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Weld Residual Stress Finite
Element Analysis Validation
Part II—Proposed Validation Procedure
Final Report
Office of Nuclear Regulatory Research
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NUREG-2228
Weld Residual Stress Finite
Element Analysis Validation
Part II—Proposed Validation Procedure
Final Report
Manuscript Completed: January 2020
Date Published: July 2020
Prepared by:
M. Benson
P. Raynaud
J. Wallace
U.S. Nuclear Regulatory Commission
Michael Benson, NRC Project Manager
Office of Nuclear Regulatory Research
ABSTRACT
Under a Memorandum of Understanding, the U.S. Nuclear Regulatory Commission and the
Electric Power Research Institute conducted a research program aimed at gathering data on weld
residual stress modeling. As described in NUREG-2162, “Weld Residual Stress Finite Element
Analysis Validation: Part I—Data Development Effort,” issued March 2014, this program
consisted of round robin measurement and modeling studies on various mockups. At that time,
the assessment of the data was qualitative. This report describes an additional residual stress
round robin study and a methodology for capturing residual stress uncertainties. This quantitative
approach informed the development of guidelines and a validation methodology for finite element
prediction of weld residual stress. For example, comparisons of modeling results to
measurements provided a basis for establishing guidance on a material hardening approach for
residual stress models. The proposed validation procedure involves an analyst modeling a known
case (the Phase 2b round robin mockup) and comparing results to two proposed quality metrics.
These recommendations provide a potential method by which analysts can bolster confidence in
their modeling practices for regulatory applications.
iii
TABLE OF CONTENTS
ABSTRACT ................................................................................................................... iii
LIST OF FIGURES......................................................................................................... ix
LIST OF TABLES .......................................................................................................... xi
EXECUTIVE SUMMARY ............................................................................................. xiii
ACKNOWLEDGMENTS ............................................................................................... xv
ABBREVIATIONS AND ACRONYMS ........................................................................ xvii
1
2
INTRODUCTION ................................................................................................... 1-1
1.1
Phase 2b Effort............................................................................................................... 1-1
1.2
Scope of This Report ..................................................................................................... 1-2
PHASE 2B ROUND ROBIN STUDY ..................................................................... 2-1
2.1
Purpose .......................................................................................................................... 2-1
2.2
Mockup Fabrication ........................................................................................................ 2-1
2.3
Round Robin Participants .............................................................................................. 2-4
2.4
Weld Residual Stress Measurements ........................................................................... 2-4
2.5
Modeling Guidance ........................................................................................................ 2-7
2.6
Results ............................................................................................................................ 2-9
2.7
3
2.6.1
Measurement Results ..................................................................................... 2-9
2.6.2
Modeling Results ........................................................................................... 2-10
2.6.3
Discussion ..................................................................................................... 2-12
Conclusions .................................................................................................................. 2-12
UNCERTAINTY QUANTIFICATION METHODOLOGY ........................................ 3-1
3.1
Motivation ....................................................................................................................... 3-1
3.2
Methodology ................................................................................................................... 3-2
3.3
3.2.1
Functional Data ............................................................................................... 3-2
3.2.2
Screening of Outlier Predictions ...................................................................... 3-2
3.2.3
Data Smoothing ............................................................................................... 3-2
3.2.4
Amplitude and Phase Variability ..................................................................... 3-3
3.2.5
Modeling Amplitude and Phase Variability...................................................... 3-4
3.2.6
Bootstrapping................................................................................................... 3-5
3.2.7
Uncertainty Characterization of the Measurement Data ................................ 3-5
3.2.8
Tolerance Bounds versus Confidence Bounds .............................................. 3-6
Results ............................................................................................................................ 3-6
v
3.4
4
5
3.3.1
Uncertainty Quantification for the Prediction Data .......................................... 3-6
3.3.2
Uncertainty Quantification for the Deep Hole Drilling Measurement
Data ............................................................................................................... 3-11
3.3.3
Uncertainty Quantification for the Contour Measurement Data.................... 3-15
Conclusions .................................................................................................................. 3-16
WRS IMPACT ON FLAW GROWTH CALCULATIONS........................................ 4-1
4.1
Regulatory Application ................................................................................................... 4-1
4.2
Inputs .............................................................................................................................. 4-3
4.3
Superposition of Stresses .............................................................................................. 4-3
4.4
Stress Intensity Factor and Crack Growth ..................................................................... 4-4
4.5
Flaw Growth Results ...................................................................................................... 4-6
4.6
Discussion ...................................................................................................................... 4-8
4.7
Conclusion .................................................................................................................... 4-11
VALIDATION PROCEDURE AND FINITE ELEMENT GUIDELINES ................... 5-1
5.1
Introduction ..................................................................................................................... 5-1
5.2
Material Hardening Law ................................................................................................. 5-1
5.2.1
Difference in Means and Root Mean Square Error Functions ....................... 5-1
5.2.2
Assessment of Prediction Trends ................................................................... 5-2
5.2.3
Assessment of Root Mean Square Error ........................................................ 5-8
5.2.4
Hardening Law Recommendation................................................................. 5-10
5.3
Modeling Guidelines ..................................................................................................... 5-10
5.4
Proposed Validation Scheme....................................................................................... 5-12
5.4.1
Overview of Approach ................................................................................... 5-12
5.4.2
Benchmark..................................................................................................... 5-13
5.4.3
Circumferential Flaw Growth – Isotropic Hardening ..................................... 5-14
5.4.4
Circumferential Flaw Growth – Average Hardening ..................................... 5-16
5.4.5
Axial Flaw Growth – Isotropic Hardening ...................................................... 5-17
5.4.6
Axial Flaw Growth – Average Hardening ...................................................... 5-20
5.4.7
Overview of Quality Metrics........................................................................... 5-21
5.4.8
Quality Metrics for Axial Stress Predictions .................................................. 5-22
5.4.9
Recommended Acceptance Measures – Axial Residual Stress .................. 5-24
5.4.10
Quality Metrics for Hoop Stress Predictions ................................................. 5-24
5.4.11
Recommended Acceptance Measures – Hoop Residual Stress ................. 5-28
5.5
Summary of Validation Procedure ............................................................................... 5-28
5.6
Modeling a Nuclear Plant Application .......................................................................... 5-30
vi
5.7
5.6.1
Applicability of Validation Scheme and Acceptance Measures .................... 5-30
5.6.2
Welding Process............................................................................................ 5-31
5.6.3
Hardening Law .............................................................................................. 5-31
5.6.4
Best Practices for a Plant Application ........................................................... 5-32
Conclusion .................................................................................................................... 5-32
6
CONCLUSIONS .................................................................................................... 6-1
7
REFERENCES ...................................................................................................... 7-1
APPENDIX A
MODEL-MEASUREMENT COMPARISONS ..................................... A-1
APPENDIX B
MATERIAL PROPERTIES ................................................................. B-1
APPENDIX C
DATA FOR VALIDATION PROCESS ............................................... C-1
APPENDIX D
ANALYSIS OF VALIDATION METRICS FOR AVERAGE
HARDENING ...................................................................................... D-1
APPENDIX E
ANALYSIS OF VALIDATION METRICS FOR ISOTROPIC
HARDENING ...................................................................................... E-1
vii
LIST OF FIGURES
Figure 2-1
Figure 2-2
Figure 2-3
Figure 2-4
Figure 2-5
Figure 2-6
Figure 2-7
Figure 2-8
Figure 2-9
Figure 2-10
Figure 2-11
Figure 2-12
Figure 2-13
Figure 3-1
Figure 3-2
Figure 3-3
Figure 3-4
Figure 3-5
Figure 3-6
Figure 3-7
Figure 3-8
Figure 3-9
Figure 3-10
Figure 3-11
Figure 3-12
Figure 3-13
Figure 3-14
Figure 3-15
Figure 4-1
Figure 4-2
Figure 4-3
Figure 4-4
Figure 4-5
Figure 4-6
Figure 4-7
Figure 4-8
Figure 4-9
Figure 5-1
Figure 5-2
Phase 2b Mockup Geometry (Dimensions in inches [millimeters]) ....................... 2-2
ID Backweld ............................................................................................................2-3
Participating Organizations .....................................................................................2-4
Deep Hole Drilling Measurement Setup .................................................................2-5
Hole Drilling Measurements around Circumference .............................................. 2-5
Contour Measurement Setup .................................................................................2-6
Cuts to Extract Contour Specimen .........................................................................2-7
Hole Drilling Measurement: (a) Axial, (b) Hoop...................................................... 2-9
Hoop Stress—Contour Measurement ....................................................................2-9
Axial Stress—Contour Measurement ...................................................................2-10
Example Mesh ......................................................................................................2-11
Processed Isotropic Hardening Results: (a) Axial, (b) Hoop ...............................2-12
Processed Nonlinear Kinematic Hardening Results: (a) Axial, (b) Hoop ............ 2-12
Axial Isotropic Data after Smoothing ......................................................................3-3
Amplitude and Phase Variability .............................................................................3-3
Axial Isotropic Data after Alignment .......................................................................3-4
100 Sampled WRS Curves Based upon Round Robin Modeling Data ................. 3-5
Contour Axial Stress Data ......................................................................................3-6
Constructing Confidence Bounds on the Mean (Axial, Isotropic Case): (a)
30 of the 1,000 Bootstrap Sample Means and (b) Resulting Confidence
Bounds ....................................................................................................................3-7
Constructing Tolerance Bounds (Axial, Isotropic Case): (a) 30 of the 1,000
Bootstrap 2.5th and 97.5th Quantiles and (b) Resulting Tolerance Bounds ........... 3-8
Bootstrap Tolerance Bounds on Isotropic Hoop Stress Predictions: (a) With
Potential Outlier and (b) Without Potential Outlier ...............................................3-10
Data Smoothing for Axial DHD Data: (a) Raw Data, (b) Smoothed Data,
and (c) Residuals ..................................................................................................3-12
Confidence Bounds on the Mean (Axial DHD Data) ............................................3-13
Tolerance Bounds (Axial DHD Data)....................................................................3-13
Confidence Bounds on Mean (Hoop DHD Data) .................................................3-14
Tolerance Bounds (Hoop DHD Data) ...................................................................3-14
50 Extracted Stress Profiles .................................................................................3-15
Tolerance Bounds for Axial Contour Data ............................................................3-16
ASME Code Flaw Disposition Procedure...............................................................4-1
Analytical Flaw Evaluation Procedure ....................................................................4-2
Loads from Various Sources ..................................................................................4-3
Superposition of Membrane, Crack-Face Pressure, and Weld Residual
Stresses ..................................................................................................................4-4
SIF at Two Locations along Crack Front ................................................................4-5
(a) K90 and (b) Growth in Depth Direction ...............................................................4-7
(a) K0 and (b) growth in length direction .................................................................4-8
Flaw Growth after 20 Years ....................................................................................4-9
(a) Membrane Stresses, (b) Area under the Curve..............................................4-10
Nonlinear Kinematic Hardening Predictions against the DHD
Measurements: (a) Actual Data and (b) Mean Difference Function and
Tolerance Bounds ...................................................................................................5-3
Isotropic Hardening Predictions against the DHD Measurements: (a) Actual
Data and (b) Mean Difference Function and Tolerance Bounds ........................... 5-5
ix
Figure 5-3
Figure 5-4
Figure 5-5
Figure 5-6
Figure 5-7
Figure 5-8
Figure 5-9
Figure 5-10
Figure 5-11
Figure 5-12
Figure 5-13
Figure 5-14
Figure 5-15
Figure 5-16
Figure 5-17
Figure 5-18
Figure 5-19
Figure 5-20
Figure 5-21
Average Hardening Predictions against the DHD Measurements: (a) Actual
Data and (b) Mean Difference Function and Tolerance Bounds ........................... 5-6
Root Mean Square Error for Axial Stress Predictions: (a) DHD Benchmark
and (b) Contour Benchmark ...................................................................................5-9
Root Mean Square Error for Hoop Stress Predictions: (a) DHD Benchmark
and (b) Contour Benchmark .................................................................................5-10
Comparison of DHD and Contour Axial Stress Predictions (a) Raw Data
and (b) Difference in Means .................................................................................5-14
Smoothed Axial WRS Profiles and Mean, Isotropic Hardening ...........................5-15
Circumferential Flaw Growth, Isotropic Hardening ..............................................5-15
Smoothed Axial WRS Profiles, Average Hardening ............................................5-16
Circumferential Flaw Growth, Average Hardening...............................................5-17
Smoothed Hoop WRS Profiles, Isotropic Hardening ...........................................5-18
Axial Flaw Growth, Isotropic Hardening ...............................................................5-19
Hoop WRS Profiles, Average Hardening .............................................................5-20
Axial Flaw Growth, Average Hardening ...............................................................5-20
Stress Intensity Factor: (a) Isotropic Hardening and (b) Average Hardening...... 5-21
Prediction C (Isotropic) against the Mean Prediction ...........................................5-23
Hoop Stress Prediction from Participant G ..........................................................5-26
Hoop Stress Prediction from Participant C ...........................................................5-27
Hoop Stress Prediction from Participant D ...........................................................5-27
A Partial Arc Weld Repair .....................................................................................5-30
EWR Mockup ........................................................................................................5-31
x
LIST OF TABLES
Table 2-1
Table 2-2
Table 4-1
Table 4-2
Table 5-1
Table 5-2
Table 5-3
Table 5-4
Table 5-5
Table 5-6
Table 5-7
Table 5-8
Table 5-9
Table 5-10
Table 5-11
Mockup Fabrication Steps ......................................................................................2-3
Model Guidance ......................................................................................................2-8
Inputs for Flaw Growth Calculations .......................................................................4-3
Symbol Definition for Equation 4-3 .........................................................................4-6
Benchmark Cases and Their Location in Appendix A ........................................... 5-7
Qualitative Assessment of Prediction Bias .............................................................5-8
RMSE for DHD Benchmark ....................................................................................5-9
RMSE for Contour Benchmark ...............................................................................5-9
Time to Through-Wall ...........................................................................................5-19
Quality Metrics Applied to Phase 2b Axial Isotropic Predictions..........................5-23
Quality Metrics Applied to the Phase 2b Axial Average Hardening
Predictions ............................................................................................................5-24
Quality Metrics Applied to Phase 2b Hoop Isotropic Predictions .........................5-25
Quality Metrics Applied to Phase 2b Hoop Average Hardening Predictions ....... 5-28
Acceptance Measures for Axial Stresses .............................................................5-29
Acceptance Measures for Hoop Stresses ............................................................5-29
xi
EXECUTIVE SUMMARY
Weld residual stress (WRS) is known to be an important driver of primary water stress corrosion
cracking in safety-related nuclear piping. For this reason, it is desirable to formalize finite element
modeling procedures for residual stress prediction. The U.S. Nuclear Regulatory Commission
(NRC) and the Electric Power Research Institute have conducted joint research programs on
residual stress prediction under a memorandum of understanding. These studies have involved
modeling and measurement of WRS in various mockups. The latest of these studies, Phase 2b,
is discussed in this document.
The Phase 2b mockup was prototypic of a pressurizer surge nozzle dissimilar metal weld, which
forms part of the reactor coolant pressure boundary. Two sets of residual stress measurement
data were obtained on the Phase 2b mockup: deep hole drilling and contour. Both these
methods are strain-relief techniques. In addition to the measurements, 10 independent analysts
submitted finite element modeling results of the residual stresses in the mockup. Each participant
was provided the same set of modeling guidelines, with the aim of reducing analyst-to-analyst
scatter as much as possible. These measurement and modeling data were then used to develop
an uncertainty quantification methodology.
The residual stress uncertainty methodology consisted of constructing a statistical model of the
data and using bootstrapping methods to calculate relevant 95/95 tolerance bounds and
confidence bounds. This methodology improves on past work (e.g., NUREG-2162, “Weld
Residual Stress Finite Element Analysis Validation: Part I—Data Development Effort,” issued
March 2014), which described uncertainty in WRS predictions only in qualitative terms.
Furthermore, the results of the uncertainty quantification effort informed the development of a
validation approach of residual stress finite element models.
The uncertainty quantification work provided methods to compare measurements to models,
which in turn led to recommendations on hardening law (see Section 5.2). The validation method
is a step-by-step procedure for comparing independent finite element modeling results of the
Phase 2b mockup to the acceptance measures. If an analyst meets the criteria, then the
modeling procedure may be applied with greater confidence to a real case. This procedure is
intended as a recommendation rather than a regulatory requirement. It provides a means to
demonstrate proficiency in finite element modeling of WRS.
The validation methodology is aimed at 2D axisymmetric WRS predictions for deterministic flaw
growth evaluations. The nuclear industry often performs flaw evaluations when seeking
alternatives to established inspection and repair/replacement rules. These evaluations require a
WRS assumption. If that assumption is based on finite element results, then following the
validation procedure offers the industry one method to strengthen its case when seeking NRC
approval. This document also investigated how differences in residual stress can affect these flaw
evaluations. Important features of the stress profiles include the inner diameter stress, the stress
magnitude at the initial flaw depth, and the depths at which the stress profile crosses zero.
Decision-makers can review these aspects of submitted stress profiles as another option for
gaining confidence in residual stress predictions.
xiii
ACKNOWLEDGMENTS
The authors would like to thank the following.
•
•
•
John Broussard of Dominion Engineering, Inc., Paul Crooker of the Electric Power
Research Institute (EPRI), and Michael Hill of University of California, Davis for technical
cooperation in the joint NRC-EPRI research program.
Dusty Brooks, Remy Dingreville, and John Lewis of Sandia National Laboratory for their
excellent work on developing an uncertainty quantification scheme for the round robin
dataset (see Chapter 3).
The round robin modeling participants for contributing their work to this effort, as described
in Chapter 2.
xv
ABBREVIATIONS AND ACRONYMS
ASME Code
American Society of Mechanical Engineers Boiler and Pressure Vessel Code
DHD
deep hole drilling
EPRI
Electric Power Research Institute
EWR
excavate and weld repair
FE
finite element
fPCA
functional principal components analysis
ID
inner diameter
mm
millimeter
MPa
megapascal
NDE
nondestructive examination
NRC
U.S. Nuclear Regulatory Commission
OD
outer diameter
PWSCC
primary water stress corrosion cracking
RMSE
root mean square error
SIF
stress intensity factor
WRS
weld residual stress
i
reference index
k
reference index
µk
mean at the kth position through the wall thickness
σk
standard deviation at the kth position through the wall thickness
wi
weighting factor for the ith WRS profile
xi,kWRS
stress magnitude of the ith profile at the kth position through the thickness
f
a function
r
radial position through the wall thickness
t
wall thickness of the weld or pipe
d
normalized distance through the wall thickness, d = r/t
γ
warping function
T
operating temperature
P
operating pressure
τ
time
KI
mode I stress intensity factor
a
half-depth of a flaw
xvii
c
half-length of a flaw
σm
membrane stress
σb
bending stress
σcfp
crack face pressure stress
Gb
influence coefficient for global bending
Q
flaw shape parameter
h(x,a)
weight function for the Universal Weight Function Method
s(x)
stress variation along the crack face
da/dτ
flaw growth with respect to time
KIth
stress intensity factor threshold
Qg
activation energy
Rg
ideal gas constant
Tabs
absolute operating temperature
Tref
empirical reference temperature
φ
tabulated crack growth coefficient
η
tabulated crack growth coefficient
K90
SIF at the deepest point along the crack front
K0
SIF at the surface point along the crack front
g
a function
L
number of locations through the wall thickness where a WRS magnitude is known
ne
number of sampled measurement WRS profiles
np
number of sampled prediction WRS profiles
s
reference index
h(d)
difference in means function
RMSEWRS
quality metric on the root mean square error of stress magnitude
WRSmean
benchmark WRS (the mean of the isotropic predictions from the Phase 2b study)
D1
first derivative of the WRS magnitude with respect to through-wall position
h
interval between two positions through the wall thickness
diffavg
quality metric on the average difference between the prediction WRS and the
benchmark value
xviii
1
INTRODUCTION
The U.S. Nuclear Regulatory Commission (NRC) and the Electric Power Research Institute
(EPRI) initiated a long-term research program on understanding and reducing uncertainty in the
numerical prediction of weld residual stress (WRS) in safety-related nuclear components. Part 1
of this report [1] discusses the background and past work of the program in detail (also see [2]).
The through-wall WRS profile is an important input to deterministic flaw growth calculations.
These calculations may form the technical basis for regulatory relief requests to modify
repair/replacement or nondestructive examination requirements in nuclear components subject to
primary water stress corrosion cracking (PWSCC). Probabilistic fracture mechanics calculations
[3]–[6] rely on well characterized uncertainty for important inputs. For these reasons, it is
important to develop sound approaches for reaching best estimates of WRS and the associated
uncertainty.
The past research was categorized according to four phases:
(1)
(2)
(3)
(4)
Phase 1: small-scale scientific specimens
Phase 2a: fabricated prototypic pressurizer surge line nozzle
Phase 3: pressurizer surge line nozzles from a canceled plant
Phase 4: optimized weld overlay on a prototypic cold-leg nozzle
These four phases consisted of double-blind measurement and modeling studies on the mockups
of varying geometry. In general, the work showed that axisymmetric finite element (FE) models
provided reasonable estimations of the measurements, but that relatively large analyst-to-analyst
uncertainty existed in the predictions. NUREG-2162 [1] lists the following recommendations for
future work in the WRS Validation Program:
•
Develop specific validation criteria for comparing WRS measurement and modeling
results.
•
Establish guidelines for WRS input development for deterministic flaw evaluations,
including FE best practices.
•
Develop additional guidance for accounting for uncertainty in WRS inputs for flaw
evaluations.
•
Focus future FE round robin studies on reducing model-to-model variability, given lessons
learned in FE modeling best practices.
•
Apply more robust methods to quantify modeling uncertainty in future round robin efforts.
The final phase of this research, dubbed Phase 2b, aimed at addressing these issues.
1.1
Phase 2b Effort
Phase 2b was a second double-blind round robin measurement and FE modeling study involving
a pressurizer surge line nozzle mockup. This mockup was similar to, but not exactly the same as,
the Phase 2a mockup discussed in [1]–[2]. Modeling guidelines [7], based on lessons learned
from the previous research phases, were developed with the intent of reducing the uncertainty
observed in the past work. The measurement program consisted of deep hole drilling (DHD) and
the contour methods. Ten international participants submitted independent modeling results.
1-1
The dataset from the Phase 2b study was intended to address the items for future work identified
in NUREG-2162. An unbiased view of expected modeling uncertainty is important for developing
acceptance measures and WRS input guidelines. These efforts also require more quantitative
approaches to describing the data.
1.2
Scope of This Report
This report is intended to document the development of a 2D axisymmetric FE validation
approach for prediction of WRS for flaw evaluation applications. The validation scheme proposed
here draws on the results of the Phase 2b round robin and an uncertainty quantification
methodology. Chapter 2 summarizes the Phase 2b round robin study and the resulting dataset.
Chapter 3 discusses the mathematical methods developed to characterize uncertainty in the
Phase 2b dataset. Chapter 4 presents the impacts of WRS assumptions on flaw growth
calculations. Chapter 5 develops guidelines for WRS inputs for flaw growth calculations and the
proposed validation approach. Finally, Chapter 6 contains overall conclusions of the work.
1-2
2
2.1
PHASE 2B ROUND ROBIN STUDY
Purpose
The purpose of this effort was to conduct a second FE round robin, similar to the Phase 2a study
[1], [8]–[10], with improved FE modeling guidance. The modeling guidance aimed at reducing
analyst-to-analyst scatter. Determining appropriate scatter bands for FE predictions is important
for formulating acceptance measures and modeling guidelines. WRS measurements have
uncertainties as well. Chapter 3 discusses quantification of both measurement and modeling
uncertainties. Both uncertainties must be accounted for when deciding what constitutes an
appropriate FE prediction. This chapter discusses the research effort designed to collect the
measurement and modeling data. An NRC technical letter report [7] presents more detailed
information about the Phase 2b effort.
2.2
Mockup Fabrication
The geometry chosen for the Phase 2b round robin study was representative of a pressurizer
surge nozzle. Figure 2-1 shows the overall geometry of the mockup.
2-1
2-2
Figure 2-1
Phase 2b Mockup Geometry (Dimensions in inches [millimeters])
Figure 2-2 emphasiz es that a backweld was performed on the mockup after ID machining.
F ig ure 2 - 2
I D B ackweld
Table 2-1 provides an overview of the fabrication process. More detailed fabrication information,
including welding parameters and bead map drawings, is found in [ 7] .
Table 2 - 1
M ockup F abrication Steps
Step D es cription
Purpos e
1
A36 flange welded to SA182 noz z le
Simulates noz z le stiffness in service; not modeled
2
Alloy 82 buttering applied to noz z le
Allows for post-weld heat treat of low alloy steel and
prepares dissimilar metal weld
3
Post weld heat treatment
Tempers martensite in low alloy steel and relieves
residual stress
5
Buttered noz z le welded to F316L
safe end with Alloy 182 filler metal
Backchip and reweld
6
Safe end welded to TP316 pipe
4
Simulates shop weld
Simulates repair weld at inner diameter
Simulates field closure weld
2-3
2 .3
Round Robin Participants
Ten participants representing 12 organiz ations submitted FE results to the round robin study, as
represented in Figure 2-3.
F ig ure 2 - 3
Participating O rg aniz ations
These participants represent a cross section of international industry, government, academic, and
private contractor organiz ations. The study was double blind, so the modelers did not have
access to the measurement data. Likewise, the measurement practitioners did not have access
to the modelers’ results.
2 .4
W eld Res idual Stres s M eas urements
Additional background on residual stress measurement is given in Section 2.2 of [ 1] . VEQ TER,
Ltd., in Bristol, United K ingdom, and H ill Engineering, LLC, in Rancho Cordova, CA, performed
the Phase 2b residual stress measurements. Two sets of measurements were carried out: hole
drilling and contour (see Section 2.2.2 of [ 1] ). The DH D and contour methods were chosen over
x -ray or neutron diffraction, because they are more reliable in a dissimilar weld. In fact, x -ray and
neutron diffraction are known to have larger uncertainties associated with d0 calibration, grain
siz e, tex ture, and chemical dilution effects present in welds. The hole drilling measurements
consisted of a combination of DH D and incremental DH D. Figure 2-4 shows the ex perimental
setup of the hole drilling measurements. Four hole drilling measurements were taken starting at
location B shown in Figure 2-4. Location B was located 22° from the weld start location. The
other three measurements were made 90° apart from one another (see Figure 2-5). Care was
taken to avoid weld start/ stop locations around the circumference.
2-4
F ig ure 2 - 4
D eep H ole D rilling M eas urement Setup
F ig ure 2 - 5
H ole D rilling M eas urements around C ircumference
The contour measurements involved several cuts, including one cut each for the ax ial residual
stress measurement and the hoop residual stress measurement (see Figure 2-6).
2-5
Figure 2-6
Contour Measurement Setup
The final calculation of residual stress accounted for the release of stress at each sectioning
operation. The hole drilling measurements were made before the destructive contour
measurements. Each of the required cuts was made with the hole drilling measurements in mind,
as shown in Figure 2-7.
2-6
Figure 2-7
Cuts to Extract Contour Specimen
The section outlined in red in Figure 2-6 is represented by the “Cut out section” cuts shown in
Figure 2-7 (i.e., Cut 3). At each of these cuts, strain gauge measurements are made for the final
stress calculation. The cuts represented by the blue, green, and yellow lines in Figure 2-6 were
then made. A laser profilometer measured displacements along the relevant cross sections.
2.5
Modeling Guidance
The round robin participants were tasked with creating an axisymmetric FE model to predict the
residual stress distribution of the Phase 2b mockup. The written problem statement provided to
the round robin participants is provided in [7]. Table 2-2 summarizes this guidance, which was
based on modeling experience gained in previous work [1], [2], [11].
2-7
Table 2 - 2
M odel G uidance
2-8
2.6
Results
This section reports the basic set of results from the Phase 2b FE round robin study. The raw
data are reported in both graphs and tables in [7].
2.6.1
Measurement Results
Figure 2-8 shows the hole drilling measurement results. In this report, r/t=0 represents the inner
surface of the pipe wall, and r/t=1 represents the outer surface of the pipe wall.
Figure 2-8
Hole Drilling Measurement: (a) Axial, (b) Hoop
Figure 2-9 and Figure 2-10, respectively, show the hoop and axial stress measurements for the
contour method.
Figure 2-9
Hoop Stress—Contour Measurement
2-9
Figure 2-10 Axial Stress—Contour Measurement
The DHD and contour datasets are, by nature, different. The DHD data are a one-dimensional
profile of the stress variation along a straight path through the weld centerline. In contrast, the
contour method gives a two-dimensional representation of the stress variation on an entire cross
section. As described in Table 2-2, the modeling data were collected as one-dimensional path
data extracted from the FE results along the centerline of the weld, so as to be compared with the
DHD data. Therefore, the contour data must be processed to extract appropriate path data to
compare to the modeling data. The extracted contour data should be a one-dimensional stress
profile and represent the stress along a straight path through the weld centerline, normal to the
inside surface.
2.6.2
Modeling Results
Figure 2-11 shows an example mesh from one of the FE round robin participants.
2-10
F ig ure 2 - 1 1
Ex ample M es h
The red line in Figure 2-11 represents the path along which the participant ex tracted the data (i.e.,
the weld centerline). The figure also illustrates major geometry features modeled by the
participants. Figure 2-12 and Figure 2-13 show the isotropic and nonlinear kinematic hardening
results of the Phase 2b round robin study, respectively. A qualitative look at the modeling results
reveals the following observations:
• Individual predictions may potentially be considered outliers (e.g., participant J), when
compared to the rest of the sample.
• Nonlinear kinematic and isotropic results show different through-wall trends.
• The nonlinear kinematic results show smaller stress magnitudes than the isotropic results.
• The nonlinear kinematic results generally ex hibit less scatter than the isotropic results.
2-11
Figure 2-12 Processed Isotropic Hardening Results: (a) Axial, (b) Hoop
Figure 2-13 Processed Nonlinear Kinematic Hardening Results: (a) Axial, (b) Hoop
2.6.3
Discussion
Comparison of the measurement and modeling data requires careful thought. Both the modeling
data and the measurement data exhibit uncertainties. Chapter 3 of this document focuses on
quantitatively evaluating both modeling and measurement uncertainty. The end goal is to develop
a procedure to objectively judge FE models of WRS, as discussed in Chapter 5.
One final technical topic to be resolved is the choice of hardening law. When providing guidance
on hardening law, it is important to avoid biases in the model prediction in addition to minimizing
prediction errors. One simplified approach suggested elsewhere [12] involves use of the average
of the nonlinear kinematic and isotropic predictions. Section 5.2 discusses the choice of
hardening law.
2.7
Conclusions
This chapter summarizes the Phase 2b round robin study. The WRS modeling and measurement
results from this study constitute the dataset analyzed in the uncertainty analysis in Chapter 3.
Ten analysts participated in the modeling portion of the round robin, resulting in 10 isotropic and
10 nonlinear kinematic WRS predictions for both axial and hoop stresses. Two measurement
2-12
vendors performed strain relief-based WRS measurements on the mockup. This resulted in four
DHD measurements of axial and hoop stresses along the weld centerline. The contour
measurement resulted in a two-dimensional representation of the WRS along a given cross
section. This dataset, viewed in the context of the Chapter 3 analysis, will be used to develop the
guidelines and validation scheme presented in Chapter 5.
2-13
3
3.1
UNCERTAINTY QUANTIFICATION METHODOLOGY
Motivation
In [1], the NRC documented the need to apply more sophisticated data analysis techniques to
residual stress measurement and modeling data. At that stage, only qualitative judgments were
applied to describe modeling uncertainty and measurement-to-model comparisons. To define an
objective validation process for WRS predictions, it is necessary to use a quantitative analysis.
The quantitative analysis described in this section of the NUREG does not need to be repeated by
analysts trying to use the WRS prediction validation scheme presented in this report.
Previous work in this area includes the development of a sampling scheme for WRS in a
probabilistic fracture mechanics code [5]–[6]. The baseline dataset was four FE WRS profiles
obtained by different analysts for a given weld configuration. Estimates of skewness and kurtosis
were used to assign an appropriate uncertainty distribution type for the WRS FE data at each
point through the pipe thickness. In determining the final estimates for the mean WRS and
standard deviation, the analysts introduced a weighting approach that decreased the importance
of a particular WRS prediction the further away it was from the other predictions. This is
represented mathematically as:
∑ wx
=
∑ w
n
µk
i =1
WRS
i i, k
n
and
i
i =1
Equation 3-1
σk =
∑
n
i =1
(
wi xiWRS
,k − µ k
∑
n
w
i =1 i
)
2
,
where µk and σk are the mean and standard deviation of the distribution at the kth position through
the weld thickness, respectively; wi is the weight for the ith WRS profile based on the differences
in stress predictions between two WRS profiles; and xi,k is the value of the stress prediction for the
ith profile at the kth position through the wall thickness. With the uncertainty in WRS thus defined,
Kurth et al. [5]–[6] conceived a sampling strategy that accounted for point-to-point smoothness
and static equilibrium requirements. In this way, the probabilistic fracture mechanics analysis may
account for the uncertainty in the WRS profile for a given weld configuration.
The approach presented here is aimed at deterministic fitness-for-service calculations, where a
residual stress assumption is required (see Chapter 4). This approach involves a range of
mathematical tools aimed at defining uncertainty bands on the round robin measurement and
modeling results. The results then form the basis for recommended modeling practices and
model validation approaches. The methodology described here is documented in greater detail in
[13], and Section 3.2 provides only a summary.
3-1
3.2
3.2.1
Methodology
Functional Data
The round robin WRS modeling and measurement dataset is discrete by nature. Even the
contour measurement is based on a finite number of measurements along the surface of the part.
It is also functional data, in that the WRS magnitude for a given stress component, WRS, depends
on the spatial location along the pipe cross section, as:
WRS = f (d )
Equation 3-2
where d=r/t is the normalized distance from the inside surface to the outside surface of the weld.
References [15] and [16] point out that it is useful to consider such data as a continuous function.
They introduce the mathematical methods that can be applied to functional data. This section
describes a statistical model constructed to represent the round robin dataset, based on the
methods of [15] and [16]. This approach enables bootstrapping to estimate confidence bounds
and tolerance bounds on both the measurement and modeling data [17]–[18]. The final outcome
of the work is an objective process for validating WRS FE modeling (see Chapter 5).
3.2.2
Screening of Outlier Predictions
As described in Chapter 2, the idea behind the Phase 2b round robin study was to assess the
prediction uncertainty of a group of analysts modeling the same problem under a given set of
guidelines. The 10 submissions were screened for potential outlying results that may not have
been obtained in strict accordance with the modeling guidance. Tran et al. [19] described two
outlier predictions in the round robin dataset and the reasons behind them. One participant used
incorrect material property data, and the other incorrectly modeled the heat input of the stainless
steel closure weld. A third outlier involving incorrect weld thickness was identified in [13]. These
three predictions were screened out for the purposes of uncertainty quantification. As discussed
in Section 3.3.1, one other prediction was removed from the hoop stress profiles because of the
undue influence it had on the bootstrap tolerance bound results.
3.2.3
Data Smoothing
The WRS measurements were reported at discrete spatial locations. Similarly, the round robin
modeling participants provided stress magnitudes at discrete depths through the wall thickness.
The actual WRS distribution is expected to be a continuous function of spatial location. Data
smoothing was applied here to arrive at a smooth, continuous representation of the WRS profile.
The smoothing was accomplished via cubic splines, which are a series of third degree
polynomials connected together at a given number of nodes [20]. The optimal number of nodes
to achieve an acceptable fit was determined by an algorithm discussed in [13]. As an example,
the axial stress predictions assuming isotropic hardening are shown after smoothing in Figure 3-1
(compare with Figure 2-12a; outliers removed).
3-2
Figure 3-1
3.2.4
Axial Isotropic Data after Smoothing
Amplitude and Phase Variability
Functional data can exhibit two types of variability: amplitude and phase variability. Considering
a sinusoidal function, the amplitude variability is the result of differences in peak height of two
curves, and phase variability is the result of a horizontal shift of one curve relative to the other
(see Figure 3-2).
Figure 3-2
Amplitude and Phase Variability
Both types of uncertainty are present in the round robin dataset, as shown in Figure 3-1. For the
case of WRS, amplitude variability is equivalent to variability in stress magnitude at corresponding
local extrema. Phase variability has a spatial context (e.g., how the depth at the local maximum
differs among various predictions). The methodology proposed in this section accounts for these
two types of variability.
3-3
3.2.5
Modeling Amplitude and Phase Variability
Registration is the process of aligning the data horizontally and thus removing phase variability.
Specifically, this process aligns the local extrema. Registration is accomplished through the use
of warping functions, γ [13]. Warping functions are chosen such that the boundaries of the
original functions are preserved. For the case of WRS, this means that the inner diameter (ID)
and outer diameter (OD) stresses of the smoothed data are retained in the transformed functions.
are differentiable, so that the
The other requirement for warping functions is that both γ and γ
transformed function is smooth and can be mapped back to the original function [13]. Removing
the phase variability via the warping functions allows characterization of the amplitude variability.
The warping functions themselves provide a useful characterization of the phase variability.
−1
Figure 3-3
Axial Isotropic Data after Alignment
Functional principal components analysis (fPCA) is a dimension reduction technique to model the
dominant modes of variation in the aligned data. The mathematical details and fundamental
concepts of fPCA are better described elsewhere [13]–[16]. fPCA was applied to both the
registered data and the warping functions to construct a statistical model of the residual stress
data. The model allows for statistical sampling. Figure 3-4 shows 100 sampled profiles from the
model constructed from the seven isotropic hardening axial stress predictions. The black curves
in Figure 3-4 are the smoothed WRS predictions from the Phase 2b study. The modeled profiles
demonstrate amplitude and phase variability similar to those of the original sample.
3-4
Figure 3-4
3.2.6
100 Sampled WRS Curves Based upon Round Robin Modeling Data
Bootstrapping
Bootstrapping is a statistical sampling technique that provides a method to estimate uncertainty in
distribution parameters, such as the mean. Further details of bootstrapping are described in [13]–
[16]. This technique is applied here, along with the model described in Sections 3.2.1–3.2.5, to
determine confidence bounds and tolerance bounds related to the Phase 2b round robin dataset,
with mathematical details provided in [13]. The results are applied in Chapter 5 to draw
conclusions about modeling recommendations and to inform development of a validation
procedure for FE prediction of WRS.
3.2.7
Uncertainty Characterization of the Measurement Data
The statistical model summarized in Sections 3.2.1–3.2.5, while presented in the context of the
modeling data, can also be applied to the measurement data. The DHD data consisted of four
measurements around the circumference of the mockup, 90° apart from one another. The
statistical model for bootstrapping was constructed from the four measured stress profiles.
Figure 3-5 shows the axial contour data again. As the figure suggests, many one-dimensional
stress profiles through the weld centerline can be extracted from the axial contour dataset.
Because of this unique feature of the axial contour data, it was not necessary to construct a
statistical model or to perform bootstrapping.
3-5
Figure 3-5
3.2.8
Contour Axial Stress Data
Tolerance Bounds versus Confidence Bounds
Bootstrapping was employed to construct both tolerance bounds and confidence bounds related
to the round robin dataset. A more rigorous treatment of these statistical bounds is found
elsewhere [21]. For the purposes of this document, a high-level definition of the statistical bounds
determined in this work will suffice. Confidence bounds provide intervals within which a particular
statistic is expected to lie. For instance, the confidence bounds on the mean prediction indicate
the interval within which the true mean lies, with 95 % statistical confidence. Figure 3-6 gives
examples of bootstrapped confidence bounds on the mean. Confidence bounds can be
constructed on other statistics, such as quantiles.
Tolerance bounds are intervals within which 95 % of the residual stress data (either measurement
or prediction) are expected to lie. The upper tolerance bound is constructed using the upper
confidence bound on the 0.975 quantile. The lower tolerance bound is constructed using the
lower confidence bound on the 0.025 quantile. The choice of the 0.975 and 0.025 quantiles leads
to a coverage level of 95 %. Figure 3-7 shows examples of bootstrapped tolerance bounds.
3.3
Results
This section presents example results that illustrate the outcomes of the methods described in
Section 3.2. Comprehensive results are presented in [13].
3.3.1
Uncertainty Quantification for the Prediction Data
Figure 3-6(a) shows bootstrap sample means compared against the smoothed axial stress FE
prediction data for isotropic hardening. These results lead to 95 % confidence bounds on the
prediction mean for these data, which are shown in Figure 3-6(b). Figure 3-7 shows similar plots
for constructing the 95/95 tolerance bounds.
3-6
Figure 3-6
Constructing Confidence Bounds on the Mean (Axial, Isotropic Case): (a) 30
of the 1,000 Bootstrap Sample Means and (b) Resulting Confidence Bounds
3-7
Figure 3-7
Constructing Tolerance Bounds (Axial, Isotropic Case): (a) 30 of the 1,000
Bootstrap 2.5th and 97.5th Quantiles and (b) Resulting Tolerance Bounds
3-8
As introduced in Section 3.2.2, the analysis of the seven isotropic hoop stress predictions that
passed the initial screening revealed that one prediction was strongly influencing the upper
tolerance bound results. Figure 3-8 shows the potential outlier and demonstrates the significant
impact it has on the 95/95 tolerance bounds. Specifically, the potential outlier suggests a roughly
constant hoop stress prediction through the thickness. Since a majority of this dataset, including
measurements (see Figure 2-8 and Figure 2-9) and modeling results, indicates some variation of
hoop stress through the weld thickness, it may be appropriate to screen out this potential outlier
when determining tolerance bounds for validation purposes.
3-9
Figure 3-8
Bootstrap Tolerance Bounds on Isotropic Hoop Stress Predictions: (a) With
Potential Outlier and (b) Without Potential Outlier
3-10
3.3.2
Uncertainty Quantification for the Deep Hole Drilling Measurement Data
Figure 3-9 demonstrates the data smoothing process for axial DHD data. It shows that the
smoothing residuals increase beyond a normalized depth of 0.6. This is because the
measurement data were obtained at relatively coarse spatial increments near the OD. This data
fitting issue may lead to less confidence in the bootstrap quantities determined for all DHD data
beyond r/t=0.6.
3-11
Figure 3-9
Data Smoothing for Axial DHD Data: (a) Raw Data, (b) Smoothed Data, and (c)
Residuals
3-12
Figure 3-10 and Figure 3-11 show the bootstrap confidence bounds and tolerance bounds for the
DHD axial stress measurement data, respectively.
Figure 3-10 Confidence Bounds on the Mean (Axial DHD Data)
Figure 3-11 Tolerance Bounds (Axial DHD Data)
3-13
Figure 3-12 and Figure 3-13 show the bootstrap confidence bounds and tolerance bounds for the
DHD hoop stress measurement data, respectively.
Figure 3-12 Confidence Bounds on Mean (Hoop DHD Data)
Figure 3-13 Tolerance Bounds (Hoop DHD Data)
3-14
3.3.3
Uncertainty Quantification for the Contour Measurement Data
As mentioned in Section 3.2.7, the axial stress contour data did not require construction of a
statistical model and bootstrapping to quantify the uncertainty in the data. The cross section
shown in Figure 3-5 is located entirely along the dissimilar metal weld centerline (also see the
illustration in Figure 2-6). This centerline data can be used to directly compare to the prediction
stress profiles by extracting path data, as illustrated in Figure 3-5. The linear paths defined for
data extraction should be perpendicular to the boundaries of the data. Figure 3-14 shows
example stress profiles extracted from the contour data, along with the profiles provided by the
measurement vendor.
Figure 3-14 50 Extracted Stress Profiles
While Figure 3-14 shows 50 example stress profiles, the analysis procedure used 500 extracted
profiles in order to have adequate statistics to determine confidence bounds and tolerance
bounds. Repeating the analysis with 5,000 profiles did not change the results. Figure 3-15
compares the tolerance bounds based on two different methods. One method is based on the
five stress profiles provided by the contour measurement vendor. The other method is based on
extracting 500 stress profiles without modeling and bootstrapping. Figure 3-15 shows that using
the extracted 500 stress profiles leads to tighter tolerance bounds.
3-15
Figure 3-15 Tolerance Bounds for Axial Contour Data
3.4
Conclusions
This chapter presented a methodology for quantifying uncertainty in the round robin WRS dataset.
A statistical model was constructed for the modeling data and the DHD measurement data. The
statistical model enabled bootstrap estimates of confidence bounds and tolerance bounds. For
the axial contour measurement data, it was not necessary to employ bootstrap techniques.
Instead, 500 curvilinear stress profiles were extracted from the contour measurements. Use of
these profiles allowed direct determination of the mean and tolerance bounds. This method
improves on the previous work [1], where uncertainty was described only in subjective terms. The
results of this analysis will help inform the development of a validation process in Chapter 5.
3-16
4
4 .1
W RS I M PAC T O N F L AW
G RO W TH C AL C U L ATI O NS
Reg ulatory Application
NRC regulations require owners of nuclear power plants to periodically perform nondestructive
ex aminations of safety-related piping according to the American Society of Mechanical Engineers
Boiler and Pressure Vessel Code (ASME Code), Section X I [ 22] , and ASME Code Case N-770
[ 23] . If the ex am discovers an indication, then the geometry of the potential flaw is compared to
the acceptance standards of Section X I, IWB-3500. If the flaw is not allowable, then the licensee
must either repair or replace the piping or perform an analytical evaluation according to IWB-3600
for temporary acceptance of the flaw. Connected flaws on the inner surface are generally not
allowed in service because of PWSCC [ 22] . The NRC has granted short-term regulatory relief to
licensees, usually in cases where inservice inspection requirements present a demonstrated
hardship to the plant owner (see [ 24] as one ex ample). Figure 4-1 summariz es this process.
F ig ure 4 - 1
ASM E C ode F law D is pos ition Procedure
An analytical evaluation per ASME Code, Section X I, IWB-3600, involves an engineering estimate
of the growth of the flaw within an established timeframe. ASME Code, Section X I,
Nonmandatory Appendix C, provides guidance and equations for many aspects of a flaw
evaluation in piping, including the following:
4-1
•
•
•
•
•
pipe stress
acceptance measures
screening for failure mode
flaw stability
crack growth rate laws [ 25]
Nonmandatory Appendix A of Section X I, paragraph A-3000, contains stress intensity factor (SIF)
solutions that may be applied to piping. Figure 4-2 outlines the basic procedure of a flaw growth
evaluation.
F ig ure 4 - 2
Analy tical F law Ev aluation Procedure
As Figure 4-2 shows, the analytical evaluation requires an assumption about WRS. In many
licensee submittals, an FE model serves as the basis behind the assumed WRS. The focus of
this chapter is to ex amine how the WRS input affects the flaw growth calculation.
4-2
4.2
Inputs
This work draws on a series of flaw growth calculations documented in the technical letter report
on the Phase 2b study [7]. Table 4-1 shows the inputs for this work.
Table 4-1
OD [mm]
381
Inputs for Flaw Growth Calculations
t [mm]
36.07
Weld Width [mm]
26.48
a 0 [mm]
2c 0 [mm]
T [oC]
3.607
7.214
315.6
OD – outer diameter
t – pipe wall thickness
T – operating temperature
2c0 – initial flaw length
σm – operating membrane stress
p [MPa] σ m [MPa] σ b [MPa]
15.5
60
100
a0 – initial flaw depth
p – operating pressure
σb – operating bending stress
The dimensions in Table 4-1 are consistent with the Phase 2b mockup geometry given in Chapter
2. This chapter will examine the case of a circumferential flaw subjected to the WRS profiles
determined by the axial WRS measurements in the Phase 2b study, as shown in Figure 2-8a and
Figure 2-10.
4.3
Superposition of Stresses
Figure 4-3 shows the residual stress profiles, with σm and σcfp (the crack face pressure stress)
overlayed on the figure. A representative contour measurement of axial residual stress is also
included on the figure.
Figure 4-3
Loads from Various Sources
4-3
In the method applied here, σm, σcfp, and WRS were superimposed for the purpose of calculating
the SIF. Figure 4-4 shows the results of this superposition, along with annotations of salient
features of the curves.
The ID stresses (i.e., at r/t=0) were compressive for each curve, with the contour measurement
being the most compressive at -166 MPa. The stress at the initial crack depth (i.e., r/t=0.1) was
tensile for the DHD curves and compressive for the contour measurement. The highest stress at
r/t=0.1 was observed in the 112° curve at 18 MPa.
Three of the four DHD curves crossed zero for the second time at r/t=0.32, with very little spread.
The 292° curve deviated from this trend by crossing zero at r/t = 0.37. The local maximum around
r/t=0.2 for the 292° curve, however, was less than that of the other three DHD curves. The DHD
curves crossed zero a third time at roughly r/t=0.64, although there was noticeable spread about
this value. Finally, the contour curve crossed zero only one time, beyond the mid-thickness of the
pipe and just ahead of the DHD curves.
Figure 4-4
4.4
Superposition of Membrane, Crack-Face Pressure, and Weld Residual
Stresses
Stress Intensity Factor and Crack Growth
The total SIF was the sum of that stemming from global bending stress and that stemming from
the remaining stresses (i.e., σm, σcfp, and WRS). The SIF for σb was calculated with influence
coefficients for global bending [26], according to Equation 4-1.
K I = σ b Gb
πa
Q
4-4
Equation 4-1
where Gb is the influence coefficient and Q is the flaw shape parameter. In this case, σb was
considered to be the max imum bending stress occurring at the top dead-center location of the
pipe as a result of the applied bending moment.
The SIF for the remaining stresses was determined with the Universal Weight Function Method
[ 27] -[ 28] . The superimposed stress profiles shown in Figure 4-4 were input into the calculation as
discrete arrays. As such, there was no need for a polynomial fit of the stress profile, as is
sometimes the practice. The basic form for this SIF solution is shown in Equation 4-2.
a
K I = ∫ h(x, a )σ (x )dx
0
Eq uation 4 - 2
where h(x,a) is the weight function. In this work the SIF was calculated at both the deepest point
(K90) and the surface point (K0) of the flaw, as shown in Figure 4-5.
F ig ure 4 - 5
SI F at Two L ocations along C rack F ront
SIF calculations are necessary to use the established crack growth law, which is based on
laboratory crack growth ex periments on compact tension fracture mechanics specimens [ 25] . The
equation describing the crack growth rate is given in Equation 4-3.
Qg
da
= exp −
dτ
R g
1
1
−
T
abs Tref
Table 4-2 defines the symbols in Equation 4-3.
4-5
φ (K I − K Ith )η
Eq uation 4 - 3
Table 4 - 2
4 .5
Sy mbol D efinition for Eq uation 4 - 3
F law G rowth Res ults
Figure 4-6 shows K90 and a/ t versus time for the flaw growth calculation. No growth resulted from
the calculation based on the contour measurement. The DH D SIF curves ex hibited similar trends
of increasing to a peak early in time and subsequently decreasing to a plateau. The 112° and
292° curves showed a rapid increase in SIF later in time. Correspondingly, the 112° and 292°
curves showed through-wall crack growth, while the 22° and 202° curves demonstrated crack
arrest for the time period analyz ed. H owever, a typical relief request submitted to the NRC is only
concerned with timeframes of less than 20 years or 240 months, as discussed further in Section
4.6.
The 112° curve started at the highest K90 value and peaked the earliest in time at about
100 months and 14 MPa√m. It plateaued at a value of roughly 3.5 MPa√m, which was noticeably
higher than the other three curves. After 400 months, the SIF steadily increased, followed by a
rapid increase at 480 months. The flaw growth in the depth direction responded to the trends in
SIF just described. H ence, when the SIF plateaued at 3.5 MPa√m, the flaw grew linearly in time.
The other three DH D curves ex hibited similar trends, but with a few differences. The peak in SIF
was shifted to later times relative to the 112° measurement. This peak was also of a slightly lower
magnitude, but still in the range of 12– 13 MPa√m. The 22° and 202° curves did not ex hibit the
sharp increase in SIF in the time period analyz ed here.
4-6
F ig ure 4 - 6
( a) K90 and ( b) G rowth in D epth D irection
Figure 4-7 shows K0 and 2c/ C, where C is the inner circumference of the pipe, versus time for the
analytical flaw evaluation. The SIF value was highest for the 112° measurement, followed by
292° , 202° , and 22° . The crack length grew relatively slowly during the first 100 months for the
112° measurement. After this time, the length steadily grew to 6 % of the circumference. The
other curves ex hibited similar trends but did not grow to the same ex tent.
4-7
F ig ure 4 - 7
4 .6
( a) K0 and ( b) g rowth in leng th direction
D is cus s ion
Figure 4-6 shows the apparent uncertainty in flaw growth calculations resulting from the residual
stress assumption. H owever, regulatory relief submittals to the NRC do not evaluate 720 months
of operation. In fact, evaluation periods may ex tend only one or two refueling outages (1.5 to
3 years). Figure 4-8 shows that the uncertainty in the results decreases for shorter evaluation
periods.
4-8
F ig ure 4 - 8
F law G rowth after 2 0 Y ears
H owever, it is still useful to ex amine the reasons behind the apparent sensitivity to residual stress
assumption evident in Figure 4-6. As Figure 4-4 shows, the 112° case remained within 50 MPa of
the other DH D measurements for the first 30 % of the wall thickness. Even so, the calculated flaw
growth roughly doubled the others with a/ t= 0.3 at 100 months for the 112° curve. Only the 112°
and 292° cases showed through-wall crack growth. The 292° residual stress led to through-wall
growth despite peaking below 100 MPa at r/t= 0.2, which is in stark contrast to the remaining
curves. This section will seek to ex plain how features of the assumed residual stress profile may
affect the calculated flaw growth behavior.
The SIF for membrane and residual stresses is given by the integral shown in Equation 4-2. As a
first approx imation, this integral is similar to the area under the curves of Figure 4-4. Figure 4-9
shows the area under these curves as calculated by the trapez oidal rule.
4-9
Figure 4-9
(a) Membrane Stresses, (b) Area under the Curve
The area under the curve in Figure 4-9(b) reveals greater contrasts among the various
measurements than the stress profiles in Figure 4-9(a). The area under the 112° curve remains
clearly highest until about half way through the wall thickness, where the 292° curve becomes
slightly dominant. The 292° curve, despite peaking at a lower stress magnitude than the other
measurements, remains tensile for the greatest depth. This fact keeps the SIF high enough
through the calculation to allow for through-wall growth by 720 months. Once the crack tip
reaches the compressive zone (e.g., r/t=0.32), the 22° and 202° measurements show a steeper
drop in both stress magnitude and area under the curve. This leads to crack arrest in these two
cases.
4-10
4.7
Conclusion
This chapter confirms previous work showing that flaw growth calculations are sensitive to the
residual stress input [29]. Features of the residual stress curves that affected the results included
the following:
•
•
•
•
the ID stress magnitude
the stress magnitude at initial flaw depth
the location through the wall thickness where the compressive zone started
the slope of the curve in the compressive zone
While Chapter 5 presents a validation methodology for WRS FE models, this chapter
demonstrates that simplified approaches for judging the adequacy of residual stress inputs may
also be applied. Where confirmatory analyses of residual stress predictions are practical, the
listed features can be compared and contrasted to provide confidence in assumed inputs.
Estimating the area under the curve may also provide additional insights. While current
approaches to flaw evaluation appear to be adequate, a validation methodology for WRS FE
models may ease regulatory uncertainty and review times for relief requests.
4-11
5
5.1
VALIDATION PROCEDURE AND FINITE ELEMENT GUIDELINES
Introduction
This chapter draws on the uncertainty quantification methodology described in Chapter 3 to
develop a validation procedure for FE predictions of WRS. Accompanying the validation
procedure are guidelines for creating FE models of WRS. Together, the validation method and
guidelines may increase confidence in WRS inputs in relief requests.
The procedure proposed here involves two aspects: (1) establishing and justifying modeling
guidelines (Sections 5.2 and 5.3) and (2) proposing a series of quality metrics an analyst can
calculate to objectively assess the quality of an individual FE prediction of WRS (Section 5.4).
The discussion in Sections 5.2 and 5.3 assesses the Phase 2b dataset as a whole, while Section
5.4 focuses on a particular analyst seeking to validate an FE methodology.
The approach presented here requires an individual analyst to construct an FE model of the
Phase 2b mockup, according to the guidelines given in [7]. A series of quality metrics and
acceptance measures are proposed to validate the analyst’s prediction. Since this process is
aimed at deterministic fitness-for-service calculations (see Chapter 4), the metrics were designed
to ensure acceptable predictions of flaw growth.
The Phase 2b dataset is directly applicable to axisymmetric (i.e., two-dimensional) models of
dissimilar metal butt weld geometry. Extending this procedure to other geometries may require
additional work, such as fabrication of a mockup and measurement of residual stress. Section 5.6
discusses further the validation of residual stress predictions in other applications.
5.2
Material Hardening Law
One topic identified in earlier work was the need to establish guidance on hardening law choice
because of the significant impact this assumption has on the FE results [1]. This section
describes measurement-model comparisons with the goal of making informed judgments about
the appropriate approach to modeling material hardening during thermal cycling that occurs during
welding operations. This section first describes a methodology to account for modeling and
measurement uncertainty when making measurement-model comparisons, as developed in [13].
Sections 5.2.3 and 5.2.4 compare and contrast three approaches (isotropic, nonlinear kinematic,
and the average of isotropic and kinematic) to hardening law.
5.2.1
Difference in Means and Root Mean Square Error Functions
One method used here to investigate the performance of various hardening law approaches is to
examine the difference in means between the predictions and the measurements in the Phase 2b
dataset. The methodology, described in [13], is summarized as follows.
1. Sample ne measurement WRS functions and np prediction WRS functions from the model
described in Chapter 3 on a fine grid of L values of d, where d is the normalized distance
through the pipe wall thickness. This results in samples of WRS functions from both
measurements and predictions representing uncertainty in both. Let f be a function
representing sampled measurement WRS profiles and g be a function representing
sampled prediction WRS profiles, as follows.
5-1
f i (d k
) , k = 1,2,…,L and i = 1,2,…,ne
g i (d k
) , k = 1,2,…,L and i = 1,2,…,np
where i represents the ιth sample. The two functions are sampled independently.
2. Compute hs = (hs (d1 ), hs (d 2 ),..., hs (d k )) where
hs (d k ) =
1
ne
ne
∑
i =1
f i (d k ) −
1
np
np
∑ g (d )
i =1
i
k
Equation 5-1
The subscript s represents the sth difference in means calculation.
3. As a measure of prediction quality, calculate the root mean square error (RMSE) of hs,
which is defined as
RMSE s =
1 L
2
∑ h(d k )
L k =1
Equation 5-2
4. Repeat steps 1, 2, and 3 S times. In this study, S = 1,000.
5. Compute the pointwise 0.975 and 0.025 quantiles of hs(dk) and RMSE over the S samples
for each k. These quantiles form a pointwise 95 % bootstrap confidence bound for the
population difference of means function and the RMSE.
This methodology allows consideration of both measurement and modeling uncertainty when
assessing the quality of predictions.
5.2.2
Assessment of Prediction Trends
WRS predictions should capture variations in stress magnitude with spatial position. Within the
context of the procedure outlined in Section 5.2.1, this means that the estimated confidence
bounds on the mean difference function, hs(dk), should encompass zero. If the confidence bounds
on the mean difference function do not encompass zero, this implies a prediction bias. Figure 5-1
shows how well the nonlinear kinematic hardening FE results for axial stress predict the DHD
measurements. Figure 5-1 clearly illustrates where the nonlinear kinematic predictions
systematically over- and under-predict the DHD data in a statistically significant manner. In Figure
5-1(b), a positive mean difference implies an underprediction of the measurements.
5-2
Figure 5-1
Nonlinear Kinematic Hardening Predictions against the DHD Measurements:
(a) Actual Data and (b) Mean Difference Function and Tolerance Bounds
5-3
Figure 5-2 and Figure 5-3 show similar figures for isotropic hardening and average hardening,
respectively. Overall, the mean difference functions for these two cases remain closer to zero
throughout the wall thickness than was observed for the nonlinear kinematic predictions. There
are locations through the wall thickness where the tolerance bounds do not encompass zero,
indicating certain trends that are not captured by the FE. In Figure 5-2(b) and Figure 5-3(b), for
instance, the isotropic and average hardening results consistently underpredict the DHD
measurements around r/t=0.75. A result such as that in Figure 5-2(b), while not perfect, indicates
that the predictions are reasonable in a qualitative sense.
5-4
Figure 5-2
Isotropic Hardening Predictions against the DHD Measurements: (a) Actual
Data and (b) Mean Difference Function and Tolerance Bounds
5-5
Figure 5-3
Average Hardening Predictions against the DHD Measurements: (a) Actual
Data and (b) Mean Difference Function and Tolerance Bounds
Appendix A shows other plots similar to Figure 5-1, Figure 5-2, and Figure 5-3 for all. Table 5-1
lists all the relevant cases and the corresponding location of the data plots.
5-6
Table 5 - 1
B ench mark C as es and Th eir L ocation in Appendix A
Table 5-2 contains a qualitative description of the prediction quality for each of the cases in Table
5-1 along with a description of where measurement trends are not well captured by the model
predictions. The mean RMSE is included in Table 5-2 for a quantitative point of reference. Table
5-2 indicates that RMSE alone may not be an adequate indicator of prediction quality. For case 3
(ax ial stress, average hardening, DH D benchmark) RSME= 48.14 MPa, which is a relatively low
value for this dataset. H owever, the average hardening approach consistently underpredicted the
DH D measurements at r/ t= 0.25 and 0.75 for this case. In contrast, case 5 (hoop stress, isotropic
hardening, DH D benchmark) had a relatively high RSME but no evident prediction bias. In other
words, the confidence bounds encompassed z ero throughout the entire thickness for case 5
(Figure A-5). Case 9 (ax ial stress, average hardening, contour benchmark) is perhaps the best
agreement between measurements and models obtained in this study. Figure A-9 shows that,
where systematic prediction biases ex isted, the tolerance bounds were very close to
encompassing z ero. This case also had the lowest mean RMSE (36.77 MPa). Overall, this work
(and past work [ 1] -[ 2] ) demonstrates that FE provides reasonable predictions of residual stress.
Although, it is evident that established guidelines on hardening law approach are needed.
5-7
Table 5 - 2
5 .2 .3
Q ualitativ e As s es s ment of Prediction B ias
As s es s ment of Root M ean Sq uare Error
RMSE (Equation 5-2) may be one potential indicator of prediction quality. This value provides a
general measure of how well stress magnitudes are predicted. Table 5-3 and Table 5-4 show
RMSE for the various comparison cases for the DH D benchmark and the contour benchmark,
respectively.
5-8
Table 5-3
RMSE for DHD Benchmark
Table 5-4
RMSE for Contour Benchmark
Figure 5-4 and Figure 5-5 show these values plotted as bar charts, including the appropriate
confidence bounds, for the axial stress predictions and the hoop stress predictions, respectively.
In most cases, the models that used the nonlinear kinematic assumption demonstrated the
highest RSME. For the axial stress predictions with contour benchmark [Figure 5-4(b)], the
nonlinear kinematic and isotropic models were indistinguishable. This is also the only case where
the average hardening approach showed clearly superior predictions than the isotropic models. In
all other cases, the apparent improvement in prediction agreement of average over isotropic was
within the confidence bounds on RMSE.
Figure 5-4
Root Mean Square Error for Axial Stress Predictions: (a) DHD Benchmark and
(b) Contour Benchmark
5-9
Figure 5-5
5.2.4
Root Mean Square Error for Hoop Stress Predictions: (a) DHD Benchmark and
(b) Contour Benchmark
Hardening Law Recommendation
This study investigated three hardening law approaches: nonlinear kinematic, isotropic, and the
average of the kinematic result and isotropic result. Two aspects that should be assessed when
evaluating different hardening law approaches are prediction bias and RMSE. Table 5-2 indicates
that the through-wall trends were predicted well for three cases: 2, 5, and 9. Overall, the analysis
in this section indicates that the nonlinear kinematic hardening models were the least accurate.
While the uncertainties were large, there was some indication that the averaging approach
provides better predictions than isotropic. Given these considerations, the authors recommend
use of the averaging approach. Based on this study, the NRC considers that the mean of the
average models is validated against the measured data, and therefore it is acceptable to validate
further models against the mean of the average models.
This study did not consider the Lemaitre-Chaboche hardening law, also known as mixed
hardening [34]. Real materials exhibit both isotropic and kinematic hardening characteristics. The
experimental material data needed to develop a mixed hardening law were not available at the
time the Phase 2b round robin was began. In the future, it may be valuable to repeat a round
robin modeling study with mixed hardening to investigate how well this hardening law compares
with the measurement data presented here. The same statistical methods should be applied to
account for both modeling and measurement uncertainty and develop tolerance bounds on the
mean difference function.
5.3
Modeling Guidelines
While hardening law may be the most important factor, guidelines regarding other modeler
choices have been developed through industry experience [11] and through the round robin
studies [1]. These guidelines were provided to the Phase 2b modelers as instructions to
participate in the round robin analysis effort, as described in [7]. As indicated by the comparisons
of measurements and models in Section 5.2.2 and Appendix A, there is a general, qualitative
sense that the FE predictions of residual stress from the Phase 2b study are reasonable (e.g., see
Figure A-5 and Figure A-9). Thus, the modeling guidelines of [11] are adopted here. The
recommendations for axisymmetric models are shown in the following list.
5-10
•
•
•
•
•
Weld Bead Geometry Definition
o Modeling the precise bead shape is unnecessary.
o Weld beads may be approximated as trapezoids.
o The total number of weld beads, the number of weld layers, and the number of beads
in each layer should approximate the real weld configuration as closely as possible.
o Given current computational capabilities, modeling a realistic number of weld passes
in an axisymmetric model is feasible. Approximations resulting from pass-lumping
should be avoided.
o Where fabrication records are lacking, the assumed weld configuration should be
based on common industry practice and knowledge of the component (e.g., nominal
wall thickness).
o The cross-sectional area of each weld bead should be approximately equal to aid in
heat input tuning.
Bead and Process Sequence
o Residual stress FE models are path-dependent. The analyst should model the actual
fabrication process as closely as possible, including bead sequencing, number and
size of weld beads, and other relevant processes (e.g., repairs and PWSCC
mitigation).
o The analyst should explicitly model the application of the butter and associated
postweld heat treatment.
Element birth and death methodology for welding process modeling
o Several different methodologies may be acceptable to model the addition of material
as the weld is deposited.
o One method consists of removing the weld elements at the beginning of the
simulation, and then adding one weld pass at a time in a stress-free configuration but
with active temperature dependent material properties, heating it and cooling it, and so
on.
o Another method consists of having the elements present at all times in the model, but
only activation their temperature dependent material properties once a pre-determined
temperature is reached (perhaps the melting temperature).
Heat Input Model Tuning
o Tuning the heat input to match known quantities, such as expected interpass
temperature, is acceptable but not required.
o In all cases, the analyst should visually confirm that the entire weld bead reaches the
melting temperature. The material surrounding the weld bead should also reach the
melting temperature, such that the size of the fusion zone in the actual weld is
matched as closely as possible.
Structural and Thermal Boundary Conditions
o In general, boundary conditions should represent the physical situation being modeled.
As such, they can change from application to application.
o For a typical dissimilar metal butt weld, axial displacement in one node, located away
from the weld at the edge of the model, should be constrained (see [11] for more
information).
5-11
For a weld to infinitely-long straight pipe, the modeled pipe length should be 4 times
the ID to avoid edge effects.
o For the nozzle and pipe cross-sections typically of interest in reactor coolant pressure
boundary welds, heat convection at the surface is negligible compared to conduction
through the part.
Material Properties
o The average of isotropic and nonlinear kinematic is the recommended hardening
approach.
o The temperature-dependent material properties provided to the Phase 2b round-robin
participants (see Appendix B) may be used, subject to the following constraints.
The material property inputs should accurately reflect the materials in the real
situation, including temperature dependence.
The scope of material property inputs depends on the material behavior required to
successfully execute all aspects of the model. For example, creep properties are
required to approximate stress relaxation during postweld heat treatment.
Element Selection and Mesh
o Quadrilateral, linear elements should be applied.
o Quadratic and triangular elements should be avoided in all regions of the model,
because they tend to result in slight discontinuities in profiles due to more complex
interpolations between Gauss points. Mesh refinement with linear elements is
preferred over the use of quadratic or triangular elements.
o A fine mesh is recommended in the weld regions of the model. Approximate element
size for the weld passes should be 1.00 mm2. This corresponds to 10-25 elements in
a typical weld pass in reactor coolant pressure boundary nozzle welds.
o The mesh may coarsen away from the weld regions.
o
•
•
5.4
Proposed Validation Scheme
The purpose of this Section is to develop a process to judge the quality of a particular WRS
prediction. To apply the method proposed here, an analyst must create an FE model of the
Phase 2b mockup according to the guidelines of Section 5.3 and reference [7]. Then, the analyst
calculates a series of quality metrics to judge how well the analyst’s prediction agrees with the
round robin dataset. This validation methodology was developed assuming that the end
application is a deterministic flaw growth calculation. As such, flaw growth studies were
performed here to inform development of the metrics and acceptance measures. Section 5.5
summarizes the entire validation process, while Section 5.6 gives additional recommendations on
modeling a real application.
5.4.1
Overview of Approach
As discussed in [19], validation of a model determines how well the model reflects the physical
system being approximated. A validation approach requires a benchmark, a set of metrics, and
acceptance measures. Section 5.4.2 describes the recommended benchmark and associated
justification. The recommended metrics and acceptance measures were based on a flaw growth
argument. The concept applied here was to find a set of metrics that were relevant to flaw growth
predictions. The metrics should, therefore, interrogate features of a residual stress curve that are
5-12
important to flaw growth. Two metrics are proposed here for validating FE predictions of residual
stress:
1. RMSE on WRS magnitude through the entire wall thickness
2. Average difference up to the initial crack depth
Sections 5.4.7, 5.4.8, and 5.4.10 describe and develop these metrics in detail. Each metric
will have associated acceptance measures. The proposed acceptance measures are based
on values of the metric that lead to a reasonable crack growth prediction, given the chosen
benchmark. Sections 5.4.9 and 5.4.11 develop these acceptance measures. Additional
analysis of potential metrics and acceptance measures appears in Appendix D. While the
recommended acceptance measures were developed assuming an average hardening
benchmark, corresponding criteria assuming an isotropic hardening benchmark are included
in Sections 5.4.8 and 5.4.10 for illustration purposes. Additional analysis of potential metrics
and acceptance measures for isotropic hardening appears in Appendix E.
5.4.2
Benchmark
Validation first requires choice of a benchmark that reflects the real world. The predictions of the
model can then be quantitatively compared to the benchmark to assess the robustness of the
modeling approach. Often, physical measurements are a natural choice of benchmark. In the
application of concern here, there are four possibilities for a benchmark.
1.
2.
3.
4.
DHD measurement data
contour measurement data
average of 1 and 2
mean of the Phase 2b models
In the case of WRS, the “measurement data” is part physical measurement and part model, since
residual stress cannot be directly measured. This fact complicates the rigorous selection of a
benchmark for validating residual stress predictions. Lewis and Brooks [13] compared the Phase
2b DHD and contour measurement results to each other using the same methodology outlined in
Section 5.2.1 for comparing models to measurements. Figure 5-6 suggests that the two
measurements are significantly different than each other, especially near the inner surface. The
flaw growth calculations in Chapter 4 show that the differences between the contour and DHD
measurements lead to different flaw growth results. Averaging the two measurements (option 3)
may not be a valid option, since there is no reason to believe that the two datasets belong to the
same population. While a benchmark based upon the measurements may be ideal, parsing out
which measurement is most correct (option 1 or option 2) requires a more thorough investigation
than was performed in this work. The analysis in Section 5.2 suggests that the average hardening
modeling results provide reasonable predictions of the measurement data. Given the
complications associated with the measurement benchmark and the analysis in Section 5.2, the
mean of the average hardening models from the Phase 2b study was chosen as a benchmark for
demonstrating how a validation process may be developed.
5-13
Figure 5-6
5.4.3
Comparison of DHD and Contour Axial Stress Predictions (a) Raw Data and
(b) Difference in Means
Circumferential Flaw Growth – Isotropic Hardening
Figure 5-7 shows the seven smoothed isotropic WRS profiles that passed the initial outlier
screening discussed in Section 3.2.2. These profiles represent the axial WRS predictions from
the Phase 2b study, along with the mean WRS profile. The mean profile is the mean of six of the
predictions in Figure 5-7. Although prediction B was screened out of certain calculations, as
5-14
described in Section 3.2.8, it is included in this discussion for illustrative purposes. The mean
profile in Figure 5-7 is the cross-sectional mean, as described in [13].
Figure 5-7
Smoothed Axial WRS Profiles and Mean, Isotropic Hardening
The flaw growth study in Chapter 4 (see Table 4-1) was repeated assuming a circumferential flaw
and the residual stress profiles in Figure 5-7. The results are shown in Figure 5-8.
Figure 5-8
Circumferential Flaw Growth, Isotropic Hardening
5-15
As mentioned in Chapter 4, 720 months is much longer than a typical industry flaw analysis.
However, decreasing the evaluation period did not affect the metrics and acceptance measures
(see Appendix E). The mean WRS profile exhibited flaw growth to 45 % through-wall in
250 months, followed by flaw arrest. Three other residual stress predictions (D, E, and G) led to
arrest at 45-50 % through-wall. The other four WRS profiles (A, B, C, and F) led to different crack
growth behavior. Using participant B’s WRS profile, the flaw grew through-wall in under
200 months. The remaining calculations showed negligible growth throughout the evaluation
period.
This qualitative discussion of the different flaw growth predictions aids in establishing quantitative
acceptance measures for the three quality metrics proposed in Sections 5.4.8 and 5.4.10. WRS
predictions D, E, and G are considered reasonable, given that they result in similar end-of-life
flaws as the proposed benchmark.
5.4.4
Circumferential Flaw Growth – Average Hardening
Figure 5-9 shows the seven smoothed average hardening WRS profiles that passed the initial
outlier screening discussed in Section 3.2.2. These profiles represent the axial WRS predictions
from the Phase 2b study, along with the cross-sectional mean WRS profile.
Figure 5-9
Smoothed Axial WRS Profiles, Average Hardening
The flaw growth results are shown in Figure 5-10. Once again, as mentioned in Chapter 4,
720 months is much longer than a typical industry flaw analysis. However, decreasing the
evaluation period did not affect the metrics and acceptance measures (see Appendix D).
5-16
Figure 5-10 Circumferential Flaw Growth, Average Hardening
In comparison to the isotropic case in Figure 5-8, there is more uncertainty in the flaw growth
prediction. It is also apparent that the average hardening WRS predictions led to through-wall
flaw growth, while three cases arrested for the isotropic curves. The averaging approach tends to
decrease the magnitude of tensile and compressive stress peaks, relative to isotropic hardening.
On the one hand, the shallower compressive troughs of average hardening speeds up crack
growth and leads to more cases that go through-wall. On the other hand, arresting of the flaws
with isotropic hardening decreases uncertainty, since the exact magnitude of the compressive
trough is only important in cases where the flaw has not arrested. As will be shown in Section
5.4.9, these variances lead to different approaches to determining appropriate acceptance
measures. Therefore, the exact values of the acceptance measures are dependent upon the
choice of the benchmark in the approach adopted here.
5.4.5
Axial Flaw Growth – Isotropic Hardening
Figure 5-11 shows the seven smoothed hoop WRS profiles that passed the initial screening
described in Section 3.2.2, assuming isotropic hardening. The hoop stresses are distinct from the
axial stresses in that there is no force balance requirement through the wall thickness, given path
data extracted from an axisymmetric analysis. While the area under the curve of an axial stress
profile extracted from an axisymmetric FE model will be roughly zero, that of a hoop stress profile
may be non-zero. These differences are apparent in the smoothed curves of Figure 5-7 and
Figure 5-11.
5-17
Figure 5-11 Smoothed Hoop WRS Profiles, Isotropic Hardening
Figure 5-12 shows the axial flaw growth results, assuming the residual stress profiles of Figure
5-11. The results appear much more consistent among the various stress inputs than was
observed for the circumferential flaw growth study in Figure 5-8. This result is a consequence of
the hoop residual stresses being either positive or only slightly negative in the initial flaw depth
zone. With the axial stresses being compressive in the initial flaw depth zone, the exact
magnitude of the compressive stress can have a large impact on the flaw growth early in time.
Whereas the axial flaw will always exhibit early growth in the depth direction, the circumferential
flaw may or may not, depending on the interaction of the residual stresses with operating loads.
5-18
F ig ure 5 - 1 2
Ax ial F law G rowth , I s otropic H ardening
Prediction A is unique in that it led to crack arrest at 50 % through-wall for an ax ial flaw. All other
residual stress predictions led to through-wall growth in under 200 months. Table 5-5 shows the
time to through-wall for each prediction, with a value of 1,000 months assigned to prediction A to
represent “essentially infinite.” These observations will inform quality metrics for hoop stress
predictions using isotropic hardening.
Table 5 - 5
Time to Th roug h - W all
5-19
5.4.6
Axial Flaw Growth – Average Hardening
Figure 5-13 shows the seven smoothed hoop WRS profiles that passed the initial screening
described in Section 3.2.2, assuming average hardening.
Figure 5-13 Hoop WRS Profiles, Average Hardening
Figure 5-14 shows the corresponding crack growth results.
Figure 5-14 Axial Flaw Growth, Average Hardening
As was seen for the circumferential crack case, the uncertainty in time to through-wall increases
for average hardening relative to the isotropic results shown in Figure 5-12. This increase in
5-20
uncertainty is due to the fact that the averaging process broadened the tensile concave-down
region in the first 50 % of the wall thickness. The effect is illustrated more clearly in Figure 5-15,
which shows the stress intensity factors as a function of time for the first 200 months. The time of
the initial peak in stress intensity factor is much more variable for the average hardening law case.
Figure 5-15 Stress Intensity Factor: (a) Isotropic Hardening and (b) Average Hardening
5.4.7
Overview of Quality Metrics
To complete the validation procedure, an analyst must complete an FE model of the Phase 2b
mockup with isotropic hardening. The analyst then extracts WRS data from a weld centerline path
(because that is the location at which the DHD and contour data were measured for comparison).
The analyst may also smooth the extracted profile, provided that the smoothed WRS profile is
representative of the original. The extracted profile and the cross-sectional mean profiles shown
5-21
in Figure 5-7 and Figure 5-11 are then used to calculate two quality metrics. This section
develops the recommended metrics.
The first quality metric involves RMSE averaged through the entire wall thickness, as described in
Equation 5-3.
RMSEWRS =
1 L
WRS k − WRS kmean
∑
L k =1
(
)
2
Equation 5-3
where k and L are the same as in Equation 5-2, WRSk is the analyst’s predicted stress magnitude
at the kth position through the wall thickness, and WRSkmean is the cross-sectional mean prediction
(see [13]). For the analyst to complete this step, the WRS values from the analyst’s FE
calculation should be extracted at the same spatial intervals as those shown in Appendix C (i.e.,
every 1 % of the thickness). This quality metric provides an overall measure of how close the
predicted stress magnitudes are to the mean of the Phase 2b isotropic hardening predictions.
The second quality metric is average difference over the initial flaw depth. This metric is related to
the importance of the stress prediction over the distance corresponding to the initial flaw depth
(the first 10 % of wall thickness, in this case). The stress magnitude and trends in this area have
a profound impact on circumferential flaw growth behavior, as discussed further in Section 5.4.8.
This metric is calculated according to Equation 5-6.
diff avg
1 L0.1
=
WRS k − WRS kmean
∑
L0.1 k =1
(
)
Equation 5-6
where L0.1 is the number of spatial locations where WRS is determined up to xnorm=0.1.
5.4.8
Quality Metrics for Axial Stress Predictions
This section explores how the proposed metrics can be applied to discriminate between axial
residual stress predictions, with reference to the circumferential flaw growth studies presented in
Sections 5.4.3 and 5.4.4. As such, two approaches are considered here: one using the isotropic
hardening predictions and the other using the average hardening predictions. This discussion
leads to proposing acceptance measures for each of the metrics. Final recommendations on
acceptance measures are presented in Section 5.4.9.
Table 5-6 shows the two quality metrics applied to the seven isotropic predictions. The entries are
sorted according to how the corresponding circumferential crack growth prediction compared with
the mean prediction (see Figure 5-8). The prediction was deemed to yield “similar” flaw growth
results as the benchmark if the final flaw depth was within 10 % of the wall thickness (i.e., within
a/t=0.1) of the benchmark case. Thus, predictions D, E, and G were deemed to yield acceptable
flaw growth results.
5-22
Table 5-6
Quality Metrics Applied to Phase 2b Axial Isotropic Predictions
Participant
RMSE WRS [MPa] diff avg [MPa]
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
52
48
74
109
78
43
67
43
50
67
76
109
8
22
-1
120
-84
-55
-16
-84
-36
-1
15
120
Crack Growth
Similar
Similar
Similar
Too Fast
Too Slow
Too Slow
Too Slow
Table 5-6 demonstrates that RMSEWRS alone is not a sufficient measure of the quality of the
prediction. For example, RMSEWRS for prediction C was the minimum of the seven predictions at
43 MPa. However, diffavg for prediction C was -55 MPa. Figure 5-16 shows a plot of prediction C
compared against the mean WRS. The fact that prediction C was below the mean for the first
10 % of the wall thickness severely impacted the flaw growth calculation, such that the flaw never
grew. Therefore, it is important that diffavg be closer to zero in order to obtain the expected crack
growth behavior.
Figure 5-16 Prediction C (Isotropic) against the Mean Prediction
5-23
Since the flaw growth results were more variable for average hardening (see Figure 5-10),
discriminating between good and bad predictions requires a different thought process. In the case
of average hardening, the authors chose the 3 predictions (roughly, the top 50 %) that matched
closest to the benchmark as acceptable WRS predictions. For this case, predictions D, E, and G
were again considered acceptable. Table 5-7 shows the values of the quality metrics for each of
the average hardening predictions.
Table 5-7
Quality Metrics Applied to the Phase 2b Axial Average Hardening Predictions
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
5.4.9
RMSE WRS [MPa] diff avg [MPa] Delta from Mean Time to leakage
29
32
52
79
59
28
46
28
30
46
56
79
-10
11
14
92
-55
-31
-20
-55
-26
-10
12
92
71
-181
-86
-338
559
408
559
512
260
355
103
1000
849
1000
103
308
512
925
1000
Recommended Acceptance Measures – Axial Residual Stress
While the analysis in Section 5.4.8 was applied to both the isotropic hardening and average
hardening datasets, the final recommended acceptance measures should be based on one or the
other. The discussion in Section 5.2 indicated that the averaging approach may be preferable to
isotropic hardening. Therefore, the recommend acceptance measures is based upon the analysis
of the average hardening dataset in Table 5-7. In general, the recommended acceptance
measures are based upon values of the metric that (1) screen out those WRS curves that lead to
unreasonable crack growth predictions and (2) screen in those curves that lead to reasonable
crack growth predictions. For RMSEWRS, the recommended criterion is RMSEWRS≤55 MPa. The
recommended acceptance criterion for diffavg is -15 MPa≤ diffavg≤15 MPa.
5.4.10 Quality Metrics for Hoop Stress Predictions
Table 5-8 shows the two quality metrics for the seven isotropic hoop stress predictions, sorted
according to time to through-wall (see Figure 5-12).
5-24
Table 5-8
Quality Metrics Applied to Phase 2b Hoop Isotropic Predictions
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
RMSE WRS [MPa] diff avg [MPa] Delta from Mean Time to leakage
70
49
82
114
86
53
59
49
56
70
84
114
95
23
23
156
-68
-71
1
-71
-34
23
59
156
-51
-7
-29
-61
895
23
33
54
98
76
44
1000
128
138
44
65
98
133
1000
The time to leakage assuming the mean hoop residual stress profile in Figure 5-11 was 105 months.
This value was considered as a benchmark for this discussion. The times to leakage assuming
Participant G, E, and C’s residual stress profiles were each within 36 months of this value. Thirtysix months corresponds to the length of time for two refueling outages. Participant E’s case is
straightforward, in each quality metrics were relatively low in magnitude. Participant G, on the other
hand, had a high RMSEWRS value of 82 MPa. Figure 5-17 compares Participant G’s prediction
against the mean curve. The high value of the WRS for 0.11, some false positives occur for hoop WRS.
Consequently, the weighted metrics do not offer any improvement over the basic metrics
proposed in this report (see Sections 5.4.8 and 5.4.10).
E-19
W=1
Axial Stress / Circ Flaw Growth
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
Weighted
(RMSE WRS ) W (RMSE D1 ) W diff avg
37
38
55
68
57
30
51
30
38
51
56
68
540
581
645
555
429
342
889
342
485
555
613
889
8
22
-1
120
-84
-55
-16
-84
-36
-1
15
120
Crack Growth
Similar
Similar
Similar
Too Fast
Too Slow
Too Slow
Too Slow
Hoop Stress / Axial Flaw Growth
W=1
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
Axial WRS
Metric
(RMSE WRS ) W
≤
(RMSE D1 ) W
≤
diff avg
≥
Weighted
(RMSE WRS ) W (RMSE D1 ) W diff avg Time to leakage
55
36
65
81
66
40
53
36
46
55
66
81
531
530
719
839
778
443
828
443
531
719
803
839
Acceptance
55
645
-5
95
23
23
156
-68
-71
1
-71
-34
23
59
156
Too Short
Acceptable
Acceptable
Too Short
Too Long
Acceptable
Acceptable
Hoop WRS
Metric
(RMSE WRS ) W ≤
(RMSE D1 ) W
≤
diff avg
≥
diff avg
≤
Acceptance
65
830
-75
75
Figure E-15 Weighted Quality Metrics for Phase 2b Isotropic Predictions (left), and
Acceptance measures for the Proposed Metrics (right), for W=1
Assessment of weighted metrics for W=1:
•
•
Acceptance measures:
o Narrower (improved) (RMSEWRS)T acceptance criterion for axial and hoop WRS
o Narrower (improved) (RMSED1)T acceptance criterion for axial and hoop WRS
o Overall significant improvement in acceptance measures
Ability to distinguish between good and bad predictions:
o Good for axial and hoop WRS
o Same ability to distinguish as basic metrics proposed in 5.4.8 and 5.4.10
E-20
W=2
Axial Stress / Circ Flaw Growth
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
Weighted
(RMSE WRS ) W (RMSE D1 ) W diff avg
30
33
44
55
46
25
44
25
32
44
45
55
442
541
538
408
334
257
814
257
371
442
539
814
8
22
-1
120
-84
-55
-16
-84
-36
-1
15
120
Crack Growth
Similar
Similar
Similar
Too Fast
Too Slow
Too Slow
Too Slow
Hoop Stress / Axial Flaw Growth
W=2
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
Axial WRS
Metric
(RMSE WRS ) W
≤
(RMSE D1 ) W
≤
diff avg
≥
Weighted
(RMSE WRS ) W (RMSE D1 ) W diff avg Time to leakage
46
30
57
66
53
33
48
30
39
48
55
66
443
445
638
662
620
337
782
337
444
620
650
782
Acceptance
45
550
-5
95
23
23
156
-68
-71
1
-71
-34
23
59
156
Too Short
Acceptable
Acceptable
Too Short
Too Long
Acceptable
Acceptable
Hoop WRS
Metric
(RMSE WRS ) W ≤
(RMSE D1 ) W
≤
diff avg
≥
diff avg
≤
Acceptance
60
790
-75
75
Figure E-16 Weighted Quality Metrics for Phase 2b Isotropic Predictions (left), and
Acceptance measures for the Proposed Metrics (right), for W=2
Assessment of weighted metrics for W=2:
•
•
Acceptance measures:
o Narrower (improved) (RMSEWRS)T acceptance criterion for axial and hoop WRS
o Narrower (improved) (RMSED1)T acceptance criterion for axial and hoop WRS
o Overall significant improvement in acceptance measures
Ability to distinguish between good and bad predictions:
o Good for axial WRS
o False positive for prediction A for hoop WRS
o Less ability to distinguish than basic metrics proposed in 5.4.8 and 5.4.10
E-21
W=5
Axial Stress / Circ Flaw Growth
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
Weighted
(RMSE WRS ) W (RMSE D1 ) W diff avg
19
26
28
42
33
20
34
19
23
28
34
42
302
474
386
259
200
141
698
141
230
302
430
698
8
22
-1
120
-84
-55
-16
-84
-36
-1
15
120
Crack Growth
Similar
Similar
Similar
Too Fast
Too Slow
Too Slow
Too Slow
Hoop Stress / Axial Flaw Growth
W=5
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
Axial WRS
Metric
(RMSE WRS ) W
≤
(RMSE D1 ) W
≤
diff avg
≥
Weighted
(RMSE WRS ) W (RMSE D1 ) W diff avg Time to leakage
32
21
42
50
34
25
38
21
28
34
40
50
318
337
525
452
395
189
704
189
328
395
489
704
Acceptance
30
475
-5
95
23
23
156
-68
-71
1
-71
-34
23
59
156
Too Short
Acceptable
Acceptable
Too Short
Too Long
Acceptable
Acceptable
Hoop WRS
Metric
(RMSE WRS ) W ≤
(RMSE D1 ) W
≤
diff avg
≥
diff avg
≤
Acceptance
45
710
-75
75
Figure E-17 Weighted Quality Metrics for Phase 2b Isotropic Predictions (left), and
Acceptance measures for the Proposed Metrics (right), for W=5
Assessment of weighted metrics for W=5:
•
•
Acceptance measures:
o Narrower (improved) (RMSEWRS)T acceptance criterion for axial and hoop WRS
o Narrower (improved) (RMSED1)T acceptance criterion for axial and hoop WRS
o Overall significant improvement in acceptance measures
Ability to distinguish between good and bad predictions:
o Good for axial WRS
o False positive for prediction A for hoop WRS
o Less ability to distinguish than basic metrics proposed in 5.4.8 and 5.4.10
E-22
W=10
Axial Stress / Circ Flaw Growth
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
Weighted
(RMSE WRS ) W (RMSE D1 ) W diff avg
12
19
17
34
26
16
26
12
16
19
26
34
208
411
276
204
132
70
607
70
168
208
344
607
8
22
-1
120
-84
-55
-16
-84
-36
-1
15
120
Crack Growth
Similar
Similar
Similar
Too Fast
Too Slow
Too Slow
Too Slow
Hoop Stress / Axial Flaw Growth
W=10
Participant
D
E
G
B
A
C
F
Min
25th percentile
Median
75th percentile
Max
Axial WRS
Metric
(RMSE WRS ) W
≤
(RMSE D1 ) W
≤
diff avg
≥
Weighted
(RMSE WRS ) W (RMSE D1 ) W diff avg Time to leakage
26
14
27
42
23
20
28
14
21
26
28
42
239
273
428
338
252
114
628
114
245
273
383
628
Acceptance
20
415
-5
95
23
23
156
-68
-71
1
-71
-34
23
59
156
Too Short
Acceptable
Acceptable
Too Short
Too Long
Acceptable
Acceptable
Hoop WRS
Metric
(RMSE WRS ) W ≤
(RMSE D1 ) W
≤
diff avg
≥
diff avg
≤
Acceptance
30
630
-75
75
Figure E-18 Weighted Quality Metrics for Phase 2b Isotropic Predictions (left), and
Acceptance measures for the Proposed Metrics (right), for W=10
Assessment of weighted metrics for W=10:
•
•
Acceptance measures:
o Narrower (improved) (RMSEWRS)T acceptance criterion for axial and hoop WRS
o Narrower (improved) (RMSED1)T acceptance criterion for axial and hoop WRS
o Overall significant improvement in acceptance measures
Ability to distinguish between good and bad predictions:
o Good for axial WRS
o False positive for prediction A for hoop WRS
o Less ability to distinguish than basic metrics proposed in 5.4.8 and 5.4.10
E-23
NU REG - 2 2 2 8
W eld Res idual Stres s F inite Element Analy s is V alidation
Part II—Proposed Validation Procedure
July
2 0 2 0
a
Michael L. Benson
Patrick A. C. Raynaud
Jay S. Wallace
Technical
Division of Engineering, Office of Research
U.S. Nuclear Regulatory Commission
Washington, DC 20555-0001
Same as above
Under a Memorandum of Understanding, the U.S. Nuclear Regulatory Commission and the Electric Power
Research Institute conducted a research program aimed at gathering data on weld residual stress modeling.
As described in NUREG-2162, “Weld Residual Stress Finite Element Analysis Validation: Part I—Data
Development Effort,” issued March 2014, this program consisted of round robin measurement and modeling
studies on various mockups. At that time, the assessment of the data was qualitative. This report describes
an additional residual stress round robin study and a methodology for capturing residual stress uncertainties.
This quantitative approach informed the development of guidelines and a validation methodology for finite
element prediction of weld residual stress. For ex ample, comparisons of modeling results to measurements
provided a basis for establishing guidance on a material hardening approach for residual stress models.
The proposed validation procedure involves an analyst modeling a known case (the Phase 2b round robin
mockup) and comparing results to two proposed quality metrics. These recommendations provide a
potential method by which analysts can bolster confidence in their modeling practices for regulatory
applications.
weld residual stress, finite element modeling, model validation, flaw evaluation,
uncertainty characteriz ation
NUREG-2228
Final
Weld Residual Stress Finite Element Analysis Validation
Part II—Proposed Validation Procedure
July 2020
| File Type | application/pdf |
| File Title | Weld Residual Stress Finite Element Analysis Validation |
| Author | Machalek, Woody |
| File Modified | 2021-09-29 |
| File Created | 2020-07-27 |