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 
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 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 
 

 iii 

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.
 

 iv 
 

 v 

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 
 

 vi 
 

 vii 

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 Historical Review........................................................................................................... 5

2.2 Current NRC Regulatory Approach to PTS............................................................... 6

2.3 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
 

 viii 

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
 

 ix 

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(τ

n

)(i,j,k) is the increase in cpi(τ

n

)(i,j,k) that occurs during each discrete time step. When 

the maximum value of cpi(τ

n

)(i,j,k) is reached, negative values of Δcpi(τ

n

)(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 
 

 x 

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 RT RT RT LB NDT = −Δ 0 LB ............................................................................................100
 

 xi 

37. Comparison of cumulative distribution functions developed for RTNDT0-T0 and 

RTNDT0-RTLB . ..........................................................................................................................102

38. Lognormal distribution of 0 ARREST KIa Δ =− RT TT 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
 

 xii 

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 
 

 xiii 

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. 

 

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 

Summary Report – NUREG-1806

• 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 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.

• 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.

• 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.

• Sensitivity Studies: Arcieri, W.C., et al., 

“RELAP5/MOD3.2.2Gamma Results for 

Palisades 1D Downcomer Sensitivity 

Study”

• Consistency Check: Junge, M., “PTS 

Consistency Effort”

• 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.

• Beaver: Whitehead, D.W., et al., “Beaver 

Valley PTS PRA”

• Oconee: Kolaczkowski, A.M., et al., “Oconee 

PTS PRA”

• Palisades: Whitehead, D.W., et al., 

“Palisades 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”

Results Models, Validation, & Procedures

PFM TH PRA

This Report
 

 xiv 

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. 
 

 xv 

Abbreviations 

ASME American Society of Mechanical Engineers 

ASTM American Society for Testing and Materials 

BNL Brookhaven National Laboratory 

CCA compact crack-arrest test specimen 

C(T) compact tension fracture-toughness test specimen 

CDF cumulative distribution function 

CPI conditional probability of initiation 

CPF conditional probability of failure (as indicated by through-wall cracking) 

CRP copper-rich precipitate 

CVN Charpy V-Notch test specimen 

DTE differential-thermal expansion 

EFPY effective full-power years 

EPFM elastic-plastic fracture mechanics 

EPRI Electric Power Research Institute 

EOL end-of-licensing 

FAVOR Fracture Analysis of Vessels – Oak Ridge 

FEM finite-element method 

HAZ heat-affected zone 

HSST Heavy Section Steel Technology Program 

IPTS Integrated Pressurized Thermal Shock Program 

LEFM linear-elastic fracture mechanics 

LOCA loss-of-coolant accident 

NESC Network for Evaluating Structural Components 

NIST National Institute for Standards and Technology 

NRC United States Nuclear Regulatory Commission 

ORNL Oak Ridge National Laboratory 

PDF probability density function 

PFM probabilistic fracture mechanics 

PNNL Pacific Northwest National Laboratory 

PRA Probabilistic Risk Assessment 

PTS pressurized thermal shock 

PWHT post-weld heat treatment 

PWR pressurized water reactor 

RCW recirculating cooling water 

RG1.99 NRC Regulatory Guide 1.99, Revision 2, Ref. [12] 

RG1.154 NRC Regulatory Guide 1.154, Ref. [11] 

RPV reactor pressure vessel 

RVID Reactor Vessel Integrity Database, Version 2, Ref. [129] 

SIFIC stress-intensity influence coefficients 

SMD stable matrix defect 

10CFR50.61 Title 10 of the Code of Federal Regulations, Part 50, Section 50.61, Ref. [10] 

TMI Three-Mile-Island nuclear reactor 

T-E thermo-elastic 
 

 xvi 

T-H thermal-hydraulic 

UMD unstable matrix defect 

WOL wedge-open loading test specimen for fracture toughness 

WPS warm prestressing 
 

 xvii 

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. 
 

 xviii 
 

 1

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. 
 

 2

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]). 
 

 3

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 
 

 4

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. 
 

 5

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. 
 

 6

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 
 

 7

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 6 5 10− × 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 [141-17] was initiated in 1989 through 

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 
 

 8

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. 
 

 9

Table 1. Large-Scale PTS Experiments and Performing Organizations 

ID No. Experiment Title Research Organization Country Refs. 

Tests with Pressurized Vessels 

ITV 1-8 Intermediate Test Vessels Oak Ridge National Laboratory USA 19-2 2 2 2 225 

PTSE-1 Pressurized Thermal Shock 

Experiments 

Oak Ridge National Laboratory USA 26 

PTSE-2 Pressurized Thermal Shock 

Experiments 

Oak Ridge National Laboratory USA 27 

PTS I/6 Pressurized Thermal Shock 

Experiment I/6 

Central Research Institute for 

Structural Materials (CRISM) 

Russia 28, 29 

Tests with Cylindrical Specimens 

NKS-3 Thermal Shock 

Experiment 3 

Materialprüfungsanstalt (MPA) Germany 30 

NKS-4 Thermal Shock 

Experiment 4 

Materialprüfungsanstalt (MPA) Germany 30 

NKS-5 Thermal Shock 

Experiment 5 

Materialprüfungsanstalt (MPA) Germany 31 

NKS-6 Thermal Shock 

Experiment 6 

Materialprüfungsanstalt (MPA) Germany 29, 31 

SC-1 Spinning Cylinder PTS 

Experiment 1 

AEA Technology UK 32 

SC-2 Spinning Cylinder PTS 

Experiment 2 

AEA Technology UK 32 

SC-4 Spinning Cylinder PTS 

Experiment 4 

AEA Technology UK 33 

TSE-6 Thermal Shock Cylinders 

(Cylinder with Short Flaws) 

Oak Ridge National Laboratory 

(ORNL) 

USA 34 

TSE-7 Thermal Shock Cylinders 

(Clad Cylinder) 

Oak Ridge National Laboratory 

(ORNL) 

USA 35 

TSE-8 Thermal Shock Cylinders 

(Clad Cylinder) 

Oak Ridge National Laboratory 

(ORNL) 

USA 35 

NESC-1 NESC-1 Spinning Cylinder 

PTS Experiment 

Network for Evaluating Steel 

Components (NESC) 

International 

Network 

36

Tests with Plate Specimens 

PTS Step B Wide-Plate PTS Step B 

Experiment 

Japan Power and Engineering 

Inspection Corporation 

(JAPEIC) 

Japan 37 

WP-1 & 2 Wide-Plate Crack Arrest 

Tests of A533B and LUS 

Steels 

Oak Ridge National Laboratory 

(ORNL) 

USA 38, 39 

GP-1 Wide Plate Test Materialprüfungsanstalt (MPA) Germany 40 

Tests with Beam Specimens 

DD-2 & 

DSR-3 

Clad-beam experiments Electricité de France (EdF) France 29, 41 

SE(B) RPV 

Steel 

Full-Thickness Clad Beam 

Experiments 

National Institute of Standards 

and Testing (NIST) and ORNL 

USA 42, 43 

CB Cruciform Beam (CB) 

Experiments 

Oak Ridge National Laboratory 

(ORNL) 

USA 44 
 

 10

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]. 
 

 11

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. 
 

 12

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. 
 

 13

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 
 

 14

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. 
 

 15

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 1 ≤ ≤ CPI ) and conditional probability of failure ( 0 1 ≤ ≤ CPF ) by through-wall cracking 

are calculated for each flaw subjected to each transient. 
 

 16

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 [48-450] of RPV steels. Each of the embrittlement-related 

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. 
 

 17

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. 
 

 18

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

submodel3

 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 surface-

 

3

 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. 
 

 19

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. 
 

 20

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 fastneutron fluence, 

0f , product form, and chemistry content,   ) Cu Ni P , , and 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. 
 

 21

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. 
 

 22

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 [53-5 5 5 5 5 5 6 662]. The technical basis for the inclusion of warm prestressing effects in FAVOR is 

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. 
 

 23

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 ( / 0 I dK dτ ≤ ), 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 / 0 I dK dτ ≤ 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. 
 

 24

(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). 
 

 25

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 N i ← 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, log log Λ(,) μ σ ; 

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 ; 

1 ( | , ) ; 

0 ; 

U

x a

f x ab a x b

b a

x b

⎧ <

⎪

⎪ = ≤≤ ⎨ − ⎪

⎪ > ⎩

CDF: 

0 ; 

Pr( ) ( | , ) ; 

1 ; 

U

x a

x a X x F x ab a x b

b a

x b

⎧ <

⎪

⎪ − ≤ = = ≤≤ ⎨ − ⎪

⎪ > ⎩

Moments: 

Mean 

2

a b

μ

+ =

Variance 

2

2 ( )

12

b a

σ − =
 

 26

Sampling from a two-parameter Uniform Distribution: (0,1) U U i ←

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 18 2.3 10 × . 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

PDF: 

2

2

1 () ( | , ) exp ; 2 2 Nx f x x μ μ σ

σ π σ

⎡ ⎤ − = − −∞< <+ ⎢ ⎥ ∞ ⎢ ⎥ ⎣ ⎦

CDF: 

2 1 Pr( ) ( ) exp ; ; 2 2

z

x Xx z d z x ξ μ ξ

π σ −∞

⎛ ⎞ − ≤ = Φ = − = − ∞ < < +∞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ∫

Moments: 

Mean μ

Variance 2 σ

Sampling from a two-parameter Normal Distribution: (,) X N i ← μ σ

Earlier versions of FAVOR used the Box-Müller Transformation Method [68-670] to sample from a 

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, Zi , is then scaled 

to the required random normal deviate with mean, μ , and standard deviation, σ , by. 

(0,1) i

i i

Z N

X Z σ 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 
 

 27

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: 

234

01 2 3 4

234

01 2 3 4

1 for 

2

1 1 for 

2

2ln( )

1

sgn 2 p

p p

x

p p

y x

a ay a y ay a y Z py

b by b y by b y

⎧ < ⎪⎪ = ⎨

⎪ − ≥ ⎪⎩

= −

⎛ ⎞⎛ ⎞ ++ + +

= −+ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ++ + +

 (2) 

where 

1 if 0

sgn( )

1 if 0

x

x

x

⎧− < ⎪ = ⎨

⎪⎩+ ≥

and the coefficients of the rational function are: 

0 0

1 1

2 2

3 3

4 4

-0.3222324310880000 0.0993484626060

-1.0000000000000000 0.5885815704950

-0.3422420885470000 0.5311034623660

-0.0204231210245000 0.1035377528500

-0.0000453642210148 0.0038560700634

a b

a b

a b

a b

a b

= =

= =

= =

= =

= =

The standard normal deviate is then scaled to obtain the required quantile 

X Z p p = + σ μ (3) 

Lognormal Distribution – ( ) log log Λ μ σ,

PDF: ( )2

log log log

2 log log

 0 ; 0

(| , ) ln 1

exp ; 0

2 2

x

f x x

x

x

μ σ μ

σ π σ

Λ

⎧ ≤

⎪

⎪ ⎡ ⎤ = − ⎨ ⎢ ⎥

− < < ∞ ⎪ ⎢ ⎥ ⎪ ⎢ ⎥ ⎩ ⎣ ⎦

CDF: 2 log

log

0 ; 0

Pr( ) ( ) 1 ln

exp ; , 0

2 2

z

x

Xx z x

d z x ξ μ ξ

π σ −∞

⎧ ≤

⎪⎪ ≤ =Φ = ⎨ ⎛ ⎞ − ⎜ ⎟ − = < <∞ ⎪ ⎜ ⎟ ⎪⎩ ⎝ ⎠ ∫

Moments: 

Mean 

2

log

log exp 2

σ

μ μ

⎛ ⎞ ⎜ ⎟ = + ⎜ ⎟

⎝ ⎠
 

 28

Variance 2 2

log log σ ωω μ ω σ = − ( 1)exp(2 ); exp( ) =

Sampling from a two-parameter Lognormal Distribution: log log) ( , Xi ← Λ μ σ

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. 

log log (,)

exp( )

i

i i

Y N

X Y

← μ σ

= (4) 

Weibull Distribution – W(a,b,c) 

(a = location parameter, b = scale parameter, c = shape parameter) 

PDF: ( )

0 ; 

( ,,) 1exp ; ( ( ) / , , , 0) W

x a

f xabc c c c y y y x a bx a bc b

⎧ ≤

⎪ = ⎨ − ⎪ − =− > > ⎩

CDF: 

0 ; 

Pr( ) ( | , , ) 1 exp ; ( ( ) / , , , 0) W c

x a

X x F x abc

y y x a bx a bc

⎧ ≤ ⎪ ≤= = ⎨ ⎡ ⎤ − − =− > > ⎪⎩ ⎢ ⎥ ⎣ ⎦

Moments: 

Mean 1 a b 1

c

μ ⎛ ⎞ =+Γ + ⎜ ⎟ ⎝ ⎠

Variance 22 2 2 1 b 1 1

c c

σ

⎡ ⎤ ⎛⎞ ⎛⎞ = Γ + −Γ + ⎢ ⎥ ⎜⎟ ⎜⎟ ⎣ ⎦ ⎝⎠ ⎝⎠

where ( ) Γ x is Euler’s gamma function. 

Sampling from a three-parameter Weibull Distribution: (,,) Xi ←W abc

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. 

[ ]1/

(0,1)

ln(1 )

i

c

i i

U U

X ab U

←

=+ − −

 (5) 
 

 29

Logistic Distribution – ( , ) L α β

PDF: 2 ( | , ) ; exp , 

(1 ) L

z x f x z x

z

α α β

β β

⎡ ⎤ ⎛ ⎞ − = =− − ⎢ ⎥ ⎜ ⎟ ∞ < < ∞

+ ⎣ ⎦ ⎝ ⎠

CDF: 1 Pr( ) ( | , ) ; exp , 1 L

x X x Fx z x

z

α α β β

⎡ ⎤ ⎛ ⎞ − ≤ = = = − −∞< <∞ ⎢ ⎥ ⎜ ⎟ + ⎣ ⎦ ⎝ ⎠

Moments: 

Mean μ =α

Variance 

2 2

2

3

π β σ =

Sampling from a two-parameter Logistic Distribution (,) X L i ← 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. 

(0,1)

1 ln 1

i

i

i

U U

X

U α β

←

⎛ ⎞ =− − ⎜ ⎟ ⎝ ⎠

 (6) 

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. 
 

 30

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-SR4

 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: 

( ) 1

0 ; 

( ,,)

exp ; ( ( ) / , , , 0) W c c

x a

f xabc c y y y x a bx a bc 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. 
 

 31

 

 

( ) 19.35 8.335exp 0.02254( ) [ksi in.]

( ) 15.61 50.132exp 0.008( ) [ksi in.]

4

Ic

Ic

Ic

RELATIVE RELATIVE K

RELATIVE RELATIVE K

K

a T T

b T T

c

Δ =+ Δ ⎡ ⎤ ⎣ ⎦

Δ =+ Δ ⎡ ⎤ ⎣ ⎦

=

 (8) 

where  ( () )  Δ =− T Tt R RELATIVE TNDT 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 ( ) ( )

0 ; ( )

( ) 1 exp ; ( )

Ic

KIc

Ic

Ic

Ic

n Ic I i j k i j k

n I i jk K

c n I i jk K n I i jk K

K

K K cpi

K a

K a

K a

b

τ τ

τ

τ

τ

≤ ==

⎧ ≤ ⎪

⎪⎪ ⎧ ⎫ ⎨ ⎡ ⎤ − ⎪ ⎪ ⎪ − −⎨ ⎬ ⎢ ⎥ > ⎪ ⎪ ⎪ ⎣ ⎦ ⎪⎩ ⎩ ⎭

 (9) 

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, KIc

a ) 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 KIc

a 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. 
 

 32

Fig. 11. Interaction of the applied KI time history and the Weibull KIc statistical model for a 

postulated flaw. 
 

 33

Table 2. Illustration of Computational Procedure to Determine CPI and CPF for a 

Postulated Flaw (Warm Prestress Not Included) 

Time(τn ) T (τn ) RT NDT T (τn )-RT NDT a bc KI

(τn ) cpi (τn ) Δcpi (τn ) P ( F |I ) Δcpf (τn ) cpf (τn )

(min) (°F) (°F) (°F) (ksi√in) (ksi√in) (-) (ksi√in) (-) (-) (-) (-) (-)

8 360.68 270.0 90.68 83.70 119.16 4 50.90 0 0 000

10 328.28 270.0 58.28 50.35 95.52 4 55.70 9.82E-06 9.82E-06 000

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

KIc Weibull Parameters

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(t

n

) 

cpf(τ n

) = instantaneous conditional probability of failure by through-wall cracking 

CPI = sup-norm5

 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 

(, , ) { }(, , ) ( ) for 1 m i jk i jk CPI cpi τ m n

∞

= ≤ ≤ (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 
 

 34

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(τ

n

)(i,j,k) is the increase in cpi(τ

n

)(i,j,k) that occurs during each discrete time step. 

When the maximum value of cpi(τ

n

)(i,j,k) is reached, negative values of Δcpi(τ

n

)(i,j,k) are 

set to zero. (Note: i = transient index, j = RPV trial index, and k = flaw index) 
 

 35

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 1⁄2-inch in depth is assumed to 

become a 1⁄2-inch deep 360-degree circumferential flaw. An embedded flaw 1⁄2-inch in depth with its 

inner crack tip located at 1⁄2-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 Flaw Category 1 Flaw Category 2 Flaw Category 3 

axial weld circumferential axial axial 

circumferential weld circumferential circumferential circumferential 

plate/forging circumferential 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 
 

 36

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. 
 

 37

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 

PF I ( |)is the conditional probability of failure given initiation; ( | ) PF 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 ( | ) PF I and 

Δcpi(τ

 n). The instantaneous value of the conditional probability of failure at time τ n, cpf(τ n), is 

therefore 

max max

1 1

( ) ( |) ( ) ( )

n n n mm

m m

cpf P F I cpi τ ττ cpf

= =

= ×Δ = Δ ∑ ∑ (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. 
 

 38

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: 

(, , )

1 (, )

( , ,1) ( , ,2) (, , )

Conditional probability

 = (1 )

of non-initiation

(1 )(1 ) (1 )

nflaw

i jk

k i j

i j i j i j nflaw

CPI

CPI CPI CPI

=

⎧ ⎫ ⎪ ⎪ ⎨ ⎬ −

⎪ ⎪ ⎩ ⎭

=− − −

∏

K

 (12) 

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): 

( )

(, ) (, , )

1

( , ,1) ( , ,2) (, , )

 =1- (1 )

1 1 (1 ) (1 )

nflaw

RPV i j i j k

k

i j i j i j nflaw

CPI CPI

CPI CPI CPI

=

−

=− − − ⎡ ⎤ − ⎣ ⎦

∏

K

 (13) 

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. 
 

 39

Table 4. Sources for Ductile-Tearing Data [26, 27, 78, 79, 80, 116] 

Materials Reference

61-67W NUREG/CR-3506

Midland Weld NUREG/CR-5736

P02, 68-71W NUREG/CR-4880

PTSE-1 Post Test NUREG/CR-4106

PTSE-2 Post Test NUREG/CR-4888

W8A & W9A NUREG/CR-5492

Table 5. Chemical Composition of Materials Used in the Ductile-Tearing Model 

Development 

HSST Weld Flux

ID Lot ID C Mn P S Si Cr Ni Mo Cu V

Plate 02 (-) 0.230 1.550 0.009 0.014 0.200 0.040 0.670 0.530 0.140 0.003

Midland Beltine Linde 80 0.083 1.607 0.017 0.006 0.622 0.100 0.574 0.410 0.256 0.006

Midland Nozzle Linde 80 0.083 1.604 0.016 0.007 0.605 0.110 0.574 0.390 0.290 0.008

W8A Linde 80 0.083 1.330 0.011 0.016 0.770 0.120 0.590 0.470 0.390 0.003

W9A Linde 0091 0.190 1.240 0.010 0.008 0.230 0.100 0.700 0.490 0.390

68W Linde 0091 0.150 1.380 0.008 0.009 0.160 0.040 0.130 0.600 0.040 0.007

69W Linde 0091 0.140 1.190 0.010 0.009 0.190 0.090 0.100 0.540 0.120 0.005

70W Linde 0124 0.100 1.480 0.011 0.011 0.440 0.130 0.630 0.470 0.056 0.004

71W Linde 80 0.120 1.580 0.011 0.011 0.540 0.120 0.630 0.450 0.046 0.005

61W Linde 80 btwn A533B 0.090 1.480 0.020 0.014 0.570 0.160 0.630 0.370 0.280 0.005

62W Linde 80 btwn A508 0.083 1.510 0.160 0.007 0.590 0.120 0.537 0.377 0.210 0.010

63W Linde 80 btwn A508 0.098 1.650 0.016 0.011 0.630 0.095 0.685 0.427 0.299 0.011

64W Linde 80 btwn A508 0.085 1.590 0.014 0.015 0.520 0.092 0.660 0.420 0.350 0.007

65W Linde 80 btwn A508 0.080 1.450 0.015 0.015 0.480 0.088 0.597 0.385 0.215 0.006

66W Linde 80 btwn A508 0.092 1.630 0.018 0.009 0.540 0.105 0.595 0.400 0.420 0.009

67W Linde 80 btwn A508 0.082 1.440 0.011 0.012 0.500 0.089 0.590 0.390 0.265 0.007

Chemistry Composition (wt %)

Table 6. Summary of Ductile-Tearing Data Used in the Ductile-Tearing Model Development 

Material Size Fluence Temp. J Ic Avg. T R Avg. USE Material Size Fluence Temp. J Ic Avg. T R Avg. USE Material Size Fluence Temp. J Ic Avg. T R Avg. USE

ID 1019 n/cm2 (°C) (kJ/m2

) (-) (ft-lbf) ID 1019 n/cm2 (°C) (kJ/m2) (-) (ft-lbf) ID 1019 n/cm2 (°C) (kJ/m2) (-) (ft-lbf)

61W 0.8 0 75 142.3 89 62 64W 0.5 0.582 177 119.1 36 75 Mid-Belt NA 0 21 167.4 71 65

61W 0.5 0 75 143.4 106 62 64W 4 0.66 200 78.7 50 75 Mid-Belt NA 0 21 116.4 84 65

61W 0.8 0 121 123.9 74 62 64W 4 0.64 200 94.9 49 75 Mid-Belt NA 0 21 131.4 76 65

61W 0.5 0 121 130.6 90 62 64W 1.6 0.623 200 57.3 46 75 Mid-Belt NA 0 21 164.7 70 65

61W 4 0 200 97.4 100 62 64W 1.6 0.671 200 80.2 50 75 Mid-Belt NA 0 150 133.4 41 65

61W 4 0 200 128.1 72 62 64W 0.8 0.773 200 101.9 31 75 Mid-Belt NA 0 150 125.1 44 65

61W 1.6 0 200 78.3 70 62 64W 0.5 0.672 200 99.4 23 75 Mid-Belt NA 0 150 141.1 60 65

61W 0.8 0 200 89.5 52 62 64W 0.8 0.773 288 46 15 75 Mid-Belt NA 0 288 86.4 32 65

61W 0.5 0 200 89.1 66 62 64W 0.5 0.672 288 66.3 18 75 Mid-Belt NA 0 288 103.3 33 65

61W 1.6 0 288 57.7 68 62 65W 1.6 0 132 123.4 120 108 Mid-Nozz NA 0 21 126.6 47 64

61W 0.8 0 288 66.1 47 62 65W 0.8 0 132 147.2 97 108 Mid-Nozz NA 0 21 113.0 57 64

61W 0.5 0 288 75 53 62 65W 0.5 0 132 118.5 130 108 Mid-Nozz NA 0 150 102.8 39 64

61W 0.5 0 288 76.5 53 62 65W 4 0 177 80.4 138 108 Mid-Nozz NA 0 150 89.9 43 64

61W 0.8 1.1 121 103.1 51 52 65W 0.8 0 177 117.6 76 108 Mid-Nozz NA 0 288 69.1 32 64

61W 1.6 1.3 121 83 41 52 65W 0.5 0 177 114.8 102 108 Mid-Nozz NA 0 288 64.5 39 64

61W 0.5 1.6 121 76.4 22 52 65W 4 0 200 69.3 114 108 Mid-Nozz NA 0 288 64.3 37 64

61W 0.5 1 200 96.4 60 52 65W 1.6 0 200 104.1 72 108 Plate 02 NA 0 50 117.3 197 105

61W 4 1.1 200 52.4 38 52 65W 0.8 0 200 128.9 84 108 Plate 02 NA 0 50 189.9 164 105

61W 1.6 1.2 200 63.6 31 52 65W 0.5 0 200 94.8 111 108 Plate 02 NA 0 50 191.8 154 105

61W 0.8 1.2 200 69.5 44 52 65W 4 0 288 120.1 73 108 Plate 02 NA 0 50 205.1 141 105

61W 4 1.4 200 61.3 30 52 65W 1.6 0 288 71.9 73 108 Plate 02 NA 0 50 218.9 153 105

61W 0.8 1.1 288 46.4 15 52 65W 1.6 0 288 74.2 69 108 Plate 02 NA 0 121 111.0 156 105

61W 0.5 1.4 288 44.6 17 52 65W 0.8 0 288 73.5 56 108 Plate 02 NA 0 121 137.1 178 105

62W 0.5 0 75 121.7 119 93 65W 0.5 0 288 83.8 69 108 Plate 02 NA 0 121 161.7 147 105

62W 1.6 0 149 114.5 124 93 65W 1.6 0.67 132 106.2 77 72 Plate 02 NA 0 121 168.3 133 105

62W 0.8 0 149 150.1 139 93 65W 0.8 0.744 132 113.6 54 72 Plate 02 NA 0 121 171.4 138 105

62W 0.5 0 149 91.4 99 93 65W 0.5 0.767 132 110.3 48 72 Plate 02 NA 0 204 132.1 118 105

62W 4 0 177 107.6 154 93 65W 4 0.74 177 53.1 89 72 Plate 02 NA 0 204 134.7 99 105

62W 0.8 0 177 160.3 115 93 65W 0.8 0.744 177 104.8 45 72 Plate 02 NA 0 204 139.2 115 105

62W 0.5 0 177 101 94 93 65W 0.5 0.629 177 114.7 47 72 Plate 02 NA 0 204 140.4 113 105

62W 4 0 200 145.5 140 93 65W 4 0.61 200 85.6 61 72 Plate 02 NA 0 204 181.0 100 105

62W 1.6 0 200 154.4 117 93 65W 1.6 0.62 200 70.4 56 72 Plate 02 NA 0 288 111.8 81 105

62W 1.6 0 200 128.7 133 93 65W 0.8 0.756 200 91.5 41 72 Plate 02 NA 0 288 112.1 73 105

62W 0.8 0 200 150.8 99 93 65W 0.5 0.629 200 107 54 72 Plate 02 NA 0 288 118.1 92 105

62W 0.5 0 200 78.4 83 93 65W 0.8 0.756 288 41 23 72 Plate 02 NA 0 288 121.9 73 105

62W 0.5 0 200 113.8 87 93 65W 0.5 0.767 288 43.9 32 72 Plate 02 NA 0 288 132.6 89 105

62W 4 0 288 87.3 112 93 66W 0.5 0 100 94.4 41 76 68W NA 0 23 160.1 219 147

62W 1.6 0 288 101 118 93 66W 1.6 0 200 67 55 76 68W NA 0 121 151.1 204 147

62W 0.8 0 288 93.8 59 93 66W 0.8 0 200 103.6 50 76 68W NA 0 121 196.9 204 147

62W 0.5 0 288 83.6 59 93 66W 0.5 0 200 73 42 76 68W NA 0 200 223.5 111 147

62W 0.5 0 288 85 84 93 66W 0.8 0 288 73.8 40 76 68W NA 0 288 121.3 132 147
 

 40

Table 6. (cont.) Summary of Ductile-Tearing Data Used in the Ductile-Tearing Model 

Development 

Material Size Fluence Temp. J Ic Avg. T R Avg. USE Material Size Fluence Temp. J Ic Avg. T R Avg. USE Material Size Fluence Temp. J Ic Avg. T R Avg. USE

ID 1019 n/cm2 (°C) (kJ/m2

) (-) (ft-lbf) ID 1019 n/cm2 (°C) (kJ/m2) (-) (ft-lbf) ID 1019 n/cm2 (°C) (kJ/m2) (-) (ft-lbf)

62W 1.6 1.4 149 118.3 60 80 66W 0.5 0 288 61.9 25 76 68W NA 0 288 190.7 138 147

62W 0.8 1.3 149 118.7 91 80 66W 1.6 0.854 200 68.4 31 58 69W NA 0 50 143.0 87 147

62W 0.5 1.6 149 96.2 32 80 66W 1.6 0.944 200 66.4 29 58 69W NA 0 50 147.9 80 147

62W 0.5 1.3 176 94.1 50 80 66W 0.8 1.022 200 75.2 22 58 69W NA 0 50 163.7 70 147

62W 4 1.4 177 105.9 62 80 66W 0.5 0.896 200 67.4 18 58 69W NA 0 121 139.5 89 147

62W 0.8 1.5 177 127.4 45 80 66W 0.8 1.03 288 42.8 17 58 69W NA 0 121 141.7 93 147

62W 0.5 0.8 177 95.9 34 80 66W 0.5 0.896 288 51.6 16 58 69W NA 0 121 142.7 82 147

62W 4 1.5 200 90 62 80 67W 1.6 0 100 130.4 164 103 69W NA 0 121 158.9 88 147

62W 1.6 1.6 200 85 52 80 67W 0.8 0 100 166.5 112 103 69W NA 0 200 174.5 54 147

62W 0.8 1.3 200 115.9 69 80 67W 0.5 0 100 132.8 98 103 69W NA 0 204 98.9 76 147

62W 0.5 1 200 63.3 29 80 67W 4 0 200 97.4 121 103 69W NA 0 204 117.5 61 147

62W 0.8 1.5 288 60.9 24 80 67W 1.6 0 200 84.1 116 103 69W NA 0 288 89.7 56 147

62W 0.5 1.5 288 61.9 24 80 67W 0.8 0 200 118 85 103 69W NA 0 288 94.1 49 147

63W 1.6 0 100 118 120 87 67W 0.5 0 200 102.1 76 103 69W NA 0 288 103.8 56 147

63W 0.8 0 100 141.2 95 87 67W 0.5 0 200 92 69 103 69W NA 0 288 129.4 56 147

63W 0.5 0 100 131.1 86 87 67W 4 0 288 97.9 58 103 70W NA 0 50 106.2 188 74

63W 4 0 171 148.4 100 87 67W 1.6 0 288 63.4 83 103 70W NA 0 50 177.8 163 74

63W 1.6 0 171 103.5 97 87 67W 0.8 0 288 82.6 56 103 70W NA 0 121 127.5 159 74

63W 0.8 0 171 112.4 77 87 67W 0.5 0 288 80 51 103 70W NA 0 121 131.1 148 74

63W 0.5 0 171 113.2 88 87 67W 4 0.86 200 67.3 45 73 70W NA 0 121 142.8 140 74

63W 4 0 200 77.7 113 87 67W 4 0.96 200 56.7 57 73 70W NA 0 204 103.3 108 74

63W 1.6 0 200 79.6 94 87 67W 0.8 1.022 200 76.3 45 73 70W NA 0 204 112.0 133 74

63W 0.8 0 200 120.3 69 87 67W 0.5 0.834 200 92.2 32 73 70W NA 0 204 121.0 110 74

63W 0.5 0 200 89.2 70 87 67W 0.8 1.03 288 58.6 23 73 70W NA 0 288 89.0 79 74

63W 0.5 0 200 98.4 80 87 67W 0.5 0.617 288 80 24 73 70W NA 0 288 105.6 93 74

63W 4 0 288 88.4 62 87 W8A 1 0 0 104.4 72 58 70W NA 0 288 106.2 88 74

63W 1.6 0 288 122.4 64 87 W8A 1 0 75 94.4 81 58 71W NA 0 30 128.0 186 81

63W 0.8 0 288 66.8 57 87 W8A 1 0 200 79.7 57 58 71W NA 0 50 97.9 144 81

63W 0.5 0 288 59.1 55 87 W8A 1 0 288 58.6 34 58 71W NA 0 50 121.0 98 81

63W 0.5 0 288 66.7 52 87 W8A 1 2.1 125 69.9 16 36 71W NA 0 121 110.8 153 81

63W 0.5 1.1 149 68.4 43 68 W8A 1 2.1 200 54.1 14 36 71W NA 0 121 126.7 105 81

63W 1.6 1.3 171 79.2 49 68 W8A 1 2.1 288 38.6 9 36 71W NA 0 121 131.0 155 81

63W 0.8 1.1 171 89.7 32 68 W8A 1 1.5 30 80.8 54 40 71W NA 0 204 77.6 66 81

63W 0.5 1.3 171 78.9 27 68 W8A 1 1.5 75 84.6 28 40 71W NA 0 204 84.7 87 81

63W 4 1.25 200 72.7 16 68 W8A 1 1.5 200 60 17 40 71W NA 0 204 115.4 90 81

63W 1.6 1.4 200 62.2 29 68 W8A 1 1.5 200 57.4 18 40 71W NA 0 288 64.5 72 81

63W 0.8 1.1 200 75.8 33 68 W8A 1 1.5 288 41.6 11 40 71W NA 0 288 77.4 71 81

63W 0.5 0.9 200 77 49 68 W9A 1 0 -40 207.4 NA 115 71W NA 0 288 80.2 61 81

63W 0.5 1 204 56.3 42 68 W9A 1 0 0 255 173 115

63W 0.8 1.4 288 42.7 19 68 W9A 1 0 75 195.9 170 115

63W 0.5 1.2 288 51.5 23 68 W9A 1 0 200 147.9 130 115

64W 1.6 0 100 105.7 148 100 W9A 1 0 288 92.9 120 115

64W 0.8 0 100 160.4 105 100 W9A 1 0 288 116 97 115

64W 0.5 0 100 116 89 100 W9A 1 2.1 75 156.2 42 74

64W 4 0 177 117.4 146 100 W9A 1 2.1 200 124.1 37 74

64W 1.6 0 177 134.6 103 100 W9A 1 2.1 200 147.7 40 74

64W 0.8 0 177 114.9 83 100 W9A 1 2.1 288 81.5 31 74

64W 0.5 0 177 125 73 100 W9A 1 1.5 75 167.7 52 84

64W 4 0 200 161.4 96 100 W9A 1 1.5 200 146.4 46 84

64W 1.6 0 200 67.8 97 100 W9A 1 1.5 200 127.2 47 84

64W 0.8 0 200 118.8 76 100 W9A 1 1.5 288 96.1 36 84

64W 0.5 0 200 115.8 54 100 PTSE-2 NA 0 100 64 120 46.4

64W 4 0 288 85.5 96 100 PTSE-2 NA 0 100 55.6 145 46.4

64W 1.6 0 288 76.6 83 100 PTSE-2 NA 0 175 58.3 106 46.4

64W 0.8 0 288 75.9 54 100 PTSE-2 NA 0 175 68.4 105 46.4

64W 0.5 0 288 74.2 44 100 PTSE-2 NA 0 250 52.8 67 46.4

64W 0.8 0.773 177 92.9 37 75 PTSE-2 NA 0 250 52.2 61 46.4
 

 41

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: 

 () m

R J = Δ C a (14) 

where the tearing resistance is characterized by the material’s local tearing modulus, TR, defined by 

( 1)

2 2

R m

R

f f

E E dJ T m C a

σ σ da

− ⎛⎞ ⎛⎞ ⎛ ⎞ = = ××× ⎜⎟ ⎜⎟ ⎜ ⎟ Δ ⎝ ⎠ ⎝⎠ ⎝⎠

 (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 

(0.2 mm offset) 0 2( ) f J = Δ −Δ σ a a (16) 
 

 42

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 0 Δ = a 0.2 mm (0.008 in) . Therefore, with an estimate of JIc and the 

power-law exponent, m, the power-law coefficient, C, is 

0 0

0

2( ) 2

 

2

m Ic

Ic m

Ic

Ic f

f

Ic

m

Ic

f

J J Ca C

a

J J aa a a

J C

J

a

σ

σ

σ

= Δ ⇒ = Δ

= Δ −Δ ⇒ Δ = +Δ

∴ =

⎛ ⎞ ⎜ ⎟ + Δ

⎝ ⎠

 (17) 
 

 43

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) 

)

 ( )

[ ] 0

1

2.25

0

27.82 122.4 ln(1 ) 32

 C

1.8

NDT DT RT PT

T

−

⎧ ⎫ ⎪ ⎪ ⎡ ⎤

+ − −− − ⎨ ⎬ ⎢ ⎥ ⎪ ⎪ ⎩ ⎭ ⎣ ⎦ = ° (18) 

Where  ( )   RT r RT RT r NDT DT − , , K K = +Δ NDT 0 NDT ( ) , (see Eq. (91)) with 

RT NDT 0 equal to the 

sampled unirradiated value of RTNDT, 



ΔRT r NDT ( ) ,K equal to the shift due to radiation embrittlement, 

and T0 P = 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 

  T T US =+ ° 50.1 0.794 C ( ) 0 [ ] (19) 
 

 44

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] 

{ ( ) } ( )

( ) [ ]

2 2

0

( ) 2

1000 30 70exp 0.019 1 kJ m

where

207200 57.1 MPa and 0.3

US

c med

US

T T

J

E

E T

ν

ν

+ −− ⎡ ⎤ ⎣ ⎦ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦

= − =

 (20) 

Step 4. – Calculate an Estimate for Ic ΔJ at TUS

Using the relationship derived in [83] to characterize the temperature dependence of JIc

{ } ( ) ( ) ()

288 C

1 2 3

1

-1 -1

2

-1

3

 2.09 exp 273.15 273.15 ln

1033 MPa

where 0.00698 K 0.0004 sec

0.000415 K 3.3318 MPa

meas

Ic Ic Ic

US US ref

ref

JJ J

C CT CT

C

C

C

ε σ

ε

σ

° Δ= − =

⎡ ⎤ −+ ++ − ⎣ ⎦

=

= =

= =

&

&

 (21) 

Step 5. – Calculate an Estimated Mean and Standard Deviation for the Aleatory Uncertainty in JIc 

At a given wall temperature, T Rt wall( , ) C[° ], an estimated mean value for JIc can now be estimated 

by 

{ } ( ) ( ) ()

( )

12 3 2

kJ 2.09 exp 273.15 273.15 ln 

m

Ic c med Ic

wall wall ref

JJ J

C CT CT ε σ

= −Δ +

⎡ ⎤ ⎡ ⎤ −+ + + − ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ & (22) 

Where an estimate for the standard deviation is given in [83] by 

( ) 2

kJ 62.023exp 0.0048 Ic m

σ J w T all

⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦

 (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 

)

( ) 2

kJ , Ic m

Ic Ic J J NJ σ ⎡ ⎤ ← ⎢ ⎥ ⎣ ⎦

 (24) 
 

 45

where the sampled value is truncated at  2 2 Ic Ic Ic J Ic Ic J J JJ − ≤≤+ σ σ 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 ( ) m

R J Ca = Δ ) is estimated based on the 

sampled value of JIc and the local value of the wall temperature, ( ,) T Rt wall , from the following 

equation (developed from the data given in [83]) 

[ ] 

( )3 2

-09

std-error

2

C

exp kJ/m

0.1117 5.8701 10

0.4696 -758.19

0.08425

0.2992

wall

Ic

T

mab c J

d

a c

b d

R

σ

⎛ ⎞ ° = + + ⎡ ⎤ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

= =×

= =

=

=

 (25) 

The J-R curve exponent m with aleatory uncertainty can then be sampled from the following normal 

distribution: )

m Nm ← ( ,0.08425) (26) 

The J-R curve coefficient, C, then follows from 

) 



)

) (at )

(at )

0 2

wall

wall

Ic T

m

Ic T

f

J

C

J

a

σ

=

⎛ ⎞ ⎜ ⎟ + Δ ⎜ ⎟ ⎝ ⎠

 (27) 

where 

σ f is the sampled flow stress and 0 Δ = a 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) 
 

 46

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 

( ) ( ) 0.2223

1.456 0.8894

() ( ) 19 0.0570 17.5 1 1.17 305 [ft-lbf] 10 i u

t USE A USE f Cu Ni P ⎛ ⎞ φ =+ ⋅ − ⋅ ⋅+ + ⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠ (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 %; φt 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

61.0 for plates

66.3 for forgings

1 1 0.138 tanh

2 2 0.0846

A

Cu f Cu

⎧

⎪ = ⎨

⎪

⎩

⎡ − ⎤ = + ⎢ ⎥ ⎣ ⎦

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, USE( ) u μ . 

Step 2. At this value of USE( ) u μ , sample a value for the standard deviation from a normal 

distribution given by 

) ( ) ( ) ( )

( ) ( )

2

( )

( )

4.3296 0.0857 0.0012

( ,2.2789)

u u u

u u

USE mean USE USE

USE N USE mean

σ μ μ

σ σ

=− +

←

 (29) 
 

 47

Step 3. Sample a value for the unirradiated upper-shelf energy, 

USE( ) u , from the following 

normal distribution 

 )

( ) ( ) ( ) (, ) u u USE USE N u U ← μ σ SE (30) 

Step 4. The irradiated value for the upper-shelf energy is then estimated from Eq. (28), or, 

applying sampling notation: 

  

( )  ( ) )  0.2223

1.456 0.8894

( ) ( ) 19 0.0570 17.5 1 1.17 305 [ft-lbf] 10 i u

t USE A USE f Cu Ni P ⎡ ⎤⎛ ⎞ φ =+ ⋅ − ⋅ ⋅+ + ⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎣ ⎦⎝ ⎠

 (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 φt to 

estimate a value of 

USE( )i using Eq. (31). 

Step 3. Convert this estimate of 

USE( )i value to a value of  ( )(at 550 F) Ic KJ i ° at 550°F using the mean 

curve established in [84], where the uncertainty in  ( )(at 550 F) Ic KJ i ° is not sampled, 

  ( )(at 550 F) 70.855 0.5784 [ksi in ] ( ) ( ) Ic K U J i ° =+ × SE i (32) 

Step 4. Convert the  ( )(at 550 F) Ic KJ i ° value to a  ( )(at ) Ic wall KJi T value at the wall temperature of interest 

using the mean curve from [84]: 

 

( )

(at 550 F) (at )

459.69 0.000415 ln 0.0004 

1.8

1.35 1033 exp [ksi in ] 459.69 0.00698

1.8

Ic Ic Ic wall J J J T

wall

ref

wall

KK K

T

T

σ

Δ= − = °

⎧ ⎫ ⎡ ⎤ ⎛ ⎞ + ⎪ ⎪ ⋅ ⎢ ⎥ ⎜ ⎟ ⎪ ⎝⎠ ⎪ = ⋅ ⎨ ⎬ −

⎪ ⎪ ⎛ ⎞ + − ⎜ ⎟ ⎩ ⎭ ⎣ ⎦ ⎝ ⎠

 (33) 

 where σ ref is 

( ) 550 459.69 0.000415 ln 0.0004 

1.8

1033 exp = 3.331798

550 459.69 0.00698

1.8

σ ref

⎡ ⎤ ⎛ ⎞ + ⋅ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ = ⋅ ⎛ ⎞ + − ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

 (34) 

and Twall is the wall temperature at the crack tip in °F. Therefore 
 

 48

   (at ) (at 550 F) [ksi in ] Ic wall Ic Ic KK K JT J = +Δ ° J (35) 

 The required sampled value of JIc follows from the plane strain conversion 

)  2 2 2 (at ) (at )

1 [in-kips/in ] wall Ic wall J K Ic T J T

E

⎛ ⎞ −ν = ⎜ ⎟ ⎝ ⎠

 (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 φt 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 U mat i( )(at 550 F) ° =+ × 3.9389 0.5721 ( ) SE( )i (37) 

Step 4. Convert the )

T mat i( )(at 550 F) ° value to a ) ( )(at ) T mat i Twall value at the wall temperature of interest 

using the mean curve from [84]: 

 )

( )

( )(at 550 F) ( )(at )

459.69 0.000415 ln 0.0004 

1.8

1.38 1033 exp [-] 459.69 0.00698

1.8

wall mat mat i T mat i

wall

ref

wall

TT T

T

T

σ

Δ= − = °

⎧ ⎫ ⎡ ⎤ ⎛ ⎞ + ⎪ ⎪ ⋅ ⎢ ⎥ ⎜ ⎟ ⎪ ⎝⎠ ⎪ = ⋅ ⎨ ⎬ −

⎪ ⎪ ⎛ ⎞ + − ⎜ ⎟ ⎩ ⎭ ⎣ ⎦ ⎝ ⎠

 (38) 

 where σ ref is 

( ) 550 459.69 0.000415 ln 0.0004 

1.8

1033 exp = 3.331798

550 459.69 0.00698

1.8

σ ref

⎡ ⎤ ⎛ ⎞ + ⋅ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ = ⋅ ⎛ ⎞ + − ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

 (39) 

 and Twall is the wall temperature at the crack tip in °F. Therefore 
 

 49

) )  ( )(at ) ( )(at 550 F) [-] TT T mat i T mat i wall = + ° Δ 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 T =+ × 0.3214 0.0019 ( ) mat i( ) (41) 

Step 6. Calculate a value for the JR power-law coefficient, C, from the definition of JIc in ASTM 

E-1820 

) )

)

)

) ( )(at )

( )(at )

0

 

2

wall

m

wall

Ic i T

Ic i T

f

J C

J

a

σ

=

⎛ ⎞ ⎜ ⎟ + Δ

⎝ ⎠

 (42) 

 where 0 Δ = a 0.2 mm (0.008 in) and 

σ f is the sampled flow stress. 
 

 50

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 d / 0 cpi dt > , then the flaw becomes a candidate for propagation 

through the wall. (This submodel will be described in detail in the following.) If 

d /d 0 cpi t ≤ , 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., N CPI STEP NSTEP > , branch to Step G9. If the 

transient is not finished, cycle to Step G3. Note that N CPI STEP NSTEP = at which 

cpi t cpi t CPI () () ∞ = = . 

 

6

 The value of Pf represents the percentile used in sampling ΔRT ARREST (see Step 11 in Sect. 4.5) and KIa

(see Step 15 in Sect. 4.5) in Step P6 and in sampling 

KIc 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. 
 

 51

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. 
 

 52

(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. 
 

 53

(b)

Fig. 17 (continued) (b) IGA Propagation submodel to test for Stable Arrest (no failure) and 

Vessel Failure. 
 

 54

(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. 
 

 55

(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: 

(, , ) (, , )

1

( ) ( |)

( ) ( |)

NSTEPCPI

m m i jk i jk

m

m

CPF cpi t P F I

NFAIL m PF I

NTRIALS

=

= Δ

=

∑

 (43) 

 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. 
 

 56

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, σflow u( ) . After each resampling of 

ΔT30 , the 

flow stress will have been adjusted by the following relation: 

 

( ) 30

0.112 ksi/ F for welds

 where 

0.131 ksi/ F for plates flow flow u T ⎧ ° σ = σ + γΔ γ = ⎨

⎩ °

This sampled value of 

σflow 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 

( )

( )

( ) 1 hoop stress ( ) ; 2 axial stress

i i m o i

p Ra t

R Ra

τ σ β β+ ⎧ = = ⎨ − − ⎩

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 
 

 57



if REINITIATED_BY_DUCTILE_TEARING is TRUE then

or

_ is TRUE if then 

or

 vessel failure = TRUE during ductile tearing

 

m flow

o i

FAIL UDT

a FAILCR

R R

⎧ ⎫ σ σ> ⎪ ⎪

⎨ ⎬

⎪ ⎪

⎛ ⎞ ⎜ ⎟ >

⎝ ⎠ − ⎩ ⎭



 return to Step G5 in Model

elseif then 

 vessel failure = TRUE during flaw growth by cleavage

 return to Step G5 in Mode

m flow

o i

IGA

or

a FAILCR

R R

IGA

σ σ

⎧ ⎫

⎪ ⎪ >

⎨ ⎬

⎪ ⎪ ⎛ ⎞ ⎜ ⎟ > − ⎩ ⎭ ⎝ ⎠

l

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. 

 ( ) 1

2.177 45.586 131.27 ln 1 [ F]

14.4 [ F]

epistemic

epistemic arrest epistemic

RT

RT RT −

Δ =− + − −Φ ° ⎡ ⎤ ⎢ ⎥ ⎣ ⎦

Δ = Δ −°



 
 

 58

Retrieve the previously sampled unirradiated value of RTNDTo

 for this subregion and the 

sampled value of the irradiation shift for this flaw,  (, ) ΔRT r NDT 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 



ln( ) ln( ) ( , ) [ F] ARREST ARREST ARREST RT RT μ σ RT Δ Δ ←Λ Δ °  

where (see Appendix F for the development of this protocol) 

( ) 0

2

ln( )

( ) ln( )

( ) 0

0

2 ln( )

ln

2

44.122exp 0.005971 [ C]

32 /1.8 [ C]

ln exp 0.38998

ARREST

ARREST

ARREST

RT

ARREST mean RT

ARREST mean

NDT epist arrest

RT

RT

RT T

T RT RT

σ

μ

σ

Δ

Δ

−

Δ

=Δ − ⎡ ⎤ ⎣ ⎦

Δ = − ×° ⎡ ⎤ ⎣ ⎦

= −Δ − °

=

  

 

  

 { } ( ) ( ) ( ) 0

2

0

0 0 0

0

2ln( ) var 2ln

(12.778) for 35.7 C

var( ) 99.905972 -1.7748073 for -35.7 C 56 C

0 for 56 C

RT ARREST mean T RT ARREST mean

T

T TT

T

⎡ ⎤ +Δ − − Δ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

⎧ <− ° ⎪

⎪ = ° ⎨ ≤ ≤ °

⎪

⎪ > ° ⎩

  



  



Calculate the estimated arrest temperature7

 by Eq. (127) in Step 12 of Sect. 4.5 

  0 ARREST (, ) NDT epist arrest ARREST (, ) RT r RT RT RT RT r NDT = −Δ +Δ +Δ − K 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 

  Δ =− T r T r t RT r RELATIVE (, ) (,) (, ) K K ARREST

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, KI−initiation Φ , 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 



ln( )

ln( )

1 ln( ) ( ) erf 1

2 2

Ia

I initiation

Ia

RELATIVE I initiation K

K

K

K T μ

σ −

− ⎡ ⎤ ⎛ ⎞ − Δ

Φ = ⎢ ⎥ ⎜ ⎟ +

⎣ ⎦ ⎝ ⎠

where 

 

7

 The subregion value of RTNDTo

 is not re-sampled in this step.
 

 59

 

2

0

(mean)

ln( )

2 erf( ) error function = exp( ) d ; erf( ) erf( )

if _Model is equal to 1

( ) 27.302 69.962exp 0.006057( ) [ksi in.]

0.18

else if _Model is equal to 2

Ia

x

Ia

RELATIVE RELATIVE Ia

K

Ia

x x x

K

K T T

K

K

ξ ξ

π

σ

= − − = −

Δ =+ ⎡ ⎤ Δ ⎣ ⎦

=

∫

 

 

(mean)

ln( )

2

ln( )

ln( ) (mean)

( ) 27.302 70.6998exp 0.008991( ) [ksi in.]

0.34

( ) ln ( ) 2

Ia

Ia

Ia

RELATIVE RELATIVE Ia

K

K

RELATIVE RELATIVE K Ia

T T

T KT

σ

σ

μ

Δ =+ ⎡ ⎤ Δ ⎣ ⎦

=

Δ= Δ − ⎡ ⎤ ⎣ ⎦

In the above relation for KI −initiation Φ , ln( ) KIa μ is calculated at the location of the initiation of 

the flaw. For this flaw, the value of KI−initiation Φ 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 KI−initiation Φ (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) 

 ( )( ) K fK Ia I initiation P − Φ= Φ

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 KIa Φ fractile, draw a value of KIa from its lognormal distribution as given by 

Eq. (132) of Step 15 in Sect. 4.5 

 

ln( ) ln( ) ( , ) exp ( )

standard normal deviate corresponding

 to the fractile 

Ia Ia Ia Ia

KIa

Ia

RELATIVE RELATIVE Ia K K K K

K

KT Z T

Z

σ μ Φ

Φ

ΦΔ = ⎡ + Δ ⎤ ⎣ ⎦

=

Φ

In the above relation for KIa , ln( ) KIa μ 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 KIa

ZΦ has been determined for this IGA trial, the arrest 

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 
 

 60

  

( )  ( ) )  0.2223

1.456 0.8894

( ) ( ) 19 0.0570 17.5 1 1.17 305 [ft-lbf] 10 i u

t USE A USE f Cu Ni P ⎡ ⎤⎛ ⎞ φ =+ ⋅ − ⋅ ⋅+ + ⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎣ ⎦⎝ ⎠

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 then

 the flaw has arrested

 proceed to Step P8

else

 the flaw has not arrested

 proceed to Step P2 

K K I Ia <

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 

  Δ =− T r T r RT r RELATIVE (, ) (, ) (, ) K K τ RTNDT

Now calculate the parameters of the KIc model 

 

 

( ) 19.35 8.335exp 0.02254( ) [ksi in.]

( ) 15.61 50.132exp 0.008( ) [ksi in.]

4

Ic

Ic

Ic

RELATIVE RELATIVE K

RELATIVE RELATIVE K

K

a T T

b T T

c

Δ =+ Δ ⎡ ⎤ ⎣ ⎦

Δ =+ Δ ⎡ ⎤ ⎣ ⎦

=

with KIc in ksi√in and ΔT = (T-RTNDT) in °F. 

The static initiation toughness, KIc, is calculated from its Weibull distribution by 

    



(max)

1/

( ) ( ) ( ) ln(1 ) 

for ( )

KIc

Ic Ic

Ic

RELATIVE RELATIVE RELATIVE Ic K K f

RELATIVE K Ic Ic

c KT a T b T P

a T KK

Δ = Δ + Δ −− ⎡ ⎤ ⎣ ⎦

Δ ≤≤

Proceed to Step P9. 
 

 61

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 and _ and _ are both FALSE then

 No re-initiation.

 Proceed to Step P10.

else if _ is on and is TRUE then

 No re-initiation

 Proceed to Step P10

else if 

K K STABLE DT FAIL UDT I Ic

WPS OPTION WPS

F

<

_ is TRUE then

 the vessel has failed by unstable ductile tearing

 set vessel failure to TRUE

 return to Step G5 of model

else if _ is TRUE and is less than then

 the fla

Ic J Ic

AIL UDT

IGA

STABLE DT K K

0

w has re-initiated by a ductile-tearing event

 REINITIATED_BY_DUCTILE_TEARING = TRUE

 the current level of tearing is set by the ductile-tearing model

 Proceed to Step P3

else

 The flaw has re-i

Δa

0

nitiated by a cleavage event.

 REINITIATED_BY_DUCTILE_TEARING = FALSE

 Reset the current level of tearing 0

 Proceed to Step P2 and advance the flaw

Δ = a

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 
 

 62

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 *

RJ will be known, where *RJ is a measure of the current 

deformation state due to tearing. 

if then

 _ FALSE

 _ FALSE

 Return to Step P3 or P9 of Submodel 

else

 Proceed to Step D2

T T wall DT

FAIL UDT

STABLE DT

IGA

<

=

=

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. 



) )

2

2 2

 from either Ductile-Tearing Model No. 1 or 2

 , and from either Ductile-Tearing Model No. 1 or 2

(1 ) [in-kips/in ]

get 

get

Proceed to Step D3

applied I applied

Ic

C m

J K

E

J

ν −− =

Step D3. The submodel then compares the Japplied to the estimated value of JIc obtained in Step D2 

and the known value of *

RJ . If this is the first entry into the model or if a cleavage 

reinitiation has occurred since the last entry into the model, then * 0 RJ = . *

R J 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 *

R J . 

* if ( ) or ( ) then

 _ FALSE

 _ FALSE

 Return to Step P3 or P9 of Submodel 

else

 Proceed to Step D4

 

applied Ic applied R J JJ J

FAIL UDT

STABLE DT

IGA

< ≤

=

=

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. 
 

 63

( )

0

*

*

* 1

2 2

*

ln( ) ln( ) exp , [in]

*

R applied

R

R m

R

flow a flow

J J

J C

a

m

aa a

E E dJ T m C a

σ σ da

−

Δ

=

⎡ ⎤ − Δ = ⎢ ⎥

⎢ ⎥ ⎣ ⎦

= +Δ

⎛⎞ ⎛⎞

= =× ⎜⎟ ⎜⎟ × × Δ

⎝⎠ ⎝⎠

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. 

( )

2

1 1 2

1

1

1

2 *

( 1) , O ( 1)

where

applied n nn

n n

n n

n n

applied

applied

flow a a

dJ J JJ

x

da x

xx x

x x

x x

E dJ

T

da

α α

α α

α

σ

+ −

−

+

−

=

+− − ≈ Δ

+ Δ

Δ= −

− = −

⎛ ⎞

= ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

Step D5. 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. 

0

0

if then

 _ = TRUE

 _ = FALSE

 Return to Step P3 or Step P9 in the Submodel

else

 _ = FALSE

 _ = TRUE

 

 *

 Return to Step P3 or Step

T T applied R

FAIL UDT

STABLE DT

IGA Propagation

FAIL UDT

STABLE DT

a a

a a

>

Δ =Δ

=

P9 in the IGA Propagation Submodel
 

 64

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. 
 

 65

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). 



( ) (, ) ( ) of transient- initiating frequency φ E CDF i i ← i j

End of Transient Loop 

The above loop generates a vector of transient-initiating frequencies for this 

vessel simulation, 

{ }(1 )

( )

NTRAN

φ E × . 

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. 





( ) ( )

1

( ) ( )

1

( ) ( ) (, )

( ) ( ) (, )

TRAN

TRAN

N

j i

i

N

j i

i

I E PFMI i j

F E PFMF i j

φ

φ

=

=

Φ =

Φ =

∑

∑

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, 

( ) ( ) j Φ I . 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 
 

 66

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, { } 1 ( ) NSIM

I × Φ , and the frequency of vessel failure, { } 1 ( ) NSIMF × Φ . The above process is 

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. 
 

 67

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. 
 

 68

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: 

1 () () pT T c T kT r

rr r

ρ

τ

∂∂ ∂ ⎡ ⎤ = ⎢ ⎥ ∂∂ ∂ ⎣ ⎦

 (44) 

where ρ is the mass density, ( ) p c 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 

1 1 ( ) 0 for ; (0, ) T T Tr r

rr r

λ τ

τ

∂∂ ∂ ⎡ ⎤

− =∈ ∈ ∞ ⎢ ⎥ ∂∂ ∂ ⎣ ⎦  (45) 

where the property grouping () () () λ T kT c T = ρ p 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

( ,0) for Tr T R r R = ≤≤ initial i o

 (46) 

( )

Boundary Conditions

( , ) ( ) ( ) ( , ) at 

( , ) 0 at 

i i i

o o

qR t ht T t T R t r R

qR t r R

=− = ∞

= =

 (47) 

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 and Ri o R 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 ∞ . 
 

 69

Fig. 21. Isoparametric mapping from parameter space to axisymmetric 1  Euclidean space 

using three-node quadratic basis functions. 

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 

{ }

1

1

2 2

2

3

3

( ) (1 ) (1 2)

1 1 ( ) ( ) 2(1 ) ; 4 2 2

( ) (1 ) (1 2 )

dN

d N

dN dN N N

d d

N

dN

d

ξ ξ ξξ ξ

ξξ ξ ξ ξ ξ

ξ ξξ ξ

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, i ξ , 
 

 70

{ } { } 1 4

1 1

3 2 6/5 1 1

7 2 6 6/5

3 2 6/5 1 1

7 6 2 6/ 5

( ) ( ) where ;

1 1 3 2 6/5

2 6 6/5 7

1 1

3 2 6/5 2 6 6/5

7

i i i i

i

gd g ξξ ω ξ ξ ω

+

− =

⎧ ⎫ + ⎧ ⎫ ⎪ ⎪ − − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − − ⎪ + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ≈ == ⎨ ⎬⎨ ⎬ ⎪ ⎪⎪ ⎪ − + ⎪

⎪

⎪

⎪ + − ⎪ ⎪ ⎪⎩ ⎭⎪ ⎩ ⎭

∫ ∑ (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. 
 

 71

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 ( , , ) uvw be the radial, circumferential, and axial displacements, respectively, of a material point 

in a cylindrical ( , , ) r z θ coordinate system. The general two-dimensional axisymmetric case requires 

that 

 0; 0; 0 rzr z v θθ θθ = == == τ τ γ γ (50) 

where , r z θ θ τ τ are shear stresses and , r z θ θ γ γ are engineering shear strains. The strain-displacement 

relationships for the two-dimensional case are 

0

1 0

0

rr

zz

zr

r

r u

w

z

z r

θθ

ε

ε

ε

γ

⎡ ⎤ ∂

⎢ ⎥ ∂

⎢ ⎥ ⎧ ⎫ ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪⎪ ⎢ ⎥ ⎨ ⎬ ⎨⎬ = ⎢ ⎥ ⎪ ⎪ ⎢ ⎥ ∂ ⎪⎩ ⎭⎪ ⎪ ⎪ ⎢ ⎥ ∂ ⎪ ⎪ ⎩ ⎭ ⎢ ⎥

⎢ ⎥ ∂ ∂

⎢ ⎥ ⎣ ⎦ ∂ ∂

 (51) 

For the one-dimensional axisymmetric case, ( , , ) r z θ are principal directions, and w z = ∂∂ = 0; 0;

such that 

; ; 0; 0 rr zz zr

u u w uw

r r z zr

θθ ε εε γ

∂ ∂∂ ∂

= = = = =+ =

∂ ∂∂ ∂ (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 

( ) 1 ( ) (1 )(1 2 ) 1 2 rr rr ref

E E θθ T T α

σ ν ε νε

ν ν ν

= −+− − ⎡ ⎤ ⎣ ⎦ + − − (53) 

[ ] (1 ) ( ) (1 )(1 2 ) 1 2 rr ref

E E θθ θθ T T α

σ ν ε νε

ν ν ν

= − +− −

+ − − (54) 

where 
 

 72

 

 radial normal stress

 circumferential (hoop) normal stress

 radial normal strain

 circumferential (hoop) normal strain

 wall temperature as a function of 

 thermal stress-free refe

rr

rr

ref

T r

T

θθ

θθ

σ

σ

ε

ε

=

=

=

=

=

= rence temperature

 radial position in wall

 Young's modulus of elasticity

 Poisson's ratio

 linear coefficient of thermal expansion

r

E

ν

α

=

=

=

=

For generalized plane-strain conditions, the stress in the axial direction, PS σ zz , is given by 

( )( ) PS σ νσ σ α zz r = +− − r θθ 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 

2 o

i

R PS PS

zz R f = π σ rdr ∫ (56) 

where and Ri o R 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 ν , and 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 

( )( ) PS σ νσ σ α zz j j rr j j j j j ref − −− = +− − θθ E T T (57) 

is the axial stress in each element under plane-strain conditions. To achieve a free-end condition, the 

force PS

j f [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 

j Δf is the reduction in the plane-strain force, PSj f , on element j, then 

1 2

11 2 2

nele

nele nele

f f f

AE AE A E

Δ Δ Δ = =L (58) 

and 

1

( )0

nele PS

j j

j

f f =

∑ +Δ = (59) 

where 
 

 73

( )

2 2

( )

( )

PS

j j j rr j j j j j ref

j oj ij

f A ET T

A rr

νσ σ α θθ

π

− −

− −

= +− − ⎡ ⎤ ⎣ ⎦

= −

 (60) 

where ro and ri are the outer and inner radii of element j, respectively. Let p j f − 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 

1 2

11 2 2

p p p nele

nele nele

ff f

AE AE A E

−− − = =L (61) 

and 

2

1

nele

pj o

j

f π R p −

=

∑ = (62) 

where 

j j pj f f f =Δ + −

Recalling that the uniform change in axial strain has no effect on σ rr and σθθ , Eqs. (60), (61), and 

(62) can be solved for fj after calculating values of and σ σ rr j − − θθ j ; then the axial stress is calculated 

from 

( ) PS

j j

zz j

j

f f

A

σ −

+ = (63) 

FAVOR uses a reduced-integration two-point Gauss-Legendre quadrature rule for the calculation of 

 and σ σ rr θθ in each element. The Gauss sample points and weights for two-point quadrature are: 

{ } { } 1 2

1 1

1

3 1

( ) ( ) where ;

1 1

3

i i i i

i

gd g ξξ ω ξ ξ ω

+

− =

⎧ ⎫

⎪ ⎪ − ⎧ ⎫ ⎪ ⎪ ⎪⎪ ≈ == ⎨ ⎬ ⎨⎬ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎪ +

⎩ ⎭

∫ ∑ (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]. 
 

 74

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 

 for axial flaws

2

 for circumferential flaws

2

I

I

zz

K

r

K

r

σθθ

π

σ

π

=

=

 (65) 

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 

[919-94], including the developers of OCA-I [95] and OCA-P [96]. The HSST Program at ORNL 

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. 
 

 75

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 

2 3

01 2 3 σ () ( /) ( /) ( /) a C Ca a Ca a Ca a ′ ′′ ′ =+ + + (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 ( 0123 CCCC ,,, ) 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 *

0 0

() () ( ) I Ij jj

j j

K a K a C aK a π = == = ∑ ∑ (67) 

where 

'

*

'

( ) ( ) Ij

j

j

K a

K a

C a π = (68) 

Values of ' ' ( )/ KaC a Ij j π 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 ' 1 Cj = . The dimensionless quantity *

( ) K a j is referred to as the 

influence coefficient. For semi-elliptic flaws, *( ) K a j values can be calculated for several points along 

the crack front, in which case Eq. (67) becomes 

3 *

0

( ) ( ) I jj

j

K C aK φ π φ == ∑ (69) 

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., 90 φ = ° ). 

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-
 

 76

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 ( / ) 0.5 ≤ ≤ a t 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 2 2

2 2

3(1 )

R t i

v

π

⎡ ⎤ = ⎢ ⎥ ⎣ − ⎦

l (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 / 0.5 ≤ < a t 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. 
 

 77

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. 
 

 78

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

*

1

() ( ,)

n

I i ii i

i

K a aK a a σ

=

= Δ ∑ ′ (71) 

circumferential flaws

*

1

() 2( ) ( ,)

n

I i i ii i

i

K a R a aK a a π σ =

= +Δ ∑ ′ ′ (72) 

where 

n

i=1

 an increment of about such that 

 radial distance from open end of crack to point of application of unit load,

 average crack-free stress over for equivalent problem

 o

i i i

i

i i

I

a a a a a

a

a

K

σ

Δ= Δ= ′

′ =

= Δ

=

∑

*

pening Mode I stress-intensity factor

 stress-intensity factor per unit load applied at , where load has dimensions

 of force/length for axial flaws and force for circumferential flaws

 numbe

K a I i

n

= ′

= * r of points along length of crack for which are available,

 inside radius of vessel.

Ki

R =

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 t

1/2), 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 t

3/2 ). These normalizing factors account for the 
 

 79

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 / 0.95 a a ′ ≥ 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. 
 

 80

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. 
 

 81

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 M M aQ I mm bb = + σ σπ (73) 

where: 

2 the minor axis of the elliptical subsurface flaw

 flaw shape parameter

 free-surface correction factor for membrane stresses

 free-surface correction factor for bending stresses

 membrane s

m

b

m

a

Q

M

M

σ

=

=

=

=

= tress

 bending stress σ b =

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: 
 

 82

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. 
 

 83

( ) ( ) 2 1

1 1

( ) () ˆ( / 2) /2 ( ) 2

x x

t t x x m a

σ σ

σ σ σ − = = × −+ (74) 

( ) ( ) 1 2 () ( ) ˆ(0) / 2

2 m

x x

t b a

σ σ

σσ σ − = −= × (75) 

The formal definition of the shape parameter Q is based on the complete elliptic integral of the second 

kind, E(x), 

( )

2

/ 2

2

0

2

() ()

( ) 1 sin ( ) for 0 1

1 4

Qx E x

E x xd x

a

x

L

π

θ θ

=

= − ≤≤

⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠

∫ (76) 

In ref. [103], the elliptic integral is replaced by an infinite-series approximation for Q of the form 

2 2 24 6 2 2

8 10

2

5 7 1

4(1 ) 4 64 256 128 256

mmm Q m m

m

π ⎡ ⎤ ⎛⎞ ⎛ ⎞ ≈ +++ + + ⎢ ⎥ ⎜⎟ ⎜ ⎟

+ ⎝⎠ ⎝ ⎠ ⎣ ⎦

 (77) 

where 

1 2( / )

1 2( / )

a L

m

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: 

( )( )

2468

12 3 4 5

20

6

1/2

(2 / ) (2 / ) (2 / ) (2 / )

(2 / )

1 2/ 2/

M D D at D at D at D at

m

D at

et at

=+ + + + +

⎡ ⎤ − − ⎣ ⎦

 (78) 

where: 

1 D =1

2 D = 0.5948

() () 2

3 D ea ea = ++ 1.9502 / 0.7816 / 0.4812

() () ()

( )

432

4 3.1913 / 1.6206 / 1.8806 /

0.4207 / 0.3963

D ea ea ea

e a

= +++

+

() () ()

() () ()

654

5

3 2

6.8410 / 3.6902 / 2.7301 /

1.4472 / 1.8104 / 0.3199 /

0.3354

D ea ea ea

ea ea ea

=+++

+ ++
 

 84

6 D = 0.303

The equation for the free-surface correction factor for bending stresses (Mb ) is: 

( ) ( )( )

( )( ) ( )

( ) ( )( )

( )( )

2

23 4

2

5 6

2 2

78 9

1 1/2

2 / (2 / ) 2 / 2 /

2/ 2/ 2/

2/ 2/ 2/

1 2/ 2/

b

E et E et E et at

E at et E at

E at E et at E

M E

et at

⎡ ⎤ ++ + ⎢ ⎥

+ +

+ + ⎣ ⎦ = +

⎡ ⎤ − − ⎣ ⎦

 (79) 

where: 

1 E = 0.8408685 , 2 E =1.509002 , 3 E = −0.603778 , 

4 E = −0.7731469 , 5 E = 0.1294097, 6 E = 0.8841685 , 

7 E = −0.07410377 , 8 E = 0.04428577 9 E = −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 
 

 85

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. 
 

 86

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, KI cfp − , of the crack-face pressure loading to the total stress 

intensity factor at the deepest point of the flaw ( ) φ = ° 90 by 

*

0 ( ) K aK p I cfp π τ − =

where p( ) τ is the coolant pressure in ksi at time τ in the transient. By linear superposition, the 

crack-face pressure component, KI 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

*

1

() () ( , )

n

I cfp i i i

i

K a p aK a a τ −

=

= Δ ∑ ′

for circumferential flaws

*

1

() 2 ( ) () ( , )

n

I cfp i i i i

i

K a R a p aK a a π τ −

=

= +Δ ∑ ′ ′

where n

i=1

 an increment of about such that 

 radial distance from open end of crack to point of application of unit load,

( ) coolant pressure at time uniformly applied over the crack 

i i i

i

a a a a a

a

p τ τ

Δ= Δ= ′

′ =

=

∑

*

face

 opening Mode I stress-intensity factor contribution due to crack-face pressure

 stress-intensity factor per unit load applied at , where load has dimensions

 of force/length for ax

I cfp

I i

K

K a

− =

= ′

*

ial flaws and force for circumferential flaws

 number of points along length of crack for which are available,

 inside radius of vessel.

i n K

R

=

=
 

 87

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: 

RT RT M RT PTS NDT U = + +Δ ( ) PTS (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 

2 2

( )

2

 the standard deviation for 

 the standard deviation for 

U

U NDT U

NDT

M

RT

RT

σ σ

σ

σ

Δ

Δ

= +

=

= Δ

 (81) 

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 
 

 88

10 (0.28 0.10log ( ( )))

19 2

( )()

 chemistry factor, a continuous function of copper and nickel

( ) best-estimate neutron fluence [10 n/cm ; 1 MeV] attenuated

 from the inner surface to the

f RT CF f PTS

CF

f E

δ δ

δ

− Δ =

=

= >

 clad/base metal interface

δ = distance from the inner surface to the clad/base metal interface [in.]

 (82) 

The fast-neutron fluence is attenuated through the wall by the relation 

 ( ) (0)exp( 0.24 ) f f δ δ = −× (83) 

where δ is in inches and (0) f 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 
 

 89

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

  ) 

)

( ) 

( )  ( )  ( )

30 0 exposure

0.4601 1.250

0 0

-17

-17

17

( , , , ( ), , ,product form)

19310 exp 1 110 ( ) 1 2.40 ( ) ( ) 460

8.86 10 for welds 

9.30 10 for forgings

12.7 10 for plates 

c

c

T Ni Cu P f r T

A P f r B Ni f Cu g f r Bias T

A

τ

−

Δ =

⎛ ⎞ ⎜ ⎟ + + + +

+ ⎝ ⎠

⎧ ⎫ × ⎪ ⎪

= × ⎨ ⎬

⎪

⎪ × ⎩ ⎭



 12

10 0 exposu

0

230 for welds 

132 for forgings 

206 for plates in CE vessels

 156 for other plates 

log ( ) 4.579 10 1 1 ( ( )) tanh 2 2

B

f r

gf r

τ

⎪

⎪

⎧ ⎫

⎪ ⎪

= ⎨ ⎬

⎪ ⎪

⎩ ⎭

+ ×

= + ( )



( )





( ) 

re

0.659

18.265

0.713

 0 for 0.072 wt %

-0.072 for 0.072 wt %

Cu

f Cu

Cu Cu

⎡ ⎤ − ⎢ ⎥

⎣ ⎦

⎧ ⎫ ≤ ⎪ ⎪ = ⎨ ⎬

⎪ ⎪ > ⎩ ⎭

 (84) 

subject to 

 

8

 A curved overbar, X , indicates a sampled random variate. 
 

 90

max

exposure

exposure

0.25 for welds with Linde 80 or Linde 0091 flux

0.305 for everything else 

and

0 for 97000 h

9.4 for 97000 h

Cu

Bias

τ

τ

⎧ ⎫ ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ ⎩ ⎭

⎧ ⎫ < ⎪ ⎪ = ⎨ ⎬

⎪ ⎪ ≥ ⎩ ⎭

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%, 0f ( )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, 0f (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]. 







30

30

0.99 ( , ) welds (, )

1.10 ( , ) plates and forgings

NDT

T r

RT r

T r

⎧

⎪ Δ

Δ = ⎨

⎪⎩ Δ

K K

K

 (85) 
 

 91

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]). 
 

 92

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, 



0f (0) , at the inner surface of the RPV wall. This attenuation takes the following form 

 )

0 0 f ( ) (0) exp( 0.24 ) a f = ×− a (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 

)

 )

 ) 

0

( ,)

(0) ( , )

global subregion

mean subregion global

local mean

mean local

SIGFGL fluence

f N fluence

SIGFLC f

f Nf

σ

σ

σ

σ

= ×

←

= ×

←

 (87) 

where the best-estimate fluence, subregion fluence , 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. [1121 1-116]. The extended KIc database, compiled from the amended EPRI data and from 

the ORNL survey, provided a total of 255 fracture-toughness data points from 18 materials for input 
 

 93

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 [118-1120] to augment the 

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]. 
 

 94

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. 
 

 95

(a) 

(b) 

Fig. 33. KIa databases (a) ORNL 99/27 KIa database and (b) Extended KIa database. 
 

 96

Table 7. Summary of ORNL 99/27 KIc Extended Database 

 Temp. (T-RTNDT0) No. of 

 Specimen Size Range Range Data 

 Material Source Type Range (°F) (°F) Points 

EPRI Database EPRI NP-719-SR 

1 HSST 01 subarc 

weldment 

Shabbits (1969) C(T) 1T - 6T -200 to -50 -200 to -50 8 

2 A533B Cl. 1 

subarc weld 

Shabbits (1969) C(T) 1T - 8T -200 to 0 -200 to 0 8 

3 HSST 01 Mager (1970) C(T) 1T -150 -170 17 

4 HSST 03 Mager (1970) C(T) 1T -150 -170 9 

5 A533B Cl. 1 Mager (1969) WOL 1T - 2T -320 to -150 -385 to -215 13 

6 HSST 02 Mager (1969) WOL & C(T) 1T - 2T -200 to 0 -200 to 0 41 

6 HSST 02 Shabbits (1969) C(T) 1T - 11T -250 to 50 -250 to 50 28 

7 A533B Cl. 1 

weldment 

Mager (1969) WOL 1T - 2T -320 to -200 -275 to -155 10 

8 A533 B Cl. 1 

weldment/HAZ 

Mager (1969) WOL 1T - 2T -320 to -200 -320 to -200 6 

9 A508 Cl.2 

European Forging 

Mager (1969) WOL 1T - 2T -320 to -100 -370 to -150 12 

10 A508 Class 2 unpublished C(T) 2T - 6T -150 to 0 -201 to -51 9 

11 A508 Class 2 unpublished C(T) 2T - 8T -125 to -75 -190 to -30 10 

 Total 171 

Additional Data 

12 HSSI Weld 72W NUREG/CR-5913. C(T) 1T-6T -238 to 50 -229 to 59 13 

13 HSSI Weld 73W NUREG/CR-5913 C(T) 1T-4T -238 to -58 -209 to -29 10 

14 HSST Plate 13A NUREG/CR-5788 C(T) 1⁄2T-4T -238 to -103 -229 to -94 43 

15 A508 Cl. 3 ASTM STP 803 Bx2B C(T) 1T-4T -238 to -4 -225 to 9 6 

16 Midland Nozzle 

Course Weld 

NUREG/CR-6249 C(T) 1T -148 to -58 -200 to -110 6 

17 Midland Beltline NUREG/CR-6249 C(T) 1T -148 -171 2 

18 Plate 02 4th Irr. 

Series (68-71W) 

NUREG/CR-4880 C(T) 1T -148 to -139 -148 to -139 4 

 Total 84 

Grand Total 255 

Table 8. Summary of KIa Extended Database 

Test Temp. (T-RTNDT) No. of

Specimen Size Range Range Data Points

Material Source Type Range (°F) (°F)

EPRI Database EPRI NP-719-SR

1 HSST 02 Ripling (1971) CCA crack arrest 1T-3T -150 to 121 -150 to 121 50

Additional Data Additional Data

2 HSSI Weld 72W NUREG/CR-5584 CCA crack arrest -78 to 41 -68 to 51 32

3 HSSI Weld 73W NUREG/CR-5584 CCA crack arrest -78 to 59 -48 to 89 26

4 MW15J NUREG/CR-6621 CCA crack arrest -4 to 50 -36 to 18 4

Large Specimen Data

5 HSST WP1 NUREG/CR-5330 Wide Plate Tests (-) 84 to 198 94 to 207 18

6 HSST WP2 NUREG/CR-5451 Wide Plate Tests (-) 142 tp 324 2 to 184 38

7 HSST PTSE-1 NUREG/CR-4106 Pressurized Vessel (-) 326 to 354 100 to 158 2

8 HSST PTSE-2 NUREG/CR-4888 Pressurized Vessel (-) 267 tp 325 130 to 158 3

9 HSST TSE NUREG/CR-4249 Thermally-Shocked Cylinder (-) 72 to 268 -63 to 103 10

Total = 183
 

 97

Table 9. Chemistry and Heat Treatment of Principal Materials: ORNL 99/27 Database 

 Chemistry – wt (%) Heat 

Material ID Specification Source C P Mn Ni Mo Si Cr Cu S Al Treatment 

HSST 01 A533B Cl. 1 Mager 

(1970) 

.22 .012 1.48 .68 .52 .25 - - .018 - Note 1 

HSST 02 A533B Cl. 1 Mager 

(1969) 

.22 .012 1.48 .68 .52 .25 - - .018 - Note 2 

HSST 03 A533B Cl. 1 Mager 

(1970) 

.20 .011 1.26 .56 .45 .25 .10 .13 .018 .034 Note 3 

HSST 02 A533B Cl. 1 Shabbits 

(1969) 

.22 .012 1.48 .68 .52 .25 - - .018 - Note 4 

HSST 01 

subarc weld 

A533B Cl. 1 Shabbits 

(1969) 

.12 .014 1.35 .65 .52 .23 - - .012 - Note 5 

B&W subarc 

weldment 

A533B Cl. 1 Shabbits 

(1969) 

.10 .009 1.77 .64 .42 .36 - - .015 - Note 6 

PW/PH 

weldment 

A533B Cl. 1 Mager 

(1969) 

.09 .019 1.25 1.0

8 

.52 .23 .05 .22 .13 .037 Note 7 

MD07 

European 

A508 Cl. 2 

Ring forging 

Mager 

(1969) 

.18 .009 1.16 .72 .51 .24 .28 - .10 - Note 8 

- A533B 

Cl. 1 

Mager 

(1969) 

.19 .012 1.37 .52 .45 .25 .13 .15 .016 .048 Note 9 

72W A533B weld 5788 .09 .006 1.66 .60 .58 .04 .27 .23 .006 - 

73W A533B weld 5788 .10 .005 1.56 .60 .58 .04 .25 .21 .005 - 

Notes: 

1. Normalizing: 1675 °F 4 hr, air cooled 

 Austentizing: 1600 °F 4 hr 

 Quenching: Water quench 

 Tempering: 1225 °F 4 hr, furnace cooled 

 Stress Relief: 1150 °F 40 hr, furnace cooled 

2. Normalizing: 1675 °F 4 hr, air cooled 

 Austentizing: 1600 °F 4 hr 

 Quenching: Water quench 

 Tempering: 1225 °F 4 hr, furnace cooled 

 Stress Relief: 1150 °F 40 hr, furnace cooled 

3. Normalizing: 1675 °F 12 hr, air cooled 

 Austentizing: 1575 °F 12 hr 

 Quenching: Water quench 

 Tempering: 1175 °F 12 hr, furnace cooled 

 Stress Relief: 1125 °F 40 hr, furnace cooled 

4. Normalizing: 1675 ± 25 °F 4 hr 

Austentizing: 1520 °F – 1620 °F 4 hr 

Quenching: Water quench. 

Tempering: 1200 °F – 1245 °F 4 hr, air cooled 

Stress Relief: 1150 ± 25 °F 40 hr, furnace cooled to 600 °F 

5. Post Weld: 1150 ± 25 °F 12 hr 

 Intermediate 1100 ± 25 °F 15 min 

6. Post Weld 1100 °F – 1150 °F 12 hr 

 Intermediate 1100 °F – 1150 °F 15 min 

7. 620 °C 27 hr, air cooled 

8. 925 °C 5 hr 

 Quenching: Water quench 

 650 °C 3 hr, furnace cooled 

 620 °C 24 hr, air cooled 

9. 910 °C 8 hr 

 Quenching: Water quench 

 680 °C 10 hr, furnace cooled 

 850 °C 8 hr 

Quenching: Water quench 

 690 °C 8 hr, air cooled 

 620 °C 24 hr, air cooled 
 

 98

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. 
 

 99

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. 
 

 100

Table 10. Materials Used from the ORNL 99/27 KIc Extended Database 

ID Form RTNDT (°F) T0 (°F)* RTNDT - T0 P Tq (°F)** 

HSST-03 Plate 20 31 -11 0.0455 26.1 

HSST-02 Plate 0 -17 17 0.1104 -17.4 

HSST-01 Plate 20 -1 21 0.1753 -2.9 

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 -56.7 

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 -104.4 

A533B Cl. 1 Weld -45 -151 106 0.6299 -151.5 

A508 Cl. 2 Forging 51 -60 111 0.6948 -59.9 

A508 Cl. 2 Forging 65 -55 120 0.7597 -5.8 

A533B Cl. 1 HAZ 0 -132 132 0.8247 -132.3 

A533B Cl. 1 Plate 65 -74 139 0.8896 -73.8 

A508 Cl. 2 Forging 50 -124 174 0.9545 -119.3 

*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]. 

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 RT RT RT LB NDT LB = −Δ 0 . 
 

 101

Table 11. Values of Lower-Bounding Reference Temperature 

with and without Sample-Size Adjustment: ORNL 99/27 Database 

Material ID 

No. 

Sample RTNDT0 RTLB

(k)* ΔRTLB

(k)* Size 

Correct.

RTLB

(k)** ΔRTepistemic

(k)** 

k Size, N (°F) (°F) (°F) (°F) (°F) (°F) 

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 

( )* ( ) ( )*

(0)

kk k Δ= − RT RT RT LB NDT LB

( )** k RTLB = lower-bounding reference temperature for the kth material with sample size-adjustment 

( )** ( ) ( )**

0

k kk Δ =− RT RT RT epistemic NDT LB

The adjusted ASME lower-bounding curve shown in Fig. 36 has the following form: 

K T Ic =+ − 23.65 29.56exp 0.02( ) ksi in. [ RTNDT ] (88) 

with ( ) in F. T RT − ° NDT The adjustment for sample size indicated in Table 11 assumes that Eq. (88) 

represents a 0.01 fractile. The RT T NDT 0 0 − CDF (Figs. 34 and 37) is a Weibull distribution with a 

flaw-size dependence 

( )

0 0

1/4

1

1

1

1

( ) ( ,122.4,2.25)

1.8 80 10

ln [ F] 0.019 70

27.82 F

 flaw length [in.]

 1.0 in.

NDT xT

xT T

xT T

T

xT

T

RT T W a

B B

a a

a

B

B

− ←

⎡ ⎤ − = − ⎢ ⎥ °

⎣ ⎦

=− °

=

=

 (89) 
 

 102

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) ): 



 ( ) 1/ 2.177

( 45.586,131.27,2.177)

45.586 131.27 ln 1 [ F]

where U(0,1)

epistemic

epistemic

RT W

RT

Δ ←−

Δ =− + − −Φ ° ⎡ ⎤ ⎣ ⎦

Φ ←

 (90) 

Combined with the sampled irradiation-shift term described in Sect. 4.2.2, the irradiated value of 



RTNDT is calculated by 

    RT r RT RT RT r NDT (, ) K K = −Δ +Δ NDT 0 epistemic NDT (, ) (91) 

where  0 0 0 ( ,) NDT NDT NDT RT N RT ← σ RT 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 . 
 

 103

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. 
 

 104

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 T Ia mean ( ) = + 30 70exp 0.019 [MPa m] ( ) TKIa ⎡ − ⎤ ⎣ ⎦ (92) 

where ( ) T T− KIa 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: 

( ) 44.123 exp 0.006 [ C] { 0} ARREST mean K o Ia Δ ≡ −= ⋅ − ° RT T T T (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). 
 

 105

Table 12. ORNL 99/27 KIa Database – Reference-Transition Temperatures 

Material Product Sample RTNDT0 RTLB T0 RTArrest TKIa 

ID Form Size (°F) (°F) (°F) (°F) (°F) 

HSST-02 Plate 50 0 -2.1 -17 76.8 75.2 

72W Weld 32 -9.4 -42.6 -70 49.8 8.6 

73W Weld 26 -29.2 -67.6 -78 34.1 6.8 

Midland Weld 4 32.2 -58.9 NA NA NA 

Fig. 38. Lognormal distribution of 0 ARREST KIa Δ =− RT TT as a function of T0

Fig. 39. Lognormal probability densities for ΔRTArrest as function of T0. 
 

 106

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. 

Δ =Δ ° RT RT epist arrest − 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 

  0 ARREST (, ) epist arrest ARREST (, ) RT r RT RT RT RT r NDT NDT = −Δ +Δ +Δ − K K    (95) 

where 0 , epist arrest , and ( , ) RT RT NDT RT r NDT Δ Δ −    L have not been re-sampled from their initiation values 

and 

ln( ) ln( ) (,) ARREST ARREST ARREST RT RT μ σ RT Δ Δ ←Λ Δ

  is sampled from the following lognormal distribution: 
 

 107

( ) 0

2

ln( )

( ) ln( )

0

( ) 0

ln( )

ln

2

where

32 /1.8 [ C]

44.122exp 0.005971 [ C]

ln exp 0.3

ARREST

ARREST

ARREST

RT

ARREST mean RT

NDT epist arrest

ARREST mean

RT

RT

T RT RT

RT T

σ

μ

σ

Δ

Δ

−

Δ

=Δ − ⎡ ⎤ ⎣ ⎦

= −Δ − °

Δ = − ×° ⎡ ⎤ ⎣ ⎦

=

  

  

 

 { } ( ) 2 ( ) ( ) 0

2

0

0 0 0

0

8998 2ln( ) var 2ln

where

(12.778) for 35.7 C

var( ) 99.905972 -1.7748073 for -35.7 C 56 C

0 for 56 C

RT ARREST mean T RT ARREST mean

T

T TT

T

⎡ ⎤ +Δ − − Δ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

⎧ <− ° ⎪

⎪ = ° ⎨ ≤ ≤ °

⎪

⎪ > ° ⎩

  



  



 (96) 

and 

ΔRT ARREST is sampled from (see Step 11 in Sect. 4.5) 

 

  



ln( ) 1.8exp ln( ) [ F]

(0,1); is the standard normal deviate corresponding to the fractile

(0 1) for this trial in the crack - -

Arrest f Arrest

f f

ARREST RT P RT

P P f

f

RT Z

ZN Z P

P Initiation Growth Arre

σ μ Δ Δ Δ = ⎡ ⎤ + ° ⎣ ⎦

←

< <

 

st 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 ;

Pr( ) ( | , ) ( ) 1 exp ;

( )

Ic

KIc

Ic Ic Ic Ic

Ic

Ic

I K

Ic I K I K K RELATIVE I K

K I

RELATIVE K

c

K a

K K Ka b Ka T

a K

b T

⎧ ≤

⎪

⎪ ⎡ ⎤ < =Φ = ⎨ ⎛ ⎞ − Δ ⎢ ⎥ ⎪ − −⎜ ⎟ < <∞

⎪ Δ ⎝ ⎠ ⎩ ⎣ ⎦

 (97) 

where the inverse CDF or percentile function is given by 

    

(max)

1/

( ) ( ) ( ) ln(1 ) for 0 1

for 

KIc

Ic Ic Ic Ic Ic K K K K

Ic Ic

c K Ta Tb T

aK K

Δ = Δ + Δ − −Φ ⎡ ⎤ <Φ < ⎣ ⎦

≤ ≤

 (98) 

where the bounding value of KIc(max) is input by the user to FAVOR (typically KIc(max) =

200 ksi in. ). The parameters of the distribution are 
 

 108

 

 

( ) 19.35 8.335exp 0.02254( ) [ksi in.]

( ) 15.61 50.132exp 0.008( ) [ksi in.]

4

Ic

Ic

Ic

RELATIVE RELATIVE K

RELATIVE RELATIVE K

K

a T T

b T T

c

Δ =+ Δ ⎡ ⎤ ⎣ ⎦

Δ =+ Δ ⎡ ⎤ ⎣ ⎦

=

 (99) 

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  () (, )  T T RT r RELATIVE NDT Δ =− τ 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). 
 

 109

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 

2 ln( )

2

2 (ln( ) ) /

1 () Pr{ } exp d

2 2

ln( ) 1 exp d 2 2

x

x

X x

x

x μ σ

ξ μ ξ

σ π σ

μ ς ς

σ π

−∞

−

−∞

⎡ ⎤ ⎛ ⎞ − ≤ = ⎢ ⎥ −⎜ ⎟ =

⎣ ⎦ ⎝ ⎠

⎛ ⎞ − ⎡ ⎤ Φ= − ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦

∫

∫

 (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( ) ln( )

ln( ) ln( )

ln( ) ( ) ln( ) ( ) 1 Pr erf 1

2 2

Ia Ia

Ia

Ia Ia

I K I K

Ia I K

K K

KT KT

K K μ μ

σ σ

⎛ ⎞ − Δ ⎡ ⎛ ⎞ − Δ ⎤

≤ =Φ = + ⎜ ⎟ ⎢ ⎜ ⎟ ⎥ ⎢ ⎥ ⎝ ⎠ ⎣ ⎝ ⎠ ⎦

 (101) 

where KIa Φ is now the cumulative probability of crack extension and the error function (a special 

case of the incomplete gamma function, P(a,x

2

)) is defined by 

( ) 2 2

0

2 0.5, erf( ) exp( ) d

erf( ) erf( )

x

Px x

x x

ξ ξ

π == −

− =−

∫ (102) 

The inverse CDF for the lognormal distribution allows sampling of KIa by 

  ) 

)





ln( ) ln( ) ( , ) exp ( )

 standard normal deviate 

 corresponding to the fractile 

(0,1)

Ia Ia Ia Ia

KIa

Ia

Ia

K RELATIVE K RELATIVE Ia K K

K

K

K T Z T

Z

U

σ μ Φ

Φ

ΦΔ = ⎡ + Δ ⎤ ⎣ ⎦

=

Φ

Φ ←

 (103) 
 

 110

(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. 
 

 111

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 

 

 

 

2

ln( )

ln( )

ln( )

(mean)

( ) ln ( ) 2

where

0.18

( ) 27.302 69.962exp 0.006057( ) [ksi in.]

( , ) ( , ) [ F]

Ia

Ia

Ia

K

RELATIVE RELATIVE K Ia

K

RELATIVE RELATIVE Ia

RELATIVE Arrest

T KT

K T T

T T r RT r

σ

μ

σ

τ

Δ =Δ − ⎡ ⎤ ⎣ ⎦

=

Δ =+ ⎡ ⎤ Δ ⎣ ⎦

Δ =− ° L

 (104) 

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 

 

 

 

2

ln( )

ln( )

ln( )

(mean)

( ) ln ( ) 2

where

0.34

( ) 27.302 70.6998exp 0.008991( ) [ksi in.]

( , ) ( , ) [ F]

Ia

Ia

Ia

K

RELATIVE RELATIVE K Ia

K

RELATIVE RELATIVE Ia

RELATIVE Arrest

T KT

K T T

T T r RT r

σ

μ

σ

τ

Δ =Δ − ⎡ ⎤ ⎣ ⎦

=

Δ =+ ⎡ ⎤ Δ ⎣ ⎦

Δ =− ° L

 (105) 

Model 2 will be automatically selected when the ductile-tearing model is activated, and any specified 

upper bound on KIa is ignored. 
 

 112

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 HE HE HE Cu Ni P , , and 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: 





)

1

1

1

( ,)

( ,)

( ,)

Flaw Cu Cu

Flaw Ni Ni

Flaw P P

Cu N HE

Ni N HE

P N HE

σ

σ

σ

←

←

←

 (106) 

where the recommended constant standard deviations are 

0.0073 wt%

0.0244 wt%

0.0013 wt%

Cu

Ni

P

σ =

σ =

σ =

 (107) 

The triplet ( , ,) σσσ Cu Ni P is supplied by the user in the input file for the FAVPFM module. Negative 

values of sampled   ) Cu Ni P Flaw Flaw 1 1 , , and 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. 
 

 113

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: 





)

7

1

7

1

5

1

( 3.89 10 ,0.00191)

( 1.39 10 ,0.00678)

(1.30 10 ,0.000286)

Flawx Flaw

Flawx Flaw

Flawx Flaw

Cu Cu L

Ni Ni L

P PL

−

−

−

← +− ×

← +− ×

←+×

 (108) 











7

1

7

1

5

1 3.89 10 0.00191ln 1 for (0,1)

1 1.39 10 0.00678ln 1 for (0,1)

1 1.3 10 0.000286ln 1

Cu Flawx Cu

Cu

Flawx Flaw Cu Flawx

Ni Flawx Ni

Ni

Flawx Flaw Ni Flawx

P Flawx

P

U

Cu Cu

U

Ni Ni

− −

−

− −

−

− −

⎡ ⎤ Δ =− × − − Φ ← ⎢ ⎥ ⎣ ⎦ Φ

= +Δ

⎡ ⎤ Δ =− × − − Φ ← ⎢ ⎥ ⎣ ⎦ Φ

= +Δ

⎡ Δ =× − − ⎢

⎣Φ



1

 for (0,1) P

Flawx Flaw P Flawx

U

P P −

⎤

⎥ Φ ←

⎦

= +Δ

 (109) 

Negative values of sampled   ) Cu Ni P Flawx Flawx , , and 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, CuFlaw1: 

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: 

 

 ( ) ( )

1 ( ,)

0.167 , min 0.0718 ,0.0185

Flaw Cu Cu

Cu Cu Cu

Cu N HE

N HE HE

σ

σ

←

←× ×

 (110) 
 

 114

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 N HE Flaw1 ← ( ,0.162) Ni (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%. 

 )

) 1

1

1 (, )

(0.029,0.0165)

Flaw

Flaw

Flaw Ni Ni

Ni

Ni N HE

N

σ

σ

←

←

 (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. 

)

Flaw1 ( ,0.0013) P N HE ← P (113) 

Negative values of sampled   ) Cu Ni P Flaw Flaw 1 1 , , and 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, CuFlawx : 

The local variability for Cu is determined by sampling a ΔCu value drawn from a logistic distribution 

with parameters 8 α β 6.85 10 and 0.0072 − =× = such that 





 

8

8

1

(6.85 10 ,0.0072)

1 6.85 10 0.0072ln 1 for (0,1)

Cu Flawx

Cu Flawx Cu

Cu

Flawx Flaw Cu Flawx

L

U

Cu Cu

− −

− −

−

Δ ←×

⎡ ⎤ Δ = × − − Φ← ⎢ ⎥ ⎣ ⎦ Φ

= +Δ

 (114) 
 

 115

Nickel, NiFlawx : 

The local variability for Ni is determined by sampling a ΔNi value drawn from a logistic distribution 

with parameters 0.0014 and 0.00647 α = − β = such that 

)

)

 )

1

( 0.0014,0.00647)

1 0.0014 0.00647ln 1 for (0,1)

Ni Flawx

Ni Flawx Ni

Ni

Flawx Flaw Ni Flawx

L

U

Ni Ni

−

−

−

Δ ←−

⎡ ⎤ Δ =− − − Φ ← ⎢ ⎥ ⎣ ⎦ Φ

= +Δ

 (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 ΔP value drawn from a logistic 

distribution with parameters 6 α β 3.27 10 and 0.000449 − =× = . 

)

)

 )

6

6

1

(3.27 10 ,0.000449)

1 3.27 10 0.000449ln 1 for (0,1)

P Flawx

P Flawx P

P

Flawx Flaw Ni Flawx

L

U

P P

− −

− −

−

Δ ←×

⎡ ⎤ Δ = × − − Φ← ⎢ ⎥ ⎣ ⎦ Φ

= +Δ

 (116) 

Negative values of sampled   ) Cu Ni P Flawx Flawx , , and 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. 
 

 116

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, 

CuFlawx , 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, 

ΔCu Flawx − , as determined from its logistic 

distribution, could possibly be sufficiently negative such that the perturbed value of 

CuFlawx might 

take on a value below the truncation upper bound. However, if the value of 

CuFlawx should exceed the 

upper truncation boundary, then FAVPFM will automatically truncate back to the appropriate upper 

bound. 
 

 117

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. 
 

 118

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: 

2

Number of Flaws 2 3 2

in Weld Subregions 360 8

 inner surface-breaking flaw density (per unit surface area - flaws/in )

 weld embedded-flaw density (per 

SB i EW

SB

EW

R dz d dA π ρ θ ρ

ρ

ρ

⎛ ⎞ ⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = + ⎢ ⎥⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎝ ⎠ ⎝ ⎠

=

= 2

2

unit weld-fusion area - flaws/in )

 user-input weld-fusion area (for one side of weld) (in - input by user)

 internal radius of RPV (in. - input by user)

 height of subregion element (in. - in

i

dA

R

dz

=

=

= put by user)

dθ = subtended angle of subregion element ( degrees - input by user)

 (117) 

where ρSB 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: 

( ) ( )2 2

2

Number of Flaws 2 3

in Plate Subregions 360 8 360

 inner surface-breaking flaw density (per unit surface area - flaws/in )

 plate em

SB i EP o i

SB

EP

d R dz d R R CLTH dz π θ ρ θ ρ π

ρ

ρ

⎛ ⎞ ⎡ ⎤⎡ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎤ ⎜ ⎟ = +− ⎢ ⎥⎢ ⎜ ⎟ ⎜ ⎟ − ⎜ ⎟⎥ ⎝ ⎠ ⎣ ⎦⎣ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎦

=

=

3

bedded-flaw density summed over all flaw depths 

 (flaws per unit volume - flaws/in )

 external radius of RPV (in - input by user)

 internal radius of RPV (in. - input by user)

 cladding thi

o

i

R

R

CLTH

=

=

= ckness (in. - input by user)

 height of subregion element (in. - input by user)

 subtended angle of subregion element 

 ( degrees - input by user)

dz

dθ

=

=

 (118) 

where ρSB and ρ EP are summed over all flaw depths.
 

 119

(a)

(b)

Fig. 43. Weld fusion area definitions for (a) axial-weld subregion elements and 

(b) circumferential subregion elements. 
 

 120

Fig. 43. (continued) (c) Plate subregion element. 
 

 121

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 N Ni N P σ σσ Cu Ni 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, 0 0 (0) ( (0), ) N f σ f . 

Provide a best estimate for the standard deviations, RTNDT 0 σ , of unirradiated RTNDT 0 and 

RTNDT

σ Δ of the irradiation shift model ΔRTNDT . The value of RTNDTσ Δ 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 RTRTND0 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  0 0 0 ( ,) NDT NDT RT N RT NDT ← σ RT ; then proceed 

to Step 2. 

 

9

 Note that negative values of   ) Cu Ni P , , and 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. 
 

 122

Step 2. Generate a random number, Φ , between 0 and 1 from a uniform distribution. Use this random 

number to sample10 a value of 

ΔRT epistemic from the following Weibull percentile function 

(inverse CDF): 



  ( ) 1/ 2.177

( 45.586,131.27,2.177)

45.586 131.27 ln 1 [ F]

epistemic

epistemic

RT W

RT

Δ ←−

Δ =− + − −Φ ° ⎡ ⎤ ⎣ ⎦

 (119) 



ΔRT epistemic represents the epistemic uncertainty in NDT0 RT . 

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, 0f ( )r ) ; copper 

content,  (, ) Cu N Cu ← σ Cu ; nickel content,  (, ) Ni N Ni ← σ Ni ; phosphorous content, ) (, ) P NP ← σ P ; and product form. 







30

30

0.99 ( , ) weld (, )

1.10 ( , ) plate and forgings

NDT

T r

RT r

T r

⎧

⎪ Δ

Δ = ⎨

⎪⎩ Δ

K K

K

 (120) 

where 

  ) 

)

( ) 

( )  ( )  ( )

30 0 exposure

0.4601 1.250

0 0

-17

-17

17

( , , , ( ), , ,product form)

19310 exp 1 110 ( ) 1 2.40 ( ) ( ) 460

8.86 10 for welds 

9.30 10 for forgings

12.7 10 for plates 

c

c

T Ni Cu P f r T

A P f r B Ni f Cu g f r Bias T

A

τ

−

Δ =

⎛ ⎞ ⎜ ⎟ + ++ +

+ ⎝ ⎠

⎧ ⎫ × ⎪ ⎪

= × ⎨ ⎬

⎪

⎪ × ⎩ ⎭



 12

10 0 exposu

0

230 for welds 

132 for forgings 

206 for plates in CE vessels

 156 for other plates 

log ( ) 4.579 10 1 1 ( ( )) tanh 2 2

B

f r

gf r

τ

⎪

⎪

⎧ ⎫

⎪ ⎪

= ⎨ ⎬

⎪ ⎪

⎩ ⎭

+ ×

= + ( )



( )





( ) 

re

0.659

18.265

0.713

 0 for 0.072 wt %

-0.072 for 0.072 wt %

Cu

f Cu

Cu Cu

⎡ ⎤ − ⎢ ⎥

⎣ ⎦

⎧ ⎫ ≤ ⎪ ⎪ = ⎨ ⎬

⎪ ⎪ > ⎩ ⎭

 

10 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. 
 

 123

subject to 

max

exposure

exposure

0.25 for welds with Linde 80 or Linde 0091 flux

0.305 for everything else 

and

0 for 97000 h

9.4 for 97000 h

Cu

t

Bias

t

⎧ ⎫ ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ ⎩ ⎭

⎧ ⎫ < ⎪ ⎪ = ⎨ ⎬

⎪ ⎪ ≥ ⎩ ⎭

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%, 0f ( )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 

0f ( )r . 

Step 4. Calculate the sampled, irradiated value of RTNDT from 

    RT r RT RT RT r NDT (, ) K K = −Δ +Δ NDT 0 epistemic NDT (, ) (121) 

 where 





0 0 0

0

0

( , ) if RVID2 method is 

Heat Estimate of if RVID2 method is NB-2331 or MTEB 5-2

NDT NDT NDT RT

NDT

NDT

RT N RT Generic

RT

RT

⎧

⎪ ← σ = ⎨

⎪⎩

Step 5. Calculate the normalized temperature of the vessel at the current location, r, of the crack tip 

in the RPV wall as 

  Δ =− T r T r RT r RELATIVE (, ) (,) (, ) K K τ NDT (122) 

Step 6. Calculate the parameters of the Weibull distribution of the KIc Weibull statistical distribution 

by 

 

 

( ) 19.35 8.335exp 0.02254( ) [ksi in.]

( ) 15.61 50.132exp 0.008( ) [ksi in.]

4

Ic

Ic

Ic

RELATIVE RELATIVE K

RELATIVE RELATIVE K

K

a T T

b T T

c

Δ =+ Δ ⎡ ⎤ ⎣ ⎦

Δ =+ Δ ⎡ ⎤ ⎣ ⎦

=

 (123) 

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. 
 

 124

Step 7. For a given applied KI , calculate the instantaneous conditional probability of crack initiation, 

Pr{KIc I ≤ K } with aleatory uncertainty, from the following Weibull distribution 

 



0 ;

Pr( ) ( ) 1 exp ;

( )

Ic

KIc

Ic

Ic

Ic

I K

Ic I RELATIVE I K

I K

RELATIVE K

c

K a

K K cpi Ka T

K a

b T

⎧ ≤

⎪

⎪ ⎡ ⎤ < == ⎨ ⎛ ⎞ − Δ ⎢ ⎥ ⎪ − −⎜ ⎟ >

⎪ Δ ⎝ ⎠ ⎩ ⎣ ⎦

 (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 N Ni N P σ σσ Cu Ni P .

11

Best estimate for the mean and standard deviation for a normal distribution of fluence at the 

inside surface of the vessel, 0 0 (0) ( (0), ) N f σ f .

12

Best estimate for the standard deviation, RTNDT 0 σ , 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 

 0 ( ) 0 ( ,) NDT NDT RVID RT N RT NDT ← σ RT if the RVID2 RTNDT(u) method is Generic. 

 

11 Note that negative values of chemistry content (   ) Cu Ni P , , and ) 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, 

0f (0) , are handled as nonphysical exceptions in FAVOR using 

the truncation protocol described in Sect. 3.3.4 with 0 as the truncation boundary. 
 

 125

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 NDT0 RT by applying a shift of − ° 14.4 F 

Δ =Δ ° RT RT epist arrest − 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 r NDT (, )  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 r NDT (, )  K . 

Step 11. Sample 

ln( ) ln( ) (,) ARREST ARREST ARREST RT RT μ σ RT Δ Δ ←Λ Δ

  from a lognormal distribution (see 

Appendix F) where 

( ) 0

2

ln( )

( ) ln( )

0

( ) 0

ln( )

ln

2

where

32 /1.8 [ C]

44.122exp 0.005971 [ C]

ln exp 0.3

ARREST

ARREST

ARREST

RT

ARREST mean RT

NDT epist arrest

ARREST mean

RT

RT

T RT RT

RT T

σ

μ

σ

Δ

Δ

−

Δ

=Δ − ⎡ ⎤ ⎣ ⎦

= −Δ − °

Δ = − ×° ⎡ ⎤ ⎣ ⎦

=

  

  

 

 { } ( ) 2 ( ) ( ) 0

2

0

0 0 0

0

8998 2ln( ) var 2ln

where

(12.778) for 35.7 C

var( ) 99.905972 -1.7748073 for -35.7 C 56 C

0 for 56 C

RT ARREST mean T RT ARREST mean

T

T TT

T

⎡ ⎤ +Δ − − Δ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

⎧ <− ° ⎪

⎪ = ° ⎨ ≤ ≤ °

⎪

⎪ > ° ⎩

  



  



 (126) 



ΔRT ARREST is sampled from the lognormal percentile function and then converted into F°

 

  



ln( ) 1.8exp ln( ) [ F]

(0,1); is the standard normal deviate corresponding to the fractile

(0 1) for this trial in the crack - -

Arrest f Arrest

f f

ARREST RT P RT

P P f

f

RT Z

ZN Z P

P Initiation Growth Arre

σ μ Δ Δ Δ = ⎡ ⎤ + ° ⎣ ⎦

←

< <

 

st model.

Step 12. Calculate the estimated arrest reference temperature, 

RT ARREST

  0 ARREST (, ) NDT epist arrest ARREST (, ) RT r RT RT RT RT r NDT = −Δ +Δ +Δ − K K    (127) 

Step 13. Calculate the normalized (relative to 

RT ARREST ) temperature of the vessel at the current 

location, r, in the RPV wall 

  Δ =− T r T r t RT r RELATIVE (, ) (,) (, ) K K ARREST (128) 
 

 126

Step 14. Calculate the lognormal mean, 

ln( ) ( ) Ia RELATIVE μ K ΔT , of the KIa statistical distribution by 

 

 

2

ln( )

ln( ) ( )

(mean)

ln( )

( ) ln ( ) 2

where

if _Model is equal to 1

( ) 27.302 69.962exp 0.006057( ) [ksi in.]

0.18

else if _Model is equa

Ia

Ia

Ia

K

RELATIVE RELATIVE K Ia mean

Ia

RELATIVE RELATIVE Ia

K

Ia

T KT

K

K T T

K

σ

μ

σ

Δ= Δ − ⎡ ⎤ ⎣ ⎦

Δ =+ ⎡ ⎤ Δ ⎣ ⎦

=

 

(mean)

ln( )

l to 2

( ) 27.302 70.6998exp 0.008991( ) [ksi in.]

0.34 Ia

RELATIVE RELATIVE Ia

K

K T T

σ

Δ =+ ⎡ ⎤ Δ ⎣ ⎦

=

 (129) 

Step 15. Given the current value of KI-initiation from the initiation model, we first calculate the fractile, 



ΦKI−initiation , associated with this value in the arrest model by 





ln( )

ln( )

1 ln( ) ( ) erf 1

2 2

Ia

I initiation

Ia

RELATIVE I initiation K

K

K

K T μ

σ −

− ⎡ ⎤ ⎛ ⎞ − Δ

Φ = ⎢ ⎥ ⎜ ⎟ +

⎣ ⎦ ⎝ ⎠

 (130) 

where 2

0

2 erf( ) exp( ) d

x

x ξ ξ

π ≡ − ∫ . Using the same value of Pf

 from Step 11, scale by 



ΦKI−initiation such that 

  ( )( ) Φ= Φ K K Ia Pf I initiation −

 (131) 

With this 

ΦKIa fractile, draw a value of KIa from its lognormal distribution 

  ) 

) 

ln( ) ln( ) ( , ) exp ( )

 standard normal deviate corresponding to the fractile 

K Ia Ia Ia Ia

K Ia Ia

RELATIVE RELATIVE Ia K K K

K

K T Z T

Z

σ μ Φ

Φ

ΦΔ = ⎡ ⎤ + Δ ⎣ ⎦

= Φ

 (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, σflow u( ) , in the crack Initiation-Growth-Arrest model. After each 

resampling of 

ΔT30 , the flow stress is adjusted by the following relation: 



( ) 30

0.112 ksi/ F for welds

 where 

0.131 ksi/ F for plates flow flow u T ⎧ ° σ = σ + γΔ γ = ⎨

⎩ °

This value of σflow is then used in the vessel-failure test against the pressure-induced membrane 

stress in the remaining ligament, checking for net-section plastic collapse. 
 

 127

Note on Step 11: The only random variate sampled in Step 11 is 

Pf Z . 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. 
 

 128

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. 
 

 129

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American Society for Testing and Materials, (1979) 341-364. 

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95. S. K. Iskander, R. D. Cheverton, and D. G. Ball, OCA-I, A Code for Calculating the Behavior of 

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Transients, NUREG/CR-2113 (ORNL/NUREG-84), Oak Ridge National Laboratory, Oak 

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National Laboratory, Oak Ridge, TN, May 1984. 

97. J. W. Bryson and T. L. Dickson, “Stress-Intensity-Factor Influence Coefficients for Axial and 

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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. 

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Evaluation Procedure as Specified in the ASME Code, Section XI, Appendix A, EPRI Report 

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for Inservice Inspection of Nuclear Power Plant Components, Appendix A, Analysis of Flaws, 

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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. 
 

 136

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. 

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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 

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Testing; Elevated and Low-Temperature Tests; Metallography, American Society for Testing 

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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. 
 

 137

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. 

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Information Meeting, Paper No. 9, 1971. 

122. E. J. Ripling and P. B. Crosley, “Crack Arrest Studies,” HSST 6th Annual Information Meeting, 

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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. 

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Transition Range,” E 1921-97, Annual Book of ASTM Standards Section 3: Metals Test 

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Determined Using ASTM Standard E 1921-97 for Nuclear Reactor Pressure Vessel Steels,” 

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Testing and Materials, West Conshococken, PA, (2000) 143-161. 

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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. 

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Regulatory Commission, July 1995. 

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July 6, 2000. 
 

 A-1

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-2

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) [11-113], OPERA [14], and PARISH (Probabilistic 

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 [19-121]. Experience with the 
 

 A-3

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-4

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-5

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. 
 

B-1

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-2

Table B1. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical 

Surface Flaws: R / t = 10 and a / t = 0.01 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle (deg) Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.

2:1 0.00 0.764 0.153 0.061 0.034 0.764 0.153 0.764 0.153 

 2.37 0.754 0.165 0.062 0.032 0.754 0.165 0.754 0.165 

 16.60 0.690 0.192 0.079 0.040 0.690 0.192 0.690 0.192 

 30.80 0.669 0.264 0.127 0.069 0.669 0.264 0.669 0.264 

 45.00 0.660 0.335 0.196 0.124 0.660 0.335 0.660 0.335 

 59.20 0.653 0.393 0.269 0.198 0.653 0.393 0.653 0.393 

 73.40 0.651 0.434 0.329 0.268 0.651 0.434 0.651 0.434 

 87.60 0.649 0.463 0.366 0.310 0.649 0.463 0.649 0.463 

 90.00 0.649 0.468 0.372 0.317 0.649 0.468 0.649 0.468 

6:1 0.00 0.670 0.134 0.048 0.024 0.670 0.134 0.670 0.134 

 2.37 0.667 0.134 0.043 0.019 0.667 0.134 0.667 0.134 

 16.60 0.654 0.170 0.055 0.009 0.654 0.170 0.654 0.170 

 30.80 0.741 0.269 0.109 0.029 0.741 0.269 0.741 0.269 

 45.00 0.827 0.381 0.199 0.100 0.827 0.381 0.827 0.381 

 59.20 0.893 0.481 0.302 0.197 0.893 0.481 0.893 0.481 

 73.40 0.938 0.559 0.389 0.290 0.938 0.559 0.938 0.559 

 87.60 0.970 0.594 0.435 0.341 0.970 0.594 0.970 0.594 

 90.00 0.975 0.601 0.443 0.350 0.975 0.601 0.975 0.601 

10:1 0.00 0.515 0.090 0.020 0.006 0.515 0.090 0.515 0.090 

 2.37 0.529 0.094 0.010 0.005 0.529 0.094 0.529 0.094 

 16.60 0.610 0.146 0.033 0.005 0.610 0.146 0.610 0.146 

 30.80 0.762 0.258 0.060 0.019 0.762 0.258 0.762 0.258 

 45.00 0.889 0.389 0.171 0.066 0.889 0.389 0.889 0.389 

 59.20 0.979 0.507 0.290 0.136 0.979 0.507 0.979 0.507 

 73.40 1.033 0.593 0.389 0.249 1.033 0.593 1.033 0.593 

 87.60 1.064 0.635 0.439 0.307 1.064 0.635 1.064 0.635 

 90.00 1.069 0.642 0.447 0.316 1.069 0.642 1.069 0.642 
 

B-3

Table B2. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical 

Surface Flaws: R / t = 10 and a / t = 0.0184 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle (deg) Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in. 

2:1 0.00 0.777 0.155 0.061 0.034 0.777 0.155 0.777 0.155 

 2.37 0.767 0.167 0.062 0.032 0.767 0.167 0.767 0.167 

 16.60 0.700 0.194 0.079 0.040 0.700 0.194 0.700 0.194 

 30.80 0.677 0.266 0.127 0.069 0.677 0.266 0.677 0.266 

 45.00 0.667 0.338 0.196 0.125 0.667 0.338 0.667 0.338 

 59.20 0.660 0.397 0.270 0.198 0.660 0.397 0.660 0.397 

 73.40 0.657 0.438 0.330 0.267 0.657 0.438 0.657 0.438 

 87.60 0.654 0.467 0.366 0.310 0.654 0.467 0.654 0.467 

 90.00 0.653 0.472 0.373 0.317 0.653 0.472 0.653 0.472 

6:1 0.00 0.653 0.127 0.043 0.021 0.653 0.127 0.653 0.127 

 2.37 0.654 0.128 0.038 0.016 0.654 0.128 0.654 0.128 

 16.60 0.654 0.168 0.045 0.021 0.654 0.168 0.654 0.168 

 30.80 0.758 0.271 0.099 0.026 0.758 0.271 0.758 0.271 

 45.00 0.852 0.387 0.192 0.085 0.852 0.387 0.852 0.387 

 59.20 0.920 0.492 0.298 0.187 0.920 0.492 0.920 0.492 

 73.40 0.963 0.569 0.387 0.283 0.963 0.569 0.963 0.569 

 87.60 0.994 0.609 0.434 0.335 0.994 0.609 0.994 0.609 

 90.00 0.999 0.616 0.442 0.344 0.999 0.616 0.999 0.616 

10:1 0.00 0.525 0.092 0.019 0.007 0.525 0.092 0.525 0.092 

 2.37 0.538 0.096 0.009 0.005 0.538 0.096 0.538 0.096 

 16.60 0.621 0.149 0.039 0.005 0.621 0.149 0.621 0.149 

 30.80 0.777 0.262 0.050 0.022 0.777 0.262 0.777 0.262 

 45.00 0.899 0.392 0.164 0.075 0.899 0.392 0.899 0.392 

 59.20 0.982 0.509 0.283 0.127 0.982 0.509 0.982 0.509 

 73.40 1.033 0.595 0.383 0.242 1.033 0.595 1.033 0.595 

 87.60 1.063 0.637 0.433 0.300 1.063 0.637 1.063 0.637 

 90.00 1.068 0.644 0.441 0.310 1.068 0.644 1.068 0.644 
 

B-4

Table B3. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical 

Surface Flaws: R / t = 10 and a / t = 0.05 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle (deg) Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.

2:1 0.00 0.779 0.155 0.061 0.034 0.708 0.184 0.636 0.205 

 2.37 0.769 0.166 0.062 0.031 0.701 0.194 0.624 0.213 

 16.60 0.701 0.194 0.079 0.040 0.659 0.264 0.509 0.232 

 30.80 0.678 0.267 0.128 0.070 0.581 0.340 0.246 0.124 

 45.00 0.668 0.339 0.199 0.126 0.326 0.188 0.159 0.083 

 59.20 0.661 0.398 0.273 0.201 0.233 0.127 0.128 0.067 

 73.40 0.658 0.440 0.333 0.270 0.204 0.110 0.115 0.060 

 87.60 0.656 0.469 0.370 0.313 0.185 0.099 0.106 0.055 

 90.00 0.655 0.474 0.377 0.320 0.182 0.097 0.104 0.054 

6:1 0.00 0.655 0.128 0.043 0.021 0.631 0.151 0.576 0.176 

 2.37 0.655 0.128 0.039 0.016 0.628 0.156 0.570 0.177 

 16.60 0.655 0.167 0.049 0.019 0.646 0.221 0.537 0.213 

 30.80 0.758 0.270 0.104 0.013 0.688 0.357 0.340 0.167 

 45.00 0.851 0.386 0.197 0.091 0.494 0.263 0.271 0.138 

 59.20 0.918 0.492 0.305 0.193 0.422 0.217 0.253 0.128 

 73.40 0.962 0.569 0.395 0.290 0.396 0.201 0.241 0.121 

 87.60 0.992 0.609 0.443 0.342 0.374 0.189 0.231 0.115 

 90.00 0.997 0.616 0.450 0.351 0.370 0.186 0.229 0.115 

10:1 0.00 0.523 0.092 0.021 0.005 0.533 0.119 0.496 0.149 

 2.37 0.537 0.095 0.011 0.015 0.543 0.121 0.504 0.146 

 16.60 0.622 0.147 0.033 0.050 0.631 0.149 0.547 0.199 

 30.80 0.778 0.261 0.061 0.080 0.718 0.348 0.376 0.182 

 45.00 0.898 0.391 0.171 0.065 0.550 0.286 0.349 0.156 

 59.20 0.981 0.509 0.292 0.138 0.474 0.241 0.287 0.144 

 73.40 1.034 0.596 0.392 0.252 0.444 0.224 0.273 0.136 

 87.60 1.063 0.638 0.442 0.310 0.418 0.221 0.260 0.130 

 90.00 1.068 0.645 0.450 0.320 0.414 0.221 0.257 0.128 
 

B-5

Table B4. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical 

Surface Flaws: R / t = 10 and a / t = 0.075 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle (deg) Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.

2:1 0.00 0.740 0.128 0.045 0.023 0.650 0.197 0.572 0.210 

 7.03 0.737 0.147 0.055 0.028 0.629 0.220 0.529 0.217 

 14.20 0.721 0.179 0.067 0.033 0.593 0.271 0.400 0.177 

 35.90 0.671 0.298 0.155 0.086 0.219 0.120 0.118 0.060 

 48.70 0.661 0.355 0.220 0.143 0.161 0.085 0.094 0.048 

 61.50 0.656 0.404 0.285 0.212 0.137 0.071 0.081 0.042 

 74.30 0.654 0.439 0.336 0.273 0.125 0.065 0.075 0.038 

 87.00 0.651 0.468 0.372 0.313 0.114 0.065 0.068 0.035 

 90.00 0.651 0.475 0.381 0.322 0.111 0.065 0.067 0.034 

6:1 0.00 0.650 0.098 0.029 0.013 0.591 0.170 0.527 0.188 

 2.37 0.635 0.104 0.031 0.013 0.571 0.180 0.495 0.179 

 16.60 0.672 0.140 0.040 0.014 0.590 0.243 0.441 0.187 

 30.80 0.786 0.309 0.139 0.048 0.334 0.171 0.195 0.098 

 45.00 0.862 0.410 0.229 0.125 0.294 0.149 0.180 0.090 

 59.20 0.918 0.501 0.326 0.219 0.275 0.138 0.170 0.085 

 73.40 0.952 0.566 0.404 0.303 0.265 0.133 0.164 0.082 

 87.60 0.980 0.602 0.446 0.351 0.265 0.133 0.159 0.080 

 90.00 0.987 0.611 0.456 0.362 0.265 0.132 0.157 0.079 

10:1 0.00 0.547 0.073 0.016 0.006 0.514 0.148 0.469 0.171 

 2.37 0.551 0.074 0.016 0.003 0.514 0.145 0.458 0.131 

 16.60 0.636 0.113 0.023 0.009 0.583 0.220 0.465 0.173 

 30.80 0.812 0.303 0.124 0.018 0.375 0.189 0.223 0.112 

 45.00 0.914 0.419 0.225 0.111 0.335 0.168 0.206 0.103 

 59.20 0.982 0.522 0.332 0.216 0.310 0.156 0.193 0.096 

 73.40 1.022 0.593 0.416 0.307 0.298 0.149 0.185 0.093 

 87.60 1.048 0.631 0.461 0.356 0.295 0.147 0.185 0.092 

 90.00 1.055 0.639 0.471 0.368 0.295 0.147 0.184 0.092 
 

B-6

Table B5. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical 

Surface Flaws: R / t = 10 and a / t = 0.1 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle (deg) Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.

2:1 0.00 0.729 0.124 0.044 0.023 0.596 0.195 0.519 0.205 

 5.27 0.741 0.139 0.053 0.027 0.582 0.208 0.483 0.198 

 17.10 0.722 0.230 0.096 0.048 0.366 0.213 0.168 0.086 

 31.10 0.676 0.273 0.133 0.072 0.176 0.097 0.095 0.048 

 45.10 0.664 0.339 0.201 0.127 0.122 0.064 0.072 0.037 

 59.10 0.658 0.396 0.274 0.200 0.101 0.052 0.061 0.031 

 73.10 0.655 0.436 0.333 0.268 0.091 0.047 0.056 0.028 

 87.00 0.653 0.470 0.373 0.313 0.082 0.047 0.050 0.025 

 90.00 0.652 0.477 0.382 0.323 0.080 0.047 0.049 0.025 

6:1 0.00 0.641 0.094 0.029 0.014 0.550 0.175 0.485 0.188 

 2.37 0.630 0.098 0.031 0.015 0.532 0.176 0.454 0.168 

 16.60 0.701 0.196 0.067 0.015 0.427 0.232 0.211 0.108 

 30.80 0.756 0.273 0.115 0.039 0.258 0.131 0.152 0.077 

 45.00 0.848 0.385 0.207 0.109 0.224 0.112 0.138 0.069 

 59.20 0.915 0.489 0.312 0.207 0.208 0.104 0.129 0.065 

 73.40 0.958 0.565 0.402 0.302 0.200 0.100 0.125 0.062 

 87.60 0.989 0.607 0.450 0.356 0.200 0.100 0.120 0.060 

 90.00 0.996 0.616 0.461 0.367 0.200 0.100 0.119 0.060 

10:1 0.00 0.543 0.067 0.016 0.007 0.490 0.148 0.443 0.168 

 2.37 0.536 0.069 0.016 0.006 0.479 0.144 0.421 0.138 

 16.60 0.670 0.175 0.047 0.027 0.443 0.220 0.229 0.117 

 30.80 0.778 0.269 0.102 0.030 0.291 0.143 0.176 0.088 

 45.00 0.897 0.395 0.202 0.089 0.256 0.128 0.159 0.080 

 59.20 0.979 0.512 0.318 0.199 0.236 0.118 0.147 0.074 

 73.40 1.029 0.597 0.416 0.302 0.226 0.113 0.141 0.071 

 87.60 1.060 0.640 0.466 0.358 0.224 0.111 0.140 0.070 

 90.00 1.066 0.649 0.477 0.370 0.223 0.111 0.140 0.070 
 

B-7

Table B6. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical 

Surface Flaws: R / t = 10 and a / t = 0.2 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle (deg) Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.

2:1 0.00 0.692 0.127 0.046 0.024 0.457 0.173 0.393 0.178 

 19.80 0.695 0.214 0.089 0.044 0.155 0.080 0.071 0.031 

 31.10 0.679 0.273 0.133 0.073 0.090 0.050 0.048 0.023 

 42.50 0.671 0.332 0.192 0.120 0.061 0.031 0.038 0.019 

 53.80 0.665 0.383 0.255 0.182 0.052 0.026 0.032 0.016 

 65.20 0.660 0.423 0.312 0.245 0.047 0.023 0.029 0.014 

 76.50 0.658 0.450 0.354 0.296 0.044 0.022 0.027 0.014 

 87.90 0.656 0.475 0.384 0.329 0.041 0.021 0.025 0.013 

 90.00 0.656 0.479 0.389 0.335 0.040 0.020 0.025 0.013 

6:1 0.00 0.617 0.101 0.034 0.017 0.434 0.163 0.377 0.171 

 2.37 0.699 0.194 0.066 0.019 0.180 0.090 0.093 0.043 

 16.60 0.781 0.280 0.118 0.045 0.127 0.063 0.079 0.039 

 30.80 0.856 0.375 0.195 0.101 0.116 0.058 0.072 0.036 

 45.00 0.915 0.464 0.283 0.180 0.110 0.055 0.069 0.034 

 59.20 0.958 0.538 0.366 0.265 0.106 0.053 0.066 0.033 

 73.40 0.986 0.590 0.430 0.336 0.104 0.052 0.065 0.032 

 87.60 1.010 0.619 0.464 0.373 0.102 0.051 0.064 0.032 

 90.00 1.020 0.624 0.470 0.380 0.101 0.051 0.063 0.032 

10:1 0.00 0.525 0.077 0.022 0.009 0.402 0.149 0.355 0.160 

 2.37 0.694 0.183 0.050 0.025 0.200 0.100 0.106 0.050 

 16.60 0.815 0.280 0.107 0.011 0.149 0.073 0.093 0.046 

 30.80 0.915 0.387 0.190 0.083 0.137 0.068 0.085 0.043 

 45.00 0.991 0.488 0.287 0.170 0.130 0.065 0.081 0.040 

 59.20 1.045 0.572 0.379 0.263 0.125 0.062 0.078 0.039 

 73.40 1.080 0.631 0.449 0.340 0.122 0.061 0.077 0.038 

 87.60 1.103 0.660 0.483 0.378 0.120 0.060 0.075 0.037 

 90.00 1.107 0.666 0.490 0.385 0.119 0.060 0.075 0.037 
 

B-8

Table B7. Influence Coefficients for Inside Axial and Circumferential Semi-elliptical 

Surface Flaws: R / t = 10 and a / t = 0.3 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle 

(deg) 

Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.

2:1 0.00 0.723 0.127 0.048 0.026 0.404 0.188 0.334 0.176 

 17.40 0.708 0.203 0.083 0.042 0.102 0.049 0.056 0.025 

 29.10 0.690 0.264 0.126 0.068 0.058 0.028 0.034 0.016 

 40.90 0.680 0.326 0.185 0.114 0.043 0.021 0.026 0.013 

 52.60 0.673 0.381 0.251 0.177 0.036 0.018 0.022 0.011 

 64.40 0.668 0.423 0.310 0.242 0.032 0.016 0.020 0.010 

 76.10 0.665 0.452 0.355 0.297 0.030 0.015 0.018 0.009 

 87.90 0.662 0.478 0.385 0.331 0.028 0.014 0.017 0.009 

 90.00 0.662 0.482 0.391 0.337 0.027 0.014 0.017 0.009 

6:1 0.00 0.665 0.112 0.041 0.022 0.380 0.181 0.315 0.167 

 2.37 0.715 0.190 0.068 0.027 0.117 0.054 0.069 0.032 

 16.60 0.804 0.277 0.118 0.051 0.093 0.045 0.057 0.028 

 30.80 0.886 0.376 0.194 0.104 0.085 0.042 0.053 0.026 

 45.00 0.951 0.470 0.284 0.182 0.081 0.040 0.050 0.025 

 59.20 0.998 0.549 0.372 0.270 0.078 0.039 0.049 0.024 

 73.40 1.028 0.605 0.439 0.345 0.077 0.038 0.048 0.024 

 87.60 1.053 0.635 0.475 0.384 0.075 0.038 0.047 0.024 

 90.00 1.058 0.640 0.481 0.391 0.075 0.037 0.047 0.023 

10:1 0.00 0.562 0.085 0.029 0.014 0.344 0.168 0.290 0.153 

 2.37 0.707 0.176 0.052 0.016 0.128 0.059 0.078 0.037 

 16.60 0.848 0.276 0.104 0.016 0.110 0.054 0.068 0.034 

 30.80 0.962 0.389 0.188 0.082 0.102 0.051 0.064 0.032 

 45.00 1.051 0.498 0.288 0.169 0.098 0.049 0.062 0.031 

 59.20 1.115 0.590 0.385 0.265 0.096 0.048 0.060 0.030 

 73.40 1.157 0.653 0.460 0.346 0.095 0.047 0.060 0.030 

 87.60 1.183 0.685 0.496 0.387 0.094 0.047 0.059 0.029 

 90.00 1.187 0.691 0.503 0.394 0.094 0.047 0.059 0.029 
 

B-9

Table B8. Influence Coefficients for Inside Axial Semi-elliptical Surface Flaws: R / t = 10 

and a / t = 0.5 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle (deg) Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in. 

2:1 0.00 0.736 0.132 0.053 0.029 0.327 0.162 0.272 0.150 

 15.40 0.746 0.203 0.083 0.043 0.079 0.037 0.045 0.020 

 27.50 0.719 0.263 0.124 0.067 0.042 0.020 0.025 0.012 

 39.60 0.704 0.327 0.183 0.112 0.029 0.014 0.018 0.009 

 51.70 0.693 0.383 0.249 0.175 0.023 0.012 0.015 0.007 

 63.70 0.685 0.426 0.311 0.242 0.021 0.010 0.013 0.006 

 75.80 0.681 0.456 0.357 0.299 0.019 0.009 0.012 0.006 

 87.90 0.676 0.483 0.389 0.334 0.018 0.009 0.011 0.006 

 90.00 0.676 0.488 0.395 0.340 0.017 0.009 0.011 0.005 

6:1 0.00 0.758 0.142 0.059 0.033 0.322 0.163 0.268 0.149 

 2.37 0.814 0.213 0.083 0.040 0.091 0.041 0.054 0.025 

 16.60 0.908 0.302 0.132 0.065 0.070 0.034 0.043 0.021 

 30.80 0.998 0.405 0.208 0.116 0.065 0.032 0.040 0.020 

 45.00 1.069 0.504 0.300 0.195 0.062 0.031 0.039 0.019 

 59.20 1.120 0.588 0.392 0.285 0.061 0.030 0.038 0.019 

 73.40 1.153 0.647 0.463 0.363 0.060 0.030 0.038 0.019 

 87.60 1.182 0.679 0.500 0.404 0.059 0.029 0.037 0.018 

 90.00 1.187 0.685 0.506 0.411 0.059 0.029 0.037 0.018 

10:1 0.00 0.666 0.119 0.049 0.028 0.302 0.156 0.254 0.140 

 2.37 0.822 0.208 0.077 0.033 0.097 0.044 0.060 0.028 

 16.60 0.995 0.316 0.131 0.056 0.086 0.042 0.054 0.027 

 30.80 1.138 0.440 0.216 0.112 0.083 0.041 0.052 0.026 

 45.00 1.251 0.560 0.321 0.198 0.083 0.041 0.052 0.026 

 59.20 1.335 0.662 0.425 0.298 0.083 0.041 0.052 0.026 

 73.40 1.390 0.734 0.506 0.383 0.083 0.041 0.052 0.026 

 87.60 1.423 0.770 0.546 0.427 0.083 0.041 0.052 0.026 

 90.00 1.429 0.776 0.553 0.434 0.083 0.041 0.052 0.026 
 

B-10

Table B9. Influence Coefficients for Inside Circumferential Semi-elliptical Surface 

Flaws: R / t = 10 and a / t = 0.5 

Aspect Elliptic K0 K1 K2 K3 K0 K1 K0 K1 

Ratio Angle (deg) Uniform Linear Quadratic Cubic tcl=0.25 in. tcl=0.25 in. tcl=0.156 in. tcl=0.156 in.

2:1 0.00 0.741 0.134 0.054 0.030 0.324 0.162 0.269 0.151 

 15.40 0.750 0.205 0.084 0.044 0.079 0.038 0.045 0.020 

 27.50 0.721 0.264 0.124 0.067 0.042 0.020 0.025 0.012 

 39.60 0.706 0.328 0.183 0.112 0.029 0.014 0.018 0.009 

 51.70 0.698 0.384 0.250 0.175 0.024 0.012 0.015 0.007 

 63.70 0.692 0.430 0.312 0.243 0.021 0.010 0.013 0.007 

 75.80 0.686 0.461 0.360 0.301 0.019 0.010 0.012 0.006 

 87.90 0.682 0.488 0.392 0.336 0.020 0.010 0.012 0.006 

 90.00 0.682 0.493 0.398 0.343 0.020 0.009 0.013 0.006 

6:1 0.00 0.727 0.132 0.053 0.030 0.315 0.161 0.262 0.147 

 15.40 0.786 0.205 0.079 0.037 0.087 0.039 0.052 0.024 

 27.50 0.882 0.295 0.128 0.062 0.067 0.032 0.041 0.020 

 39.60 0.974 0.398 0.205 0.114 0.062 0.031 0.038 0.019 

 51.70 1.049 0.499 0.298 0.193 0.060 0.030 0.037 0.019 

 63.70 1.103 0.584 0.390 0.284 0.058 0.029 0.036 0.018 

 75.80 1.138 0.644 0.462 0.362 0.057 0.029 0.036 0.018 

 87.90 1.166 0.676 0.499 0.403 0.058 0.029 0.036 0.018 

 90.00 1.171 0.682 0.506 0.410 0.058 0.029 0.036 0.018 

10:1 0.00 0.616 0.101 0.040 0.023 0.291 0.152 0.247 0.138 

 15.40 0.770 0.195 0.071 0.028 0.090 0.039 0.055 0.026 

 27.50 0.936 0.301 0.125 0.053 0.078 0.038 0.049 0.024 

 39.60 1.076 0.424 0.211 0.109 0.075 0.037 0.047 0.024 

 51.70 1.190 0.544 0.315 0.196 0.075 0.037 0.047 0.023 

 63.70 1.275 0.647 0.420 0.295 0.075 0.037 0.047 0.023 

 75.80 1.330 0.719 0.501 0.381 0.075 0.037 0.047 0.023 

 87.90 1.363 0.755 0.542 0.425 0.075 0.037 0.047 0.024 

 90.00 1.368 0.762 0.549 0.433 0.075 0.037 0.047 0.024 
 

B-11

Table B10. Influence Coefficients for Inside Axial Infinite-Length Surface Flaws, 

R / t = 10 

0.1 t 1/2 K* 

a' / a a/t=0.01 a/t=0.02 a/t=0.03 a/t=0.05 a/t=0.075 a/t=0.10 

0 1.434 1.029 0.846 0.667 0.565 0.511 

0.0556 1.435 1.029 0.846 0.667 0.564 0.510 

0.1111 1.436 1.029 0.846 0.666 0.563 0.508 

0.1667 1.436 1.028 0.846 0.665 0.562 0.506 

0.2222 1.438 1.029 0.846 0.665 0.561 0.505 

0.2778 1.442 1.032 0.848 0.666 0.561 0.504 

0.3333 1.450 1.037 0.852 0.669 0.563 0.505 

0.3888 1.463 1.046 0.859 0.674 0.566 0.507 

0.4444 1.482 1.058 0.869 0.682 0.571 0.511 

0.500 1.509 1.077 0.884 0.693 0.580 0.517 

0.5556 1.546 1.103 0.905 0.708 0.592 0.527 

0.6111 1.598 1.138 0.934 0.731 0.609 0.541 

0.6666 1.669 1.188 0.974 0.761 0.633 0.561 

0.7222 1.768 1.258 1.031 0.804 0.668 0.590 

0.7778 1.913 1.360 1.113 0.868 0.718 0.632 

0.8333 2.138 1.518 1.242 0.967 0.798 0.699 

0.8888 2.534 1.798 1.470 1.143 0.940 0.821 

0.9166 2.878 2.041 1.668 1.294 1.064 0.927 

0.9444 3.499 2.624 2.187 1.749 1.385 1.224 

0.9639 5.831 4.227 3.499 2.770 2.187 1.895 

0.9778 11.225 7.289 5.685 4.227 3.426 2.916 

0.9889 17.493 11.662 8.746 6.414 5.102 4.373 

 

 

a'/a a/t=0.2 a/t=0.3 a/t=0.4 a'/a a/t=0.5 

0 0.461 0.510 0.617 0 0.781 

0.0552 0.457 0.502 0.602 0.059 0.755 

0.1103 0.452 0.492 0.586 0.118 0.730 

0.1655 0.447 0.483 0.571 0.176 0.704 

0.2206 0.443 0.475 0.556 0.235 0.679 

0.2757 0.439 0.466 0.542 0.294 0.654 

0.3309 0.436 0.459 0.527 0.353 0.630 

0.3861 0.434 0.451 0.513 0.412 0.605 

0.4412 0.432 0.445 0.500 0.471 0.582 

0.4963 0.433 0.440 0.488 0.529 0.559 

0.5515 0.435 0.436 0.477 0.588 0.538 

0.6066 0.440 0.434 0.467 0.647 0.518 

0.6618 0.450 0.435 0.460 0.706 0.501 

0.7169 0.464 0.440 0.456 0.750 0.491 

0.7721 0.487 0.453 0.457 0.794 0.485 

0.8272 0.526 0.477 0.468 0.838 0.486 

0.8824 0.598 0.527 0.501 0.882 0.501 

0.9118 0.665 0.577 0.538 0.912 0.526 

0.9412 0.875 0.729 0.671 0.941 0.656 

0.9618 1.385 1.020 0.948 0.962 0.875 

0.9765 2.187 1.749 1.604 0.976 1.312 

0.9882 2.916 2.478 2.187 0.988 2.041 

 
 

B-12

Table B10. (continued) Influence Coefficients for Inside Axial Infinite-Length Surface 

Flaws, R / t = 10 

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

0.1 t

1/2 K*
 

B-13

Table B11. Influence Coefficients for Inside Circumferential 360 Degree Surface Flaws, 

R / t = 10 

10t t

1/2 K* 

a' / a a / t=0.01 a/t=0.02 a/t=0.03 a/t=0.05 a/t=0.075 a/t=0.10 

0 2.255 1.616 1.325 1.036 0.867 0.771 

0.0556 2.256 1.616 1.324 1.036 0.865 0.769 

0.1111 2.257 1.616 1.324 1.035 0.864 0.767 

0.1667 2.258 1.616 1.323 1.034 0.863 0.765 

0.2222 2.260 1.617 1.324 1.035 0.862 0.764 

0.2778 2.267 1.621 1.327 1.037 0.863 0.764 

0.3333 2.280 1.629 1.334 1.041 0.866 0.766 

0.3888 2.300 1.642 1.344 1.049 0.872 0.770 

0.4444 2.329 1.662 1.361 1.061 0.880 0.777 

0.5000 2.372 1.691 1.384 1.079 0.894 0.788 

0.5556 2.431 1.732 1.417 1.104 0.914 0.804 

0.6111 2.511 1.788 1.462 1.138 0.941 0.826 

0.6666 2.623 1.866 1.526 1.187 0.979 0.859 

0.7222 2.779 1.975 1.615 1.255 1.034 0.905 

0.7778 3.008 2.135 1.744 1.355 1.114 0.972 

0.8333 3.361 2.383 1.946 1.510 1.239 1.079 

0.8888 3.986 2.823 2.305 1.786 1.462 1.271 

0.9166 4.520 3.199 2.611 2.022 1.654 1.425 

0.9444 6.195 3.965 3.346 2.478 1.982 1.735 

0.9639 8.674 5.948 4.956 3.717 2.974 2.602 

0.9778 13.630 9.913 8.054 6.195 4.956 4.337 

0.9889 18.586 14.249 11.771 9.045 7.682 6.567 

 

 

a'/a a/t=0.2 a/t=0.3 a/t=0.4 a'/a a/t=0.5 

0 0.645 0.644 0.691 0 0.764 

0.0552 0.640 0.635 0.678 0.059 0.744 

0.1103 0.635 0.626 0.664 0.118 0.724 

0.1655 0.630 0.617 0.651 0.176 0.704 

0.2206 0.625 0.609 0.638 0.235 0.684 

0.2757 0.622 0.601 0.625 0.294 0.666 

0.3309 0.619 0.594 0.613 0.353 0.647 

0.3861 0.618 0.588 0.602 0.412 0.630 

0.4412 0.618 0.584 0.592 0.471 0.614 

0.4963 0.622 0.581 0.584 0.529 0.600 

0.5515 0.628 0.581 0.578 0.588 0.589 

0.6066 0.639 0.584 0.574 0.647 0.580 

0.6618 0.656 0.592 0.575 0.706 0.577 

0.7169 0.681 0.607 0.581 0.750 0.579 

0.7721 0.721 0.633 0.596 0.794 0.588 

0.8272 0.784 0.678 0.626 0.838 0.608 

0.8824 0.900 0.764 0.691 0.882 0.650 

0.9118 1.007 0.845 0.793 0.912 0.702 

0.9412 1.363 1.078 0.954 0.941 0.843 

0.9618 1.921 1.487 1.301 0.962 1.115 

0.9765 2.912 2.354 1.982 0.976 1.859 

0.9882 3.841 3.346 2.912 0.988 2.726 
 

B-14

Table B11. (continued) Influence Coefficients for Inside Circumferential 360 Degree 

Surface Flaws, R / t = 10 

10t 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 

0 0.852 0 0.944 0 1.028 0 1.129 

0.0564 0.827 0.057 0.913 0.058 0.995 0.058 1.099 

0.1127 0.802 0.115 0.883 0.116 0.962 0.117 1.070 

0.1691 0.778 0.172 0.853 0.174 0.929 0.175 1.041 

0.2255 0.753 0.229 0.823 0.232 0.897 0.233 1.013 

0.2819 0.729 0.286 0.794 0.289 0.866 0.292 0.986 

0.3382 0.706 0.343 0.766 0.347 0.835 0.350 0.959 

0.3946 0.684 0.401 0.739 0.405 0.805 0.409 0.932 

0.4510 0.663 0.458 0.712 0.463 0.776 0.467 0.907 

0.5074 0.642 0.515 0.687 0.521 0.748 0.526 0.882 

0.5637 0.624 0.572 0.663 0.579 0.721 0.584 0.857 

0.6201 0.608 0.630 0.641 0.637 0.695 0.643 0.832 

0.6765 0.595 0.687 0.622 0.695 0.671 0.701 0.809 

0.7328 0.586 0.744 0.607 0.753 0.651 0.759 0.786 

0.7892 0.586 0.802 0.600 0.811 0.636 0.818 0.767 

0.8456 0.601 0.859 0.608 0.869 0.637 0.876 0.757 

0.9020 0.653 0.916 0.661 0.927 0.686 0.935 0.786 

0.9265 0.703 0.937 0.709 0.945 0.729 0.951 0.820 

0.9510 0.867 0.958 0.855 0.963 0.880 0.967 0.892 

0.9681 1.140 0.973 1.155 0.976 1.128 0.979 1.115 

0.9804 1.797 0.983 1.760 0.985 1.722 0.987 1.735 

0.9902 2.602 0.992 2.602 0.993 2.466 0.993 2.478 

 
 

C-1

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-2

Table C1. Static Initiation Toughness KIc Extended Database 

Material Reference Source Specimen Type Orientation T RTNDT T - RTNDT KIc

 ID No. (°F) (°F) (°F) (ksi√in)

HSST 01 Shabbits 1T-C(T) 1 -200 0 -200 46.6 

subarc (1969) 1T-C(T) 1 -175 0 -175 55.8 

weldment 4T-C(T) 4 -150 0 -150 56.1 

 4T-C(T) 4 -125 0 -125 61.1 

 4T-C(T) 4 -100 0 -100 96.0 

 4T-C(T) 4 -75 0 -75 90.3 

 4T-C(T) 4 -75 0 -75 93.1 

 6T-C(T) 6 -50 0 -50 72.6 

A533B Class 1 Shabbits 1T-C(T) 1 -200 0 -200 35.1 

subarc (1969) 1T-C(T) 1 -200 0 -200 45.2 

weldment 1T-C(T) 1 -320 0 -320 25.9 

 1T-C(T) 1 -320 0 -320 23.7 

 4T-C(T) 4 -100 0 -100 55.2 

 4T-C(T) 4 -50 0 -50 71.6 

 4T-C(T) 4 -25 0 -25 105.9 

 8T-C(T) 8 0 0 0 113.1 

HSST 01 Mager (1969) 1T-C(T) 1 RW -150 20 -170 43.9 

 1T-C(T) 1 RW -150 20 -170 39.4 

 1T-C(T) 1 RW -150 20 -170 31.3 

 1T-C(T) 1 RW -150 20 -170 47.3 

 1T-C(T) 1 RW -150 20 -170 50.4 

 1T-C(T) 1 RW -150 20 -170 41.2 

 1T-C(T) 1 RW -150 20 -170 54.0 

 1T-C(T) 1 RW -150 20 -170 50.9 

 1T-C(T) 1 RW -150 20 -170 35.5 

 1T-C(T) 1 RW -150 20 -170 33.2 

 1T-C(T) 1 RW -150 20 -170 37.2 

 1T-C(T) 1 RW -150 20 -170 37.1 

 1T-C(T) 1 RW -150 20 -170 37.1 

 1T-C(T) 1 RW -150 20 -170 34.7 

 1T-C(T) 1 RW -150 20 -170 35.0 

 1T-C(T) 1 RW -150 20 -170 32.6 

 1T-C(T) 1 RW -150 20 -170 29.4 

HSST 03 Mager (1969) 1T-C(T) 1 RW -150 20 -170 44.0 

 1T-C(T) 1 RW -150 20 -170 31.4 

 1T-C(T) 1 RW -150 20 -170 39.3 

 1T-C(T) 1 RW -150 20 -170 31.3 

 1T-C(T) 1 RW -150 20 -170 33.0 

 1T-C(T) 1 RW -150 20 -170 38.1 

 1T-C(T) 1 RW -150 20 -170 31.1 

 1T-C(T) 1 RW -150 20 -170 44.9 

 1T-C(T) 1 RW -150 20 -170 39.4 

A533B Class 1 Mager (1969) 1X-WOL 1 RW -320 65 -385 31.6 

 1T-WOL 1 RW -320 65 -385 32.5 

 1X-WOL 1 RW -250 65 -315 40.9 

 1X-WOL 1 RW -250 65 -315 37.1 

 1X-WOL 1 RW -250 65 -315 44.0 

 1T-WOL 1 RW -250 65 -315 40.8 

 1T-WOL 1 RW -250 65 -315 31.2 

 1X-WOL 1 RW -200 65 -265 30.6 
 

C-3

Material Reference Source Specimen Type Orientation T RTNDT T - RTNDT KIc

 ID No. (°F) (°F) (°F) (ksi√in)

 1X-WOL 1 RW -200 65 -265 29.0 

 1T-WOL 1 RW -200 65 -265 35.6 

 1T-WOL 1 RW -200 65 -265 42.8 

 2T-WOL 2 RW -150 65 -215 46.9 

 2T-WOL 2 RW -150 65 -215 66.9 

HSST 02 Mager (1969) 1X-WOL 1 RW -200 0 -200 30.5 

 1X-WOL 1 RW -200 0 -200 37.5 

 1X-WOL 1 RW -200 0 -200 41.0 

 1T-WOL 1 RW -200 0 -200 31.2 

 1T-WOL 1 RW -200 0 -200 30.8 

 1T-WOL 1 RW -175 0 -175 43.5 

 1X-WOL 1 RW -150 0 -150 29.7 

 1X-WOL 1 RW -150 0 -150 31.5 

 1X-WOL 1 RW -150 0 -150 41.2 

 1X-WOL 1 RW -150 0 -150 30.5 

 1X-WOL 1 RW -125 0 -125 39.1 

 1T-WOL 1 RW -125 0 -125 48.3 

 1T-WOL 1 RW -125 0 -125 43.4 

 1T-WOL 1 RW -125 0 -125 38.1 

 2T-WOL 2 RW -100 0 -100 51.4 

 2T-WOL 2 RW -100 0 -100 59.0 

 2T-WOL 2 RW -100 0 -100 56.2 

 2T-WOL 2 RW -100 0 -100 50.2 

 2T-WOL 2 RW -50 0 -50 65.1 

 2T-WOL 2 RW -50 0 -50 65.0 

 2T-WOL 2 RW -50 0 -50 67.5 

 2T-WOL 2 RW -50 0 -50 65.0 

 1X-WOL 1 RW -250 0 -250 37.3 

 1X-WOL 1 RW -200 0 -200 44.0 

 1X-WOL 1 RW -200 0 -200 34.6 

 1X-WOL 1 RW -200 0 -200 39.9 

 1X-WOL 1 RW -200 0 -200 38.5 

 1T-C(T) 1 RW -150 0 -150 42.1 

 1T-C(T) 1 RW -150 0 -150 37.7 

 1T-C(T) 1 RW -150 0 -150 40.7 

 1T-C(T) 1 RW -100 0 -100 42.2 

 1T-C(T) 1 RW -100 0 -100 48.5 

 1T-C(T) 1 RW -100 0 -100 48.5 

 1T-C(T) 1 RW -75 0 -75 50.3 

 1T-C(T) 1 RW -75 0 -75 46.6 

 1T-C(T) 1 RW -100 0 -100 54.8 

 1T-C(T) 1 RW -100 0 -100 54.4 

 2T-WOL 2 RW -50 0 -50 56.7 

 2T-WOL 2 RW 0 0 0 66.4 

 2T-WOL 2 RW 0 0 0 93.7 

 2T-WOL 2 RW 0 0 0 83.4 

A533B Class 1 Mager (1969) 1X-WOL 1 -320 -45 -275 29.7 

weld 1X-WOL 1 -320 -45 -275 27.2 

 1X-WOL 1 -250 -45 -205 37.6 

 1X-WOL 1 -250 -45 -205 37.8 

 1T-WOL 1 -250 -45 -205 43.6 

 2T-WOL 2 -250 -45 -205 55.6 

 1T-WOL 1 -225 -45 -180 40.1 

 1T-WOL 1 -225 -45 -180 52.8 
 

C-4

Material Reference Source Specimen Type Orientation T RTNDT T - RTNDT KIc

 ID No. (°F) (°F) (°F) (ksi√in)

 2T-WOL 2 -225 -45 -180 66.2 

 2T-WOL 2 -200 -45 -155 70.7 

A533B Class 1 Mager (1969) 1X-WOL 1 -320 0 -320 30.3 

weld-HAZ 1X-WOL 1 -250 0 -250 35.2 

 1X-WOL 1 -250 0 -250 40.4 

 1T-WOL 1 -250 0 -250 30.5 

 1T-WOL 1 -250 0 -250 44.2 

 2T-WOL 2 -200 0 -200 71.2 

A508 Class 2 Mager (1969) 1X-WOL 1 -320 50 -370 39.6 

European 1X-WOL 1 -320 50 -370 27.5 

Forging 1T-WOL 1 -320 50 -370 47.5 

“ring forging” 1X-WOL 1 -250 50 -300 43.2 

 1X-WOL 1 -250 50 -300 47.9 

 1X-WOL 1 -250 50 -300 41.6 

 1T-WOL 1 -250 50 -300 51.3 

 1T-WOL 1 -200 50 -250 55.0 

 2T-WOL 2 -200 50 -250 43.3 

 2T-WOL 2 -150 50 -200 57.2 

 2T-WOL 2 -125 50 -175 56.2 

 2T-WOL 2 -100 50 -150 56.0 

HSST 02 Shabbits 6T-C(T) 6 RW 25 0 25 98.9 

 (1969) 6T-C(T) 6 RW 25 0 25 74.5 

 6T-C(T) 6 RW 25 0 25 90.5 

 6T-C(T) 6 RW 0 0 0 73.9 

 6T-C(T) 6 RW 0 0 0 66.9 

 11T-C(T) 11 RW 50 0 50 148.6 

 10T-C(T) 10 RW 50 0 50 137.3 

 10T-C(T) 10 RW 50 0 50 139.0 

 4T-C(T) 4 RW 0 0 0 87.2 

 4T-C(T) 4 RW -25 0 -25 61.0 

 4T-C(T) 4 RW -25 0 -25 58.7 

 4T-C(T) 4 RW -25 0 -25 45.9 

 10T-C(T) 10 RW 0 0 0 87.5 

 10T-C(T) 10 RW 25 0 25 110.3 

 1T-C(T) 1 RW -250 0 -250 37.3 

 1T-C(T) 1 RW -200 0 -200 44.4 

 1T-C(T) 1 RW -200 0 -200 34.6 

 1T-C(T) 1 RW -200 0 -200 39.9 

 1T-C(T) 1 RW -200 0 -200 34.8 

 1T-C(T) 1 RW -150 0 -150 44.1 

 1T-C(T) 1 RW -150 0 -150 37.4 

 1T-C(T) 1 RW -150 0 -150 41.8 

 1T-C(T) 1 RW -100 0 -100 48.3 

 1T-C(T) 1 RW -100 0 -100 48.3 

 1T-C(T) 1 RW -100 0 -100 41.9 

 2T-C(T) 2 RW -100 0 -100 49.7 

 2T-C(T) 2 RW -50 0 -50 64.6 

 2T-C(T) 2 RW -50 0 -50 64.7 

A508 Class 2 unpublished 2T-C(T) 2 -150 51 -201 52.2 

 outside of 2T-C(T) 2 -150 51 -201 45.5 

 EPRI NP-719-SR 2T-C(T) 2 -125 51 -176 46.0 

 2T-C(T) 2 -125 51 -176 64.3 

 2T-C(T) 2 -125 51 -176 50.0 

 4T-C(T) 4 -25 51 -76 45.0 
 

C-5

Material 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 

A508 Class 2 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 

HSSI Weld NUREG/CR- 1T-C(T) 1 T-L -238 -9.4 -228.6 35.09 

72W 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 

HSSI NUREG/CR- 1T-C(T) 1 T-L -238 -29.2 -208.8 34.64 

73W 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 

HSST Plate 13 NUREG/CR- 1T-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 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 25.36 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 26.18 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 29.27 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 29.45 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 30.18 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 31.00 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 32.82 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 33.82 

 1⁄2T-C(T) 0.5 L-T -238 -9.4 -228.6 36.00 

 1⁄2T-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-6

Material Reference Source Specimen Type Orientation T RTNDT T - RTNDT KIc

 ID No. (°F) (°F) (°F) (ksi√in)

 1T-C(T) 1 L-T -238 -9.4 -228.6 35.09 

 1T-C(T) 1 L-T -238 -9.4 -228.6 36.00 

 1T-C(T) 1 L-T -238 -9.4 -228.6 37.45 

 1T-C(T) 1 L-T -238 -9.4 -228.6 37.45 

 1T-C(T) 1 L-T -238 -9.4 -228.6 39.55 

 1T-C(T) 1 L-T -238 -9.4 -228.6 39.73 

 1T-C(T) 1 L-T -238 -9.4 -228.6 40.36 

 1T-C(T) 1 L-T -238 -9.4 -228.6 42.36 

 1T-C(T) 1 L-T -238 -9.4 -228.6 43.73 

 1T-C(T) 1 L-T -238 -9.4 -228.6 46.45 

 1T-C(T) 1 L-T -238 -9.4 -228.6 49.55 

 1T-C(T) 1 L-T -238 -9.4 -228.6 49.64 

 2T-C(T) 2 L-T -238 -9.4 -228.6 30.09 

 2T-C(T) 2 L-T -238 -9.4 -228.6 33.00 

 2T-C(T) 2 L-T -238 -9.4 -228.6 36.55 

 2T-C(T) 2 L-T -238 -9.4 -228.6 37.00 

 2T-C(T) 2 L-T -238 -9.4 -228.6 39.36 

 2T-C(T) 2 L-T -238 -9.4 -228.6 39.91 

 2T-C(T) 2 L-T -238 -9.4 -228.6 40.91 

 2T-C(T) 2 L-T -238 -9.4 -228.6 41.45 

 2T-C(T) 2 L-T -238 -9.4 -228.6 42.18 

 2T-C(T) 2 L-T -238 -9.4 -228.6 46.45 

 2T-C(T) 2 L-T -238 -9.4 -228.6 48.64 

 2T-C(T) 2 L-T -238 -9.4 -228.6 53.18 

A508 Class 3 Iwadate, et al. Bx2B 1 NA -238 -13 -225 37.29 

 ASTM STP Bx2B 1 NA -238 -13 -225 39.89 

 803 Bx2B 1 NA -238 -13 -225 44.22 

 Bx2B 4 NA -166 -13 -153 43.36 

 Bx2B 4 NA -76 -13 -63 63.30 

 Bx2B 3 NA -4 -13 9 69.37 

Midland Nozzle NUREG/CR- 1T-C(T) 1 -58 52 -110 49.81 

Course Weld 6249 1T-C(T) 1 -148 52 -200 45.63 

 1T-C(T) 1 -148 52 -200 44.63 

 1T-C(T) 1 -148 52 -200 42.81 

 1T-C(T) 1 -148 52 -200 33.45 

 1T-C(T) 1 -148 52 -200 32.36 

Midland Beltline NUREG/CR- 1T-C(T) 1 -148 23 -171 36.45 

 6249 1T-C(T) 1 -148 23 -171 34.91 

Plate 02 4th Irr. NUREG/CR- 1T-C(T) 1 T-L -148 0 -148 38.09 

Series 4880, 1988 1T-C(T) 1 T-L -139 0 -139 33.45 

 Plate 02 1T-C(T) 1 T-L -139 0 -139 39.27 

 (68-71W) 1T-C(T) 1 T-L -139 0 -139 40.09 
 

C-7

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-8

Table C2. Crack Arrest Toughness KIa ORNL 99/27 Database 

Material Reference Specimen Size Orientation T RTNDT T-RTNDT KIa

 Source ID No. (°F) (°F) (°F) (ksi√in) 

HSST-02 EPRI NP CCA 1.4 L-T -150 0 -150 28.0 

HSST-02 719-SR CCA 1 L-T -70 0 -70 43.0 

HSST-02 Ripling (1971) CCA 2 L-T -70 0 -70 48.0 

HSST-02 CCA 2 L-T -70 0 -70 43.0 

HSST-02 CCA 1 L-T 0 0 0 68.0 

HSST-02 CCA 1 L-T 0 0 0 58.0 

HSST-02 CCA 1 L-T 0 0 0 48.0 

HSST-02 CCA 1 L-T 0 0 0 57.0 

HSST-02 CCA 1 L-T 0 0 0 62.0 

HSST-02 CCA 1.3 L-T 0 0 0 58.0 

HSST-02 CCA 1.3 L-T 0 0 0 60.0 

HSST-02 CCA 1.3 L-T 0 0 0 65.0 

HSST-02 CCA 1.6 L-T 0 0 0 60.0 

HSST-02 CCA 1.6 L-T 0 0 0 58.0 

HSST-02 CCA 2 L-T 0 0 0 53.0 

HSST-02 CCA 2 L-T 0 0 0 58.0 

HSST-02 CCA 2 L-T 0 0 0 70.0 

HSST-02 CCA 2 L-T 0 0 0 57.0 

HSST-02 CCA 3 L-T 0 0 0 57.0 

HSST-02 CCA 3 L-T 0 0 0 61.0 

HSST-02 CCA 2 L-T 22 0 22 68.0 

HSST-02 CCA 1.4 L-T 35 0 35 59.0 

HSST-02 CCA 1.6 L-T 35 0 35 84.0 

HSST-02 CCA 2 L-T 35 0 35 62.0 

HSST-02 CCA 1.4 L-T 50 0 50 92.0 

HSST-02 CCA 2 L-T 50 0 50 73.0 

HSST-02 CCA 3 L-T 50 0 50 75.0 

HSST-02 CCA 1 L-T 75 0 75 94.0 

HSST-02 CCA 1.6 L-T 75 0 75 107.0 

HSST-02 CCA 2 L-T 75 0 75 77.0 

HSST-02 CCA 2 L-T 75 0 75 81.0 

HSST-02 CCA 2 L-T 75 0 75 91.0 

HSST-02 CCA 2 L-T 75 0 75 102.3 

HSST-02 CCA 2 L-T 80 0 80 109.0 

HSST-02 CCA 2 L-T 83 0 83 87.0 

HSST-02 CCA 3 L-T 83 0 83 94.0 

HSST-02 CCA 3 L-T 83 0 83 107.0 

HSST-02 CCA 3 L-T 83 0 83 111.0 

HSST-02 CCA 2 L-T 96 0 96 111.0 

HSST-02 CCA 2 L-T 102 0 102 117.0 

HSST-02 CCA 1.8 L-T 105 0 105 118.0 

HSST-02 CCA 2 L-T 105 0 105 103.0 

HSST-02 CCA 2 L-T 105 0 105 107.0 

HSST-02 CCA 3 L-T 105 0 105 130.0 

HSST-02 CCA 2 L-T 107 0 107 87.0 

HSST-02 CCA 2 L-T 110 0 110 88.0 

HSST-02 CCA 2 L-T 110 0 110 88.0 

HSST-02 CCA 1.1 L-T 112 0 112 112.0 

HSST-02 CCA 2 L-T 115 0 115 111.0 

HSST-02 CCA 1.1 L-T 121 0 121 116.0 

72W NUREG/CR-5584 CCA Crack -77.8 -10 -68 60.1 
 

C-9

Material Reference Specimen Size Orientation T RTNDT T-RTNDT KIa

 Source ID No. (°F) (°F) (°F) (ksi√in) 

72W CCA runs -76 -10 -66 48.2 

72W CCA in -76 -10 -66 69.2 

72W CCA welding -74.2 -10 -64.2 51.9 

72W CCA direction -52.6 -10 -42.6 61.0 

72W CCA -52.6 -10 -42.6 64.6 

72W CCA -49 -10 -39 66.4 

72W CCA -49 -10 -39 67.3 

72W CCA -49 -10 -39 69.2 

72W CCA -49 -10 -39 83.7 

72W CCA -25.6 -10 -15.6 83.7 

72W CCA -22 -10 -12 54.6 

72W CCA -22 -10 -12 55.5 

72W CCA -22 -10 -12 77.4 

72W CCA -22 -10 -12 82.8 

72W CCA -22 -10 -12 89.2 

72W CCA -22 -10 -12 94.6 

72W CCA -22 -10 -12 97.4 

72W CCA 3.2 -10 13.2 88.3 

72W CCA 5 -10 15 85.5 

72W CCA 5 -10 15 85.5 

72W CCA 5 -10 15 86.5 

72W CCA 5 -10 15 93.7 

72W CCA 6.8 -10 16.8 82.8 

72W CCA 28.4 -10 38.4 93.7 

72W CCA 30.2 -10 40.2 113.8 

72W CCA 32 -10 42 84.6 

72W CCA 32 -10 42 97.4 

72W CCA 32 -10 42 103.7 

72W CCA 33.8 -10 43.8 98.3 

72W CCA 39.2 -10 49.2 113.8 

72W CCA 41 -10 51 104.7 

73W NUREG/CR-5584 CCA Crack -77.8 -30 -47.8 62.8 

73W CCA runs -76 -30 -46 52.8 

73W CCA in -74.2 -30 -44.2 65.5 

73W CCA welding -49 -30 -19 47.3 

73W CCA direction -49 -30 -19 66.4 

73W CCA -49 -30 -19 68.3 

73W CCA -49 -30 -19 77.4 

73W CCA -47.2 -30 -17.2 64.6 

73W CCA -25.6 -30 4.4 77.4 

73W CCA -23.8 -30 6.2 68.3 

73W CCA -22 -30 8 61.0 

73W CCA -22 -30 8 72.8 

73W CCA -22 -30 8 91.0 

73W CCA -20.2 -30 9.8 70.1 

73W CCA -20.2 -30 9.8 81.0 

73W CCA 3.2 -30 33.2 100.1 

73W CCA 5 -30 35 106.5 

73W CCA 5 -30 35 111.9 

73W CCA 5 -30 35 112.8 

73W CCA 10.4 -30 40.4 102.3 

73W CCA 23 -30 53 91.9 

73W CCA 41 -30 71 97.4 

73W CCA 41 -30 71 101.9 
 

C-10

Material Reference Specimen Size Orientation T RTNDT T-RTNDT KIa

 Source ID No. (°F) (°F) (°F) (ksi√in) 

73W CCA 41 -30 71 102.8 

73W CCA 41 -30 71 108.3 

73W CCA 59 -30 89 120.1 

MW15JC NUREG/CR-6621 CCA Crack -4 32.2 -36.2 63.7 

MW15JBr CCA runs 14 32.2 -18.2 79.0 

MW15JEr1 CCA in welding 32 32.2 -0.2 97.1 

MW15JF CCA direction 50 32.2 17.8 119.7 

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-11

Table C3. Crack Arrest Toughness KIa Extended KIa Database – Large Specimen Data 

Material Reference T RT NDT T-RT NDT K Ia

Test No. Source (°F) (°F) (°F) (ksi-in1/2)

WP 1.2A NUREG/CR-4930 -9.4 143.6 153.0 385.81

WP 1.2B -9.4 197.6 207.0 623.29

WP 1.3 -9.4 129.2 138.6 213.83

WP 1.4B -9.4 140.0 149.4 352.14

WP 1.5A -9.4 132.8 142.2 210.19

WP 1.5B -9.4 161.6 171.0 463.15

WP 1.6A -9.4 129.2 138.6 250.23

WP 1.6B -9.4 176.0 185.4 361.24

WP 1.7A NUREG/CR-5330 -9.4 141.8 151.2 290.26

WP 1.7B -9.4 190.4 199.8 505.00

WP 1.8A -9.4 104.0 113.4 313.92

WP 1.8B -9.4 131.0 140.4 440.40

WP 1.8C -9.4 174.2 183.6 512.28

WP CE-1 -31.0 96.8 127.8 154.69

WP CE-2A -31.0 107.6 138.6 198.36

WP CE-2B -31.0 127.4 158.4 322.11

WP CE-2C -31.0 140.0 171.0 524.11

SP 1.3 Smirt 10 Vol F, p 37 -9.4 111.2 120.6 160.15

WP 2.1A NUREG/CR-5451 140.0 176.0 36.0 96.45

WP 2.1B 140.0 204.8 64.8 139.22

WP 2.1D 140.0 221.0 81.0 143.77

WP 2.1E 140.0 233.6 93.6 154.69

WP 2.1F 140.0 257.0 117.0 182.89

WP 2.1H 140.0 275.0 135.0 266.61

WP 2.1I 140.0 293.0 153.0 337.58

WP 2.1J 140.0 305.6 165.6 369.43

WP 2.2A 140.0 248.0 108.0 182.89

WP 2.2B 140.0 264.2 124.2 235.67

WP 2.2C 140.0 271.4 131.4 255.69

WP 2.2D 140.0 282.2 142.2 252.05

WP 2.2E 140.0 287.6 147.6 345.77

WP 2.2F 140.0 302.0 162.0 331.21

WP 2.2G 140.0 323.6 183.6 405.82

WP 2.3A 140.0 206.6 66.6 131.03

WP 2.3B 140.0 222.8 82.8 211.10

WP 2.3D 140.0 231.8 91.8 232.03

WP 2.3F 140.0 258.8 118.8 234.76

WP 2.4B 140.0 186.8 46.8 124.66

WP 2.4C 140.0 215.6 75.6 171.06

WP 2.4D 140.0 224.6 84.6 255.69

WP 2.4E 140.0 249.8 109.8 226.57

WP 2.4F 140.0 260.6 120.6 279.34

WP 2.4G 140.0 278.6 138.6 346.68

WP 2.4H 140.0 300.2 160.2 361.24

WP 2.5B 140.0 219.2 79.2 155.60

WP 2.5C 140.0 255.2 115.2 172.88

WP 2.5D 140.0 275.0 135.0 243.86

WP 2.5E 140.0 291.2 151.2 278.43

WP 2.5F 140.0 309.2 169.2 333.03

WP 2.6A 140.0 219.2 79.2 185.62

WP 2.6B 140.0 239.0 99.0 235.67

WP 2.6C 140.0 246.2 106.2 260.24

WP 2.6D 140.0 257.0 117.0 318.47

WP 2.6F 140.0 271.4 131.4 298.45

WP 2.6G 140.0 282.2 142.2 373.98

WP 2.6H 140.0 312.8 172.8 375.80

PTSE 1B NUREG/CR-4106 196.3 326.3 130.0 182.80

PTSE 1C 196.3 354.2 157.9 271.97

PTSE 2A NUREG/CR-4888 167.0 267.1 100.1 237.85

PTSE 2B 167.0 296.2 129.2 329.03

PTSE 2C 167.0 325.2 158.2 381.53

TSE 4 NUREG/CR-4249 167.0 267.8 100.8 115.56

TSE 5-1 152.6 96.8 -55.8 78.25

TSE 5-2 152.6 179.6 27.0 94.63

TSE 5-3 152.6 192.2 39.6 83.71

TSE 5A-1 50.0 71.6 21.6 69.15

TSE 5A-2 50.0 100.4 50.4 78.25

TSE 5A-3 50.0 123.8 73.8 97.36

TSE 5A-4 50.0 152.6 102.6 118.29

TSE 6-1 152.6 89.6 -63.0 57.32

TSE 6-2 152.6 145.4 -7.2 95.54
 

C-12

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. 
D-1

Appendix D – Summary of RVID2 Data for Use in FAVOR Calculations 

RTNDT(u) [oF] Com

Product Form Heat Beltline σflow(u) [ksi] RTNDT(u) Method RTNDT(u) Value σ(u) Value Cu 

Beaver Valley 1, (Designer: Westinghouse, Manufacturer: CE) Coolant Temperature = 547°F, Vessel Thickness = 7-7/8 in. 

C4381-1 INTERMEDIATE SHELL B6607-1 83.8 MTEB 5-2 43 0 0.14 C4381-2 INTERMEDIATE SHELL B6607-2 84.3 MTEB 5-2 73 0 0.14 C6293-2 LOWER SHELL B7203-2 78.8 MTEB 5-2 20 0 0.14 

PLATE 

C6317-1 LOWER SHELL B6903-1 72.7 MTEB 5-2 27 0 0.2 

305414 LOWER SHELL AXIAL WELD 20-714 75.3 Generic -56 17 0.337 

LINDE 1092 WELD 

305424 INTER SHELL AXIAL WELD 19-714 79.9 Generic -56 17 0.273 

LINDE 0091 WELD 90136 CIRC WELD 11-714 76.1 Generic -56 17 0.269 

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 78.8 MTEB 5-2 -20 0 0.11 B-8489-2 LOWER SHELL D-7207-2 80.3 MTEB 5-2 -10 0 0.11 C-4351-2 INTERMEDIATE SHELL D-7206-1 74.7 MTEB 5-2 20 0 0.11 C-4420-1 LOWER SHELL D-7207-1 78.0 MTEB 5-2 10 0 0.13 C-4441-1 INTERMEDIATE SHELL D-7206-3 78.5 ASME NB-2331 10 0 0.12 

PLATE 

C-4441-2 INTERMEDIATE SHELL D-7206-2 82.6 ASME NB-2331 -30 0 0.12 

20291/12008 INTERMEDIATE SHELL AXIAL WELD 2-203 78.8 ASME NB-2331 -50 0 0.22 

LINDE 1092 WELD 

21935 LOWER SHELL AXIAL WELD 3-203A/C 78.6 Generic -56 17 0.18 

LINDE 0091 WELD 33A277 INT. TO LOWER SHELL CIRC. WELD 9-203 78.6 ASME NB-2331 -80 0 0.24 

Oconee 1, (Designer and Manufacturer: B&W) Coolant Temperature = 556°F, Vessel Thickness = 8.44-in.

FORGING AHR54 (ZV2861) LOWER NOZZLE BELT (4) B&W Generic 3 31 0.16 

C2197-2 INTERMEDIATE SHELL (4) B&W Generic 1 26.9 0.15 C2800-1 LOWER SHELL (4) B&W Generic 1 26.9 0.11 C2800-2 LOWER SHELL 69.9 B&W Generic 1 26.9 0.11 C3265-1 UPPER SHELL 75.8 B&W Generic 1 26.9 0.1 

PLATE 

C3278-1 UPPER SHELL (4) B&W Generic 1 26.9 0.12 

1P0962 INTERMEDIATE SHELL AXIAL WELDS SA-1073 79.4 B&W Generic -5 19.7 0.21 299L44 INT./UPPER SHL CIRC WELD (OUTSIDE 39%) WF-25 (4) B&W Generic -7 20.6 0.34 61782 NOZZLE BELT/INT. SHELL CIRC WELD SA-1135 (4) B&W Generic -5 19.7 0.23 71249 INT./UPPER SHL CIRC WELD (INSIDE 61%) SA-1229 76.4 ASME NB-2331 10 0 0.23 

LINDE 80 WELD 

72445 UPPER/LOWER SHELL CIRC WELD SA-1585 (4) B&W Generic -5 19.7 0.22 
D-2

RTNDT(u) [oF] Com

Product Form Heat Beltline σflow(u) [ksi] RTNDT(u) Method RTNDT(u) Value σ(u) Value Cu 

8T1762 LOWER SHELL AXIAL WELDS SA-1430 75.5 B&W Generic -5 19.7 0.19 8T1762 UPPER SHELL AXIAL WELDS SA-1493 (4) B&W Generic -5 19.7 0.19 8T1762 LOWER SHELL AXIAL WELDS SA-1426 75.5 B&W Generic -5 19.7 0.19 

Palisades, (Designer and Manufacturer: CE) Coolant Temperature = 532°F, Vessel Thickness = 81⁄2 in.

A-0313 D-3803-2 (4) MTEB 5-2 -30 0 0.24 B-5294 D-3804-3 (4) MTEB 5-2 -25 0 0.12 C-1279 D-3803-3 (4) ASME NB-2331 -5 0 0.24 C-1279 D-3803-1 74.7 ASME NB-2331 -5 0 0.24 C-1308A D-3804-1 (4) ASME NB-2331 0 0 0.19 

PLATE 

C-1308B D-3804-2 (4) MTEB 5-2 -30 0 0.19 

LINDE 0124 WELD 27204 CIRC. WELD 9-112 76.9 Generic -56 17 0.203 

34B009 LOWER SHELL AXIAL WELD 3-112A/C 76.1 Generic -56 17 0.192 

LINDE 1092 WELD W5214 LOWER SHELL AXIAL WELDS 3-112A/C 72.9 Generic -56 17 0.213 

W5214 INTERMEDIATE SHELL AXIAL WELDS 2-112 A/C 72.9 Generic -56 17 0.213 

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 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, whi

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 

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-3

Kirk M. T. Kirk, C. S. Santos, E.D. Eason, J.E. Wright, and G. R. Odette, “Updated Embrittlement Trend Curve 

Vessel Steels,” Paper No. G01-5, Transactions of the 17th International Conference on Structural MechaniTechnology (SMiRT 17), Prague, Czech Republic, August 17-22, 2003. 
 

E-1

Appendix E – Statistical Point-Estimation Techniques for Weibull Distributions 

The three parameters for the Weibull distributions of RT T NDT − 0 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 

( ) 1 ( , , ) exp , for

( ( ) / , , , 0)

c c c w RT a b c y y b

y RT a b RT a b c

− Δ= −

=Δ − Δ > >

 (E1) 

and the corresponding likelihood function is the joint density (see Ref.[E1]) (given the location 

parameter, a) 

1

() ()

1

(, | , )

exp

c c N

i i

i

L b c RT a

c RT a RT a

b b b

−

=

Δ =

Δ− Δ− ⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎢− ⎥ ⎜ ⎟⎜ ⎟ ⎢ ⎥ ⎝ ⎠⎝ ⎠ ⎣ ⎦

∏ (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 

( ) ( )

1

( )

1

( )

1

( ) ln( )

(ln( ( )) 1 1

ln( ) 0

( )

N c i i N

i

i N c i

i

i

RT a RT a

L c

RT a

c N c

RT a

′

=

′ =

=

Δ− Δ−

∂ ′ = − Δ − − = ∂ ′ ′

Δ −

∑

∑

∑

 (E3) 

Upon obtaining a solution for c′ , the ML estimator for the scale parameter, b′ , follows directly 

from 
 

E-2

( )

1

( )

1

(ln( )) 0

c c N i

i

RT a L b

b N

′ ′

=

⎡ ⎤ Δ − ∂ ⎢ ⎥ =− = ′ ∂ ′

⎣ ⎦

∑ (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), 

* 2

2 bm c c = Γ + −Γ + /[ (1 2 / ) (1 1/ )] (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 

* * 1 a mb c = − Γ+ ′ (1 1/ ) (E6) 

where m1

′ is the 1st crude moment of the sample (the sample mean) and the sample moments are 

defined by 

( )

( )

1

1

2

1

2

1

( )

N

NDT i

i

N

NDT

i

RT

m

N

RT i m

m

N

=

=

Δ ′ =

Δ − ′ =

∑

∑

 (E7) 

From Ref. [B.2], a moment estimator for the shape parameter, *

c , also exists 

2 3 * 11 1

1

4.104683 1.148513 0.44326( ) 0.053025( )

1.139547

bb b

c

b

−+ − = +

 (E8) 

where 1b is the sample skewness. However, for sample sizes as small as 20, there will be a 

high level of uncertainty in the *** (,,) abc 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 ( ) ( , 1, ) Δ = RTi N i , calculate the sample moments, 1 2 (, ) m m′ from Eqs. (E7). 

2) Select a trial value for the location parameter, trial a where trial min( , 1,2, ) ( )i a R <Δ = Ti N K . 

3) Calculate ML estimates for (, ) c b′ ′ from Eqs. (E3)-(E4) by letting trial a a = . 
 

E-3

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 trial a and the MM estimate of * a from Step 4 

by 

* trial

trial

a a

a

δ − = (E9) 

6) Given tolerance ε , as a pre-selected convergence tolerance, if tolerance δ ε > , then select a new 

trial location parameter, trial a , and repeat Steps 3-6 until convergence, defined as tolerance δ ε ≤ . 

Upon convergence, there will be two triplets ( , , ) trial a bc ′ ′ and * * ( , ,) abc′ where in general 

* trial a a ≈ and * b b ′ ≠ although b′ was typically close to *b in this study. The triplet * ( ,,) a bc ′ ′

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. 
 

F-1

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 

( )

( ) ( )

1

( | , , ) exp , , ( , ) 0

Pr ( | , , ) 1 exp , , ( , ) 0

c c

W

c

W

c RT a RT a f RT a b c RT a b c

b b b

RT a X RT F RT a b c P RT a b c

b

−

Δ− Δ− ⎡ ⎤ ⎛⎞⎛⎞ Δ = − Δ> > ⎜⎟⎜⎟ ⎢ ⎥ ⎝⎠⎝⎠ ⎢ ⎥ ⎣ ⎦

⎡ ⎤ ⎛ ⎞ Δ − ≤Δ = Δ = = − −⎢ ⎥ ⎜ ⎟ Δ > >

⎢ ⎥ ⎝ ⎠ ⎣ ⎦

 (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

Δ =+ RT a b P P ln 1 ; 0 1 c ⎡− − ⎤ < < ⎣ ⎦ (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 

RT RT RT LB NDT = −Δ (u) (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): 

40.02 F

124.88 F

 1.96

a

b

c

=− °

= °

=

 (F4) 

based on the sample (N = 18) given in Table 8 and repeated in Table F1. 
 

F-2

Table F1. ΔRTepistemic Ranked Data with Order-Statistic Estimates of P

i ΔRT i , (°F) P i ln( - ln(1 - P i ))

1 -19.4 0.03804 -3.24970

2 -10.9 0.09239 -2.33364

3 -1.7 0.14674 -1.84080 mean = 70.67

4 2.1 0.20109 -1.49387 variance = 3669.77

5 33.2 0.25543 -1.22093 stdv = 60.58

6 38.4 0.30978 -0.99223

7 50.1 0.36413 -0.79239 Pi =(i -0.3)/(n +0.4)

8 54.6 0.41848 -0.61229

9 62.3 0.47283 -0.44594

10 64.3 0.52717 -0.28898

11 81.9 0.58152 -0.13796

12 89.4 0.63587 0.01019

13 91.5 0.69022 0.15861

14 97.8 0.74457 0.31100

15 142.2 0.79891 0.47251

16 147.6 0.85326 0.65186

17 162.4 0.90761 0.86782

18 186.2 0.96196 1.18449

Sample

From the following asymptotic relations for the mean and variance of a Weibull distribution, 

2 2 2

1

0

1 1

2 1 1 1 ,

( ) x t

a b

c

b

c c

x t e dt

μ

σ

∞ − −

⎛ ⎞ =+Γ + ⎜ ⎟ ⎝ ⎠

⎡ ⎤ ⎛⎞⎛⎞ = Γ + −Γ + ⎢ ⎥ ⎜⎟⎜⎟ ⎣ ⎦ ⎝⎠⎝⎠

Γ = ∫

 (F5) 

the mean and variance for the Weibull model for ΔRTepistemic compared to the corresponding 

sample estimators are: 

1

2 2

Model Sample

 = 70.70 F = 70.67 F

= 3473.65 = 3669.77

 = 58.94 F = 60.58 F

m

s

s

μ

σ

σ

° ° ′

° °
 

F-3

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 ( , ), 1,2, i i x yi n = K be an observed set of 

data. Assume that the values yi are a (possibly nonlinear) function of xi and a set of unknown 

parameters p β ∈R , where both yi and xi contain the uncertainties, * 1

i ε ∈R and * 1i δ ∈R , 

respectively. The superscript “*” denotes an actual but unknown value. The observed value, yi , 

can be expressed in terms of a model equation

( ) { } ( ) * * * ; 1, 2, i i ii i k y += + = ε δβ f x i n K (F6) 

for some actual values of the parameter vector ( ) { }

* ; 1,2, k β k p = K . The variables yi 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 

( ) 2 2

, , 1

min n

i i

βδε i

ε δ

=

∑ + (F7) 

subject to the constraints 

( ) { } 1,2, . i ii i i y =+ − = f x δβ ε i n K (F8) 

Since the constraints are linear in i ε , they and thus i ε can be eliminated from the minimization 

problem, obtaining 

{ }{ }

( ) { }

2

2

, 1

min n

ii i i i

i

fx y

β δ

δ β δ =

⎛ ⎞ ⎜ ⎟ ⎡ ⎤ + −+ ⎣ ⎦ ⎝ ⎠ ∑ (F9) 

The algorithm implemented in ODRPACK uses the Levenberg-Marquardt trust region method to 

iteratively solve the nonlinear minimization problem of Eq. (F9). 
 

F-4

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 

( )

() ( )

1 exp

1 exp

ln 1

ln ln 1 ln ln( )

c

c

c

RT a P

b

RT a P

b

RT a P

b

P c RT a c b

⎡ Δ − ⎤ ⎛ ⎞ =− −⎢ ⎜ ⎟ ⎥ ⎢⎣ ⎝ ⎠ ⎥⎦

⎡ Δ − ⎤ ⎛ ⎞ −= −⎢ ⎜ ⎟ ⎥ ⎢⎣ ⎝ ⎠ ⎥⎦

⎛ ⎞ Δ − − −= ⎜ ⎟ ⎝ ⎠

⎡ ⎤ − − = Δ −− ⎣ ⎦

 (F10) 

The location parameter, a, is related to the scale, b, and shape, c, parameters through its moment 

estimator 

1

1 am b 1

c

⎛ ⎞ ≈ −Γ + ′ ⎜ ⎟ ⎝ ⎠ (F11) 

where m1

′ 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, m1

′ . Introducing Eq. (F11) into Eq. (F10), the final form of the nonlinear 

model equation is 

{ } { } { } {} ( ) { }

1 12 1 2

1

1

( )

2

1 ln 1 ln( ); ( 1,2, )

where 

; ; ln ln 1

i i

i ii i

y xm i n

c

x RT y P b

β β ββ β

β β β

⎡ ⎤ ⎛ ⎞ = − + Γ+ − = ... ′ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

⎛ ⎞ ⎧ ⎫ ⎧ ⎫ ⎜ ⎟ = = =Δ = − − ⎨ ⎬ ⎨⎬ ⎡ ⎤ ⎣ ⎦ ⎝ ⎠ ⎩ ⎭ ⎩ ⎭

 (F12) 

Values for Pi can be estimated by ranking the data in Table F1 and applying the median-rank 

order statistic 

0.3

0.4 i

i P

n

− ≈ +

 (F13) 
 

F-5

ODRPACK iteratively solves for the solution vector 

1

2

1

2

n n 2

β

β

δ

δ

δ +

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭⎪

M

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 : 

( )

1

1.855

Location Parameter, -45.586 95% Confidence Intervals

Scale Parameter, 130.899 10.259 109.15 to 152.65

Shape Parameter, 1.855 0.227 1.374 to 2.337

45.586 130.899 ln 1 ; 0 1

Sample Me

ODR

a

b

c

RT P P

=

= ±

= ±

Δ = − + −− ⎡ ⎤ < < ⎣ ⎦

1

2

2

 an, 70.67

Weibull Mean, 70.667

Sample Stdv, 60.58

Weibull Stdv, 65.036

Sample Variance, 3669.77

Weibull Variance, 4229.692

 

m

s

s

μ

σ

σ

′ =

=

=

=

=

=

The 95% confidence intervals for the two parameters 1 β = c and 2 β = b are calculated by 

ODRPACK using (0.975, ) k k t β ± μ β σ where (0.975, ) t 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, βk σ , 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-6

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 (NUMBER WITH NONZERO WEIGHT = 18) 

 NQ = 1 

 M = 1 

 NP = 2 (NUMBER UNFIXED = 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 CRITERIA: 

 SSTOL = 1.49D-08 (SUM OF SQUARES STOPPING TOLERANCE) 

 PARTOL = 3.67D-11 (PARAMETER STOPPING TOLERANCE) 

 MAXIT = 50 (MAXIMUM NUMBER OF ITERATIONS) 

 --- INITIAL WEIGHTED SUM OF SQUARES = 1.15671908D+00 

 SUM OF SQUARED WEIGHTED DELTAS = 0.00000000D+00 

 SUM OF SQUARED WEIGHTED EPSILONS = 1.15671908D+00 

 *** ITERATION REPORTS FOR FIT BY METHOD OF ODR *** 

 CUM. ACT. REL. PRED. REL. 

 IT. NO. FN WEIGHTED SUM-OF-SQS SUM-OF-SQS G-N 

 NUM. EVALS SUM-OF-SQS REDUCTION REDUCTION TAU/PNORM STEP 

 ---- ------ ----------- ----------- ----------- --------- ---- 

 1 12 5.36253D-01 5.3640D-01 5.3739D-01 1.333D-01 YES 

 2 19 5.33419D-01 5.2849D-03 4.2184D-03 4.265D-02 YES 

 3 26 5.33152D-01 4.9976D-04 3.9259D-04 1.461D-02 YES 

 4 33 5.33130D-01 4.1577D-05 3.2561D-05 4.323D-03 YES 

 5 40 5.33128D-01 3.2902D-06 2.5746D-06 1.224D-03 YES 

 6 47 5.33128D-01 2.5647D-07 2.0064D-07 3.423D-04 YES 

 7 54 5.33128D-01 1.9907D-08 1.5572D-08 9.542D-05 YES 

 8 61 5.33128D-01 1.5432D-09 1.2072D-09 2.657D-05 YES 

 
 

F-7

Table F2. ODRPACK Results of ODR Analysis of ΔRTepistemic Model Equation 

(continued) 

*** FINAL SUMMARY FOR FIT BY METHOD OF ODR *** 

 --- STOPPING CONDITIONS: 

 INFO = 1 ==> SUM OF SQUARES CONVERGENCE. 

 NITER = 8 (NUMBER OF ITERATIONS) 

 NFEV = 67 (NUMBER OF FUNCTION EVALUATIONS) 

 IRANK = 0 (RANK DEFICIENCY) 

 RCOND = 1.20D-01 (INVERSE CONDITION NUMBER) 

 ISTOP = 0 (RETURNED BY USER FROM SUBROUTINE FCN) 

 --- FINAL WEIGHTED SUMS OF SQUARES = 5.33127879D-01 

 SUM OF SQUARED WEIGHTED DELTAS = 7.67684538D-04 

 SUM OF SQUARED WEIGHTED EPSILONS = 5.32360195D-01 

 --- RESIDUAL STANDARD DEVIATION = 1.82539016D-01 

 DEGREES OF FREEDOM = 16 

 --- ESTIMATED BETA(J), J = 1, ..., NP: 

 BETA S.D. BETA ---- 95% CONFIDENCE INTERVAL ---- 

 1 1.85530498D+00 2.2706D-01 1.37390691D+00 TO 2.33670305D+00 

 2 1.30899017D+02 1.0259D+01 1.09149592D+02 TO 1.52648443D+02 

 --- ESTIMATED EPSILON(I) AND DELTA(I,*), I = 1, ..., N: 

 I EPSILON(I,1) DELTA(I,1) 

 1 2.62841903D-01 -1.86361603D-02 

 2 -1.29977011D-01 6.95094427D-03 

 3 -1.86382404D-01 7.87802505D-03 

 4 -3.79012096D-01 1.47415688D-02 

 5 2.78865897D-01 -6.56742977D-03 

 6 1.68817068D-01 -3.72942044D-03 

 7 2.10949482D-01 -4.09035239D-03 

 8 1.16154880D-01 -2.15105581D-03 

 9 8.71915578D-02 -1.49943300D-03 

 10 -3.56507199D-02 6.01915026D-04 

 11 8.89342397D-02 -1.29426169D-03 

 12 4.68465281D-02 -6.43875329D-04 

 13 -7.29122682D-02 9.86768713D-04 

 14 -1.41925842D-01 1.83636941D-03 

 15 1.97009129D-01 -1.94642622D-03 

 16 7.02764840D-02 -6.74910438D-04 

 17 -8.73096746D-03 7.78822029D-05 

 18 -1.24381318D-01 9.95579717D-04 
 

F-8

F.1.3. Final Stochastic Model for ΔRTepistemic in FAVOR 

The epistemic uncertainty in RTNDT(u) is estimated in FAVOR by 

Δ=− RT RT RT epistemic NDT u LB ( ) (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( ) var( ) var( ) Δ= + RT R epistemic T R NDT TLB (F15) 

where the ( ) ( ) cov RT RT NDT u LB has been assumed to be zero. The statistical model developed for 

ΔRTepistemic using the ODR procedure contains the following four sources of uncertainty 

1. Measurement uncertainty and material variability in RTNDT(u), 2 σ (1)

2. Measurement uncertainty and material variability in RTLB , 2 σ (2)

3. Model uncertainty in RTNDT(u) , 2 σ (3)

4. Model uncertainty in RTLB , 2 σ (4)

such that the components of the variances for RTNDT(u) and RTLB are the following: 

2 2

( ) (1) (3)

2 2

(2) (4)

var( )

var( )

NDT u

LB

RT

RT

σ σ

σ σ

= +

= +

 (F16) 

Therefore, the variance (uncertainty) in the ODR-developed Weibull distribution for ΔRTepistemic

can be expressed as 

2 22 2 2

(1) (2) (3) (4) 4229.69 σ σσσσ ΔRT =+++= (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 22 σ σσ σ σσ Δ Δ RT rev RT ( ) (3) (4) (1) (2) = + = −− (F18) 

Two cases were examined: 
 

F-9

Case 1:

( )2 2

(1)

2

(2)

23 F

0

σ

σ

= °

=

Case 2:

( )

( )

2 2

(1)

2 2

(2)

23 F

23 F

σ

σ

= °

= °

The required adjustments to the Weibull model for ΔRTepistemic can be calculated by solving the 

following nonlinear system of equations 

22 2

( )

1 1 0

2 1 1 10

RT

RT rev

a b

c

b

c c

μ

σ

Δ

Δ

⎛ ⎞ −−Γ + = ⎜ ⎟ ⎝ ⎠

⎡ ⎤ ⎛⎞⎛⎞ − Γ + −Γ + = ⎢ ⎥ ⎜⎟⎜⎟ ⎣ ⎦ ⎝⎠⎝⎠

 (F19) 

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 22

( ) (1) (2)

2 2

( )

( )

4229.692 23 0 3700.692

60.83 F

RT rev RT

RT rev

RT rev

σ σ σσ

σ

σ

Δ Δ

Δ

Δ

= −−

= − −=

= °

The solutions for (b,c) are 

131.18 F

1.998

b

c

= °

=
 

F-10

Case 2:

2 2 22

( ) (1) (2)

2 2 2

( )

( )

4229.692 23 23 3171.692

56.32 F

RT rev RT

RT rev

RT rev

σ σ σσ

σ

σ

Δ Δ

Δ

Δ

= −−

= −−=

= °

The solutions for (b,c) are 

131.27 F

2.177

b

c

= °

=

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

1.998 ( ) 45.586 131.18 ln 1 ; 0 1 Δ =− + RT rev ⎡− − P ⎤ < < P ⎣ ⎦

Case 2:

( ) 1

2.177 ( ) 45.586 131.27 ln 1 ; 0 1 Δ =− + RT rev ⎡− − P ⎤ < < P ⎣ ⎦

Case 2 was selected for implementation into FAVOR.
 

F-11

F.2. Stochastic Model for ΔRTarrest in FAVOR 

F.2.1 Initial Model for ΔRTarrest

The initial stochastic model developed for FAVOR to describe the statistical distribution of 

0 Ia Δ =− RT TT arrest K was based on a lognormal distribution (see Fig. F5) with the parameters 

0 0 0

2 2

log

( ) 44.123exp( 0.006 ); [ ]

0.39 0.1521 (constant)

RT T arrest μ T Tc

σ

Δ == − °

= = (F20) 

The asymptotic relations for the log-mean and variance of the model are: 

[ ]

2

log

log 0 0

2 2

0 log 0 log

( ) ln ( ) 2

var( ) ( ) ( 1)exp 2 ( ) ; =exp( ) arrest

T T

RT T T

σ

μ μ

σ ωω μ ω σ

= −

Δ = =− ⎡ ⎤ ⎣ ⎦

 (F21) 

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 

10 2 ln( ) Δ =+ RT T arrest β β (F22) 

where, upon reversing the log-transformation, the mean value for ΔRTarrest is 

Δ = RT T arrest exp( )exp β β 2 10 ( ) (F23) 

The results of the ODR analysis are presented in Table F4 with the following ODR estimates for 

the model parameters: 

[ ]

1

2

2

0

2

log log

0.00597110744 0.00082458

3.78696343 0.065299

exp( ) 44.12221645 2.908036613

44.1222exp( 0.00597 ); C

0.389987535; 0.1520903

RT arrest T

β

β

β

σ σ

= − ±

= ±

= ±

Δ= − °

= =

 (F24) 
 

F-12

Table F3. Data Used in the Development of the ΔRTarrest Model 

N T 0 T KIa T KIa -T 0 ln(T KIa -T 0)

(°C) (°C) (°C)

1 -114 16 130 4.8675

2 131 140 9 2.1972

3 -66 13 79 4.3694

4 -78 6 84 4.4308

5 -104 -16 88 4.4773

6 -108 44 152 5.0239

7 43 113 70 4.2485

8 -20 60 80 4.3820

9 -71 -41 30 3.4012

10 -66 6 72 4.2767

11 -84 9 93 4.5326

12 -21 65 86 4.4543

13 -53 -6 47 3.8501

14 -54 18 72 4.2767

15 62 93 31 3.4340

16 -65 -12 53 3.9703

17 -100 -15 85 4.4427

18 -130 -8 122 4.8040

19 -100 -18 82 4.4067

20 -27 25 52 3.9512

21 -78 10 88 4.4773

22 -115 -25 90 4.4998

23 -68 -9 59 4.0775

24 -70 17 87 4.4659

25 -65 -25 40 3.6889

26 -51 19 70 4.2485

27 17 77 60 4.0943

28 -48 48 96 4.5643

29 -92 -26 66 4.1897

30 -70 -18 52 3.9512

31 -81 -20 61 4.1109

32 -157 -27 130 4.8675

33 67 78 11 2.3979

34 -84 9 93 4.5326

35 -67 18 85 4.4427

36 -58 -14 44 3.7842

37 35 74 39 3.6636

38 39 67 28 3.3322

39 -61 -15 46 3.8286

40 6 62 56 4.0254

41 -61 -16 45 3.8067

42 -48 8 56 4.0254

43 -24 32 56 4.0254

44 -19 10 29 3.3673

45 -85 -33 52 3.9512

46 -131 -26 105 4.6540

47 -3 33 36 3.5835

48 -95 -62 33 3.4965

49 -93 -17 76 4.3307

50 -68 -8 60 4.0943

51 184 220 36 3.5835

52 42 71 29 3.3673

53 27 68 41 3.7136
 

F-13

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 

 USER 

 SUPPLIED RELATIVE DERIVATIVE 

 DERIVATIVE WRT VALUE DIFFERENCE ASSESSMENT 

 BETA( 1) -1.57D+02 4.25D-07 VERIFIED 

 BETA( 2) 1.00D+00 7.87D-08 VERIFIED 

 DELTA( 1, 1) -5.84D-03 4.30D-07 VERIFIED 

 NUMBER OF RELIABLE DIGITS IN FUNCTION RESULTS 16 

 (ESTIMATED BY ODRPACK) 

 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 (NUMBER WITH NONZERO WEIGHT = 53) 

 NQ = 1 

 M = 1 

 NP = 2 (NUMBER UNFIXED = 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-14

Table F4. ODRPACK Results of ODR Analysis of ΔRTarrest Model Equation (continued) 

 --- STOPPING CRITERIA: 

 SSTOL = 1.49D-08 (SUM OF SQUARES STOPPING TOLERANCE) 

 PARTOL = 3.67D-11 (PARAMETER STOPPING TOLERANCE) 

 MAXIT = 50 (MAXIMUM NUMBER OF ITERATIONS) 

 --- INITIAL WEIGHTED SUM OF SQUARES = 7.76381810D+00 

 SUM OF SQUARED WEIGHTED DELTAS = 0.00000000D+00 

 SUM OF SQUARED WEIGHTED EPSILONS = 7.76381810D+00 

 *** ITERATION REPORTS FOR FIT BY METHOD OF ODR *** 

 CUM. ACT. REL. PRED. REL. 

 IT. NO. FN WEIGHTED SUM-OF-SQS SUM-OF-SQS G-N 

 NUM. EVALS SUM-OF-SQS REDUCTION REDUCTION TAU/PNORM STEP 

 ---- ------ ----------- ----------- ----------- --------- ---- 

 1 15 7.75660D+00 9.2916D-04 9.2766D-04 3.063D-02 YES 

 2 16 7.75660D+00 1.7592D-08 1.7540D-08 5.224D-05 YES 

 3 17 7.75660D+00 6.0973D-13 6.0818D-13 1.064D-06 YES 

 *** FINAL SUMMARY FOR FIT BY METHOD OF ODR *** 

 --- STOPPING CONDITIONS: 

 INFO = 1 ==> SUM OF SQUARES CONVERGENCE. 

 NITER = 3 (NUMBER OF ITERATIONS) 

 NFEV = 17 (NUMBER OF FUNCTION EVALUATIONS) 

 NJEV = 4 (NUMBER OF JACOBIAN EVALUATIONS) 

 IRANK = 0 (RANK DEFICIENCY) 

 RCOND = 1.02D-01 (INVERSE CONDITION NUMBER) 

 ISTOP = 0 (RETURNED BY USER FROM SUBROUTINE FCN) 

 --- FINAL WEIGHTED SUMS OF SQUARES = 7.75660416D+00 

 SUM OF SQUARED WEIGHTED DELTAS = 2.76544656D-04 

 SUM OF SQUARED WEIGHTED EPSILONS = 7.75632762D+00 

 --- RESIDUAL STANDARD DEVIATION = 3.89987535D-01

 DEGREES OF FREEDOM = 51 

 --- ESTIMATED BETA(J), J = 1, ..., NP: 

 BETA S.D. BETA ---- 95% CONFIDENCE INTERVAL ---- 

 1 -5.97110744D-03 8.2458D-04 -7.62651413D-03 TO -4.31570076D-03 

 2 3.78696343D+00 6.5299D-02 3.65587019D+00 TO 3.91805666D+00 

 2a 44.1222164 1.06747815 38.70118385 TO 50.30259469 
 

F-15

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 -1.43102053D-01 -8.54477100D-04 

 2 -8.47788261D-02 -5.06223103D-04 

 3 -2.40805066D-01 -1.43787185D-03 

 4 -2.61679548D-02 -1.56251554D-04 

 5 -3.99850519D-01 -2.38754864D-03 

 6 -5.92016383D-01 -3.53499080D-03 

 7 -6.93757401D-02 -4.14249691D-04 

 8 -5.85749970D-02 -3.49757341D-04 

 9 -2.26442691D-02 -1.35211263D-04 

 10 8.57680493D-01 5.12129857D-03 

 11 1.15426669D-02 6.89224532D-05 

 12 1.46645341D-01 8.75634434D-04 

 13 3.43251602D-01 2.04959067D-03 

 14 -2.44054340D-01 -1.45727360D-03 

 15 -2.44054340D-01 -1.45727360D-03 

 16 1.59743570D-01 9.53845309D-04 

 17 -1.78100642D-01 -1.06345728D-03 

 18 -2.24618999D-01 -1.34122318D-03 

 19 8.09685804D-01 4.83471734D-03 

 20 -2.60957867D-01 -1.55820631D-03 

 21 2.53688183D-01 1.51479827D-03 

 22 1.15457172D-01 6.89406666D-04 

 23 9.86506532D-02 5.89053212D-04 

 24 -2.55614517D-01 -1.52630061D-03 

 25 -1.88384618D-01 -1.12486396D-03 

 26 -9.56061927D-02 -5.70874424D-04 

 27 2.04786195D-01 1.22279946D-03 

 28 4.86188622D-01 2.90308234D-03 

 29 3.22548084D-01 1.92596784D-03 

 30 3.44526207D-01 2.05720147D-03 

 31 3.49085578D-01 2.08442594D-03 

 32 -1.67256927D-01 -9.98708341D-04 

 33 2.53275489D-01 1.51233403D-03 

 34 -1.56999738D-01 -9.37461609D-04 

 35 -4.90754110D-01 -2.93034334D-03 

 36 4.82231733D-02 2.87945535D-04 

 37 -3.06028247D-03 -1.82732618D-05 

 38 -9.50782960D-02 -5.67722299D-04 

 39 -5.41971290D-01 -3.23616640D-03 

 40 -4.75624102D-01 -2.84000050D-03 

 41 5.33099631D-01 3.18319281D-03 

 42 2.21349919D-01 1.32170317D-03 

 43 -2.74205133D-01 -1.63730709D-03 

 44 -4.08875384D-01 -2.44143703D-03 

 45 -8.78254100D-02 -5.24414570D-04 

 46 -8.55839285D-02 -5.11030452D-04 

 47 2.21877816D-01 1.32485529D-03 

 48 1.68875063D-01 1.00837040D-03 

 49 -7.18263826D-01 -4.28882729D-03 

 50 -1.72318244D-02 -1.02892998D-04 

 51 9.88968694D-01 5.90523394D-03 

 52 8.07494984D-01 4.82163573D-03 

 53 -8.95207363D-01 -5.34537537D-03 
 

F-16

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 0 Ia Δ =− RT TT arrest K is 

0 0 var( ) var( ) var( ) 2cov( ) Ia Ia Δ=+ − RT T T T T arrest K K (F25) 

In the absence of data to the contrary, we assume the statistical independence of T0 and TKIa such 

that 0 cov( ) 0 KIa T T = , and Eq. (F25) becomes 

0 var( ) var( ) var( ) Ia Δ=+ RT TT arrest K (F26) 

The variance of both the initial and ODR lognormal model is a decreasing function of increasing 

T0 

[ ]

2

( ) 0

2 2 2

0

var( ) ( )

exp(0.38998 ) exp(0.38998 ) 1 exp 2ln ( ) 0.38998

RTarrest ODR ODR T

T

σ

μ

Δ =

=× − ⎡ ⎤ ⎡ × − ⎤ ⎣ ⎦ ⎣ ⎦

 (F27) 

as shown in Fig. F7. By ( )2

0 56 C, var( ) 12.78 C . T R ≈° Δ = ° Tarrest

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 KIa T . If we assume that the ( )2 2

0 var( ) 23 F (12.778 C) T =°= ° , then 

{ }

{ }

( ) 0

2

( ) 0

2 2

log 0 0

2 2

log 0 log 0 log 0

var( ) var( ) var( ) var( )

var( ) ( )

exp(0.38998 ) exp(0.38998 ) 1 exp 2 ( ) var( )

exp ( ) exp ( ) 1 exp 2 ( )

Ia arrest rev K arrest

arrest rev rev

rev rev rev

RT T RT T

RT T

T T

TT T

σ

μ

σσ μ

Δ = =Δ −

Δ ==

× − ⎡ ⎤× −= ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤ × − × ⎡ ⎣ ⎦⎣ ⎦ ⎣ ⎤

⎦

 (F28) 

where 

( ) ( ) 2

log 0

log 0 0 ( ) ln 2

rev

rev

T

T T

σ

μ μ = − ⎡ ⎤ ⎣ ⎦

and μ ( ) T0 remains a fixed function of T0. Solving Eq. (F28) for 2

log 0 ( ) σ T rev results in 

{ ( )} [ ] 2 2

log 0 0 0 0 ( ) ln exp 0.38998 2ln( ( ) var 2ln ( ) σ μμ T T rev = + −− ⎡ ⎤ T T ⎣ ⎦ (F29) 

and solving for 2

() 0 var( ) ( ) Δ = RTarrest rev σ T rev gives 
 

F-17

{ }

{ } [ ]

22 2

0 log 0 log 0

2

0 log 0

( ) exp ( ) exp ( ) 1

exp 2ln ( ) ( )

rev rev rev

rev

TT T

T T

σσ σ

μ σ

=× − ⎡ ⎤⎡ ⎤ × ⎣ ⎦⎣ ⎦

−

 (F30) 

However, as noted earlier and indicated in Fig. F7, at 

( )2

0 0 T R ≈° Δ = = ° 56 C, var( ) var( ) 12.78 C T T arrest which would produce 2

0 () 0 σ T rev = . In order 

to prevent a nonphysical zero variance at this point, the assumed constant value of 0 var( ) T can be 

replaced by the following function with a transition region: 

2

0

0 0 0

0

(12.778) for 35.7 C

var( ) 99.905972 -1.7748073 for -35.7 C 56 C

0 for 56 C

T

T TT

T

⎧ <− ° ⎪ = ° ⎨ ≤ ≤ °

⎪ > ° ⎩

 (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, 

{ } ( ) [ ]

0 0 0

2

log 0 0 0 0

2

0

0 0 0

0

( ) 44.122exp( 0.005971 ); [°C]

( ) ln exp 0.38998 2ln( ( ) var 2ln ( )

where

(12.778) for 35.7 C

var( ) 99.905972 -1.7748073 for -35.7 C 56 C

0 for 56 C

arrest

rev

RT T T T

T T T T

T

T TT

T

μ

σ μ μ

Δ == −

= + −− ⎡ ⎤ ⎣ ⎦

⎧ <− ° ⎪ = ° ⎨ ≤ ≤ °

⎪ > ° ⎩

 (F32) 
 

F-18

(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-19

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-20

Fig. F3. Comparison of initial model in FAVOR, W0, with Case 2, W2. 
 

F-21

Fig. F4. Comparison of ODR model, WODR, with Case 2, W2. 
 

F-22

Fig. F5. Data used to develop the lognormal statistical model for ΔRTarrest as a function of 

T0 . 
 

F-23

Fig. F6. Model developed from ODR analysis of log-transformed data. 
 

F-24

Fig. F7. Variance of ODR model compared to final model. 
 

F-25

Fig. F8. Comparison of ODR model with final model. 
 

F-26

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. 
NRC FORM 335 U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER

(2-89)

3. DATE REPORT PUBLISHED

MONTH YEAR 

 August 2007

4. FIN OR GRANT NUMBER 

 Y6533 

6. TYPE OF REPORT 

 Final, technical 

7. PERIOD COVERED (Inclusive Dates)

15. NUMBER OF PAGES

16. PRICE

unclassified

(This Report)

unclassified

(This Page)

14. SECURITY CLASSIFICATION

unlimited

13. AVAILABILITY STATEMENT

 Division of Fuel, Engineering, and Radiological Research 

 Office of Nuclear Regulatory Research 

 U. S. Nuclear Regulatory Commission 

 Washington, DC 2055-0001 

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.)

Fracture Analysis of Vessels – Oak Ridge (FAVOR, v04.1), Computer Code: 

Theory and Implementation of Algorithms, Methods, and Correlations 

2. TITLE AND SUBTITLE 

 P. T. Williams, T. L. Dickson, and S. Yin 

5. AUTHOR(S)

 Heavy Section Steel Technology Program 

 Oak Ridge National Laboratory 

 P. O. Box 2008, Mail Stop 6085 

 Oak Ridge, TN 37831-6085

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.)

M. EricksonKirk, NRC Project Manager

10. SUPPLEMENTARY NOTES

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. 

1 1. ABSTRACT (200 words or less)

 pressurized thermal shock, probabilistic fracture mechanics, reactor pressure vessels 

12. KEY WORDS/DESCRIPTORS (List words or phrases that will assist researchers in locating the report.)

(Assigned by NRC, Add Vol., Supp., Rev.,

and Addendum Numbers, if anv.1 

 NUREG/CR-6854 

 ORNL/TM-2004/244

NRCM 1102, BIBLIOGRAPHIC DATA SHEET

(See instructions on the reverse)

3201,3202

NRC FORM 335 (2-89)