Report on an Assessment of the Application of EPP Results ...tests to compare with EPP strain limits...

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

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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.



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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



ANL-ART-96 vi

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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



ANL-ART-96 viii

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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



ANL-ART-96 x

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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



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

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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



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

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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



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-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


and compression 1.05% 950 500 psi 94.3 280


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 ---



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.



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



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



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.

August 2017

ANL-ART-96 11

Fig. 8. Thermal cycle, 515 to 815°C. (a) EPP results, (b) inelastic results, (c) EPP margin

(a) (b)

(c)



ANL-ART-96 12

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)

August 2017

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



ANL-ART-96 14

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.

August 2017

ANL-ART-96 15

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.”



ANL-ART-96 16

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.

August 2017

ANL-ART-96 17

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



ANL-ART-96 18

August 2017

ANL-ART-96 19

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.



ANL-ART-96 20

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.

August 2017

ANL-ART-96 21

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



ANL-ART-96 22

(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.

August 2017

ANL-ART-96 23

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.



ANL-ART-96 24

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.

August 2017

ANL-ART-96 25

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.



ANL-ART-96 26

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.

August 2017

ANL-ART-96 27

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



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.



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.

August 2017

ANL-ART-96 29

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



ANL-ART-96 30

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

August 2017

ANL-ART-96 31

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



ANL-ART-96 32

Fig. A 4. Thermal cycle, 515 to 815°C. a) EPP results, b) inelastic results, c) EPP margin

(a) (b)

(c)

August 2017

ANL-ART-96 33

Fig. A 5. Thermal cycle, 415 to 515°C. (a) EPP results, (b) inelastic results, (c) EPP margin

(a) (b)

(c)



ANL-ART-96 34

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

August 2017

ANL-ART-96 35

Fig. A 8. Finite element mesh used to simulate the response of the ORNL test article



ANL-ART-96 36

Fig. A 9. Elastic stress analysis of the specimen, figure zoomed into the critical section

August 2017

ANL-ART-96 37

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



ANL-ART-96 38

August 2017

ANL-ART-96 39

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.



ANL-ART-96 40

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

August 2017

ANL-ART-96 41

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.



ANL-ART-96 42

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.

August 2017

ANL-ART-96 43

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.



ANL-ART-96 44

References

B.1. American Society of Mechanical Engineers, “Case N-861: Satisfaction of Strain Limits for



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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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.



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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.



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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



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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.



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DISTRIBUTION LIST

Name Affiliation Email

Corwin, W. DOE-NE [email protected]

Gougar, H. INL [email protected]

Grandy, C. ANL [email protected]

Hill, R.N. ANL [email protected]

Jetter, R.I. R.I. Jetter Consulting [email protected]

Li, D. DOE-NE [email protected]

McMurtrey, M. INL [email protected]

Messner, M.C. ANL [email protected]

Natesan, K. ANL [email protected]

Robinson, B. DOE-NE [email protected]

Sham, T.-L. ANL [email protected]

Wang, Y. ORNL [email protected]

Wright, R.N. INL [email protected]

Yankeelov, J. DOE-ID [email protected]



ANL-ART-96 56

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