classification and propagation of input uncertainties and the characterization of output
uncertainties as statistical distributions.
Chapter 2 of this report provides a short historical perspective for viewing the pressurized-thermalshock problem, including a summary of events leading to the current regulations. This chapter is
followed by a full description of the analytical models employed in the FAVOR code, described in
3
Chapters 3 and 4. In that presentation, particular emphasis is given to the new features of the code
that were highlighted above. A summary and conclusions are given in Chapter 5. Appendix A gives a
summary of the development history of FAVOR and its antecedents. Appendix B presents the
database of stress-intensity-factor influence coefficients that has been implemented in FAVOR for its
surface-breaking flaw models. The database of plane-strain static initiation fracture toughness, KIc,
and plane-strain crack arrest, KIa, properties for pressure vessel steels is given in Appendix C. This
fracture-toughness database was used in the construction of the statistical models for crack initiation
and arrest that are implemented in FAVOR. Appendix D presents a summary of RVID2 data to be
used in FAVOR analyses for the PTS Re-evaluation Project. The point-estimation techniques used in
the development of the Weibull cumulative distribution functions that estimate the epistemic
uncertainty in the fracture initiation and arrest reference temperatures are given in Appendix E. The
development of the sampling protocols for the epistemic uncertainties in two important reference
temperatures is given in Appendix F.
4
2. Pressurized Thermal Shock Events
Overcooling events, where the temperature of the coolant in contact with the inner surface of the
reactor pressure vessel (RPV) wall rapidly decreases with time, produce temporally dependent
temperature gradients that induce biaxial stress states varying in magnitude through the vessel wall.
Near the inner surface and through most of the wall thickness the stresses are tensile, thus presenting
Mode I opening driving forces that can act on possible surface-breaking or embedded flaws. The
combined thermal plus mechanical loading results in a transient condition known as a pressurized
thermal shock (PTS) event.
Concern with PTS results from the combined effects of (1) simultaneous pressure and thermal-shock
loadings, (2) embrittlement of the vessel material due to cumulative irradiation exposure over the
operating history of the vessel, and (3) the possible existence of crack-like defects at the inner surface
of or embedded within the RPV heavy-section wall. The decrease in vessel temperature associated
with a thermal shock also reduces the fracture toughness of the vessel material and introduces the
possibility of flaw propagation. Inner surface-breaking flaws and embedded flaws near the inner
surface are particularly vulnerable, because at the inner surface the temperature is at its minimum and
the stress and radiation-induced embrittlement are at their maximum.
2.1
Historical Review
The designers of the first pressurized-water reactor (PWR) vessels in the late 1950s and early 1960s
were cognizant of PTS as a reactor vessel integrity issue where nonductile fracture was evaluated as a
part of the design basis using a transition-temperature approach [3]. Continued concerns about vessel
failure due to overcooling events motivated a number of advances in fracture mechanics technology
in the late 1960s and the 1970s. Before the late 1970s, it was postulated that the most severe thermal
shock challenging a PWR vessel would occur during a large-break loss-of-coolant accident (LOCA),
where room-temperature emergency core-cooling water would flood the reactor vessel within a few
minutes, rapidly cooling the wall and inducing tensile thermal stresses near the inner surface of the
vessel [4]. However, the addition of pressure loading to the thermal loading was not typically
considered, since it was expected that during a large-break LOCA the system would remain at low
pressure. Two events in the late 1970s served to raise the concern of PTS to a higher priority in the
1980s, and this concern continues to the present.
5
In 1978, the occurrence of a non-LOCA event at the Rancho Seco Nuclear Power Plant in California
showed that during some types of overcooling transients, the rapid cooldown could be accompanied
by repressurization of the primary recirculating-cooling-water (RCW) system, compounding the
effects of the thermal stresses. The Three-Mile-Island (TMI) incident in 1979, which also involved a
cooldown event at high RCW system pressure, drew additional attention to the impact of operator
action and control system effects on transient temperature and pressure characteristics for PTS events
[3].
Following these two events, the U.S. Nuclear Regulatory Commission (NRC) designated PTS as an
unresolved safety issue (A-49). Questions also arose concerning the mixing (or lack of mixing) of
cold safety injection water with reactor coolant in the vessel, leading to an amplification of the PTS
effect. In late 1980, the NRC issued NUREG 0737-Item II.K.2.13, which required that the operators
of all PWRs and all applicants for licenses evaluate reactor vessel integrity following a small-break
LOCA as part of the TMI action plan [5]. Additional potential transients were added in March of
1981. At the end of 1981, the nuclear power industry submitted its response to NUREG 0737 to the
NRC. These submittals were based primarily on deterministic analyses using conservative thermalhydraulic and fracture-mechanics models of postulated design-basis transients and the temperature
and pressure time-histories from some of the PTS events that had actually been experienced in
operating PWR plants [3]. On the basis of these analyses, the NRC concluded that no event having a
significant probability of occurring could cause a PWR vessel to fail at that time or within the next
few years. However, the NRC continued to be concerned that other events with more limiting
transient characteristics in combination with the impact of operator action and control system effects
were not being addressed. As a result, greater emphasis was placed on Probabilistic Risk Assessment
(PRA) combined with thermal-hydraulic (T-H) analysis and probabilistic fracture mechanics (PFM)
as primary vessel-integrity assessment tools.
2.2
Current NRC Regulatory Approach to PTS
During the 1980s, in an effort to establish generic limiting values of vessel embrittlement, the NRC
funded the Integrated Pressurized Thermal Shock (IPTS) Program [4, 6, 7] which developed a
comprehensive probabilistic approach to risk assessment. Current regulatory requirements are based
on the resulting risk-informed probabilistic methodology. In the early 1980s, extensive analyses were
performed by the NRC and others to estimate the likelihood of vessel failure due to PTS events in
PWRs. Though a large number of parameters governing vessel failure were identified, the single most
significant parameter was a correlative index of the material that also serves as a measure of
embrittlement. This material index is the reference nil-ductility transition temperature, RTNDT. The
NRC staff and others performed analyses of PTS risks on a conservative and generic basis to bound
6
the risk of vessel failure for any PWR reactor. The NRC staff approach to the selection of the RTNDT
screening criteria is described in SECY-82-465 [8]. Reference [9] is a short review of the derivation
of the PTS screening criteria from both deterministic and probabilistic fracture mechanics
considerations. The analyses discussed in SECY-82-465 led to the establishment of the PTS rule [10],
promulgated in Title 10 of the Code of Federal Regulations, Chapter I, Part 50, Section 50.61
(10CFR50.61), and the issuance of NRC Regulatory Guide 1.154 (RG1.154) [11].
The PTS rule specifies screening criteria in the form of limiting irradiated values of RTNDT (designated by the rule as RTPTS) of 270 °F for axially oriented welds, plates, and forgings and 300 °F for
circumferentially oriented welds. The PTS rule also prescribes a method to estimate RTPTS for
materials in an RPV in Regulatory Guide 1.99, Revision 2 [12]. For nuclear power plants to operate
beyond the time that they exceed the screening criteria, the licensees must submit a plant-specific
safety analysis to the NRC three years before the screening limit is anticipated to be reached.
Regulatory Guide 1.154 recommends the content and format for these plant-specific integrated PTS
analyses with the objective of calculating an estimate for the frequency of vessel failure caused by
PTS events. RG1.154 also presents the primary PTS acceptance criterion for acceptable failure risk
to be a mean frequency of less than 5 × 10−6 vessel failures per year.
2.3
Contributions of Large-Scale Experiments to the Technical Basis for PTS
Assessment
A number of large-scale experiments conducted internationally over the past 30 years have
contributed significantly to a better understanding of the factors influencing the behavior of RPVs
subjected to postulated PTS scenarios [13]. These experiments, several of which are summarized in
Table 1, reflect different objectives that range from studies of “separate effects” to others that
integrate several features into a single experiment. In Table 1, the experiments are organized in terms
of four specimen groups: (1) pressure-vessel specimens, (2) cylindrical specimens, (3) plate
specimens, and (4) beam specimens. The actual test specimens were fabricated from prototypical
RPV steels, including plate, forgings, and weld product forms. Some of the specimens included
prototypical cladding, and others used steels that had been heat-treated or were fabricated with a
special chemistry to simulate near-end-of-licensing (degraded properties) conditions.
In recent years, these large-scale experiments have provided a catalyst in western Europe and the
United States for intensive international collaboration and for the formation of multinational networks
to assess and extend RPV/PTS technology. Project FALSIRE [14 -17] was initiated in 1989 through
1
support provided by governmental agencies within Germany and the U. S., under sponsorship of the
OECD/Nuclear Energy Agency. Within FALSIRE, researchers from a large number of international
organizations used selected large-scale experiments to evaluate levels of conservatism in RPV
7
integrity assessment methodologies. In 1993, the Joint Research Centre of the European Commission
launched the Network for Evaluating Structural Components (NESC) to study the entire process of
RPV integrity assessment. The NESC projects brought together a large number of leading
international research organizations to evaluate all aspects of the assessment process (i.e., fracture
methodologies, material properties characterization, inspection trials, and experimental techniques)
through a large-scale PTS spinning cylinder experiment [18, 36]. Issues receiving special attention in
the NESC experiment included (1) effects of constraint, (2) effects of cladding and HAZ regions, and
(3) behavior of sub-clad flaws under simulated PTS loading.
The large-scale experimental database and extensive body of associated analytical interpretations
have provided support for the technical basis that underpins various elements of the fracture models
implemented in the FAVOR code. In particular, these results have contributed significantly to
confirming the applicability of fracture methodologies to cleavage fracture events in RPV steels,
including crack initiation and crack arrest. References [14-18, 36] (and references given therein)
provide comprehensive evaluations of RPV integrity assessment methodologies applied to a broad
selection of experiments.
Within the HSST Program, the large-scale experiments are contributing to a framework for future
integration of advanced fracture techniques into RPV integrity assessment methodology. These
advanced techniques provide a sharp contrast to the current approach to RPV integrity assessment as
exemplified by the methodology implemented in the FAVOR code (described herein). The FAVOR
code executes probabilistic defect assessments of RPVs using (1) linear-elastic stress analysis
methods and (2) conventional, high-constraint fracture-toughness data. The advanced fracturemechanics methodologies currently under development depart from the latter approach in three major
components: (1) stress analyses of cracked regions to include plasticity, (2) constraint adjustments to
material toughness values for shallow surface and embedded flaws, and (3) probabilistic descriptions
of material fracture toughness in the transition temperature region consistent with the methodologies
embodied by ASTM Standard E-1921 (i.e., the Master Curve). Development of an updated analytical
tool incorporating these advanced techniques and providing extended applicability to RPV integrity
assessments is envisioned for the near future.
8
Table 1. Large-Scale PTS Experiments and Performing Organizations
ID No.
Experiment Title
Tests with Pressurized Vessels
ITV 1-8
Intermediate Test Vessels
PTSE-1
Pressurized Thermal Shock
Experiments
PTSE-2
Pressurized Thermal Shock
Experiments
PTS I/6
Pressurized Thermal Shock
Experiment I/6
Tests with Cylindrical Specimens
NKS-3
Thermal Shock
Experiment 3
NKS-4
Thermal Shock
Experiment 4
NKS-5
Thermal Shock
Experiment 5
NKS-6
Thermal Shock
Experiment 6
SC-1
Spinning Cylinder PTS
Experiment 1
SC-2
Spinning Cylinder PTS
Experiment 2
SC-4
Spinning Cylinder PTS
Experiment 4
TSE-6
Thermal Shock Cylinders
(Cylinder with Short Flaws)
TSE-7
Thermal Shock Cylinders
(Clad Cylinder)
TSE-8
Thermal Shock Cylinders
(Clad Cylinder)
NESC-1
NESC-1 Spinning Cylinder
PTS Experiment
Tests with Plate Specimens
PTS Step B Wide-Plate PTS Step B
Experiment
WP-1 & 2
Wide-Plate Crack Arrest
Tests of A533B and LUS
Steels
Wide Plate Test
GP-1
Tests with Beam Specimens
DD-2 &
Clad-beam experiments
DSR-3
SE(B) RPV Full-Thickness Clad Beam
Steel
Experiments
CB
Cruciform Beam (CB)
Experiments
Research Organization
Country
Refs.
Oak Ridge National Laboratory
Oak Ridge National Laboratory
USA
USA
19- 25
26
Oak Ridge National Laboratory
USA
27
Russia
28, 29
Materialprüfungsanstalt (MPA)
Germany
30
Materialprüfungsanstalt (MPA)
Germany
30
Materialprüfungsanstalt (MPA)
Germany
31
Materialprüfungsanstalt (MPA)
Germany
29, 31
AEA Technology
UK
32
AEA Technology
UK
32
AEA Technology
UK
33
Oak Ridge National Laboratory
(ORNL)
Oak Ridge National Laboratory
(ORNL)
Oak Ridge National Laboratory
(ORNL)
Network for Evaluating Steel
Components (NESC)
USA
34
USA
35
USA
35
International
Network
36
Japan Power and Engineering
Inspection Corporation
(JAPEIC)
Oak Ridge National Laboratory
(ORNL)
Japan
37
USA
38, 39
Materialprüfungsanstalt (MPA)
Germany
40
France
29, 41
USA
42, 43
USA
44
Central Research Institute for
Structural Materials (CRISM)
Electricité de France (EdF)
National Institute of Standards
and Testing (NIST) and ORNL
Oak Ridge National Laboratory
(ORNL)
9
2 2 2 2 2
3. Structure and Organization of the FAVOR Code
3.1
FAVOR – Computational Modules and Data Streams
As shown in Fig. 2, FAVOR is composed of three computational modules: (1) a deterministic load
generator (FAVLoad), (2) a Monte Carlo PFM module (FAVPFM), and (3) a post-processor
(FAVPost). Figure 2 also indicates the nature of the data streams that flow through these modules.
Fig. 2. FAVOR data streams flow through three modules: (1) FAVLoad, (2) FAVPFM, and (3)
FAVPost.
The formats of the required user-input data files are discussed in detail in the companion report
FAVOR (v04.1): User’s Guide [45].
10
3.2
FAVOR Load Module (FAVLoad)
The functional structure of the FAVOR load module, FAVLoad, is shown in Fig. 3, where multiple
thermal-hydraulic transients are defined in the input data. The number of transients that can be
analyzed in a single execution of FAVLoad is dependent upon the memory capacity of the computer
being used for the analysis. For each transient, deterministic calculations are performed to produce a
load-definition input file for FAVPFM. These load-definition files include time-dependent throughwall temperature profiles, through-wall circumferential and axial stress profiles, and stress-intensity
factors for a range of axially and circumferentially oriented inner surface-breaking flaw geometries
(both infinite- and finite-length).
Fig. 3. The FAVOR load generator module FAVLoad performs deterministic analyses for a
range of thermal-hydraulic transients.
3.2.1
Thermal-Hydraulic Transient Definitions
The thermal-hydraulic (T-H) definitions required by FAVLoad are supplied by the user in the form of
digitized tables of bulk coolant temperature, convective heat-transfer coefficient, and internal
pressure, all as functions of elapsed time for the transient. Time-history data pairs can be input for
each of the three variables, allowing a very detailed definition of the thermal-hydraulic loading
imposed on the RPV internal wall. An option is also available to specify a stylized exponentially
decaying coolant temperature-time history.
11
3.2.2
Required Vessel Geometry and Thermo-Elastic Property Data
The FAVLoad module requires fundamental vessel geometry data, including the vessel’s inner radius,
wall thickness, and cladding thickness. Temperature-dependent thermo-elastic properties are also
input for the cladding and base materials. These geometric descriptions and property data for the RPV
are treated as fixed parameters in all subsequent analyses.
3.2.3
Deterministic Analyses
Finite-element analyses are carried out on a one-dimensional axisymmetric model of the vessel wall.
The transient heat conduction equation with temperature-dependent properties is solved for the
combined cladding and base materials to produce time-varying temperature profiles through the wall.
The finite-element stress analysis calculates radial displacements and then, through straindisplacement and linear-elastic stress-strain relationships, time-varying axial and hoop stress profiles
are also calculated. These stresses include the effects of thermal and mechanical loading (internal
pressure applied to the inner vessel surface and exposed crack face) along with the option of
superimposed weld-residual stress profiles developed by the HSST program. The stress discontinuity
at the clad-base interface is also captured by the finite-element stress model. Through the
specification of a selected stress-free temperature by the user, the effects of an initial thermaldifferential expansion between the cladding and base materials can also be included in the quasi-static
load path. The finite-element thermal and stress models use the same quadratic elements and gradedmesh discretization.
The finite-element method (FEM), together with the very detailed definition of the thermal-hydraulic
boundary conditions, provides the capability to generate accurate thermal, stress, and applied stressintensity factor, KI, solutions. The application of FEM in this way allows the resolution of complex
thermal-hydraulic transients that exhibit discontinuities in the boundary condition time-histories, e.g.,
transients with late repressurizations.
Time-dependent stress-intensity factors for infinite-length and finite-length (semi-elliptical) surfacebreaking flaws are calculated for a range of flaw depths, sizes, and aspect ratios. Due to its generality,
the embedded-flaw model was implemented in the FAVPFM module, rather than FAVLoad. The
details of these deterministic analyses are given in Chapter 4. See Fig. 4 for a summary of the flaw
models available in FAVOR.
12
Fig. 4. Flaw models in FAVOR include infinite-length surface breaking flaws, finite-length
semi-elliptic surface flaws (with aspect ratios L / a = 2, 6, and 10), and fully elliptic
embedded flaws. All flaw models can be oriented in either the axial or circumferential
directions.
3.2.4
Flaw Categories Used in FAVOR
As indicated in Fig. 4, three categories of flaws are available in FAVOR:
•
Category 1 – surface-breaking flaws
infinite length – aspect ratio L/a = ∞
semi-elliptic – aspect ratio L/a = 2
semi-elliptic – aspect ratio L/a = 6
semi-elliptic – aspect ratio L/a = 10
•
Category 2 – embedded flaws – fully elliptic geometry with inner crack tip located between
the clad/base interface and 1/8t from the inner surface (t = thickness of the RPV wall)
•
Category 3 – embedded flaws – fully elliptic geometry with inner crack tip located between
1/8t and 3/8t from the inner surface
13
3.3
FAVOR PFM Module (FAVPFM)
The FAVOR PFM model is based on the Monte Carlo technique, where deterministic fracture
analyses are performed on a large number of stochastically generated RPV trials or realizations. Each
vessel realization can be considered a perturbation of the uncertain condition of the specific RPV
under analysis. The condition of the RPV is considered uncertain in the sense that a number of the
vessel’s properties along with the postulated flaw population have uncertainties associated with them.
These input uncertainties are described by statistical distributions. The RPV trials propagate the input
uncertainties with their interactions through the model, thereby determining the probabilities of crack
initiation and through-wall cracking for a set of postulated PTS events at a selected time in the
vessel’s operating history. The improved PFM model also provides estimates of the uncertainties in
its outputs in terms of discrete statistical distributions. By repeating the RPV trials a large number of
times, the output values constitute a random sample from the probability distribution over the output
induced by the combined probability distributions over the several input variables [46].
The assumed fracture mechanism is stress-controlled cleavage initiation (in the lower-transitiontemperature region of the vessel material) modeled under the assumptions of linear-elastic fracture
mechanics (LEFM). The failure mechanism by through-wall cracking is the prediction of sufficient
flaw growth either (1) to produce a net-section plastic collapse of the remaining ligament or (2) to
advance the crack tip through a user-specified fraction of the wall thickness. Flaw growth can be due
to either cleavage propagation or stable ductile tearing. In addition, if the conditions for unstable
ductile tearing are satisfied, then vessel failure by through-wall cracking is assumed to occur.
The Monte Carlo method involves sampling from appropriate probability distributions to simulate
many possible combinations of flaw geometry and RPV material embrittlement subjected to transient
loading conditions. The PFM analysis is performed for the beltline of the RPV, usually assumed to
extend from one foot below the reactor core to one foot above the reactor core. The RPV beltline can
be divided into major regions such as axial welds, circumferential welds, and plates or forgings that
may have their own embrittlement-sensitive chemistries. The major regions may be further
discretized into subregions to accommodate detailed neutron fluence maps that can include
significant details regarding azimuthal and axial variations in neutron fluence. The general data
streams that flow through the FAVPFM module are depicted in Fig. 5.
14
Fig. 5. The FAVPFM module takes output from FAVLoad and user-supplied data on flaw
distributions and embrittlement of the RPV beltline and generates PFMI and PFMF
arrays.
As shown in Fig. 5, the FAVPFM module requires, as input, load-definition data from FAVLoad and
user-supplied data on flaw distributions and embrittlement of the RPV beltline. FAVPFM then
generates two matrices: (1) the conditional probability of crack initiation (PFMI) matrix and (2)
conditional probability of through-wall cracking (PFMF) matrix. The (i, j)th entry in each array
contains the results of the PFM analysis for the jth vessel simulation subjected to the ith transient.
Current PTS regulations are based on analyses from PFM models that produced a Bernoulli sequence
of boolean results for cleavage fracture initiation and RPV failure by through-wall cracking; i.e., the
outcome for each RPV trial in the Monte Carlo analysis was either crack initiation or no crack
initiation and either failure or no failure. The conditional probability of initiation, P(I|E), was
calculated simply by dividing the number of RPV trials predicted to experience cleavage fracture by
the total number of trials. Similarly, the conditional probability of failure, P(F|E), was calculated by
dividing the number of RPV trials predicted to fail by the total number of trials. The final results were
discrete values for P(I|E) and P(F|E), without any quantification of the uncertainty in the solution.
The improved PFM model in the new FAVPFM (v04.1) module provides for the calculation of
discrete probability distributions of RPV fracture and failure along with the estimation of
uncertainties in the results. In this improved PFM model, values for the conditional probability of
initiation ( 0 ≤ CPI ≤ 1 ) and conditional probability of failure ( 0 ≤ CPF ≤ 1 ) by through-wall cracking
are calculated for each flaw subjected to each transient.
15
3.3.1
FAVPFM Flowchart
Figure 6 is a flowchart illustrating the essential elements of the nested-loop structure of the PFM
Monte Carlo model – (1) RPV Trial Loop, (2) Flaw Loop, (3) Transient Loop, and (4) Timeintegration Loop. The outermost RPV Trial Loop is indexed for each RPV trial included in the
analysis, where the number of RPV trials is specified by the user in the FAVPFM input stream. Since
each RPV trial can be postulated to contain multiple flaws, the next innermost loop (the Flaw Loop)
is indexed for the number of flaws for this trial. Each postulated flaw is positioned (through sampling)
in a particular RPV beltline subregion having its own distinguishing embrittlement-related
parameters. Next, the flaw geometry (depth, length, aspect ratio, and location within the RPV wall) is
determined by sampling from appropriate distributions derived from expert judgment [47] and nondestructive and destructive examinations [48- 50] of RPV steels. Each of the embrittlement-related
4
parameters [nickel (an alloying element), copper and phosphorus (contaminants), neutron fluence,
and an estimate of the epistemic and aleatory uncertainties in the unirradiated RTNDT0] are sampled
from appropriate distributions. 1 The neutron fluence is attenuated to the crack-tip location, and a
value for the irradiated reference index, RTNDT (serving as a quantitative estimate of radiation
damage), is calculated.
A deterministic fracture analysis is then performed on the current flaw for each of the postulated PTS
transients; thus, the deterministic component of the analysis involves two inner nested loops – a
Transient Loop and a Time-integration Loop. The temporal relationship between the applied Mode I
stress intensity factor (KI) and the static cleavage fracture initiation toughness (KIc) at the crack tip is
calculated at discrete transient time steps. The fracture-toughness, KIc , statistical model is a function
of the normalized temperature, T(τ) – RTNDT, where T(τ) is the time-dependent temperature at the
crack tip. Analysis results are used to calculate the conditional probability of crack initiation (CPI) 2,
i.e., the probability that pre-existing fabrication flaws will initiate in cleavage fracture. Also, the PFM
model calculates the conditional probability of failure (CPF)2 by through-wall cracking, i.e., the
probability that an initiated flaw will propagate through the RPV wall. These probabilities are
conditional in the sense that the transients are assumed to occur and that the postulated flaws do in
fact exist. In the treatment of multiple flaws to be discussed in Sect. 3.3.10, the values of CPI and
CPF calculated for individual flaws become the statistically independent marginal probabilities used
in the construction of the joint conditional probabilities of initiation and failure.
1
The details of the protocols and statistical distributions for all sampled parameters are given in Chapter 4.
2
The notations of CPI and CPF are used here rather than the older P(I|E) and P(F|E) notations in order to
highlight the fact that a new PFM methodology is being applied.
16
Fig. 6. Flow chart for improved PFM model implemented in FAVPFM showing the four
primary nested loops – (1) RPV Trial Loop, (2) Flaw Loop, (3) Transient Loop, and
(4) Time Loop. Note: ++ notation indicates increment index by 1, e.g., i++ means i=i+1.
17
Great care was taken in the construction of the nested-loop structure shown in Fig. 6 to preclude the
introduction of a bias in the results due to the arbitrary ordering of the transients. In other words, for a
given RPV trial, flaw, and transient, the same value of CPI and CPF will be calculated irrespective of
the position of the transient (or the number of transients) in the load-definition transient stack. This
objective was accomplished by confining all random sampling to the sampling block located at the
point of entry into the flaw loop. Any sampling required in the crack Initiation-Growth-Arrest
submodel 3 draws from sets of random number sequences derived in the sampling block. These setaside random number sequences remain fixed for the current flaw and are reset to the start of the
sequence as each transient is incremented in the Transient Loop. New random number sequences are
constructed (resampled) for each increment in the Flaw Loop. The above approach involves an
implementation of a variance reduction technique called common random numbers (CRN) which, in
the terminology of classical experimental design, is a form of blocking. CRN has also been called
correlated sampling or matched streams in some statistical simulation contexts [51].
3.3.2
Beltline Configurations and Region Discretization
The FAVOR code provides the capability to model the variation of radiation damage in the beltline
region of an RPV with as much detail as the analyst considers necessary. In this section, a description
of the beltline region is given, focusing on those aspects that are relevant to a FAVOR PFM analysis.
The beltline region of an RPV is fabricated using either forged-ring segments or rolled-plate segments
[4]. The vessels are typically constructed of a specialty pressure vessel ferritic steel (e.g., A533-B,
Class 1 plate or A508, Class 2 forging) as the base material. The heavy-section steel wall is lined with
an internal cladding of austenitic stainless steel. Vessels made with forgings have only circumferential welds, and plate-type vessels have both circumferential welds and axial welds, as shown in
Fig. 7. Therefore, beltline shells of a plate-type vessel contain three major region categories to model:
(1) axial welds, (2) circumferential welds, and (3) plate segments. Only that portion of a weld that is
within the axial bounds of the core need be considered, because the fast-neutron flux (and thus the
radiation damage) experiences a steep attenuation beyond the fuel region. The extended surface
length of an axially oriented flaw in a plate segment is also limited by the height of the core but not
by the height of the shell course; therefore, the surface length of axial flaws in plate segments can be
greater than those in axial welds [4]. Circumferential flaws in circumferential welds can be assumed
to be limited by the full 360-degree arc-length of the weld. Due to the fabrication procedures for
applying the cladding on the inner surface of the vessel, FAVOR assumes all pre-existing surface3
As will be discussed in Chapter 4, resampling of weld chemistry is required in the through-wall crack growth
protocol as the crack front advances into a different weld layer.
18
breaking flaws (in plate or weld subregions) are circumferential flaws. Embedded flaws can be either
axially or circumferentially oriented.
Fig. 7. Fabrication configurations of PWR beltline shells (adapted from [3]): (a) rolled-plate
construction with axial and circumferential welds and (b) ring-forging construction
with circumferential welds only.
19
Given the above considerations, the beltline region in FAVOR is defined as that portion of the RPV
shell (including plate segments and welds) that extends from one foot below the bottom of the active
core to one foot above the core. It is this region of the RPV wall that is explicitly modeled in
FAVOR. As will be discussed in later sections, the assumption applied in the crack InitiationGrowth-Arrest submodel is that all finite-length flaws (both surface-breaking and embedded)
instantly upon initiation become infinite-length flaws at depths corresponding to the locations of their
outer crack tips at the time of initiation. This assumption that there is lateral extension of finite flaws
before they extend through the vessel wall is supported by experimental observations made during
large-scale PTS experiments (discussed in Chapter 2) conducted at ORNL in the 1980s.
Figure 8 shows a typical rollout section of the beltline region. The user is required to discretize
(subdivide) the beltline into several major regions that contain plates (or forgings), axial welds, and
circumferential welds. These major regions are further discretized into subregions for greater
resolution of the variation in radiation-induced embrittlement. An embrittlement-distribution map is
defined in the input data for FAVPFM using these major region and subregion definitions.
3.3.3
Treatment of the Fusion-Line Along Welds
The discretization and organization of major regions and subregions in the beltline includes a special
treatment of weld fusion lines These fusion lines can be visualized as approximate boundaries
between the weld subregion and its neighboring plate or forging subregions. FAVOR checks for the
possibility that the plate subregions adjacent to a weld subregion could have a higher degree of
radiation-induced embrittlement than the weld. The irradiated value of RTNDT for the weld subregion
of interest is compared to the corresponding values of the adjacent (i.e., nearest-neighbor) plate
subregions. Each weld subregion will have at most two adjacent plate subregions. The embrittlementrelated properties of the most limiting (either the weld or the adjacent plate subregion with the highest
value of irradiated RTNDT) material are used when evaluating the fracture toughness of the weld
subregion. These embrittlement-related properties include the unirradiated value of RT NDT 0 , the fast)
neutron fluence, f 0 , product form, and chemistry content, Cu, Ni, and P wt %, as discussed in
Steps 3 and 4 and Eqs. (120) and (121) of Sect. 4.5. Flaw type and pre- and post-initiation orientation
(see Sect. 3.3.8 and Table 3) of flaws are not transferred from a dominant plate subregion to a weld
subregion.
20
Fig. 8. FAVOR uses a discretization of the RPV beltline region to resolve the variation in
radiation damage in terms of plate, axial weld, and circumferential weld major regions
which are further discretized into multiple subregions.
For the Ductile Tearing Model No. 2, implemented in FAVOR, v03.1 (see the discussion in
Sect. 3.3.11), a second weld-fusion-line dependency structure is created based on the irradiated uppershelf energy, USE. This weld-fusion-line dependency structure for sampling ductile-tearing properties
is independent of the embrittlement-related dependency structure discussed above. For Ductiletearing Model No. 2, the ductile-tearing-related properties of the most limiting (either the weld or the
adjacent plate subregion with the lowest value of irradiated USE) material are used when evaluating
ductile-tearing of a flaw located in the weld subregion. As with the embrittlement-related weldfusion-line treatment, the flaw type and pre- and post-initiation orientation of flaws are not transferred
from a dominant plate subregion to a weld subregion. Ductile-Tearing Model No. 1, implemented in
FAVOR, v04.1, this second weld-fusion-line dependency structure for sampling ductile-tearing
properties is not required.
For those conditions in which plate embrittlement properties are used to characterize the weld
subregion fracture toughness, the weld chemistry re-sampling protocols continue to be applied.
21
3.3.4
Warm Prestressing
Experimental evidence for the warm prestressing (WPS) effect in ferritic steels was first reported
almost 40 years ago [52]. Since then, this phenomena has been the subject of extensive research; e.g.,
see [53- 62]. The technical basis for the inclusion of warm prestressing effects in FAVOR is
5 5 5 5 5 5 6 6
presented in detail in [63]. The following is a summary of the discussion in [63].
The WPS phenomena can be characterized as an increase in the apparent fracture toughness of a
ferritic steel after first being “prestressed” at an elevated temperature. Three mechanisms have been
identified [53, 57, 61] to produce the WPS phenomena:
1. Preloading at an elevated temperature work-hardens the material ahead of the crack tip. The
increase in yield strength with decreasing temperature “immobilizes” the dislocations in the
plastic zone [55,56]. Consequently, an increase in applied load is needed for additional plastic
flow (a prerequisite for fracture) to occur at the lower temperature.
2. Preloading at an elevated temperature blunts the crack tip, reducing the geometric stress
concentration making subsequent fracture more difficult.
3. Unloading after or during cooling from the elevated WPS temperature down to a reduced
temperature produces residual compressive stresses ahead of the crack tip. The load applied
at the reduced temperature must first overcome these compressive stresses before the loading
can produce additional material damage and possibly fracture. The residual compressive
stresses associated with the unloaded initial plastic zone can be viewed as protecting the
crack tip, since higher applied loads are required to achieve a given level of crack driving
force compared to the condition before preloading [59].
Heretofore, probabilistic fracture mechanics calculations performed in the United States have
typically not included the WPS phenomena as a part of the PFM model. This omission was based on
the following considerations:
1. Thermal-hydraulic (TH) transients were often represented as smooth temporal variations of
both pressure and coolant temperature; however, data taken from operating nuclear power
plants demonstrate that actual overcooling events are not necessarily so well behaved. This
non-smoothness of these fundamental mechanical and thermal loads created the possibility
that, due to short-duration time-dependent fluctuations of pressure and/or coolant
temperature, the criteria for WPS might be satisfied by the idealized transient but not satisfied
by the real transient.
2. Previous PRA models of human reliability (HR) were typically not sufficiently sophisticated
to capture the potential for plant operators to repressurize the primary coolant system as part
of their response to an RPV-integrity challenge. Since such a repressurization would largely
nullify the benefit of WPS, it was viewed as nonconservative to account for WPS within a
model that may also ignore the potentially deleterious effects of operator actions.
FAVOR, v04.1, addresses both of these concerns by allowing as input data (1) more realistic and
detailed representations of the postulated PTS transients and (2) more sophisticated PRA/HR models
that explicitly consider both acts of omission and commission on the part of plant operators.
22
The FAVOR WPS-modeling option implements the conservative WPS principle first proposed by
McGowan [54]. This principle states that for cleavage crack initiation to be possible the following
criteria must be met: (1) the applied-KI at the crack tip must exceed some minimum value of KIc and
(2) the applied-KI must be increasing with time (i.e., dKI / dτ > 0) when the load path first enters the
finite KIc probability space. Equivalently, a flaw is assumed by FAVOR to be in a state of WPS when
either of the two following conditions are met:
1. the time-rate-of-change of the applied-KI is nonpositive ( dK I / dτ ≤ 0 ), or
2. the applied KI is less than the maximum KI experienced by the flaw up to the current time in
the transient, where this KI(max) must be greater than the current value of KIc(min) as defined by
the location parameter of the statistical model (to be discussed in Sect. 3.3.7) for cleavagefracture initiation.
Figures 9a and b present an example of a PTS transient (Fig. 9a) applied to a flaw with its resulting
load path (Fig. 9b). At Point 1 in Fig. 9b, the load path for the flaw enters finite KIc probability space,
and, shortly thereafter, dKI / dτ becomes negative. The flaw is in a state of WPS from Point 1 to
Point 2. At Point 2, the applied-KI at the crack tip exceeds the current KI(max) (established at Point 1).
Along the load path between Points 2 and 3, the flaw is no longer in a state of WPS and has a finite
probability of crack initiation. At Point 3, a new KI(max) is established, and, since dK I / dτ ≤ 0 or KI <
KI(max) for the remainder of the load path, the flaw returns to and remains in a state of WPS. While the
WPS condition is in effect, the instantaneous conditional probability of initiation, cpi(τ), for the flaw
is set to zero, even though the applied KI of the flaw is within the finite KIc probability space
(KI > KIc(min)). To assess the impact of including WPS in the analysis, WPS has been implemented in
FAVOR as a user-set option, thus allowing cases to be run with and without WPS effects.
If the WPS option is activated, the applied KI of an arrested flaw must also be greater than the
previous maximum KI (of the arrested flaw geometry since the time of the arrest) for the flaw to
reinitiate.
23
(a)
(b)
Fig. 9. Example of warm prestressing: (a) loading history with pressure applied to the inner
surface and the temperature at the crack tip, (b) load path for a flaw showing two WPS
regions. (cpi is the instantaneous conditional probability of initiation).
24
3.3.5
Probability Distributions
The sampled variables used in FAVPFM are drawn from a range of specified statistical distributions.
The following presents general information about these distributions including, the form of their
probability density function (PDF), cumulative distribution function (CDF), first and second
moments, and sampling methods used in FAVOR. The notation X i ← N ( μ ,σ ) signifies that a
random variate is drawn as a sample from a population described by the specified distribution. In this
example, the population is described by a two-parameter normal distribution with mean, μ , and
standard deviation, σ . Other distributions applied in FAVOR include the standard uniform
distribution for a unit open interval, U(0,1); the two-parameter lognormal distribution, Λ ( μlog ,σ log ) ;
the three-parameter Weibull distribution, W(a,b,c); and the two-parameter logistic distribution,
L(α , β ) .
A standard uniform distribution on the interval U(0,1) is the starting point for all of the transformation
methods that draw random variates from nonuniform continuous distributions. A uniform distribution
is defined by the following:
Uniform Distribution – U(a,b)
PDF:
⎧ 0 ; x<a
⎪ 1
⎪
fU ( x | a , b ) = ⎨
; a≤ x≤b
⎪b − a
⎪⎩ 0 ; x > b
CDF:
; x<a
⎧ 0
⎪x −a
⎪
Pr( X ≤ x) = FU ( x | a, b) = ⎨
; a≤ x≤b
−
b
a
⎪
; x>b
⎪⎩ 1
Moments:
a+b
2
Mean
μ=
Variance
σ2 =
(b − a )2
12
25
Sampling from a two-parameter Uniform Distribution:
U i ← U (0,1)
Sampling from a standard uniform distribution, U(0,1), is accomplished computationally with a
Random Number Generator (RNG). A portable random number generator [64-66], written in Fortran,
has been implemented and tested in FAVOR. This portable generator, based on a composite of two
multiplicative linear congruential generators using 32 bit integer arithmetic, has a reported theoretical
minimum period of 2.3 × 1018 . This implementation was successfully tested by the HSST Program at
ORNL for statistical randomness using the NIST Statistical Test Suite for Random and
Pseudorandom Number Generators [67].
Normal Distribution – N ( μ ,σ )
⎡ ( x − μ )2 ⎤
1
exp ⎢ −
⎥ ; − ∞ < x < +∞
σ 2π
⎢⎣
2σ 2 ⎥⎦
PDF:
f N ( x | μ ,σ ) =
CDF:
Pr( X ≤ x) = Φ ( z ) =
1
2π
⎛ ξ2 ⎞
x−μ
exp
∫ ⎜⎜ − 2 ⎟⎟ dξ ; z = σ ; − ∞ < x < +∞
⎝
⎠
−∞
z
Moments:
Mean
μ
Variance
σ2
Sampling from a two-parameter Normal Distribution: X i ← N ( μ ,σ )
Earlier versions of FAVOR used the Box-Müller Transformation Method [68- 70] to sample from a
6
standard normal distribution, N(0,1). Beginning with FAVOR, v04.1, the more computationally
efficient Forsythe’s method (as extended by Ahrens and Dieter [71]) for sampling from a standard
normal distribution has been implemented. The sampled standard normal deviate, Z i , is then scaled
to the required random normal deviate with mean, μ , and standard deviation, σ , by.
Zi ← N (0,1)
X i = Ziσ + μ
(1)
The extended Forsythe’s method is computationally very efficient; however, one problem with the
method is that there is no direct connection between the standard normal deviate and its associated
p-value in the normal cumulative distribution function. When this relationship between the p-value
and the deviate is required, an alternative method for expressing the inverse of a standard normal
26
CDF (also known as a percentile function) is applied in FAVOR. The following rational function [72]
represents an accurate approximation of the standard normal percentile function:
1
⎧
for p <
⎪⎪ p
2
x=⎨
⎪1 − p for p ≥ 1
⎪⎩
2
y = −2ln( x)
(2)
a + a y + a2 y 2 + a3 y 3 + a4 y 4 ⎞
1 ⎞⎛
⎛
Z p = sgn ⎜ p − ⎟ ⎜ y + 0 1
⎟
2 ⎠ ⎜⎝
b0 + b1 y + b2 y 2 + b3 y 3 + b4 y 4 ⎟⎠
⎝
where
⎧⎪−1 if x < 0
sgn( x) = ⎨
⎪⎩+1 if x ≥ 0
and the coefficients of the rational function are:
a0 = -0.3222324310880000
a1 = -1.0000000000000000
a2 = -0.3422420885470000
a3 = -0.0204231210245000
a4 = -0.0000453642210148
b0 = 0.0993484626060
b1 = 0.5885815704950
b2 = 0.5311034623660
b3 = 0.1035377528500
b4 = 0.0038560700634
The standard normal deviate is then scaled to obtain the required quantile
X p = Z pσ + μ
(
Lognormal Distribution – Λ μlog ,σ log
(3)
)
PDF:
0
;
x≤0
⎧
⎪
2⎤
⎡
⎪
f Λ ( x | μlog ,σ log ) = ⎨
1
⎢ ln x − μlog ⎥
⎪ σ x 2π exp ⎢ −
⎥ ; 0< x<∞
2
σ
2
⎪ log
⎢
⎥
log
⎣
⎦
⎩
CDF:
0
;
x≤0
⎧
⎪⎪
⎛ ξ2 ⎞
ln x − μlog
Pr( X ≤ x) = Φ ( z ) = ⎨ 1 z
, 0< x<∞
⎪ 2π ∫ exp ⎜⎜ − 2 ⎟⎟ d ξ ; z = σ
log
⎝
⎠
−∞
⎩⎪
(
Moments:
Mean
2 ⎞
⎛
σ log
⎜
⎟
μ = exp μlog +
⎜
⎟
2
⎜
⎟
⎝
⎠
27
)
Variance
2 )
σ 2 = ω (ω − 1) exp(2μlog ); ω = exp(σ log
Sampling from a two-parameter Lognormal Distribution: X i ← Λ ( μlog ,σ log)
The log-transformed deviate is sampled from a normal distribution with mean equal to the lognormal
mean, μlog , and standard deviation equal to the lognormal standard deviation, σ log . The logtransformed deviate is then converted into the required random deviate by the exponential function.
Yi ← N ( μlog ,σ log )
(4)
X i = exp(Yi )
Weibull Distribution – W(a,b,c)
(a = location parameter, b = scale parameter, c = shape parameter)
PDF:
0
;
x≤a
⎧
⎪
f W ( x a , b, c ) = ⎨ c
c − 1 exp − y c ; ( y = ( x − a ) / b, x > a, b, c > 0)
⎪ y
⎩b
CDF:
0 ;
x≤a
⎧
⎪
Pr( X ≤ x) = FW ( x | a, b, c) = ⎨
1 − exp ⎢⎡ − y c ⎥⎤ ; ( y = ( x − a ) / b, x > a, b, c > 0)
⎣
⎦
⎩⎪
( )
Moments:
⎛
⎝
1⎞
⎡ ⎛
⎣ ⎝
2⎞
Mean
μ = a + b Γ ⎜1 + ⎟
c
Variance
σ 2 = b 2 ⎢Γ ⎜1 + ⎟ − Γ 2 ⎜1 + ⎟ ⎥
c
c
⎠
⎠
⎛
⎝
1 ⎞⎤
⎠⎦
where Γ( x) is Euler’s gamma function.
Sampling from a three-parameter Weibull Distribution: X i ← W (a, b, c)
A random number is drawn from a uniform distribution on the open interval (0,1) and then
transformed to a Weibull variate with the Weibull percentile function.
U i ← U (0,1)
1/ c
X i = a + b [ − ln(1 − U i )]
28
(5)
Logistic Distribution – L(α , β )
z
⎡ ⎛ x − α ⎞⎤
z = exp ⎢ − ⎜
⎟⎥ , − ∞ < x < ∞
⎣ ⎝ β ⎠⎦
PDF:
fL (x | α , β ) =
CDF:
Pr( X ≤ x) = FL ( x | α , β ) =
β (1 + z )2
;
⎡ ⎛ x − α ⎞⎤
1
; z = exp ⎢ − ⎜
⎟⎥ , − ∞ < x < ∞
1+ z
⎣ ⎝ β ⎠⎦
Moments:
Mean
μ =α
Variance
σ2 =
π 2β 2
3
Sampling from a two-parameter Logistic Distribution X i ← L(α , β )
A random number is drawn from a uniform distribution on the open interval (0,1) and then
transformed to a logistic variate by the logistic percentile function.
U i ← U (0,1)
⎛ 1
⎞
X i = α − β ln ⎜
− 1⎟
⎝ Ui
⎠
Figure 10 gives examples of PDFs for each of these continuous probability distributions.
(a)
(b)
Fig. 10. Example probability density functions for (a) normal and logistic and (b) uniform,
Weibull, and lognormal continuous distributions.
29
(6)
3.3.6
Truncation Protocol
When sampling physical variables from statistical distributions, it is sometimes necessary to truncate
the distribution to preclude the sampling of nonphysical values. When truncation is required in
FAVOR, the truncation bounds, either symmetric or one-sided, are explicitly stated in the sampling
protocols presented in Chapters 3 and 4. The truncation rule applied in FAVOR requires a sampled
variable that exceeds its truncation bounds to be replaced by the boundary value. This exceptionhandling protocol ensures that the integrated area under the truncated probability density function
remains equal to unity; however, the shape of the resulting sampled density distribution will have a
step-function rise at the truncated boundaries.
3.3.7
Conditional Probability of Initiation (CPI)
As discussed above, a deterministic fracture analysis is performed by stepping through discrete
transient time steps to examine the temporal relationship between the applied Mode I stress intensity
factor (KI) and the static cleavage fracture initiation toughness (KIc) at the crack tip. The
computational model for quantification of fracture-toughness uncertainty has been improved (relative
to the models used in the 1980s to derive the current PTS regulations) in three ways: (1) the KIc and
KIa databases were extended by 84 and 62 data values, respectively, relative to the databases in the
EPRI NP-719-SR 4 report [73]; (2) the statistical representations for KIc and KIa were derived through
the application of rigorous mathematical procedures; and (3) a method for estimating the epistemic
uncertainty in the transition-reference temperature was developed. Bowman and Williams [74]
provide details regarding the extended database and mathematical procedures employed in the
derivation of a Weibull distribution for fracture-toughness data. Listings of the extended ORNL 99/27
KIc and KIa database are given in Appendix C. A Weibull distribution, in which the parameters were
calculated by the Method of Moments point-estimation technique, forms the basis for the new
statistical model of KIc. For the Weibull distribution, there are three parameters to estimate: the
location parameter, a, of the random variate; the scale parameter, b, of the random variate; and the
shape parameter, c. The Weibull probability density, fW, is given by:
0
;
x≤a
⎧
⎪
f W ( x a , b, c ) = ⎨ c
c −1
c
⎪ y exp − y ; ( y = ( x − a) / b, x > a, b, c > 0)
⎩b
( )
(7)
where the parameters of the KIc distribution are a function of ΔT RELATIVE :
4
The fracture-toughness database given in EPRI NP-719-SR (1978) [73] served as the technical basis for the
statistical KIc / KIa distributions used in the IPTS studies of the 1980s.
30
aK (ΔT RELATIVE ) = 19.35 + 8.335exp ⎡ 0.02254(ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
Ic
bK (ΔT RELATIVE ) = 15.61 + 50.132exp ⎡ 0.008( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
Ic
cK = 4
(8)
Ic
where ΔT RELATIVE = (T (t ) − RTNDT ) in °F . The curve, “ X ”, above a variable indicates that it is a
randomly sampled value. The details of the development of Eq. (8) will be given in Chapter 4 along
with a discussion of the sampling methods for RTNDT .
For each postulated flaw, a deterministic fracture analysis is performed by stepping through the
transient time history for each transient. At each time step, τ n , for the ith transient and jth RPV trial,
an instantaneous cpi(τ n)(i,j,k) is calculated for the kth flaw from the Weibull KIc cumulative distribution
function at time, τ , to determine the fractional part (or fractile) of the distribution that corresponds to
the applied KI(τ n)(i,j,k):
(
)
Pr K Ic ≤ K I (τ n )(i, j ,k ) = cpi (τ )(i, j ,k ) =
⎧
0 ;
⎪
⎪⎪
⎧ ⎡
n
⎨
⎪ ⎢ K I (τ )(i, j ,k ) − aK Ic
⎪1 − exp ⎨−
bK Ic
⎪
⎪ ⎢⎣
⎪⎩
⎩
K I (τ n )(i , j ,k ) ≤ aK Ic
⎤
⎥
⎥
⎦
cK Ic
⎫
⎪
⎬;
⎪
⎭
(9)
K I (τ n )(i, j ,k ) > aK Ic
Here, cpi(τ n)(i,j,k) is the instantaneous conditional probability of initiation at the crack tip at time τ n.
Figure 11 illustrates the interaction of the applied KI time history and the Weibull KIc distribution for
an example case, in which an embedded flaw 0.67-in. in depth, 4.0-in. in length, with the inner crack
tip located 0.5-in. from the inner surface, is subjected to a severe PTS transient. The RTNDT of the
RPV material is 270 °F. A Weibull distribution, as a lower-bounded continuous statistical distribution, has a lower limit (referred to as the location parameter, aK Ic ) such that any value of KI below
the location parameter has a zero probability of initiation. As described in Fig. 11, the applied KI must
be greater than the local value of aK Ic before cpi > 0. The region designated as cpi > 0 in the figure
represents the finite probability KIc initiation space, and outside of this region cpi = 0.
31
Fig. 11. Interaction of the applied KI time history and the Weibull KIc statistical model for a
postulated flaw.
32
Table 2. Illustration of Computational Procedure to Determine CPI and CPF for a
Postulated Flaw (Warm Prestress Not Included)
K Ic Weibull Parameters
T ime(τ n )
T (τ n )
RT NDT
T ( τ n )-RT NDT
a
b
c
(ksi√in)
cpi ( τ n )
Δ cpi ( τ n )
P ( F |I )
Δ cpf (τ n )
cpf ( τ n )
50.90
(-)
0
(-)
0
(-)
0
(-)
0
(-)
0
K I (τ n )
(ksi√in)
8
( °F)
360.68
(°F)
270.0
( °F)
90.68
(ksi√in)
83.70
119.16
(-)
4
10
328.28
270.0
58.28
50.35
95.52
4
55.70
9.82E-06
9.82E-06
0
0
0
12
302.18
270.0
32.18
36.57
80.46
4
59.20
6.24E-03
6.23E-03
0.20
0.0012
0.0012
14
281.48
270.0
11.48
30.15
70.56
4
61.00
3.59E-02
2.96E-02
0.25
0.0074
0.0087
16
264.74
270.0
-5.26
26.75
63.68
4
61.80
8.77E-02
5.18E-02
0.30
0.0155
0.0242
18
251.24
270.0
-18.76
24.81
58.76
4
61.70
1.44E-01
5.62E-02
0.40
0.0225
0.0467
20
240.44
270.0
-29.56
23.63
55.18
4
61.10
1.91E-01
4.76E-02
0.50
0.0238
0.0705
22
231.62
270.0
-38.38
22.86
52.49
4
60.10
2.24E-01
3.24E-02
0.60
0.0194
0.0899
24
224.24
270.0
-45.76
22.32
50.37
4
58.80
2.40E-01
1.66E-02
0.70
0.0116
0.1015
26
218.12
270.0
-51.88
21.94
48.71
4
57.30
2.42E-01
2.04E-03
0.80
0.0016
0.1031
(min)
Notes:
cpi(τ n) – instantaneous conditional probability of initiation
Δcpi(τ n) – incremental change in instantaneous conditional probability of initiation
P( F|I ) - the number of flaws that propagated through the wall thickness divided by the total number of
initiated flaws
Δcpf(τ n) = P(F|I) × Δcpi(tn)
cpf(τ n) = instantaneous conditional probability of failure by through-wall cracking
CPI = sup-norm 5 of the vector {cpi(τ n)}
CPF = sup-norm of the vector {cpf(τ n)}
The transient index, i, RPV trial index, j, and flaw index, k, are implied.
Table 2 summarizes results of the PFM model for the postulated flaw. The transient index, i, RPV
trial index, j, and flaw index, k, are implied for all variables. The column headed cpi(τ n) is the
instantaneous value of the conditional probability of initiation determined from Eq. (9) (see Fig.12).
The next column headed Δcpi (τ n) is the increase in cpi(τ n) that occurred during the discrete time
step, Δτ n, as illustrated in Fig. 13. The current value of CPI(i,j,k) is
{
}(i, j,k ) ∞ for 1 ≤ m ≤ n
CPI (i, j ,k ) = cpi (τ m )
(10)
For the example flaw in Table 2, CPI = 0.242 occurs at a transient time of 26 minutes. The last three
columns in Table 2 are used in the determination of the conditional probability of vessel failure, CPF,
by through-wall cracking, as will be discussed below.
5
the sup-norm is the maximum-valued element (in absolute value) in the vector
33
Fig. 12. The parameter cpi(τ)(i,j,k) is the instantaneous conditional probability of initiation
(cleavage fracture) obtained from the Weibull KIc cumulative distribution function.
CPI(i,j,k) is the maximum value of cpi(τ)(i,j,k). (Note: i = transient index, j = RPV trial
index, and k = flaw index)
Fig. 13. Δcpi(τn)(i,j,k) is the increase in cpi(τn)(i,j,k) that occurs during each discrete time step.
When the maximum value of cpi(τn)(i,j,k) is reached, negative values of Δcpi(τn)(i,j,k) are
set to zero. (Note: i = transient index, j = RPV trial index, and k = flaw index)
34
3.3.8
Post-Initiation Flaw Geometries and Orientations
A flaw that initiates in cleavage fracture is assumed to become an infinite-length inner surfacebreaking flaw, regardless of its original geometry (see Fig. 14). This assumption is consistent with the
results of large-scale fracture experiments in which flaws, initiated in cleavage fracture, were
observed to extend in length before propagating through the wall thickness [75]. For example, a
circumferentially oriented semi-elliptical surface-breaking flaw ½-inch in depth is assumed to
become a ½-inch deep 360-degree circumferential flaw. An embedded flaw ½-inch in depth with its
inner crack tip located at ½-inch from the RPV inner surface becomes a 1-inch deep infinite-length
flaw, since it is assumed that an initiated embedded flaw first propagates through the clad, thus
becoming an infinite-length surface-breaking flaw before advancing into the vessel wall.
All surface-breaking semi-elliptic flaws in FAVOR are assumed to be pre-existing fabrication flaws
that are circumferentially oriented; see Table 3. This restriction is based on the assumption that
Category 1 flaws were created during vessel fabrication, as the austenitic stainless-steel cladding was
being applied to the inner surface of the vessel. This assumption introduces a preferred orientation for
these flaws. Embedded flaws may be oriented either axially or circumferentially. Upon initiation, the
transformed infinite-length flaws retain the orientation of the parent initiating flaw.
Table 3. Applied Flaw Orientations by Major Region
Major Region
axial weld
circumferential weld
plate/forging
Flaw Category 1
circumferential
circumferential
circumferential
Flaw Category 2
axial
circumferential
axial/circumferential*
Flaw Category 3
axial
circumferential
axial/circumferential*
Flaw Category 1 – surface-breaking flaw
Flaw Category 2 – embedded flaw in the base material between the clad/base interface and 1 8 t
Flaw Category 3 – embedded flaw in the base material between 1 8 t and 3 8 t
*Flaw Categories 2 and 3 in plates/forgings are equally divided between axial and circumferential orientations
35
Fig. 14. At the time of initiation, the three categories of flaws are transformed into infinitelength flaws: (a) Category 1 semi-elliptic surface-breaking circumferential flaws become
360 degree circumferential flaws, (b) and (c) Category 2 and 3 embedded flaws become
inifinite-length axial or 360 degree circumferential flaws at the same depth. Category 1
flaws are only oriented in the circumferential direction.
36
3.3.9
Conditional Probability of Failure (CPF) by Through-Wall Cracking
A flaw that has initiated in cleavage fracture has two possible outcomes for the time remaining in the
transient. The newly-formed infinite-length flaw either propagates through the entire wall thickness
causing RPV failure by through-wall cracking, or it experiences a stable arrest at some location in the
wall. In either case, the advancement of the crack tip through the RPV wall may involve a sequence
of initiation / arrest / re-initiation events as discussed in the following section. In the discussion in
this section, the transient index, i, RPV trial index, j, and flaw index, k, are implied for all variables.
They have been left off to simplify the notation.
Table 2 summarizes the calculation of RPV failure in the improved PFM model. The column headed
P ( F | I ) is the conditional probability of failure given initiation; P ( F | I ) is equal to the fraction of
initiated flaws that propagate through the wall thickness causing RPV failure. At the current time, τ n,
the increment in the conditional probability of failure, Δcpf(τ n), is the product of P ( F | I ) and
Δcpi(τ n). The instantaneous value of the conditional probability of failure at time τ n, cpf(τ n), is
therefore
cpf (τ n ) =
nmax
nmax
m =1
m =1
∑ P( F | I ) × Δcpi(τ m ) = ∑ Δcpf (τ m )
(11)
where nmax is the time step at which the current value of CPI occurred, i.e., the time at which the
maximum value of cpi(τ) occurred.
The fraction of flaws that would fail the RPV is determined (at each time step for each flaw) by
performing a Monte Carlo analysis of through-wall propagation of the infinite-length flaw. In each
analysis, the infinite-length flaw is incrementally propagated through the RPV wall until it either fails
the RPV or experiences a stable arrest. In each analysis, a KIa curve is sampled from the lognormal
KIa distribution (to be discussed). The applied KI for the growing infinite-length flaw is compared to
KIa as the flaw propagates through the wall. If crack arrest does not occur (KI ≥ KIa), the crack tip
advances another small increment, and again a check is made for arrest. If the crack does arrest (KI ≤
KIa), the simulation continues stepping through the transient time history checking for re-initiation of
the arrested flaw. At the end of the Monte Carlo analysis, P(F|I) is simply the number of flaws (that
initiated at time τ n) that propagated through the wall thickness causing RPV failure, divided by the
total number of simulated flaws. See Sect. 3.3.12 for details of the Initiation-Growth-Arrest (IGA)
submodel.
The sup-norm of the vector {cpf(τ n)}, CPF, occurs at the same time step as the CPI. In Table 2, for
the example flaw, CPF is 0.103 and occurs at a transient elapsed time of 26 minutes.
37
3.3.10 Multiple Flaws
The technical basis for the treatment of multiple flaws in the beltline region of an RPV is given in
[76,77]. For each jth RPV trial and ith transient, the process described above is repeated for all
postulated flaws, resulting in an array of values of CPI(i,j,k), for each kth flaw, where the value of
CPI(i,j,k) is the sup-norm of the vector {cpi(τ n)(i,j,k)} (0.242 for the example in Table 2).
If CPI(i,j,1) is the probability of initiation of a flaw in an RPV trial that contains a single flaw, then
(1-CPI(i,j,1)) is the probability of non-initiation. If CPI(i,j,1) and CPI(i,j,2) are the marginal probabilities
of initiation of two flaws in an RPV trial that contains two flaws, then (1-CPI(i,j,1)) × (1-CPI(i,j,2)) is the
total probability of non-initiation, i.e., the joint probability that neither of the two flaws will fracture.
This can be generalized to an RPV simulation with nflaw flaws, so that the total joint probability that
none of the flaws will initiate is:
nflaw
⎧⎪Conditional probability ⎫⎪
=
⎨
⎬
∏ (1 − CPI(i, j ,k ) )
⎩⎪ of non-initiation ⎭⎪(i, j ) k =1
(12)
= (1 − CPI (i, j ,1) )(1 − CPI (i, j ,2) )K (1 − CPI (i, j ,nflaw) )
Therefore, for the ith transient and jth RPV trial with nflaw flaws, the total probability that at least
one of the flaws will fracture is just the complement of Eq. (12):
nflaw
CPI RPV (i, j ) =1- ∏ (1 − CPI (i, j ,k ) )
k =1
(
(13)
)
= 1 − ⎡ 1 − CPI (i, j ,1) (1 − CPI (i, j ,2) )K (1 − CPI (i, j ,nflaw) ) ⎤
⎣
⎦
The method described here for combining the values of CPI for multiple flaws in an RPV is also used
for combining the values of nonfailure to produce CPFs for multiple flaws.
3.3.11 Ductile-Tearing Models in FAVOR
Two ductile-tearing models have been implemented into FAVOR. Ductile-Tearing Model No. 1,
implemented in the FAVOR, v04.1, is the recommended model to estimate the effects of ductile
tearing in the Initiation-Growth-Arrest model. Ductile-Tearing Model No. 2 was implemented in
FAVOR, v03.1, and is retained in the current release for the purposes of backward compatibility with
previous analyses carried out using FAVOR, v03.1.
Ductile-tearing property data were obtained from the PTSE-1 [26] and PTSE-2 [27] studies carried
out in the late 1980s along with additional data collected in [82-84] and applied in the model
development. A summary of the major materials and data sources is presented in Table 4 along with
the chemistry composition and relevant ductile-tearing properties in Tables 5 and 6.
38
Table 4. Sources for Ductile-Tearing Data [26, 27, 78, 79, 80, 116]
Materials
61-67W
Midland Weld
P02, 68-71W
PTSE-1 Post Test
PTSE-2 Post Test
W8A & W9A
Reference
NUREG/CR-3506
NUREG/CR-5736
NUREG/CR-4880
NUREG/CR-4106
NUREG/CR-4888
NUREG/CR-5492
Table 5. Chemical Composition of Materials Used in the Ductile-Tearing Model
Development
HSST
ID
Plate 02
Midland Beltine
Midland Nozzle
W8A
W9A
68W
69W
70W
71W
61W
62W
63W
64W
65W
66W
67W
Weld Flux
Lot ID
(-)
Linde 80
Linde 80
Linde 80
Linde 0091
Linde 0091
Linde 0091
Linde 0124
Linde 80
Linde 80 btwn A533B
Linde 80 btwn A508
Linde 80 btwn A508
Linde 80 btwn A508
Linde 80 btwn A508
Linde 80 btwn A508
Linde 80 btwn A508
C
0.230
0.083
0.083
0.083
0.190
0.150
0.140
0.100
0.120
0.090
0.083
0.098
0.085
0.080
0.092
0.082
Mn
1.550
1.607
1.604
1.330
1.240
1.380
1.190
1.480
1.580
1.480
1.510
1.650
1.590
1.450
1.630
1.440
P
0.009
0.017
0.016
0.011
0.010
0.008
0.010
0.011
0.011
0.020
0.160
0.016
0.014
0.015
0.018
0.011
Chemistry Composition (wt %)
S
Si
Cr
Ni
0.014
0.200
0.040
0.670
0.006
0.622
0.100
0.574
0.007
0.605
0.110
0.574
0.016
0.770
0.120
0.590
0.008
0.230
0.100
0.700
0.009
0.160
0.040
0.130
0.009
0.190
0.090
0.100
0.011
0.440
0.130
0.630
0.011
0.540
0.120
0.630
0.014
0.570
0.160
0.630
0.007
0.590
0.120
0.537
0.011
0.630
0.095
0.685
0.015
0.520
0.092
0.660
0.015
0.480
0.088
0.597
0.009
0.540
0.105
0.595
0.012
0.500
0.089
0.590
Mo
0.530
0.410
0.390
0.470
0.490
0.600
0.540
0.470
0.450
0.370
0.377
0.427
0.420
0.385
0.400
0.390
Cu
0.140
0.256
0.290
0.390
0.390
0.040
0.120
0.056
0.046
0.280
0.210
0.299
0.350
0.215
0.420
0.265
V
0.003
0.006
0.008
0.003
0.007
0.005
0.004
0.005
0.005
0.010
0.011
0.007
0.006
0.009
0.007
Table 6. Summary of Ductile-Tearing Data Used in the Ductile-Tearing Model Development
Material Size
ID
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
61W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
0.8
0.5
0.8
0.5
4
4
1.6
0.8
0.5
1.6
0.8
0.5
0.5
0.8
1.6
0.5
0.5
4
1.6
0.8
4
0.8
0.5
0.5
1.6
0.8
0.5
4
0.8
0.5
4
1.6
1.6
0.8
0.5
0.5
4
1.6
0.8
0.5
0.5
Fluence Temp. J Ic
Avg. T R Avg. USE Material Size Fluence Temp. J Ic
Avg. T R Avg. USE
1019 n/cm2 (°C) (kJ/m2)
1019 n/cm2 (°C) (kJ/m2)
(-)
(ft-lbf)
ID
(-)
(ft-lbf)
0
75
142.3
89
62
64W
0.5
0.582
177
119.1
36
75
0
75
143.4
106
62
64W
4
0.66
200
78.7
50
75
0
121
123.9
74
62
64W
4
0.64
200
94.9
49
75
0
121
130.6
90
62
64W
1.6
0.623
200
57.3
46
75
0
200
97.4
100
62
64W
1.6
0.671
200
80.2
50
75
0
200
128.1
72
62
64W
0.8
0.773
200
101.9
31
75
0
200
78.3
70
62
64W
0.5
0.672
200
99.4
23
75
0
200
89.5
52
62
64W
0.8
0.773
288
46
15
75
0
200
89.1
66
62
64W
0.5
0.672
288
66.3
18
75
0
288
57.7
68
62
65W
1.6
0
132
123.4
120
108
0
288
66.1
47
62
65W
0.8
0
132
147.2
97
108
0
288
75
53
62
65W
0.5
0
132
118.5
130
108
0
288
76.5
53
62
65W
4
0
177
80.4
138
108
1.1
121
103.1
51
52
65W
0.8
0
177
117.6
76
108
1.3
121
83
41
52
65W
0.5
0
177
114.8
102
108
1.6
121
76.4
22
52
65W
4
0
200
69.3
114
108
1
200
96.4
60
52
65W
1.6
0
200
104.1
72
108
1.1
200
52.4
38
52
65W
0.8
0
200
128.9
84
108
1.2
200
63.6
31
52
65W
0.5
0
200
94.8
111
108
1.2
200
69.5
44
52
65W
4
0
288
120.1
73
108
1.4
200
61.3
30
52
65W
1.6
0
288
71.9
73
108
1.1
288
46.4
15
52
65W
1.6
0
288
74.2
69
108
1.4
288
44.6
17
52
65W
0.8
0
288
73.5
56
108
0
75
121.7
119
93
65W
0.5
0
288
83.8
69
108
0
149
114.5
124
93
65W
1.6
0.67
132
106.2
77
72
0
149
150.1
139
93
65W
0.8
0.744
132
113.6
54
72
0
149
91.4
99
93
65W
0.5
0.767
132
110.3
48
72
0
177
107.6
154
93
65W
4
0.74
177
53.1
89
72
0
177
160.3
115
93
65W
0.8
0.744
177
104.8
45
72
0
177
101
94
93
65W
0.5
0.629
177
114.7
47
72
0
200
145.5
140
93
65W
4
0.61
200
85.6
61
72
0
200
154.4
117
93
65W
1.6
0.62
200
70.4
56
72
0
200
128.7
133
93
65W
0.8
0.756
200
91.5
41
72
0
200
150.8
99
93
65W
0.5
0.629
200
107
54
72
0
200
78.4
83
93
65W
0.8
0.756
288
41
23
72
0
200
113.8
87
93
65W
0.5
0.767
288
43.9
32
72
0
288
87.3
112
93
66W
0.5
0
100
94.4
41
76
0
288
101
118
93
66W
1.6
0
200
67
55
76
0
288
93.8
59
93
66W
0.8
0
200
103.6
50
76
0
288
83.6
59
93
66W
0.5
0
200
73
42
76
0
288
85
84
93
66W
0.8
0
288
73.8
40
76
39
Material Size
ID
Mid-Belt
Mid-Belt
Mid-Belt
Mid-Belt
Mid-Belt
Mid-Belt
Mid-Belt
Mid-Belt
Mid-Belt
Mid-Nozz
Mid-Nozz
Mid-Nozz
Mid-Nozz
Mid-Nozz
Mid-Nozz
Mid-Nozz
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
Plate 02
68W
68W
68W
68W
68W
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
Fluence Temp. J Ic
Avg. T R Avg. USE
1019 n/cm2 (°C) (kJ/m2)
(-)
(ft-lbf)
0
21
167.4
71
65
0
21
116.4
84
65
0
21
131.4
76
65
0
21
164.7
70
65
0
150
133.4
41
65
0
150
125.1
44
65
0
150
141.1
60
65
0
288
86.4
32
65
0
288
103.3
33
65
0
21
126.6
47
64
0
21
113.0
57
64
0
150
102.8
39
64
0
150
89.9
43
64
0
288
69.1
32
64
0
288
64.5
39
64
0
288
64.3
37
64
0
50
117.3
197
105
0
50
189.9
164
105
0
50
191.8
154
105
0
50
205.1
141
105
0
50
218.9
153
105
0
121
111.0
156
105
0
121
137.1
178
105
0
121
161.7
147
105
0
121
168.3
133
105
0
121
171.4
138
105
0
204
132.1
118
105
0
204
134.7
99
105
0
204
139.2
115
105
0
204
140.4
113
105
0
204
181.0
100
105
0
288
111.8
81
105
0
288
112.1
73
105
0
288
118.1
92
105
0
288
121.9
73
105
0
288
132.6
89
105
0
23
160.1
219
147
0
121
151.1
204
147
0
121
196.9
204
147
0
200
223.5
111
147
0
288
121.3
132
147
Table 6. (cont.) Summary of Ductile-Tearing Data Used in the Ductile-Tearing Model
Development
Material Size
ID
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
62W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
63W
64W
64W
64W
64W
64W
64W
64W
64W
64W
64W
64W
64W
64W
64W
64W
64W
1.6
0.8
0.5
0.5
4
0.8
0.5
4
1.6
0.8
0.5
0.8
0.5
1.6
0.8
0.5
4
1.6
0.8
0.5
4
1.6
0.8
0.5
0.5
4
1.6
0.8
0.5
0.5
0.5
1.6
0.8
0.5
4
1.6
0.8
0.5
0.5
0.8
0.5
1.6
0.8
0.5
4
1.6
0.8
0.5
4
1.6
0.8
0.5
4
1.6
0.8
0.5
0.8
Fluence Temp. J Ic
Avg. T R Avg. USE
1019 n/cm2 (°C) (kJ/m2)
(-)
(ft-lbf)
1.4
149
118.3
60
80
1.3
149
118.7
91
80
1.6
149
96.2
32
80
1.3
176
94.1
50
80
1.4
177
105.9
62
80
1.5
177
127.4
45
80
0.8
177
95.9
34
80
1.5
200
90
62
80
1.6
200
85
52
80
1.3
200
115.9
69
80
1
200
63.3
29
80
1.5
288
60.9
24
80
1.5
288
61.9
24
80
0
100
118
120
87
0
100
141.2
95
87
0
100
131.1
86
87
0
171
148.4
100
87
0
171
103.5
97
87
0
171
112.4
77
87
0
171
113.2
88
87
0
200
77.7
113
87
0
200
79.6
94
87
0
200
120.3
69
87
0
200
89.2
70
87
0
200
98.4
80
87
0
288
88.4
62
87
0
288
122.4
64
87
0
288
66.8
57
87
0
288
59.1
55
87
0
288
66.7
52
87
1.1
149
68.4
43
68
1.3
171
79.2
49
68
1.1
171
89.7
32
68
1.3
171
78.9
27
68
1.25
200
72.7
16
68
1.4
200
62.2
29
68
1.1
200
75.8
33
68
0.9
200
77
49
68
1
204
56.3
42
68
1.4
288
42.7
19
68
1.2
288
51.5
23
68
0
100
105.7
148
100
0
100
160.4
105
100
0
100
116
89
100
0
177
117.4
146
100
0
177
134.6
103
100
0
177
114.9
83
100
0
177
125
73
100
0
200
161.4
96
100
0
200
67.8
97
100
0
200
118.8
76
100
0
200
115.8
54
100
0
288
85.5
96
100
0
288
76.6
83
100
0
288
75.9
54
100
0
288
74.2
44
100
0.773
177
92.9
37
75
Material Size
ID
66W
66W
66W
66W
66W
66W
66W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
67W
W8A
W8A
W8A
W8A
W8A
W8A
W8A
W8A
W8A
W8A
W8A
W8A
W9A
W9A
W9A
W9A
W9A
W9A
W9A
W9A
W9A
W9A
W9A
W9A
W9A
W9A
PTSE-2
PTSE-2
PTSE-2
PTSE-2
PTSE-2
PTSE-2
0.5
1.6
1.6
0.8
0.5
0.8
0.5
1.6
0.8
0.5
4
1.6
0.8
0.5
0.5
4
1.6
0.8
0.5
4
4
0.8
0.5
0.8
0.5
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
NA
NA
NA
NA
NA
NA
Fluence Temp. J Ic
Avg. T R Avg. USE Material Size Fluence Temp. J Ic
Avg. T R Avg. USE
1019 n/cm2 (°C) (kJ/m2)
1019 n/cm2 (°C) (kJ/m2)
(-)
(ft-lbf)
ID
(-)
(ft-lbf)
0
288
61.9
25
76
68W
NA
0
288
190.7
138
147
0.854
200
68.4
31
58
69W
NA
0
50
143.0
87
147
0.944
200
66.4
29
58
69W
NA
0
50
147.9
80
147
1.022
200
75.2
22
58
69W
NA
0
50
163.7
70
147
0.896
200
67.4
18
58
69W
NA
0
121
139.5
89
147
1.03
288
42.8
17
58
69W
NA
0
121
141.7
93
147
0.896
288
51.6
16
58
69W
NA
0
121
142.7
82
147
0
100
130.4
164
103
69W
NA
0
121
158.9
88
147
0
100
166.5
112
103
69W
NA
0
200
174.5
54
147
0
100
132.8
98
103
69W
NA
0
204
98.9
76
147
0
200
97.4
121
103
69W
NA
0
204
117.5
61
147
0
200
84.1
116
103
69W
NA
0
288
89.7
56
147
0
200
118
85
103
69W
NA
0
288
94.1
49
147
0
200
102.1
76
103
69W
NA
0
288
103.8
56
147
0
200
92
69
103
69W
NA
0
288
129.4
56
147
0
288
97.9
58
103
70W
NA
0
50
106.2
188
74
0
288
63.4
83
103
70W
NA
0
50
177.8
163
74
0
288
82.6
56
103
70W
NA
0
121
127.5
159
74
0
288
80
51
103
70W
NA
0
121
131.1
148
74
0.86
200
67.3
45
73
70W
NA
0
121
142.8
140
74
0.96
200
56.7
57
73
70W
NA
0
204
103.3
108
74
1.022
200
76.3
45
73
70W
NA
0
204
112.0
133
74
0.834
200
92.2
32
73
70W
NA
0
204
121.0
110
74
1.03
288
58.6
23
73
70W
NA
0
288
89.0
79
74
0.617
288
80
24
73
70W
NA
0
288
105.6
93
74
0
0
104.4
72
58
70W
NA
0
288
106.2
88
74
0
75
94.4
81
58
71W
NA
0
30
128.0
186
81
0
200
79.7
57
58
71W
NA
0
50
97.9
144
81
0
288
58.6
34
58
71W
NA
0
50
121.0
98
81
2.1
125
69.9
16
36
71W
NA
0
121
110.8
153
81
2.1
200
54.1
14
36
71W
NA
0
121
126.7
105
81
2.1
288
38.6
9
36
71W
NA
0
121
131.0
155
81
1.5
30
80.8
54
40
71W
NA
0
204
77.6
66
81
1.5
75
84.6
28
40
71W
NA
0
204
84.7
87
81
1.5
200
60
17
40
71W
NA
0
204
115.4
90
81
1.5
200
57.4
18
40
71W
NA
0
288
64.5
72
81
1.5
288
41.6
11
40
71W
NA
0
288
77.4
71
81
0
-40
207.4
NA
115
71W
NA
0
288
80.2
61
81
0
0
255
173
115
0
75
195.9
170
115
0
200
147.9
130
115
0
288
92.9
120
115
0
288
116
97
115
2.1
75
156.2
42
74
2.1
200
124.1
37
74
2.1
200
147.7
40
74
2.1
288
81.5
31
74
1.5
75
167.7
52
84
1.5
200
146.4
46
84
1.5
200
127.2
47
84
1.5
288
96.1
36
84
0
100
64
120
46.4
0
100
55.6
145
46.4
0
175
58.3
106
46.4
0
175
68.4
105
46.4
0
250
52.8
67
46.4
0
250
52.2
61
46.4
40
In conjunction with the ductile-tearing model development, a revised fracture arrest toughness
stochastic model has also been implemented in FAVOR. A discussion of this new arrest model is
given in Sect. 4.2.8.
One of the constraints in developing a ductile-tearing model for FAVOR is that the required material
properties should currently be available for the four plants being studied in the PTS Re-evaluation
project. The relevant information available from RVID2 [129] includes Cu, Ni, and P content; the
upper-shelf Charpy V-notch (CVN) energy, USE; and the unirradiated flow stress of the RPV steels.
Consequently, all ductile fracture toughness properties used in FAVOR need to be derived from this
information.
The following models are required:
•
a model for the variation of ductile crack initiation toughness, JIc , with temperature and
irradiation, and
•
a model for the variation of ductile-tearing resistance as a function of temperature, irradiation,
and accumulated ductile tearing, Δa .
These two models are connected in that they both can be derived from a JR curve, expressed in a
power-law model form by:
J R = C (Δa m )
(14)
where the tearing resistance is characterized by the material’s local tearing modulus, TR, defined by
⎛ E
TR = ⎜ 2
⎜σ f
⎝
⎞ ⎛ dJ
⎟⎟ ⎜ R
⎠ ⎝ da
⎞ ⎛ E
⎟ = ⎜⎜ 2
⎠ ⎝σ f
⎞
( m −1)
⎟⎟ × m × C × Δa
⎠
(15)
Given the elastic modulus, E, and sampled irradiated flow stress, σ f , the remaining three variables
required by the ductile-tearing model are JIc, C, and m, where all three are a function of temperature
and level of irradiation damage.
Applying the definition of JIc in ASTM E-1820 [81], estimates of two of the variables allows the
calculation of the third. In Fig. 15, the ductile-tearing initiation toughness, JIc , is defined in ASTM
E-1820 as the intersection of the JR curve with a 0.2 mm offset blunting line given by
J (0.2 mm offset) = 2σ f (Δa − Δa0 )
41
(16)
Fig. 15. Given a JR curve in power-law model form and current flow stress, σ f , the initiation
toughness, JIc , and local tearing modulus, TR, are uniquely defined (see ASTM E-1820
[81]).
where the prescribed offset is Δa0 = 0.2 mm (0.008 in) . Therefore, with an estimate of JIc and the
power-law exponent, m, the power-law coefficient, C, is
J Ic = C Δa m ⇒ C =
J Ic
Δa m
J Ic = 2σ f (Δa − Δa0 ) ⇒ Δa =
∴ C=
J Ic
⎛ J Ic
⎞
+ Δa0 ⎟
⎜⎜
⎟
⎝ 2σ f
⎠
42
m
J Ic
+ Δa0
2σ f
(17)
The local tearing modulus then follows from Eq. (15). The focus of model development was,
therefore, placed on providing methods of estimating the initiation fracture toughness, JIc, and the
power-law exponent, m, as a function of temperature and irradiation damage.
3.3.11.1 Ductile-Tearing Model No. 1 (implemented in FAVOR, v04.1)
The recommended Ductile-Tearing Model No. 1 was developed from the research described in
[82,83]. The following is a summary of the model described in these references.
A model of ferritic steel toughness that accounts for fracture mode transition behavior, upper shelf
behavior, and the interaction between these two different fracture modes can be constructed based on
Wallin’s Master Curve [124], the relationship between the upper-shelf temperature, TUS, the Master
Curve reference temperature, T0, and the upper-shelf Master Curve. Using these relationships it is
possible, as described below, to estimate the complete variation of initiation fracture toughness, JIc,
with temperature in both the transition regime and on the upper shelf based only on an estimate of T0.
The following sampling protocols are taken from [83]:
Step 1. – Estimate a Value for T0
Given a sampled value of RT NDT 0 [°F] , adjusted for the effects of irradiation damage, an estimate for
T0 (for a reference size of 1T) can be sampled using Eq. (89) (see Sect. 4.2.5)
⎧⎪
⎡
RT NDT − DT + 27.82 − ⎨122.4 ⎢ − ln(1 − PT0 )
)
⎪⎩
⎣
T0 =
1.8
(
)
1
2.25
⎤ ⎫⎪
⎥ ⎬ − 32
⎦ ⎪⎭
Where RT NDT − DT ( r ,K) = RT NDT 0 + ΔRT NDT ( r ,K) , (see Eq. (91)) with
[°C]
(18)
RT NDT 0 equal to the
sampled unirradiated value of RTNDT, ΔRT NDT ( r ,K) equal to the shift due to radiation embrittlement,
and PT0 = Φ is the fractile drawn for the epistemic uncertainty in RTNDT in Eq. (90).
Step 2. – Estimate a Value for the Upper-Shelf Temperature, TUS
From the relationship developed in [83], an estimate for the upper-shelf temperature associated with
this sampled value for T0 can be calculated from
(
TUS = 50.1 + 0.794 T0
43
)
[°C]
(19)
Step 3. – Calculate a Value for JC Using the Master Curve at TUS
Using a plane strain conversion from KJc to Jc, we have, from the Master Curve model [124]
J c ( med ) =
{
} (1 −ν )
1000 30 + 70 exp ⎣⎡ 0.019 (TUS − T0 ) ⎦⎤
2
2
E
⎡ kJ ⎤
⎢⎣ m 2 ⎥⎦
where
(20)
E = 207200 − ( 57.1 TUS )
[ MPa ] and ν = 0.3
Step 4. – Calculate an Estimate for ΔJ Ic at TUS
Using the relationship derived in [83] to characterize the temperature dependence of JIc
ΔJ Ic = J Icmeas − J Ic288°C =
{
2.09 C1 exp ⎡⎣ −C2 (TUS + 273.15 ) + C3 (TUS + 273.15 ) ln ( ε& ) ⎤⎦ − σ ref
C1 = 1033 MPa
where C2 = 0.00698 K -1
C3 = 0.000415 K -1
}
(21)
ε& = 0.0004 sec-1
σ ref = 3.3318 MPa
Step 5. – Calculate an Estimated Mean and Standard Deviation for the Aleatory Uncertainty in JIc
At a given wall temperature, Twall ( R, t )
[°C] , an estimated mean value for JIc can now be estimated
by
J Ic = J c ( med ) − ΔJ Ic +
{
2.09 C1 exp ⎡⎣ −C2 (Twall + 273.15 ) + C3 (Twall + 273.15 ) ln ( ε& ) ⎤⎦ − σ ref
}
⎡ kJ ⎤
⎢⎣ m 2 ⎥⎦
(22)
Where an estimate for the standard deviation is given in [83] by
⎡ kJ ⎤
σ J = 62.023exp ( −0.0048 Twall ) ⎢ 2 ⎥
⎣m ⎦
Ic
(23)
Step 6. – Sample a Value for JIc from a Normal Distribution
The aleatory uncertainty in JIc is now estimated by sampling from the following normal distribution
)
J Ic ← N J Ic , σ J Ic
(
44
)
⎡ kJ ⎤
⎢⎣ m 2 ⎥⎦
(24)
where the sampled value is truncated at J Ic − 2σ J Ic ≤ J Ic ≤ J Ic + 2σ J Ic using the truncation protocol
of Sect. 3.3.6.
Step 7. – Calculate an Estimate for the Power-Law Exponent, m, and Coefficient, C
(
)
The mean value of the J-R curve exponent m (as in J R = C Δa m ) is estimated based on the
sampled value of JIc and the local value of the wall temperature, Twall ( R, t ) , from the following
equation (developed from the data given in [83])
⎛ T [°C] ⎞
2
m = a + b exp ⎜ wall
⎟ + c J Ic ⎡⎣ kJ/m ⎤⎦
d
⎝
⎠
(
)
3
a = 0.1117 c = 5.8701 × 10-09
b = 0.4696
d = -758.19
σ std-error = 0.08425
(25)
R 2 = 0.2992
The J-R curve exponent m with aleatory uncertainty can then be sampled from the following normal
distribution:
)
m ← N (m, 0.08425)
(26)
The J-R curve coefficient, C, then follows from
)
C=
J Ic (at Twall )
⎛ J Ic
⎞
⎜ (at)Twall ) + Δa0 ⎟
⎜ 2σ f
⎟
⎝
⎠
)
m
(27)
where σ f is the sampled flow stress and Δa0 = 0.2 mm .
3.3.11.2 Ductile-Tearing Model No. 2 (implemented in FAVOR, v03.1)
Pursuant to the proposal in [84], a preliminary ductile-tearing model was developed and implemented
into FAVOR, v03.1, for a scoping study of the effects of tearing resistance associated with RPV
materials.
3.3.11.2.1 Upper-Shelf Irradiation Effects Model
The following discussion is taken from [84]:
To date, efforts to trend the effects of irradiation damage on RPV steels have focused predominantly
on predicting the joint effects of radiation (as quantified by the fast-neutron fluence, energy > 1 MEv)
45
and chemical composition on the energy absorbed by a Charpy V-notch (CVN) specimen on the
upper shelf (i.e., the upper shelf energy, or USE). This focus occurs because CVN specimens are
placed into surveillance capsules that are used to assess the effect of irradiation damage on the RPV
steel. It should be emphasized that the USE is not the initiation fracture toughness (JIc) or the tearing
modulus (TR) information needed by FAVOR to assess the probability of through-wall cracking of the
RPV arising from a PTS event. Nevertheless, without significant additional research the only way to
predict the effect of irradiation on JIc and TR is to first predict the effect of irradiation on USE and
then correlate JIc and TR with USE.
In 1998, Eason, Wright, and Odette [85, 86] proposed the following relation between USE, chemical
composition, and fluence based on the USE data available from domestic nuclear RPV surveillance
programs at that time (692 data records) (NUREG/CR-6551) [86]. This model is given by the
following equation
1.456
(u )
USE( i ) = A + 0.0570 ⋅ USE
⎛ φt ⎞
− ⎡⎣17.5 ⋅ f ( Cu ) ⋅ (1 + 1.17 Ni 0.8894 ) + 305 P ⎤⎦ ⎜ 19 ⎟
⎝ 10 ⎠
0.2223
[ft-lbf]
(28)
where USEu is the unirradiated upper-shelf energy in ft-lbf; Cu, Ni, and P are the copper, nickel, and
phosphorous content in wt %; φ t is the fast-neutron fluence in neutrons/cm2; A is a product-form
constant; and f(Cu) is a function of copper content defined as
⎧ 55.4 for welds
⎪
A = ⎨ 61.0 for plates
⎪ 66.3 for forgings
⎩
f ( Cu ) =
1 1
⎡ Cu − 0.138 ⎤
+ tanh ⎢
⎥
2 2
⎣ 0.0846 ⎦
Reference [84] proposes the following method to simulate upper-shelf energies and address
uncertainties in USE(u):
Step 1. Input a best-estimate value for the unirradiated upper-shelf energy for a given major
region in the FAVOR embrittlement map of the beltline. Treat this value as the mean of a normal
distribution of USE(u) values, μUSE( u ) .
Step 2. At this value of μUSE( u ) , sample a value for the standard deviation from a normal
distribution given by
σ USE
)
( u ) ( mean )
2
= 4.3296 − 0.0857 μUSE( u ) + 0.0012 μUSE
(u )
σ USE ← N (σ USE
(u )
( u ) ( mean )
, 2.2789)
46
(29)
Step 3. Sample a value for the unirradiated upper-shelf energy, USE( u ) , from the following
normal distribution
)
USE( u ) ← N ( μUSE( u ) ,σ USE( u ) )
(30)
Step 4. The irradiated value for the upper-shelf energy is then estimated from Eq. (28), or,
applying sampling notation:
USE ( i ) = A + 0.0570 ⋅ USE
1.456
(u )
)
( )(
) ⎛ φt ⎞
0.8894
− ⎡17.5 ⋅ f Cu ⋅ 1 + 1.17 Ni
+ 305P ⎤ ⎜ 19 ⎟
⎢⎣
⎥⎦ ⎜ 10 ⎟
⎝
⎠
0.2223
[ft-lbf]
(31)
where the chemistry and attenuated fluence have been previously sampled.
3.3.11.2.2 Model for Initiation Ductile Fracture Toughness, JIc
The sampling protocol for JIc developed in [84] is as follows:
Step 1. Determine a value of USE ( u ) using the sampling protocol outlined in Sect. 3.3.11.2.1 and
Eqs. (29) and (30).
)
Step 2. Apply this sampled value of USE (u ) along with sampled values of Cu , Ni , P and φ t to
estimate a value of USE (i ) using Eq. (31).
Step 3. Convert this estimate of USE (i ) value to a value of K J Ic (i )(at 550° F) at 550°F using the mean
curve established in [84], where the uncertainty in K J Ic (i )(at 550° F) is not sampled,
(
K J Ic (i )(at 550° F) = 70.855 + 0.5784 × USE (i )
)
[ksi in ]
(32)
Step 4. Convert the K J Ic (i )(at 550° F) value to a K J Ic (i )(at Twall ) value at the wall temperature of interest
using the mean curve from [84]:
ΔK J Ic = K J Ic (at Twall ) − K J Ic (at 550° F) =
⎧
⎡
⎛ Twall + 459.69 ⎞
⎪
⎟ ⋅ ln ( 0.0004
⎢ 0.000415 ⎜
1.8
⎪
⎝
⎠
⎢
= 1.35 ⎨1033 ⋅ exp
⎢
T
+
459.69
⎛
⎞
⎪
− 0.00698 ⎜ wall
⎢
⎟
⎪
1.8
⎝
⎠
⎣
⎩
⎫
⎪
⎥ − σ ⎬⎪ [ksi in ]
ref
⎥
⎪
⎥
⎪
⎦
⎭
⎤
)⎥
(33)
where σ ref is
σ ref
⎡
⎛ 550 + 459.69 ⎞
⎟ ⋅ ln ( 0.0004
⎢ 0.000415 ⎜
1.8
⎝
⎠
= 1033 ⋅ exp ⎢
⎢
⎛ 550 + 459.69 ⎞
− 0.00698 ⎜
⎢
⎟
1.8
⎝
⎠
⎣
⎤
)⎥
⎥ = 3.331798
⎥
⎥
⎦
and Twall is the wall temperature at the crack tip in °F. Therefore
47
(34)
K J Ic (at Twall ) = K J Ic (at 550° F) + ΔK J Ic
[ksi in ]
(35)
The required sampled value of JIc follows from the plane strain conversion
)
⎛ 1 −ν 2 ⎞ 2
2
J Ic (at Twall ) = ⎜
⎟ K J Ic (at Twall ) [in-kips/in ]
E
⎝
⎠
(36)
3.3.11.2.3 Model for Normalized Average Tearing Resistance, Tmat , and JR Curve Power-Law
Exponent, m
In the analysis of ductile-tearing data in [84], the exponent, m, of the JR power-law curve (see
Eq. (14)) has been correlated with the material’s estimated value for the average tearing modulus,
Tmat , which is the normalized linear slope of all the J-Δa data between the 0.15 and 1.5 mm exclusion
lines in the ASTM E-1820 determination of JIc.
The sampling protocol for estimating a value for Tmat is the following:
Step 1. Determine a value of USE ( u ) using the sampling protocol outlined in Sect. 3.3.11.2.1 and
Eqs. (29) and (30).
)
Step 2. Apply this sampled value of USE ( u ) along with sampled values of Cu , Ni , P and φ t to
estimate a value of USE ( i ) using Eq. (31).
)
Step 3. Convert this estimate of USE ( i ) value to a value of T mat ( i )(at 550° F) at 550 °F using the mean
)
curve established in [84], where the uncertainty in T mat ( i )(at 550° F) is not sampled
)
T mat ( i )(at 550° F) = 3.9389 + 0.5721 × USE ( i )
(
)
(37)
)
)
Step 4. Convert the T mat ( i )(at 550° F) value to a T mat ( i )(at Twall ) value at the wall temperature of interest
using the mean curve from [84]:
)
ΔT mat = Tmat ( i )(at Twall ) − T mat ( i )(at 550° F) =
⎧
⎡
⎛ Twall + 459.69 ⎞
⎪
⎟ ⋅ ln ( 0.0004
⎢0.000415 ⎜
1.8
⎪
⎝
⎠
= 1.38 ⎨1033 ⋅ exp ⎢
⎢
⎛ T + 459.69 ⎞
⎪
− 0.00698 ⎜ wall
⎢
⎟
⎪
1.8
⎝
⎠
⎣
⎩
⎫
⎪
⎥ − σ ⎬⎪
ref
⎥
⎪
⎥
⎪
⎦
⎭
⎤
)⎥
[-]
(38)
where σ ref is
σ ref
⎡
⎛ 550 + 459.69 ⎞
⎟ ⋅ ln ( 0.0004
⎢ 0.000415 ⎜
1.8
⎝
⎠
⎢
= 1033 ⋅ exp
⎢
⎛ 550 + 459.69 ⎞
− 0.00698 ⎜
⎢
⎟
1.8
⎝
⎠
⎣
and Twall is the wall temperature at the crack tip in °F. Therefore
48
⎤
)⎥
⎥ = 3.331798
⎥
⎥
⎦
(39)
)
)
T mat ( i )(at Twall ) = T mat (i )(at 550° F) + ΔT mat
[-]
(40)
Step 5. Calculate an estimated value of the JR power-law exponent, m, using the correlation
)
developed in [84], where the uncertainty in m is not sampled.
)
)
m = 0.3214 + 0.0019 × T mat ( i )
(
)
(41)
Step 6. Calculate a value for the JR power-law coefficient, C, from the definition of JIc in ASTM
E-1820
)
)
J Ic ( i )(at Twall )
)
C=
(42)
m
)
⎛ J Ic ( i )(at Twall )
⎞
)
+ Δa0 ⎟
⎜
⎝ 2σ f
⎠
where Δa0 = 0.2 mm (0.008 in) and σ f is the sampled flow stress.
49
3.3.12 Initiation-Growth-Arrest (IGA) Submodel
As shown in Fig. 16, after the value of CPI has been calculated for the current flaw and transient, the
conditional probability of vessel failure, CPF, by through-wall cracking is determined by the flaw
Initiation-Growth-Arrest (IGA) submodel. The IGA submodel may be viewed as a small Monte Carlo
model nested within the larger PFM Monte Carlo model. The following steps in the IGA submodel
are shown in Fig. 17a:
Step G1.
The IGA submodel is entered from the PFM model with a given flaw and transient. The
IGA trial counter, NTRIAL, is initialized to zero. The pointer to the vector holding the
random number sequence containing the values of Pf 6 is reset to 1. Each transient for this
flaw will start with the same random number sequence for internal sampling; however,
each flaw has a different vector of random numbers. Go to Step G2.
Step G2.
The NTRIAL counter is incremented; the time-step counter NSTEP is initialized to zero;
and a random number Pf is drawn from a uniform distribution on the open interval (0,1).
Go to Step G3.
Step G3.
The time-step counter is incremented up to the time step corresponding to when CPI
occurred; time advances to the next time step. Go to Step G4.
Step G4.
For the given flaw, subjected to the current transient, the change in cpi with respect to
time is checked. If dcpi / dt > 0 , then the flaw becomes a candidate for propagation
through the wall. (This submodel will be described in detail in the following.) If
dcpi / dt ≤ 0 , then control branches to Step G8.
Step G5.
The IGA Propagation submodel is entered for this flaw, providing the submodel with the
current time step, flaw depth, and value of Pf. Go to Step G6.
Step G6.
Control returns from the IGA Propagation submodel with the fate of the flaw, either a
vessel failure or a stable arrest (no failure). If a vessel failure occurred, control is
transferred to Step G7. If a stable arrest occurred, control is transferred to Step G8.
Step G7.
The vessel failure counter, NFAIL(NSTEP), for this time step is incremented. Go to
Step G8.
Step G8.
If the transient has completed, i.e., NSTEP > NSTEPCPI , branch to Step G9. If the
transient is not finished, cycle to Step G3. Note that NSTEPCPI = NSTEP at which
cpi (t ) = cpi (t )
6
∞
= CPI .
The value of Pf represents the percentile used in sampling ΔRT ARREST (see Step 11 in Sect. 4.5) and K Ia
(see Step 15 in Sect. 4.5) in Step P6 and in sampling K Ic in Step P8 of the IGA Propagation Submodel, and
is used to ensure that the calculated initiation and failure probabilities are not affected by the order in which
transients are analyzed. The IGA Propagation Submodel is an embedded Monte Carlo model that is repeated a
user-set number of times using a different value of Pf each time. See the discussion in the final paragraph of
Sect. 3.3.1.
50
Fig. 16. Flowchart for PFM model – the Initiation-Growth-Arrest (IGA) submodel can be
viewed as a Monte Carlo model nested within the larger PFM Monte Carlo model. For a
given flaw, the IGA submodel is called after the CPI for the current transient has been
calculated. Note: ++ notation indicates increment index by 1; e.g., i++ means i=i+1.
51
(a)
Fig 17. (a) Flow chart for Initiation-Growth-Arrest Submodel – The IGA Propagation
submodel is only called for flaws with increasing CPIs. The weld-layering
scheme is also shown for Initiation-Growth-Arrest Model. No through-wall
resampling is carried out for plates or forgings.
52
(b)
Fig. 17 (continued) (b) IGA Propagation submodel to test for Stable Arrest (no failure) and
Vessel Failure.
53
(c)
Fig. 17 (continued) (c) Unstable-Ductile-Tearing submodel to test for either stable tearing to a
new flaw position, a*, or unstable ductile tearing that fails the vessel.
54
(a)
(b)
(c)
(d)
Fig. 18. An example Category 2 flaw (a) initiates, (b) expands into an infinite-length flaw,
(c) advances to new weld layer and resamples chemistry content to calculate new RTNDT,
(d) continues growth until either failure by net-section plastic collapse of remaining
ligament or stable crack arrest. The potential for arrest and subsequent re-initiation is
also modeled.
Step G9.
A check is made to see if the required number of trials has been completed. If there are
more NTRIALS to be run, control is transferred to Step G2. If the IGA submodel has
completed its sample trials for the current transient, then control is transferred to
Step G10.
Step G10. The CPF(i,j,k) for the ith transient, and jth RPV trial, and kth flaw is calculated by the
following:
CPF(i, j ,k ) =
NSTEPCPI
∑
Δcpi (t m )(i, j ,k ) P ( F | I )m
m =1
(43)
NFAIL(m)
P( F | I )m =
NTRIALS
where NSTEPCPI is the time step at which the value of CPI(i,j,k) was calculated for
this ith transient, jth RPV trial, and kth flaw.
Steps G2 through G9 are repeated NTRIAL cycles through the IGA submodel.
55
Figure 17b presents the control structure of the IGA Propagation submodel. This submodel proceeds
in the following manner:
IGA Propagation Submodel
Step P1.
Enter the submodel with the initiating time step, NSTEP, and the flaw depth. Transform
the Category 1, 2, or 3 flaw into its corresponding infinite-length flaw, and calculate the
applied stress-intensity factor, KI, for the transformed flaw at this time and designate it
KI-initiation. This value of KI will be higher than the KI for the finite-flaw at initiation. Go to
Step P2.
Step P2.
Advance the infinite-length flaw to its next position in the IGA mesh (see Fig. 18).
Proceed to Step P3.
Step P3.
Check for vessel failure by through-wall cracking. At this new flaw depth and current
time, calculate the current sampled estimate for the flow stress of the material. The current
sampled value of ΔT30 (to be discussed in Chapter 4) is also used to estimate the effects
of irradiation on the unirradiated flow stress, σ flow( u ) . After each resampling of ΔT30 , the
flow stress will have been adjusted by the following relation:
⎧0.112 ksi/ °F for welds
σ flow = σ flow( u ) + γΔT30 where γ = ⎨
⎩ 0.131 ksi/ °F for plates
This sampled value of σ flow is then used in the vessel-failure test against the pressureinduced membrane stress in the remaining ligament, checking for net-section plastic
collapse. The membrane stress is equal to
σ m (t ) =
pi (τ ) ( Ri + a )
⎧1 hoop stress
; β =⎨
β ( Ro − Ri − a )
⎩ 2 axial stress
where pi (τ) is the time-dependent internal pressure, Ri and Ro are the inner and outer
vessel radii, respectively, and a is the current flaw depth.
For the initial entry into the IGA Propagation submodel, the flaw is growing due to a
cleavage initiation; therefore, the ductile-tearing model will not be applied until the flaw
has experienced its first arrest event. After the flaw has arrested, the ductile-tearing model
is called at this point to check for unstable ductile tearing. This check for unstable tearing
is made only if the flaw has re-initiated in ductile tearing. If the flaw has re-initiated as a
cleavage event, the ductile-tearing submodel is not called. If the conditions for unstable
ductile tearing are encountered, the logical variable FAIL_UDT is set to TRUE in the
ductile-tearing submodel and returned to the IGA Propagation Submodel.
The vessel failure criterion is
56
if REINITIATED_BY_DUCTILE_TEARING is TRUE then
⎧
⎫
σ m > σ flow
⎪
⎪
⎪
⎪
or
⎪
⎪
⎪ FAIL _ UDT is TRUE ⎪
if ⎨
⎬ then
or
⎪
⎪
⎪⎛
⎪
⎞
⎪ ⎜ a ⎟ > FAILCR ⎪
⎪⎩ ⎝ Ro − Ri ⎠
⎪⎭
vessel failure = TRUE during ductile tearing
return to Step G5 in IGA Model
⎧
⎫
⎪
⎪
σ m > σ flow
⎪
⎪
⎪
⎪
elseif ⎨
or
⎬ then
⎪
⎪
⎪⎛⎜ a ⎞⎟ > FAILCR ⎪
⎪⎩⎝ Ro − Ri ⎠
⎪⎭
vessel failure = TRUE during flaw growth by cleavage
return to Step G5 in IGA Model
else
vessel failure = FALSE
proceed to Step P4
where 0.25 ≤ FAILCR ≤ 0.95 is a user-supplied failure criterion.
Step P4.
If the material is a plate or forging product form, proceed directly to Step P6. If the
material is a weld, check to see if the flaw has advanced into a new weld layer. Weld
subregions are sectioned into through-wall quadrants to simulate, in an approximate
manner, multiple weld layers. As the flaw advances from one weld-layer quadrant into the
next, the weld chemistry will be resampled with the attenuated fluence. If the flaw has just
advanced into a new weld layer, go to Step P5. If not, then proceed to Step P6.
Step P5.
Resample the weld chemistry (Cu, Ni, and P) using the sampling distributions given in
Chapter 4. Update the irradiation shift, ΔRTNDT , and the irradiated value of the upper
shelf energy, USE(i ) , using the resampled weld chemistry. If the weld-layer-resampling
option is turned on and the flaw has just entered layer 2, 3, or 4, then resample for a new
value of Pf to replace the value of Pf sampled in Step G2 of the IGA submodel. The
random iterate Pf is drawn from a uniform distribution on the open interval U(0,1).
Step P6.
Using the current chemistry content and current value of Pf , recalculate the arrest
reference temperature. The details are given in Chapter 4; however, the equations are
given here for completeness. Calculate the epistemic uncertainty in the arrest reference
temperature by Eqs. (119) and (125) given in Sect. 4.5.
ΔRT epistemic
(
)
1
2.177
= −45.586 + 131.27 ⎡⎢ − ln 1 − Φ ⎤⎥
[°F]
⎣
⎦
ΔRT epistemic − arrest = ΔRT epistemic − 14.4 [°F]
57
Retrieve the previously sampled unirradiated value of RTNDTo for this subregion and the
sampled value of the irradiation shift for this flaw, ΔRTNDT (r ,K) , determined from the
embrittlement model applied for this flaw at its current position in the RPV wall or from
weld-chemistry resampling if Step P5 was executed. Calculate the shift in the arrest
reference temperature, relative to the initiation reference temperature using Eqs. (126) in
Step 11 of Sect. 4.5
ΔRT ARREST ← Λ( μ ln( ΔRTARREST ) ,σ ln( ΔRTARREST ) ) [°F]
where (see Appendix F for the development of this protocol)
2
μ ln( ΔRT
ARREST
)
σ ln( ΔRTARREST )
= ln ⎡ ΔRT ARREST ( mean ) ⎤ −
⎣
⎦
2
ΔRT ARREST ( mean ) = 44.122exp ⎡ −0.005971 × T0 ⎤ [°C]
⎣
⎦
(
)
T0 = RT NDT0 − ΔRT epist − arrest − 32 /1.8 [°C]
σ ln( ΔRT
ARREST
)
{
( )} − 2ln ⎣⎡ΔRT
= ln exp ⎡ 0.389982 + 2ln(ΔRT ARREST ( mean ) ) ⎤ − var T0
⎣
⎦
⎧
(12.778) 2
⎪
⎪
var(T0 ) = ⎨99.905972 -1.7748073T0
⎪
0
⎪⎩
ARREST ( mean )
⎤
⎦
for T0 < −35.7 °C
for -35.7 °C ≤ T0 ≤ 56 °C
for T0 > 56 °C
Calculate the estimated arrest temperature 7 by Eq. (127) in Step 12 of Sect. 4.5
RT ARREST (r ,K) = RT NDT0 − ΔRT epist − arrest + ΔRT ARREST + ΔRTNDT ( r ,K)
Calculate the normalized (relative to RT ARREST ) temperature of the vessel at the current
location, r, in the RPV wall by Eq. (128) in Step 13 of Sect. 4.5
ΔT RELATIVE (r ,K) = T (r , t ) − RT ARREST (r ,K)
If this is the first pass through the submodel for this flaw, calculate (by Eqs. (129) and
(130) in Steps 14 and 15 in Sect. 4.5) the fractile, Φ K I −initiation , associated with this value of
KI-initiation from the arrest model, given the current value of the applied KI-initiation from the
infinite-length flaw in the IGA submodel
Φ K I −initiation =
1 ⎡ ⎛ ln( K I −initiation ) − μln( K Ia ) (ΔT RELATIVE ) ⎞ ⎤
⎢erf ⎜
⎟ + 1⎥
⎟ ⎥
2⎢ ⎜
σ
2
ln(
)
K
Ia
⎠ ⎦
⎣ ⎝
where
7
The subregion value of RTNDTo is not re-sampled in this step.
58
2
erf( x) = error function =
π
∫
x
0
exp(−ξ 2 ) dξ ; erf(− x) = − erf( x)
if K Ia _Model is equal to 1
K Ia (mean ) (ΔT RELATIVE ) = 27.302 + 69.962 exp ⎡0.006057(ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
σ ln( K Ia ) = 0.18
else if K Ia _Model is equal to 2
K Ia (mean ) (ΔT RELATIVE ) = 27.302 + 70.6998exp ⎡ 0.008991(ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
σ ln( K Ia ) = 0.34
μln( K Ia ) (ΔT RELATIVE ) = ln ⎡ K Ia (mean ) (ΔT RELATIVE ) ⎤ −
⎣
⎦
2
σ ln(
K
Ia
)
2
In the above relation for Φ K I −initiation , μln( K Ia ) is calculated at the location of the initiation of
the flaw. For this flaw, the value of Φ K I −initiation remains fixed in the IGA Propagation
submodel until Pf is resampled in Step G2 of the IGA submodel. Using the current value
of Pf , scale by Φ K I −initiation (if this is the weld layer in which the crack initiation originally
occurred) such that (from Eq. (131) in Step 15 of Sect. 4.5)
Φ K Ia = ( Pf )(Φ K I −initiation )
For subsequent weld layers do not perform the above scaling. When the flaw advances
into a new weld layer, any linkage between the flaw’s initiation and its continued
propagation is assumed to be broken.
With this Φ K Ia fractile, draw a value of KIa from its lognormal distribution as given by
Eq. (132) of Step 15 in Sect. 4.5
K Ia (Φ K Ia , ΔT RELATIVE ) = exp ⎡σ ln( K Ia ) Z Φ K Ia + μln( K Ia ) (ΔT RELATIVE ) ⎤
⎣
⎦
Z Φ K = standard normal deviate corresponding
Ia
to the Φ K Ia fractile
In the above relation for K Ia , μln( K Ia ) is calculated at the current location of the flaw. The
scaling procedure in Step P6 ensures that the initial value of KIa, calculated immediately
after initiation, does not exceed the initiating value of KI-initiation, thus producing an initial
extension. Once the value of Z Φ K has been determined for this IGA trial, the arrest
Ia
toughness during flaw advancement through the wall changes due to changes in
ΔT RELATIVE only. These changes are caused by variations in T(r,t) and RTArrest (due to
the resampling of the weld chemistry when passing into new weld layers).
For Ductile-Tearing Model No. 2, update the current value of the irradiated upper-shelf
energy by
59
USE ( i ) = A + 0.0570 ⋅ USE
1.456
(u )
( )(
)
) ⎛ φt ⎞
0.8894
− ⎡17.5 ⋅ f Cu ⋅ 1 + 1.17 Ni
+ 305P ⎤ ⎜ 19 ⎟
⎣⎢
⎦⎥ ⎜⎝ 10 ⎟⎠
0.2223
[ft-lbf]
Go to Step P7.
Step P7.
Check the current applied KI for the advancing flaw against the current value of the arrest
fracture toughness KIa.
if K I < K Ia then
the flaw has arrested
proceed to Step P8
else
the flaw has not arrested
proceed to Step P2
Step P8.
Hold the flaw at this position, and advance the time to check for re-initiation or new
ductile tearing.
NSTEP = NSTEP + 1
For this new time station, bring up the wall temperature, T(r,τ), at this position along with
the current irradiated and attenuated value of RTNDT to calculate
ΔT RELATIVE (r ,K) = T (r ,τ ) − RT RTNDT (r ,K)
Now calculate the parameters of the KIc model
aK (ΔT RELATIVE ) = 19.35 + 8.335exp ⎡0.02254( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
Ic
bK (ΔT RELATIVE ) = 15.61 + 50.132exp ⎡ 0.008( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
Ic
cK = 4
Ic
with KIc in ksi√in and ΔT = (T-RTNDT) in °F.
The static initiation toughness, KIc, is calculated from its Weibull distribution by
1/ cK Ic
K Ic (ΔT RELATIVE ) = aK Ic (ΔT RELATIVE ) + bK Ic (ΔT RELATIVE ) ⎡⎣ − ln(1 − Pf ) ⎤⎦
for aK Ic ( ΔT RELATIVE ) ≤ K Ic ≤ K Ic (max)
Proceed to Step P9.
60
Step P9.
If the warm prestressing (WPS) analysis option has been turned on by the user (see
Sect. 3.3.4 for details on WPS effects as implemented in FAVOR), check to see if the flaw
is in a state of WPS. If the ductile-tearing option is turned on, then call the ductile-tearing
model to determine if there is stable or unstable ductile tearing. If the WPS option is on
and WPS = TRUE, go to Step P10. If the WPS option is off or WPS = FALSE, check the
current applied KI for re-initiation by the test
if K I < K Ic and STABLE _ DT and FAIL _ UDT are both FALSE then
No re-initiation.
Proceed to Step P10.
else if WPS _ OPTION is on and WPS is TRUE then
No re-initiation
Proceed to Step P10
else if FAIL _ UDT is TRUE then
the vessel has failed by unstable ductile tearing
set vessel failure to TRUE
return to Step G5 of IGA model
else if STABLE _ DT is TRUE and K J Ic is less than K Ic then
the flaw has re-initiated by a ductile-tearing event
REINITIATED_BY_DUCTILE_TEARING = TRUE
the current level of tearing Δa0 is set by the ductile-tearing model
Proceed to Step P3
else
The flaw has re-initiated by a cleavage event.
REINITIATED_BY_DUCTILE_TEARING = FALSE
Reset the current level of tearing Δa0 = 0
Proceed to Step P2 and advance the flaw
Step P10. If there are time steps remaining in the transient, proceed to Step P8 and advance the time.
If the transient is complete, set vessel failure = FALSE, and return to Step 5 of the IGA
submodel.
Note that in the IGA Propagation submodel, the flaw is assumed to advance instantaneously; i.e., the
time station remains fixed during flaw growth. Time will advance only if the flaw is in a state of
arrest. If the flaw remains in arrest until the end of the transient, then the flaw is said to have
experienced a Stable Arrest.
3.3.13 Ductile-Tearing Submodel
Figure 17c presents a flowchart of the Ductile-Tearing Submodel.
Step D1.
The program enters the submodel with the current position and orientation of the crack tip
and the time within the selected transient. The submodel first checks the current wall
61
temperature at the crack tip with the ductile-tearing transition temperature, TDT. Based on
a previous study, the value of TDT is set to 200 °F . If this is not the first entry into the
model, a current value of J R* will be known, where J R* is a measure of the current
deformation state due to tearing.
if Twall < TDT then
FAIL _ UDT = FALSE
STABLE _ DT = FALSE
Return to Step P3 or P9 of IGA Submodel
else
Proceed to Step D2
Step D2.
Given the location and orientation of the flaw tip, the submodel converts the known value
of KI-applied to Japplied using a plane-strain conversion. The submodel then proceeds to
calculate/sample estimates for the JR-curve parameters, JIc , C, and m.
J applied =
(1 − ν 2 ) 2
K I −applied [in-kips/in 2 ]
E
get JIc
from either Ductile-Tearing Model No. 1 or 2
)
)
C , and m from either Ductile-Tearing Model No. 1 or 2
get
Proceed to Step D3
Step D3.
The submodel then compares the Japplied to the estimated value of JIc obtained in Step D2
and the known value of J R* . If this is the first entry into the model or if a cleavage
reinitiation has occurred since the last entry into the model, then J R* = 0 . J R* is the value
of Japplied corresponding to a previous time step at which a stable ductile tear ocurred. For a
ductile tear to occur at the current time, it is necessary for Japplied to be equal to or greater
than the current value of J R* .
if (J applied < J Ic ) or (J applied ≤ J R* ) then
FAIL _ UDT = FALSE
STABLE _ DT = FALSE
Return to Step P3 or P9 of IGA Submodel
else
Proceed to Step D4
Step D4.
The submodel then advances the position of the flaw, a0, using the known value of Japplied,
and then calculates the local tearing modulus, TR, characterizing the tearing resistance of
the material.
62
J R* = J applied
⎡ ln( J R* ) − ln(C ) ⎤
Δa = exp ⎢
⎥ , [in]
m
⎢⎣
⎥⎦
a* = a0 + Δa
⎛ E ⎞ dJ *
TR = ⎜ 2 ⎟ R
⎜ σ flow ⎟ da
⎝
⎠
⎛ E ⎞
m −1
= ⎜ 2 ⎟ × m × C × ( Δa )
⎜σ
⎟
Δa* ⎝ flow ⎠
The IGA Propagation submodel mesh is searched to find the closest node point, node n, to
the current flaw position. The flaw is then repositioned to this node point (see Fig. 19).
Based on the new position of the flaw, the applied tearing modulus is estimated from a
second-order finite-difference ratio.
J n +1 + (α − 1) J n − α 2 J n −1
, O Δx 2
α (α + 1)Δx
da
where
Δx = xn − xn−1
dJ applied
( )
≈
α=
xn +1 − xn
xn − xn −1
⎛ E ⎞ dJ applied
Tapplied = ⎜ 2 ⎟
⎜ σ flow ⎟ da
⎝
⎠
Step D5.
a = a*
A check is now made for unstable ductile tearing. If the applied tearing modulus is greater
than TR , then a state of unstable ductile tearing is declared.
if Tapplied > TR then
FAIL _ UDT = TRUE
STABLE _ DT = FALSE
Return to Step P3 or Step P9 in the IGA Propagation Submodel
else
FAIL _ UDT = FALSE
STABLE _ DT = TRUE
Δa0 = Δa
a0 = a *
Return to Step P3 or Step P9 in the IGA Propagation Submodel
63
Fig. 19. IGA Propagation submodel mesh used to estimate dJapplied / da using a second-order
central finite-difference ratio.
3.3.14 Ductile Tearing as an Initiating Event
The ductile-tearing model, as implemented, should have no effect on the values of CPI produced by
FAVOR, and this was verified in a preliminary scoping study. However, a counter was implemented
into FAVOR at the point where the conditional probability of initiation, cpi, by cleavage is calculated
to determine if initiation of flaw growth by ductile tearing was a potential issue. In all of the studies
carried out to date using the ductile-tearing models described in Sect. 3.3.11, no ductile-tearing
initiating events were discovered.
64
3.4
FAVOR Post Module – FAVPost
The distribution of the transient initiating frequencies obtained from PRA studies, the values of
conditional probability of fracture (contained in the FAVPFM-generated matrix PFMI), and the
values of the conditional probability of vessel failure (contained in the FAVPFM-generated matrix
PFMF) are combined in the FAVPost module to generate discrete distributions of the frequency of
vessel initiation, Φ ( I ) , and frequency of vessel failure, Φ ( F ) . This process is described by the
following pseudo code:
For j = 1, NSIM vessel simulations, increment by 1
For i = 1, NTRAN transients, increment by 1
Sample the discrete cumulative distribution function of the transientinitiating frequency for this transient to generate a sample initiating
frequency (in events per reactor year).
φ ( E )(i ) ← CDF(i, j ) of transient-i initiating frequency
End of Transient Loop
The above loop generates a vector of transient-initiating frequencies for this
{ }
vessel simulation, φ ( E )
(1× NTRAN )
.
For the jth vessel, take the inner product of the transient initiating frequencies
vector times the jth column-vectors in the PFMI and PFMF matrices.
Φ ( I )( j ) =
NTRAN
Φ ( F )( j ) =
∑ φ ( E )(i ) PFMI (i, j )
i =1
NTRAN
∑ φ ( E )(i ) PFMF (i, j )
i =1
End of Vessel Simulation Loop
The inner product of the row-vector of the sampled transient initiating frequencies and the jth
column-vector of PFMI produces the frequency of crack initiation for the jth vessel simulation,
Φ ( I )( j ) . Likewise, the inner product of the row-vector of sampled transient initiating frequencies and
the jth column-vector of PFMF results in the frequency of vessel failure for the jth vessel simulation,
Φ ( F )( j ) . The (i, j) entry in matrix PFMI represents the conditional probability of crack initiation of
65
the jth vessel simulation subjected to the ith transient. The units are crack initiations per event.
Therefore, the frequency of crack initiation, as determined from the inner product of the transientinitiating frequency and the conditional probability of crack initiation, is the number of crack
initiations per reactor year. Likewise, the frequency of vessel failure, as determined from the inner
product of the transient-initiating frequency and the conditional probability of vessel failure is the
number of vessel failures per reactor year.
At the end of this process, there are discrete distributions of sample size NSIM for the frequency of
crack initiation, {Φ ( I )} N ×1 , and the frequency of vessel failure, {Φ ( F )} N ×1 . The above process is
SIM
SIM
described in Fig. 20.
Fig. 20. The FAVOR post-processor FAVPost combines the distributions of conditional
probabilities of initiation and failure calculated by FAVPFM with initiating frequency
distributions for all of the transients under study to create distributions of frequencies
of RPV fracture and failure.
66
4. Probabilistic Fracture Mechanics
A central feature of modern PRA/PFM analyses is an explicit treatment of model uncertainties with
two types being distinguished, aleatory and epistemic [87]. Aleatory uncertainties arise due to the
randomness inherent in any physical or human process, whereas epistemic uncertainties are caused by
a limitation in the current state of knowledge (or understanding) of that process. Epistemic
uncertainties can therefore, in principle, be reduced by an increased state of knowledge, whereas
aleatory uncertainties are fundamentally irreducible. Playing a central role in the PTS Re-evaluation
Project, the identification and classification of epistemic and aleatory uncertainties is a crucial aspect
of PRA/PFM analyses, because the mathematical procedures used to account for them are different. A
major effort in the development of improved fracture mechanics models for FAVOR has been the
attempt to identify and classify the uncertainties in these models. Sections 4.2 through 4.5 will present
the results of this effort. The deterministic analyses carried out to create a loading definition for each
PTS transient are first discussed in Section 4.1.
It should be noted that during the investigation of new models for the FAVOR code, the basic
requirements of the PTS Re-evaluation Project played a key role in the development process. To
enable all commercial operators of pressurized water reactors to assess the state of their RPV relative
to the new PTS screening criteria without the need to make new material property measurements, the
initiation fracture toughness of the RPV needs to be estimated using only currently available RTNDT
values. Moreover, to be consistent with the LEFM principals on which the FAVOR code is based,
this RTNDT -based model needs to estimate KIc values. These restrictions suggested that only very
limited information, specifically a value of RTNDT, would be available to define the initiation fracturetoughness model appropriate to a given steel in a plant-specific RPV.
4.1
Deterministic Analyses
The FAVLoad module carries out both thermal and stress analyses of a one-dimensional
axisymmetric model of the RPV wall. The time-dependent temperature and stress distributions
through the wall constitute the thermal and mechanical loading that will be applied to postulated
flaws. In addition, Mode I stress-intensity factors are generated for a range of axially
and
circumferentially oriented infinite-length and finite-length (semi-elliptical) flaw geometries (flaw
depths and lengths). The following subsections describe how these deterministic calculations are
carried out in the FAVLoad module. The embedded-flaw model to be discussed has been
implemented in the FAVPFM module.
67
4.1.1
Thermal Analysis
The temperature time-history, T(r,τ), for the vessel is determined by modeling the RPV wall as an
axisymmetric one-dimensional structure with the temperature profile being dependent on the radial
position, r, and elapsed time, τ, in the transient. In the absence of internal heat generation, the
transient heat conduction equation is a second-order parabolic partial differential equation:
ρ c p(T )
∂T 1 ∂ ⎡
∂T ⎤
k (T ) r
=
⎢
∂τ r ∂r ⎣
∂r ⎥⎦
(44)
where ρ is the mass density, c p (T ) is the temperature-dependent mass-specific heat capacity, and
k(T) is the temperature-dependent thermal conductivity. Note that any temperature dependencies in
the mass density should be included in the characterization of the mass-specific heat capacity, leaving
the mass density as a constant in the problem formulation. Equation (44) can be expressed in the
following canonical form
∂T 1 ∂ ⎡
∂T ⎤
−
λ (T ) r
= 0 for r ∈
⎢
∂τ r ∂r ⎣
∂r ⎥⎦
1
;τ ∈ (0, ∞)
(45)
where the property grouping λ (T ) = k (T ) ρ c p (T ) is the temperature-dependent thermal diffusivity of
the material. For Eq. (45) to be well posed, initial and boundary conditions must be applied.
Initial Condition
T (r ,0) = Tinitial for Ri ≤ r ≤ Ro
(46)
Boundary Conditions
q( Ri , t ) = h(t ) (T∞ (t ) − T ( Ri , t ) ) at r = Ri
(47)
q( Ro , t ) = 0 at r = Ro
where in Eqs. (46)-(47), q is a prescribed boundary heat flux, h(τ ) is the time-dependent convective
film coefficient, T∞ (τ ) is the time-dependent bulk coolant temperature, and Ri and Ro are the inner
and outer radii of the vessel wall, respectively. Input data to the thermal model include the mesh
definition, property data, and prescribed time-histories for h(τ) and T∞ (τ ) .
68
Fig. 21. Isoparametric mapping from parameter space to axisymmetric
using three-node quadratic basis functions.
1
Euclidean space
Eqs. (45)-(47) can be solved using the finite-element method, where the variational formulation for
the transient heat conduction equation is given in Ref. [88]. The fundamental decisions required to
implement the finite-element method are (1) choice of basis functions, (2) choice of mapping, and (3)
choice of method for element integration. As shown in Fig. 21, FAVOR uses an isoparametric
mapping with 3-node quadratic cardinal basis functions, specifically
⎧ N1 (ξ ) ⎫
⎧ −ξ (1 − ξ ) ⎫
⎪⎪
⎪⎪ 1 ⎪⎪
⎪
{ N (ξ )} = ⎨ N 2 (ξ ) ⎬ = ⎨ 2(1 − ξ 2 ) ⎬⎪ ;
⎪
⎪ 2⎪
⎪
⎪⎩ N 3 (ξ ) ⎪⎭
⎪⎩ ξ (1 + ξ ) ⎪⎭
⎧ dN1 ⎫
⎪ dξ ⎪
⎪
⎪
⎧(−1 + 2ξ ) ⎫
⎪
⎪
⎧ dN ⎫ ⎪ dN 2 ⎪ 1 ⎪⎪
⎪⎪
⎨
⎬=⎨
⎬ = ⎨ −4ξ ⎬
⎩ dξ ⎭ ⎪ dξ ⎪ 2 ⎪
⎪
⎪
⎪
⎪⎩ (1 + 2ξ ) ⎪⎭
⎪ dN 3 ⎪
⎪⎩ d ξ ⎪⎭
(48)
The elements of the thermal stiffness matrix [88] are calculated using a full-integration fourth-order
Gauss-Legendre quadrature rule with the following weights, ωi , and Gauss sampling points, ξi ,
69
⎧
⎪−
⎪
⎪
⎪
⎪−
+1
4
⎪
ωi g (ξi ) where {ξi } = ⎨
∫ g (ξ )dξ ≈ ∑
i =1
−1
⎪
⎪
⎪
⎪
⎪
⎪
⎩
3+ 2 6/5 ⎫
⎧1
⎪
⎪2 − 6
7
⎪
⎪
⎪
⎪1
⎪
3− 2 6/5
⎪ +
⎪
⎪⎪ 2 6
7
⎪
⎬ ;{ωi } = ⎨
⎪1
3− 2 6/5 ⎪
⎪
⎪2 + 6
7
⎪
⎪
⎪
⎪1
⎪ −
3+ 2 6/5 ⎪
⎪⎩ 2 6
⎪
7
⎭
1 ⎫
6 / 5 ⎪⎪
1 ⎪
⎪
6 / 5 ⎪⎪
⎬
1 ⎪
6/5 ⎪
⎪
1 ⎪
⎪
6 / 5 ⎪⎭
(49)
In FAVOR, a graded mesh (see Fig. 22) is generated through the wall thickness using ten three-noded
quadratic isoparametric axisymmetric elements (21 nodes). Note that the FEM model does not use the
same discretization applied in the IGA submodel. The first two elements represent the cladding, and
the remaining eight elements model the base material. Explicit forward time integration is employed
with a fixed time step of 1.0 second. Temperature and hoop-stress profiles are plotted in Fig. 22 for a
fixed time in an example transient.
Fig. 22. One-dimensional axisymmetric finite-element model used in FAVOR to calculate both
temperature and stress histories through the wall of an RPV.
70
4.1.2
Stress Analysis
FAVLoad carries out a displacement-based finite-element analysis of the vessel using a onedimensional axisymmetric model of the vessel wall. The calculated displacements are converted into
strains using strain-displacement relationships, and the associated stresses are then calculated using
linear-elastic stress-strain relationships. At each time station during the transient, the structure is in a
state of static equilibrium; thus the load history is considered quasi-static.
Let (u , v, w) be the radial, circumferential, and axial displacements, respectively, of a material point
in a cylindrical (r ,θ , z ) coordinate system. The general two-dimensional axisymmetric case requires
that
v = 0;τ rθ = τθ z = 0; γ rθ = γ θ z = 0
(50)
where τ rθ ,τθ z are shear stresses and γ rθ , γ θ z are engineering shear strains. The strain-displacement
relationships for the two-dimensional case are
⎡∂
⎤
⎢ ∂r 0 ⎥
⎥
⎧ ε rr ⎫ ⎢⎢
⎥
1
⎪
⎪
0⎥
⎪⎪εθθ ⎪⎪ ⎢⎢ r
⎥ ⎪⎧ u ⎫⎪
(51)
⎨
⎬=⎢
⎥⎨ ⎬
∂ ⎥ ⎪⎩ w⎪⎭
⎪ ε zz ⎪ ⎢
⎪
⎪ ⎢ 0 ∂z ⎥
⎥
⎩⎪ γ zr ⎭⎪ ⎢
∂⎥
⎢∂
⎢⎣ ∂z ∂r ⎥⎦
For the one-dimensional axisymmetric case, (r ,θ , z ) are principal directions, and w = 0; ∂ ∂z = 0;
such that
ε rr =
∂u
u
∂w
∂u ∂w
= 0; γ zr =
+
=0
; εθθ = ; ε zz =
∂r
r
∂z
∂z ∂r
(52)
For the case of a long cylinder with free ends and no axial or circumferential variations in temperature
or material properties and with no radial variation in material properties, the radial and
circumferential stresses for the one-dimensional axisymmetric case are calculated from the strains by
σ rr =
E
αE
⎡(1 − ν ) ε rr + νε θθ ⎤⎦ −
(T − Tref )
(1 + ν )(1 − 2ν ) ⎣
1 − 2ν
(53)
σ θθ =
E
αE
(T − Tref )
[(1 −ν )εθθ + νε rr ] −
(1 + ν )(1 − 2ν )
1 − 2ν
(54)
where
71
σ rr
σ θθ
ε rr
ε θθ
= radial normal stress
= circumferential (hoop) normal stress
= radial normal strain
= circumferential (hoop) normal strain
T = wall temperature as a function of r
Tref = thermal stress-free reference temperature
r=
E=
ν=
α=
radial position in wall
Young's modulus of elasticity
Poisson's ratio
linear coefficient of thermal expansion
For generalized plane-strain conditions, the stress in the axial direction, σ zzPS , is given by
σ zzPS = ν (σ rr + σ θθ ) − α E (T − Tref )
(55)
To obtain the axial stresses with the ends free (assuming no cap load), it is necessary to remove the
net end force associated with the plane-strain condition. This net load is
Ro
f PS = 2π ∫ σ zzPS rdr
Ri
(56)
where Ri and Ro are the inner and outer radii of the cylinder.
In FAVOR, the radial and hoop stresses are calculated using the finite-element method in which
Eqs. (53) and (54) apply to each finite element, and thus radial variations in the material properties E,
α , and ν can be included by letting the properties vary from one element material group to another.
To account for radial variations in properties when calculating the axial stresses, Eq. (55) is applied to
each element j such that
σ zzPS− j = ν j (σ rr − j + σ θθ − j ) − α j E j (T j − Tref )
(57)
is the axial stress in each element under plane-strain conditions. To achieve a free-end condition, the
force f jPS [Eq. (56)] must be released in such a manner that the change in axial strain (displacement)
is the same for each element, because it is assumed that initial planes remain in plane under load. If
Δf j is the reduction in the plane-strain force, f jPS , on element j, then
Δf nele
Δf1
Δf 2
=
=L
A1 E1 A2 E2
Anele Enele
(58)
and
nele
∑( f
PS
j
+ Δf j ) = 0
j =1
where
72
(59)
f jPS = Aj ⎡⎣ν j (σ rr − j + σ θθ − j ) − α j E j (T j − Tref ) ⎤⎦
(60)
Aj = π (r
2
o− j
2
i− j
−r )
where ro and ri are the outer and inner radii of element j, respectively. Let f p − j be the axial forces that
are the result of adding internal pressure, p. Specifying that the axial displacements for each element
be the same gives
f p −1
A1 E1
=
f p−2
A2 E2
=L
f p − nele
(61)
Anele Enele
and
nele
∑f
j =1
p− j
= π Ro2 p
(62)
where
f j = Δf j + f p − j
Recalling that the uniform change in axial strain has no effect on σ rr and σ θθ , Eqs. (60), (61), and
(62) can be solved for fj after calculating values of σ rr − j and σ θθ − j ; then the axial stress is calculated
from
σ zz − j
(f
=
PS
j
+ fj
)
(63)
Aj
FAVOR uses a reduced-integration two-point Gauss-Legendre quadrature rule for the calculation of
σ rr and σ θθ in each element. The Gauss sample points and weights for two-point quadrature are:
⎧
⎪−
2
⎪
ωi g (ξi ) where {ξi } = ⎨
∫−1 g (ξ )dξ ≈ ∑
i =1
⎪
⎪+
⎩
+1
1⎫
⎪
3⎪
⎪⎧1⎪⎫
⎬ ;{ωi } = ⎨ ⎬
1⎪
⎩⎪1⎭⎪
⎪
3⎭
(64)
For the calculation of the axial stresses, each of the elements is divided into two sub-elements, each
containing one of the two Gauss points, and the axial stresses are calculated at each of the Gauss
points. Stresses at the nodes of the finite-element mesh are obtained by interpolation and
extrapolation using a cubic spline fit of the stresses at the Gauss points. The stress analysis uses the
same mesh and quadratic elements that are applied in the thermal analysis described in the previous
section. Details regarding the formation and assembly of the stiffness matrix and load vector for a
static stress analysis are given in any text on finite-element methods. See, for example, ref. [89].
73
4.1.3
Linear-Elastic Fracture Mechanics (LEFM)
The FAVOR code’s linear-elastic stress model treats axial flaws exposed to a one-dimensional
axisymmetric stress field and circumferential flaws exposed to a generalized-plane-strain stress field.
These flaws are, therefore, assumed to experience only a Mode I loading, where the principal load is
applied normal to the crack plane, thus tending to open the crack. It is also assumed that the plastic
zone around the crack tip is fully contained, and the overall deformation-load response of the
structure is linear. For these high-constraint conditions, the principles of linear-elastic fracture
mechanics (LEFM) apply when calculating driving forces for the crack.
4.1.3.1
Mode I Stress-Intensity Factors
For the cracked structure under LEFM conditions, the singular stress field in the vicinity of the crack
tip can be characterized by a single parameter. This one-parameter model has the form
σ θθ =
σ zz =
KI
2π r
KI
2π r
for axial flaws
(65)
for circumferential flaws
where r is the radial distance from the crack tip, and the crack plane is assumed to be a principal
plane. The critical fracture parameter in Eq. (65) is the Mode I stress-intensity factor, KI. When the
conditions for LEFM are met, the problem of calculating the stress-intensity factor can be formulated
solely in terms of the flaw geometry and the stress distribution of the uncracked structure.
FAVOR, v04.1, has an extensive stress-intensity-factor-influence coefficient (SIFIC) database for
finite- and infinite-length surface flaws that has been implemented in the FAVLoad module for Ri /t =
10 only. The HSST program at ORNL has also developed a similar database for Ri /t = 20, which was
implemented in earlier versions of FAVOR and could be re-installed for future releases if the need
arises.
4.1.3.2
Inner Surface-Breaking Flaw Models –Semi-Elliptic and Infinite Length
For inner surface-breaking flaws, the stress-intensity-factor, KI, is calculated in FAVOR using a
weighting-function approach originally introduced by Bückner [90] and applied by other researchers
[91 -94], including the developers of OCA-I [95] and OCA-P [96]. The HSST Program at ORNL
9
generated a database of SIFICs for axial infinite-length [97] and axial semi-elliptical [98] surface
flaws along with circumferential 360-degree [97] and circumferential semi-elliptical [99] surface
flaws. These databases have been implemented in the FAVLoad module.
74
Semi-Elliptic Finite Surface Flaws
As mentioned above, the stress-intensity factor, KI, is calculated by a linear superposition technique
proposed by Bückner [90], where, instead of analyzing the cracked structure using actual loads, the
analysis is performed with a distributed pressure loading applied to the crack surfaces only. This
pressure is opposite in sign, but equal in magnitude and distribution, to the stresses along the crack
line that are calculated for the uncracked structure with the actual loads applied. For an arbitrary
stress distribution and for the case of a three-dimensional semi-elliptical surface flaw, the truncated
stress distribution can be approximated by a third-order polynomial of the form
σ (a′) = C0 + C1 (a′ / a ) + C2 (a′ / a )2 + C3 (a′ / a)3
(66)
where σ (a′) is the stress normal to the crack plane at radial position, a′ . The variables a′ and a are
defined in Fig. 23, and the coefficients ( C0 , C1 , C2 , C3 ) are calculated by a generalized least squares
regression analysis in the FAVLoad module for the stress distribution calculated for the uncracked
structure across the crack depth. The KI values are determined for each of the individual terms (stress
distributions) in Eq. (66) and then added to obtain the total KI value as follows:
3
3
j =0
j =0
K I (a) = ∑ K Ij (a ) = ∑ C j π aK *j (a)
(67)
where
K *j (a ) =
Values of K Ij' ( a) / C 'j π a
K Ij' (a)
C 'j π a
(68)
were calculated for each of the normalized stress distributions
corresponding to each term in Eq. (66) (see Fig. 24), using three-dimensional finite-element analysis
results and an arbitrary value of C 'j = 1 . The dimensionless quantity K *j ( a) is referred to as the
influence coefficient. For semi-elliptic flaws, K *j ( a) values can be calculated for several points along
the crack front, in which case Eq. (67) becomes
3
K I (φ ) = ∑ C j π aK *j (φ )
(69)
j =0
where φ is the elliptical angle denoting the point on the crack front, and the crack-depth notation (a)
has been dropped. Although SIFICs are available in the database for a range of elliptical angles, this
baseline release of FAVOR only calculates the value of KI at the deepest point along the flaw front
(i.e., ϕ = 90° ).
The presence of a thin layer of stainless steel cladding on the inner surface of reactor pressure vessels
has a significant effect on the KI values for inner-surface flaws because of very high thermal stresses
generated in the cladding during a thermal transient. When using influence coefficients for three-
75
dimensional flaws, it is necessary to represent the stress distribution in the uncracked cylinder with a
third-order polynomial, and thus the discontinuity in the thermal stress at the clad-base material
interface presents a problem. To accommodate the stress discontinuity associated with the cladding,
influence coefficients were calculated for the cladding stresses alone; the corresponding KI value can
then be superimposed on the KI value due to the stresses in the base material. This is accomplished by
first calculating a KI value for a continuous-function stress distribution obtained by a linear
extrapolation of the stress distribution in the base material to the clad-base interface. Then a KI value
is calculated for the stress distribution in the cladding by subtracting the extrapolated distribution
from the actual assumed-linear distribution in the cladding. The total KI value is simply the sum of the
two. Because the stress distribution in the cladding is essentially linear, only a first-order polynomial
is used for the cladding stress-intensity-factor-influence coefficients.
The influence coefficients implemented in FAVOR were calculated using the ABAQUS [100] finiteelement code. Three-dimensional finite-element models were generated for a range of relative crack
depths (a / t) and aspect ratios (L / a) (see Fig. 23). The analysis matrix included relative crack depths
of 0.01 ≤ (a / t ) ≤ 0.5 and aspect ratios of L / a = 2,6,10 . In the process of calculating the SIFICs,
careful attention was paid to using adequately converged finite-element meshes and an appropriate
cylinder length. The number of elements in the circumferential and axial directions and around the
crack front was increased, one at a time, until the addition of one element changed the value of KI by
less than one percent. With regard to cylinder length, a minimum incremental length of the cylinder
that could be added to the length of the flaw to negate end effects was estimated from Eq. (70) [101]
1/ 4
⎡ R2 t 2 ⎤
l = 2π ⎢ i 2 ⎥
⎣ 3(1 − v ) ⎦
(70)
where ν is Poisson’s ratio, Ri is the inner radius of the vessel, and t is the wall thickness.
The analysis results in Ref. [99] demonstrated that there were essentially no differences in SIFICs
between the axial and circumferential orientations for relative flaw depths of 0.01 ≤ a / t < 0.5 and
flaw aspect ratios of L / a = 2, 6, and 10. This important finding implies that SIFICs for axial flaws
can be used for circumferential flaws up to a relative flaw depth of 0.5 with very little error. The
greatest difference ( 5% ) between the two orientations occurs for flaw geometries with an a/t = 0.5
and L/a = 10. In Appendix B, SIFICs for both axial and circumferential orientations for relative flaw
depths of a/t = 0.01, 0.0184, 0.05, 0.075, 0.1, 0.2, and 0.3 are presented in Tables B1-B7,
respectively. Table B8 presents the SIFICs for an axial flaw with a/t = 0.5, and Table B9 presents the
SIFICs for a circumferential flaw with a/t = 0.5.
76
Fig. 23. Influence coefficients, K*, have been calculated for finite semi-elliptical flaws with
aspect ratios L / a = 2, 6, and 10 for Ri / t = 10.
Fig. 24. Crack-surface loading cases for determining finite 3D flaw influence coefficients:
(a) uniform unit load, (b) linear load, (c) quadratic load, and (d) cubic load.
77
Infinite-Length Surface Flaws
Figure 25 shows the geometries for the axial and circumferential infinite-length flaws. Figure 26
illustrates the decomposition of a cracked structure under actual loads into an equivalent problem
with two components. One component is an uncracked structure under actual loads for which KI = 0 ,
since there is no crack. The second component is a cracked structure having a crack face loading
equal in magnitude and opposite in direction to the stress distribution in the uncracked structure at the
location of the crack. Therefore, the problem of interest reduces to the calculation of the KI for the
second component. This calculation can be accomplished by computing K* values for each of several
unit loads applied at specified points along the crack face (see Fig. 27) and then weighting them by
the truncated crack-free stress distribution associated with the equivalent problem [95]. The procedure
can be summarized as follows:
axial flaws
n
K I (a ) = ∑ σ i Δai K i* (ai′, a)
(71)
i =1
circumferential flaws
n
K I (a ) = ∑ 2π ( R + ai′)σ i Δai K i* (ai′, a )
(72)
i =1
where
Δai = an increment of a about ai′ such that
n
∑ Δa
i
=a
i=1
ai′ = radial distance from open end of crack to point of application of unit load,
σ i = average crack-free stress over Δai for equivalent problem
K I = opening Mode I stress-intensity factor
K I* = stress-intensity factor per unit load applied at ai′ , where load has dimensions
of force/length for axial flaws and force for circumferential flaws
n = number of points along length of crack for which K i* are available,
R = inside radius of vessel.
The ABAQUS (version 4.9.1) finite-element code was used to calculate the influence coefficients
presented in Appendix B. The general procedure consisted of developing a finite-element model for
each crack depth and then individually applying unit loads at corner nodes located along the crack
face. The axial stress-intensity-factor influence coefficients given in Table B10 have been
nondimensionalized by multiplying by the factor (0.1 t1/2), where t is the wall thickness, and the
circumferential stress-intensity-factor influence coefficients given in Table B11 have been
nondimensionalized by multiplying by the factor (10 t3/2 ). These normalizing factors account for the
78
fact that the applied load in the generalized plane-strain analyses for axial flaws is 1.0 kip/in. of
model thickness, and the applied load in the axisymmetric analyses of the circumferential flaws is a
1.0 kip total “ring” load. For both orientations, the range of relative flaw depths are a / t = {0.01, 0.02,
0.03, 0.05, 0.075, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95}. It should be noted that values in
Tables B10 and B11 for a′ / a ≥ 0.95 represent “fitted” or extrapolated values rather than directly
computed ones. ABAQUS version 4.9.1 did not correctly compute the J-integral for J-paths in which
the load on the crack face was contained within the contour itself.
Finally, it should be pointed out that, as with the finite-surface flaws, great care was exercised in
developing finite-element meshes that would produce converged solutions. Higher-order meshes were
employed throughout the modeling. Starter finite-element meshes for each crack depth were
examined for convergence by approximately doubling the mesh refinement, i.e., the number of nodes
and elements, and performing a representative K * calculation with the more refined model. This
procedure was repeated until the difference in K * values between successive models was less than
one percent, at which time the more refined model was selected for the final computation.
Fig. 25. Influence coefficients have been computed for both infinite axial and 360-degree
circumferential flaws.
79
Fig. 26. Superposition allows the use of an equivalent problem to compute the stress intensity
factor.
Fig. 27. Influence coefficients, K*, represent stress intensity factor per unit load applied to the
crack face.
80
4.1.3.3
Embedded Flaw Model
The computational methodology implemented in FAVOR for calculating Mode I stress-intensity
factors, KI , for embedded flaws [102] is the EPRI NP-1181 analytical interpretation [103] of the
ASME Section XI-Appendix A [104] model for embedded (or “subsurface” in the nomenclature of
Ref. [104]) flaws. Figure 28 is a schematic of the ASME embedded flaw model with the relevant
descriptive variables.
The procedure for calculating Mode I stress-intensity factors, KI , is based on the resolution of
nonlinear applied stresses through the RPV wall thickness into the linear superposition of
approximate membrane and bending stress components. The KI factor is thus computed from the
following relation:
K I = ( M mσ m + M bσ b ) π a / Q
(73)
where:
2a = the minor axis of the elliptical subsurface flaw
Q = flaw shape parameter
M m = free-surface correction factor for membrane stresses
M b = free-surface correction factor for bending stresses
σ m = membrane stress
σ b = bending stress
The stress-linearization procedure, depicted in Fig. 29 for a concave upward nonlinear stress profile,
involves the interpolation of the applied stresses at two points on the flaw crack front – point 1 at a
distance x1 from the inner surface and point 2 at a distance x2 from the inner surface. A straight line is
fitted through these two points which represents a linear approximation, σˆ ( x) , of the original
nonlinear stress profile, σ(x), where x is the distance from the inner surface. The effective membrane
stress, σm , is located at x = t/2 along this line, and the bending stress, σb, is the stress at the inner
surface (x = 0) minus the membrane stress. The nonlinear stress profile, σ(x), is resolved into the
linear superposition of the membrane stress (σm) and bending stress (σb) (see Fig. 29) as follows:
81
Fig. 28. Geometry and nomenclature used in embedded-flaw model.
Fig. 29. Resolution of computed nonlinear stress profile into the linear superposition of effective
membrane and bending stresses.
82
σ m = σˆ (t / 2) =
(σ ( x2 ) − σ ( x1 ) )
2a
σ b = σˆ (0) − σ m =
× ( t / 2 − x1 ) + σ ( x1 )
(σ ( x1 ) − σ ( x2 ) )
2a
×
(74)
(t / 2)
(75)
The formal definition of the shape parameter Q is based on the complete elliptic integral of the second
kind, E(x),
Q( x) = E 2 ( x)
E ( x) =
π /2
∫
0
(1 − x sin 2 (θ ) )dθ for 0 ≤ x ≤ 1
⎛a⎞
x = 1− 4⎜ ⎟
⎝L⎠
(76)
2
In ref. [103], the elliptic integral is replaced by an infinite-series approximation for Q of the form
2
⎡ m 2 m 4 m6 ⎛ 5 ⎞ 2
⎤
8 ⎛ 7 ⎞
⎢
+
+
+⎜
Q≈
1+
m +⎜
m10 ⎥
⎟
⎟
4
64 256 ⎝ 128 ⎠
⎝ 256 ⎠
⎥⎦
4(1 + m)2 ⎢⎣
π2
2
(77)
where
m=
1 − 2(a / L)
1 + 2(a / L)
Equation (77) has been implemented in FAVOR. The equation for the free-surface correction factor
for the membrane stress (Mm ) is as follows:
M m = D1 + D2 (2a / t ) 2 + D3 (2a / t ) 4 + D4 (2a / t )6 + D5 (2a / t )8 +
D6 (2a / t ) 20
⎡⎣1 − ( 2e / t ) − ( 2a / t ) ⎤⎦
(78)
1/ 2
where:
D1 = 1
D2 = 0.5948
2
D3 = 1.9502 ( e / a ) + 0.7816 ( e / a ) + 0.4812
4
3
2
5
4
D4 = 3.1913 ( e / a ) + 1.6206 ( e / a ) + 1.8806 ( e / a ) +
0.4207 ( e / a ) + 0.3963
6
D5 = 6.8410 ( e / a ) + 3.6902 ( e / a ) + 2.7301( e / a ) +
3
2
1.4472 ( e / a ) + 1.8104 ( e / a ) + 0.3199 ( e / a ) +
0.3354
83
D6 = 0.303
The equation for the free-surface correction factor for bending stresses (Mb ) is:
⎡ E ( 2e / t ) + E (2e / t )2 + E ( 2e / t )( 2a / t ) + ⎤
3
4
⎢ 2
⎥
⎢ E ( 2a / t )( 2e / t )2 + E ( 2a / t ) +
⎥
6
⎢ 5
⎥
⎢ E 2 a / t 2 + E 2e / t 2 a / t 2 + E
⎥
)
)(
)
8(
9
⎢ 7(
⎣
⎦⎥
M b = E1 +
1/ 2
⎡⎣1 − ( 2e / t ) − ( 2a / t ) ⎤⎦
(79)
where:
E1 = 0.8408685 , E2 = 1.509002 , E3 = −0.603778 ,
E4 = −0.7731469 , E5 = 0.1294097, E6 = 0.8841685 ,
E7 = −0.07410377 , E8 = 0.04428577 E9 = −0.8338377
4.1.3.4
Inclusion of Residual Stresses in Welds
The through-wall weld residual stress distribution was derived in the HSST program from a
combination of experimental measurements taken from an RPV shell segment made available from a
cancelled pressurized-water reactor plant and finite-element thermal and stress analyses [105,106].
The residual stresses in an RPV structural weld stem from (a) the clad-shell differential thermal
expansion (DTE) and (b) the residual stresses, generated by the structural welding process, that are
not completely relaxed by the post-weld heat-treatment [107]. Data required for calculation of these
residual stresses were obtained by cutting a radial slot in the longitudinal weld in a shell segment
from an RPV and then measuring the deformation of the slot width after cutting. The measured slot
openings were assumed to be the sums of the openings due to the clad-base material differential
thermal expansion (DTE) and the weld residual stresses. To evaluate the residual stresses in an RPV
structural weld, a combined experimental and analytical process was used. Slot opening
measurements were made during the machining of full-thickness clad beam specimens with twodimensional flaws. The blanks measured 54-inches long (circumferential direction), 9-inches wide
(longitudinal direction), and 9-inches thick (radial direction). The blanks were cut so as to have a
segment of a longitudinal seam weld from the original RPV at the mid-length of the blank. Using the
wire-EDM process, a slot was cut along the weld centerline in a radial direction from the inside (clad)
surface of the blank. Measurements were made on three specimens having final slot depths of 0.045
inches, 0.90 inches, or 4.50 inches, respectively. After machining, the widths of the slots were
84
measured along each radial face of the blanks. Finite-element analyses were used to develop a
through-thickness stress distribution that gave a deformation profile matching the measured values.
This distribution is shown in Fig. 30, where the contributions from clad and base DTE have been
removed. The residual stress profile is modified to apply to an analysis of a vessel that has a wall
thickness other than the one from which the stress distribution is derived. The through-wall weld
residual stress distribution retains the shape and magnitude as derived from experiment/analysis;
however, it is compressed or expanded to fit the current wall thickness by modifying the residual
profile data by the ratio of the current RPV wall thickness to 8.936, i.e., the wall thickness from
which the stress distribution was derived. The user has the option in the input deck for FAVLoad [45]
to specify whether or not the weld residual stress profile will be superimposed on either the axial or
circumferential through-wall stress distributions, or both.
Fig. 30. Weld residual stress through-thickness distribution developed for use in RPV integrity
analyses.
85
4.1.3.5
Inclusion of Crack-Face Pressure Loading for Surface-Breaking Flaws
Crack-face pressure loading on the exposed faces of internal surface-breaking flaws is included as an
option in the mechanical loading of the family of surface-breaking flaws in a FAVLoad deterministic
analysis. The Mode I Stress Intensity Factor database provides a simple but accurate mechanism for
including the effects of crack-face pressure loading.
Semi-Elliptic Finite Surface Flaws
For semi-elliptic finite surface flaws, the uniform unit-load 3D-flaw influence coefficients can be
applied to calculate the contribution, K I −cfp , of the crack-face pressure loading to the total stress
intensity factor at the deepest point of the flaw (φ = 90° ) by
K I − cfp = π a K 0* p(τ )
where p (τ ) is the coolant pressure in ksi at time τ in the transient. By linear superposition, the
crack-face pressure component, K I −cfp , is then added to the total stress intensity factor.
Infinite-Length Surface Flaws
A similar procedure can be followed for infinite-length surface flaws.
for axial flaws
n
K I −cfp (a) = ∑ p(τ ) Δai K i* (ai′, a)
i =1
for circumferential flaws
n
K I −cfp (a) = ∑ 2π ( R + ai′) p (τ ) Δai K i* (ai′, a )
i =1
where
Δai = an increment of a about ai′ such that
n
∑ Δa
i
=a
i=1
ai′ = radial distance from open end of crack to point of application of unit load,
p (τ ) = coolant pressure at time τ uniformly applied over the crack face
K I − cfp = opening Mode I stress-intensity factor contribution due to crack-face pressure
K I* = stress-intensity factor per unit load applied at ai′ , where load has dimensions
of force/length for axial flaws and force for circumferential flaws
n = number of points along length of crack for which K i* are available,
R = inside radius of vessel.
86
4.2
Sampled LEFM Material and Correlative Properties
A detailed description of the technical bases for the models in this section is presented in Ref. [108].
A summary of the material in [108] is presented here with emphasis on the implementation of these
models into FAVOR.
4.2.1
Reference Nil-Ductility Transition Temperature, RTPTS, at EOL Fluence
For each major region, FAVOR calculates and reports a value of RTNDT (designated as RTPTS). The
value of RTPTS that is reported for each major region corresponds to the subregion within that major
region that has the highest value of RTPTS. This value of RTPTS is not sampled from a distribution and
is reported for comparison purposes only and is not used in any subsequent analyses.
There are two minor differences between the definition of RTPTS as utilized in FAVOR and as
currently defined by 10CFR50.61 [10], where RTPTS is defined as follows:
RTPTS = RTNDT (U ) + M + ΔRTPTS
(80)
where M is the margin term added to account for uncertainties in the values of the unirradiated
RTNTD(U) and ΔRTPTS . The margin term, M, is determined by
M = 2 σ U2 + σ Δ2
σ U = the standard deviation for RTNDT (U )
(81)
σ Δ = the standard deviation for ΔRTNDT
In 10CFR50.61 [10], ΔRTPTS. is the mean value of the transition temperature shift due to irradiation at
the EOL (end-of-licensing) fast-neutron fluence attenuated to the clad-base interface; whereas, in
FAVOR, ΔRTPTS .is the mean value of the transition temperature shift due to the irradiation
corresponding to the attenuated neutron fluence at the time in the operating life of the vessel for
which the PFM analysis is being performed.
Currently, in 10CFR50.61, ΔRTPTS is calculated from the irradiation shift model taken from
Regulatory Guide 1.99, revision 2 [12], where
87
ΔRTPTS = (CF ) f (δ )(0.28−0.10 log10 ( f (δ )))
CF = chemistry factor, a continuous function of copper and nickel
f (δ ) = best-estimate neutron fluence [1019 n/cm 2 ; E > 1 MeV] attenuated
from the inner surface to the clad/base metal interface
(82)
δ = distance from the inner surface to the clad/base metal interface [in.]
The fast-neutron fluence is attenuated through the wall by the relation
f (δ ) = f (0) exp(−0.24 × δ )
(83)
where δ is in inches and f (0) is the neutron fluence at the inner surface. Look-up tables for the
chemistry factor, CF, taken from 10CFR50.61 [10], are included in FAVOR for the calculation of
RTPTS.
In FAVOR, ΔRTPTS may be calculated using either Regulatory Guide 1.99, Rev 2 (RG 1.99, Rev 2)
[12], as defined above, or by ΔT30 (see Eq. (84)) as calculated by the Eason and Wright irradiationshift model [86] to be discussed in the following section. The intent is to provide for the generality of
using the current RG 1.99, Rev 2 or the Eason and Wright irradiation-shift model [86]. It is
anticipated that the Eason and Wright model [86] may supersede the current RG 1.99, Rev 2 model
discussed above.
4.2.2
Radiation Embrittlement
Irradiation damage of RPV steels in U.S. PWRs occurs as a consequence of two hardening
mechanisms: matrix hardening and age hardening. Details of these mechanisms are taken from
[108]:
Matrix Hardening – Matrix damage develops continuously during irradiation, producing
hardening that has a square root dependence on fluence. Matrix damage can be divided
into two components: unstable matrix defects (UMD), and stable matrix defects (SMD).
Unstable matrix defects are formed at relatively low fluence and are small vacancy or
interstitial clusters, complexed with solutes such as phosphorous. UMDs are produced in
displacement cascades. Increasing flux causes increasing hardening due to these defects,
but they occur relatively independently of alloy composition. In low copper alloys, at low
fluence and high flux, UMD is the dominant source of hardening; however, in high copper
steels, these defects delay the copper-rich precipitate contribution to hardening by
reducing the efficiency of radiation-enhanced diffusion. Stable matrix features form at
high fluence and include nanovoids and more highly complexed clusters. These defects
cause hardening that increases with the square root of exposure and is especially important
at high fluence levels.
Age Hardening – Radiation accelerates the precipitation of copper held in solid solution,
forming copper-rich precipitates (CRPs) that inhibit dislocation motion and, thereby,
harden the material. This hardening rises to a peak value and is then unaffected by
subsequent irradiation because no copper remains in solid solution to precipitate out and
88
cause damage. The magnitude of this peak depends on the amount of copper initially in
solution. This copper is available for subsequent precipitation. Post-weld heat treatment
(PWHT) performed before the RPV is placed into service can also precipitate copper,
removing its ability to cause further damage during irradiation. Thus, different materials
are expected to have different peak hardening values due to differing pre-service thermal
treatments. Additionally, the presence of nickel in the alloy further enhances its agehardening capacity. Nickel precipitates together with copper, forming larger second-phase
particles that present greater impediments to dislocation motion and, thereby, produce a
greater hardening effect.
These physical insights helped to establish the functional form of a relationship between basic
material composition, irradiation-condition variables, and measurable quantities such as yieldstrength increase, Charpy-transition-temperature shift, and toughness-transition-temperature shift. A
quantitative relationship was developed from the database of Charpy shift values, ΔT30 , generated in
US commercial reactor surveillance programs. Eason and Wright [86] recently developed the
following physically motivated fit from these data. 8
)
ΔT30 ( Ni, Cu , P, f 0 (r ),τ exposure , Tc , product form) =
)
⎛ 19310 ⎞
A exp ⎜
⎟ 1 + 110 P
⎝ Tc + 460 ⎠
(
) ( f (r ) )
0.4601
0
(
1.250
+ B 1 + 2.40 Ni
⎧8.86 × 10-17 for welds ⎫
⎪
⎪
⎪
⎪
A = ⎨ 9.30 × 10-17 for forgings ⎬
⎪
⎪
−17
⎪⎩12.7 × 10 for plates ⎭⎪
⎧ 230 for welds
⎫
⎪
⎪
⎪⎪ 132 for forgings
⎪⎪
B=⎨
⎬
⎪ 206 for plates in CE vessels ⎪
⎪
⎪
⎪⎩ 156 for other plates
⎪⎭
(
) f (Cu) g ( f (r)) + Bias
0
)
⎡ log f (r ) + 4.579 × 1012τ
⎤
10
0
exposure − 18.265
1 1
⎥
g ( f 0 (r )) = + tanh ⎢
⎢
⎥
2 2
0.713
⎣
⎦
⎧
0
⎪
f Cu = ⎨
⎪ Cu -0.072
⎩
( )
(
)
0.659
for Cu ≤ 0.072 wt % ⎫
⎪
⎬
for Cu > 0.072 wt % ⎪
⎭
subject to
8
A curved overbar, X , indicates a sampled random variate.
89
(84)
⎧⎪ 0.25 for welds with Linde 80 or Linde 0091 flux ⎫⎪
Cumax = ⎨
⎬
⎪⎩0.305 for everything else
⎪⎭
and
⎧⎪ 0 for τ exposure < 97000 h ⎫⎪
Bias = ⎨
⎬
⎩⎪9.4 for τ exposure ≥ 97000 h ⎭⎪
)
where Cu is the sampled copper content in wt%, Ni is the sampled nickel content in wt%, P is the
)
sampled phosphorous content in wt%, f 0 (r ) is the sampled and then attenuated neutron fluence in
n/cm2, r is the position from the inner surface of RPV wall, τexposure is exposure time in hours (input to
FAVOR in EFPY), and Tc is coolant temperature in °F . The fast-neutron fluence at the inner surface
of the vessel, f 0 (0) , is sampled using the protocol given in Sect. 4.2.3. The sampled neutron fluence
for the flaw is then attenuated (but not resampled) as the crack grows through the wall. The sampling
distributions and protocols for plate and weld chemistry are presented in Sect. 4.2.9.
Reference [108] recommends that the uncertainty in the CVN transition shift values, ΔT30 , be treated
as epistemic. Having used information concerning composition and irradiation conditions to estimate
the CVN transition temperature shift using Eq. (84), it is necessary to transform these ΔT30 values
into shifts in the fracture-toughness transition temperature. Figure 31 provides an empirical basis for
the following least-squares fits for ΔRT NDT using data extracted from the literature as discussed in
[108].
⎧⎪0.99ΔT30 (r ,K) welds
ΔRT NDT (r ,K) = ⎨
⎪⎩1.10ΔT30 (r ,K) plates and forgings
90
(85)
Fig. 31. Relationship between the change in the fracture-toughness index temperature
( ΔT0 ≈ ΔRTNDT ) change in the 30 ft-lbf CVN transition temperature ( ΔT30 ) for welds and
plates/forgings produced by irradiation. The difference in the best-fit slopes is
statistically significant (from [108]).
91
4.2.3
Fast-Neutron Fluence Attenuation and Sampling Distribution
The sampled fast-neutron fluence at the crack tip is attenuated from its sampled reference value,
f 0 (0) , at the inner surface of the RPV wall. This attenuation takes the following form
)
f 0 (a) = f 0 (0) × exp(−0.24a)
(86)
where a is the position of the flaw tip (in inches) relative to the inner surface.
The inner surface fluence is sampled from two normal distributions such that
σ global = SIGFGL × fluencesubregion
)
f mean ← N ( fluencesubregion , σ global )
)
σ local = SIGFLC × f mean
)
f 0 (0) ← N ( f mean , σ local )
(87)
where the best-estimate fluence, fluencesubregion , is input by the user at the subregion level. The global
SIGFGL and local SIGFLC multipliers are supplied as input by the user. Recommended values are
SIGFGL = 0.056 and SIGFLC = 0.118. Negative values of sampled fast-neutron fluence are handled
as nonphysical exceptions in FAVOR using the truncation protocol described in Sect. 3.3.6, with 0.0
as a one-sided truncation boundary.
4.2.4
ORNL 99/27 KIc and KIa Databases
The EPRI KIc database [73] as amended by Nanstad et al. [109] consists of 171 data points and
includes data from 11 unirradiated pressure-vessel steels. These data were taken using compact
tension C(T) and wedge-open-loading (WOL) test specimens ranging in size from 1T to 11T. A
survey was recently conducted by ORNL to identify additional KIc and KIa data to augment the EPRI
database. The result of this survey has been designated as the ORNL 99/27 extended KIc/KIa database
[74].
The candidate KIc data were evaluated using the following criteria: (a) satisfaction of validity
requirements given in ASTM Standard E 399 [110] to maintain consistency with the LEFM driving
forces applied in the fracture model, (b) availability in tabular form, and (c) availability of
unirradiated RTNDT0, determined according to the ASME Boiler and Pressure Vessel Code, Section III,
NB-2331 [111]. The ORNL survey produced an additional 84 KIc fracture-toughness values obtained
from Refs. [112 -116]. The extended KIc database, compiled from the amended EPRI data and from
1 1
the ORNL survey, provided a total of 255 fracture-toughness data points from 18 materials for input
92
to the statistical model development procedures described in Ref. [74] and applied herein. A plot of
the extended KIc database versus (T – RTNDT0) is given in Fig. 32; the complete tabulation of the
database is included in Appendix C of this report with a summary presented in Table 7.
A similar survey was carried out to compile an extended KIa database that would include those data in
the EPRI report (see Fig. 33a). Because the ASTM Standard E 1221 [117] is relatively new, many of
the existing data were generated before the adoption of the standard. Thus, it was agreed that
candidate KIa data would be evaluated in a more general context, including engineering judgment of
acknowledged experts and general acceptance by the nuclear technology community. The ORNL
survey produced an additional 62 fracture-toughness, KIa, data points [118- 120] to augment the
1
existing 50 data points [121,122] in EPRI NP-719-SR. A complete tabulation of the 112 fracturetoughness values is given in Appendix C of this report with a summary presented in Table 8. A
description of the chemistry and heat treatment of the principal steels in the ORNL 99/27 database is
shown in Table 9.
In conjunction with the development of a ductile-tearing model, arrest data from large-specimen
experiments carried out in the 1980s were also added to the KIa database (see Fig. 33b). These
additional large-specimen arrest data came from the HSST Wide Plate test program (WP-1 [38] and
WP2 [39]), the HSST Pressurized Thermal Shock Experiments (PTSE-1[26] and PTSE-2 [27]), and
the HSST Thermal Shock Experiments (TSE) [123].
93
Fig. 32. ORNL 99/27 KIc database including modified ASME KIc curve that served as a lowerbounding reference curve in the development of a new transition index temperature.
94
(a)
(b)
Fig. 33. KIa databases (a) ORNL 99/27 KIa database and (b) Extended KIa database.
95
Table 7. Summary of ORNL 99/27 KIc Extended Database
EPRI
1
2
3
4
5
6
6
7
8
9
10
11
Additional
12
13
14
15
16
17
18
Material
Database
HSST 01 subarc
weldment
A533B Cl. 1
subarc weld
HSST 01
HSST 03
A533B Cl. 1
HSST 02
HSST 02
A533B Cl. 1
weldment
A533 B Cl. 1
weldment/HAZ
A508 Cl.2
European Forging
A508 Class 2
A508 Class 2
Data
HSSI Weld 72W
HSSI Weld 73W
HSST Plate 13A
A508 Cl. 3
Midland Nozzle
Course Weld
Midland Beltline
Plate 02 4th Irr.
Series (68-71W)
Specimen
Type
Size
Range
Temp.
Range
(°F)
(T-RTNDT0)
Range
(°F)
No. of
Data
Points
C(T)
1T - 6T
-200 to -50
-200 to -50
8
Shabbits (1969)
C(T)
1T - 8T
-200 to 0
-200 to 0
8
Mager (1970)
Mager (1970)
Mager (1969)
Mager (1969)
Shabbits (1969)
Mager (1969)
C(T)
C(T)
WOL
WOL & C(T)
C(T)
WOL
1T
1T
1T - 2T
1T - 2T
1T - 11T
1T - 2T
-150
-150
-320 to -150
-200 to 0
-250 to 50
-320 to -200
-170
-170
-385 to -215
-200 to 0
-250 to 50
-275 to -155
17
9
13
41
28
10
Mager (1969)
WOL
1T - 2T
-320 to -200
-320 to -200
6
Mager (1969)
WOL
1T - 2T
-320 to -100
-370 to -150
12
unpublished
unpublished
C(T)
C(T)
2T - 6T
2T - 8T
-150 to 0
-125 to -75
-201 to -51
-190 to -30
Total
9
10
171
NUREG/CR-5913.
NUREG/CR-5913
NUREG/CR-5788
ASTM STP 803
NUREG/CR-6249
C(T)
C(T)
C(T)
Bx2B C(T)
C(T)
1T-6T
1T-4T
½T-4T
1T-4T
1T
-238 to 50
-238 to -58
-238 to -103
-238 to -4
-148 to -58
-229 to 59
-209 to -29
-229 to -94
-225 to 9
-200 to -110
13
10
43
6
6
NUREG/CR-6249
NUREG/CR-4880
C(T)
C(T)
1T
1T
-148
-148 to -139
-171
-148 to -139
2
4
Total
Grand Total
84
255
Source
EPRI NP-719-SR
Shabbits (1969)
Table 8. Summary of KIa Extended Database
Material
EPRI
Database
1
HSST 02
Additional Data Additional Data
2
HSSI Weld 72W
3
HSSI Weld 73W
4
MW15J
Large Specimen Data
5
HSST WP1
6
HSST WP2
7
HSST PTSE-1
8
HSST PTSE-2
9
HSST TSE
Source
EPRI NP-719-SR
Ripling (1971)
Specimen
Type
Size
Range
Test Temp. (T-RTNDT )
No. of
Range
Range
Data Points
(°F)
(°F)
CCA crack arrest
1T-3T
-150 to 121 -150 to 121
NUREG/CR-5584
NUREG/CR-5584
NUREG/CR-6621
CCA crack arrest
CCA crack arrest
CCA crack arrest
NUREG/CR-5330
NUREG/CR-5451
NUREG/CR-4106
NUREG/CR-4888
NUREG/CR-4249
Wide Plate Tests
Wide Plate Tests
Pressurized Vessel
Pressurized Vessel
Thermally-Shocked Cylinder
96
(-)
(-)
(-)
(-)
(-)
50
-78 to 41
-78 to 59
-4 to 50
-68 to 51
-48 to 89
-36 to 18
32
26
4
84 to 198
142 tp 324
326 to 354
267 tp 325
72 to 268
94 to 207
2 to 184
100 to 158
130 to 158
-63 to 103
Total =
18
38
2
3
10
183
Table 9. Chemistry and Heat Treatment of Principal Materials: ORNL 99/27 Database
Material ID
HSST 01
Specification
A533B Cl. 1
HSST 02
A533B Cl. 1
HSST 03
A533B Cl. 1
HSST 02
A533B Cl. 1
HSST 01
subarc weld
B&W subarc
weldment
PW/PH
weldment
MD07
European
-
A533B Cl. 1
72W
73W
Notes:
1.
2.
3.
4.
5.
6.
7.
8.
A533B Cl. 1
A533B Cl. 1
A508 Cl. 2
Ring forging
A533B
Cl. 1
A533B weld
A533B weld
Normalizing:
Austentizing:
Quenching:
Tempering:
Stress Relief:
Normalizing:
Austentizing:
Quenching:
Tempering:
Stress Relief:
Normalizing:
Austentizing:
Quenching:
Tempering:
Stress Relief:
Normalizing:
Austentizing:
Quenching:
Tempering:
Stress Relief:
Post Weld:
Intermediate
Post Weld
Intermediate
Quenching:
9.
Quenching:
Quenching:
Source
Mager
(1970)
Mager
(1969)
Mager
(1970)
Shabbits
(1969)
Shabbits
(1969)
Shabbits
(1969)
Mager
(1969)
Mager
(1969)
Mager
(1969)
5788
5788
Chemistry – wt (%)
C
P
Mn
Ni
.22 .012 1.48 .68
Mo
.52
Si
.25
Cr
-
Cu
-
S
.018
Al
-
Heat
Treatment
Note 1
.22
.012
1.48
.68
.52
.25
-
-
.018
-
Note 2
.20
.011
1.26
.56
.45
.25
.10
.13
.018
.034
Note 3
.22
.012
1.48
.68
.52
.25
-
-
.018
-
Note 4
.12
.014
1.35
.65
.52
.23
-
-
.012
-
Note 5
.10
.009
1.77
.64
.42
.36
-
-
.015
-
Note 6
.09
.019
1.25
.52
.23
.05
.22
.13
.037
Note 7
.18
.009
1.16
1.0
8
.72
.51
.24
.28
-
.10
-
Note 8
.19
.012
1.37
.52
.45
.25
.13
.15
.016
.048
Note 9
.09
.10
.006
.005
1.66
1.56
.60
.60
.58
.58
.04
.04
.27
.25
.23
.21
.006
.005
-
1675 °F
1600 °F
Water quench
1225 °F
1150 °F
1675 °F
1600 °F
Water quench
1225 °F
1150 °F
1675 °F
1575 °F
Water quench
1175 °F
1125 °F
1675 ± 25 °F
1520 °F – 1620 °F
Water quench.
1200 °F – 1245 °F
1150 ± 25 °F
1150 ± 25 °F
1100 ± 25 °F
1100 °F – 1150 °F
1100 °F – 1150 °F
620 °C
925 °C
Water quench
650 °C
620 °C
910 °C
Water quench
680 °C
850 °C
Water quench
690 °C
620 °C
4 hr, air cooled
4 hr
4 hr, furnace cooled
40 hr, furnace cooled
4 hr, air cooled
4 hr
4 hr, furnace cooled
40 hr, furnace cooled
12 hr, air cooled
12 hr
12 hr, furnace cooled
40 hr, furnace cooled
4 hr
4 hr
4 hr, air cooled
40 hr, furnace cooled to 600 °F
12 hr
15 min
12 hr
15 min
27 hr, air cooled
5 hr
3 hr, furnace cooled
24 hr, air cooled
8 hr
10 hr, furnace cooled
8 hr
8 hr, air cooled
24 hr, air cooled
97
4.2.5
Index Temperature RTNDT – Uncertainty Classification and Quantification
Values of RTNDT are uncertain both due to epistemic and aleatory causes. The epistemic uncertainty is
due to the conservative bias implicit in the ASME NB-2331 [111] definition of RTNDT, the variety of
inconsistent transition temperature metrics used to define RTNDT, the lack of prescription in the test
methods used to define RTNDT, and the fact that the CVN and NDT values used to define RTNDT do not
themselves measure fracture toughness. Aleatory uncertainties are due to material variability. It is
expected that epistemic uncertainty sources outnumber aleatory ones [108]; however, this expectation
alone is inadequate to classify the uncertainty in RTNDT as being primarily aleatory or primarily
epistemic. To make this distinction, a comparison of the RTNDT index temperature to an exemplar
index temperature (such as the Master Curve index T0 [124]) associated with a physically motivated
model of crack initiation toughness is needed.
The Master Curve index temperature T0 is estimated directly from fracture-toughness data, and, by
definition, it is therefore associated with the same location on the transition temperature curve of
every steel, suggesting that the sources of epistemic uncertainty that are associated with RTNDT do not
influence T0. Thus, the uncertainty in T0 is expected to be primarily aleatory, and a comparison
between T0 and RTNDT values can be used to quantify the epistemic uncertainty in RTNDT. The
numerical difference between RTNDT and T0 has been used to quantify how far away from measured
fracture-toughness data RTNDT positions a model of fracture toughness for a given heat of steel [108].
Figure 34 shows a cumulative distribution function (CDF) constructed from the difference between
values of RTNDT and T0 reported in the literature [125] for the RPV steels in the ORNL 99/27
database. See Appendix E for a description of the statistical procedures applied in the construction of
this CDF. These data (see Table 10) demonstrate that the epistemic uncertainty in RTNDT almost
always produces a high estimate of the actual fracture-toughness transition temperature.
Even though it quantifies the epistemic uncertainty in RTNDT, the CDF illustrated in Fig. 34 cannot be
used directly in FAVOR because of inconsistencies between T0 and the requirements of the PTS reevaluation project. Consequently, an alternative CDF (see Fig. 35) was developed that avoids the
explicit treatment of size effects and the use of elastic-plastic fracture mechanics (EPFM) toughness
data, but retains the important concept from the Master Curve that the index temperature should be
quantitatively linked to the measured toughness data. This alternative CDF was determined based on
the temperature shift values (∆RTepistemic in Table 11) needed to make a NB-2331 RTNDT-positioned KIc
curve lower-bound the ASTM E-399 valid KIc data for each of the 18 heats of RPV steel in the ORNL
99/27 database. See Fig. 36 for an example of this lower-bounding shift procedure for HSST Plate 02.
98
Fig. 34. Cumulative distribution function of the observed difference in RTNDT 0 and To (with a
size of 1T) using data in the ORNL 99/27 database.
Fig. 35. Cumulative distribution function of the difference (designated as ΔRTepistemic ) between
RTNDT0 and a new lower-bounding reference index designated RTLB.
99
Table 10. Materials Used from the ORNL 99/27 KIc Extended Database
ID
Form
RTNDT (°F)
T0 (°F)*
RTNDT - T0
P
HSST-03
Plate
20
31
-11
0.0455
HSST-02
Plate
0
-17
17
0.1104
HSST-01
Plate
20
-1
21
0.1753
A508 Cl. 3
Forging
-13
-46
33
0.2403
73W
Weld
-29.2
-78
48.8
0.3052
A533B Cl. 1
Weld
0
-57
57
0.3701
72W
Weld
-9.4
-70
60.6
0.4351
A533B Cl. 1
Plate
-9.4
-109
99.6
0.5000
HSST-01
Weld
0
-105
105
0.5649
A533B Cl. 1
Weld
-45
-151
106
0.6299
A508 Cl. 2
Forging
51
-60
111
0.6948
A508 Cl. 2
Forging
65
-55
120
0.7597
A533B Cl. 1
HAZ
0
-132
132
0.8247
A533B Cl. 1
Plate
65
-74
139
0.8896
A508 Cl. 2
Forging
50
-124
174
0.9545
*T0 values reported in [125]. Calculated using ASTM E-1921 valid data.
**Provisional Tq values calculated using ASTM E-399 valid KIc data in [74].
Tq (°F)**
26.1
-17.4
-2.9
-56.7
-104.4
-151.5
-59.9
-5.8
-132.3
-73.8
-119.3
Fig. 36. The ΔRTLB for HSST Plate 02. The lower-bounding transition reference temperature,
RTLB , was developed from 18 materials in the ORNL 99/27 database, where for each
material RTLB = RTNDT 0 − ΔRTLB .
100
Table 11. Values of Lower-Bounding Reference Temperature
with and without Sample-Size Adjustment: ORNL 99/27 Database
Material ID Sample RTNDT0 RTLB (k)* ΔRTLB (k)*
Size RTLB (k)** ΔRTepistemic(k)**
No.
Correct.
Size, N
(°F)
(°F)
(°F)
(°F)
(°F)
(°F)
k
1
8
0.0
-75.2
75.2
10.9
-64.3
64.3
2
8
0.0
0.0
0
10.9
10.9
-10.9
3
17
20.0
-82.4
102.4
4.6
-77.8
97.8
4
9
20.0
-81.1
101.1
9.6
-71.5
91.5
5
13
65.0
-127.6
192.6
6.4
-121.2
186.2
6
69
0.0
-2.1
2.1
0
-2.1
2.1
7
10
-45.0
-195.7
150.7
8.5
-187.2
142.2
8
6
0.0
-176.9
176.9
14.5
-162.4
162.4
9
12
50.0
-104.5
154.5
6.9
-97.6
147.6
10
9
51.0
-8.7
59.7
9.6
0.9
50.1
11
10
65.0
1.9
63.1
8.5
10.4
54.6
12
13
-9.4
3.6
-13.0
6.4
10.0
-19.4
13
10
-29.2
-76.1
46.9
8.5
-67.6
38.4
14
43
-9.4
-43.5
34.1
0.9
-42.6
33.2
15
6
-13.0
-25.8
12.8
14.5
-11.3
-1.7
16
6
52.0
-51.9
103.9
14.5
-37.4
89.4
17
2
23.0
-99.7
122.7
40.8
-58.9
81.9
18
4
0.0
-83.8
83.8
21.5
-62.3
62.3
( k )*
RTLB = lower-bounding reference temperature for the kth material without sample size-adjustment
(k )
( k )*
ΔRTLB( k )* = RTNDT
(0) − RTLB
RTLB( k )** = lower-bounding reference temperature for the kth material with sample size-adjustment
( k )**
(k )
( k )**
ΔRTepistemic
= RTNDT
0 − RTLB
The adjusted ASME lower-bounding curve shown in Fig. 36 has the following form:
K Ic = 23.65 + 29.56exp [ 0.02(T − RTNDT ) ] ksi in.
(88)
with (T − RTNDT ) in °F. The adjustment for sample size indicated in Table 11 assumes that Eq. (88)
represents a 0.01 fractile. The RTNDT 0 − T0 CDF (Figs. 34 and 37) is a Weibull distribution with a
flaw-size dependence
( RTNDT 0 − T0 ) ← W (axT ,122.4, 2.25)
1/ 4
⎡ 80 ( B
− 10 ⎤
1.8
xT B1T )
⎢
⎥ [°F]
axT = a1T −
ln
0.019 ⎢
70
⎥
⎣
⎦
a1T = −27.82 °F
BxT = flaw length [in.]
B1T = 1.0 in.
101
(89)
The lower-bounding CDF, Eq. (90), quantifies the epistemic uncertainty in RTNDT in a manner fully
consistent with the constraints placed on the toughness models used in the PTS re-evaluation effort.
In Fig. 37, we also compare this quantification of epistemic uncertainty with that based on the Master
Curve. This comparison illustrates that the implicit treatment of size effects adopted when developing
the alternative CDF using ASTM E 399 valid data produces a result quite similar in form to that
based on the Master Curve. The similarity of the alternative CDF to the Master Curve-based CDF
provides a link between the RTLB concept developed to conform to the requirements of the PTS reevaluation and the physical and empirical underpinnings of the Master Curve, thereby demonstrating
that aleatory and epistemic uncertainties can be reasonably distinguished using RTLB and ΔRTLB. The
epistemic uncertainty in the unirradiated value of RTNDT0 is estimated by sampling from the following
Weibull distribution (see Appendix F for details on the development of Eq. (90) ):
ΔRT epistemic ← W (−45.586,131.27, 2.177)
1/ 2.177
ΔRT epistemic = −45.586 + 131.27 ⎡⎣ − ln (1 − Φ ) ⎤⎦
[°F]
(90)
where Φ ← U(0,1)
Combined with the sampled irradiation-shift term described in Sect. 4.2.2, the irradiated value of
RTNDT is calculated by
RT NDT (r ,K) = RT NDT 0 − ΔRT epistemic + ΔRT NDT (r ,K)
(91)
where RT NDT 0 ← N ( RT NDT 0 ,σ RTNDT 0 ) and RTNDT is a function of the position of the crack tip due to
the attenuation of the fast-neutron fluence at position r in the vessel wall.
Fig. 37. Comparison of cumulative distribution functions developed for RTNDT0-T0 and
RTNDT0-RTLB .
102
4.2.6
Index Temperature RTArrest – Uncertainty Classification and Quantification
To enable all commercial operators of pressurized water reactors to assess the state of their RPV
relative to new PTS screening criteria without the need to make new material property measurements,
the arrest fracture toughness of the RPV needs to be estimated using only currently available
unirradiated RTNDT0 values. These restrictions suggest that very limited information, specifically a
value of RTNDT0, is available to define the arrest fracture-toughness model appropriate to a particular
steel in a particular RPV. Consequently, the temperature dependency and uncertainty of the arrest
fracture-toughness model will either have to be demonstrated or assumed to be invariant over a wide
range of conditions because sufficient information is not available to establish these features on a
heat-specific basis [108].
The information presented in [108] suggests that a relevant arrest reference temperature can be
defined based on (a) an index temperature that defines the position of the plane-strain crack arrest
toughness, KIa , transition curve on the temperature axis and (b) a relationship between the index
temperatures for the initiation and arrest fracture-toughness curves (assuming such a relationship
exists). For this study, the temperature dependency of KIa data was assumed to be universal to all
reactor pressure vessel steels, or, more specifically, within this class of materials the temperature
dependency was assumed to be insensitive to all individual and combined effects of alloying, heat
treatment (and other thermal processing), mechanical processing, and irradiation. These material
variables only influence the temperature range over which a particular steel experiences a transition
from brittle behavior (at low temperatures) to ductile behavior (at higher temperatures), this being
quantified by a heat-specific index temperature value. Furthermore, the information presented in
[108] suggests that the relationship between the index temperatures for crack initiation and crack
arrest toughness is also not expected to be influenced strongly by heat-specific factors.
From [108]:
Crack arrest occurs when dislocations can move faster than the crack
propagates, resulting in crack tip blunting and arrest. Dislocation mobility
therefore controls the ability of a ferritic steel to arrest a running cleavage
crack, and thus its crack arrest toughness. The atomic lattice structure is the
only feature of the material that controls the temperature dependence of the
material properties that are controlled by dislocation motion. Consequently, as
was the case for crack initiation toughness, the temperature dependency of
crack arrest toughness depends only on the short-range barriers to dislocation
motion established by the BCC lattice structure. Other features that vary with
steel composition, heat treatment, and irradiation include grain size/boundaries,
point defects, inclusions, precipitates, and dislocation substructures. These
features all influence dislocation motion, and thereby both strength and
toughness, but their large inter-barrier spacing relative to the atomic scale
associated with the lattice structure makes these effects completely athermal.
103
This understanding suggests that the myriad of metallurgical factors that can
influence absolute strength and toughness values, and thereby the transition
temperature, exert no control over the temperature dependency of arrest
toughness in fracture mode transition. Additionally, since KIc and KIa both
depend on the ability of the material to absorb energy via dislocation motion, KIc
and KIa are both expected to exhibit a similar temperature dependence.
As described in [108], a strong physical basis supports a temperature dependency in arrest fracturetoughness data that is universal to all ferritic steels; this temperature dependence has a similar
functional form to that of crack-initiation toughness. Mathematically, Wallin and co-workers
proposed [126,127]:
K Ia ( mean ) = 30 + 70exp ⎣⎡ 0.019 (T − TKIa ) ⎤⎦ [MPa m]
(92)
where (T − TKIa ) is in °C . Equation (92) describes the temperature (T) dependency of the mean
arrest toughness (KIa(mean)). In this equation, temperature is normalized to the index temperature TKIa,
where TKIa is defined as the temperature at which the mean arrest toughness is 100 MPa m
( 91 ksi in. ). Wallin found that a lognormal distribution having a lognormal standard deviation of
0.18 fits the extensive database used in his study.
The physical understanding of the relationship between crack initiation and crack arrest presented in
[108] suggests that the temperature separation between the KIc and KIa transition curves should
progressively diminish as the material is hardened (e.g. by cold work, irradiation, etc.). Available
empirical evidence supports this expectation, as illustrated in Fig. 38. An exponentially decaying
functional form for the mean was selected to represent these data, because this relationship had the
mathematical form anticipated from physical considerations (i.e. the separation between the KIc and
KIa curves diminishes as To increases). This nonlinear regression fit was:
ΔRTARREST ( mean ) ≡ TK Ia − To = 44.123 ⋅ exp {−0.006T0 } [°C]
(93)
where ΔRTARREST is distributed lognormally about the mean given by Eq. (93), with an estimated lognormal standard deviation of 0.39 (see Fig. 39). Table 12 presents several reference-transition temperature indices for the steels in the ORNL 99/27 KIa database including RT Arrest calculated from
Eq. (93).
104
Table 12. ORNL 99/27 KIa Database – Reference-Transition Temperatures
Material
ID
HSST-02
72W
73W
Midland
Product
Form
Plate
Weld
Weld
Weld
Sample
Size
50
32
26
4
RTNDT0
(°F)
0
-9.4
-29.2
32.2
RTLB
(°F)
-2.1
-42.6
-67.6
-58.9
T0
(°F)
-17
-70
-78
NA
RTArrest
(°F)
76.8
49.8
34.1
NA
TKIa
(°F)
75.2
8.6
6.8
NA
Fig. 38. Lognormal distribution of ΔRTARREST = TK Ia − T0 as a function of T0
Fig. 39. Lognormal probability densities for ΔRTArrest as function of T0.
105
Fig. 40. Proposed adjustment to RTLB arises from observed offset between Δ RTLB CDF and
RTNDT – T0 CDF at median (P = 0.5).
An approximate connection between T0 and the initiation reference temperature RTLB can be
established from the observed offset of 14.4 °F between the medians of the RTNDT0-T0 CDF and the
ΔRTLB CDF in Fig. 40. This observation allows us to apply Eq. (93) to develop a distribution for the
epistemic uncertainty in the arrest reference temperature linked to the epistemic uncertainty in the
initiation reference temperature.
ΔRT epist − arrest = ΔRT epistemic -14.4 [°F]
(94)
where ΔRT epistemic has been sampled from the distribution given by Eq. (90). The sampled arrest
reference temperature can now be calculated by
RT ARREST (r ,K) = RTNDT 0 − ΔRT epist − arrest + ΔRT ARREST + ΔRTNDT (r ,K)
(95)
where RTNDT 0 , ΔRT epist − arrest , and ΔRTNDT ( r ,L) have not been re-sampled from their initiation values
and ΔRT ARREST ← Λ ( μ ln( ΔRTARREST ) ,σ ln( ΔRTARREST ) ) is sampled from the following lognormal distribution:
106
2
μ ln( ΔRT
ARREST
)
σ ln( ΔRT
= ln ⎡ ΔRT ARREST ( mean ) ⎤ −
⎣
⎦
ARREST
)
2
where
(
)
T0 = RT NDT0 − ΔRT epist − arrest − 32 /1.8 [°C]
ΔRT ARREST ( mean ) = 44.122exp ⎡ −0.005971 × T0 ⎤ [°C]
⎣
⎦
σ ln( ΔRT
ARREST
)
(96)
{
( )} − 2ln ⎡⎣ΔRT
= ln exp ⎡0.389982 + 2ln(ΔRT ARREST ( mean ) ) ⎤ − var T0
⎣
⎦
ARREST ( mean )
⎤
⎦
where
⎧
(12.778) 2
⎪
⎪
var(T0 ) = ⎨99.905972 -1.7748073T0
⎪
0
⎪⎩
for T0 < −35.7 °C
for -35.7 °C ≤ T0 ≤ 56 °C
for T0 > 56 °C
and ΔRT ARREST is sampled from (see Step 11 in Sect. 4.5)
ΔRT ARREST = 1.8exp ⎡σ ln( ΔRTArrest ) Z Pf + μ ln( ΔRTArrest ) ⎤ [°F]
⎣
⎦
Z Pf ← N (0,1); Z Pf is the standard normal deviate corresponding to the Pf fractile
(0 < Pf < 1) for this trial in the crack Initiation - Growth - Arrest model.
See Appendix F for the details of the development of Eq. (96).
4.2.7
Plane-Strain Static Cleavage Initiation Toughness – KIc
Using the KIc data in the ORNL 99/27 fracture-toughness database (see Fig. 41) and the new lowerbounding reference temperature, RTLB, a statistical model based on a Weibull distribution was
developed by applying the statistical procedures given in [74]. The cumulative distribution function
(CDF) for the Weibull model has the following form:
0;
K I ≤ aK Ic
⎧
⎪
cK Ic
⎪
⎡ ⎛
⎤
Pr( K Ic < K I ) = Φ K Ic ( K I | aK Ic , bK Ic ) = ⎨
K
−
a
(
Δ
T
RELATIVE ) ⎞
I
K
Ic
⎢
⎥
⎜
⎟
−
−
1
exp
; aK Ic < K I < ∞
⎪
⎢ ⎜ b (ΔT RELATIVE ) ⎟ ⎥
⎪
K Ic
⎠ ⎦⎥
⎣⎢ ⎝
⎩
(97)
where the inverse CDF or percentile function is given by
1/ cK Ic
K Ic (ΔT ) = aK Ic (ΔT ) + bK Ic (ΔT ) ⎡⎣ − ln(1 − Φ K Ic ) ⎤⎦
for a ≤ K Ic ≤ K Ic (max)
for 0 < Φ K Ic < 1
(98)
where the bounding value of K Ic (max) is input by the user to FAVOR (typically K Ic (max) =
200 ksi in. ). The parameters of the distribution are
107
aK (ΔT RELATIVE ) = 19.35 + 8.335exp ⎡ 0.02254( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
Ic
bK (ΔT RELATIVE ) = 15.61 + 50.132exp ⎡ 0.008( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
Ic
cK = 4
(99)
Ic
Fig. 41. Weibull statistical distribution for plane-strain cleavage initiation fracture toughness,
KIc, with prescribed validity bounds. The ORNL 99/27 KIc database was used in the
construction of the model.
with KIc in ksi√in and ΔTRELATIVE = T (τ ) − RTNDT ( r ,L) in °F. Note that this Weibull statistical
model describes the aleatory uncertainty in the plane-strain static initiation fracture toughness, since
it is assumed that the epistemic uncertainty has been reduced by the sampled ΔRT epistemic in Eq. (90).
108
4.2.8
Plane-Strain Crack Arrest Toughness – KIa
Two lognormal distributions (see Fig. 42) are available in FAVOR to describe the aleatory
uncertainty in the plane-strain crack arrest toughness, KIa. For a lognormal distribution with random
variate, x, the cumulative distribution function is expressed by
Pr{ X ≤ x} =
1
σ x 2π
∫
ln( x )
−∞
1
⎛ ln( x) − μ ⎞
Φ⎜
⎟=
σ
2π
⎝
⎠
∫
⎡ ⎛ (ξ − μ ) 2 ⎞ ⎤
exp ⎢ − ⎜
⎟ ⎥ dξ =
2
⎠⎦
⎣ ⎝ 2σ
(ln( x ) − μ ) / σ
−∞
⎡ ς2 ⎤
exp ⎢ − ⎥ dς
⎣ 2⎦
(100)
The function Φ can be evaluated numerically through its relation to the error function, erf(x), such
that for a given applied stress intensity factor, KI, and normalized temperature, ΔT = T-RTArrest,
⎛ ln( K I ) − μln( K Ia ) (ΔT ) ⎞ 1 ⎡ ⎛ ln( K I ) − μln( K Ia ) (ΔT ) ⎞ ⎤
⎟ + 1⎥
Pr { K Ia ≤ K I } = Φ K Ia ⎜
⎟ = ⎢erf ⎜
⎜
⎟ 2⎢ ⎜
⎟ ⎥
σ ln( K Ia )
σ
2
ln(
)
K
⎝
⎠
Ia
⎠ ⎦
⎣ ⎝
(101)
where Φ K Ia is now the cumulative probability of crack extension and the error function (a special
case of the incomplete gamma function, P(a,x2)) is defined by
P ( 0.5, x 2 ) = erf( x) =
2
x
exp(−ξ
π ∫
2
) dξ
0
(102)
erf(− x) = − erf( x)
The inverse CDF for the lognormal distribution allows sampling of KIa by
)
K Ia (Φ K Ia , ΔT RELATIVE ) = exp ⎡σ ln( K Ia ) Z Φ K Ia + μln( K Ia ) ( ΔT RELATIVE ) ⎤
⎣
⎦
)
Z Φ KIa = standard normal deviate
corresponding to the Φ K Ia fractile
Φ K Ia ← U (0,1)
109
(103)
(a)
(b)
Fig. 42. Lognormal statistical distribution for plane-strain crack arrest fracture toughness, KIa,
constructed using the (a) Model 1: ORNL 99/27 KIa database normalized by the arrest
reference temperature, RTArrest and (b) Model 2: Extended KIa database normalized by
the arrest reference temperature, RTArrest.
110
Model 1 is based on the ORNL 99/27 KIa database of 112 data points which were taken using CCA
specimens. The parameters of the Model 1 KIa lognormal distribution, shown in Fig. 42a, are
μln( K Ia ) (ΔT RELATIVE ) = ln ⎡ K Ia (ΔT RELATIVE ) ⎤ −
⎣
⎦
where
σ ln( K Ia ) = 0.18
2
σ ln(
K
Ia )
2
(104)
K Ia (mean ) (ΔT RELATIVE ) = 27.302 + 69.962exp ⎡0.006057( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
ΔT RELATIVE = T (r ,τ ) − RTArrest (r ,L) [°F]
The equation for the mean was developed by nonlinear regression of the data shown in Fig. 42a.
Model 1 is recommended to be used when the ductile-tearing model is not activated, and an upper
bound for KIa of 200 ksi in. should be set in the FAVPFM input file.
Model 2 is based on the Extended KIa database of 183 data points which were taken using both CCA
specimens and Large-Specimen experiments. The parameters of the Model 2 KIa lognormal distribution, shown in Fig. 42b, are
μln( K ) (ΔT RELATIVE ) = ln ⎡ K Ia (ΔT RELATIVE ) ⎤ −
⎣
Ia
⎦
2
σ ln(
K
Ia
)
2
where
σ ln( K
Ia
)
(105)
= 0.34
K Ia (mean ) (ΔT RELATIVE ) = 27.302 + 70.6998exp ⎡ 0.008991(ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
ΔT RELATIVE = T (r ,τ ) − RTArrest (r ,L) [°F]
Model 2 will be automatically selected when the ductile-tearing model is activated, and any specified
upper bound on KIa is ignored.
111
4.2.9
Material Chemistry –Sampling Protocols
The sampling protocol used by FAVOR, v04.1, requires estimated chemistry (Cu, Ni, and P) content
values for each weld and plate subregion used to model the beltline shells of the vessel. The user will,
therefore, input best-heat estimates designated as HECu , HE Ni , and HEP in wt%.
FAVOR treats the vessel beltline as a collection of major regions of plates, forgings, and welds.
These major regions are then discretized into subregions, where within a given subregion flaws are
analyzed through Monte Carlo realizations of the RPV subjected to the PTS transients under study.
The sampling protocols for plate and weld chemistry distinguish between the first flaw simulated in a
subregion, designated as Flaw1, and all subsequent flaws in the subregion, designated as Flawx. The
plate or weld chemistry for the set of Flawx’s will be perturbations of the sampled Flaw1 chemistry
for this subregion. This variation in chemistry is intended to simulate local variability in the
subregion chemistry.
Plate Subregion Chemistry
Flaw1
The Cu, Ni, and P content (expressed in wt%) for the first flaw in a subregion are sampled from the
following normal distributions:
Cu Flaw1 ← N ( HECu ,σ Cu )
Ni Flaw1 ← N ( HE Ni ,σ Ni )
)
P Flaw1 ← N ( HEP ,σ P )
(106)
where the recommended constant standard deviations are
σCu = 0.0073 wt%
σ Ni = 0.0244 wt%
(107)
σ P = 0.0013 wt%
The triplet (σCu , σ Ni , σ P ) is supplied by the user in the input file for the FAVPFM module. Negative
)
values of sampled Cu Flaw1 , Ni Flaw1 , and P Flaw1 are handled as nonphysical exceptions in FAVOR using
the truncation protocol described in Sect. 3.3.6, with 0.0 applied as a one-sided truncation boundary.
112
Flawx – local variability
All subsequent flaws in a given subregion should contain small local variability in Cu, Ni, and P
content. This local variability is determined by sampling values from the following logistic
distributions:
Cu Flawx ← Cu Flaw1 + L(−3.89 × 10−7 ,0.00191)
Ni Flawx ← NiFlaw1 + L(−1.39 × 10−7 ,0.00678)
)
P Flawx ← PFlaw1 + L(1.30 × 10−5 ,0.000286)
(108)
⎡ 1
⎤
ΔCu − Flawx = −3.89 × 10−7 − 0.00191ln ⎢
− 1⎥ for ΦCu ← U (0,1)
⎣ ΦCu
⎦
Cu Flawx = Cu Flaw1 + ΔCu − Flawx
⎡ 1
⎤
Δ Ni − Flawx = −1.39 × 10−7 − 0.00678ln ⎢
− 1⎥ for Φ Ni ← U (0,1)
⎣ Φ Ni ⎦
(109)
NiFlawx = NiFlaw1 + Δ Ni − Flawx
⎡ 1
⎤
Δ P − Flawx = 1.3 × 10−5 − 0.000286ln ⎢
− 1⎥ for Φ P ← U (0,1)
⎣ ΦP ⎦
PFlawx = PFlaw1 + Δ P − Flawx
)
Negative values of sampled Cu Flawx , Ni Flawx , and P Flawx are handled as nonphysical exceptions in
FAVOR using the truncation protocol described in Sect. 3.3.6, with 0.0 applied as a one-sided
truncation boundary.
Through-thickness sampling for Plates
There is no resampling protocol for flaws growing through the thickness of plate subregions.
Weld Subregion Chemistry
Flaw1
Copper, Cu Flaw1 :
The Cu content for the first flaw in a weld subregion is sampled from a normal distribution with mean
equal to the heat estimate for Cu and a sampled standard deviation:
Cu Flaw1 ← N ( HECu ,σ Cu )
σ Cu ← N ( 0.167 × HECu , min ( 0.0718 × HECu ,0.0185 ) )
113
(110)
Nickel, NiFlaw1 :
Ni-addition welds (heats 34B009 and W5214)
The Ni content for the first flaw in a weld subregion is sampled from a normal distribution with mean
equal to the heat estimate for Ni and standard deviation equal to 0.162.
Ni Flaw1 ← N ( HE Ni ,0.162)
(111)
All other heats
The Ni content for the first flaw in a weld subregion is sampled from a normal distribution with mean
equal to the heat estimate for Ni and standard deviation sampled from a normal distribution with
mean equal to 0.029 wt% and standard deviation equal to 0.0165 wt%.
)
Ni Flaw1 ← N ( HE Ni ,σ NiFlaw1 )
)
σ NiFlaw1 ← N (0.029,0.0165)
(112)
Phosphorous, PFlaw1 :
The phosphorous content for the first flaw in a weld subregion is sampled from a normal distribution
with mean equal to the heat estimate for phosphorous and standard deviation equal to 0.0013.
)
P Flaw1 ← N ( HEP ,0.0013)
(113)
)
Negative values of sampled Cu Flaw1 , Ni Flaw1 , and P Flaw1 are handled as nonphysical exceptions in
FAVOR using the truncation protocol described in Sect. 3.3.6, with 0.0 applied as a one-sided
truncation boundary.
Flawx – local variability
All subsequent flaws in a given weld subregion should contain small local variability in Cu, Ni, and P
content.
Copper, Cu Flawx :
The local variability for Cu is determined by sampling a ΔCu value drawn from a logistic distribution
with parameters α = 6.85 × 10−8 and β = 0.0072 such that
ΔCu − Flawx ← L(6.85 × 10−8 ,0.0072)
⎡ 1
⎤
ΔCu − Flawx = 6.85 × 10−8 − 0.0072ln ⎢
− 1⎥ for ΦCu ← U (0,1)
⎣ ΦCu
⎦
Cu Flawx = Cu Flaw1 + ΔCu − Flawx
114
(114)
Nickel, NiFlawx :
The local variability for Ni is determined by sampling a Δ Ni value drawn from a logistic distribution
with parameters α = −0.0014 and β = 0.00647 such that
)
Δ Ni − Flawx ← L(−0.0014,0.00647)
)
⎡ 1
⎤
Δ Ni − Flawx = −0.0014 − 0.00647 ln ⎢
− 1⎥ for Φ Ni ← U (0,1)
⎣ Φ Ni ⎦
)
NiFlawx = NiFlaw1 + Δ Ni − Flawx
(115)
The same local variability samplings are applied to Ni-addition and non-Ni-addition welds.
Phosphorous, PFlawx :
The local variability for phosphorous is determined by sampling a Δ P value drawn from a logistic
distribution with parameters α = 3.27 × 10−6 and β = 0.000449 .
)
Δ P − Flawx ← L(3.27 × 10−6 ,0.000449)
)
⎡ 1
⎤
Δ P − Flawx = 3.27 × 10−6 − 0.000449ln ⎢
− 1⎥ for Φ P ← U (0,1)
⎣ ΦP ⎦
)
PFlawx = PFlaw1 + Δ Ni − Flawx
(116)
)
Negative values of sampled Cu Flawx , Ni Flawx , and P Flawx are handled as nonphysical exceptions in
FAVOR using the truncation protocol described in Sect. 3.3.6, with 0.0 applied as a one-sided
truncation boundary.
Through-thickness re-sampling for Weld Layers
Due to their thickness, RPV welds were typically constructed using multiple coils of weld wire. The
variability in chemistry from one coil or weld layer to another is resampled in FAVOR as a given
crack grows through the wall and enters a new weld layer. The weld-layer thickness in which this
variability is imposed is every 1/4T of the RPV. In general, when a flaw has initiated, the weld
chemistry content is not resampled for each growth increment. However, if the inner crack tip of the
flaw has moved from one 1/4T of the vessel wall thickness to an adjoining 1/4T region, then the
chemistry of the weld is sampled as if the flaw had advanced into a new material.
115
Additional Comments on Chemistry Sampling in Plate and Weld Subregions
When a sampled chemistry value for the first flaw in a subregion (for the current RPV trial) is
truncated internally by FAVPFM, the non-truncated chemistry value for Flaw1 continues to be used
as the basis for subsequent local variability perturbation samplings. As an example, for a given RPV
trial and first flaw in a given subregion, the sampled value of CuFlaw1 might be truncated back to 0.25
for Linde welds or to 0.305 for all other welds, plates, and forgings, when applying the Eason and
Wright correlation [86] to calculate ΔRTNDT. However, FAVPFM will utilize the non-truncated value
for CuFlaw1 in the determination of the local variability copper content, Cu Flawx , for all subsequent
flaws located in this subregion for the current RPV trial. The rationale for this procedure is that the
local variability random perturbation sampled for copper, Δ Cu − Flawx , as determined from its logistic
distribution, could possibly be sufficiently negative such that the perturbed value of Cu Flawx might
take on a value below the truncation upper bound. However, if the value of Cu Flawx should exceed the
upper truncation boundary, then FAVPFM will automatically truncate back to the appropriate upper
bound.
116
4.3
NRC RVID2 Database
The Reactor Vessel Integrity Database, RVID [128] , developed following the NRC staff review of
licensee responses to Generic Letter (GL) 92-01, Revision 1, provides a key source of input data for
FAVOR. The most recent update of the database, RVID2 [129], was released in July of 2000. The
RIVD2 summarizes the properties of the reactor vessel beltline materials for each operating
commercial nuclear power plant. The RVID includes four tables for each plant: (1) background
information table, (2) chemistry data table, (3) upper-shelf energy table, and (4) pressure-temperature
limits or pressurized thermal shock table. References and notes follow each table to document the
source(s) of data and to provide supplemental information. Appendix D presents a selection of
RVID2 data relevant to FAVOR for the four power plants included in the PTS Re-evaluation Project.
As of this writing, they are: (1) Beaver Valley 1, (2) Calvert Cliffs 1, (3) Oconee 1, and
(4) Pallisades 1.
4.4
Discrete Flaw Density and Size Distributions
The method used to quantify the uncertainty in the flaw characterization is to include 1000 flawcharacterization records in each of the three data files: (1) inner surface-breaking flaws (2) embedded
flaws in weld material, and (3) embedded flaws in plate material. The flaw-characterization file for
inner surface- breaking flaws is applicable to weld and plate material. Each of these records contains
separate discrete flaw-density and flaw-size distributions.
During the Monte Carlo PFM analysis, the RPV flaw-characterization data for the first stochastically
generated RPV trial are taken from the first group of records, i.e., the first inner surface-breaking
record, the first embedded-flaw weld material record, and the first embedded-flaw plate material
record. The RPV flaw characterization for the second stochastically generated RPV trial is
determined from the second group of records, etc. The RPV trials cycle through the flawcharacterization records sequentially up to 1000, and then restart at the first record.
Inner surface-breaking flaw density data are expressed in flaws per unit RPV-inner-surface area and
weld subregion embedded flaws are flaws per unit area on the fusion line between the weld and
adjacent plate subregions. These conventions are consistent with the physical model utilized by
Pacific Northwest National Laboratory to derive the flaw characterization data input to FAVOR.
Embedded flaws in plate regions are expressed on a volumetric basis.
117
Figures 43a and 43b illustrate axial and circumferential weld subregion elements, respectively. The
number of flaws in each of these weld elements is calculated (internally by FAVOR) as the sum of
the number of inner- surface breaking flaws and the number of embedded flaws as follows:
⎛ Number of Flaws ⎞
⎡⎛ 2π ⎞
⎤
⎡ ⎛3⎞ ⎤
⎜
⎟ = ρ SB ⎢⎜
⎟ Ri dz dθ ⎥ + ρ EW ⎢ 2 ⎜ ⎟ dA⎥
in
Weld
Subregions
⎣⎝ 360 ⎠
⎦
⎣ ⎝8⎠ ⎦
⎝
⎠
ρ SB = inner surface-breaking flaw density (per unit surface area - flaws/in 2 )
ρ EW = weld embedded-flaw density (per unit weld-fusion area - flaws/in 2 )
dA = user-input weld-fusion area (for one side of weld) (in 2 - input by user)
Ri = internal radius of RPV (in. - input by user)
(117)
dz = height of subregion element (in. - input by user)
dθ = subtended angle of subregion element ( degrees - input by user)
where ρ SB and ρ EW are summed over all flaw depths.
For axial welds, the fusion lines are on the sides of the weld, whereas for circumferential welds, the
fusion lines are on the top and bottom of the welds. In the term {2 (3/8) dA }, the factor of 2 accounts
for the fact that the user input data is the area on one side of the fusion line whereas flaws reside in
fusion lines on both sides of the welds. The (3/8) accounts for the fact that embedded flaws that reside
beyond the first 3/8 of the base metal are not included in a PTS analysis. All flaw densities are
assumed to be uniform through the RPV wall thickness.
Figure 43c illustrates a plate subregion element. The number of flaws in each of these plate elements
is calculated (internally by FAVOR) as the sum of the number of inner surface-breaking flaws and the
number of embedded flaws as follows:
⎛ Number of Flaws ⎞
⎡⎛ 2π ⎞
⎤
⎡⎛ 3 ⎞
2
⎛ dθ ⎞ ⎤
Ri dz dθ ⎥ + ρ EP ⎢⎜ ⎟ π Ro2 − ( Ri − CLTH ) dz ⎜
⎜
⎟ = ρ SB ⎢⎜
⎟
⎟⎥
⎝ 360 ⎠ ⎦
⎣⎝ 360 ⎠
⎦
⎣⎝ 8 ⎠
⎝ in Plate Subregions ⎠
(
)
ρ SB = inner surface-breaking flaw density (per unit surface area - flaws/in 2 )
ρ EP = plate embedded-flaw density summed over all flaw depths
(flaws per unit volume - flaws/in 3 )
Ro = external radius of RPV (in - input by user)
Ri = internal radius of RPV (in. - input by user)
CLTH = cladding thickness (in. - input by user)
dz = height of subregion element (in. - input by user)
dθ = subtended angle of subregion element
( degrees - input by user)
where ρ SB and ρ EP are summed over all flaw depths.
118
(118)
(a)
(b)
Fig. 43. Weld fusion area definitions for (a) axial-weld subregion elements and
(b) circumferential subregion elements.
119
Fig. 43. (continued) (c) Plate subregion element.
120
4.5
Summary of Sampling Distributions and Protocols
Plane-Strain Static Initiation
The following sampling distribution and protocols have been implemented in the FAVOR code
(FAVPFM) to represent (for a given flaw at a given time in the specific PTS transient under study)
the epistemic and aleatory uncertainties in the plane-strain static initiation fracture-toughness values
used in determining the probability of cleavage initiation:
Step 1. For plate, forging, and weld product forms, provide the following input to FAVOR:
Provide best estimates for the mean and standard deviation for normal distributions of copper,
nickel, and phosphorous content, N (Cu, σ Cu ), N ( Ni, σ Ni ), N ( P, σ P ) . 9
Provide a best estimate for the mean and standard deviation for a normal distribution of
fluence at the inside surface of the vessel, N ( f 0 (0), σ f (0) ) .
0
Provide a best estimate for the standard deviations, σ RTNDT 0 , of unirradiated RTNDT 0 and
σ ΔRTNDT of the irradiation shift model ΔRTNDT . The value of σ ΔRTNDT is used only to
calculate the regulatory value of RTPTS for reporting purposes.
Provide the coolant temperature, Tc in °F , and RPV exposure time in EFPY, where Tc is the
temperature of the coolant on the inner surface of the RPV beltline region (adjacent to the
active core) at the time the transient originates (at time = 0).
Determine the current regulatory estimate of the mean value of the unirradiated RTNDT from
the Reactor Vessel Integrity Database (RVID2) [129] for the material of interest (see
Appendix D).
a) If this RTNDT value was determined using either the ASME NB-2331 or MTEB 5-2
methods, designate the value of RTNDT ( RVID ) from RVID as RTRTND 0 and proceed directly to
Step 2.
b) If this RTNDT value was determined using the Generic method, assign RT NDT 0 as –8 °F for
welds and 0 °F for plates and forgings; sample RTNDT 0 ← N ( RT NDT 0 ,σ RTNDT 0 ) ; then proceed
to Step 2.
9
)
Note that negative values of Cu , Ni, and P sampled from normal distributions are handled as nonphysical
exceptions in FAVOR using the truncation protocol described in Sect. 3.3.6 with 0.0 as the truncation
boundary.
121
Step 2. Generate a random number, Φ , between 0 and 1 from a uniform distribution. Use this random
number to sample 10 a value of ΔRT epistemic from the following Weibull percentile function
(inverse CDF):
ΔRT epistemic ← W (−45.586,131.27, 2.177)
(
)
1/ 2.177
ΔRT epistemic = −45.586 + 131.27 ⎡ − ln 1 − Φ ⎤
⎣
⎦
(119)
[°F]
ΔRT epistemic represents the epistemic uncertainty in RTNDT .
0
Step 3. Sample the irradiation shift, ΔRT NDT , using the Eason and Wright [86] embrittlement
)
correlation from sampled values (sampled for each flaw) of neutron fluence, f 0 (r ) ; copper
content, Cu ← N (Cu , σ Cu ) ; nickel content, Ni ← N ( Ni, σ Ni ) ; phosphorous content,
)
P ← N ( P, σ P ) ; and product form.
⎧⎪0.99ΔT30 (r ,K) weld
ΔRT NDT ( r ,K) = ⎨
⎪⎩1.10ΔT30 ( r ,K) plate and forgings
(120)
where
)
ΔT30 ( Ni, Cu , P, f 0 (r ),τ exposure , Tc , product form) =
)
⎛ 19310 ⎞
A exp ⎜
⎟ 1 + 110 P
⎝ Tc + 460 ⎠
(
) ( f (r ) )
0.4601
0
(
1.250
+ B 1 + 2.40 Ni
⎧ 8.86 × 10-17 for welds ⎫
⎪
⎪
⎪
⎪
A = ⎨ 9.30 × 10-17 for forgings ⎬
⎪
⎪
−17
⎪⎩12.7 × 10 for plates ⎭⎪
⎧ 230 for welds
⎫
⎪
⎪
⎪⎪ 132 for forgings
⎪⎪
B=⎨
⎬
⎪ 206 for plates in CE vessels ⎪
⎪
⎪
⎪⎩ 156 for other plates
⎪⎭
(
) f (Cu) g ( f (r)) + Bias
0
)
⎡ log f (r ) + 4.579 × 1012τ
⎤
10
0
exposure − 18.265
1 1
⎢
⎥
g ( f 0 (r )) = + tanh
⎢
⎥
2 2
0.713
⎣
⎦
⎧
0
⎪
f Cu = ⎨
⎪ Cu -0.072
⎩
( )
10
(
)
0.659
for Cu ≤ 0.072 wt % ⎫
⎪
⎬
for Cu > 0.072 wt % ⎪
⎭
A curved overbar, X , indicates a sampled random variate. A braced overbar, X , indicates that sampling has
occurred in a prior step but not in the current step.
122
subject to
⎧⎪ 0.25 for welds with Linde 80 or Linde 0091 flux ⎫⎪
Cumax = ⎨
⎬
⎩⎪0.305 for everything else
⎭⎪
and
⎧⎪ 0 for texposure < 97000 h ⎫⎪
Bias = ⎨
⎬
⎪⎩9.4 for texposure ≥ 97000 h ⎪⎭
)
where Cu is the sampled copper content in wt%, Ni is the sampled nickel content in wt%, P
)
is the sampled phosphorous content in wt%, f 0 (r ) is the sampled and then attenuated neutron
fluence in n/cm2, r is the position from the inner surface of RPV wall, τexposure is exposure
time in hours (input to FAVOR in EFPY), and Tc is coolant temperature in °F . The fastneutron fluence at the inner surface of the vessel is sampled using the protocol described in
Sect. 4.2.3. The sampled neutron fluence for the flaw is then attenuated (but not resampled)
as the crack grows through the wall to produce f 0 ( r ) .
Step 4. Calculate the sampled, irradiated value of RTNDT from
RT NDT (r ,K) = RT NDT 0 − ΔRT epistemic + ΔRT NDT (r ,K)
(121)
⎧⎪
RT NDT 0 ← N ( RT NDT 0 ,σ RTNDT 0 ) if RVID2 method is Generic
where RT NDT 0 = ⎨
⎪⎩Heat Estimate of RT NDT 0 if RVID2 method is NB-2331 or MTEB 5-2
Step 5. Calculate the normalized temperature of the vessel at the current location, r, of the crack tip
in the RPV wall as
ΔT RELATIVE (r ,K) = T (r ,τ ) − RT NDT (r ,K)
(122)
Step 6. Calculate the parameters of the Weibull distribution of the KIc Weibull statistical distribution
by
aK (ΔT RELATIVE ) = 19.35 + 8.335exp ⎡ 0.02254( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
Ic
bK (ΔT RELATIVE ) = 15.61 + 50.132exp ⎡ 0.008( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
Ic
cK = 4
Ic
with KIc in ksi√in and ΔT = (T-RTNDT) in °F.
Note that this Weibull statistical model describes the aleatory uncertainty in plane-strain
static initiation.
123
(123)
Step 7. For a given applied KI , calculate the instantaneous conditional probability of crack initiation,
Pr { K Ic ≤ K I } with aleatory uncertainty, from the following Weibull distribution
0;
K I ≤ aK Ic
⎧
⎪
cK Ic
⎪
⎡ ⎛ K − a (ΔT
⎤
Pr( K Ic < K I ) = cpi = ⎨
RELATIVE ) ⎞
I
K
Ic
⎢
⎟ ⎥ ; K I > aK Ic
⎪1 − exp − ⎜
⎢
⎜
bK Ic (ΔT RELATIVE ) ⎟⎠ ⎥
⎪
⎢⎣ ⎝
⎥⎦
⎩
(124)
If the flaw is determined to be in a warm-prestressing state (and the WPS option has been
turned on by the user), then the conditional probability of initiation is set to zero. See
Sect. 3.3.4 for a complete discussion of warm prestressing.
Plane-Strain Static Crack Arrest
Assuming that the given flaw at a given time (for the specific PTS transient under study) has a finite
conditional probability of initiation that is increasing with time, the following protocol has been
implemented in FAVOR as a part of the Initiation-Growth-Arrest (IGA) submodel (see Sect. 3.3.12)
to represent the epistemic and aleatory uncertainties in plane-strain crack arrest fracture-toughness
values.
Step 8. For plate, forging, and weld product forms, the following input will have been provided to
FAVOR:
Best estimates for the mean and standard deviation for normal distributions of copper, nickel,
and phosphorous content: N (Cu, σ Cu ), N ( Ni, σ Ni ), N ( P, σ P ) . 11
Best estimate for the mean and standard deviation for a normal distribution of fluence at the
inside surface of the vessel, N ( f 0 (0), σ f (0) ) . 12
0
Best estimate for the standard deviation, σ RT
NDT 0
, of unirradiated RTNDT .
The coolant temperature, Tc in °F , and RPV exposure time in EFPY.
From the initiation procedure for this flaw, the current regulatory estimate of the unirradiated
RTNDT will have already been determined from the Reactor Vessel Integrity Database
(RVID2) [129] for the material of interest (see Appendix D) and designated as either RTNDT 0 if
the RVID2 RTNDT(u) method is NB-2331 or MTEB 5-2 or sampled from a normal distribution
RTNDT 0 ← N ( RT NDT ( RVID ) , σ RTNDT 0 ) if the RVID2 RTNDT(u) method is Generic.
)
11
Note that negative values of chemistry content ( Cu , Ni, and P ) sampled from normal distributions are
handled as nonphysical exceptions in FAVOR using the truncation protocol described in Sect. 3.3.4 with 0 as
the truncation boundary.
12
Note that sampled negative values of fluence, f 0 (0) , are handled as nonphysical exceptions in FAVOR using
the truncation protocol described in Sect. 3.3.4 with 0 as the truncation boundary.
124
Step 9. Retrieve the value of ΔRT epistemic determined from Step 2 in the initiation procedure applied for
this flaw and adjust the epistemic uncertainty in RTNDT by applying a shift of −14.4 °F
0
ΔRT epist − arrest = ΔRT epistemic -14.4 [°F]
(125)
Note that this step does not involve a resampling of ΔRT epistemic .
Step 10. Retrieve the sampled value of the irradiation shift for this flaw, ΔRT NDT (r ,K) , determined
from Step 3 in the initiation procedure applied for this flaw at its current position in the RPV
wall. Note that this step does not involve a resampling of ΔRT NDT (r ,K) .
Step 11. Sample ΔRT ARREST ← Λ ( μ ln( ΔRTARREST ) ,σ ln( ΔRTARREST ) ) from a lognormal distribution (see
Appendix F) where
2
μ ln( ΔRT
ARREST
)
σ ln( ΔRTARREST )
= ln ⎡ ΔRT ARREST ( mean ) ⎤ −
⎣
⎦
2
where
(
)
T0 = RT NDT0 − ΔRT epist − arrest − 32 /1.8 [°C]
ΔRT ARREST ( mean ) = 44.122exp ⎡ −0.005971 × T0 ⎤ [°C]
⎣
⎦
σ ln( ΔRT
ARREST
)
{
(126)
( )} − 2ln ⎡⎣ΔRT
= ln exp ⎡0.389982 + 2ln(ΔRT ARREST ( mean ) ) ⎤ − var T0
⎣
⎦
ARREST ( mean )
⎤
⎦
where
⎧
(12.778) 2
⎪
⎪
var(T0 ) = ⎨99.905972 -1.7748073T0
⎪
0
⎪⎩
for T0 < −35.7 °C
for -35.7 °C ≤ T0 ≤ 56 °C
for T0 > 56 °C
ΔRT ARREST is sampled from the lognormal percentile function and then converted into °F
ΔRT ARREST = 1.8exp ⎡σ ln( ΔRTArrest ) Z Pf + μ ln( ΔRTArrest ) ⎤ [°F]
⎣
⎦
Z Pf ← N (0,1); Z Pf is the standard normal deviate corresponding to the Pf fractile
(0 < Pf < 1) for this trial in the crack Initiation - Growth - Arrest model.
Step 12. Calculate the estimated arrest reference temperature, RT ARREST
RT ARREST (r ,K) = RT NDT0 − ΔRT epist − arrest + ΔRT ARREST + ΔRTNDT ( r ,K)
(127)
Step 13. Calculate the normalized (relative to RT ARREST ) temperature of the vessel at the current
location, r, in the RPV wall
ΔT RELATIVE ( r ,K) = T (r , t ) − RT ARREST (r ,K)
125
(128)
Step 14. Calculate the lognormal mean, μln( K ) (ΔT RELATIVE ) , of the KIa statistical distribution by
Ia
μln( K ) (ΔT RELATIVE ) = ln ⎡ K Ia ( mean ) (ΔT RELATIVE ) ⎤ −
⎣
Ia
⎦
2
σ ln(
K
Ia )
2
where
if K Ia _Model is equal to 1
K Ia (mean ) ( ΔT RELATIVE ) = 27.302 + 69.962exp ⎡0.006057(ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
σ ln( K Ia ) = 0.18
(129)
else if K Ia _Model is equal to 2
K Ia (mean ) ( ΔT RELATIVE ) = 27.302 + 70.6998exp ⎡0.008991( ΔT RELATIVE ) ⎤ [ksi in.]
⎣
⎦
σ ln( K Ia ) = 0.34
Step 15. Given the current value of KI-initiation from the initiation model, we first calculate the fractile,
Φ K I −initiation , associated with this value in the arrest model by
Φ K I −initiation =
where erf( x) ≡
2
1 ⎡ ⎛ ln( K I −initiation ) − μln( K Ia ) (ΔT RELATIVE ) ⎞ ⎤
⎢erf ⎜
⎟ + 1⎥
⎟ ⎥
2⎢ ⎜
σ
2
ln(
)
K
Ia
⎠ ⎦
⎣ ⎝
(130)
x
exp(−ξ
π ∫
2
) dξ . Using the same value of Pf from Step 11, scale by
0
Φ K I −initiation such that
Φ K Ia = ( Pf )(Φ K I −initiation )
(131)
With this Φ K Ia fractile, draw a value of KIa from its lognormal distribution
)
K Ia (Φ K Ia , ΔT RELATIVE ) = exp ⎡σ ln( K Ia ) Z Φ K Ia + μln( K Ia ) (ΔT RELATIVE ) ⎤
⎣
⎦
)
Z Φ K Ia = standard normal deviate corresponding to the Φ K Ia fractile
(132)
Notes:
Note on Step 3: The current sampled value of ΔT30 is also used to estimate the effects of irradiation
on the unirradiated flow stress, σ flow( u ) , in the crack Initiation-Growth-Arrest model. After each
resampling of ΔT30 , the flow stress is adjusted by the following relation:
⎧0.112 ksi/ °F for welds
σ flow = σ flow( u ) + γΔT30 where γ = ⎨
⎩ 0.131 ksi/ °F for plates
This value of σ flow is then used in the vessel-failure test against the pressure-induced membrane
stress in the remaining ligament, checking for net-section plastic collapse.
126
Note on Step 11: The only random variate sampled in Step 11 is Z Pf . All other variates have been
sampled in previous steps.
Note on Step 15: The scaling procedure in Step 15 ensures that the initial value of KIa , calculated
immediately after initiation, does not exceed the initiating value of KI, thus ensuring an initial
extension. For welds, the scaling procedure of Eq. (131) is used only in the weld layer in which the
flaw originally initiated. If the flaw advances into other weld layers, then this scaling is not applied,
since it is assumed that any linkage between the original initiation event and crack arrest is thereby
broken.
For either an initiated (cpi > 0) surface-breaking or embedded flaw, the flaw is first assumed to
extend to become an infinite-length flaw before it is allowed to advance through the RPV wall. It is
the applied KI of the infinite-length flaw (designated as KI-initiation in Step 15, Eq. (130)) that is taken as
the operative initiating KIc to establish the required scaling factor and not the applied KI of the
surface-breaking or embedded flaw at initiation. It was determined that scaling by the lower embedded-flaw KI at initiation was an overly restrictive constraint.
127
5. Summary and Conclusions
This report has provided a detailed description of the theory, algorithms, methods, and correlations
that have been implemented in this baseline release of the FAVOR, v04.1, computer code for
performing probabilistic fracture mechanics analyses of nuclear reactor pressure vessels subjected to
pressurized thermal shock and other pressure-thermal events. In support of the PTS Re-evaluation
Project, the following advanced technologies and new capabilities have been incorporated into
FAVOR, v04.1:
•
the ability to incorporate new detailed flaw-characterization distributions from NRC
research (with Pacific Northwest National Laboratory, PNNL),
•
the ability to incorporate detailed neutron fluence regions – detailed fluence maps from
Brookhaven National Laboratory, BNL,
•
the ability to incorporate warm-prestressing effects into the analysis,
•
the ability to include temperature-dependencies in the thermo-elastic properties of base and
cladding,
•
the ability to include crack-face pressure loading for surface-breaking flaws,
•
a new ductile-fracture model simulating stable and unstable ductile tearing,
•
a new embrittlement correlation,
•
the ability to include multiple transients in one execution of FAVOR,
•
input from the Reactor Vessel Integrity Database, Revision 2, (RVID2) of relevant RPV
material properties,
•
fracture-toughness models based on extended databases and improved statistical
distributions,
•
a variable failure criterion, i.e., how far must a flaw propagate into the RPV wall for the
vessel simulation to be considered as “failed” ?
•
semi-elliptic surface-breaking and embedded-flaw models,
•
through-wall weld residual stresses, and an
•
improved PFM methodology that incorporates modern PRA procedures for the
classification and propagation of input uncertainties and the characterization of output
uncertainties as statistical distributions.
The companion report Fracture Analysis of Vessels – Oak Ridge, FAVOR, v04.1 Computer Code:
User’s Guide [45] gives complete details on input requirements and execution of FAVOR, v04.1.
128
References
1.
T. L. Dickson, S. N. M. Malik, J. W. Bryson, and F. A. Simonen, “Revisiting the Integrated
Pressurized Thermal Shock Studies of an Aging Pressurized Water Reactor,” ASME PVPVolume 388, Fracture, Design Analysis of Pressure Vessels, Heat Exchangers, Piping
Components, and Fitness for Service, ASME Pressure Vessels and Piping Conference, August,
1999.
2.
A. R. Foster and R. L. Wright, Jr., Basic Nuclear Engineering, 2nd ed., Allyn and Bacon, Inc.,
Boston, 1973.
3.
K. Balkey, F. J. Witt, and B. A. Bishop, Documentation of Probabilistic Fracture Mechanics
Codes Used for Reactor Pressure Vessels Subjected to Pressurized Thermal Shock Loading,
Parts 1 and 2, EPRI TR-105001, Westinghouse Electric Corporation, Pittsburgh, PA, June 1995.
4.
D. L. Selby, et al., Pressurized Thermal Shock Evaluation of the H.B. Robinson Nuclear Power
Plant, NUREG/CR-4183 (ORNL/TM-9567), September 1985.
5.
Classification of TMI Action Plan Requirements, U.S. Nuclear Regulatory Commission,
NUREG-0737, November 1980.
6.
D. L. Selby, et al., Pressurized Thermal Shock Evaluation of the Calvert Cliffs Unit 1 Nuclear
Power Plant, NUREG/CR-4022 (ORNL/TM-9408), Oak Ridge National Laboratory, Oak
Ridge, TN, September 1985.
7.
T. J. Burns, et al., Preliminary Development of an Integrated Approach to the Evaluation of
Pressurized Thermal Shock as Applied to the Oconee Unit 1 Nuclear Power Plant,
NUREG/CR-3770 (ORNL/TM-9176), May 1986.
8.
Policy Issue from J. W. Dircks to NRC Commissioners, Enclosure A: NRC Staff Evaluation of
Pressurized Thermal Shock, November 1982, SECY-82-465, November 23, 1982, Division of
Nuclear Reactor Regulation, U.S. Nuclear Regulatory Commission, Washington, D.C.
9.
T. L. Dickson, Review of Pressurized Thermal Shock Screening Criteria for Embrittled
Pressurized Water Reactor Pressure Vessels, ORNL/NRC/LTR-95/39, December, 1995.
10. U.S. Code of Federal Regulations, Title 10, Part 50, Section 50.61 and Appendix G.
11. U.S. Nuclear Regulatory Commission, Regulatory Guide 1.154 (1987), Format and Content of
Plant-Specific Pressurized Thermal Shock Safety Analysis Reports for Pressurized Water
Reactors.
12. U. S. Nuclear Regulatory Commission, Regulatory Guide 1.99, Revision 2 (1988), Radiation
Embrittlement of Reactor Vessel Materials.
13. C. E. Pugh and B. R. Bass, A Review of Large-Scale Fracture Experiments Relevant to
Pressure Vessel Integrity Under Pressurized Thermal Shock Conditions, NUREG/CR-6699
(ORNL/TM-2000/360), Oak Ridge National Laboratory, January 2001.
14. B. R. Bass, C. E. Pugh, J. Keeney-Walker, H. Schulz, and J. Sievers, CSNI Project for Future
Analyses of Large-Scale International Reference Experiments (Project FALSIRE), NUREG/CR5997 (ORNL/TM-12307), Oak Ridge National Laboratory, December 1992.
15. B. R. Bass, C. E. Pugh, J. Keeney-Walker, H. Schulz, and J. Sievers, CSNI Project for Future
Analyses of Large-Scale International Reference Experiments (FALSIRE II), NUREG/CR-6460
(ORNL/TM-13207), Oak Ridge National Laboratory, April 1996.
16. J. Sievers and B. R. Bass, “Comparative Assessment of Project FALSIRE-Results,” Journal
Nucl. Engr. Des. 152, (1994) 19-38.
129
17. B. R. Bass, et al., “CSNI Project for Fracture Analysis of Large-Scale International Reference
Experiments (FALSIRE II),” Proceedings of the International Conference on Nuclear Engineering-4 (ICONE-4), Vol. 1, Part A, American society of Mechanical Engineers, (1996) 149162.
18. B. R. Bass, C. E. Pugh, S. Crutzen, and R. Murgatroyd, Relationship of NESC-1 to Other LargeScale International Projects, ORNL/NRC/LTR-99/20, Oak Ridge National Laboratory, Oak
Ridge, TN, 1999.
19. J. G. Merkle, G. D. Whitman, and R. H. Bryan, An Evaluation of the HSST Program
Intermediate Pressure Vessel Tests in Terms of Light-Water-Reactor Pressure Vessel Safety,
ORNL/TM-5090, Oak Ridge National Laboratory, Oak Ridge, TN, November 1975.
20. G. D. Whitman, Historical Summary of the Heavy-Section Steel Technology Program and Some
Related Activities in Light-Water-Reactor Pressure Vessel Safety Research, NUREG/CR-4489
(ORNL-6259), Oak Ridge National Laboratory, Oak Ridge, TN, March 1986.
21. R. W. Derby, et al., Test of 6-Inch-Thick Pressure Vessels, Series 1: Intermediate Test Vessels
V-1 and V-2, ORNL-4895, Oak Ridge National Laboratory, Oak Ridge, TN, February 1974.
22. R. H. Bryan, et al., Test of 6-Inch-Thick Pressure Vessels, Series 2: Intermediate Test Vessels
V-3, V-4, and V-6, ORNL-5059, Oak Ridge National Laboratory, Oak Ridge, TN, November
1975.
23. R. H. Bryan, et al., Test of 6-Inch-Thick Pressure Vessels, Series 3: Intermediate Test Vessels
V-7B, NUREG/CR-0309 (ORNL/NUREG-38), Oak Ridge National Laboratory, Oak Ridge, TN,
October 1978.
24. J. G. Merkle, et al., Test of 6-Inch-Thick Pressure Vessels, Series 4: Intermediate Test Vessels
V-5 and V-9 with Nozzle Corner Cracks, ORNL/NUREG-7, Oak Ridge National Laboratory,
Oak Ridge, TN, August 1977.
25. R. H. Bryan, et al., Test of 6-Inch-Thick Pressure Vessels, Series 3: Intermediate Test Vessel
V-8, NUREG/CR-0675 (ORNL/NUREG-58), Oak Ridge National Laboratory, Oak Ridge, TN,
December 1979.
26. R. H. Bryan, et al., Pressurized-Thermal Shock Test of 6-Inch-Thick Pressure Vessel, PTSE-1:
Investigations of Warm Prestressing and Upper-Shelf Arrest, NUREG/CR-4106 (ORNL-6135),
Oak Ridge National Laboratory, Oak Ridge, TN, April 1985.
27. R. H. Bryan, et al., Pressurized Thermal Shock Test of 6-Inch-Thick Pressure Vessel PTSE-2:
Investigation of Low Tearing Resistance and Warm Prestressing, NUREG/CR-4888
(ORNL-6377), Oak Ridge National Laboratory, Oak Ridge, TN, December 1987.
28. H. Keinanen, et al., “Pressurized Thermal Shock Tests with Model Pressure Vessels Made of
VVER-440 Reactor Steel,” IAEA/CSNI Specialists’ Meeting on Fracture Mechanics Verification
by Large-Scale Testing, Oak Ridge, Tennessee, October 26-29, 1992, NUREG/CP-0131
(ORNL/TM-12413), October 1993, 275-288.
29. B. R. Bass, et al., CSNI Project for Fracture Analyses of Large-Scale International Reference
Experiments (FALSIRE II), NUREG/CR-6460 (ORNL/TM-13207), Oak Ridge National
Laboratory, Oak Ridge, TN, April 1996.
30. L. Stumpfrock, “FALSIRE Results for NKS-3 and NKS-4,” IAEA/CSNI Specialists’ Meeting on
Fracture Mechanics Verification by Large-Scale Testing, Oak Ridge, Tennessee, October 26-29,
1992, NUREG/CR-0131 (ORNL/TM-12413), October 1993, 151-188.
130
31. L. Stumpfrock, et al., “Fracture Mechanics Investigations on Cylindrical Large-Scale Specimens
under Thermal-Shock Loading,” Journal of Nuclear Engineering Design 144, (1993) 31-44.
32. E. Morland, “Spinning Cylinder Experiments SC-I and SC-II: A Review of Results and
Analyses Provided to the FALSIRE Project,” IAEA/CSNI Specialists’ Meeting on Fracture
Mechanics Verification by Large-Scale Testing, Oak Ridge, Tennessee, October 26-29, 1992,
NUREG/CR-0131 (ORNL/TM-12413), October 1993, 39-74.
33. D. J. Lacey, et al., Spinning Cylinder Test 4: An Investigation of Transition Fracture Behavior
for Surface Breaking Defects in Thick-Section Steel Specimens, AEA Technology Report AEA
TRS 4098, June 1991.
34. R. D. Cheverton, et al., “Review of Pressurized-Water-Reactor-Related Thermal Shock
Studies,” Fracture Mechanics: Nineteenth Symposium (June 30-July 2, 1986), ASTM STP-969,
American Society for Testing and Materials, (1988) 752-766.
35. R. D. Cheverton, “Thermal Shock Experiments with Flawed Clad Cylinders,” Journal of
Nuclear Engineering and Design 124, (1990) 109-119.
36. B. R. Bass, J. Wintle, R. C. Hurst, and N. Taylor (eds), NESC-1 Project Overview, EUR
19051EN, European Commission, Directorate-General Joint Research Centre, Institute for
Advanced Materials, Petten, The Netherlands, 2001.
37. H. Okumura, et al., “PTS Integrity Study in Japan,” Ninth International Conference on
Structural Mechanics in Reactor Technology, Vol. F, (1989) 7-12.
38. D. J. Naus, et al., High-Temperature Crack Arrest Behavior in 152-mm-Thick SEN Wide Plates
of Quenched and Tempered A533 Grade B Class 1 Steel, NUREG/CR-5330 (ORNL-11083),
Oak Ridge National Laboratory, Oak Ridge, TN, April 1989.
39. D. J. Naus, et al., Crack-Arrest Behavior in SEN Wide Plates of Low-Upper-Shelf Base Metal
Tested Under Nonisothermal Conditions: WP-2 Series, NUREG/CR-5451 (ORNL-6584), Oak
Ridge National Laboratory, Oak Ridge, TN, April 1989.
40. K. Kussmaul, R. Gillot, and T. Elenz, “Full Thickness Crack Arrest Investigations on Compact
Specimens and a Heavy-Wide Plate,” IAEA/CSNI Specialists’ Meeting on Fracture Mechanics
Verification by Large-Scale Testing, Oak Ridge, Tennessee, October 26-29, 1992,
NUREG/CR-0131 (ORNL/TM-12413), October 1993, 551-572.
41. D. Moinerau, et al., “Cleavage Fracture of Specimens Containing an Underclad Crack, PVP
Vol. 233, Pressure Vessel Fracture, Fatigue and Life Management, American Society of
Mechanical Engineers, 1992.
42. J. A. Keeney, B. R. Bass, and W. J. McAfee, “Fracture Assessment of Weld Material from a
Full-Thickness Clad RPV Shell Segment,” Fatigue and Fracture Mechanics, 20th Volume,
ASTM STP-1321, eds., J. H. Underwood, B. D. McDonald, and M. R. Mitchell, American
Society of Materials and Testing, 1997.
43. J. A. Keeney and P. T. Williams, “Fracture Analysis of Ductile Crack Growth in Weld Material
from a Full-Thickness Clad RPV Shell Segment,” Fatigue and Fracture Mechanics, 29th
Volume, ASTM STP-1332, eds. T. L. Panontin and S. D. Sheppard, American Society of
Materials and Testing, 1998.
44. B. R. Bass, et al., “Evaluation of Constraint Methodologies Applied to Shallow-Flaw Cruciform
Bend Specimens Tested under Biaxial Loading Conditions,” PVP Vol. 365, Proceedings of the
1998 ASME Pressure Vessel and Piping Conference, San Diego, CA, July 1998, 11-26.
131
45. T. L. Dickson, P. T. Williams, and S. Yin, Fracture Analysis of Vessels – FAVOR (v04.1):
User’s Guide, NUREG/CR-6855 (ORNL/TM-2004/245), Oak Ridge National Laboratory, Oak
Ridge, TN, 2004.
46. M. G. Morgan and M. Henrion, Uncertainty – A Guide to Dealing with Uncertainty in
Quantitative Risk and Policy Analysis, Cambridge University Press, Cambridge, UK, 1990.
47. F.A. Simonen, S.R. Doctor, G.J. Schuster, and P.G. Heasler, “A Generalized Procedure for
Generating Flaw Related Inputs for the FAVOR Code,” NUREG/CR-6817, Rev. 1, U.S. Nuclear
Regulatory Commission, October 2003.
48. G. J. Schuster, S. R. Doctor, S. L. Crawford, and A. F. Pardini, Characterization of Flaws in
U.S. Reactor Pressure Vessels: Density and Distribution of Flaw Indications in
PVRUF, USNRC Report NUREG/CR-6471, Vol. 1, U.S. Nuclear Regulatory Commission,
Washington, D.C. (1998).
49. G. J. Schuster, S. R. Doctor, and P. G. Heasler, Characterization of Flaws in U.S. Reactor
Pressure Vessels: Validation of Flaw Density and Distribution in the Weld Metal of the PVRUF
Vessel, USNRC Report NUREG/CR-6471, Vol. 2, U.S. Nuclear Regulatory Commission,
Washington, D.C. (2000).
50. G. J. Schuster, S. R. Doctor, S. L. Crawford, and A. F. Pardini, Characterization of Flaws in
U.S. Reactor Pressure Vessels: Density and Distribution of Flaw Indications in the Shoreham
Vessel, USNRC Report NUREG/CR-6471, Vol. 3, U.S. Nuclear Regulatory Commission,
Washington, D.C. (1999).
51. A. M. Law and W. D. Kelton, Simulation Modeling and Analysis, 3rd ed., McGraw-Hill, New
York, NY, (2000).
52. A. J. Brothers and S. Yukawa, “The Effect of Warm Prestressing on Notch Fracture Strength,”
Journal of Basic Engineering, Transaction of the ASME, Series D, 85(1), (1963) 87-104.
53. R. W. Nichols, “The Use of Overstressing Techniques to Reduce the Risk of Subsequent Brittle
Fracture: Parts 1 and 2,” British Welding Journal, January and February 1968.
54. J. J. McGowan, “An Assessment of the Beneficial Effects of Warm Prestressing on the Fracture
Properties of Nuclear Reactor Pressure Vessels Under Severe Thermal Shock,” Westinghouse
Electric Company, WCAP-9178, March 1978.
55. G. C. Chell, “A Theory of Warm Prestressing: Experimental Validation and the Implication for
Elastic-Plastic Failure Criteria,” CERL Lab Note RD/L/N78/79, September 1979.
56. G. C. Chell, “Some Fracture Mechanics Applications of Warm Prestressing to Pressure
Vessels,” 4th International Conference on Pressure Vessel Technology, Paper C22/80, London,
May 1980.
57. B. W. Pickles and A. Cowan, “A Review of Warm Prestressing Studies,” International Journal
of Pressure Vessels and Piping 14, (1983) 95-131.
58. D. Lidbury and P. Birkett, “Effect of Warm Prestressing on the Transition Toughness Behaviour
of an A533 Grade B Class 1 Pressure Vessel Steel,” ASTM STP 1074, Fracture Mechanics: 21st
Symposium, American Society for Testing and Materials, Philadelphia, PA, (1990) 264-285.
59. F. M. Burdekin and D. P. G. Lidbury, “Views of TAGSI on the Current Position with Regard to
Benefits of Warm Prestressing,” International Journal of Pressure Vessels and Piping 76,
(1999) 885-890.
132
60. H. Kordisch, R. Böschen, J. G. Blauel, W. Schmitt, and G. Nagel, “Experimental and Numerical
Investigation of the Warm-Prestressing (WPS) Effect Considering Different Load Paths,”
Nuclear Engineering Design 198, (2000) 89-96.
61. J. H. Chen, V. B. Wang, G. Z. Wang, and X. Chen, “Mechanism of Effects of Warm
Prestressing on Apparent Toughness of Precracked Specimens of HSLA Steels,” Engineering
Fracture Mechanics 68, (2001) 1669-1689.
62. W. Lefevre, G. Barbier, R. Masson, and G. Rousselier, “A Modified Beremin Model to Simulate
the Warm Pre-Stress Effect,” Nuclear Engineering and Design 216, (2002) 27-42.
63. M. T. Kirk, “Inclusion of Warm Prestressing Effects in Probabilistic Fracture Mechanics
Calculations Performed to Assess the Risk of RPV Failure Produced by Pressurized Thermal
Shock Events: An Opinion,” presented at the NATO Advanced Research Workshop – Scientific
Fundamentals for the Life Time Extension of Reactor Pressure Vessels, Kiev, Ukraine, April
22-25, 2002.
64. B. W. Brown and J. Lovato, RANLIB – A Library of Fortran Routines for Random Number
Generation, Department of Biomathematics, University of Texas, Houston, TX, 1994.
65. P. L’Ecuyer, “Efficient and Portable Combined Random Number Generators,” Communications
of the ACM 31(6), (1988) 742-774.
66. P. L’Ecuyer and S. Cote, “Implementing a Random Number Package with Splitting Facilities.”
ACM Transactions on Mathematical Software 17, (1991) 98-111.
67. A. Rukhin, et al., A Statistical Test Suite for Random and Pseudorandom Number Generators
for Cryptographic Applications, NIST Special Publication 800-22, National Institute of
Standards and Technology, Gaithersburg, MD, May 15, 2001.
68. G. E. P. Box and M. E. Müller, “A Note on the Generation of Random Normal Deviates,”
Annals of Mathematical Statistics, 29 (1958), 610-611.
69. M. E. Müller, “An Inverse Method for the Generation of Random Normal Deviates on Large
Scale Computers,” Math. Tables Aids Comp. 63, (1958) 167-174.
70. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications,
Inc., New York, (1972) 931.
71. J. H. Ahrens and U. Dieter, “Extensions of Forsythe’s Method for Random Sampling from the
Normal Distribution,” Math. Comput. 27(124), (1973) 927-937.
72. Kennedy and Gentle, Statistical Computing, Marcel Dekker, NY, (1980) 95.
73. EPRI Special Report, 1978, Flaw Evaluation Procedures: ASME Section XI, EPRI NP-719-SR,
Electric Power Research Institute, Palo Alto, CA.
74. K. O. Bowman and P. T. Williams, Technical Basis for Statistical Models of Extended KIc and
KIa Fracture Toughness Databases for RPV Steels, ORNL/NRC/LTR-99/27, Oak Ridge
National Laboratory, Oak Ridge, TN, February, 2000.
75. R. D. Cheverton, et al., Pressure Vessel Fracture Studies Pertaining to the PWR Thermal-Shock
Issue: Experiment TSE-7, NUREG/CR-4304 (ORNL/TM-6177), Oak Ridge National
Laboratory, Oak Ridge, TN, August 1985.
76. F. Li, Assessment of Pressure Vessel Failure Due to Pressurized Thermal Shock with
Consideration of Probabilistic-Deterministic Screening and Uncertainty Analysis, Ph.D.
Dissertation, Center for Technology Risk Studies, University of Maryland, 2001.
133
77. T. Fang and M. Modarres, “Probabilistic and Deterministic Treatments for Multiple Flaws in
Reactor Pressure Vessel Safety Analysis,” Transactions of the 17th International Conference on
Structural Mechanics in Reactor Technology (SMiRT 17), August 17-22, 2003, Prague, Czech
Republic.
78. A. L. Hiser, et al., “J-R Curve Characterization of Irradiated Low Upper Shelf Welds,”
NUREG/CR-3506, U. S. Nuclear Regulatory Commission, 1984.
79. D. E. McCabe, R. K. Nanstad, S. K. Iskander, D. W. Heatherly, and R. L. Swain, Evaluation of
WF-70 Weld Metal From the Midland Unit 1 Reactor Vessel, USNRC Report NUREG/CR-5736
(ORNL/TM-13748), Oak Ridge National Laboratory, Oak Ridge, TN, November 2000.
80. J. R. Hawthorne, J.R., “Investigations of Irradiation-Anneal-Reirradiation (IAR) Properties
Trends of RPV Welds: Phase 2 Final Report,” NUREG-CR/5492, U. S. Nuclear Regulatory
Commission, 1990.
81. Standard Test Method for Measurement of Fracture Toughness, ASTM E 1820-1, Annual Book
of ASTM Standards 2002, Section Three, Metals Test Methods and Analytical Procedures,
Volume 03.01, American Society for Testing and Materials, West Conshohocken, PA (2002).
82. M. EricksonKirk and M. T. EricksonKirk, “The Relationship Between the Transition and Uppershelf Fracture Toughness of Ferritic Steels,” Fatigue Fract Engng Mater Struct 29, (2006) 1–13.
83. M. T. EricksonKirk, B. R. Bass, T. L. Dickson, C. E. Pugh, T. Santos, and P. T. Williams,
“Probabilistic Fracture Mechanics: Models, Parameters, and Uncertainty Treatment Used in
FAVOR Version 04.1,” U.S. Nuclear Regulatory Commission, NUREG-1807, 2006.
84. M. T. EricksonKirk and M. EricksonKirk, “An Upper-Shelf Fracture Toughness Master Curve
for Ferritic Steels,” to appear in the International Journal of Pressure Vessels and Piping, 2006.
85. E. D. Eason, J. E. Wright, and G. R. Odette, “Improved Embrittlement Correlations for Reactor
Pressure Vessel Steels,” NUREG/CR-6551, U. S. Nuclear Regulatory Commission,
Washington, DC, 1998.
86. M. T. Kirk, C. S. Santos, E.D. Eason, J.E. Wright, and G. R. Odette, “Updated Embrittlement
Trend Curve for Reactor Pressure Vessel Steels,” Paper No. G01-5, Transactions of the 17th
International Conference on Structural Mechanics in Reactor Technology (SMiRT 17), Prague,
Czech Republic, August 17-22, 2003.
87. N. Siu, S. Malik, D. Bessette, R. Woods, A. Mosleh, and M. Modarres, “Treating Aleatory and
Epistemic Uncertainties in Analyses of Pressurized Thermal Shock,” Proceedings of PSAM 5,
International Conference on Probabilistic Safety Assessment and Management, Osaka, Japan,
(2000) 377-382.
88. R. W. Lewis, K. Morgan, H. R. Thomas, and K. N. Seetharamu, The Finite-Element Method in
Heat Transfer Analysis, John Wiley & Sons, New York, 1996.
89. R. D. Cook, D. S. Malkus, and M. E. Plesha, Concepts and Applications of Finite Element
Analysis, 3rd ed., John Wiley & Sons, New York, 1989.
90. H. F. Bückner, “A Novel Principle for the Computation of Stress Intensity Factors,” Z. angew.
Math. Mech. 50, (1970) 529-546.
91. C. B. Buchalet and W. H. Bamford, “Stress Intensity Factor Solutions for Continuous Surface
Flaws in Reactor Pressure Vessels,” ASTM STP 590, Mechanics of Crack Growth, American
Society for Testing and Materials, (1976) 385-402.
134
92. J. Heliot, R. C. Labbens, and A. Pellissier-Tanon, “Semi-Elliptical Cracks in a Cylinder
Subjected to Stress Gradients,” ASTM STP 677, Fracture Mechanics, ed. C. W. Smith,
American Society for Testing and Materials, (1979) 341-364.
93. J. J. McGowan and M. Raymund, “Stress Intensity Factor Solutions for Internal Longitudinal
Semi-Elliptical Surface Flaws in a Cylinder Under Arbitrary Loadings,” ASTM STP 677,
Fracture Mechanics, ed. C. W. Smith, American Society for Testing and Materials, (1979)
365-380.
94. I. S. Raju and J. C. Newman, Jr., “Stress Intensity Factor Influence Coefficients for Internal and
External Surface Cracks in Cylindrical Vessels,” PVP Vol. 58, Aspects of Fracture Mechanics
in Pressure Vessels and Piping, ASME Pressure Vessels and Piping Conference, (1982) 37-48.
95. S. K. Iskander, R. D. Cheverton, and D. G. Ball, OCA-I, A Code for Calculating the Behavior of
Flaws on the Inner Surface of a Pressure Vessel Subjected to Temperature and Pressure
Transients, NUREG/CR-2113 (ORNL/NUREG-84), Oak Ridge National Laboratory, Oak
Ridge, TN, 1981.
96. R. D. Cheverton and D. G. Ball, OCA-P, A Deterministic and Probabilistic Fracture Mechanics
Code for Application to Pressure Vessels, NUREG/CR-3618 (ORNL-5991), Oak Ridge
National Laboratory, Oak Ridge, TN, May 1984.
97. J. W. Bryson and T. L. Dickson, “Stress-Intensity-Factor Influence Coefficients for Axial and
Circumferential Flaws in Reactor Pressure Vessels,” PVP Vol. 250, ASME Pressure Vessels and
Piping Conference, (1993) 77-88.
98. T. L. Dickson, J. A. Keeney, and J. W. Bryson, “Validation of FAVOR Code Linear-Elastic
Fracture Solutions for Finite-Length Flaw Geometries,” PVP-Vol. 304, Fatigue and Fracture
Mechanics in Pressure Vessels and Piping, ASME Pressure Vessels and Piping Conference,
(1995) 51-58.
99. J. W. Bryson and T. L. Dickson, Stress-Intensity-Factor Influence Coefficients for
Circumferentially Oriented Semielliptical Inner Surface Flaws in Clad Pressure Vessels,
ORNL/NRC/LTR-94/8, Oak Ridge National Laboratory, Oak Ridge, TN, April, 1994.
100. ABAQUS Theory Manual, Version 4.8, Hibbit, Karlson, and Sorenesen, Inc., Providence, RI,
1989.
101. S. Timoshenko, Theory of Plates and Shells, McGraw-Hill, New York, 1940.
102. T. L. Dickson, B. R. Bass, and P. T. Williams, “Validation of a Linear-Elastic Fracture
Methodology for Postulated Flaws Embedded in the Wall of a Nuclear Reactor Pressure
Vessel,” PVP-Vol. 403, Severe Accidents and Other Topics in RPV Design, American Society
of Mechanical Engineering Pressure Vessels and Piping Conference, (2000) 145-151.
103. R.C. Cipolla, et al., Failure Analysis Associates, Computational Method to Perform the Flaw
Evaluation Procedure as Specified in the ASME Code, Section XI, Appendix A, EPRI Report
NP-1181, September, 1979.
104. American Society of Mechanical Engineers Boiler and Pressure Vessel Code, Section XI, Rules
for Inservice Inspection of Nuclear Power Plant Components, Appendix A, Analysis of Flaws,
Article A-3000, Method for KI Determination, American Society of Mechanical Engineers, New
York, 1998.
105. T. L. Dickson, B. R. Bass, and W. J. McAfee, “The Inclusion of Weld Residual Stress In
Fracture Margin Assessments of Embrittled Nuclear Reactor Pressure Vessels,” PVP-Vol. 373,
Fatigue, Fracture, and Residual Stresses, ASME Pressure Vessels and Piping Conference,
(1998) 387-395.
135
106. T.L .Dickson, W.J. McAfee, W.E. Pennell, and P.T. Williams, Evaluation of Margins in the
ASME Rules for Defining the P-T Curve for an RPV, NUREG/CP-0166, Oak Ridge National
Laboratory, Oak Ridge, TN, Proceedings of the Twenty-Sixth Water Reactor Safety Meeting 1,
(1999) 47-72.
107. J. A. Keeney, et al., Preliminary Assessment of the Fracture Behavior of Weld Material in FullThickness Clad Beams, NUREG/CR-6228 (ORNL/TM-12735), Oak Ridge National Laboratory,
Oak Ridge, TN, October 1994.
108. M. T. EricksonKirk, B. R. Bass, T. L. Dickson, C. E. Pugh, T. Santos, and P. T. Williams,
“Probabilistic Fracture Mechanics: Models, Parameters, and Uncertainty Treatment Used in
FAVOR Version 04.1,” U.S. Nuclear Regulatory Commission, NUREG-1807, 2006.
109. R. K. Nanstad, J. A. Keeney , and D. E. McCabe, Preliminary Review of the Bases for the KIc
Curve in the ASME Code, ORNL/NRC/LTR-93/15, Oak Ridge National Laboratory, Oak Ridge,
TN, 1993.
110. Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials, E 399-90,
Annual Book of ASTM Standards - Section 3: Metals Test Methods and Analytical Procedures,
vol. 03.01, Metals – Mechanical Testing; Elevated and Low-Temperature Tests; Metallography,
American Society for Testing and Materials, West Conshohocken, PA, 1998.
111. ASME Boiler and Pressure Vessel Code, Section III, Article NB-2331, American Society of
Mechanical Engineers, New York, NY, (1998).
112. R. K. Nanstad, F. M. Haggag, and D. E. McCabe, Irradiation Effects on Fracture Toughness of
Two High-Copper Submerged-Arc Welds, HSSI Series 5, USNRC Report NUREG/CR-5913
(ORNL/TM-12156/V1 and V2) Vols. 1 and 2, Oak Ridge National Laboratory, Oak Ridge, TN,
October 1992.
113.D. E. McCabe, A Comparison of Weibull and βIc Analysis of Transition Range Fracture
Toughness Data, USNRC Report NUREG/CR-5788 (ORNL/TM-11959), Oak Ridge National
Laboratory, Oak Ridge, TN, January 1992.
114.T. Iawadate, Y. Tanaka, S. Ono, and J. Watanabe, “An Analysis of Elastic-Plastic Fracture
Toughness Behavior for JIc Measurements in the Transition Region,” Elastic-Plastic Fracture:
Second Symposium, Vol. II-Fracture Resistance Curves and Engineering Applications, (edited
by C. F. Shih and J. P. Gudas) ASTM STP 803, (1983) II531-II561.
115.D. E. McCabe, R. K. Nanstad, S. K. Iskander, and R. L. Swain, Unirradiated Material Properties
of Midland Weld WF-70, USNRC Report NUREG/CR-6249 (ORNL/TM-12777), Oak Ridge
National Laboratory, Oak Ridge, TN, October 1994.
116.J. J. McGowan, R. K. Nanstad, and K. R. Thoms, Characterization of Irradiated CurrentPractice Welds and A533 Grade B Class 1 Plate for Nuclear Pressure Vessel Service, USNRC
Report NUREG/CR-4880 (ORNL-6484/V1 and V2), Oak Ridge National Laboratory, Oak
Ridge, TN, July 1988.
117.American Society for Testing and Materials, Standard Test Method for Determining Plane-Strain
Crack Arrest Toughness, KIa, of Ferritic Steels, E 1221-88, Annual Book of ASTM Standards,
Section 3: Metals Test Methods and Analytical Procedures, vol. 03.01, Metals – Mechanical
Testing; Elevated and Low-Temperature Tests; Metallography, American Society for Testing
and Materials, West Conshohocken, PA, 1998.
118. S. K. Iskander, W. R. Corwin, R. K. Nanstad, Results of Crack-Arrest Tests on Two Irradiated
High-Copper Welds, USNRC Report NUREG/CR-5584 (ORNL/TM-11575), Oak Ridge
National Laboratory, Oak Ridge, TN, December 1990.
136
119. S. K. Iskander, C. A. Baldwin, D. W. Heatherly, D. E. McCabe, I. Remec, and R. L. Swain,
Detailed Results of Testing Unirradiated and Irradiated Crack-Arrest Toughness Specimens
from the Low Upper-Shelf Energy, High Copper Weld, WF-70, NUREG/CR-6621
(ORNL/TM-13764). 2007
120. S. K. Iskander, R. K. Nanstad, D. E. McCabe, and R. L. Swain, “Effects of Irradiation on CrackArrest Toughness of a Low Upper-Shelf Energy, High-Copper Weld,” Effects of Radiation on
Materials: 19th International Symposium, ASTM STP 1366, M. L. Hamilton, A. S. Kumar, S. T.
Rosinski, and M. L. Grossbeck, eds., American Society for Testing and Materials, to be published.
121. E. J. Ripling and P. B. Crosley, “Strain Rate and Crack Arrest Studies,” HSST 5th Annual
Information Meeting, Paper No. 9, 1971.
122. E. J. Ripling and P. B. Crosley, “Crack Arrest Studies,” HSST 6th Annual Information Meeting,
Paper No. 10, 1972.
123. R. D. Cheverton, D. G. Ball, S. E. Bolt, S. K. Iskander, and R. K. Nanstad, Pressure Vessel
Fracture Studies Pertaining to the PWR Thermal-Shock Issue: Experiments TSE-5, TSE-5A, and
TSE-6, NUREG/CR-4249 (ORNL-6163), Oak Ridge National Laboratory, Oak Ridge, TN, June
1985.
124. “Standard Test Method for Determination of Reference Temperature, T0, for Ferritic Steels in the
Transition Range,” E 1921-97, Annual Book of ASTM Standards Section 3: Metals Test
Methods and Analytical Procedures, vol. 03.01, Metals – Mechanical Testing; Elevated and
Low-Temperature Tests; Metallography, American Society for Testing and Materials, West
Conshohocken, PA, 1998.
125. M. Kirk, R. Lott, W. L. Server, R. Hardies, and S. Rosinski, “Bias and Precision of T0 Values
Determined Using ASTM Standard E 1921-97 for Nuclear Reactor Pressure Vessel Steels,”
Effects of Radiation on Materials: 19th International Symposium, ASTM STP 1366, M. L.
Hamilton, A. S. Kumar, S. T. Rosinski, and M. L. Grossbeck, Eds., American Society for
Testing and Materials, West Conshococken, PA, (2000) 143-161.
126. K. Wallin, “Master Curve Based Correlation Between Static Initiation Toughness, KIc, and Crack
Arrest Toughness, KIa,” presented at 24th MPA Seminar, Stuttgart, Germany, October 8 and 9,
1998.
127. K. Wallin, “Application of the Master Curve Method to Crack Initiation and Crack Arrest,”
Fracture, Fatigue, and Weld Residual Stress, PVP Vol. 393, ASME Pressure Vessels and Piping
Conference, Boston, MA, August 1-5, 1999.
128.Reactor Vessel Integrity Database (RVID), User’s Manual, Version 1.1, U. S. Nuclear
Regulatory Commission, July 1995.
129. Reactor Vessel Integrity Database (RVID), Version 2.1.1, U. S. Nuclear Regulatory Commission
July 6, 2000.
137
Appendix A – Background and Antecedents of FAVOR, v04.1
An important element of the PTS plant-specific analysis is the calculation of the conditional
probability of failure of the vessel by performing probabilistic fracture mechanics (PFM)
analyses. The term conditional refers here to two assumed preconditions: (1) the specific PTS
event under study has in fact occurred, and (2) the postulated flaws do exist on the surface or
embedded within the RPV wall. Combined with an estimate of the frequency of occurrence for
the event, a predicted frequency of vessel failure can then be calculated. OCA-P [1] and
VISA-II [2] are PTS PFM computer programs, independently developed at Oak Ridge National
Laboratory (ORNL) and Pacific Northwest National Laboratory (PNNL), respectively, in the
1980s with NRC funding, that are currently referenced in Regulatory Guide 1.154 as acceptable
codes for performing plant-specific analyses.
There have also been other proprietary and public-domain PTS PFM codes independently
developed in the US and internationally by reactor vendors and research laboratories. The
development of the OCA-P code [1] (and its deterministic predecessors, OCA-I [3], and OCA-II
[4]) and the VISA II code [2] was preceded by two earlier probabilistic computer programs
developed by the NRC, specifically OCTAVIA [5] (Operationally Caused Transients and Vessel
Integrity Analysis) and a second unnamed code developed by Gamble and Strosnider [6].
OCTAVIA [5] was developed in the mid-1970s to calculate the probability of RPV failure from
operationally caused pressure transients which can occur in a PWR vessel at low operating
temperatures. OCTAVIA computed the pressure at which the vessel would fail for different-sized
flaws existing in the beltline region, where only axially oriented flaws in the vessel beltline were
considered. The probability of vessel failure was then calculated as the product of two factors: the
probability that the maximum-sized flaw in the beltline is of a given size, and the probability that
the transient would occur and would have a pressure exceeding the vessel failure pressure
associated with the flaw size. The probabilities of vessel failure were summed over the various
sizes to obtain the total vessel failure probability.
The code developed by Gamble and Strosnider [6] calculates the probability of flaw-induced
failure in the vessel beltline region using mathematical relationships based on linear-elastic
fracture mechanics to model variable interaction and to estimate a failure rate. The RPV failure
criterion was based on a comparison of the driving-force stress-intensity factor, KI, with the static
initiation toughness, KIc, of the material. Monte Carlo methods were used to simulate
independently each of the several variables and model their interaction to obtain values of KI and
KIc to predict the probabilities of vessel failure. Near the end of this study, an importance-
A-1
sampling scheme was developed and incorporated into the computer code to increase the code’s
efficiency for performing calculations in the transition-temperature region and to allow greater
accuracy for analyzing conditions associated with low-failure probabilities (see Appendix B of
ref. [6]).
An early version of the VISA code [7] was used in the NRC staff evaluation of PTS as described
in SECY-82-465 [8]. VISA is a simulation model, which means that the failure probability is
assessed by performing a large number of deterministic evaluations with random variables
selected for various parameters. The user can specify the thermal transient with either a
polynomial representation or an exponential decay model, and the pressure transient can be
specified with a polynomial function. The deterministic analysis in VISA assumes linear-elastic
material behavior, implying that the total maximum stresses are less than the yield strength of the
material. This assumption of linear-elastic deformation response allows stress components to be
added through linear superposition, and the principles of linear-elastic fracture mechanics
(LEFM) can be applied. For rapid thermal transients, high stresses (potentially above the yield
strength of the cladding) can occur locally at the inside surface of the vessel wall; however,
acceptable stress distributions can still be obtained over the remaining section if the overstressed
region is relatively thin. Stress intensity factors are calculated from influence coefficients
developed by Heliot, Labbens, and Pellissier-Tanon [9, 10].
Examples of internationally developed PFM/PTS codes include PASCAL (PFM Analysis of
Structural Components in Aging LWR) [11- 13], OPERA [14], and PARISH (Probabilistic
1
Assessment of Reactor Integrity under pressurized thermal SHock) [15]. In addition, other PFM
codes such as PRAISE [16] and STAR6 [17] have been developed to calculate failure
probabilities considering the aged condition of RCW piping systems allowing for factors such as
fatigue crack growth, stress corrosion crack growth, and changes in mechanical properties.
The above codes perform PFM/PTS analyses using Monte Carlo techniques to estimate the
increase in failure probability as the vessel accumulates radiation damage over its operating life.
The results of such analyses, when compared with the limit of acceptable failure probability,
provide an estimate of the residual life of a reactor pressure vessel. Also results of such analyses
can be used to evaluate the potential benefits of plant-specific mitigating actions designed to
reduce the probability of reactor vessel failure, thus potentially extending the operating life of the
vessel [18].
Previous efforts at obtaining the same probabilistic solutions to a specified PTS problem using
different PFM codes have met with varying degrees of success [19- 21]. Experience with the
1
A-2
application of OCA-P, VISA-II, and other PFM codes as well as advancements in the science of
probabilistic risk assessment (PRA) over the past 15 years have provided insights into areas
where the PTS PFM methodology could be improved. The FAVOR computer code was initially
developed at ORNL in the early 1990s [22] (see Fig. A1) in an effort to combine the best
attributes of OCA-P and VISA-II. In the ensuing years, the NRC-funded FAVOR code has
continued its advancement with the goal of providing a computational platform for incorporating
additional capabilities and new developments in relevant fracture-related disciplines, as illustrated
in Fig. A1.
Fig. A1. Depiction of the development history of the FAVOR code
A-3
References to Appendix A
1.
R. D. Cheverton and D. G. Ball, OCA-P, A Deterministic and Probabilistic Fracture
Mechanics Code for Application to Pressure Vessels, NUREG/CR-3618 (ORNL-5991), Oak
Ridge National Laboratory, Oak Ridge, TN, May 1984.
2.
F. A. Simonen, et al., VISA-II – A Computer Code for Predicting the Probability of Reactor
Pressure Vessel Failure, NUREG/CR-4486 (PNL-5775), Pacific Northwest Laboratory,
Richland, Washington, March 1986.
3.
S. K. Iskander, R. D. Cheverton, and D. G. Ball, OCA-I, A Code for Calculating the
Behavior of Flaws on the Inner Surface of a Pressure Vessel Subjected to Temperature and
Pressure Transients, NUREG/CR-2113 (ORNL/NUREG-84), Oak Ridge National
Laboratory, Oak Ridge, TN, 1981.
4.
D. G. Ball, R. D. Cheverton, J. B. Drake, and S. K. Iskander, OCA-II, A Code for
Calculating the Behavior of 2-D and 3-D Surface Flaws in a Pressure Vessel Subjected to
Temperature and Pressure Transients, NUREG/CR-3491 (ORNL-5934), Oak Ridge
National Laboratory, Oak Ridge, TN, 1983.
5.
W. E. Vesely, E. K. Lynn, and F. F. Goldberg, The Octavia Computer Code: PWR Reactor
Pressure Vessel Failure Probabilities Due to Operationally Caused Pressure Transients,
NUREG-0258, U. S. Nuclear Regulatory Commission, Washington, D.C., 1978.
6.
R. M. Gamble and J. Strosnider, Jr., An Assessment of the Failure Rate for the Beltline
Region of PWR Pressure Vessels During Normal Operation and Certain Transient
Conditions, NUREG-0778, U. S. Nuclear Regulatory Commission, Washington, D.C., 1981.
7.
D. L. Stevens, F. A. Simonen, J. Strosnider, Jr., R. W. Klecker, D. W. Engel, and K. I.
Johnson, VISA – A Computer Code for Predicting the Probability of Reactor Pressure
Vessel Failure, NUREG/CR-3384, (PNL-4774), Pacific Northwest Laboratory, Richland,
Washington, 1983.
8.
Policy Issue from J. W. Dircks to NRC Commissioners, Enclosure A: NRC Staff Evaluation
of Pressurized Thermal Shock, November 1982, SECY-82-465, November 23, 1982,
Division of Nuclear Reactor Regulation, U.S. Nuclear Regulatory Commission,
Washington, D.C.
9.
R. Labbens, A. Pellissier-Tanon, and J. Heliot, “Practical Method for Calculating StressIntensity Factors Through Weight Functions,” ASTM STP-590, Mechanics of Crack
Growth, American Society for Testing and Materials, (1976) 368-384.
10. J. Heliot, R. C. Labbens, and A. Pellissier-Tanon, “Semi-Elliptical Cracks in the Mendional
Plane of a Cylinder Subjected to Stress Gradients, Calculation of Stress Intensity Factors by
the Boundary Integral Equations Method,” XIth National Symposium on Fracture
Mechanics, Blacksburg, VA, 1978.
11. G. Yagawa, Y. Kanto, S. Yoshimura, H. Machida, and K. Shibata, “Probabilistic Fracture
Mechanics Analysis of Nuclear Structural Components: A Review of Recent Japanese
Activities,” Nuclear Engineering and Design 207, (2001) 269-286.
12. K. Shibata, D. Kato, and Y. Li, “Development of a PFM Code for Evaluating Reliability of
Pressure Components Subject to Transient Loading,” Nuclear Engineering and Design 208,
(2001) 1-13.
A-4
13. Y. Li, D. Kato, K. Shibata, and K. Onizawa, “Improvements to a Probabilistic Fracture
Mechanics Code for Evaluating the Integrity of an RPV Under Transient Loading,”
International Journal of Pressure Vessels and Piping 78, (2001) 271-282.
14. M. Persoz, S. Hugonnard-Bruyere, et al., “Deterministic and Probabilistic Assessment of the
Reactor Pressure Vessel Structural Integrity with a User-Friendly Software,” PVP-Vol. 403,
presented at the ASME Pressure Vessels and Piping Conference, (2000) 137-144.
15. B. K. Dutta, H. S. Kushwaha, and V. V. Raj, “Probabilistic Assessment of Reactor Pressure
Vessel Integrity Under Pressurised Thermal Shock,” International Journal of Pressure
Vessels and Piping 76, (1999) 445-453.
16. D. O. Harris, E. Y. Lim, et al., “Probability of Pipe Fracture in the Primary Coolant Loop of
a PWR Plant,” NUREG/CR-2189, 1981.
17. R. Wilson and R. A. Ainsworth, “A Probabilistic Fracture Mechanics Assessment
Procedure,” SMiRT 11, vol. G, (1991) 325-330.
18. T. L. Dickson and F. A. Simonen, “The Application of Probabilistic Fracture Analysis to
Residual Life Evaluation of Embrittled Reactor Vessels,” AD-Vol. 28, Reliability
Technology, American Society of Mechanical Engineers, (1992) 43-55.
19. B. A. Bishop, T. L. Dickson, and F. A. Simonen, Benchmarking of Probabilistic Fracture
Mechanics Analyses of Reactor Vessels Subjected to Pressurized Thermal Shock (PTS)
Loading, Research Project 2975-5, Final Report, February 1993.
20. T. L. Dickson and R. D. Cheverton, Review of Reactor Pressure Vessel Evaluation Report
for Yankee Rowe Nuclear Power Station (YAEC No. 1735), Appendix D, NUREG/CR-5799
(ORNL/TM-11982), Oak Ridge National Laboratory, Oak Ridge, TN, March 1992.
21. B. R. Bass, et al., International Comparative Assessment Study of Pressurized Thermal
Shock in Reactor Pressure Vessels, NUREG/CR-6651 (ORNL/TM-1999/231), Oak Ridge
National Laboratory, Oak Ridge, TN, December 1999.
22. T. L. Dickson, FAVOR: A Fracture Analysis Code for Nuclear Reactor Pressure Vessels,
Release 9401, ORNL/NRC/LTR/94/1, Oak Ridge National Laboratory, Oak Ridge, TN,
1994.
A-5
Appendix B – Stress-Intensity Factor Influence Coefficients
Table B1. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t =10 and a/t=0.01
Table B2. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t =10 and a/t=0.0184
Table B3. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t =10 and a/t=0.05
Table B4. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t =10 and a/t=0.075
Table B5. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t =10 and a/t=0.1
Table B6. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t =10 and a/t=0.2
Table B7. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t =10 and a/t=0.3
Table B8. Influence Coefficients for Inside Axial Semi-elliptical Surface Flaws: R / t =10 and
a/t=0.5
Table B9. Influence Coefficients for Inside Circumferential Semi-elliptical Surface Flaws:
R / t =10 and a/t=0.5
Table B10. Influence Coefficients for Inside Axial Infinite-Length Surface Flaws: R / t =10
Table B11. Influence Coefficients for Inside Circumferential 360-Degree Surface Flaws:
R / t =10
B-1
Table B1. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t = 10 and a / t = 0.01
Aspect
Ratio
2:1
6:1
10:1
Elliptic
K0
Angle (deg) Uniform
0.00
0.764
2.37
0.754
16.60
0.690
30.80
0.669
45.00
0.660
59.20
0.653
73.40
0.651
87.60
0.649
90.00
0.649
0.00
0.670
2.37
0.667
16.60
0.654
30.80
0.741
45.00
0.827
59.20
0.893
73.40
0.938
87.60
0.970
90.00
0.975
0.00
0.515
2.37
0.529
16.60
0.610
30.80
0.762
45.00
0.889
59.20
0.979
73.40
1.033
87.60
1.064
90.00
1.069
K1
Linear
0.153
0.165
0.192
0.264
0.335
0.393
0.434
0.463
0.468
0.134
0.134
0.170
0.269
0.381
0.481
0.559
0.594
0.601
0.090
0.094
0.146
0.258
0.389
0.507
0.593
0.635
0.642
K2
Quadratic
0.061
0.062
0.079
0.127
0.196
0.269
0.329
0.366
0.372
0.048
0.043
0.055
0.109
0.199
0.302
0.389
0.435
0.443
0.020
0.010
0.033
0.060
0.171
0.290
0.389
0.439
0.447
B-2
K3
Cubic
0.034
0.032
0.040
0.069
0.124
0.198
0.268
0.310
0.317
0.024
0.019
0.009
0.029
0.100
0.197
0.290
0.341
0.350
0.006
0.005
0.005
0.019
0.066
0.136
0.249
0.307
0.316
K0
K1
K0
K1
tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.
0.764
0.153
0.764
0.153
0.754
0.165
0.754
0.165
0.690
0.192
0.690
0.192
0.669
0.264
0.669
0.264
0.660
0.335
0.660
0.335
0.653
0.393
0.653
0.393
0.651
0.434
0.651
0.434
0.649
0.463
0.649
0.463
0.649
0.468
0.649
0.468
0.670
0.134
0.670
0.134
0.667
0.134
0.667
0.134
0.654
0.170
0.654
0.170
0.741
0.269
0.741
0.269
0.827
0.381
0.827
0.381
0.893
0.481
0.893
0.481
0.938
0.559
0.938
0.559
0.970
0.594
0.970
0.594
0.975
0.601
0.975
0.601
0.515
0.090
0.515
0.090
0.529
0.094
0.529
0.094
0.610
0.146
0.610
0.146
0.762
0.258
0.762
0.258
0.889
0.389
0.889
0.389
0.979
0.507
0.979
0.507
1.033
0.593
1.033
0.593
1.064
0.635
1.064
0.635
1.069
0.642
1.069
0.642
Table B2. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t = 10 and a / t = 0.0184
Aspect
Ratio
2:1
6:1
10:1
Elliptic
Angle (deg)
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
K0
Uniform
0.777
0.767
0.700
0.677
0.667
0.660
0.657
0.654
0.653
0.653
0.654
0.654
0.758
0.852
0.920
0.963
0.994
0.999
0.525
0.538
0.621
0.777
0.899
0.982
1.033
1.063
1.068
K1
Linear
0.155
0.167
0.194
0.266
0.338
0.397
0.438
0.467
0.472
0.127
0.128
0.168
0.271
0.387
0.492
0.569
0.609
0.616
0.092
0.096
0.149
0.262
0.392
0.509
0.595
0.637
0.644
K2
Quadratic
0.061
0.062
0.079
0.127
0.196
0.270
0.330
0.366
0.373
0.043
0.038
0.045
0.099
0.192
0.298
0.387
0.434
0.442
0.019
0.009
0.039
0.050
0.164
0.283
0.383
0.433
0.441
B-3
K3
Cubic
0.034
0.032
0.040
0.069
0.125
0.198
0.267
0.310
0.317
0.021
0.016
0.021
0.026
0.085
0.187
0.283
0.335
0.344
0.007
0.005
0.005
0.022
0.075
0.127
0.242
0.300
0.310
K0
tcl=0.25 in.
0.777
0.767
0.700
0.677
0.667
0.660
0.657
0.654
0.653
0.653
0.654
0.654
0.758
0.852
0.920
0.963
0.994
0.999
0.525
0.538
0.621
0.777
0.899
0.982
1.033
1.063
1.068
K1
tcl=0.25 in.
0.155
0.167
0.194
0.266
0.338
0.397
0.438
0.467
0.472
0.127
0.128
0.168
0.271
0.387
0.492
0.569
0.609
0.616
0.092
0.096
0.149
0.262
0.392
0.509
0.595
0.637
0.644
K0
K1
tcl=0.156 in. tcl=0.156 in.
0.777
0.155
0.767
0.167
0.700
0.194
0.677
0.266
0.667
0.338
0.660
0.397
0.657
0.438
0.654
0.467
0.653
0.472
0.653
0.127
0.654
0.128
0.654
0.168
0.758
0.271
0.852
0.387
0.920
0.492
0.963
0.569
0.994
0.609
0.999
0.616
0.525
0.092
0.538
0.096
0.621
0.149
0.777
0.262
0.899
0.392
0.982
0.509
1.033
0.595
1.063
0.637
1.068
0.644
Table B3. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t = 10 and a / t = 0.05
Aspect
Ratio
Elliptic
Angle (deg)
K0
Uniform
K1
Linear
K2
Quadratic
K3
Cubic
K0
tcl=0.25 in.
2:1
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.779
0.769
0.701
0.678
0.668
0.661
0.658
0.656
0.655
0.655
0.655
0.655
0.758
0.851
0.918
0.962
0.992
0.997
0.523
0.537
0.622
0.778
0.898
0.981
1.034
1.063
1.068
0.155
0.166
0.194
0.267
0.339
0.398
0.440
0.469
0.474
0.128
0.128
0.167
0.270
0.386
0.492
0.569
0.609
0.616
0.092
0.095
0.147
0.261
0.391
0.509
0.596
0.638
0.645
0.061
0.062
0.079
0.128
0.199
0.273
0.333
0.370
0.377
0.043
0.039
0.049
0.104
0.197
0.305
0.395
0.443
0.450
0.021
0.011
0.033
0.061
0.171
0.292
0.392
0.442
0.450
0.034
0.031
0.040
0.070
0.126
0.201
0.270
0.313
0.320
0.021
0.016
0.019
0.013
0.091
0.193
0.290
0.342
0.351
0.005
0.015
0.050
0.080
0.065
0.138
0.252
0.310
0.320
0.708
0.701
0.659
0.581
0.326
0.233
0.204
0.185
0.182
0.631
0.628
0.646
0.688
0.494
0.422
0.396
0.374
0.370
0.533
0.543
0.631
0.718
0.550
0.474
0.444
0.418
0.414
6:1
10:1
B-4
K1
K0
K1
tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.
0.184
0.194
0.264
0.340
0.188
0.127
0.110
0.099
0.097
0.151
0.156
0.221
0.357
0.263
0.217
0.201
0.189
0.186
0.119
0.121
0.149
0.348
0.286
0.241
0.224
0.221
0.221
0.636
0.624
0.509
0.246
0.159
0.128
0.115
0.106
0.104
0.576
0.570
0.537
0.340
0.271
0.253
0.241
0.231
0.229
0.496
0.504
0.547
0.376
0.349
0.287
0.273
0.260
0.257
0.205
0.213
0.232
0.124
0.083
0.067
0.060
0.055
0.054
0.176
0.177
0.213
0.167
0.138
0.128
0.121
0.115
0.115
0.149
0.146
0.199
0.182
0.156
0.144
0.136
0.130
0.128
Table B4. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t = 10 and a / t = 0.075
Aspect
Ratio
Elliptic
Angle (deg)
K0
Uniform
K1
Linear
K2
Quadratic
K3
Cubic
K0
tcl=0.25 in.
2:1
0.00
7.03
14.20
35.90
48.70
61.50
74.30
87.00
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.740
0.737
0.721
0.671
0.661
0.656
0.654
0.651
0.651
0.650
0.635
0.672
0.786
0.862
0.918
0.952
0.980
0.987
0.547
0.551
0.636
0.812
0.914
0.982
1.022
1.048
1.055
0.128
0.147
0.179
0.298
0.355
0.404
0.439
0.468
0.475
0.098
0.104
0.140
0.309
0.410
0.501
0.566
0.602
0.611
0.073
0.074
0.113
0.303
0.419
0.522
0.593
0.631
0.639
0.045
0.055
0.067
0.155
0.220
0.285
0.336
0.372
0.381
0.029
0.031
0.040
0.139
0.229
0.326
0.404
0.446
0.456
0.016
0.016
0.023
0.124
0.225
0.332
0.416
0.461
0.471
0.023
0.028
0.033
0.086
0.143
0.212
0.273
0.313
0.322
0.013
0.013
0.014
0.048
0.125
0.219
0.303
0.351
0.362
0.006
0.003
0.009
0.018
0.111
0.216
0.307
0.356
0.368
0.650
0.629
0.593
0.219
0.161
0.137
0.125
0.114
0.111
0.591
0.571
0.590
0.334
0.294
0.275
0.265
0.265
0.265
0.514
0.514
0.583
0.375
0.335
0.310
0.298
0.295
0.295
6:1
10:1
B-5
K1
K0
K1
tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.
0.197
0.220
0.271
0.120
0.085
0.071
0.065
0.065
0.065
0.170
0.180
0.243
0.171
0.149
0.138
0.133
0.133
0.132
0.148
0.145
0.220
0.189
0.168
0.156
0.149
0.147
0.147
0.572
0.529
0.400
0.118
0.094
0.081
0.075
0.068
0.067
0.527
0.495
0.441
0.195
0.180
0.170
0.164
0.159
0.157
0.469
0.458
0.465
0.223
0.206
0.193
0.185
0.185
0.184
0.210
0.217
0.177
0.060
0.048
0.042
0.038
0.035
0.034
0.188
0.179
0.187
0.098
0.090
0.085
0.082
0.080
0.079
0.171
0.131
0.173
0.112
0.103
0.096
0.093
0.092
0.092
Table B5. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t = 10 and a / t = 0.1
Aspect
Ratio
Elliptic
Angle (deg)
K0
Uniform
K1
Linear
K2
Quadratic
K3
Cubic
K0
tcl=0.25 in.
2:1
0.00
5.27
17.10
31.10
45.10
59.10
73.10
87.00
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.729
0.741
0.722
0.676
0.664
0.658
0.655
0.653
0.652
0.641
0.630
0.701
0.756
0.848
0.915
0.958
0.989
0.996
0.543
0.536
0.670
0.778
0.897
0.979
1.029
1.060
1.066
0.124
0.139
0.230
0.273
0.339
0.396
0.436
0.470
0.477
0.094
0.098
0.196
0.273
0.385
0.489
0.565
0.607
0.616
0.067
0.069
0.175
0.269
0.395
0.512
0.597
0.640
0.649
0.044
0.053
0.096
0.133
0.201
0.274
0.333
0.373
0.382
0.029
0.031
0.067
0.115
0.207
0.312
0.402
0.450
0.461
0.016
0.016
0.047
0.102
0.202
0.318
0.416
0.466
0.477
0.023
0.027
0.048
0.072
0.127
0.200
0.268
0.313
0.323
0.014
0.015
0.015
0.039
0.109
0.207
0.302
0.356
0.367
0.007
0.006
0.027
0.030
0.089
0.199
0.302
0.358
0.370
0.596
0.582
0.366
0.176
0.122
0.101
0.091
0.082
0.080
0.550
0.532
0.427
0.258
0.224
0.208
0.200
0.200
0.200
0.490
0.479
0.443
0.291
0.256
0.236
0.226
0.224
0.223
6:1
10:1
B-6
K1
K0
K1
tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.
0.195
0.208
0.213
0.097
0.064
0.052
0.047
0.047
0.047
0.175
0.176
0.232
0.131
0.112
0.104
0.100
0.100
0.100
0.148
0.144
0.220
0.143
0.128
0.118
0.113
0.111
0.111
0.519
0.483
0.168
0.095
0.072
0.061
0.056
0.050
0.049
0.485
0.454
0.211
0.152
0.138
0.129
0.125
0.120
0.119
0.443
0.421
0.229
0.176
0.159
0.147
0.141
0.140
0.140
0.205
0.198
0.086
0.048
0.037
0.031
0.028
0.025
0.025
0.188
0.168
0.108
0.077
0.069
0.065
0.062
0.060
0.060
0.168
0.138
0.117
0.088
0.080
0.074
0.071
0.070
0.070
Table B6. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t = 10 and a / t = 0.2
Aspect
Ratio
Elliptic
Angle (deg)
K0
Uniform
K1
Linear
K2
Quadratic
K3
Cubic
K0
tcl=0.25 in.
2:1
0.00
19.80
31.10
42.50
53.80
65.20
76.50
87.90
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.692
0.695
0.679
0.671
0.665
0.660
0.658
0.656
0.656
0.617
0.699
0.781
0.856
0.915
0.958
0.986
1.010
1.020
0.525
0.694
0.815
0.915
0.991
1.045
1.080
1.103
1.107
0.127
0.214
0.273
0.332
0.383
0.423
0.450
0.475
0.479
0.101
0.194
0.280
0.375
0.464
0.538
0.590
0.619
0.624
0.077
0.183
0.280
0.387
0.488
0.572
0.631
0.660
0.666
0.046
0.089
0.133
0.192
0.255
0.312
0.354
0.384
0.389
0.034
0.066
0.118
0.195
0.283
0.366
0.430
0.464
0.470
0.022
0.050
0.107
0.190
0.287
0.379
0.449
0.483
0.490
0.024
0.044
0.073
0.120
0.182
0.245
0.296
0.329
0.335
0.017
0.019
0.045
0.101
0.180
0.265
0.336
0.373
0.380
0.009
0.025
0.011
0.083
0.170
0.263
0.340
0.378
0.385
0.457
0.155
0.090
0.061
0.052
0.047
0.044
0.041
0.040
0.434
0.180
0.127
0.116
0.110
0.106
0.104
0.102
0.101
0.402
0.200
0.149
0.137
0.130
0.125
0.122
0.120
0.119
6:1
10:1
B-7
K1
K0
K1
tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.
0.173
0.080
0.050
0.031
0.026
0.023
0.022
0.021
0.020
0.163
0.090
0.063
0.058
0.055
0.053
0.052
0.051
0.051
0.149
0.100
0.073
0.068
0.065
0.062
0.061
0.060
0.060
0.393
0.071
0.048
0.038
0.032
0.029
0.027
0.025
0.025
0.377
0.093
0.079
0.072
0.069
0.066
0.065
0.064
0.063
0.355
0.106
0.093
0.085
0.081
0.078
0.077
0.075
0.075
0.178
0.031
0.023
0.019
0.016
0.014
0.014
0.013
0.013
0.171
0.043
0.039
0.036
0.034
0.033
0.032
0.032
0.032
0.160
0.050
0.046
0.043
0.040
0.039
0.038
0.037
0.037
Table B7. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical
Surface Flaws: R / t = 10 and a / t = 0.3
Aspect
Ratio
2:1
6:1
10:1
Elliptic
Angle
(deg)
0.00
17.40
29.10
40.90
52.60
64.40
76.10
87.90
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
K0
Uniform
K1
Linear
K2
Quadratic
K3
Cubic
K0
tcl=0.25 in.
0.723
0.708
0.690
0.680
0.673
0.668
0.665
0.662
0.662
0.665
0.715
0.804
0.886
0.951
0.998
1.028
1.053
1.058
0.562
0.707
0.848
0.962
1.051
1.115
1.157
1.183
1.187
0.127
0.203
0.264
0.326
0.381
0.423
0.452
0.478
0.482
0.112
0.190
0.277
0.376
0.470
0.549
0.605
0.635
0.640
0.085
0.176
0.276
0.389
0.498
0.590
0.653
0.685
0.691
0.048
0.083
0.126
0.185
0.251
0.310
0.355
0.385
0.391
0.041
0.068
0.118
0.194
0.284
0.372
0.439
0.475
0.481
0.029
0.052
0.104
0.188
0.288
0.385
0.460
0.496
0.503
0.026
0.042
0.068
0.114
0.177
0.242
0.297
0.331
0.337
0.022
0.027
0.051
0.104
0.182
0.270
0.345
0.384
0.391
0.014
0.016
0.016
0.082
0.169
0.265
0.346
0.387
0.394
0.404
0.102
0.058
0.043
0.036
0.032
0.030
0.028
0.027
0.380
0.117
0.093
0.085
0.081
0.078
0.077
0.075
0.075
0.344
0.128
0.110
0.102
0.098
0.096
0.095
0.094
0.094
B-8
K1
K0
K1
tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.
0.188
0.049
0.028
0.021
0.018
0.016
0.015
0.014
0.014
0.181
0.054
0.045
0.042
0.040
0.039
0.038
0.038
0.037
0.168
0.059
0.054
0.051
0.049
0.048
0.047
0.047
0.047
0.334
0.056
0.034
0.026
0.022
0.020
0.018
0.017
0.017
0.315
0.069
0.057
0.053
0.050
0.049
0.048
0.047
0.047
0.290
0.078
0.068
0.064
0.062
0.060
0.060
0.059
0.059
0.176
0.025
0.016
0.013
0.011
0.010
0.009
0.009
0.009
0.167
0.032
0.028
0.026
0.025
0.024
0.024
0.024
0.023
0.153
0.037
0.034
0.032
0.031
0.030
0.030
0.029
0.029
Table B8. Influence Coefficients for Inside Axial Semi-elliptical Surface Flaws: R / t = 10
and a / t = 0.5
Aspect
Ratio
Elliptic
Angle (deg)
K0
Uniform
K1
Linear
K2
Quadratic
K3
Cubic
K0
tcl=0.25 in.
K1
tcl=0.25 in.
K0
tcl=0.156 in.
K1
tcl=0.156 in.
2:1
0.00
15.40
27.50
39.60
51.70
63.70
75.80
87.90
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.00
2.37
16.60
30.80
45.00
59.20
73.40
87.60
90.00
0.736
0.746
0.719
0.704
0.693
0.685
0.681
0.676
0.676
0.758
0.814
0.908
0.998
1.069
1.120
1.153
1.182
1.187
0.666
0.822
0.995
1.138
1.251
1.335
1.390
1.423
1.429
0.132
0.203
0.263
0.327
0.383
0.426
0.456
0.483
0.488
0.142
0.213
0.302
0.405
0.504
0.588
0.647
0.679
0.685
0.119
0.208
0.316
0.440
0.560
0.662
0.734
0.770
0.776
0.053
0.083
0.124
0.183
0.249
0.311
0.357
0.389
0.395
0.059
0.083
0.132
0.208
0.300
0.392
0.463
0.500
0.506
0.049
0.077
0.131
0.216
0.321
0.425
0.506
0.546
0.553
0.029
0.043
0.067
0.112
0.175
0.242
0.299
0.334
0.340
0.033
0.040
0.065
0.116
0.195
0.285
0.363
0.404
0.411
0.028
0.033
0.056
0.112
0.198
0.298
0.383
0.427
0.434
0.327
0.079
0.042
0.029
0.023
0.021
0.019
0.018
0.017
0.322
0.091
0.070
0.065
0.062
0.061
0.060
0.059
0.059
0.302
0.097
0.086
0.083
0.083
0.083
0.083
0.083
0.083
0.162
0.037
0.020
0.014
0.012
0.010
0.009
0.009
0.009
0.163
0.041
0.034
0.032
0.031
0.030
0.030
0.029
0.029
0.156
0.044
0.042
0.041
0.041
0.041
0.041
0.041
0.041
0.272
0.045
0.025
0.018
0.015
0.013
0.012
0.011
0.011
0.268
0.054
0.043
0.040
0.039
0.038
0.038
0.037
0.037
0.254
0.060
0.054
0.052
0.052
0.052
0.052
0.052
0.052
0.150
0.020
0.012
0.009
0.007
0.006
0.006
0.006
0.005
0.149
0.025
0.021
0.020
0.019
0.019
0.019
0.018
0.018
0.140
0.028
0.027
0.026
0.026
0.026
0.026
0.026
0.026
6:1
10:1
B-9
Table B9. Influence Coefficients for Inside Circumferential Semi-elliptical Surface
Flaws: R / t = 10 and a / t = 0.5
Aspect
Ratio
Elliptic
Angle (deg)
K0
Uniform
K1
Linear
K2
Quadratic
K3
Cubic
K0
tcl=0.25 in.
2:1
0.00
15.40
27.50
39.60
51.70
63.70
75.80
87.90
90.00
0.00
15.40
27.50
39.60
51.70
63.70
75.80
87.90
90.00
0.00
15.40
27.50
39.60
51.70
63.70
75.80
87.90
90.00
0.741
0.750
0.721
0.706
0.698
0.692
0.686
0.682
0.682
0.727
0.786
0.882
0.974
1.049
1.103
1.138
1.166
1.171
0.616
0.770
0.936
1.076
1.190
1.275
1.330
1.363
1.368
0.134
0.205
0.264
0.328
0.384
0.430
0.461
0.488
0.493
0.132
0.205
0.295
0.398
0.499
0.584
0.644
0.676
0.682
0.101
0.195
0.301
0.424
0.544
0.647
0.719
0.755
0.762
0.054
0.084
0.124
0.183
0.250
0.312
0.360
0.392
0.398
0.053
0.079
0.128
0.205
0.298
0.390
0.462
0.499
0.506
0.040
0.071
0.125
0.211
0.315
0.420
0.501
0.542
0.549
0.030
0.044
0.067
0.112
0.175
0.243
0.301
0.336
0.343
0.030
0.037
0.062
0.114
0.193
0.284
0.362
0.403
0.410
0.023
0.028
0.053
0.109
0.196
0.295
0.381
0.425
0.433
0.324
0.079
0.042
0.029
0.024
0.021
0.019
0.020
0.020
0.315
0.087
0.067
0.062
0.060
0.058
0.057
0.058
0.058
0.291
0.090
0.078
0.075
0.075
0.075
0.075
0.075
0.075
6:1
10:1
B-10
K1
K0
K1
tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.
0.162
0.038
0.020
0.014
0.012
0.010
0.010
0.010
0.009
0.161
0.039
0.032
0.031
0.030
0.029
0.029
0.029
0.029
0.152
0.039
0.038
0.037
0.037
0.037
0.037
0.037
0.037
0.269
0.045
0.025
0.018
0.015
0.013
0.012
0.012
0.013
0.262
0.052
0.041
0.038
0.037
0.036
0.036
0.036
0.036
0.247
0.055
0.049
0.047
0.047
0.047
0.047
0.047
0.047
0.151
0.020
0.012
0.009
0.007
0.007
0.006
0.006
0.006
0.147
0.024
0.020
0.019
0.019
0.018
0.018
0.018
0.018
0.138
0.026
0.024
0.024
0.023
0.023
0.023
0.024
0.024
Table B10. Influence Coefficients for Inside Axial Infinite-Length Surface Flaws,
R / t = 10
a' / a
0
0.0556
0.1111
0.1667
0.2222
0.2778
0.3333
0.3888
0.4444
0.500
0.5556
0.6111
0.6666
0.7222
0.7778
0.8333
0.8888
0.9166
0.9444
0.9639
0.9778
0.9889
a/t=0.01
1.434
1.435
1.436
1.436
1.438
1.442
1.450
1.463
1.482
1.509
1.546
1.598
1.669
1.768
1.913
2.138
2.534
2.878
3.499
5.831
11.225
17.493
a/t=0.02
1.029
1.029
1.029
1.028
1.029
1.032
1.037
1.046
1.058
1.077
1.103
1.138
1.188
1.258
1.360
1.518
1.798
2.041
2.624
4.227
7.289
11.662
0.1 t 1/2 K*
a/t=0.03
0.846
0.846
0.846
0.846
0.846
0.848
0.852
0.859
0.869
0.884
0.905
0.934
0.974
1.031
1.113
1.242
1.470
1.668
2.187
3.499
5.685
8.746
a'/a
0
0.0552
0.1103
0.1655
0.2206
0.2757
0.3309
0.3861
0.4412
0.4963
0.5515
0.6066
0.6618
0.7169
0.7721
0.8272
0.8824
0.9118
0.9412
0.9618
0.9765
0.9882
a/t=0.2
0.461
0.457
0.452
0.447
0.443
0.439
0.436
0.434
0.432
0.433
0.435
0.440
0.450
0.464
0.487
0.526
0.598
0.665
0.875
1.385
2.187
2.916
a/t=0.3
0.510
0.502
0.492
0.483
0.475
0.466
0.459
0.451
0.445
0.440
0.436
0.434
0.435
0.440
0.453
0.477
0.527
0.577
0.729
1.020
1.749
2.478
a/t=0.4
0.617
0.602
0.586
0.571
0.556
0.542
0.527
0.513
0.500
0.488
0.477
0.467
0.460
0.456
0.457
0.468
0.501
0.538
0.671
0.948
1.604
2.187
B-11
a/t=0.05
0.667
0.667
0.666
0.665
0.665
0.666
0.669
0.674
0.682
0.693
0.708
0.731
0.761
0.804
0.868
0.967
1.143
1.294
1.749
2.770
4.227
6.414
a/t=0.075
0.565
0.564
0.563
0.562
0.561
0.561
0.563
0.566
0.571
0.580
0.592
0.609
0.633
0.668
0.718
0.798
0.940
1.064
1.385
2.187
3.426
5.102
a/t=0.10
0.511
0.510
0.508
0.506
0.505
0.504
0.505
0.507
0.511
0.517
0.527
0.541
0.561
0.590
0.632
0.699
0.821
0.927
1.224
1.895
2.916
4.373
a'/a
0
0.059
0.118
0.176
0.235
0.294
0.353
0.412
0.471
0.529
0.588
0.647
0.706
0.750
0.794
0.838
0.882
0.912
0.941
0.962
0.976
0.988
a/t=0.5
0.781
0.755
0.730
0.704
0.679
0.654
0.630
0.605
0.582
0.559
0.538
0.518
0.501
0.491
0.485
0.486
0.501
0.526
0.656
0.875
1.312
2.041
Table B10. (continued) Influence Coefficients for Inside Axial Infinite-Length Surface
Flaws, R / t = 10
0.1 t
1/2
K*
a'/a
a/t=0.6 a'/a a/t=0.7 a'/a a/t=0.8 a'/a a/t=0.9 a/t=0.95
0
1.021
0
1.35
0
1.739
0
1.952
1.902
0.0564 0.983 0.057 1.294 0.058 1.661 0.058 1.866
1.827
0.1127 0.946 0.115 1.238 0.116 1.583 0.117 1.779
1.752
0.1691 0.908 0.172 1.182 0.174 1.506 0.175 1.694
1.678
0.2255 0.871 0.229 1.127 0.232 1.428 0.233 1.608
1.604
0.2819 0.834 0.286 1.071 0.289 1.351 0.292 1.523
1.529
0.3382 0.798 0.343 1.016 0.347 1.275
0.35
1.438
1.456
0.3946 0.761 0.401 0.961 0.405 1.198 0.409 1.354
1.381
0.451
0.725 0.458 0.906 0.463 1.122 0.467
1.27
1.308
0.5074
0.69
0.515 0.852 0.521 1.047 0.526 1.186
1.234
0.5637 0.655 0.572 0.799 0.579 0.971 0.584 1.102
1.162
0.6201 0.622
0.63
0.747 0.637 0.897 0.643 1.019
1.088
0.6765
0.59
0.687 0.696 0.695 0.824 0.701 0.936
1.017
0.7328 0.561 0.744 0.648 0.753 0.752 0.759 0.854
0.947
0.7892 0.536 0.802 0.604 0.811 0.685 0.818 0.773
0.878
0.8456 0.521 0.859 0.569 0.869 0.627 0.876 0.699
0.815
0.902
0.528 0.916 0.562 0.927 0.598 0.935 0.651
0.768
0.9265 0.549 0.937 0.575 0.945 0.607 0.951 0.654
0.766
0.951
0.671 0.958 0.729 0.963
0.7
0.967 0.729
0.781
0.9681 0.933 0.973
1.02
0.976
1.02
0.979 0.875
0.826
0.9804 1.399 0.983 1.458 0.985 1.458 0.987 1.166
0.911
0.9902 2.041 0.992 2.041 0.993 2.041 0.993 1.749
1.093
B-12
Table B11. Influence Coefficients for Inside Circumferential 360 Degree Surface Flaws,
R / t = 10
a' / a
0
0.0556
0.1111
0.1667
0.2222
0.2778
0.3333
0.3888
0.4444
0.5000
0.5556
0.6111
0.6666
0.7222
0.7778
0.8333
0.8888
0.9166
0.9444
0.9639
0.9778
0.9889
a / t=0.01
2.255
2.256
2.257
2.258
2.260
2.267
2.280
2.300
2.329
2.372
2.431
2.511
2.623
2.779
3.008
3.361
3.986
4.520
6.195
8.674
13.630
18.586
a/t=0.02
1.616
1.616
1.616
1.616
1.617
1.621
1.629
1.642
1.662
1.691
1.732
1.788
1.866
1.975
2.135
2.383
2.823
3.199
3.965
5.948
9.913
14.249
10t t1/2 K*
a/t=0.03
1.325
1.324
1.324
1.323
1.324
1.327
1.334
1.344
1.361
1.384
1.417
1.462
1.526
1.615
1.744
1.946
2.305
2.611
3.346
4.956
8.054
11.771
a'/a
0
0.0552
0.1103
0.1655
0.2206
0.2757
0.3309
0.3861
0.4412
0.4963
0.5515
0.6066
0.6618
0.7169
0.7721
0.8272
0.8824
0.9118
0.9412
0.9618
0.9765
0.9882
a/t=0.2
0.645
0.640
0.635
0.630
0.625
0.622
0.619
0.618
0.618
0.622
0.628
0.639
0.656
0.681
0.721
0.784
0.900
1.007
1.363
1.921
2.912
3.841
a/t=0.3
0.644
0.635
0.626
0.617
0.609
0.601
0.594
0.588
0.584
0.581
0.581
0.584
0.592
0.607
0.633
0.678
0.764
0.845
1.078
1.487
2.354
3.346
a/t=0.4
0.691
0.678
0.664
0.651
0.638
0.625
0.613
0.602
0.592
0.584
0.578
0.574
0.575
0.581
0.596
0.626
0.691
0.793
0.954
1.301
1.982
2.912
B-13
a/t=0.05
1.036
1.036
1.035
1.034
1.035
1.037
1.041
1.049
1.061
1.079
1.104
1.138
1.187
1.255
1.355
1.510
1.786
2.022
2.478
3.717
6.195
9.045
a/t=0.075
0.867
0.865
0.864
0.863
0.862
0.863
0.866
0.872
0.880
0.894
0.914
0.941
0.979
1.034
1.114
1.239
1.462
1.654
1.982
2.974
4.956
7.682
a/t=0.10
0.771
0.769
0.767
0.765
0.764
0.764
0.766
0.770
0.777
0.788
0.804
0.826
0.859
0.905
0.972
1.079
1.271
1.425
1.735
2.602
4.337
6.567
a'/a
0
0.059
0.118
0.176
0.235
0.294
0.353
0.412
0.471
0.529
0.588
0.647
0.706
0.750
0.794
0.838
0.882
0.912
0.941
0.962
0.976
0.988
a/t=0.5
0.764
0.744
0.724
0.704
0.684
0.666
0.647
0.630
0.614
0.600
0.589
0.580
0.577
0.579
0.588
0.608
0.650
0.702
0.843
1.115
1.859
2.726
Table B11. (continued) Influence Coefficients for Inside Circumferential 360 Degree
Surface Flaws, R / t = 10
a'/a
0
0.0564
0.1127
0.1691
0.2255
0.2819
0.3382
0.3946
0.4510
0.5074
0.5637
0.6201
0.6765
0.7328
0.7892
0.8456
0.9020
0.9265
0.9510
0.9681
0.9804
0.9902
a/t=0.6
0.852
0.827
0.802
0.778
0.753
0.729
0.706
0.684
0.663
0.642
0.624
0.608
0.595
0.586
0.586
0.601
0.653
0.703
0.867
1.140
1.797
2.602
a'/a
0
0.057
0.115
0.172
0.229
0.286
0.343
0.401
0.458
0.515
0.572
0.630
0.687
0.744
0.802
0.859
0.916
0.937
0.958
0.973
0.983
0.992
10t t1/2 K*
a/t=0.7
a'/a
0.944
0
0.913
0.058
0.883
0.116
0.853
0.174
0.823
0.232
0.794
0.289
0.766
0.347
0.739
0.405
0.712
0.463
0.687
0.521
0.663
0.579
0.641
0.637
0.622
0.695
0.607
0.753
0.600
0.811
0.608
0.869
0.661
0.927
0.709
0.945
0.855
0.963
1.155
0.976
1.760
0.985
2.602
0.993
B-14
a/t=0.8
1.028
0.995
0.962
0.929
0.897
0.866
0.835
0.805
0.776
0.748
0.721
0.695
0.671
0.651
0.636
0.637
0.686
0.729
0.880
1.128
1.722
2.466
a'/a
0
0.058
0.117
0.175
0.233
0.292
0.350
0.409
0.467
0.526
0.584
0.643
0.701
0.759
0.818
0.876
0.935
0.951
0.967
0.979
0.987
0.993
a/t=0.9
1.129
1.099
1.070
1.041
1.013
0.986
0.959
0.932
0.907
0.882
0.857
0.832
0.809
0.786
0.767
0.757
0.786
0.820
0.892
1.115
1.735
2.478
Appendix C – Listings of KIc And KIa Extended Databases
Table C1 – Static Initiation Toughness KIc Extended Database
Table C2 - Crack Arrest Toughness KIa ORNL 99/27 Database
Table C3. Crack Arrest Toughness KIa Extended KIa Database – Large Specimen Data
C-1
Table C1. Static Initiation Toughness KIc Extended Database
Material
HSST 01
subarc
weldment
A533B Class 1
subarc
weldment
HSST 01
HSST 03
A533B Class 1
Reference Source Specimen Type Orientation
T
ID
No.
(°F)
Shabbits
1T-C(T)
1
-200
(1969)
1T-C(T)
1
-175
4T-C(T)
4
-150
4T-C(T)
4
-125
4T-C(T)
4
-100
4T-C(T)
4
-75
4T-C(T)
4
-75
6T-C(T)
6
-50
Shabbits
1T-C(T)
1
-200
(1969)
1T-C(T)
1
-200
1T-C(T)
1
-320
1T-C(T)
1
-320
4T-C(T)
4
-100
4T-C(T)
4
-50
4T-C(T)
4
-25
8T-C(T)
8
0
Mager (1969)
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
Mager (1969)
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
Mager (1969)
1X-WOL
1
RW
-320
1T-WOL
1
RW
-320
1X-WOL
1
RW
-250
1X-WOL
1
RW
-250
1X-WOL
1
RW
-250
1T-WOL
1
RW
-250
1T-WOL
1
RW
-250
1X-WOL
1
RW
-200
C-2
RTNDT T - RTNDT
KIc
(°F)
(°F)
(ksi√in)
0
-200
46.6
0
-175
55.8
0
-150
56.1
0
-125
61.1
0
-100
96.0
0
-75
90.3
0
-75
93.1
0
-50
72.6
0
-200
35.1
0
-200
45.2
0
-320
25.9
0
-320
23.7
0
-100
55.2
0
-50
71.6
0
-25
105.9
0
0
113.1
20
-170
43.9
20
-170
39.4
20
-170
31.3
20
-170
47.3
20
-170
50.4
20
-170
41.2
20
-170
54.0
20
-170
50.9
20
-170
35.5
20
-170
33.2
20
-170
37.2
20
-170
37.1
20
-170
37.1
20
-170
34.7
20
-170
35.0
20
-170
32.6
20
-170
29.4
20
-170
44.0
20
-170
31.4
20
-170
39.3
20
-170
31.3
20
-170
33.0
20
-170
38.1
20
-170
31.1
20
-170
44.9
20
-170
39.4
65
-385
31.6
65
-385
32.5
65
-315
40.9
65
-315
37.1
65
-315
44.0
65
-315
40.8
65
-315
31.2
65
-265
30.6
Material
HSST 02
A533B Class 1
weld
Reference Source Specimen Type Orientation
T
ID
No.
(°F)
1X-WOL
1
RW
-200
1T-WOL
1
RW
-200
1T-WOL
1
RW
-200
2T-WOL
2
RW
-150
2T-WOL
2
RW
-150
Mager (1969)
1X-WOL
1
RW
-200
1X-WOL
1
RW
-200
1X-WOL
1
RW
-200
1T-WOL
1
RW
-200
1T-WOL
1
RW
-200
1T-WOL
1
RW
-175
1X-WOL
1
RW
-150
1X-WOL
1
RW
-150
1X-WOL
1
RW
-150
1X-WOL
1
RW
-150
1X-WOL
1
RW
-125
1T-WOL
1
RW
-125
1T-WOL
1
RW
-125
1T-WOL
1
RW
-125
2T-WOL
2
RW
-100
2T-WOL
2
RW
-100
2T-WOL
2
RW
-100
2T-WOL
2
RW
-100
2T-WOL
2
RW
-50
2T-WOL
2
RW
-50
2T-WOL
2
RW
-50
2T-WOL
2
RW
-50
1X-WOL
1
RW
-250
1X-WOL
1
RW
-200
1X-WOL
1
RW
-200
1X-WOL
1
RW
-200
1X-WOL
1
RW
-200
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-100
1T-C(T)
1
RW
-100
1T-C(T)
1
RW
-100
1T-C(T)
1
RW
-75
1T-C(T)
1
RW
-75
1T-C(T)
1
RW
-100
1T-C(T)
1
RW
-100
2T-WOL
2
RW
-50
2T-WOL
2
RW
0
2T-WOL
2
RW
0
2T-WOL
2
RW
0
Mager (1969)
1X-WOL
1
-320
1X-WOL
1
-320
1X-WOL
1
-250
1X-WOL
1
-250
1T-WOL
1
-250
2T-WOL
2
-250
1T-WOL
1
-225
1T-WOL
1
-225
C-3
RTNDT T - RTNDT
KIc
(°F)
(°F)
(ksi√in)
65
-265
29.0
65
-265
35.6
65
-265
42.8
65
-215
46.9
65
-215
66.9
0
-200
30.5
0
-200
37.5
0
-200
41.0
0
-200
31.2
0
-200
30.8
0
-175
43.5
0
-150
29.7
0
-150
31.5
0
-150
41.2
0
-150
30.5
0
-125
39.1
0
-125
48.3
0
-125
43.4
0
-125
38.1
0
-100
51.4
0
-100
59.0
0
-100
56.2
0
-100
50.2
0
-50
65.1
0
-50
65.0
0
-50
67.5
0
-50
65.0
0
-250
37.3
0
-200
44.0
0
-200
34.6
0
-200
39.9
0
-200
38.5
0
-150
42.1
0
-150
37.7
0
-150
40.7
0
-100
42.2
0
-100
48.5
0
-100
48.5
0
-75
50.3
0
-75
46.6
0
-100
54.8
0
-100
54.4
0
-50
56.7
0
0
66.4
0
0
93.7
0
0
83.4
-45
-275
29.7
-45
-275
27.2
-45
-205
37.6
-45
-205
37.8
-45
-205
43.6
-45
-205
55.6
-45
-180
40.1
-45
-180
52.8
Material
A533B Class 1
weld-HAZ
A508 Class 2
European
Forging
“ring forging”
HSST 02
A508 Class 2
Reference Source Specimen Type Orientation
T
ID
No.
(°F)
2T-WOL
2
-225
2T-WOL
2
-200
Mager (1969)
1X-WOL
1
-320
1X-WOL
1
-250
1X-WOL
1
-250
1T-WOL
1
-250
1T-WOL
1
-250
2T-WOL
2
-200
Mager (1969)
1X-WOL
1
-320
1X-WOL
1
-320
1T-WOL
1
-320
1X-WOL
1
-250
1X-WOL
1
-250
1X-WOL
1
-250
1T-WOL
1
-250
1T-WOL
1
-200
2T-WOL
2
-200
2T-WOL
2
-150
2T-WOL
2
-125
2T-WOL
2
-100
Shabbits
6T-C(T)
6
RW
25
(1969)
6T-C(T)
6
RW
25
6T-C(T)
6
RW
25
6T-C(T)
6
RW
0
6T-C(T)
6
RW
0
11T-C(T) 11
RW
50
10T-C(T) 10
RW
50
10T-C(T) 10
RW
50
4T-C(T)
4
RW
0
4T-C(T)
4
RW
-25
4T-C(T)
4
RW
-25
4T-C(T)
4
RW
-25
10T-C(T) 10
RW
0
10T-C(T) 10
RW
25
1T-C(T)
1
RW
-250
1T-C(T)
1
RW
-200
1T-C(T)
1
RW
-200
1T-C(T)
1
RW
-200
1T-C(T)
1
RW
-200
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-150
1T-C(T)
1
RW
-100
1T-C(T)
1
RW
-100
1T-C(T)
1
RW
-100
2T-C(T)
2
RW
-100
2T-C(T)
2
RW
-50
2T-C(T)
2
RW
-50
unpublished
2T-C(T)
2
-150
outside of
2T-C(T)
2
-150
EPRI NP-719-SR 2T-C(T)
2
-125
2T-C(T)
2
-125
2T-C(T)
2
-125
4T-C(T)
4
-25
C-4
RTNDT T - RTNDT
KIc
(°F)
(°F)
(ksi√in)
-45
-180
66.2
-45
-155
70.7
0
-320
30.3
0
-250
35.2
0
-250
40.4
0
-250
30.5
0
-250
44.2
0
-200
71.2
50
-370
39.6
50
-370
27.5
50
-370
47.5
50
-300
43.2
50
-300
47.9
50
-300
41.6
50
-300
51.3
50
-250
55.0
50
-250
43.3
50
-200
57.2
50
-175
56.2
50
-150
56.0
0
25
98.9
0
25
74.5
0
25
90.5
0
0
73.9
0
0
66.9
0
50
148.6
0
50
137.3
0
50
139.0
0
0
87.2
0
-25
61.0
0
-25
58.7
0
-25
45.9
0
0
87.5
0
25
110.3
0
-250
37.3
0
-200
44.4
0
-200
34.6
0
-200
39.9
0
-200
34.8
0
-150
44.1
0
-150
37.4
0
-150
41.8
0
-100
48.3
0
-100
48.3
0
-100
41.9
0
-100
49.7
0
-50
64.6
0
-50
64.7
51
-201
52.2
51
-201
45.5
51
-176
46.0
51
-176
64.3
51
-176
50.0
51
-76
45.0
Material
A508 Class 2
HSSI Weld
72W
HSSI
73W
HSST Plate 13
Reference Source Specimen Type Orientation
T
RTNDT T - RTNDT
KIc
ID
No.
(°F)
(°F)
(°F)
(ksi√in)
6T-C(T)
6
0
51
-51
107.0
2T-C(T)
2
-125
51
-176
45.6
2T-C(T)
2
-125
51
-176
68.0
unpublished
2T-C(T)
2
-75
65
-140
52.0
outside of
2T-C(T)
2
-75
65
-140
64.6
EPRI NP-719-SR 2T-C(T)
2
-75
65
-140
56.6
2T-C(T)
2
-25
65
-90
64.7
2T-C(T)
2
-25
65
-90
62.4
8T-C(T)
8
35
65
-30
81.0
2T-C(T)
2
-125
65
-190
47.2
2T-C(T)
2
-125
65
-190
40.9
2T-C(T)
2
-125
65
-190
42.5
2T-C(T)
2
-125
65
-190
42.5
NUREG/CR1T-C(T)
1
T-L
-238
-9.4
-228.6
35.09
5913
1T-C(T)
1
T-L
-238
-9.4
-228.6
35.45
1T-C(T)
1
T-L
-238
-9.4
-228.6
37.82
1T-C(T)
1
T-L
-149.8 -9.4
-140.4
42.55
1T-C(T)
1
T-L
-112
-9.4
-102.6
45.09
2T-C(T)
2
T-L
-112
-9.4
-102.6
58.73
2T-C(T)
2
T-L
-112
-9.4
-102.6
67.64
2T-C(T)
2
T-L
-58
-9.4
-48.6
63.27
4T-C(T)
4
T-L
-58
-9.4
-48.6
73.82
4T-C(T)
4
T-L
-58
-9.4
-48.6
90.91
4T-C(T)
4
T-L
-22
-9.4
-12.6
93.45
4T-C(T)
4
T-L
5
-9.4
14.4
74.64
NUREG/CR1T-C(T)
1
T-L
-238 -29.2
-208.8
34.64
5913
1T-C(T)
1
T-L
-238 -29.2
-208.8
37.82
1T-C(T)
1
T-L
-238 -29.2
-208.8
38.18
1T-C(T)
1
T-L
-238 -29.2
-208.8
39.45
2T-C(T)
2
T-L
-112 -29.2
-82.8
58.18
2T-C(T)
2
T-L
-112 -29.2
-82.8
60.64
2T-C(T)
2
T-L
-112 -29.2
-82.8
65.55
2T-C(T)
2
T-L
-58
-29.2
-28.8
66.09
4T-C(T)
4
T-L
-58
-29.2
-28.8
75.55
4T-C(T)
4
T-L
-58
-29.2
-28.8
76.45
NUREG/CR1T-C(T)
1
L-T
-103
-9.4
-93.6
32.64
5788 (A533B
2T-C(T)
2
L-T
-103
-9.4
-93.6
55.82
Plate 13A)
4T-C(T)
4
L-T
-103
-9.4
-93.6
53.73
4T-C(T)
4
L-T
-103
-9.4
-93.6
62.09
4T-C(T)
4
L-T
-103
-9.4
-93.6
70.82
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
25.36
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
26.18
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
29.27
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
29.45
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
30.18
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
31.00
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
32.82
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
33.82
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
36.00
½T-C(T) 0.5
L-T
-238
-9.4
-228.6
36.36
1T-C(T)
1
L-T
-238
-9.4
-228.6
32.09
1T-C(T)
1
L-T
-238
-9.4
-228.6
33.73
1T-C(T)
1
L-T
-238
-9.4
-228.6
34.27
1T-C(T)
1
L-T
-238
-9.4
-228.6
34.91
C-5
Material
A508 Class 3
Midland Nozzle
Course Weld
Midland Beltline
Plate 02 4th Irr.
Series
Reference Source Specimen Type Orientation
T
ID
No.
(°F)
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
1T-C(T)
1
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
2T-C(T)
2
L-T
-238
Iwadate, et al.
Bx2B
1
NA
-238
ASTM STP
Bx2B
1
NA
-238
803
Bx2B
1
NA
-238
Bx2B
4
NA
-166
Bx2B
4
NA
-76
Bx2B
3
NA
-4
NUREG/CR1T-C(T)
1
-58
6249
1T-C(T)
1
-148
1T-C(T)
1
-148
1T-C(T)
1
-148
1T-C(T)
1
-148
1T-C(T)
1
-148
NUREG/CR1T-C(T)
1
-148
6249
1T-C(T)
1
-148
NUREG/CR1T-C(T)
1
T-L
-148
4880, 1988
1T-C(T)
1
T-L
-139
Plate 02
1T-C(T)
1
T-L
-139
(68-71W)
1T-C(T)
1
T-L
-139
C-6
RTNDT T - RTNDT
KIc
(°F)
(°F)
(ksi√in)
-9.4
-228.6
35.09
-9.4
-228.6
36.00
-9.4
-228.6
37.45
-9.4
-228.6
37.45
-9.4
-228.6
39.55
-9.4
-228.6
39.73
-9.4
-228.6
40.36
-9.4
-228.6
42.36
-9.4
-228.6
43.73
-9.4
-228.6
46.45
-9.4
-228.6
49.55
-9.4
-228.6
49.64
-9.4
-228.6
30.09
-9.4
-228.6
33.00
-9.4
-228.6
36.55
-9.4
-228.6
37.00
-9.4
-228.6
39.36
-9.4
-228.6
39.91
-9.4
-228.6
40.91
-9.4
-228.6
41.45
-9.4
-228.6
42.18
-9.4
-228.6
46.45
-9.4
-228.6
48.64
-9.4
-228.6
53.18
-13
-225
37.29
-13
-225
39.89
-13
-225
44.22
-13
-153
43.36
-13
-63
63.30
-13
9
69.37
52
-110
49.81
52
-200
45.63
52
-200
44.63
52
-200
42.81
52
-200
33.45
52
-200
32.36
23
-171
36.45
23
-171
34.91
0
-148
38.09
0
-139
33.45
0
-139
39.27
0
-139
40.09
References for Table C1
EPRI Special Report, 1978, Flaw Evaluation Procedures: ASME Section XI, EPRI NP-719-SR,
Electric Power Research Institute, Palo Alto, CA.
W. O. Shabbits, W. H. Pryle, and E. T. Wessel, Heavy Section Fracture Toughness Properties of
A533, Grade B, Class-1 Steel Plate and Submerged Arc Weldments, HSST Technical Report
6, WCAP-7414, December 1969.
T. R. Mager, F. O. Thomas, and W. S. Hazelton, Evaluation by Linear Elastic Fracture
Mechanics of Radiation Damage to Pressure Vessel Steels, HSST Technical Report 5,
WCAP-7328, Revised, October 1969.
T. R. Mager, Fracture Toughness Characterization Study of A533, Grade B, Class-1 Steel, HSST
Technical Report 10, WCAP-7578, October 1970.
R. K. Nanstad, F. M. Haggag, and D. E. McCabe, Irradiation Effects on Fracture Toughness of
Two High-Copper Submerged-Arc Welds, HSSI Series 5, USNRC Report NUREG/CR-5913
(ORNL/TM-12156/V1 and V2) Vol. 1 and 2, Oak Ridge National Laboratory, Oak Ridge,
TN, October 1992.
D. E. McCabe., A Comparison of Weibull and βIc Analysis of Transition Range Fracture
Toughness Data, USNRC Report NUREG/CR-5788 (ORNL/TM-11959), Oak Ridge
National Laboratory, Oak Ridge, TN, January 1992.
T. Iawadate, Y. Tanaka, S. Ono, and J. Watanabe, “An Analysis of Elastic-Plastic Fracture
Toughness Behavior for JIc Measurements in the Transition Region,” Elastic-Plastic
Fracture: Second Symposium, Vol. II-Fracture Resistance Curves and Engineering
Applications, ASTM STP 803, (1983) II531-II561.
D. E. McCabe, R. K. Nanstad, S. K. Iskander, R. L. Swain, Unirradiated Material Properties of
Midland Weld WF-70, USNRC Report NUREG/CR-6249 (ORNL/TM-12777), Oak Ridge
National Laboratory, Oak Ridge, TN, October 1994.
J. J. McGowan, R. K. Nanstad, and K. R. Thoms, Characterization of Irradiated CurrentPractice Welds and A533 Grade B Class 1 Plate for Nuclear Pressure Vessel Service,
USNRC Report NUREG/CR-4880 (ORNL-6484/V1 and V2), Oak Ridge National
Laboratory, Oak Ridge, TN, July 1988.
C-7
Table C2. Crack Arrest Toughness KIa ORNL 99/27 Database
Material
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
HSST-02
72W
Reference
Source
EPRI NP
719-SR
Ripling (1971)
NUREG/CR-5584
Specimen
ID
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
Size
No.
1.4
1
2
2
1
1
1
1
1
1.3
1.3
1.3
1.6
1.6
2
2
2
2
3
3
2
1.4
1.6
2
1.4
2
3
1
1.6
2
2
2
2
2
2
3
3
3
2
2
1.8
2
2
3
2
2
2
1.1
2
1.1
Orientation
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
L-T
Crack
C-8
T
(°F)
-150
-70
-70
-70
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
22
35
35
35
50
50
50
75
75
75
75
75
75
80
83
83
83
83
96
102
105
105
105
105
107
110
110
112
115
121
-77.8
RTNDT
(°F)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-10
T-RTNDT
(°F)
-150
-70
-70
-70
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
22
35
35
35
50
50
50
75
75
75
75
75
75
80
83
83
83
83
96
102
105
105
105
105
107
110
110
112
115
121
-68
KIa
(ksi√in)
28.0
43.0
48.0
43.0
68.0
58.0
48.0
57.0
62.0
58.0
60.0
65.0
60.0
58.0
53.0
58.0
70.0
57.0
57.0
61.0
68.0
59.0
84.0
62.0
92.0
73.0
75.0
94.0
107.0
77.0
81.0
91.0
102.3
109.0
87.0
94.0
107.0
111.0
111.0
117.0
118.0
103.0
107.0
130.0
87.0
88.0
88.0
112.0
111.0
116.0
60.1
Material
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
72W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
73W
Reference
Source
NUREG/CR-5584
Specimen
ID
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
CCA
Size
No.
Orientation
runs
in
welding
direction
Crack
runs
in
welding
direction
C-9
T
(°F)
-76
-76
-74.2
-52.6
-52.6
-49
-49
-49
-49
-25.6
-22
-22
-22
-22
-22
-22
-22
3.2
5
5
5
5
6.8
28.4
30.2
32
32
32
33.8
39.2
41
-77.8
-76
-74.2
-49
-49
-49
-49
-47.2
-25.6
-23.8
-22
-22
-22
-20.2
-20.2
3.2
5
5
5
10.4
23
41
41
RTNDT
(°F)
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
-30
T-RTNDT
(°F)
-66
-66
-64.2
-42.6
-42.6
-39
-39
-39
-39
-15.6
-12
-12
-12
-12
-12
-12
-12
13.2
15
15
15
15
16.8
38.4
40.2
42
42
42
43.8
49.2
51
-47.8
-46
-44.2
-19
-19
-19
-19
-17.2
4.4
6.2
8
8
8
9.8
9.8
33.2
35
35
35
40.4
53
71
71
KIa
(ksi√in)
48.2
69.2
51.9
61.0
64.6
66.4
67.3
69.2
83.7
83.7
54.6
55.5
77.4
82.8
89.2
94.6
97.4
88.3
85.5
85.5
86.5
93.7
82.8
93.7
113.8
84.6
97.4
103.7
98.3
113.8
104.7
62.8
52.8
65.5
47.3
66.4
68.3
77.4
64.6
77.4
68.3
61.0
72.8
91.0
70.1
81.0
100.1
106.5
111.9
112.8
102.3
91.9
97.4
101.9
Material
73W
73W
73W
MW15JC
MW15JBr
MW15JEr1
MW15JF
Reference
Source
NUREG/CR-6621
Specimen
ID
CCA
CCA
CCA
CCA
CCA
CCA
CCA
Size
No.
Orientation
Crack
runs
in welding
direction
T
(°F)
41
41
59
-4
14
32
50
RTNDT
(°F)
-30
-30
-30
32.2
32.2
32.2
32.2
T-RTNDT
(°F)
71
71
89
-36.2
-18.2
-0.2
17.8
KIa
(ksi√in)
102.8
108.3
120.1
63.7
79.0
97.1
119.7
References for Table C2
EPRI Special Report, 1978, Flaw Evaluation Procedures: ASME Section XI, EPRI NP-719-SR,
Electric Power Research Institute, Palo Alto, CA.
E. J. Ripling and P. B. Crosley, “Strain Rate and Crack Arrest Studies,” HSST 5th Annual
Information Meeting, Paper No. 9, 1971.
S. K. Iskander, W. R. Corwin, R. K. Nanstad, Results of Crack-Arrest Tests on Two Irradiated
High-Copper Welds, USNRC Report NUREG/CR-5584 (ORNL/TM-11575), Oak Ridge
National Laboratory, Oak Ridge, TN, December 1990.
S. K. Iskander, C. A. Baldwin, D. W. Heatherly, D. E. McCabe, I. Remec, and R. L. Swain,
Detailed Results of Testing Unirradiated and Irradiated Crack-Arrest Toughness Specimens
from the Low Upper-Shelf Energy, High Copper Weld, WF-70, NUREG/CR-6621
(ORNL/TM-13764) under preparation.
S. K. Iskander, R. K. Nanstad, D. E. McCabe, and R. L. Swain, “Effects of Irradiation on CrackArrest Toughness of a Low Upper-Shelf Energy, High-Copper Weld,” Effects of Radiation
on Materials: 19th International Symposium, ASTM STP 1366, M. L. Hamilton, A. S.
Kumar, S. T. Rosinski, and M. L. Grossbeck, eds., American Society for Testing and
Materials, 2000.
C-10
Table C3. Crack Arrest Toughness KIa Extended KIa Database – Large Specimen Data
Material
Test No.
WP 1.2A
WP 1.2B
WP 1.3
WP 1.4B
WP 1.5A
WP 1.5B
WP 1.6A
WP 1.6B
WP 1.7A
WP 1.7B
WP 1.8A
WP 1.8B
WP 1.8C
WP CE-1
WP CE-2A
WP CE-2B
WP CE-2C
SP 1.3
WP 2.1A
WP 2.1B
WP 2.1D
WP 2.1E
WP 2.1F
WP 2.1H
WP 2.1I
WP 2.1J
WP 2.2A
WP 2.2B
WP 2.2C
WP 2.2D
WP 2.2E
WP 2.2F
WP 2.2G
WP 2.3A
WP 2.3B
WP 2.3D
WP 2.3F
WP 2.4B
WP 2.4C
WP 2.4D
WP 2.4E
WP 2.4F
WP 2.4G
WP 2.4H
WP 2.5B
WP 2.5C
WP 2.5D
WP 2.5E
WP 2.5F
WP 2.6A
WP 2.6B
WP 2.6C
WP 2.6D
WP 2.6F
WP 2.6G
WP 2.6H
PTSE 1B
PTSE 1C
PTSE 2A
PTSE 2B
PTSE 2C
TSE 4
TSE 5-1
TSE 5-2
TSE 5-3
TSE 5A-1
TSE 5A-2
TSE 5A-3
TSE 5A-4
TSE 6-1
TSE 6-2
Reference
Source
NUREG/CR-4930
NUREG/CR-5330
Smirt 10 Vol F, p 37
NUREG/CR-5451
NUREG/CR-4106
NUREG/CR-4888
NUREG/CR-4249
T
RT NDT
T-RT NDT
K Ia
(°F)
-9.4
-9.4
-9.4
-9.4
-9.4
-9.4
-9.4
-9.4
-9.4
-9.4
-9.4
-9.4
-9.4
-31.0
-31.0
-31.0
-31.0
-9.4
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
140.0
196.3
196.3
167.0
167.0
167.0
167.0
152.6
152.6
152.6
50.0
50.0
50.0
50.0
152.6
152.6
(°F)
143.6
197.6
129.2
140.0
132.8
161.6
129.2
176.0
141.8
190.4
104.0
131.0
174.2
96.8
107.6
127.4
140.0
111.2
176.0
204.8
221.0
233.6
257.0
275.0
293.0
305.6
248.0
264.2
271.4
282.2
287.6
302.0
323.6
206.6
222.8
231.8
258.8
186.8
215.6
224.6
249.8
260.6
278.6
300.2
219.2
255.2
275.0
291.2
309.2
219.2
239.0
246.2
257.0
271.4
282.2
312.8
326.3
354.2
267.1
296.2
325.2
267.8
96.8
179.6
192.2
71.6
100.4
123.8
152.6
89.6
145.4
(°F)
153.0
207.0
138.6
149.4
142.2
171.0
138.6
185.4
151.2
199.8
113.4
140.4
183.6
127.8
138.6
158.4
171.0
120.6
36.0
64.8
81.0
93.6
117.0
135.0
153.0
165.6
108.0
124.2
131.4
142.2
147.6
162.0
183.6
66.6
82.8
91.8
118.8
46.8
75.6
84.6
109.8
120.6
138.6
160.2
79.2
115.2
135.0
151.2
169.2
79.2
99.0
106.2
117.0
131.4
142.2
172.8
130.0
157.9
100.1
129.2
158.2
100.8
-55.8
27.0
39.6
21.6
50.4
73.8
102.6
-63.0
-7.2
(ksi-in )
C-11
1/2
385.81
623.29
213.83
352.14
210.19
463.15
250.23
361.24
290.26
505.00
313.92
440.40
512.28
154.69
198.36
322.11
524.11
160.15
96.45
139.22
143.77
154.69
182.89
266.61
337.58
369.43
182.89
235.67
255.69
252.05
345.77
331.21
405.82
131.03
211.10
232.03
234.76
124.66
171.06
255.69
226.57
279.34
346.68
361.24
155.60
172.88
243.86
278.43
333.03
185.62
235.67
260.24
318.47
298.45
373.98
375.80
182.80
271.97
237.85
329.03
381.53
115.56
78.25
94.63
83.71
69.15
78.25
97.36
118.29
57.32
95.54
References for Table C3
D. J. Naus, et al., High-Temperature Crack Arrest Behavior in 152-mm-Thick SEN Wide Plates of
Quenched and Tempered A533 Grade B Class 1 Steel, NUREG/CR-5330 (ORNL-11083),
Oak Ridge National Laboratory, Oak Ridge, TN, April 1989.
D. J. Naus, et al., Crack-Arrest Behavior in SEN Wide Plates of Low-Upper-Shelf Base Metal
Tested Under Nonisothermal Conditions: WP-2 Series, NUREG/CR-5451 (ORNL-6584),
Oak Ridge National Laboratory, Oak Ridge, TN, April 1989.
R. H. Bryan, et al., Pressurized-Thermal Shock Test of 6-Inch-Thick Pressure Vessel, PTSE-1:
Investigations of Warm Prestressing and Upper-Shelf Arrest, NUREG/CR-4106
(ORNL-6135), Oak Ridge National Laboratory, Oak Ridge, TN, April 1985.
R. H. Bryan, et al., Pressurized Thermal Shock Test of 6-Inch-Thick Pressure Vessel PTSE-2:
Investigation of Low Tearing Resistance and Warm Prestressing, NUREG/CR-4888
(ORNL-6377), Oak Ridge National Laboratory, Oak Ridge, TN, December 1987.
R. D. Cheverton, D. G. Ball, S. E. Bolt, S. K. Iskander, and R. K. Nanstad, Pressure Vessel
Fracture Studies Pertaining to the PWR Thermal-Shock Issue: Experiments TSE-5, TSE-5A,
and TSE-6, NUREG/CR-4249 (ORNL-6163), Oak Ridge National Laboratory, Oak Ridge,
TN, June 1985.
C-12
Appendix D – Summary of RVID2 Data for Use in FAVOR Calculations
Product Form
Heat
RTNDT(u) [oF]
σflow(u)
Beltline
[ksi]
Beaver Valley 1, (Designer: Westinghouse, Manufacturer: CE)
Coolant Temperature = 547°F, Vessel Thickness = 7-7/8 in.
C4381-1
INTERMEDIATE SHELL B6607-1
C4381-2
INTERMEDIATE SHELL B6607-2
PLATE
C6293-2
LOWER SHELL B7203-2
C6317-1
LOWER SHELL B6903-1
305414
LOWER SHELL AXIAL WELD 20-714
LINDE 1092 WELD
305424
INTER SHELL AXIAL WELD 19-714
LINDE 0091 WELD 90136
CIRC WELD 11-714
Calvert Cliffs 1, (Designer and Manufacturer: CE)
Coolant Temperature = 545°F, Vessel Thickness = 8 5/8-in.
B-8489-1
LOWER SHELL D-7207-3
B-8489-2
LOWER SHELL D-7207-2
C-4351-2
INTERMEDIATE SHELL D-7206-1
PLATE
C-4420-1
LOWER SHELL D-7207-1
C-4441-1
INTERMEDIATE SHELL D-7206-3
C-4441-2
INTERMEDIATE SHELL D-7206-2
20291/12008 INTERMEDIATE SHELL AXIAL WELD 2-203
LINDE 1092 WELD
21935
LOWER SHELL AXIAL WELD 3-203A/C
LINDE 0091 WELD 33A277
INT. TO LOWER SHELL CIRC. WELD 9-203
Oconee 1, (Designer and Manufacturer: B&W)
Coolant Temperature = 556°F, Vessel Thickness = 8.44-in.
AHR54
FORGING
LOWER NOZZLE BELT
(ZV2861)
C2197-2
INTERMEDIATE SHELL
C2800-1
LOWER SHELL
PLATE
C2800-2
LOWER SHELL
C3265-1
UPPER SHELL
C3278-1
UPPER SHELL
LINDE 80 WELD
1P0962
INTERMEDIATE SHELL AXIAL WELDS SA-1073
299L44
INT./UPPER SHL CIRC WELD (OUTSIDE 39%) WF-25
61782
NOZZLE BELT/INT. SHELL CIRC WELD SA-1135
71249
INT./UPPER SHL CIRC WELD (INSIDE 61%) SA-1229
72445
UPPER/LOWER SHELL CIRC WELD SA-1585
D-1
RTNDT(u) Method
Composition(2)
RTNDT(u)
Value
σ(u)
Value
Cu
Ni
P
USE0
(ft-lbf)
83.8
84.3
78.8
72.7
75.3
79.9
76.1
MTEB 5-2
MTEB 5-2
MTEB 5-2
MTEB 5-2
Generic
Generic
Generic
43
73
20
27
-56
-56
-56
0
0
0
0
17
17
17
0.14
0.14
0.14
0.2
0.337
0.273
0.269
0.62
0.62
0.57
0.54
0.609
0.629
0.07
0.015
0.015
0.015
0.01
0.012
0.013
0.013
90
84
84
80
98
112
144
78.8
80.3
74.7
78.0
78.5
82.6
78.8
78.6
78.6
MTEB 5-2
MTEB 5-2
MTEB 5-2
MTEB 5-2
ASME NB-2331
ASME NB-2331
ASME NB-2331
Generic
ASME NB-2331
-20
-10
20
10
10
-30
-50
-56
-80
0
0
0
0
0
0
0
17
0
0.11
0.11
0.11
0.13
0.12
0.12
0.22
0.18
0.24
0.53
0.56
0.55
0.54
0.64
0.64
0.83
0.72
0.16
0.008
0.009
0.011
0.01
0.011
0.011
0.01
0.015
0.014
81
90
90
77
112
81
110
109
160
B&W Generic
3
31
0.16
0.65
0.006
109
B&W Generic
B&W Generic
B&W Generic
B&W Generic
B&W Generic
B&W Generic
B&W Generic
B&W Generic
ASME NB-2331
B&W Generic
1
1
1
1
1
-5
-7
-5
10
-5
26.9
26.9
26.9
26.9
26.9
19.7
20.6
19.7
0
19.7
0.15
0.11
0.11
0.1
0.12
0.21
0.34
0.23
0.23
0.22
0.5
0.63
0.63
0.5
0.6
0.64
0.68
0.52
0.59
0.54
0.008
0.012
0.012
0.015
0.01
0.025
(3)
0.011
0.021
0.016
81
81
119
108
81
70
81
80
67
65
(4)
(4)
(4)
69.9
75.8
(4)
79.4
(4)
(4)
76.4
(4)
Product Form
Heat
σflow(u)
Beltline
Composition(2)
RTNDT(u) Method
RTNDT(u)
Value
σ(u)
Value
Cu
Ni
P
USE0
(ft-lbf)
75.5
(4)
75.5
B&W Generic
B&W Generic
B&W Generic
-5
-5
-5
19.7
19.7
19.7
0.19
0.19
0.19
0.57
0.57
0.57
0.017
0.017
0.017
70
70
70
(4)
(4)
(4)
74.7
(4)
(4)
76.9
76.1
72.9
72.9
MTEB 5-2
MTEB 5-2
ASME NB-2331
ASME NB-2331
ASME NB-2331
MTEB 5-2
Generic
Generic
Generic
Generic
-30
-25
-5
-5
0
-30
-56
-56
-56
-56
0
0
0
0
0
0
17
17
17
17
0.24
0.12
0.24
0.24
0.19
0.19
0.203
0.192
0.213
0.213
0.52
0.55
0.5
0.51
0.48
0.5
1.018
0.98
1.01
1.01
0.01
0.01
0.011
0.009
0.016
0.015
0.013
(3)
0.019
0.019
87
73
102
102
72
76
98
111
118
118
[ksi]
8T1762
LOWER SHELL AXIAL WELDS SA-1430
8T1762
UPPER SHELL AXIAL WELDS SA-1493
8T1762
LOWER SHELL AXIAL WELDS SA-1426
Palisades, (Designer and Manufacturer: CE)
Coolant Temperature = 532°F, Vessel Thickness = 8½ in.
A-0313
D-3803-2
B-5294
D-3804-3
C-1279
D-3803-3
PLATE
C-1279
D-3803-1
C-1308A
D-3804-1
C-1308B
D-3804-2
LINDE 0124 WELD 27204
CIRC. WELD 9-112
34B009
LOWER SHELL AXIAL WELD 3-112A/C
LINDE 1092 WELD W5214
LOWER SHELL AXIAL WELDS 3-112A/C
W5214
INTERMEDIATE SHELL AXIAL WELDS 2-112 A/C
RTNDT(u) [oF]
Notes:
(1) Information taken directly from the July 2000 release of the NRCs Reactor Vessel Integrity [RVID2] database.
(2) These composition values are as reported in RVID2. In FAVOR calculations these values should be treated as the central tendency of
the Cu, Ni, and P distributions.
(3) No values of phosphorus are recorded in RVID2 for these heats. A generic value of 0.012 should be used, which is the mean of 826
phosphorus values taken from the surveillance database used to calibrate the embrittlement trend curve [Kirk].
(4) No values strength measurements are available in PREP4 for these heats [PREP]. A value of 77 ksi should be used, which is the mean
of other flow strength values reported in this Appendix.
References:
RVID2
U.S. Nuclear Regulatory Commission Reactor Vessel Integrity Database, Version 2.1.1, July 6, 2000.
PREP
PREP4: Power Reactor Embrittlement Program, Version 1.0," EPRI, Palo Alto, CA: 1996. SW-106276
D-2
Kirk
M. T. Kirk, C. S. Santos, E.D. Eason, J.E. Wright, and G. R. Odette, “Updated Embrittlement Trend Curve for Reactor Pressure
Vessel Steels,” Paper No. G01-5, Transactions of the 17th International Conference on Structural Mechanics in Reactor
Technology (SMiRT 17), Prague, Czech Republic, August 17-22, 2003.
D-3
Appendix E – Statistical Point-Estimation Techniques for Weibull Distributions
The three parameters for the Weibull distributions of RTNDT − T0 and ΔRTLB 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
( )
c c −1
y exp − y c , for
b
( y = (ΔRT − a ) / b, ΔRT > a, b, c > 0)
w(ΔRT a, b, c) =
(E1)
and the corresponding likelihood function is the joint density (see Ref.[E1]) (given the location
parameter, a)
L(b, c | ΔRT , a) =
c ⎛ ΔRT(i ) − a ⎞
∏b⎜ b ⎟
⎠
i =1 ⎝
N
c −1
⎡ ⎛ ΔRT − a ⎞c ⎤
(i )
exp ⎢ − ⎜
⎟ ⎥
b
⎢ ⎝
⎠ ⎥⎦
⎣
(E2)
The maximum likelihood (ML) 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 ML estimators are, therefore,
calculated by finding the stationary point of Eq. (E2). Upon taking the logarithm of Eq. (E2), 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
∂ (ln( L (c′))
=
∂c′
∑ (ΔRT(i ) − a)c ln(ΔRT(i ) − a)
′
i =1
N
∑ (ΔRT(i ) − a)
−
c′
1
N
1
∑ ln(ΔRT(i ) − a) − c′ = 0
N
(E3)
i =1
i =1
Upon obtaining a solution for c′ , the ML estimator for the scale parameter, b′ , follows directly
from
E-1
1
c′ ⎤ c′
⎡N
Δ
RT
−
a
(
)
i
∂ (ln( L))
⎢
⎥
= b′ − ⎢
⎥ =0
N
∂b′
=
1
i
⎢⎣
⎥⎦
∑
(
)
(E4)
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* = m2 /[Γ(1 + 2 / c) − Γ 2 (1 + 1/ c)]
(E5)
where m2 is the second moment of the sample about the sample mean and Γ is Euler’s gamma
function. The MM estimator for the location parameter, a* , follows from
a* = m1′ − b*Γ(1 + 1/ c)
(E6)
where m1′ is the 1st crude moment of the sample (the sample mean) and the sample moments are
defined by
N
ΔRTNDT (i )
i =1
N
m1′ = ∑
N
( ΔRTNDT (i) − m1′ )2
i =1
N
m2 = ∑
(E7)
From Ref. [B.2], a moment estimator for the shape parameter, c* , also exists
*
c =
where
4.104683 − 1.148513 b1 + 0.44326( b1 )2 − 0.053025( b1 )3
b1 + 1.139547
(E8)
b1 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. [B.2]).
The three parameters for the Weibull distribution of ΔRT were estimated through the following
iterative sequence:
1) For the discrete set (ΔRT(i ) , i = 1, N ) , calculate the sample moments, (m1′ , m2 ) from Eqs. (E7).
2) Select a trial value for the location parameter, atrial where atrial < min(ΔRT(i ) , i = 1, 2,K N ) .
3) Calculate ML estimates for (c′, b′) from Eqs. (E3)-(E4) by letting a = atrial .
E-2
4) Calculate MM estimates for ( a* , b* ) from Eqs. (E5)-(E6) by letting c = c′ as determined in
Step 3.
5) Calculate a relative deviation between the trial atrial and the MM estimate of a* from Step 4
by
δ=
atrial − a*
atrial
(E9)
6) Given ε tolerance , as a pre-selected convergence tolerance, if δ > ε tolerance , then select a new
trial location parameter, atrial , and repeat Steps 3-6 until convergence, defined as δ ≤ ε tolerance .
Upon convergence, there will be two triplets (atrial , b′, c′) and (a* , b* , c′) where in general
atrial ≈ 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 ΔRT .
References
E1. A. Ghosh, “A FORTRAN Program for Fitting Weibull Distribution and Generating
Samples,” Computers & Geosciences 25, (1999) 729-738.
E2. K. O. Bowman and P. T. Williams, Technical Basis for Statistical Models of Extended KIc
and KIa Fracture Toughness Databases for RPV Steels, ORNL/NRC/LTR-99/27, Oak Ridge
National Laboratory, Oak Ridge, TN, February 2000.
E-3
Appendix F – Development of Stochastic Models for ΔRTepistemic and ΔRTarrest
F.1 Stochastic Model for ΔRTepistemic
F.1.1 Initial Weibull Model for ΔRTepistemic
Initially, the epistemic uncertainty in the unirradiated value for RTNDTo was modeled by a
continuous 3-parameter Weibull distribution of the form
⎡ ⎛ ΔRT − a ⎞c ⎤
exp ⎢ − ⎜
( ΔRT > a, (b, c) > 0 )
⎟ ⎥,
b
⎠ ⎥⎦
⎢⎣ ⎝
⎡ ⎛ ΔRT − a ⎞c ⎤
Pr ( X ≤ ΔRT ) = FW (ΔRT | a, b, c) = P = 1 − exp ⎢ − ⎜
⎟ ⎥ , ( ΔRT > a, (b, c) > 0 )
b
⎠ ⎦⎥
⎣⎢ ⎝
c ⎛ ΔRT − a ⎞
fW (ΔRT | a, b, c) = ⎜
⎟
b⎝
b
⎠
c −1
(F1)
where fW is the probability density function (PDF), FW is the cumulative distribution function
(CDF), and a, b, and c are the location, scale, and shape parameters, respectively, of the Weibull
distribution. In FAVOR, the epistemic uncertainty term is sampled using the inverse CDF
1
ΔRT = a + b ⎡⎣ − ln (1 − P ) ⎤⎦ c ; 0 < P < 1
(F2)
where P is randomly sampled from a uniform distribution on the open interval (0,1). The
epistemic uncertainty in RTNDT(u) can then be reduced by
RTLB = RTNDT (u) − ΔRT
(F3)
Using a combination of the Maximum Likelihood and Method of Moments point-estimation
procedures (as described in Appendix E, the following values were determined for the three
Weibull parameters in Eqs. (F1) and (F2):
a = −40.02 °F
b = 124.88 °F
c = 1.96
based on the sample (N = 18) given in Table 8 and repeated in Table F1.
F-1
(F4)
Table F1. ΔRTepistemic Ranked Data with Order-Statistic Estimates of P
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
ΔRT i , (°F)
-19.4
-10.9
-1.7
2.1
33.2
38.4
50.1
54.6
62.3
64.3
81.9
89.4
91.5
97.8
142.2
147.6
162.4
186.2
ln( - ln(1 - P i ))
-3.24970
-2.33364
-1.84080
-1.49387
-1.22093
-0.99223
-0.79239
-0.61229
-0.44594
-0.28898
-0.13796
0.01019
0.15861
0.31100
0.47251
0.65186
0.86782
1.18449
Pi
0.03804
0.09239
0.14674
0.20109
0.25543
0.30978
0.36413
0.41848
0.47283
0.52717
0.58152
0.63587
0.69022
0.74457
0.79891
0.85326
0.90761
0.96196
Sample
mean = 70.67
variance = 3669.77
stdv = 60.58
P i =(i -0.3)/(n +0.4)
From the following asymptotic relations for the mean and variance of a Weibull distribution,
⎛
⎝
1⎞
μ = a + b Γ ⎜1 + ⎟
c
⎡ ⎛
⎣ ⎝
⎠
2⎞
⎛
⎝
1 ⎞⎤
σ 2 = b 2 ⎢Γ ⎜1 + ⎟ − Γ 2 ⎜1 + ⎟ ⎥
c
c
⎠
⎠⎦
,
(F5)
∞
Γ( x) = ∫ t x −1e −t dt
0
the mean and variance for the Weibull model for ΔRTepistemic compared to the corresponding
sample estimators are:
Model
Sample
μ = 70.70 °F m1′ = 70.67 °F
σ 2 = 3473.65 s 2 = 3669.77
σ = 58.94 °F
s = 60.58 °F
F-2
F.1.2 New Model Developed Using Orthogonal Distance Regression (ODR)
The initial statistical model for ΔRTepistemic was developed using point-estimation procedures that
did not take into account any uncertainty in the data sample of Table F1. An analytical procedure,
called orthogonal distance regression (ODR), can be employed to solve the errors-in-variables
problem in which uncertainties are assumed to exist in the data. The computational procedure
implemented into the software package, ODRPACK [F1], can be used to fit a model equation to
data using orthogonal distance regression.
The explicit ODR problem is defined as follows. Let ( xi , yi ), i = 1, 2,K n be an observed set of
data. Assume that the values yi are a (possibly nonlinear) function of xi and a set of unknown
parameters β ∈ℜ p , where both yi and xi contain the uncertainties, ε i* ∈ℜ1 and δ i* ∈ ℜ1 ,
respectively. The superscript “*” denotes an actual but unknown value. The observed value, yi ,
can be expressed in terms of a model equation
(
)
yi + ε i* = f i xi + δ i* {β k*} ;
( i = 1, 2,K n )
(F6)
({β } ; k = 1, 2,K p ) . The variables y
*
k
for some actual values of the parameter vector
i
are
sometimes referred to as the dependent or response variables, and xi are the independent
(regressor or explanatory) variables.
{ }
The explicit orthogonal distance regression problem approximates β * by finding the estimate
{β }
for which the sum of the squares of the n orthogonal distances from the curve f ( x ; {β })
to the n data points is minimized [F1]. This can be accomplished by the following minimization
problem
n
∑ (ε
min
βδ ε
, ,
i =1
2
i
+ δ i2 )
(F7)
subject to the constraints
(
yi = f i xi + δ i
{β } ) − ε i
i = 1, 2,K n.
(F8)
Since the constraints are linear in ε i , they and thus ε i can be eliminated from the minimization
problem, obtaining
⎛⎡ f x +δ β − y ⎤
{ }) ⎦
⎜⎣ (
∑
min
⎝
{β } {δ }
n
i
,
i
i
i =1
i
2
+ δ i2 ⎞⎟
⎠
(F9)
The algorithm implemented in ODRPACK uses the Levenberg-Marquardt trust region method to
iteratively solve the nonlinear minimization problem of Eq. (F9).
F-3
Derivation of the Model Equation Form
To proceed, the form of the problem-specific model equation must be derived. The CDF in
Eq.(F1) can be rewritten as
⎡ ⎛ ΔRT − a ⎞c ⎤
P = 1 − exp ⎢ − ⎜
⎟ ⎥
b
⎠ ⎦⎥
⎣⎢ ⎝
⎡ ⎛ ΔRT − a ⎞c ⎤
1 − P = exp ⎢ − ⎜
⎟ ⎥
b
⎠ ⎥⎦
⎢⎣ ⎝
(F10)
c
⎛ ΔRT − a ⎞
− ln (1 − P ) = ⎜
⎟
b
⎝
⎠
ln ⎡⎣ − ln (1 − P ) ⎤⎦ = c ln ( ΔRT − a ) − c ln(b)
The location parameter, a, is related to the scale, b, and shape, c, parameters through its moment
estimator
⎛ 1⎞
a ≈ m′1 − b Γ ⎜1 + ⎟
⎝ c⎠
(F11)
where m′1 is the 1st crude moment of the sample (or sample mean). The use of the Eq. (F11) as a
constraint in the model equation forces the mean of the resulting Weibull model to be identical to
the sample mean, m′1 . Introducing Eq. (F11) into Eq. (F10), the final form of the nonlinear
model equation is
⎡
⎛
1 ⎞⎤
yi = β1 ln ⎢ xi − m′1 + β 2 Γ ⎜1 + ⎟ ⎥ − β1 ln( β 2 ); (i = 1, 2, … n)
⎝ β1 ⎠ ⎦
⎣
where
⎛
⎧ β1 ⎫ ⎧ c ⎫
⎜ {β } = ⎨ ⎬ = ⎨ ⎬ ;
⎩ β 2 ⎭ ⎩b ⎭
⎝
(F12)
⎞
{ xi } = {ΔRT(i ) } ;{ yi } = {ln ⎡⎣ − ln (1 − Pi ) ⎤⎦} ⎟
⎠
Values for Pi can be estimated by ranking the data in Table F1 and applying the median-rank
order statistic
Pi ≈
i − 0.3
n + 0.4
F-4
(F13)
⎧ β1 ⎫
⎪β ⎪
⎪ 2⎪
⎪⎪ δ ⎪⎪
ODRPACK iteratively solves for the solution vector ⎨ 1 ⎬
⎪δ 2 ⎪
⎪M ⎪
⎪ ⎪
⎪⎩δ n ⎭⎪n + 2
The results of the ODRPACK analysis are presented in Table F2. In summary, the ODR analysis
produced the following estimates for the Weibull model for ΔRTepistemic :
Location Parameter, a =
Scale Parameter, b =
Shape Parameter, c =
-45.586
130.899 ± 10.259
1.855 ± 0.227
ΔRTODR =
−45.586 + 130.899 ⎡⎣ − ln (1 − P ) ⎤⎦ 1.855 ;
Sample Mean, m′1 =
Weibull Mean, μ =
Sample Stdv, s =
Weibull Stdv, σ =
Sample Variance, s 2 =
Weibull Variance, σ 2 =
95% Confidence Intervals
109.15 to 152.65
1.374 to 2.337
1
0 < P <1
70.67
70.667
60.58
65.036
3669.77
4229.692
The 95% confidence intervals for the two parameters β1 = c and β 2 = b are calculated by
ODRPACK using β k ± t(0.975, μ )σ βk where t(0.975, μ ) is the appropriate value for constructing a
two-sided confidence interval using Student’s t distribution with μ degrees of freedom. The
computational procedure used by ODRPACK to calculate the standard deviations for the
parameters, σ βk , is given in [F2]. See Fig. F1 for a comparison of the initial Weibull model and
the model produced by the ODR analysis. The application of ODR has resulted in an increase in
the Weibull model’s standard deviation from 58.94 °F to 65.04 °F compared to the sample’s
standard deviation of 60.58 °F .
F-5
Table F2. ODRPACK Results of ODR Analysis of ΔRTepistemic Model Equation
*******************************************************
* ODRPACK VERSION 2.01 OF 06-19-92 (DOUBLE PRECISION) *
*******************************************************
ODR Analysis of DRTLB Weibull Model Parameters
BETA(1) = c >> Shape Parameter
BETA(2) = b >> Scale Parameter
a = M1 - b*Gamma[1 + 1/c]
*** INITIAL SUMMARY FOR FIT BY METHOD OF ODR ***
--- PROBLEM SIZE:
N =
18
NQ =
1
M =
1
NP =
2
(NUMBER WITH NONZERO WEIGHT =
(NUMBER UNFIXED =
18)
2)
--- CONTROL VALUES:
JOB = 00010
= ABCDE, WHERE
A=0 ==> FIT IS NOT A RESTART.
B=0 ==> DELTAS ARE INITIALIZED TO ZERO.
C=0 ==> COVARIANCE MATRIX WILL BE COMPUTED USING
DERIVATIVES RE-EVALUATED AT THE SOLUTION.
D=1 ==> DERIVATIVES ARE ESTIMATED BY CENTRAL
DIFFERENCES.
E=0 ==> METHOD IS EXPLICIT ODR.
NDIGIT =
16
(ESTIMATED BY ODRPACK)
TAUFAC =
1.00D+00
--- STOPPING
SSTOL
PARTOL
MAXIT
CRITERIA:
=
1.49D-08
=
3.67D-11
=
50
(SUM OF SQUARES STOPPING TOLERANCE)
(PARAMETER STOPPING TOLERANCE)
(MAXIMUM NUMBER OF ITERATIONS)
--- INITIAL WEIGHTED SUM OF SQUARES
SUM OF SQUARED WEIGHTED DELTAS
SUM OF SQUARED WEIGHTED EPSILONS
=
=
=
1.15671908D+00
0.00000000D+00
1.15671908D+00
*** ITERATION REPORTS FOR FIT BY METHOD OF ODR ***
IT.
NUM.
----
CUM.
NO. FN
EVALS
------
WEIGHTED
SUM-OF-SQS
-----------
ACT. REL.
SUM-OF-SQS
REDUCTION
-----------
PRED. REL.
SUM-OF-SQS
REDUCTION
-----------
TAU/PNORM
---------
G-N
STEP
----
1
2
3
4
5
6
7
8
12
19
26
33
40
47
54
61
5.36253D-01
5.33419D-01
5.33152D-01
5.33130D-01
5.33128D-01
5.33128D-01
5.33128D-01
5.33128D-01
5.3640D-01
5.2849D-03
4.9976D-04
4.1577D-05
3.2902D-06
2.5647D-07
1.9907D-08
1.5432D-09
5.3739D-01
4.2184D-03
3.9259D-04
3.2561D-05
2.5746D-06
2.0064D-07
1.5572D-08
1.2072D-09
1.333D-01
4.265D-02
1.461D-02
4.323D-03
1.224D-03
3.423D-04
9.542D-05
2.657D-05
YES
YES
YES
YES
YES
YES
YES
YES
F-6
Table F2. ODRPACK Results of ODR Analysis of ΔRTepistemic Model Equation
(continued)
*** FINAL SUMMARY FOR FIT BY METHOD OF ODR ***
--- STOPPING
INFO
NITER
NFEV
IRANK
RCOND
ISTOP
CONDITIONS:
=
1 ==> SUM OF SQUARES CONVERGENCE.
=
8
(NUMBER OF ITERATIONS)
=
67
(NUMBER OF FUNCTION EVALUATIONS)
=
0
(RANK DEFICIENCY)
=
1.20D-01
(INVERSE CONDITION NUMBER)
=
0
(RETURNED BY USER FROM SUBROUTINE FCN)
--- FINAL WEIGHTED SUMS OF SQUARES
=
SUM OF SQUARED WEIGHTED DELTAS
=
SUM OF SQUARED WEIGHTED EPSILONS =
5.33127879D-01
7.67684538D-04
5.32360195D-01
--- RESIDUAL STANDARD DEVIATION
DEGREES OF FREEDOM
1.82539016D-01
16
=
=
--- ESTIMATED BETA(J), J = 1, ..., NP:
BETA
1
2
1.85530498D+00
1.30899017D+02
S.D. BETA
---- 95%
2.2706D-01
1.0259D+01
CONFIDENCE INTERVAL ----
1.37390691D+00 TO
1.09149592D+02 TO
--- ESTIMATED EPSILON(I) AND DELTA(I,*), I = 1, ..., N:
I
EPSILON(I,1)
DELTA(I,1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
2.62841903D-01
-1.29977011D-01
-1.86382404D-01
-3.79012096D-01
2.78865897D-01
1.68817068D-01
2.10949482D-01
1.16154880D-01
8.71915578D-02
-3.56507199D-02
8.89342397D-02
4.68465281D-02
-7.29122682D-02
-1.41925842D-01
1.97009129D-01
7.02764840D-02
-8.73096746D-03
-1.24381318D-01
-1.86361603D-02
6.95094427D-03
7.87802505D-03
1.47415688D-02
-6.56742977D-03
-3.72942044D-03
-4.09035239D-03
-2.15105581D-03
-1.49943300D-03
6.01915026D-04
-1.29426169D-03
-6.43875329D-04
9.86768713D-04
1.83636941D-03
-1.94642622D-03
-6.74910438D-04
7.78822029D-05
9.95579717D-04
F-7
2.33670305D+00
1.52648443D+02
F.1.3. Final Stochastic Model for ΔRTepistemic in FAVOR
The epistemic uncertainty in RTNDT(u) is estimated in FAVOR by
ΔRTepistemic = RTNDT (u ) − RTLB
(F14)
where RTNDT(u) is the unirradiated reference nil-ductility transition temperature and RTLB is a new
temperature index developed for FAVOR analyses. If we assume that RTNDT(u) and RTLB are
statistically independent and, therefore, uncorrelated, then the variance of ΔRTepistemic is
var(ΔRTepistemic ) = var( RTNDT ) + var( RTLB )
(F15)
where the cov ( RTNDT ( u ) RTLB ) has been assumed to be zero. The statistical model developed for
ΔRTepistemic using the ODR procedure contains the following four sources of uncertainty
2
1. Measurement uncertainty and material variability in RTNDT(u), σ (1)
2
2. Measurement uncertainty and material variability in RTLB , σ (2)
2
3. Model uncertainty in RTNDT(u) , σ (3)
2
4. Model uncertainty in RTLB , σ (4)
such that the components of the variances for RTNDT(u) and RTLB are the following:
2
2
var( RTNDT ( u ) ) = σ (1)
+ σ (3)
2
2
+ σ (4)
var( RTLB ) = σ (2)
(F16)
Therefore, the variance (uncertainty) in the ODR-developed Weibull distribution for ΔRTepistemic
can be expressed as
2
2
2
2
σ Δ2RT = σ (1)
+ σ (2)
+ σ (3)
+ σ (4)
= 4229.69
(F17)
As a result of the sampling protocols in FAVOR, the uncertainties associated with sources (1) and
(2) have already been accounted for at the point in FAVOR where ΔRTepistemic is sampled. The
Weibull model for ΔRTepistemic can be revised such that it reflects the uncertainties associated
with sources (3) and (4) only, specifically
2
2
2
2
σ Δ2RT ( rev ) = σ (3)
+ σ (4)
= σ Δ2RT − σ (1)
− σ (2)
Two cases were examined:
F-8
(F18)
Case 1:
2
σ (1)
= ( 23°F )
2
2
σ (2)
=0
Case 2:
2
σ (1)
= ( 23°F )
2
2
σ (2)
= ( 23°F )
2
The required adjustments to the Weibull model for ΔRTepistemic can be calculated by solving the
following nonlinear system of equations
⎛
⎝
1⎞
μΔRT − a − b Γ ⎜1 + ⎟ = 0
c
σ
2
ΔRT ( rev )
⎠
⎡ ⎛ 2⎞
⎛ 1 ⎞⎤
− b ⎢Γ ⎜1 + ⎟ − Γ 2 ⎜1 + ⎟ ⎥ = 0
⎝ c ⎠⎦
⎣ ⎝ c⎠
(F19)
2
for the new parameters b and c, where μ ΔRT = 70.67 °F and the location parameter for the ODRdeveloped model, a = -45.586 °F, remain fixed. Equations (F19) are the asymptotic relations for
the mean and variance of a Weibull distribution.
Case 1:
2
2
σ Δ2RT ( rev ) = σ Δ2RT − σ (1)
− σ (2)
σ Δ2RT ( rev ) = 4229.692 − 232 − 0 = 3700.692
σ ΔRT ( rev ) = 60.83 °F
The solutions for (b,c) are
b = 131.18 °F
c = 1.998
F-9
Case 2:
2
2
σ Δ2RT ( rev ) = σ Δ2RT − σ (1)
− σ (2)
σ Δ2RT ( rev ) = 4229.692 − 232 − 232 = 3171.692
σ ΔRT ( rev ) = 56.32 °F
The solutions for (b,c) are
b = 131.27 °F
c = 2.177
See Fig. F2 for a comparison of the ODR-derived model with the revised models of Cases 1 and
2. Figure F3 compares the CDF of the initial Weibull model to that of Case 2 with emphasis
placed on the lower-left tail. Note that Case 2 produces a more negative ΔRTepistemic adjustment
than the initial model for cumulative probabilities less than approximately 3.5%. A comparison
between the ODR-derived model and Case 2 is shown in Fig. F4. For cumulative probabilities
less than approximately 60%, Case 2 produces more positive values of ΔRTepistemic than the ODR
model.
In summary the revised Weibull models for Cases (1) and (2) are:
Summary:
Case 1:
1
ΔRT( rev ) = −45.586 + 131.18 ⎡⎣ − ln (1 − P ) ⎤⎦ 1.998 ; 0 < P < 1
Case 2:
1
ΔRT( rev ) = −45.586 + 131.27 ⎡⎣ − ln (1 − P ) ⎤⎦ 2.177 ; 0 < P < 1
Case 2 was selected for implementation into FAVOR.
F-10
F.2. Stochastic Model for
F.2.1 Initial Model for
ΔRTarrest
in FAVOR
ΔRTarrest
The initial stochastic model developed for FAVOR to describe the statistical distribution of
ΔRTarrest = T0 − TK Ia was based on a lognormal distribution (see Fig. F5) with the parameters
ΔRT arrest = μ (T0 ) = 44.123exp(−0.006 T0 ); T0 [°c]
2
σ log
= 0.392 = 0.1521 (constant)
(F20)
The asymptotic relations for the log-mean and variance of the model are:
μlog (T0 ) = ln [ μ (T0 ) ] −
2
σ log
2
2
var(ΔRTarrest ) = σ (T0 ) = ω (ω − 1) exp ⎡⎣ 2μlog (T0 ) ⎤⎦ ; ω =exp(σ log
)
(F21)
2
The initial model was derived from an ordinary least squares regression analysis using the logtransformed data shown in Table F3.
F.2.2 Model Developed Using Orthogonal Distance Regression (ODR)
The ORDPACK program was used to reanalyze the following model equation
ln(ΔRT arrest ) = β1T0 + β 2
(F22)
where, upon reversing the log-transformation, the mean value for ΔRTarrest is
ΔRT arrest = exp( β 2 ) exp ( β1T0 )
(F23)
The results of the ODR analysis are presented in Table F4 with the following ODR estimates for
the model parameters:
β1 = −0.00597110744 ± 0.00082458
β 2 = 3.78696343 ± 0.065299
exp( β 2 ) = 44.12221645 ± 2.908036613
ΔRT arrest = 44.1222exp(−0.00597T0 );
σ log = 0.389987535; σ
2
log
F-11
= 0.1520903
(F24)
[°C]
Table F3. Data Used in the Development of the ΔRTarrest Model
N
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
T0
(°C)
-114
131
-66
-78
-104
-108
43
-20
-71
-66
-84
-21
-53
-54
62
-65
-100
-130
-100
-27
-78
-115
-68
-70
-65
-51
17
-48
-92
-70
-81
-157
67
-84
-67
-58
35
39
-61
6
-61
-48
-24
-19
-85
-131
-3
-95
-93
-68
184
42
27
T KIa
(°C)
16
140
13
6
-16
44
113
60
-41
6
9
65
-6
18
93
-12
-15
-8
-18
25
10
-25
-9
17
-25
19
77
48
-26
-18
-20
-27
78
9
18
-14
74
67
-15
62
-16
8
32
10
-33
-26
33
-62
-17
-8
220
71
68
F-12
T KIa -T 0
(°C)
130
9
79
84
88
152
70
80
30
72
93
86
47
72
31
53
85
122
82
52
88
90
59
87
40
70
60
96
66
52
61
130
11
93
85
44
39
28
46
56
45
56
56
29
52
105
36
33
76
60
36
29
41
ln(T KIa -T 0)
4.8675
2.1972
4.3694
4.4308
4.4773
5.0239
4.2485
4.3820
3.4012
4.2767
4.5326
4.4543
3.8501
4.2767
3.4340
3.9703
4.4427
4.8040
4.4067
3.9512
4.4773
4.4998
4.0775
4.4659
3.6889
4.2485
4.0943
4.5643
4.1897
3.9512
4.1109
4.8675
2.3979
4.5326
4.4427
3.7842
3.6636
3.3322
3.8286
4.0254
3.8067
4.0254
4.0254
3.3673
3.9512
4.6540
3.5835
3.4965
4.3307
4.0943
3.5835
3.3673
3.7136
Table F4. ODRPACK Results of ODR Analysis of ΔRTarrest Model Equation
*******************************************************
* ODRPACK VERSION 2.01 OF 06-19-92 (DOUBLE PRECISION) *
*******************************************************
ODR Analysis of DARTarrest Lognormal Model
BETA(1) = slope
BETA(2) = intercept of log-transformed data
LN(DRTarrest) = BETA(1)*T0 + BETA(2)
DRTArrest = EXP(BETA(2))*EXP(BETA(1)*T0)
*** DERIVATIVE CHECKING REPORT FOR FIT BY METHOD OF ODR ***
FOR RESPONSE
1 OF OBSERVATION
1
DERIVATIVE WRT
USER
SUPPLIED
VALUE
RELATIVE
DIFFERENCE
BETA( 1)
BETA( 2)
DELTA( 1, 1)
-1.57D+02
1.00D+00
-5.84D-03
4.25D-07
7.87D-08
4.30D-07
DERIVATIVE
ASSESSMENT
VERIFIED
VERIFIED
VERIFIED
NUMBER OF RELIABLE DIGITS IN FUNCTION RESULTS
(ESTIMATED BY ODRPACK)
16
NUMBER OF DIGITS OF AGREEMENT REQUIRED BETWEEN
USER SUPPLIED AND FINITE DIFFERENCE DERIVATIVE FOR
USER SUPPLIED DERIVATIVE TO BE CONSIDERED VERIFIED
4
ROW NUMBER AT WHICH DERIVATIVES WERE CHECKED
1
-VALUES OF THE EXPLANATORY VARIABLES AT THIS ROW
X( 1, 1) -1.57000000D+02
*******************************************************
* ODRPACK VERSION 2.01 OF 06-19-92 (DOUBLE PRECISION) *
*******************************************************
*** INITIAL SUMMARY FOR FIT BY METHOD OF ODR ***
--- PROBLEM SIZE:
N =
53
NQ =
1
M =
1
NP =
2
(NUMBER WITH NONZERO WEIGHT =
(NUMBER UNFIXED =
53)
2)
--- CONTROL VALUES:
JOB = 00020
= ABCDE, WHERE
A=0 ==> FIT IS NOT A RESTART.
B=0 ==> DELTAS ARE INITIALIZED TO ZERO.
C=0 ==> COVARIANCE MATRIX WILL BE COMPUTED USING
DERIVATIVES RE-EVALUATED AT THE SOLUTION.
D=2 ==> DERIVATIVES ARE SUPPLIED BY USER.
DERIVATIVES WERE CHECKED.
RESULTS APPEAR CORRECT.
E=0 ==> METHOD IS EXPLICIT ODR.
NDIGIT =
16
(ESTIMATED BY ODRPACK)
TAUFAC =
1.00D+00
F-13
Table F4. ODRPACK Results of ODR Analysis of ΔRTarrest Model Equation (continued)
--- STOPPING
SSTOL
PARTOL
MAXIT
CRITERIA:
=
1.49D-08
=
3.67D-11
=
50
(SUM OF SQUARES STOPPING TOLERANCE)
(PARAMETER STOPPING TOLERANCE)
(MAXIMUM NUMBER OF ITERATIONS)
--- INITIAL WEIGHTED SUM OF SQUARES
SUM OF SQUARED WEIGHTED DELTAS
SUM OF SQUARED WEIGHTED EPSILONS
=
=
=
7.76381810D+00
0.00000000D+00
7.76381810D+00
*** ITERATION REPORTS FOR FIT BY METHOD OF ODR ***
IT.
NUM.
----
CUM.
NO. FN
EVALS
------
WEIGHTED
SUM-OF-SQS
-----------
ACT. REL.
SUM-OF-SQS
REDUCTION
-----------
PRED. REL.
SUM-OF-SQS
REDUCTION
-----------
TAU/PNORM
---------
G-N
STEP
----
1
2
3
15
16
17
7.75660D+00
7.75660D+00
7.75660D+00
9.2916D-04
1.7592D-08
6.0973D-13
9.2766D-04
1.7540D-08
6.0818D-13
3.063D-02
5.224D-05
1.064D-06
YES
YES
YES
*** FINAL SUMMARY FOR FIT BY METHOD OF ODR ***
--- STOPPING
INFO
NITER
NFEV
NJEV
IRANK
RCOND
ISTOP
CONDITIONS:
=
1 ==> SUM OF SQUARES CONVERGENCE.
=
3
(NUMBER OF ITERATIONS)
=
17
(NUMBER OF FUNCTION EVALUATIONS)
=
4
(NUMBER OF JACOBIAN EVALUATIONS)
=
0
(RANK DEFICIENCY)
=
1.02D-01
(INVERSE CONDITION NUMBER)
=
0
(RETURNED BY USER FROM SUBROUTINE FCN)
--- FINAL WEIGHTED SUMS OF SQUARES
=
SUM OF SQUARED WEIGHTED DELTAS
=
SUM OF SQUARED WEIGHTED EPSILONS =
--- RESIDUAL STANDARD DEVIATION
DEGREES OF FREEDOM
=
=
7.75660416D+00
2.76544656D-04
7.75632762D+00
3.89987535D-01
51
--- ESTIMATED BETA(J), J = 1, ..., NP:
BETA
1 -5.97110744D-03
2
3.78696343D+00
2a 44.1222164
S.D. BETA
---- 95%
8.2458D-04
6.5299D-02
1.06747815
-7.62651413D-03 TO -4.31570076D-03
3.65587019D+00 TO 3.91805666D+00
38.70118385
TO 50.30259469
F-14
CONFIDENCE INTERVAL ----
Table F4. ODRPACK Results of ODR Analysis of ΔRTarrest Model Equation (continued)
--- ESTIMATED EPSILON(I) AND DELTA(I,*), I = 1, ..., N:
I
EPSILON(I,1)
DELTA(I,1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
-1.43102053D-01
-8.47788261D-02
-2.40805066D-01
-2.61679548D-02
-3.99850519D-01
-5.92016383D-01
-6.93757401D-02
-5.85749970D-02
-2.26442691D-02
8.57680493D-01
1.15426669D-02
1.46645341D-01
3.43251602D-01
-2.44054340D-01
-2.44054340D-01
1.59743570D-01
-1.78100642D-01
-2.24618999D-01
8.09685804D-01
-2.60957867D-01
2.53688183D-01
1.15457172D-01
9.86506532D-02
-2.55614517D-01
-1.88384618D-01
-9.56061927D-02
2.04786195D-01
4.86188622D-01
3.22548084D-01
3.44526207D-01
3.49085578D-01
-1.67256927D-01
2.53275489D-01
-1.56999738D-01
-4.90754110D-01
4.82231733D-02
-3.06028247D-03
-9.50782960D-02
-5.41971290D-01
-4.75624102D-01
5.33099631D-01
2.21349919D-01
-2.74205133D-01
-4.08875384D-01
-8.78254100D-02
-8.55839285D-02
2.21877816D-01
1.68875063D-01
-7.18263826D-01
-1.72318244D-02
9.88968694D-01
8.07494984D-01
-8.95207363D-01
-8.54477100D-04
-5.06223103D-04
-1.43787185D-03
-1.56251554D-04
-2.38754864D-03
-3.53499080D-03
-4.14249691D-04
-3.49757341D-04
-1.35211263D-04
5.12129857D-03
6.89224532D-05
8.75634434D-04
2.04959067D-03
-1.45727360D-03
-1.45727360D-03
9.53845309D-04
-1.06345728D-03
-1.34122318D-03
4.83471734D-03
-1.55820631D-03
1.51479827D-03
6.89406666D-04
5.89053212D-04
-1.52630061D-03
-1.12486396D-03
-5.70874424D-04
1.22279946D-03
2.90308234D-03
1.92596784D-03
2.05720147D-03
2.08442594D-03
-9.98708341D-04
1.51233403D-03
-9.37461609D-04
-2.93034334D-03
2.87945535D-04
-1.82732618D-05
-5.67722299D-04
-3.23616640D-03
-2.84000050D-03
3.18319281D-03
1.32170317D-03
-1.63730709D-03
-2.44143703D-03
-5.24414570D-04
-5.11030452D-04
1.32485529D-03
1.00837040D-03
-4.28882729D-03
-1.02892998D-04
5.90523394D-03
4.82163573D-03
-5.34537537D-03
F-15
Comparison of Eqs. (F20) with Eqs. (F24) indicates that the ODR analysis produced essentially
the same model as resulted from the ordinary least squares analysis (see Fig. F6).
F.2.3 Final Model for ΔRTarrest
The variance of ΔRTarrest = T0 − TK Ia is
var(ΔRTarrest ) = var(T0 ) + var(TK Ia ) − 2 cov(T0TK Ia )
(F25)
In the absence of data to the contrary, we assume the statistical independence of T0 and TKIa such
that cov(T0TK Ia ) = 0 , and Eq. (F25) becomes
var(ΔRTarrest ) = var(T0 ) + var(TK Ia )
(F26)
The variance of both the initial and ODR lognormal model is a decreasing function of increasing
T0
2
var(ΔRTarrest (ODR ) ) = σ ODR
(T0 )
= exp(0.389982 ) × ⎡⎣ exp(0.389982 ) − 1⎤⎦ × exp ⎡⎣ 2 ln [ μ (T0 )] − 0.389982 ⎤⎦
(F27)
as shown in Fig. F7. By T0 ≈ 56 °C, var(ΔRTarrest ) = (12.78 °C ) .
2
The variance for T0 has been accounted for in a separate sampling protocol prior to the sampling
of ΔRTarrest , and the statistical model for ΔRTarrest should, therefore, reflect only the remaining
variance in TK Ia . If we assume that the var(T0 ) = ( 23 °F ) = (12.778 °C) 2 , then
2
var(ΔRTarrest ( rev ) ) = var(TK Ia ) = var(ΔRTarrest ) − var(T0 )
2
var(ΔRTarrest ( rev ) ) = σ rev
(T0 ) =
{exp(0.38998 ) × ⎡⎣exp(0.38998 ) − 1⎤⎦ × exp ⎡⎣2μ (T )⎤⎦} − var(T ) =
exp ⎣⎡σ (T ) ⎦⎤ × {exp ⎣⎡σ (T ) ⎦⎤ − 1} × exp ⎣⎡ 2 μ (T ) ⎤⎦
2
2
log
2
log
0 rev
2
log
0
0 rev
where
μlog (T0 ) rev = ln ⎡⎣ μ (T0 ) ⎤⎦ −
(F28)
0
log
0 rev
2
σ log
(T0 )rev
2
2
and μ (T0 ) remains a fixed function of T0. Solving Eq. (F28) for σ log
(T0 ) rev results in
{
}
2
σ log
(T0 ) rev = ln exp ⎡⎣0.389982 + 2 ln( μ (T0 ) ⎤⎦ − var (T0 ) − 2 ln [ μ (T0 ) ]
and solving for var(ΔRTarrest ( rev ) ) = σ 2 (T0 ) rev gives
F-16
(F29)
{
}
2
2
σ 2 (T0 )rev = exp ⎡⎣σ log
(T0 ) rev ⎤⎦ × exp ⎡⎣σ log
(T0 ) rev ⎤⎦ − 1 ×
2
exp {2 ln [ μ (T0 ) ] − σ log
(T0 ) rev }
However,
as
noted
earlier
and
indicated
in
(F30)
Fig. F7,
at
T0 ≈ 56 °C, var(ΔRTarrest ) = var(T0 ) = (12.78 °C ) which would produce σ (T0 ) rev = 0 . In order
2
2
to prevent a nonphysical zero variance at this point, the assumed constant value of var(T0 ) can be
replaced by the following function with a transition region:
⎧
(12.778) 2
⎪
var(T0 ) = ⎨99.905972 -1.7748073T0
⎪
0
⎩
for T0 < −35.7 °C
for -35.7 °C ≤ T0 ≤ 56 °C
for T0 > 56 °C
(F31)
Figure F7 plots Eq. (F30) as the final model variance with Eq. (F31) used in Eq. (F29) to produce
the final log-variance as a function of T0. Figure F8 compares the 1% and 99% percentiles of the
ODR and final models for ΔRTarrest .
Summary of Stochastic Model for ΔRTarrest
The lognormal model for ΔRTarrest is, therefore,
ΔRT arrest = μ (T0 ) = 44.122 exp(−0.005971T0 ); T0 [°C]
{
}
σ log (T0 ) rev = ln exp ⎡⎣0.389982 + 2 ln( μ (T0 ) ⎤⎦ − var (T0 ) − 2 ln [ μ (T0 )]
where
(F32)
⎧
(12.778)
⎪
var(T0 ) = ⎨99.905972 -1.7748073T0
⎪
0
⎩
2
F-17
for T0 < −35.7 °C
for -35.7 °C ≤ T0 ≤ 56 °C
for T0 > 56 °C
(a)
(b)
Fig. F1. Comparison of the initial Weibull model, W0, for ΔRTepistemic with the ODR model:
(a) probability density functions and (b) cumulative distribution functions.
F-18
Fig. F2. Comparison of ODR Weibull model, WODR, for ΔRTepistemic with the models for
Case 1 (W1) and Case 2 (W2): (a) probability density functions and (b) cumulative
distribution functions.
F-19
Fig. F3. Comparison of initial model in FAVOR, W0, with Case 2, W2.
F-20
Fig. F4. Comparison of ODR model, WODR, with Case 2, W2.
F-21
Fig. F5. Data used to develop the lognormal statistical model for ΔRTarrest as a function of
T0 .
F-22
Fig. F6. Model developed from ODR analysis of log-transformed data.
F-23
Fig. F7. Variance of ODR model compared to final model.
F-24
Fig. F8. Comparison of ODR model with final model.
F-25
REFERENCES
F1. P. T. Boggs, R. H. Byrd, J. E. Rogers, R. B. Schnabel, User’s Reference Guide for
ODRPACK Version 2.01: Software for Weighted Orthogonal Distance Regression, NISTIR
92-4834, National Institute of Standards and Technology, Gaithersburg, MD, 1992.
F2. P. T. Boggs and J. E. Rogers, “The Computation and Use of the Asymptotic Covariance
Matrix for Measurement Error Models,” Internal Report 89-4102, Applied and
Computational Mathematics Division, National Institute of Standards and Technology,
Gaithersburg, MD, 1990.
F-26
U.S. NUCLEAR REGULATORY COMMISSION
NRC FORM 335
(2-89)
NRCM 1102,
3201,3202
BIBLIOGRAPHIC DATA SHEET
1. REPORT NUMBER
(Assigned by NRC, Add Vol., Supp., Rev.,
and Addendum Numbers, if anv.1
NUREG/CR-6854
ORNL/TM-2004/244
(See instructions on the reverse)
2. TITLE AND SUBTITLE
Fracture Analysis of Vessels – Oak Ridge (FAVOR, v04.1), Computer Code:
Theory and Implementation of Algorithms, Methods, and Correlations
3.
DATE REPORT PUBLISHED
MONTH
YEAR
August
2007
4. FIN OR GRANT NUMBER
Y6533
5. AUTHOR(S)
6. TYPE OF REPORT
P. T. Williams, T. L. Dickson, and S. Yin
Final, technical
7. PERIOD COVERED (Inclusive Dates)
8. PERFORMING ORGANIZATION - NAME AND ADDRESS (If NRC, provide Division, Office or Region, U.S. Nuclear Regulatory Commission, and mailing address; if contractor,
provide name and mailing address.)
Heavy Section Steel Technology Program
Oak Ridge National Laboratory
P. O. Box 2008, Mail Stop 6085
Oak Ridge, TN 37831-6085
9. SPONSORING ORGANIZATION - NAME AND ADDRESS (If NRC, type "Same as above"; if contractor, provide NRC Division, Office or Region, U.S. Nuclear Regulatory Commission,
and mailing address.)
Division of Fuel, Engineering, and Radiological Research
Office of Nuclear Regulatory Research
U. S. Nuclear Regulatory Commission
Washington, DC 2055-0001
10. SUPPLEMENTARY NOTES
M. EricksonKirk, NRC Project Manager
1 1. ABSTRACT (200 words or less)
The current regulations to insure that nuclear reactor pressure vessels (RPVs) maintain their structural integrity when subjected to
transients such as pressurized thermal shock (PTS) events were derived from computational models developed in the early-to-mid
1980s. Since that time, advancements and refinements in relevant technologies that impact RPV integrity assessment have led to an
effort by the NRC to re-evaluate its PTS regulations. Updated computational methodologies have been developed through
interactions between experts in the relevant disciplines of thermal hydraulics, probabilistic risk assessment, materials embrittlement,
fracture mechanics, and inspection (flaw characterization). Contributors to the development of these methodologies include the NRC
staff, their contractors, and representatives from the nuclear industry. These updated methodologies have been integrated into the
Fracture Analysis of Vessels – Oak Ridge (FAVOR, v04.1) computer code developed for the NRC by the Heavy Section Steel
Technology (HSST) program at Oak Ridge National Laboratory (ORNL). The FAVOR, v04.1, code represents the baseline NRCselected applications tool for re-assessing the current PTS regulations. This report is intended to document the technical bases for the
assumptions, algorithms, methods, and correlations employed in the development of the FAVOR, v04.1, code.
12. KEY WORDS/DESCRIPTORS (List words or phrases that will assist researchers in locating the report.)
13. AVAILABILITY STATEMENT
unlimited
pressurized thermal shock, probabilistic fracture mechanics, reactor pressure vessels
14. SECURITY CLASSIFICATION
(This Page)
unclassified
(This Report)
unclassified
15. NUMBER OF PAGES
16. PRICE
NRC FORM 335 (2-89)
Fly UP