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ANL-ART-96
Report on an Assessment of the Application of EPP
Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT
Methodology
Nuclear Engineering Division
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ANL-ART-96
Report on an Assessment of the Application of EPP
Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT
Methodology
prepared by R. I. Jetter, R.I. Jetter Consulting M. C. Messner and T.-L. Sham, Argonne National Laboratory Y. Wang, Oak Ridge National Laboratory August 2017
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 ii
August 2017
ANL-ART-96 iii
ABSTRACT
The goal of the proposed integrated Elastic Perfectly-Plastic (EPP) and Simplified Model Test (SMT)
methodology is to incorporate an SMT data based approach for creep-fatigue damage evaluation into the
EPP methodology to avoid the separate evaluation of creep and fatigue damage and eliminate the
requirement for stress classification in current methods; thus greatly simplifying evaluation of elevated
temperature cyclic service. This methodology should minimize over-conservatism while properly
accounting for localized defects and stress risers. To support the implementation of the proposed
methodology and to verify the applicability of the code rules, analytical studies and evaluation of
thermomechanical test results continued in FY17. This report presents the results of those studies.
An EPP strain limits methodology assessment was based on recent two-bar thermal ratcheting test results
on 316H stainless steel in the temperature range of 405 to 7050C. Strain range predictions from the EPP
evaluation of the two-bar tests were also evaluated and compared with the experimental results. The role
of sustained primary loading on cyclic life was assessed using the results of pressurized SMT data from
tests on Alloy 617 at 9500C. A viscoplastic material model was used in an analytic simulation of two-bar
tests to compare with EPP strain limits assessments using isochronous stress strain curves that are
consistent with the viscoplastic material model. A finite element model of a prior 304H stainless steel
Oak Ridge National Laboratory (ORNL) nozzle-to-sphere test was developed and used for an EPP strain
limits and creep-fatigue code case damage evaluations. A theoretical treatment of a recurring issue with
convergence criteria for plastic shakedown illustrated the role of computer machine precision in EPP
calculations.
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 iv
August 2017
ANL-ART-96 v
TABLE OF CONTENTS
Abstract ........................................................................................................................................................ iii Table of Contents .......................................................................................................................................... v List of figures .............................................................................................................................................. vii List of Tables ............................................................................................................................................... ix List of Figures – Appendix A ...................................................................................................................... xi List of Tables – Appendix A ....................................................................................................................... xii List of Figures – Appendix B ..................................................................................................................... xiii List of Tables – Appendix B ...................................................................................................................... xiv 1 Background ............................................................................................................................................ 1 2 Experimental based development ........................................................................................................... 3
2.1 Pressurized SMT tests .................................................................................................................. 3
2.2 Two-bar tests ................................................................................................................................ 5
2.2.1 Strain limits evaluation .................................................................................................... 6
2.2.2 Strain range evaluation .................................................................................................... 8 3 Analytical based development ................................................................................................................ 9
3.1 EPP code cases ............................................................................................................................. 9
3.1.1 Inelastic analysis simulations .......................................................................................... 9
3.1.2 Component test comparison .......................................................................................... 13
3.2 Shakedown criteria ..................................................................................................................... 15 4 Summary .............................................................................................................................................. 16 References ................................................................................................................................................... 17 Appendix A ................................................................................................................................................. 19 Appendix B: ................................................................................................................................................ 39 Acknowledgments ....................................................................................................................................... 53 Distribution List .......................................................................................................................................... 55
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 vi
August 2017
ANL-ART-96 vii
LIST OF FIGURES
Fig. 1. SMT methodology. (a) Shell structure with stress concentration and elastic follow-up, (b)
design curve, and (c) hold time creep–fatigue test with elastic follow-up ................................ 1 Fig. 2. Updated flow-chart on the development of the EPP-SMT approach ................................................ 2 Fig. 3. Maximum and minimum stresses in the necked region vs. cycles for tension hold only
pressurized SMT tests on Alloy 617 at 950oC ........................................................................... 4 Fig. 4. Allowable pressure vs design life for Alloy 617 pressurized tube at 950oC...................................... 4 Fig. 5. Pressurized cylinder with radial thermal gradient represented by a two-bar model .......................... 5 Fig. 6. Two-bar thermal cycle ....................................................................................................................... 6 Fig. 7. EPP Strain limits envelope and test data for 405 to 705oC temperature range (the green
symbol ☺ passed 1% strain limits, while the red symbol ⨂ did not pass 1% strain
limits) ........................................................................................................................................ 7 Fig. 8. Thermal cycle, 515 to 815°C. (a) EPP results, (b) inelastic results, (c) EPP margin ...................... 11 Fig. 9. Thermal cycle, 415 to 515°C. (a) EPP results, (b) inelastic results, (c) EPP margin ...................... 12 Fig. 10. Schematic diagram of the axisymmetric specimen with key locations labeled ............................. 13 Fig. 11. Combined figure showing the original experimental loading and key times in the EPP
analysis of the nozzle-to-sphere test article ............................................................................. 14
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 viii
August 2017
ANL-ART-96 ix
LIST OF TABLES
Table 1. Tubular SMT pressurization for Alloy 617..................................................................................... 3 Table 2. Comparison of the pressurization SMT on Alloy 617 .................................................................... 3 Table 3. Parameters of the two-bar experiment and EPP analysis ................................................................ 6 Table 4. Experimental results for 405 to 705oC temperature range .............................................................. 7 Table 5. Experimental and analytical strain ranges ...................................................................................... 8 Table 6. Loading parameters for the two bar simulations ........................................................................... 10
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 x
August 2017
ANL-ART-96 xi
LIST OF FIGURES – APPENDIX A
Fig. A 1. Example consistent isochronous curves for 500°C ...................................................................... 30 Fig. A 2. Interpretation of a two-bar experiment as probing the response of the extreme fibers of a
thin-walled pressure vessel under constant pressure and cyclic thermal load ......................... 30 Fig. A 3. Thermal cycle used for the two-bar simulations. The delay on the cooling end of the cycle
induces thermal strain in the two-bar system .......................................................................... 31 Fig. A 4. Thermal cycle, 515 to 815°C. a) EPP results, b) inelastic results, c) EPP margin ...................... 32 Fig. A 5. Thermal cycle, 415 to 515°C. (a) EPP results, (b) inelastic results, (c) EPP margin .................. 33 Fig. A 6. Schematic diagram of the axisymmetric nozzle-to-sphere test article with key locations
labeled ..................................................................................................................................... 34 Fig. A 7. Combined figure showing the original experimental loading and key times in the EPP
analysis of the nozzle-to-sphere test article ............................................................................. 34 Fig. A 8. Finite element mesh used to simulate the response of the ORNL test article .............................. 35 Fig. A 9. Elastic stress analysis of the specimen, figure zoomed into the critical section .......................... 36 Fig. A 10. Composite cycle used in the EPP analysis of the ORNL test .................................................... 37 Fig. A 11. Simulation result showing reversing ratcheting. Initially the two-bar system seems to be
approaching some saturated, compressive ratcheting rate only for the system to reverse
the direction of ratcheting ........................................................................................................ 37
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 xii
LIST OF TABLES – APPENDIX A
Table A 1. Parameters for inelastic constitutive model .............................................................................. 29 Table A 2. Loading parameters for the two sets of two-bar simulations .................................................... 29
August 2017
ANL-ART-96 xiii
LIST OF FIGURES – APPENDIX B
Fig. B 1. Schematic of the four classical cyclic plasticity deformation regimes. a) elastic response,
b) elastic shakedown, c) plastic shakedown, and d) ratcheting. .............................................. 46 Fig. B 2. Example two-bar system. ............................................................................................................. 47 Fig. B 3. Bree diagram for the classical two bar problem. Points A and B are used to test methods
for determining the shakedown boundaries from numerical finite element analysis of
the system. ............................................................................................................................... 48 Fig. B 4. Simulated two-bar stress/strain history. a) elastic response, b) elastic shakedown, c) plastic
shakedown, d) ratcheting. ........................................................................................................ 49 Fig. B 5. Apparent ratcheting strain increment per cycle for plastic shakedown loading conditions. ........ 50
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 xiv
LIST OF TABLES – APPENDIX B
Table B 1. Properties used in the example two bar simulations. ................................................................ 45 Table B 2. Convergence series for the elastic and plastic shakedown criteria varying the shakedown
residual tolerance ..................................................................................................................... 45 Table B 3. Two notional examples showing how the initial error can affect the final convergence of
Newton's method, when implemented with floating point arithmetic ..................................... 51 Table B 4. Convergence series for the elastic and plastic shakedown criteria varying the Newton
tolerance .................................................................................................................................. 51
August 2017
ANL-ART-96 1
1 Background The goal of the integrated Elastic, Perfectly Plastic (EPP) and Simplified Model Test (SMT) approach is
to incorporate an SMT data based approach for creep-fatigue (CF) damage evaluation into the EPP
methodology to avoid the use of the creep-fatigue interaction diagram, the so-called “D” diagram, and to
minimize over-conservatism while properly accounting for localized defects and stress risers. There are
two approaches of interest to the proposed integrated evaluation of cyclic service life that have received
attention over the last several years. One of these approaches is identified as the EPP methodology and
the other is identified as the SMT methodology. The EPP cyclic service methodology greatly simplifies
the design evaluation procedure by eliminating the need for stress classification that is the basis of the
current rules. EPP based design methods have already been qualified for ASME Section III Division 5
applications via two approved code cases: N-861 for the evaluation of strain limits and N-862 for the
evaluation of creep-fatigue damage, both for 304H and 316H stainless steel Class A components [1,2].
However, the EPP methodology for evaluation of creep-fatigue damage still requires the separate
evaluation of creep damage and fatigue damage by placing a limit on the allowable combined damage, the
“D” diagram based on calculating individual creep and fatigue damages.
The basic concept of the SMT methodology is shown in Fig. 1 [3,4]. The component design is
represented by a stepped cylinder with a stress concentration at the shoulder fillet radius. The component
has a global elastic follow-up, nq which is due to the interaction between the two cylindrical sections, and
a local follow-up, Lq which is due to the local stress concentration. Fig. 1(a) illustrates the damage from a
strain, ,E comp that is applied, held, and then cycled back to zero and reapplied. The damage is evaluated
from a design curve, Fig. 1(b), based on data from the simplified model test, Fig. 1(c). The SMT
specimen has a follow-up sized to bound the follow-up in representative components. The evaluation
procedure is essentially the same as that used in Division 1 Subsection NB, where the damage fraction is
determined as the ratio of actual number of cycles, n , to the allowed number of cycles, N . The design
curve envelopes the effects of hold time duration and follow-up magnitude without being excessively
conservative. It is developed from SMT data that is plotted as strain versus observed cycles to failure, Fig.
1(b).
Fig. 1. SMT methodology. (a) Shell structure with stress concentration and elastic follow-
up, (b) design curve, and (c) hold time creep–fatigue test with elastic follow-up
A detailed plan in developing this methodology has identified the key issues, assumptions and the
proposed path to resolution and verification of the EPP-SMT approach. Fig. 2 is an updated flow chart
from the initial plan [5,6] and it shows the impact of recent test results from pressurization SMT and the
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 2
EPP strain range analysis. The major elements in this flow chart include: three key assumptions that have
been made to move forward; the near term test and evaluation actions required to validate these
assumptions; and the long term test and analytical development required depending upon the outcome of
the near term validation efforts.
Fig. 2. Updated flow-chart on the development of the EPP-SMT approach
Shown under the category of “Near term test and evaluation” for assumption 1, is a comparison of tension
hold data with data from tests with alternating tension and compression hold times. The reason for this is
twofold. First, it would be desirable to base the validation on the more conservative data. However,
perhaps the more important reason is to minimize the barreling effect that clouds the interpretation of the
tension-hold only test data.
Pressurized SMT hollow cylinder tests are being used to assess the second assumption that the stress level
associated with primary loading will be small compared with the secondary and peak stress levels and
should not have a significant effect on the total life. In addition to the pressurized SMT data, the modified
two-bar test with one bar as SMT specimen at constant temperature and the second bar a smooth gage
geometry, “Long term tests” will provide valuable data for verification of the effects of superimposed
primary loading. The advantage of this two-bar modified configuration is that all the relevant test
parameters can be measured directly. If it turns out that the effect of primary loading is significant, then
the proposed solution, as shown under the long term test and analytic development column, is to develop
mean stress type design curves analogous to the mean stress correction curves for the fatigue evaluation
of some materials below the creep regime.
The third assumption, that the EPP strain range determination captures the creep-fatigue degradation due
to follow-up effects, will be evaluated using results from both the Type 1 and 2 SMT test specimens to
determine if adjustment factors will be required for the SMT based design curves.
Longer term tests are required to develop the SMT design curves and to support the development of
adjustment factors to account for such effects as sustained primary loading and retardation of stress
relaxation due to follow-up, if needed.
August 2017
ANL-ART-96 3
2 Experimental based development
2.1 Pressurized SMT tests
SMT pressurization tests are being used to assess whether the effects of stress levels associated with
sustained primary loads will be small when compared to the cyclic secondary and peak levels for the
development of SMT creep-fatigue based design curve. In the SMT pressurization test, the pressure and
temperature are held constant and the displacement is periodically cycled with a specified hold time. The
hollow cylindrical test specimen has a thinner necked region in series with a thicker driver section that are
sized to provide the desired follow-up. In FY17, two additional SMT pressurization tests were performed
on Alloy 617 at ORNL. The testing parameters and the results of these two tests are highlighted in Table
1 and Table 2 along with previous pressurized SMT results for comparison. The details of the test
procedures and results are described in [7].
Table 1. Tubular SMT pressurization for Alloy 617
Specimen
ID
Elastically
calculated strain
range
Loading condition
Initial
strain
range
Test
temperature oC
Internal
pressure
Life
time,
hr
Cycles
to
failure
INC617-P01 0.3% Tension hold 600 s 0.8% 950 2 psi 37.4 220
INC617-P02 0.3% Tension hold 600 s 0.8% 959 200 psi 37.4 220
INC617-P04 0.3% Tension hold 600 s 0.8% 957 500 psi 34.0 200
INC617-P03 0.3% Tension hold 600 s 0.75% 958 750 psi 25.5 150
INC617-P06 0.3% Tension hold 600 s 0.8% 950 750 psi 23.8 140
INC617-P09 0.2% Tension hold 600 s --- 953 750 psi 54.4 320
INC617-P05 0.3% Combined tension
and compression 1% 955 2 psi 107.7 320
INC617-P08 0.3% Combined tension
and compression 1.05% 950 500 psi 94.3 280
INC617-P07 0.3% Combined tension
and compression 1.05% 950 750 psi 60.6 180
Table 2. Comparison of the pressurization SMT on Alloy 617
Specimen
ID
Internal
pressure Original
ID/OD
Original
wall
thickness
Max OD after
testing
Wall
thickness
after failure
Failure location
Elastic
follow-up
factor
Tension hold SMT
INC617-P01 2 psi 0.5/0.62 in 60 mil ~0.68 in ~68 mil Center ~3.8
INC617-P02 200 psi 0.5/0.62 in 60 mil ~0.72 in ~62 mil Center ~3.8
INC617-P04 500 psi 0.5/0.62 in 60 mil ~0.75 in ~54mil Center ~4.0
INC617-P03 750 psi 0.5/0.62 in 60 mil ~0.81 in ~41 mil Transition radius ~4.1
INC617-P06 750 psi 0.5/0.62 in 60 mil ~0.80 in ~42 mil Transition radius ~4.1
INC617-P09 750 psi 0.5/0.62 in 60 mil ~0.78 in ~42 mil Transition radius ---
Combined Tension and Compression hold SMT
INC617-P05 2 psi 0.5/0.62 in 60 mil ~0.64 in 59 to 62 mil Center ---
INC617-P08 550 psi 0.5/0.62 in 60 mil ~0.92 in 50 to 35 mil All over ---
INC617-P07 750 psi 0.5/0.62 in 60 mil ~0.95 in 43 to 34 mil All over ---
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 4
The combined tension and compression tests lasted longer than the comparable tension only tests. Fig. 3
is a plot of the maximum and minimum stresses in the necked region versus cycles for tension only tests
at 950oC with 0.3% elastic strain range and internal pressures as noted.
Fig. 3. Maximum and minimum stresses in the necked region vs. cycles for tension hold only
pressurized SMT tests on Alloy 617 at 950oC
Based on the results tabulated in Table 1 and shown in Fig. 3, it is clear that higher pressures can reduce
the number of cycles to failure. However, those pressures may be unrealistically high for normal
operating pressure limitations. This potential limitation was assessed by determining the allowable life for
pressurized cylinders using the Alloy 617 allowable stress values from a proposed Code Case for Alloy
617 (ASME C&S Connect Record No. 16-994 and 16-1001). The primary membrane stress, mP , in a
cylindrical shell can be closely approximated by the expression, ( ) /m m wP pR t , where p is internal
pressure, wt is thickness and mR is mean radius. Rearrange and let ( )m mP S t with ( )mS t the allowable
stress results in ( ) /m w mp S t t R . For the pressurized tube tests 0.060 inwt and 0.28 inmR . ( )mS t is
a function of time so the allowable internal pressure, p , can be plotted as a function of time from the
allowable stress values. A plot of the allowable pressure vs design life for Alloy 617 at 9500C for this
testing geometry is shown in Fig. 4, with three test pressures and corresponding design life highlighted in
red.
Fig. 4. Allowable pressure vs design life for Alloy 617 pressurized tube at 950oC
August 2017
ANL-ART-96 5
At 200 psi the allowable life is approximately 22,000 hr, at 500 psi – 320 hr, and only about 60 hr at
750 psi. This supports the argument that, for normal design lives of about 30 years with allowable
pressure of ~150 psi, the primary stress evaluation will screen out high pressures that would compromise
cyclic life. Clearly additional testing at other strain ranges, temperatures and hold times will be required,
but the initial results are encouraging.
2.2 Two-bar tests
Two-bar thermal ratcheting tests were performed on 316H stainless steel to assess the material response
to cyclic thermal loading under two-bar testing conditions at the intermediate temperature range of 405 to
7050C. The details of the test procedures and results are described in [7].
A two-bar model is a simplified analysis of a vessel under a combination of a constant, primary pressure
load and a secondary, alternating thermal cycle. The two bars represent two extreme material fibers (see
Fig. 5). The two bars are constrained to have the same total displacement under a constant applied load
and a cyclic thermal loading.
Fig. 5. Pressurized cylinder with radial thermal gradient represented by a two-bar model
The two-bar thermal cycle is shown in Fig. 6. The thermal strains in the specimen are a function of the
time delay between position 3 and 5, the cooling rate and the total temperature difference. The heating
and cooling rates were 300C/min. Experiments were performed on a set of specimens at the temperature
range of 405 to 7050C with different combinations of total load levels and time delays. The changing of
loading conditions was performed at 4050C without unloading the specimens to zero load from the
previous condition.
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 6
Fig. 6. Two-bar thermal cycle
The test parameters are summarized in Table 3.
Table 3. Parameters of the two-bar experiment and EPP analysis
Material 316H stainless steel
Specimen diameter 0.25 in
Temperature range 405 to 705oC
Hold time ② to ③ 60 min
Heating and cooling rate
① to ②; ③to ④ and ⑤ to ① 30oC/min
Time delay, ③ to ⑤ From 1 to 10 min
Applied total load From -1,000 to 1,000 lbs
2.2.1 Strain limits evaluation
To assess the conservatism of the strain limits code case [1], the two-bar configuration was evaluated
using the EPP methodology. The EPP method assumes that the average deformations computed for a
system with an elastic-perfectly plastic analysis using a material’s elastic properties and a pseudo-yield
stress less than the minimum yield stress the material experiences over its design life will be greater than
the actual, experimental deformations. This temperature dependent pseudo-yield stress based on the
isochronous stress/strain curves for the material defined in the ASME Division 5 code. The strain limits
code case uses this bounding EPP analysis to evaluate designs against the inelastic strain limits
established in the ASME Division 5 code. Essentially the code case sets two conditions on the results of
the analysis: the structure must shake down under cyclic loading, as to establish steady cyclic
deformation; and, if the structure shakes down, the inelastic strain computed via the elastic-perfectly
plastic analysis meet criteria designs to ensure the structure will pass the Division 5 strain limits criteria.
If a structure meets both these criteria then, by the bounding principal outlined above, the true structure
will both shake down and accumulate less than 1% inelastic strain over its design life.
Table 4 summarizes the experimental results and Fig. 7 is the envelope of loading conditions that pass the
EPP strain limits code case. The ordinate is the time delay in minutes and the abscissa is the total applied
load in lb for the two bars with 0.25 in diameter. The analytic boundaries shown in red and blue are
incremental solutions to this two-bar problem using the strain limits code case procedure. The red
August 2017
ANL-ART-96 7
boundary is determined from incremental changes in the applied total load that reduce ratcheting and the
blue boundary is determined from incremental changes in the applied load from inside the non-ratcheting
regime that show no ratchet. The difference is due to the size of the incremental change in applied load.
Also shown in Fig. 7 are the locations of the test point coordinates. The circled red cross points are those
that did not pass the 1% strain limits criteria and the circled green smile points are those that did.
Table 4. Experimental results for 405 to 705oC temperature range
Nominal total load (lbs) Time delay (min) Ratcheting strain in
200hrs, %
Pass/fail 1% strain
limits
-550 3 0.20% Pass
-550 5 -8.79% Fail
-250 3 -0.07% Pass
-250 5 -0.47% Pass
-100 5 -0.19% Pass
-100 10 -2.31% Fail
-100 8 -1.15% Fail
50 10 -0.56% Pass
150 10 -0.08% Pass
300 10 0.47% Pass
450 10 5.19% Fail
450 7 4.18% Fail
450 5 0.29% Pass
Fig. 7. EPP Strain limits envelope and test data for 405 to 705oC temperature range (the green
symbol ☺ passed 1% strain limits, while the red symbol ⨂ did not pass 1% strain limits)
There are no measured strains greater than 1% that fall within the analytically determined strain limits
boundary determined from the EPP analysis. There are experimental points that fall outside the EPP
boundary, thus indicating their conservatism. But, generally, it is shown that for points farther outside the
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 8
boundary, the strain limits are not satisfied, thus indicating that the EPP boundary is not over-
conservative.
2.2.2 Strain range evaluation
As stated above, incorporation of the SMT data based approach for creep-fatigue damage evaluation into
the EPP methodology will avoid the use of the “D” diagram and minimize over-conservatism while
properly accounting for localized defects and stress risers. The plan is to use the strain range results from
the EPP strain limits procedure as input to the strain range based SMT design curve. The two-bar test
results can also be used to address the validity of this approach. The strain ranges of interest are those that
first satisfy the EPP strain limits code case. From Fig. 7 and Table 4 the applicable strain ranges are those
at a 3 min delay time at loads of -250 and 550 lb load and a 10 min delay at loads of 50, 150 and 300 lb.
Table 5 is a comparison of the experimentally measured strain range for these loading conditions to the
corresponding analytically determined strain range from the EPP strain limits evaluation.
Table 5. Experimental and analytical strain ranges
As seen from Table 5, for the three cases at 10 min delay, the analytic strain range is greater than the
experimental range for Bar 2, the thermal lagging bar with the largest strain range. The margin is
conservative but not excessive. Conversely, the experimental strain range in Bar 1, the thermal leading
bar with the lower strain range, is larger than the analytic strain range, which is unconservative.
For the two cases with lower strain ranges as a result of the lower delay time, the analytic strain range is
slightly unconservative in Bar 2 and more unconservative in Bar 1. Interestingly, the experimental data
show that Bar 2 had larger mechanical strain ranges while loaded at higher thermal stress with 10 min
time delay, it showed smaller strain ranges than Bar 1 at the lower loaded 3 min delay, Bar 1 experiences
the greater strain range than Bar 2. The agreement between experimental and analytical strain ranges is
not unreasonably conservative for the higher load cases and larger strain ranges, but non-conservatism at
lower loads and strain ranges warrants additional investigation
Nominal Total
Load (lbs)
Delay time,
(min)
Mechanical strain range, Bar
1 (%)
Mechanical strain range, Bar
2 (%)
Exp. Anal. Exp. Anal.
-250 3 0.17 0.096 0.11 0.099
550 3 0.17 0.098 0.11 0.099
50 10 0.32 0.183 0.41 0.490
150 10 0.30 0.183 0.41 0.490
300 10 0.30 0.184 0.42 0.493
August 2017
ANL-ART-96 9
3 Analytical based development The current analytical activities focused on the verification and implementation of the EPP code case
methodology. In one case, there were two activities that were related to the assessment of the
conservatism with respect to both a hypothetical inelastic evaluation of the two-bar test configuration and
actual test results from a prior ORNL nozzle-to-sphere test under sustained pressure loading. Those
results are more comprehensively documented in the enclosed Appendix A “Verification and validation
of the EPP strain limits and creep-fatigue through inelastic analysis and comparison to experimental
results.” The second case addressed a recurring problem with EPP analyses in general, how to define
shakedown. Those results are documented in Appendix B, “Establishing shakedown criteria for the EPP
strain limits code case.” Discussed below are the overall finding from these activities. The figures and
tables shown below are taken from the Appendices.
3.1 EPP code cases
The EPP code cases are verified by comparison to both full inelastic structural simulations and direct
experimental observations. Inelastic simulations describe the relevant deformation phenomenon in high
temperature reactor structural components: coupled, temperature and rate dependent creep and plastic
deformation. Direct EPP comparisons to such simulations have the advantages of eliminating
experimental error and material batch variation in both the test and in the ASME Division 5 isochronous
stress/strain curves used to select the EPP pseudo-yield stress. Therefore, a comparison to inelastic
analysis directly tests the EPP bounding theorem. However, ultimately the EPP method will be used in
conjunction with the isochronous stress/strain curves and the “D” diagrams in the Division 5. This section
and the corresponding appendix also compare the EPP method directly to experimental data from a
component test conducted at ORNL in the 1980s [8]. This comparison demonstrates the conservatism of
the EPP method for realistic component geometries and loading conditions.
3.1.1 Inelastic analysis simulations
This work uses a modified inelastic constitutive model based on Hyde et al. [9] to describe the
deformation of 316H stainless steel. The model follows a Chaboche form [10] and the authors of the
original work provided material constants at 300, 500, 550 and 600C. For this work the constants
describing the Chaboche backstress evolution were extended to 815C by extrapolation. The original
model does not creep sufficiently at high temperatures so dynamic recovery was added to the model to
capture high temperature creep [11]. The dynamic recovery parameters were tuned to approximately
match the 316H isochronous stress/strain curve in Division 5 at 200 hours life and 800C.
Consistent isochronous curves for the application of the EPP strain limits code case can be developed
from the model implementation by simulating a series of creep tests. First, the method requires simulating
creep tests at different stresses for each temperature of interest. An isochronous stress/strain curve is then
interpolated from this data by finding the locus of all (strain, stress) tuples at a given time t*. This curve is
the isochronous stress/strain curve for a design life of t*. The full set of consistent curves includes
temperatures between 300 and 825C at 25C intervals. Intermediate values are obtained by interpolating
between theses curves, if necessary. The complete series of isochronous stress/strain curves is the
database required to evaluate the EPP strain limits code case.
Two-bar tests are designed to mimic the loading conditions generated at the extreme fibers of a thin-
walled pressure vessel. This section considers two bar setups of the kind described above.
Simulations of this two-bar system were carried out using a custom material point computer code. A
material point integration of the model equations provides the stress and history update for each bar as a
function of strain, time, and temperature. For design life lifet for the full inelastic simulations the primary
load P is ramped up over a short period of time and then thermal cycles are simulated. At the end of n
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 10
cycles if the largest residual, inelastic strain in either of the two bars is less than 1% then the two-bar
system passes the Division 5 strain limits design check. This method can simulate thousands of two-bar
experiments in a short time.
For each inelastic simulation of a two-bar experiment a corresponding EPP calculation requires the
consistent isochronous stress/strain curve developed above for the inelastic material model. The values of
this curve at lifet and target inelastic strain, x , provide the pseudo-yield stress used in the elastic, perfectly
plastic analysis, iterating on the target strain as described in the code case. The EPP simulations use the
same material point method described for the inelastic simulations. The material point update function is
now temperature-dependent perfect plasticity with the yield stress set to the EPP pseudo yield stress. The
code case provides a procedure to determine if the system passes or fails the Division 5 strain limits
design criteria.
Table 6 describes loading parameters for two sets of two bar simulations. In this section a value of lifet of
200 hr was used to be consistent with a previous set of experimental results [12] that were extrapolated to
this design life.
Table 6. Loading parameters for the two bar simulations
Case 1 2
T (°C/s) 30 30
aT (°C) 515 415
bT (°C) 815 515
ht (min) 60 60
Fig. 8 and Fig. 9 summarize the results of a large series of viscoplastic and corresponding EPP design
calculations on the two bar systems. Subfigure (a) describes the results of the EPP method and
subfigure (b) the results of the full inelastic analysis. In these subfigures a blue region indicates the
system passed the relevant strain limits design check at the corresponding cooling delay and primary load
conditions. Subfigure (c) shows how conservative the EPP design check is relative to the full, inelastic
numerical experiment. This figure colors a region blue if the inelastic calculation passes the design check
and the EPP result fails the design check, red if the EPP check passes but the inelastic check fails, and
white if both checks fail or both checks pass. So red regions would indicate loading conditions for which
the EPP method is non-conservative. For these two bar simulations the EPP method is always
conservative, so no red regions appear in Fig. 8 or Fig. 9.
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Fig. 8. Thermal cycle, 515 to 815°C. (a) EPP results, (b) inelastic results, (c) EPP margin
(a) (b)
(c)
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Fig. 9. Thermal cycle, 415 to 515°C. (a) EPP results, (b) inelastic results, (c) EPP margin
The general trend for both the inelastic and EPP analysis is a triangle of low strain accumulation for
combinations of low primary load and low secondary load. The inelastic simulation results, and the
corresponding experiments, show a “stovepipe” of low strain accumulation extending from the apex of
this triangle to the right of the plots. For these consistent EPP calculations this stovepipe does not result in
non-conservatism – in contrast to the previous experimental results – because the EPP results either do
not have a stovepipe or have a stovepipe entirely contained inside the inelastic analysis stovepipe.
In the actual two-bar tests and in the corresponding inelastic analysis, when the hold time is at very high
temperature with maximum creep response, there is a diagonal stovepipe in the ratcheting behavior of the
two-bar system that results from the interplay of two interactive deformation mechanisms:
1. During the elevated temperature hold the two bars will creep in the direction of the applied load.
2. The thermal cycles cause ratcheting.
The amount of creep deformation is proportional to the applied primary load and the hold time and the
ratcheting is proportional to the temperature difference between the two bars, here controlled by the
(a) (b)
(c)
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ANL-ART-96 13
cooling delay. Therefore, the system experiences the least net ratcheting for low primary loads and/or low
delay times.
In these analytic simulations, the EPP strain limits evaluation did not permit the higher delay times
associated with the presence of a stovepipe. This is different than the EPP strain limits evaluation of
earlier, very high temperature two bar tests [13] which did permit high delay times with a resultant stove
pipe. However, in those cases the EPP determined stovepipe was vertical as opposed to the diagonally
skewed stovepipe observed in these analytic simulations and the experimental data. This difference,
skewed versus vertical stovepipe, results in regimes of unconservative predictions for the strain limits
code case as applied to very high temperature two-bar tests. (Note that subsequent evaluations of more
realistic distributed structures demonstrated the conservatism of the EPP strain limits procedure and
resulted in a restriction preventing the applicability of the strain limits code case to skeletal structures.)
Interestingly, when the hold time is at a lower temperature both the experimental and EPP strain limits
stovepipes are vertical as discussed earlier in this report; see Fig. 7, the EPP strain limits envelope and test
data for two-bar tests with a 405 to 705oC temperature range for 316H stainless steel. Additional work is
required to determine why the consistent simulations of two-bar tests and the validation simulations
comparing to full scale component tests cannot reproduce the non-conservatism found in the previous two
bar results at very high temperatures.
3.1.2 Component test comparison
The two-bar geometry is only an approximation to the relevant component geometry of a thin walled
pressure vessel. For example, it does not represent stress concentration caused by nozzles nor the follow
up caused by impinging piping systems. Fig. 10 shows the schematic diagram of a 304H stainless steel
representative component tested at ORNL over several years in the 1980s. This key feature test article
consists of a hemispherical pressure vessel connected to a flange, leading to a pipe. The vessel, flange,
and pipe were pressurized and the entire specimen heated in a furnace. A regulator controlled the pressure
in the system and a thermocouple feeding back to the furnace control set the temperature at the critical
section. This control scheme can impose to time-dependent thermal and pressure loading cycles on the
component.
Fig. 10. Schematic diagram of the axisymmetric specimen with key locations labeled
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
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During the test the circumferential strains along three rings of strain gauges, labeled in Fig. 10, were
monitored with a ring of strain gauges equally spaced around the flange fillet. Periodically the vessel was
unloaded and cooled so that a rubber cast of the critical section could be made. This rubber cast was then
examined with optical microscopy to check for the presence of voids or microcracks. Therefore, the
experiment also measured the growth of, or at least detected the development of, creep-fatigue damage.
The ORNL data determines when the structure first exceeded 1% circumferential strain – the strain
accumulation limit – and when creep-fatigue damage first became detectable – the condition the Division
5 creep-fatigue criteria guard against. Fig. 11 shows the relevant portion of the specimen loading history
imposed over the course of the experiment. The specimen was subject to combined pressure/temperature
cycles as the pressure was released and the specimen cooled to room temperature at the times indicated by
the diagram. Additionally, the figure shows when the specimen exceeded 1% circumferential strain at the
critical section and when damage was first observed with the rubber cast method.
The Division 5 code provides isochronous stress/strain curves for 304H stainless steel. The formulas for
these curves in the background document were used to set the EPP pseudo-yield stress, interpolating
between curves where required. Fig. 11 shows the composite loading cycle used in the analysis. Because
the actual specimen exceeded the Division 5 strain limits and developed creep-fatigue damage relatively
early in the loading history. This composite cycle is based on the first few experimental loading cycles.
Fig. 11. Combined figure showing the original experimental loading and key times in the EPP
analysis of the nozzle-to-sphere test article
Unlike actual plant components, experimental specimens of this kind do not have intended design lives.
Instead, an iterative procedure was used to generate EPP design lives – one each for strain limits and
creep fatigue – for the component geometry and loading history. Fig. 11 also shows the design lives
computed for strain accumulation and creep fatigue for the system using the EPP code case procedures.
Both the strain limits and creep-fatigue procedures return conservative bounds on the actual
experimentally measured lives. Therefore, this full validation test of the EPP methods shows that both
code cases are conservative for this particular geometry and set of loading conditions.
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3.2 Shakedown criteria
The following brief discussion and recommendation has been abstracted from the much more
comprehensive discussion in Appendix B.
The EPP strain limits code case requires an elastic perfectly-plastic analysis of the component using a
pseudo-yield stress selected by a procedure referencing the material’s isochronous stress/strain curves and
yield stress and loading set by a composite cycle incorporating the key features of all relevant design load
cases. The pass/fail EPP check has two components:
1. The system must shake down. For the strain limits code case plastic shakedown is acceptable.
2. Criteria on the accumulated inelastic strain before shakedown, designed to ensure the system will
pass the Division 5 strain limits requirements.
A key aspect of the EPP strain limits code is then establishing whether or not a particular analysis, likely
to be a numerical finite element (FE) analysis, shakes down. Establishing this behavior from numerical
results can be challenging, requiring a procedure or at least guidance for designers using the EPP method.
Problems develop because FE methods solve for the system response at discrete time steps using iterative
methods, usually Newton’s method. Because computers use limited machine precision, floating point
arithmetic there is a limit to how accurately Newton’s method, implemented on a computer, can solve a
system of nonlinear equations.
The consequence of this is some bound on the accuracy of an analysis invoking the nonlinear solver.
Newton iterations are only performed if the system response is nonlinear, i.e. if the steady state behavior
is plastic shakedown or ratcheting. Therefore, it can be difficult to distinguish these states using nonlinear
FE analysis.
Many designers may not be aware of the limitations of nonlinear finite element analysis as a tool for
analyzing cyclic plasticity and so the following warning has been proposed as an interim step for
incorporation into the strain limits Code Case:
“The strain limits EPP assessment requires the identification of non-ratcheting for an acceptable
load cycle.
Classification of an analysis as non-ratcheting requires that the deflections become cyclic. This
implies both the total strains and plastic strains also become cyclic. This steady state behavior may
develop after some initial number of load cycles that produce increasing deflections. History plots of
the deflections or strains may be used to identify a non-ratcheting response.
The numerical methods used in finite element analysis may produce noise in the deflection and strain
fields. This noise appears as small-magnitude, random variation about some constant average (non-
ratcheting) or non-constant but steadily increasing (ratcheting) response. This numerical noise
should be ignored when classifying a finite element analysis as ratcheting or non-ratcheting.”
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4 Summary The development of the integrated EPP combined SMT creep-fatigue damage evaluation approach is
reported. The goal of the proposed approach is to combine the advantage of the EPP strain limits
methodology that avoids stress classification with the advantage of the SMT method for evaluating creep-
fatigue damage without deconstructing the cyclic history into separate fatigue and creep damage
evaluations.
Two-bar thermal ratcheting tests were conducted on 316H stainless steel specimens with an applied
temperature transient from 405 to 705oC at several constant applied combined total loads. These test data
points were then evaluated analytically using the EPP strain limits procedures and compared to the 1%
EPP strain limits boundary. There were no measured strains greater than 1% that fell within the
analytically determined strain limits boundary. Although there were experimental points that fell outside
the EPP boundary, generally, it was shown that the EPP boundary was not over-conservative. Regarding
the strain range assessment, the agreement between experimental and analytical strain ranges is not
unreasonably conservative for the higher load cases and larger strain ranges, but non-conservatism at
lower loads and strain ranges warrants additional investigation.
Pressurized SMT specimens on Alloy 617 were used to assess the role of sustained primary loading on
cyclic life. Although high pressures reduced cyclic life, it was shown that such high pressure levels would
not be permissible under normal design limitations on pressure as a function of service life.
The viscoplastic material model used in an analytic simulation of two-bar tests compared favorably with
EPP strain limits assessments using consistent isochronous stress strain curves. The general trend for both
inelastic and EPP analysis is a triangle of low strain accumulation for combinations of low primary load
and low secondary load. The inelastic simulation results and the corresponding experiments show a
“stovepipe” of low strain accumulation extending from the apex of this triangle to the right of the plots.
For these consistent EPP calculations this stovepipe does not result in non-conservatism – in contrast to
the previous experimental results – because the EPP results either do not have a stovepipe or have a
stovepipe entirely contained inside the inelastic analysis stovepipe. Additional work is required to
determine why the consistent simulations of two-bar tests and the validation simulations comparing to full
scale component tests do not reproduce the non-conservatism found in the previous two bar results at very
high temperatures.
The FEA model of a prior 304H stainless steel ORNL nozzle-to-sphere test was used for an EPP strain
limits and creep-fatigue code case damage evaluation. Both the creep-fatigue and strain limits procedures
returned conservative bounds on the actual experimentally measured lives. Therefore, this full validation
test of the EPP methods shows that both code cases are conservative for this particular geometry and set
of loading conditions.
The theoretical treatment of convergence criteria for plastic shakedown illustrated the role of machine
precision. Problems occur because FE methods use iterative methods to solve the global nonlinear force
balance equations and the floating point arithmetic used is not exact. Therefore, numerical solutions may
exhibit small amounts of fictitious ratcheting strain, even if an analytical solution would shake down
exactly. A warning has been proposed as an interim step for incorporation into the strain limits Code
Case.
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REFERENCES
1. ASME B&PV Code Case N-861 “Satisfaction of Strain Limits for Division 5 Class A Components at
Elevated Temperature Service Using Elastic-Perfectly Plastic Analysis.”
2. ASME B&PV Code Case N-862 “Calculation of Creep-Fatigue for Division 5 Class A Components
at Elevated Temperature Service Using Elastic-Perfectly Plastic Analysis Section III, Division 5.”
3. R.I. Jetter, (1998), “An Alternate Approach to Evaluation of Creep-Fatigue Damage for High
Temperature Structural Design Criteria,” PVP-Vol. 5 Fatigue, Fracture and High Temperature Design
Methods in Pressure Vessel and Piping, Book No. H01146 – 1998, American Society of Mechanical
Engineers Press, New York, NY.
4. Y. Wang, R.I. Jetter, S.T. Baird, C. Pu, and T.-L. Sham, (2015), “Simplified Model Test (SMT)
creep-fatigue testing,” ORNL/TM-2015/300, Oak Ridge National Laboratory, Oak Ridge, TN
5. Y. Wang, R.I. Jetter, and T.-L. Sham, (2016), “Preliminary Test Results in Support of Integrated EPP
and SMT Design Methods Development,” ORNL/TM-2016/76, Oak Ridge National Laboratory, Oak
Ridge, TN.
6. Y. Wang, R.I. Jetter, M.C. Messner, S. Mohanty, and T.-L. Sham, (2017), “Combined Load and
Displacement Controlled Testing to Support Development of Simplified Component Design Rules for
Elevated Temperature Service,” Proceedings of the ASME 2017 Pressure Vessels and Piping
Conference, 2017, PVP2017-65455, pp. 1-6.
7. Y. Wang, R.I. Jetter, and T.-L. Sham, (2017), “Report on FY17 Testing in Support of Integrated EPP-
SMT Design Methods Development,” ORNL/TM-2017/351, Oak Ridge National Laboratory, Oak
Ridge, TN.
8. J.M. Corum and R. L. Battiste,(1993), “Predictability of Long-Term Creep and Rupture in a Nozzle-
to-Sphere Vessel Model,” Journal of Pressure Vessel Technology, vol. 115, pp. 122-127.
9. C. J. Hyde, W. Sun, and S. B. Leen, (2010), “Cyclic thermo-mechanical material modelling and
testing of 316 stainless steel,” Int. J. Press. Vessel. Pip., vol. 87, no. 6, pp. 365–372.
10. J. L. Chaboche, (2008), “A review of some plasticity and viscoplasticity constitutive theories,” Int. J.
Plast., vol. 24, no. 10, pp. 1642–1693.
11. J. L. Chaboche, (1989), “Constitutive equations for cyclic plasticty and cyclic viscoplasticity,” Int. J.
Plast., vol. 5, pp. 247–302.
12. T.-L. Sham, R. I. Jetter, and Y. Wang, (2016), “Elevated temperature cyclic service evaluation based
on elastic-perfectly plastic analysis and integrated creep-fatigue damage,” in Proceedings of the
ASME 2016 Pressure Vessels and Piping Conference, 2016, PVP2016-63730, pp. 1–10.
13. Y. Wang, R.I. Jetter, S.T. Baird, C. Pu, and T.-L. Sham, (2015), “Report on FY15 Two-Bar Thermal
Ratcheting Test Results”, ORNL/TM-2015/284, Oak Ridge National Laboratory, Oak Ridge, TN
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APPENDIX A
Verification and validation of the EPP strain limits and creep fatigue code cases through inelastic
analysis and comparison to component tests
Introduction
Two code cases [A.1], [A.2] establish the elastic perfectly-plastic (EPP) methodology for checking
designs against the ASME Section III, Division 5 criteria for strain limits and creep-fatigue damage in
high temperature reactor structures. These code cases rely on a theorem developed by Carter [A.3]–[A.7]
from previous work by Ainsworth [A.8], Frederick and Armstrong [A.9], and others that bounds
deformation and creep dissipation – and hence creep-fatigue damage – in a creeping structure with a
simplified, elastic-perfectly plastic analysis using a selected pseudo-yield stress. These EPP methods have
advantages over the Division 5 simplified methods that rely on elastic analysis: they do not require stress
classification and they extend to the regime where creep and plasticity are coupled deformation
mechanisms. However, recent work [A.10] identifies potential non-conservatism when the strain limits
code case is applied to two bar systems designed to represent thin walled pressure vessels under constant
primary pressure load superimposed with a cyclic, through-wall temperature gradient.
This section verifies the EPP code cases by comparison to both full inelastic structural simulations and
direct experimental observations. Inelastic simulations describe the relevant deformation phenomenon in
high temperature reactor structural components: coupled, temperature and rate dependent creep and
plastic deformation. Direct EPP comparisons to such simulations have the advantages of eliminating
experimental error and material batch variation in both the test and in the Division 5 isochronous
stress/strain curves, used in conjunction with the Code tensile yield strength to select the EPP pseudo-
yield stress. Therefore, a comparison to inelastic analysis directly validates the EPP bounding theorem.
However, ultimately the EPP method will be used in conjunction with the isochronous stress/strain curves
and creep-fatigue “D” diagrams in Division 5. This section also compares the EPP method directly to
experimental data from a component test conducted at Oak Ridge National Laboratory (ORNL) in the
1980s [A.11]. This comparison demonstrates the conservatism of the EPP method for realistic component
geometries and loading conditions.
These verification tests demonstrate the conservatism of the EPP design method. They supplement
previous computational verification tests [A.12] and validation experiments that show the EPP checks are
adequately conservative tools, suitable for checking reactors designs against the Section III, Division 5
design criteria. This section discusses potential reasons why, despite the numerous conservatism
demonstrated by the validation checks described here, sets of high temperature, two-bar experimental
tests show non-conservative behavior compared to corresponding EPP analysis. Finally, the overall
results of this validation effort are summarized.
Inelastic models and consistent isochronous curves
This work uses a modified inelastic constitutive model based on Hyde et al. [A.13] to describe the
deformation of 316H stainless steel. The model follows a Chaboche form [A.14] and the authors of the
original work provided material constants at 300, 500, 550, and 600C. For this work the constants
describing the Chaboche backstress evolution were extended to 815C by extrapolation. The original
model does not creep sufficiently at high temperatures so dynamic recovery was added to the model to
capture high temperature creep [A.15]. The dynamic recovery parameters were tuned to approximately
match the 316H isochronous stress/strain curve described in the ASME Code at 200 hours life and 800C.
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The equations:
: ( ) e th pC (A.1)
th TI (A.2)
p s Xp
s X (A.3)
2 / 3 2 / 33
2 2 / 3
n
s X R kp (A.4)
12 2 3
3 3 2
ia
i i i i i i i
s XX C X p A X X
s X (A.5)
( ) R b Q R p R (A.6)
describe the rate form of the full model. In these expressions is the stress rate tensor, is the total
strain rate tensor, th is the thermal strain rate tensor, p is the plastic strain rate tensor, eC is an
isotropic fourth-rank elasticity tensor parameterized by Young’s modulus E and Poisson’s ratio , is
the thermal expansion coefficient, T the current temperature rate, s the stress deviatoric tensor, X the
deviatoric composite backstress tensor, k the initial yield stress, R the isotropic hardening stress, and ,
n , iC , i
, iA ,
ia , b and Q are all model parameters. In this derivation the norm y indicates the 2-
norm, not the 2J metric, defined as :y y . This difference requires the various square root factors in
front of the constants to maintain agreement with [A.13] and [A.14]. Table A 1 lists the values of these
parameters at 300, 500, 550, 600, and 800C. The final inelastic model is an implicit, backward Euler
integration of these rate equations.
Consistent isochronous curves can be developed from the inelastic model implementation by simulating a
series of creep tests. First, the method requires simulating creep tests at different stresses for each
temperature of interest. An isochronous stress/strain curve is then interpolated from these data by finding
the locus of all (strain, stress) tuples at a given time t . This curve is the isochronous curve for a design
life of t . Fig. A 1 plots the isochronous curves generated for this inelastic model, using this procedure at
500C as an example. The full set of consistent curves includes temperatures between 300 and 825C at
25C intervals. Data are interpolated between theses curves, if necessary.
Two-bar verification simulations
The complete series of isochronous stress/strain curves is the database required to evaluate the EPP strain
limits code case. This section compares a large series of inelastic simulations of two-bar tests to the EPP
strain limits code case.
Two-bar tests are designed to mimic the loading conditions generated at the extreme fibers of a thin-
walled pressure vessel (see Fig. A 2). A traditional experiment applies some constant, primary load to the
coupled two-bar system and then cycles the temperature of one of the two bars to impose an alternating,
secondary thermal load on the system. A variant used here (Fig. A 3) instead heats and cools both bars but
delays the cooling of one bar relative to the other to cause cyclic, secondary, thermal strain.
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Simulations of this two-bar system were carried out using a custom material point computer code.
Consider a general constitutive model and thermo-mechanical responses at two consecutive time steps n
and 1n . Let be a function, developed based on an implicit time integration of the constitutive model,
that returns stresses 1n , history variables 1nh , and tangent matrix 1nM at the 1n step; using as input
temperature nT , stresses n , strains n and history variables nh at time step n , and strains 1n and
temperature 1nT at step 1n .
This update can be symbolically represented as
1 1 1 1 1, , , ,, ,,n n n n n nn n nh T h TM (A.7)
An implicit integration of equations (A.1) to (A.6) above forms a suitable function but the two-bar
simulation program is general. Any constitutive model written in the form described by (A.7) can be used,
including the elastic perfectly-plastic model used in the EPP calculations.
To reduce general, tensor constitutive equations to the uniaxial two-bar system requires the introduction
of a vector notation for symmetric tensors. Here define the components of a strain tensor as
(1) (2) (3) (4) (5) (6) (A.8)
and similarly for stress tensor. Let the response of bar #1 be indicated by subscript 1 and bar #2 by
subscript 2. The following equations are written for the 1n time step and the subscript indicating the
step is dropped. The following notation does not explicitly indicate it but the history vectors must be
transferred through each model from time step to time step.
The two-bar system solves the equilibrium equation:
(1) (1)
1 1 2 2 A A P (A.9)
the uniaxial condition:
( ) ( )
1 2 0, 2,3,4,5,6 i i i (A.10)
and the displacement compatibility condition for equal-length bars
(1) (1)
1 2 (A.11)
in terms of the unknown strain components. The bars are assumed to have different cross sectional areas
1A and 1A . P gives the total load on the two-bar system. Further loading is introduced due to the
constraint condition in equation (A.11) if the temperatures of the two bars differ, as is the case in the
current two-bar simulations. This system represents a nonlinear system of 12 unknowns and 12 equations
in the 1n time step.
Consider the residual vector
(2) (3) (4) (5) (6) (2) (3) (4) (5) (6)
1 1 1 1 1
(1) (1) (1) (1)
1 1 2 2 2 2 2 2 12 2, , , , , , , , , , , R A A P (A.12)
Reducing this residual to zero satisfies equations (A.9), (A.10), and (A.11) simultaneously. Let the vector
of unknowns be
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(2) (3) (4) (5) (6) (1(1)
2
) (2) (3) (4) (5) (6)
2 21 1 1 1 2 21 21, , , , , , , , , , , x (A.13)
Solving this system of nonlinear equations via Newton’s method requires the Jacobian:
R
x (A.14)
which is a matrix rearrangement of components of the algorithmic tangents returned by the function
describing the two bars. In the final implementation one equation is condensed out from the system, using
the constraint equation (A.11), leaving 11 equations and unknowns.
Once the two-bar areas and material models are defined the primary load P and the temperatures of each
bar, 1T and
2T control the system. Combinations of temperature and load are applied slightly differently
in the full inelastic and EPP calculations.
For design life lifet for the full inelastic simulations the primary load P is ramped up over a short period
of time and then / lifen t thermal cycles are simulated. At the end of n cycles consider the residual
strains in the two bars 1 and
2 ordered so that 1 2| | | | . If
1| | 1% then the two-bar system
passes the Division 5 strain limits design check. This method can simulate thousands of two-bar
experiments in a short time.
For each inelastic simulation of a two-bar experiment a corresponding EPP calculation requires the
consistent isochronous stress/strain curve developed above for the inelastic material model. The smaller
of the Code value of the material yield stress and the value of this curve at lifet and target inelastic strain
x provide the pseudo-yield stress used in the EPP analysis, iterating on the target strain as described in
the code case. The EPP simulations use the same material point method described for the inelastic
simulations. The material point update function is now temperature-dependent perfect plasticity with the
yield stress set to the EPP pseudo yield stress. The code case provides a procedure to determine if the
system passes or fails the Division 5 strain limits design criteria. Note the EPP method does not require
the full / lifen t cycles to be simulated. Here the definition of the EPP composite cycle is identical to
the thermal cycle shown in Fig. A 3, but the elastic-perfectly plastic solution establishes steady state
behavior in fewer than 10 cycles. Therefore, the EPP simulations only repeat the composite cycle 10
times.
Table A 2 describes loading parameters for two sets of two-bar simulations. In this section a life of
200 hrlifet was used to be consistent with a previous set of experimental results [10] that were
extrapolated to this design life.
Fig. A 4 and Fig. A 5 summarize the results of a large series of viscoplastic and corresponding EPP
design calculations on the two-bar systems. Subfigure a) describes the results of the EPP method and
subfigure b) the results of the full inelastic analysis. In these subfigures a blue region indicates the system
passed the relevant strain limits design check at the corresponding cooling delay and primary load
conditions. Subfigure c) shows how conservative the EPP design check is relative to the full, inelastic
numerical experiment. This figure colors a region blue if the inelastic calculation passes the design check
and the EPP result fails the design check, red if the EPP check passes but the inelastic check fails, and
white if both checks fail or both checks pass. So red regions would indicate loading conditions for which
the EPP method is non-conservative. For these two-bar simulations the EPP method is always
conservative, so no red regions appear in Fig. A 4 or Fig. A 5.
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The general trend for both the inelastic and EPP analysis is a triangle of low strain accumulation for
combinations of low primary load and low secondary load. The inelastic simulation results, and the
corresponding experiments, show a “stovepipe” of low strain accumulation extending from the apex of
this triangle to the right of the plots. This stovepipe is discussed below. Note that for these consistent EPP
calculations this stovepipe does not result in non-conservatism – in contrast to the previous experimental
results – because the EPP results either do not have a stovepipe or have a stovepipe entirely contained
inside the inelastic analysis stovepipe.
ORNL component test validation simulations
Each two-bar simulation is computationally inexpensive, allowing for direct comparisons between
inelastic and EPP analysis of the same geometry for a large range of representative loading conditions.
However, the two-bar geometry is only an approximation to the relevant component geometry of a thin-
walled pressure vessel. Furthermore, a thin-walled pressure vessel under constant primary and alternating
secondary load does not represent the full range of plant component geometries. For example, it does not
represent stress concentration caused by nozzles nor the follow up caused by impinging piping systems.
Fig. A 6 shows the schematic diagram of a 304H stainless steel representative component tested at ORNL
over several years in the 1980s. This test article consists of a hemispherical pressure vessel connected to a
flange, leading to a pipe. The vessel, flange, and pipe could be pressurized and the entire test article
heated in a furnace. A regulator controlled the pressure in the system and a thermocouple feeding back to
the furnace control set the temperature at the critical section. This control scheme can impose time-
dependent thermal and pressure loading cycles on the component.
During the test the circumferential strains along three rings of strain gauges, labeled in Fig. A 6, were
monitored with a ring of strain gauges equally spaced around the flange fillet. Additionally, periodically
the vessel was unloaded and cooled so that a rubber cast of the critical section could be made. This rubber
cast was then examined with optical microscopy to check for the presence of voids or microcracks.
Therefore, the experiment also measured the growth of, or at least detected the development of, creep-
fatigue damage.
The ORNL test protocol provides sufficient information to test the EPP strain limits and creep-fatigue
code cases. The ORNL data determine when the structure first exceeded 1% circumferential strain – the
strain limit – and when creep-fatigue damage first became detectable – the condition the Division 5 creep-
fatigue criteria guard against. Fig. A 7 shows the relevant portion of the specimen loading history
imposed over the course of the experiment. The test article was subject to combined pressure/temperature
cycles as the pressure was released and the test article cooled to room temperature at the times indicated
by the diagram. Additionally, the figure shows when the test article exceeded 1% circumferential strain at
the critical section and when damage was first observed with the rubber cast method.
An EPP analysis of this test requires:
1. Temperature dependent isochronous curves and the Code material yield strengths to set the EPP
pseudo yield stress;
2. A composite loading cycle representative of the actual loading conditions, imposed via appropriate
boundary conditions on the finite element model;
3. A design life to select the pseudo-yield stress from the isochronous curves;
4. An elastic perfectly-plastic finite element analysis of the component geometry.
Addressing each requirement in turn:
1. Division 5 provides isochronous stress/strain curves and the material yield strengths for 304
stainless steel. The formulas given in the background document for the construction of these
isochronous stress/strain curves were recovered for use in this analyses.
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2. Fig. A 8 shows the finite element model used to represent the component geometry. The model
does not resolve details of the vessel connection to the supporting skirt or the details of the pipe
connection to the pressure system. The specimen was designed so that the critical stresses and
strains would develop at the nozzle fillet, far away from both the connections. Fig. A 9 shows the
results of an elastic analysis of the system, showing that the highest elastic stresses and strains do
develop at this critical location. Therefore, the analysis neglects the details of the connections.
3. Fig. A 10 shows the composite loading cycle used in the analysis. Because the actual test article
exceeded the Division 5 strain limit and developed creep-fatigue damage relatively early in the
loading history this composite cycle is based on the first few experimental loading cycles.
4. Unlike actual plant components, experimental test articles of this kind do not have intended design
lives. Instead, an iterative procedure was used to generate EPP design lives – one each for strain
limits and creep fatigue – for the component geometry and loading history. The EPP code cases are
setup as pass/fail checks: given a design life and the analysis resulting from the code case procedure
the code cases will either indicate the system passes or fails the strain limits or creep-fatigue design
criteria. The iterative procedure then seeks the longest design life such that the relevant EPP code
case procedure passes. This design life might reasonably be called the EPP design life of the system
and can be compared to the experimentally observed life.
Fig. A 7 shows the design lives computed for strain limits and creep fatigue for the system using the EPP
code case procedures. Both the strain limits and creep-fatigue procedures return conservative bounds on
the actual experimentally measured lives. Therefore, this full validation test of the EPP methods shows
that both code cases are conservative for this particular geometry and set of loading conditions.
Discussion
Conservatism of the EPP method
The two-bar verification simulations and the validation comparison to the ORNL nozzle-to-sphere
experiment both demonstrate the conservatism of the EPP methods. In both cases the method tends to be
very conservative, as illustrated by the EPP margin shown in Fig. A 4 and Fig. A 5 and by the difference
between the experimental and EPP design lives shown in Fig. A 7. This conservatism reflects the purpose
of the EPP methods as fast screening criteria – failing an EPP check does not mean a design will not
ultimately be safe but rather that further analysis is required.
Stovepipe behavior in two-bar tests
In the actual material and in the corresponding inelastic analysis a diagonal stovepipe in the ratcheting
behavior of the two-bar system results from the interplay of two, potentially offsetting deformation
mechanisms:
1. During the elevated temperature hold the two bars will creep in the direction of the applied load.
2. The thermal cycles cause ratcheting. The loading described in Fig. A 3 causes compressive
ratcheting.
The amount of creep deformation is proportional to the applied primary load and the hold time and the
ratcheting is proportional to the temperature difference between the two bars, here controlled by the
cooling delay. Therefore, the system experiences the least net ratcheting for low primary loads and/or low
delay times – explaining the triangular-shaped region in Fig. A 4. The two mechanisms can also combine,
leading to additional ratcheting and causing the triangular region to offset slightly towards the
compressive side of the load axes, where the two mechanisms stack. There is also a region where the two
mechanisms exactly offset – compressive ratcheting cause by the thermal strain and tensile strain caused
by creep cancel. This is the offset “stovepipe” behavior illustrated by Fig. A 4.
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As shown by Fig. A 5, EPP analysis can also produce a stovepipe. A similar mechanism is at play,
however creep does not explicitly appear in EPP analysis. One competing mechanism remains ratcheting
caused by the cyclic thermal strain. We might envision this as some induced thermal stress causing plastic
deformation to accumulate during each cycle. The competing mechanism in the EPP analysis is the
applied primary load. If this applied primary load offsets the thermal stress so that neither bar
accumulates additional plastic deformation, then the system will shake down. If it does not offset the
ratcheting mechanism than the EPP method will indicate the system does not pass the design check, as the
procedure disallows ratcheting.
Previous analysis of two-bar tests using the Division 5 isochronous curves show a non-conservative
stovepipe: the experimental data has a “diagonal” stovepipe like the behavior shown in Fig. A 4 but the
corresponding EPP analysis has a “vertical” stovepipe of the type shown in Fig. A 5. At higher delay
times/thermal stresses these stovepipes do not intersect, leading to a region of non-conservatism. The
current verification simulations do not show this behavior. A critical question is then why the inelastic
simulations/EPP calculations with consistent isochronous curves cannot reproduce this behavior.
Possibilities fall into three categories: problems with the original experimental procedures, problems with
the code isochronous curves, or problems with the inelastic constitutive model that forms the basis of the
work discussed in this section.
One possibility is a problem with the extrapolation procedure used to extend the experimental results to
200 hours of life. This requires the system to achieve some steady-state cyclic behavior from which a
steady ratcheting rate can be determined. If a steady-state ratcheting rate is reached, then the total strain
accumulation after 200 hours will be the ratcheting rate per cycle times 200 hours divided by the cycle
period. Fig. A 11 shows simulations results illustrating a potential problem with this procedure. This
simulation of a two-bar test seems to be stabilizing after 20 cycles but in fact does not actually stabilize
for more than 140 cycles. Furthermore, the apparent steady ratcheting rate after 20 cycles has the opposite
sign as the true steady ratcheting rate. If the initial, compressive ratcheting rate was misidentified as the
steady-state response of the system the extrapolated results would be very different than the true system
response after 200 hours. Additionally, the experimental procedure did not use fresh samples for each
loading condition. Instead the same two samples were used for many combinations of primary load and
delay time. The samples therefore accumulated substantial prior history before the later loading
conditions were tested. This sample history may have substantially altered the creep response of the
material, leading to substantially different effective isochronous curves. This would affect the EPP
analysis, possibly leading to the observed inconsistency when comparing the experimental results to the
EPP analysis.
Another possibility is inaccuracy in the code isochronous curves at high temperatures or substantial batch
variation causing the material used in the experiments to vary from the code reference isochronous
curves. The Division 5 isochronous curves are intended to represent material average response from a
wide variety of sources of experimental data, material heats and product forms. However, heat variation
can be considerable and the higher temperature isochronous curves in Division 5 are currently less used in
engineering practice than the lower temperature curves and so inaccuracies may exist.
Additional work is required to determine why the consistent simulations of two-bar tests and the
validation simulations comparing to full scale component tests cannot reproduce the non-conservatism
found in the previous two bar results.
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Conclusions
These verification and validation simulations illustrate the conservatism of the EPP strain limits and creep
fatigue methodology. A very large series of numerical two-bar experiments demonstrates that the EPP
strain limits code case conservatively bounds creep and ratcheting deformation in a realistic simplified
test article when using consistent isochronous curves derived directly from the reference inelastic model
for 316H stainless steel. A large-scale validation test compares the EPP strain limits and creep-fatigue
code cases to a scaled nozzle-to-sphere test article tested at ORNL. Again, both EPP methods are
conservative for this geometry and set of loading conditions. Overall then, this set of simulations provides
additional confidence that the EPP methodology can be successfully used as a screening tool to test
designs for compliance against the ASME, Section III, Division 5 strain limits and creep-fatigue criteria.
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References
A.1. American Society of Mechanical Engineers, “Case N-861: Satisfaction of Strain Limits for
Division 5 Class A Components at Elevated Temperature Service Using Elastic-Perfectly Plastic
Analysis,” in ASME Boiler and Pressure Vessel Code, Nuclear Component Code Cases, 2015.
A.2. American Society of Mechanical Engineers, “Case N-862: Calculation of Creep-Fatigue for
Division 5 Class A Components at Elevated Temperature Service Using Elastic-Perfectly Plastic
Analysis,” in ASME Boiler and Pressure Vessel Code, Nuclear Component Code Cases, 2015.
A.3. P. Carter, “Analysis of cyclic creep and rupture. Part 1: Bounding theorems and cyclic reference
stresses,” Int. J. Press. Vessel. Pip., vol. 82, no. 1, pp. 15–26, 2005.
A.4. P. Carter, “Analysis of cyclic creep and rupture. Part 2: Calculation of cyclic reference stresses
and ratcheting interaction diagrams,” Int. J. Press. Vessel. Pip., vol. 82, no. 1, pp. 27–33, 2005.
A.5. P. Carter, R. I. Jetter, and T.-L. Sham, “Application of shakedown analysis to evaluation of creep-
fatigue limits,” in Proceedings of the ASME 2012 Pressure Vessel & Piping Division Conference,
2012, PVP2012-78083, American Society of Mechanical Engineers, New York, NY, pp. 1–10.
A.6. P. Carter, T.-L. Sham, and R. I. Jetter, “Simplified analysis methods for primary load designs at
elevated temperatures,” in Proceedings of the ASME 2011 Pressure Vessel & Piping Division
Conference, 2011, PVP2011-57074, American Society of Mechanical Engineers, New York, NY,
pp. 1-12.
A.7. P. Carter, T.-L. Sham, and R. I. Jetter, “Elevated temperature primary load design method using
pseudo-elastic perfectly plastic model,” in Proceedings of the ASME 2012 Pressure Vessels &
Piping Division Conference, 2012, PVP2012-78081, American Society of Mechanical Engineers,
New York, NY, pp. 1-10.
A.8. R. A. Ainsworth, “A note on bounding solutions for creeping structures subjected to load
variations above the shakedown limit,” Int. J. Solids Struct., vol. 15, no. 12, pp. 981–986, 1979.
A.9. C. O. Frederick and P. J. Armstrong, “Convergent internal stresses and steady cyclic states of
stress,” J. Strain Anal. Eng. Des., vol. 1, no. 2, pp. 154–159, 1966.
A.10. T.-L. Sham, R. I. Jetter, and Y. Wang, “Elevated temperature cyclic service evaluation based on
elastic-perfectly plastic analysis and integrated creep-fatigue damage,” in Proceedings of the
ASME 2016 Pressure Vessels and Piping Conference, 2016, PVP2016-63730, American Society
of Mechanical Engineers, New York, NY, pp. 1–10.
A.11. J.M. Corum and R. L. Battiste, “Predictability of Long-Term Creep and Rupture in a Nozzle-to-
Sphere Vessel Model,” Journal of Pressure Vessel Technology, vol. 115, pp. 122-127, 1993.
A.12. P. Carter, R.I. Jetter, and T.-L. Sham, “Verification of Elastic-Perfectly Plastic Methods for
Evaluation of Strain Limits - Analytical Comparisons,” in Proceedings of the ASME 2014
Pressure Vessels and Piping Division Conference, 2014, PVP2014-28352, American Society of
Mechanical Engineers, New York, NY, pp. 1-8.
A.13. C. J. Hyde, W. Sun, and S. B. Leen, “Cyclic thermo-mechanical material modelling and testing of
316 stainless steel,” Int. J. Press. Vessel. Pip., vol. 87, no. 6, pp. 365–372, 2010.
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
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ANL-ART-96 28
A.14. J. L. Chaboche, “A review of some plasticity and viscoplasticity constitutive theories,” Int. J.
Plast., vol. 24, no. 10, pp. 1642–1693, Oct. 2008.
A.15. J. L. Chaboche, “Constitutive equations for cyclic plasticty and cyclic viscoplasticity,” Int. J.
Plast., vol. 5, pp. 247–302, 1989.
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Table A 1. Parameters for inelastic constitutive model
Parameter 300C 500C 550C 600C 800C
E (MPa) 155000 144500 142000 139000 129000 0.27 0.27 0.27 0.27 0.27
(1/°C) 1.91e-5 2.02e-5 2.06e-5 2.11e-5 1.97e-5
k (MPa) 39.0 32.4 31.1 30.0 26.6
(MPa) 179 175 173 170 142
n 10.0 10.0 10.0 10.0 10.0
1C (MPa) 710000 617000 594000 571000 478000
1 5900 6700 6800 7000 7800
1A 1.0e-8 5.0e-5 0.002 0.008 0.012
1a 1 1 1 1 1
2C (MPa) 109000 111000 110000 108000 92000
2 1000 980 960 930 750
2A 0 0 0 0 0
2a 0 0 0 0 0
b 40 33 31 29 16
Q (MPa) 33 29 28 28 24
Table A 2. Loading parameters for the two sets of two-bar simulations
Case 1 2
T (°C/s) 30 30
aT (°C) 515 415
bT (°C) 815 515
ht (min) 60 60
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Fig. A 1. Example consistent isochronous curves for 500°C
Fig. A 2. Interpretation of a two-bar experiment as probing the response of the extreme fibers of a
thin-walled pressure vessel under constant pressure and cyclic thermal load
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Fig. A 3. Thermal cycle used for the two-bar simulations. The delay on the cooling end of the cycle
induces thermal strain in the two-bar system
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Fig. A 4. Thermal cycle, 515 to 815°C. a) EPP results, b) inelastic results, c) EPP margin
(a) (b)
(c)
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Fig. A 5. Thermal cycle, 415 to 515°C. (a) EPP results, (b) inelastic results, (c) EPP margin
(a) (b)
(c)
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Fig. A 6. Schematic diagram of the axisymmetric nozzle-to-sphere test article with key locations
labeled
Fig. A 7. Combined figure showing the original experimental loading and key times in the EPP
analysis of the nozzle-to-sphere test article
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Fig. A 8. Finite element mesh used to simulate the response of the ORNL test article
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Fig. A 9. Elastic stress analysis of the specimen, figure zoomed into the critical section
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Fig. A 10. Composite cycle used in the EPP analysis of the ORNL test
Fig. A 11. Simulation result showing reversing ratcheting. Initially the two-bar system seems to be
approaching some saturated, compressive ratcheting rate only for the system to reverse the
direction of ratcheting
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APPENDIX B:
Establishing shakedown criteria for the EPP strain limits code case
Introduction
The elastic perfectly-plastic (EPP) strain limits code case [B.1] provides an alternate procedure for
checking a design against the ASME Section III, Division 5 strain limits criteria. Compared to the
traditional approach relying on elastic analysis the method has several advantages:
1. It does not require stress classification and so is more amenable to finite element (FE) analysis.
2. The method remains applicable even at high temperatures where creep and plasticity are coupled.
3. The code case is written as a simple pass/fail check so that designers can quickly evaluate design
using the method as a screening test.
Previous work establishes the theory behind the method and validates it against both numerical and actual
experiments [B.2]–[B.6], demonstrating its utility as a conservative design tool.
The EPP strain limits code case requires an elastic perfectly-plastic analysis of the component using a
pseudo-yield stress selected by a procedure referencing the Division 5 isochronous stress/strain curves
and yield strengths for 304H and 316H stainless steels, and loading set by a composite cycle
incorporating the key features of all relevant design load cases. The pass/fail EPP check has two
components:
1. The system must shake down. For the strain limits code case plastic shakedown is acceptable.
2. Criteria on the accumulated inelastic strain before shakedown, designed to ensure the system will
pass the Division 5 strain limits criteria.
A key aspect of the EPP strain limits code is then establishing whether or not a particular analysis, likely
to be a numerical finite element analysis, shakes down. As described below, establishing this behavior
from numerical results can be challenging, requiring a procedure or at least guidance for designers using
the EPP method.
This section first describes the analytic cyclic behavior of elastic perfectly-plastic structures. Then the
results of example finite element analyses are discussed to show the behavior of structural systems
undergoing periodic load when analyzed with finite element methods. These results show that nonlinear
FE analyses do not exactly shake down. This section then describes two methods for determining plastic
shakedown from numerical analysis results: one graphical and one numeric. A proposed modification to
the EPP strain limits code case is discussed. Finally, the reasons why FE analysis fails to perfectly shake
down are discussed and the overall conclusions of this section are summarized.
Steady state cyclic behavior
Consider some structure with homogeneous displacement boundary conditions undergoing cyclic
tractions and temperatures with period As first demonstrated by Frederick and Armstrong [B.7] under
these conditions and provided the structure does not collapse it will eventually come to some steady state
where the stresses , strain rates , and plastic strain rates p all become periodic with period .
Classically, this steady state behavior is divided into four categories:
1. Elastic response: the structure never deforms plastically.
2. Elastic shakedown: the steady state strain field is periodic and, after some initial plasticity, the
steady state response is purely elastic.
3. Plastic shakedown: the steady state strain field is periodic and the steady state response involves
plastic deformation.
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4. Ratcheting: the steady state strain field is not periodic – the structure continues to accumulate
deformation without bound.
Fig. B 1 schematically illustrates each of these four response categories. An additional category often
included with these four steady state responses is:
5. Plastic collapse: the structure exceeds its limit load and collapses.
This final category clearly does not describe steady state behavior, but it is a possible outcome for an
arbitrary elastic perfectly-plastic structure under periodic loading.
These five regions are often plotted on Bree diagrams, named after the original diagram developed by
Bree for an elastic perfectly-plastic, open-ended, cylindrical pressure vessel under constant pressure load
and a periodic, linear, through-wall temperature gradient [B.8].
It will be useful to have mathematical criteria uniquely distinguishing each of the four regions of steady
state response. Here consider three quantities: the strain field in the structure , tx , the internal energy
in the whole structure : t V
E dVdtt , and the plastic dissipation in the whole structure
: p p
t V
W dVdtt . In terms of these three quantities, the regions of behavior can be defined as:
1. Elastic response: 0pW t .
2. Elastic shakedown: E t E t .
3. Plastic shakedown: 0 V
dVt t
4. Ratcheting: 1-3 not met.
These checks must be applied in order. Here || || is an appropriate norm, for example the von Mises
equivalent strain.
One way to think of the EPP strain limits provision requiring an analysis shake down is that the EPP
method is attempting to bound the deformation of the true structure over a given design life. If the EPP
analysis ratchets it continues to accumulate deformation without bound – therefore not imposing any
restrictions on the strain accumulation in the true structure. Establishing shakedown is then a critical
feature of the EPP method.
Example finite element analysis
Fig. B 2 shows the example used in this section: a two-bar system. The two bars have different cross-
sectional areas and are held rigidly in parallel so that the total strain in the two bars is identical. The bars
share an applied, primary load. The temperature of one bar cycles between aT and
bT while the
temperature of the other bar remains constant at aT . This temperature difference provides a secondary,
thermal load. This system can be thought of as representative of the extreme fibers of a thin-walled
pressure vessel under primary, constant pressure loading and an alternating, through-wall thermal
gradient. One advantage of this classical two-bar system is that its Bree diagram can be found
analytically. Fig. B 3 shows this diagram, with regions of elastic, elastic shakedown, plastic shakedown,
ratcheting, and collapse behavior.
Fig. B 4 shows the stress/strain hysteresis loops for both bars, using the geometric and loading parameters
described in Table B 1 and different loading conditions described in the figure caption. These loading
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conditions are designed to place the system in a different response regime in each subfigure: a) elastic
response, b) elastic shakedown, c) plastic shakedown, and d) ratcheting.
Figure B.5 shows the strain accumulation per cycle for the plastic shakedown case by plotting the
ratcheting rate per cycle, i.e. the difference between the strains at two corresponding points in the load
cycle between cycle 1 and 2, 2 and 3, etc.
Establishing shakedown
Mathematically, the criteria listed above clearly define the different regimes of behavior. However, the
implementation of the nonlinear finite element method used to analyze the two bar system means that
these mathematical criteria do not perfectly separate ratcheting from shakedown. Consider two
boundaries shown on Fig. B 3: the first between elastic shakedown and plastic shakedown (A) and the
second between plastic shakedown and ratcheting (B).
To define the transition between elastic shakedown and plastic shakedown we might use a restatement of
the criterion defined above. FE methods solve for the system response at discrete time steps. Consider two
time steps at equivalent times in the loading cycle but in two different, adjacent load cycles. Call the time
step in cycle nn t and the step in cycle ( 1) nn t . We then might restate the elastic shakedown criteria
as: a model shakes down elastically if
1 1 0 elastic n n n nR EE t t E E . (B.1)
Similarly, we might define the plastic shakedown criteria as: a model shakes down plastically if
1 1 0 plasti
V V
c n n n nR dt dVt V . (B.2)
However, computer floating point arithmetic used to represent operations on the real numbers is not
exact. These residual quantities will never be exactly zero. Furthermore, the amount of imprecision due to
floating point arithmetic will depend on the magnitude of the quantities themselves, which in turn will
depend on the units used in the finite element calculation. To avoid having floating point precision and
the physical units of the calculation influencing the procedure for determining shakedown, apply a
relative tolerance to the residual quantities, rather than requiring they become identically zero:
1 1 elastic n n rel nR E E E (B.3)
1 1 plastic n n re
V V
l nR dV dV (B.4)
The remaining task is to set the value of the relative tolerance parameter rel. Ideally, decreasing the value
of this tolerance would increase the accuracy of the shakedown/ratcheting determination. Selecting the
tolerance would then become a matter of engineering judgment: a designer could select an appropriate
tolerance based on their judgment of the consequences of misidentifying a ratcheting analysis as non-
ratcheting and vice-versa.
Table B 2 shows a convergence study on the criteria described by equations (B.3) and (B.4) using the
analytic location of the boundaries (A) and (B) from Fig. B 3 as reference points. The table shows that the
criterion for elastic shakedown converges to the analytic solution: as the tolerance is decreased the
method becomes more accurate at separating elastic from plastic shakedown. However, the table shows
that the criterion for separating ratcheting from non-ratcheting does not converge – decreasing the
tolerance does not decrease the error.
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Discussion
Fig. B 5 shows why the ratcheting/non-ratcheting criterion fails to converge. Even in the region where,
analytically, the system should shake down plastically the model still continues to accumulate small
amounts of strain per cycle. Tellingly, these small ratcheting strains are not present in the elastic or elastic
shakedown regimes.
The details of nonlinear finite element analysis explain this anomalous ratcheting. A nonlinear finite
element method is typically posed as a nonlinear, vector-valued residual equation equating the internal
model forces, due to stress in the material, to the external model forces due to the boundary conditions. In
the general case, this nonlinear equation cannot be solved analytically. Instead iterative methods are used
to reduce the residual to some acceptable threshold. This acceptable threshold is set by user-configurable
tolerance parameters. The FE results presented here use the smallest possible value of this tolerance.
However, the iterative method used in the FE analysis program here, and in the vast majority of
commercial and non-commercial finite element packages, is Newton’s method. Newton’s method
converges quadratically, which means that the error at iteration 1i of the method is proportional to the
square of the error at iteration i .
However, floating point arithmetic sets a lower bound on the smallest representable, non-zero number –
often called the machine precision. Say machine precision is 10-16. Table B 3 then shows two notional
convergence histories, representing possible outcomes of Newton iteration solving a system of nonlinear
finite element residual equations. The column labeled “low error” converges to close to machine precision
whereas the column labeled “high error” does not. The distinction between the two cases arises because
the method cannot take a step that would result in a residual below machine precision. Both behaviors are
plausible for an actual FE code and which type of behavior occurs is entirely dependent on the starting
value of the residual, which is essentially random. Furthermore, a nonlinear FE method solving multiple
load steps, as with the periodic loadings considered here, applies Newton’s method many times, meaning
that it is extremely likely to encounter the “high error” behavior.
The consequence of this is some bound on the accuracy an analysis invoking the nonlinear solver.
Newton iterations are only performed if the system response is nonlinear, i.e. if the steady state behavior
is plastic shakedown or ratcheting. Therefore, when solving for a nonlinear, plastic step there will always
be some error in the FE strains, relative to the analytic solution. This error can manifest as some fictitious
ratcheting strain present in configurations that should analytically shake down. These fictitious strains
will tend to have magnitudes that depend on the initial guess used in the Newton iteration process and the
details of convergence of Newton’s method, described above. Because Newton’s method is extremely
sensitive to the initial guess, the fictitious strains will tend to be chaotic, in the sense of dynamical
systems. The fictitious ratcheting strains will then usually appear as random noise, with the possibility of
some systemic trend developing depending on the system.
This suggests that the tolerance used to determine convergence in the nonlinear Newton iterations is a
critical factor for determining shakedown from numerical FE results. Table B 4 proves this is the case.
This table fixes the tolerance used in the shakedown criteria and varies the Newton tolerance used in the
FE solver. Below a critical value the Newton tolerance controls the accuracy of the shakedown
determination. Above this critical value tightening the Newton tolerance does not improve the solution
accuracy, for the reason described above.
Finally, Fig. B 5 suggests an alternative, visual method of separating ratcheting from shakedown. This
figure shows the “apparent” incremental ratcheting strain per cycle for a FE analysis of a two-bar system
that should, analytically, achieve plastic shakedown. Despite the noise caused by the numerical methods
used to solve the FE problem, clearly the mean ratcheting rate is on the order of 10-11 per cycle –
approximately zero. For a ratcheting case the mean ratcheting rate, ignoring the noise in the solution,
would be much higher. By this method the two cases can be visually distinguished.
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Conclusions
This section proposes a criterion for determining whether a nonlinear finite element analysis shakes down
or ratchets. This distinction is important for the EPP strain limits code case, as the EPP analysis must
shake down to bound the deformation of the true, creeping structure. This method is based on comparing
the strains from cycle to cycle and comparing the difference in the average strain fields to a small number,
controlled by a relative tolerance. However, the details of nonlinear finite element methods set a
minimum achievable error using this procedure and so it tends not to converge to an analytic solution as
the relative tolerance decreases.
Whether or not this error is acceptable for a particular analysis will be a matter of engineering judgment.
Plots of the ratcheting rate like Fig. B 5 can also be helpful in distinguishing between ratcheting and non-
ratcheting responses. However, many designers may not be aware of the limitations of nonlinear finite
element analysis as a tool for analyzing cyclic plasticity and so the following warning has been proposed
for incorporation into the strain limits Code Case:
“The strain limits EPP assessment requires the identification of non-ratcheting for an acceptable
load cycle.
Classification of an analysis as non-ratcheting requires that the deflections become cyclic. This
implies both the total strains and plastic strains also become cyclic. This steady state behavior may
develop after some initial number of load cycles that produce increasing deflections. History plots of
the deflections or strains may be used to identify a non-ratcheting response.
The numerical methods used in finite element analysis may produce noise in the deflection and strain
fields. This noise appears as small-magnitude, random variation about some constant average (non-
ratcheting) or non-constant but steadily increasing (ratcheting) response. This numerical noise
should be ignored when classifying a finite element analysis as ratcheting or non-ratcheting.”
This warning will ensure users of the EPP method will be aware of these issues and can make an
informed judgment as to whether a particular analysis achieves shake down, as required by the EPP strain
limits code case.
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 44
References
B.1. American Society of Mechanical Engineers, “Case N-861: Satisfaction of Strain Limits for
Division 5 Class A Components at Elevated Temperature Service Using Elastic-Perfectly Plastic
Analysis,” in ASME Boiler and Pressure Vessel Code, Nuclear Component Code Cases, 2015.
B.2. P. Carter, “Analysis of cyclic creep and rupture. Part 1: Bounding theorems and cyclic reference
stresses,” Int. J. Press. Vessel. Pip., vol. 82, no. 1, pp. 15–26, 2005.
B.3. P. Carter, “Analysis of cyclic creep and rupture. Part 2: Calculation of cyclic reference stresses and
ratcheting interaction diagrams,” Int. J. Press. Vessel. Pip., vol. 82, no. 1, pp. 27–33, 2005.
B.4. P. Carter, T.-L. Sham, and R. I. Jetter, “Elevated temperature primary load design method using
pseudo-elastic perfectly plastic model,” in Proceedings of the ASME 2012 Pressure Vessels &
Piping Division Conference, 2012, PVP2012-78081, pp. 1-10.
B.5. P. Carter, R.I. Jetter, and T.-L. Sham, “Verification of Elastic-Perfectly Plastic Methods for
Evaluation of Strain Limits - Analytical Comparisons,” in Proceedings of the ASME 2014 Pressure
Vessels and Piping Division Conference, 2014, PVP2014-28352, American Society of Mechanical
Engineers, New York, NY, pp. 1-8.
B.6. R. I. Jetter, Y. Wang, P. Carter, and T.-L. Sham, “Simplified methods for elevated temperature
structural design - an overview of some current activities,” in Proceedings of the ASME Symposium
on Elevated Temperature Application of Materials for Fossil, Nuclear, and Petrochemical
Industries, 2014, pp. 1–10.
B.7. C. O. Frederick and P. J. Armstrong, “Convergent internal stresses and steady cyclic states of
stress,” J. Strain Anal. Eng. Des., vol. 1, no. 2, pp. 154–159, 1966.
B.8. J. Bree, “Elastic-plastic behaviour of thin tubes subjected to internal pressure and intermittent high-
heat fluxes with application to fast-nuclear-reactor fuel elements,” J. Strain Anal. Eng. Des., vol.
2, no. 3, pp. 226–238, 1967.
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Table B 1. Properties used in the example two bar simulations.
A y E
1.0 1.5 150 100000 10-5
Table B 2. Convergence series for the elastic and plastic shakedown criteria varying the shakedown
residual tolerance
Tolerance Elastic shakedown error Plastic shakedown error
1.0e-1 8.35e-2 1.65e-3
1.0e-2 4.46e-3 1.28e-2
1.0e-3 4.27e-4 1.28e-2
1.0e-4 4.26e-5 1.28e-2
1.0e-5 4.26e-6 1.28e-2
1.0e-6 4.26e-7 1.28e-2
1.0e-7 4.26e-8 1.28e-2
1.0e-8 4.26e-9 1.28e-2
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the Prediction of Cyclic Life Based on the SMT Methodology
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Fig. B 1. Schematic of the four classical cyclic plasticity deformation regimes. a) elastic response, b)
elastic shakedown, c) plastic shakedown, and d) ratcheting.
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Fig. B 2. Example two-bar system.
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
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Fig. B 3. Bree diagram for the classical two bar problem. Points A and B are used to test methods
for determining the shakedown boundaries from numerical finite element analysis of the
system.
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a)
b)
c)
d)
Fig. B 4. Simulated two-bar stress/strain history. a) elastic response, b) elastic shakedown, c) plastic
shakedown, d) ratcheting.
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
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Fig. B 5. Apparent ratcheting strain increment per cycle for plastic shakedown loading conditions.
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Table B 3. Two notional examples showing how the initial error can affect the final convergence of
Newton's method, when implemented with floating point arithmetic
Iteration Low error High error
1 1.0e-1 1.0e-3
2 1.0e-2 1.0e-6
3 1.0e-4 1.0e-12
4 1.0e-8 1.0e-12
5 1.0e-16 1.0e-12
6 1.0e-16 1.0e-12
Table B 4. Convergence series for the elastic and plastic shakedown criteria varying the Newton
tolerance
Tolerance Elastic shakedown error Plastic shakedown error
1.0e-4 5.51e-1 1.28e-2
1.0e-5 5.51e-1 1.28e-2
1.0e-6 4.26e-9 1.28e-2
1.0e-7 4.26e-9 1.28e-2
1.0e-8 4.26e-9 1.28e-2
1.0e-9 4.26e-9 1.28e-2
1.0e-10 4.26e-9 1.28e-2
1.0e-11 4.26e-9 1.28e-2
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 52
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ACKNOWLEDGMENTS
This research was sponsored by the U.S. Department of Energy (DOE), Office of Nuclear Energy (NE),
Advanced Reactor Technologies (ART) Program. We gratefully acknowledge the support provided by
Brian Robinson of DOE-NE, Advanced Reactor Technologies, ART Program Manager; William Corwin
of DOE-NE, ART Materials Technology Lead; Hans Gougar of Idaho National Laboratory (INL), ART
Co-National Technical Director; and Richard Wright of INL, Technical Lead, High Temperature
Materials.
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 54
August 2017
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DISTRIBUTION LIST
Name Affiliation Email
Corwin, W. DOE-NE william.corwin@nuclear.energy.gov
Gougar, H. INL Hans.Gougar@inl.gov
Grandy, C. ANL cgrandy@anl.gov
Hill, R.N. ANL bobhill@anl.gov
Jetter, R.I. R.I. Jetter Consulting bjetter@sbcglobal.net
Li, D. DOE-NE Diana.Li@nuclear.energy.gov
McMurtrey, M. INL michael.mcmurtrey@inl.gov
Messner, M.C. ANL messner@anl.gov
Natesan, K. ANL natesan@anl.gov
Robinson, B. DOE-NE Brian.Robinson@Nuclear.Energy.gov
Sham, T.-L. ANL ssham@anl.gov
Wang, Y. ORNL wangy3@ornl.gov
Wright, R.N. INL richard.wright@inl.gov
Yankeelov, J. DOE-ID yankeeja@id.doe.gov
Report on an Assessment of the Application of EPP Results from the Strain Limit Evaluation Procedure to
the Prediction of Cyclic Life Based on the SMT Methodology
ANL-ART-96 56
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