Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

PUBLICATION NO. FHWA-HRT-22-113 JUNE 2023 

Ruggedness and Interlaboratory 

Studies for Asphalt Mixture 

Performance Tester (AMPT) 

Cyclic Fatigue Test: Phase II Report
FOREWORD

The standards for asphalt mixture performance tester (AMPT) cyclic fatigue testing (American 

Association of State Highway and Transportation Officials (AASHTO) TP 107-22 and 

AASHTO TP 133-21) enable pavement engineers to predict asphalt mixture performance over a 

wide range of loading and climate conditions.(1,2) These standards are an important component of 

ongoing Federal Highway Administration efforts to increase pavement life through fundamental 

testing and mechanistically based predictive models. For the widespread implementation of 

AMPT cyclic fatigue testing procedures to take place, precision statements that define the 

repeatability and reproducibility of the test standards are needed. This report details an 

interlaboratory study of AMPT cyclic fatigue testing standards that was undertaken to meet this 

need. Researchers analyzed the results of the interlaboratory study experiments to define the 

allowable within- and between-laboratory variations in the test results. They developed 

corresponding precision statements for both AASHTO TP 107-22 and AASHTO TP 133-21 and 

assessed the implications of the precision statements on the uncertainty in practical applications 

of the test. Draft revised versions of both standards have also been developed as part of this 

effort.

Mark E. Swanlund 

Acting Director, Office of Infrastructure

Research and Development

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

(USDOT) in the interest of information exchange. The U.S. Government assumes no liability for 

the use of the information contained in this document.

The U.S. Government does not endorse products or manufacturers. Trademarks or 

manufacturers’ names appear in this report only because they are considered essential to the 

objective of the document. 

Quality Assurance Statement

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. Standards 

and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its 

information. FHWA periodically reviews quality issues and adjusts its programs and processes to 

ensure continuous quality improvement. 
TECHNICAL REPORT DOCUMENTATION PAGE

1. Report No.

FHWA-HRT-22-113

2. Government Accession No. 3. Recipient’s Catalog No.

4. Title and Subtitle

Ruggedness and Interlaboratory Studies for Asphalt Mixture 

Performance Tester (AMPT) Cyclic Fatigue Test: Phase II

Report

5. Report Date

June 2023

6. Performing Organization Code

7. Author(s)

Cassie Castorena (ORCID: 0000-0002-5915-0084), 

B. Shane Underwood (ORCID: 0000-0002-7223-3968), 

Y. Richard Kim (ORCID: 0000-0003-3295-977X), Jing Ding

Kangjin Lee (ORCID: 0000-0003-3458-0747), Nam Tran

(ORCID: 0000-0001-8183-4741), and Adam Taylor

8. Performing Organization Report No.

9. Performing Organization Name and Address

North Carolina State University

Department of Civil, Construction, and Environmental 

Engineering

Campus Box 7908

Raleigh, NC 27695

10. Work Unit No.

11. Contract or Grant No.

DTFH6117C00037

12. Sponsoring Agency Name and Address

Office of Infrastructure Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101

13. Type of Report and Period Covered

Phase II Report; 

September 2017–October 2021

14. Sponsoring Agency Code

HRDI-10

15. Supplementary Notes

The Contracting Officer’s Representative was David J. Mensching, Ph.D., P.E. (HRDI-10; ORCID: 0000-0003-

2460-2586).

16. Abstract

This report highlights findings from Phase II of a project designed to develop precision statements for cyclic 

fatigue testing using an asphalt mixture performance tester (AMPT). These standards include American 

Association of State Highway and Transportation Officials (AASHTO) TP 107-22 and AASHTO TP 133-21, 

which apply to 100-mm-diameter and 38-mm-diameter test specimens, respectively.(1,2) Seven laboratories 

participated in an interlaboratory study of the AMPT cyclic fatigue tests designed according to ASTM E691-20 

and ASTM C670-15(4,5)

. Researchers in these laboratories conducted dynamic modulus and cyclic fatigue testing 

of three mixtures for each specimen geometry. The results were used to establish repeatability and reproducibility 

precision limits for damage characteristic curve and failure criterion results of the AMPT cyclic fatigue tests; these 

limits reflect the variation in test results that will be exceeded with a probability of 5 percent if the test is executed 

properly. The researchers introduced a refined functional data metric to capture the variation in damage 

characteristic curve results. All precision limits were defined as a function of the mixture nominal maximum 

aggregate size (NMAS) except the failure criterion reproducibility, which did not follow a consistent trend with 

respect to NMAS. The established repeatability limits quantify the acceptable variation among three test 

determinations (specimens) obtained within a laboratory on a single material. The reproducibility limits quantify

the acceptable variation between average test results of two laboratories conducted on the same material.

17. Key Words

Cyclic fatigue testing, asphalt mixture performance 

tester, AMPT, asphalt mixture, interlaboratory 

study, repeatability, reproducibility, precision

asphalt mixture performance testing standards

18. Distribution Statement

No restrictions. This document is available to the public 

through the National Technical Information Service, 

Springfield, VA 22161.

45TUhttps://www.ntis.gov

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

98

22. Price

N/A

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized.
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iii

TABLE OF CONTENTS

CHAPTER 1. INTRODUCTION ................................................................................................ 1

CHAPTER 2. METHODOLOGY ............................................................................................... 3

Participating Laboratories ..................................................................................................... 3

ILS Protocol............................................................................................................................. 3

Materials .................................................................................................................................. 4

Pilot Testing ............................................................................................................................. 6

Full-scale Testing .................................................................................................................... 6

Test Analysis Procedures ....................................................................................................... 7

Dynamic Modulus Testing .................................................................................................. 7

Cyclic Fatigue Testing ........................................................................................................ 9

Statistical Analysis ................................................................................................................ 11

Single-Point Measure Analysis Conducted Using ASTM E691 and ASTM C670 .......... 11

Advanced Analysis of Damage Characteristic Curve Repeatability and Reproducibility 

Using a Functional Data Metric ........................................................................................ 16

Analysis of the Precision Statement Implications on Uncertainty in the Test Results’ 

Practical Applications ........................................................................................................... 21

Uncertainty in FlexPAVETM %Damage Predictions ........................................................ 21

Uncertainty in Apparent Damage Capacity ...................................................................... 25

CHAPTER 3. RESULTS ............................................................................................................ 31

ILS Results Overview ........................................................................................................... 31

ILS Results Statistical Analysis ........................................................................................... 43

Consistency Statistics........................................................................................................ 43

Single-Point Measure Repeatability and Reproducibility Analysis Conducted Following 

ASTM E691 ...................................................................................................................... 44

Advanced Analysis of Damage Characteristic Curve Repeatability and Reproducibility 

Using a Functional Data Metric ........................................................................................ 49

Evaluation of Repeatability and Reproducibility Trends for NMAS ............................... 51

Establishment of the Recommended Precision Statements .............................................. 55

Proposed Precision Statements ............................................................................................ 58

Analysis of the Precision Statement Implications on Uncertainty in Practical 

Applications of the Test Results........................................................................................... 59

Data Excluded From the Analysis .................................................................................... 59

Uncertainty in FlexPAVE %Damage Predictions ............................................................ 61

Uncertainty in Apparent Damage Capacity ...................................................................... 64

CHAPTER 4. CONCLUSIONS AND RECOMMENDED CHANGES TO THE 

STANDARDS FOR AMPT CYCLIC FATIGUE TESTING ................................................. 75

APPENDIX. INITIAL PILOT TEST RESULTS EVALUATION AND RESOLUTION OF 

PROCEDURAL DISCREPANCIES......................................................................................... 77

Final Pilot Test Results ......................................................................................................... 80

Full-Scale ILS Dynamic Modulus Test Results .................................................................. 81

REFERENCES ............................................................................................................................ 87
iv

LIST OF FIGURES

Figure 1. Graph. Sieve size 0.45-mm power chart for gradations. ................................................. 5

Figure 2. Graph. DR failure criterion. ............................................................................................ 11

Figure 3. Graph. An example of C value and shape factor. .......................................................... 12

Figure 4. Graphs. An example for calculating repeatability vnorm. ............................................... 19

Figure 5. Graphs. An example for calculating reproducibility vnorm. ............................................ 20

Figure 6. Flowchart. The procedure to assess uncertainty in %Damage predictions under 

repeatability conditions. ...................................................................................................... 24

Figure 7. Flowchart. The procedure to assess uncertainty in %Damage predictions under 

reproducibility conditions. .................................................................................................. 25

Figure 8. Flowchart. The procedure to assess uncertainty in Sapp values under repeatability 

conditions. ........................................................................................................................... 27

Figure 9. Flowchart. The procedure to assess uncertainty in Sapp values under 

reproducibility conditions. .................................................................................................. 29

Figure 10. Graph. Individual specimen damage characteristic curves for 9.5-mm small 

specimens. ........................................................................................................................... 31

Figure 11. Graph. Fitted damage characteristic curves for 9.5-mm small specimens. ................. 32

Figure 12. Graph. Cumulative (1−C) results from individual tests for 9.5-mm small 

specimens. ........................................................................................................................... 32

Figure 13. Graph. DR results for 9.5-mm small specimens........................................................... 33

Figure 14. Graph. Individual specimen damage characteristic curves for 9.5-mm large 

specimens. ........................................................................................................................... 33

Figure 15. Graph. Fitted damage characteristic curves for 9.5-mm large specimens. .................. 34

Figure 16. Graph. Cumulative (1−C) results from individual tests for 9.5-mm large 

specimens. ........................................................................................................................... 34

Figure 17. Graph. DR results for 9.5-mm large specimens. .......................................................... 35

Figure 18. Graph. Individual specimen damage characteristic curves for 12.5-mm small 

specimens. ........................................................................................................................... 35

Figure 19. Graph. Fitted damage characteristic curves for 12.5-mm small specimens. ............... 36

Figure 20. Graph. Cumulative (1−C) results from individual tests for 12.5-mm small 

specimens. ........................................................................................................................... 36

Figure 21. Graph. DR results for 12.5-mm small specimens......................................................... 37

Figure 22. Graph. Individual specimen damage characteristic curves for 12.5-mm large 

specimens. ........................................................................................................................... 37

Figure 23. Graph. Fitted damage characteristic curves for 12.5-mm large specimens. ................ 38

Figure 24. Graph. Cumulative (1−C) results from individual tests for 12.5-mm large 

specimens. ........................................................................................................................... 38

Figure 25. Graph. DR results for 12.5-mm large specimens. ........................................................ 39

Figure 26. Graph. Individual specimen damage characteristic curves for 19-mm small 

specimens. ........................................................................................................................... 39

Figure 27. Graph. Fitted damage characteristic curves for 19-mm small specimens. .................. 40

Figure 28. Graph. Cumulative (1−C) results from individual tests for 19-mm small 

specimens. ........................................................................................................................... 40

Figure 29. Graph. DR results for 19-mm small specimens. ........................................................... 41
v 

Figure 30. Graph. Individual specimen damage characteristic curves for 25-mm large 

specimens. ........................................................................................................................... 41

Figure 31. Graph. Fitted damage characteristic curves for 25-mm large specimens. ................... 42

Figure 32. Graph. Cumulative (1−C) results from individual tests for 25-mm large 

specimens. ........................................................................................................................... 42

Figure 33. Graph. DR results for 25-mm large specimens. ........................................................... 43

Figure 34. Graphs. Sensitivity analysis of repeatability and reproducibility COV on 

single-point C value selection. ............................................................................................ 47

Figure 35. Graphs. Sensitivity analysis of repeatability and reproducibility COV on 

single-point shape factor selection. ..................................................................................... 48

Figure 36. Graphs. Repeatability analysis comparison between single-point measure and 

vnorm approach. ..................................................................................................................... 53

Figure 37. Graphs. Reproducibility analysis comparison between single-point measure and 

vnorm approach. ..................................................................................................................... 55

Figure 38. Graphs. Histograms of DR difference with the fitted normal distribution. .................. 57

Figure 39. Graph. Measured Sapp values for the 9.5-mm mixture small-specimen geometry. ...... 65

Figure 40. Graph. Measured Sapp values for the 9.5-mm mixture large-specimen geometry. ...... 65

Figure 41. Graph. Measured Sapp values for the 12.5-mm mixture small-specimen 

geometry. ............................................................................................................................. 66

Figure 42. Graph. Measured Sapp values for the 12.5-mm mixture large-specimen geometry. .... 66

Figure 43. Graph. Measured Sapp values for the 19-mm mixture small-specimen geometry. ....... 67

Figure 44. Graph. Measured Sapp values for the 25-mm mixture large-specimen geometry. ....... 67

Figure 45. Graph. Comparison of the maximum Sapp COV% values measured among test 

determinations and calculated among test results from the uncertainty analysis. ............... 69

Figure 46. Graph. Comparison of the maximum Sapp COV% values measured among test 

determinations and calculated among test results from the uncertainty analysis. ............... 72

Figure 47. Graphs. Comparison between uncertainty of %Damage and uncertainty of Sapp. ...... 74

Figure 48. Graph. Example of actuator strain results without a loose connection. ...................... 79

Figure 49. Graph. Example of actuator strain standard error results indicating a loose 

connection in the cyclic fatigue setup. ................................................................................ 79

Figure 50. Graph. Dynamic modulus master curve for pilot test results (log-log scale). ............. 80

Figure 51. Graph. Dynamic modulus master curve for pilot test results (semilog scale). ............ 80

Figure 52. Graph. Damage characteristic curves for pilot test results. ......................................... 81

Figure 53. Graph. Failure criterion for pilot test results. .............................................................. 81

Figure 54. Graph. Dynamic modulus test results for 9.5-mm mixture large specimen. ............... 82

Figure 55. Graph. Dynamic modulus test results for 9.5-mm mixture small specimen. .............. 82

Figure 56. Graph. Dynamic modulus test results for 12.5-mm mixture large specimen. ............. 83

Figure 57. Graph. Dynamic modulus test results for 12.5-mm mixture small specimen. ............ 83

Figure 58. Graph. Dynamic modulus test results for 19-mm mixture small specimen. ............... 84

Figure 59. Graph. Dynamic modulus test results for 25-mm mixture large specimen. ................ 84
vi 

LIST OF TABLES

Table 1. ILS participants. ................................................................................................................ 3

Table 2. Selected mixtures for the ILS. .......................................................................................... 4

Table 3. Multiplier for repeatability statistic. ............................................................................... 16

Table 4. Percentage of data exceeding critical consistency value. ............................................... 44

Table 5. Repeatability and reproducibility COV. ......................................................................... 45

Table 6. Repeatability and reproducibility limits (COV) for C value at (1−C̿)=0.7. .................... 48

Table 7. Repeatability and reproducibility limits (COV) for shape factor at (1−C̿)=0.7.............. 49

Table 8. Repeatability and reproducibility precision statements (standard deviation) of DR. ...... 49

Table 9. Repeatability vnorm for different laboratories and different materials. ............................ 50

Table 10. Reproducibility vnorm for different laboratories and different materials. ...................... 50

Table 11. Repeatability and reproducibility precision statements with vnorm. .............................. 50

Table 12. DR reproducibility analysis results aggregating NMAS for different geometry. .......... 56

Table 13. Recommended repeatability and reproducibility precision statements (maximum 

allowable difference) of DR. ................................................................................................ 58

Table 14. Single-operator precision for damage characteristic curve and DR results. .................. 59

Table 15. Multilaboratory precision for damage characteristic curve and DR results. ................. 59

Table 16. Data failing repeatability limits. ................................................................................... 60

Table 17. Data excluded from reproducibility analysis. ............................................................... 60

Table 18. Repeatability standard deviation of %Damage for all the laboratories and 

mixtures. .............................................................................................................................. 61

Table 19. Reproducibility standard deviation of %Damage for all the laboratories and 

mixtures. .............................................................................................................................. 62

Table 20. Maximum repeatability variation for each mixture. ..................................................... 62

Table 21. Maximum reproducibility variation for each mixture. ................................................. 63

Table 22. Median repeatability variation for each mixture........................................................... 63

Table 23. Median reproducibility variation for each mixture. ...................................................... 63

Table 24. Measured repeatability COV% of Sapp among test determinations for all the 

laboratories and mixtures. ................................................................................................... 68

Table 25. Repeatability COV% of Sapp results for all the laboratories and mixtures from the 

uncertainty analysis. ............................................................................................................ 68

Table 26. Measured reproducibility COV% of Sapp for all the laboratories and mixtures. ........... 70

Table 27. Reproducibility COV% of Sapp for all the laboratories and mixtures from the 

uncertainty analysis. ............................................................................................................ 71

Table 28. Maximum and median COV% of Sapp for each mixture from the uncertainty 

analysis. ............................................................................................................................... 73

Table 29. Dynamic modulus data used for ILS cyclic fatigue analysis. ....................................... 85
vii 

LIST OF ABBREVIATIONS AND VARIABLES

Abbreviations

2S2P1D two springs, two parabolic elements, and one dashpot 

AASHTO American Association of State Highway and Transportation Officials

AMPT asphalt mixture performance tester

COV coefficient of variation

FHWA Federal Highway Administration

ILS interlaboratory study

LVDT linear variable differential transformer

MCMC Markov Chain Monte Carlo 

NMAS nominal maximum aggregate size

PG performance grade 

PID proportional, integral, derivative 

RAP reclaimed asphalt pavement

SMA stone matrix asphalt

SPT simple performance tester

S-VECD simplified viscoelastic continuum damage

Variables 

α damage growth rate

an multiplier affected by the number of test determinations, n, listed in table 3

αT time–temperature shift factor at a given temperature

aTotal sum of the contributions from uncertainty in the |E*| model, damage 

characteristic model, and failure criterion 

β 2S2P1D fitting coefficient

C pseudostiffness

C11 fitting coefficient for power law damage-characteristic curve model 

C12 fitting coefficient for power law damage-characteristic curve model

C11_new fitted parameters for damage-characteristic curve model after normalization

C12_new fitted parameters for damage-characteristic curve model after normalization

C11-i_new fitted parameters for specimen i after normalization

C12-i_new fitted parameters for specimen i after normalization

C̅11-i_new fitted parameters for laboratory i after normalization

C̅12-i_new fitted parameters for laboratory i after normalization

C̿11_new fitted parameters for both laboratory results after normalization

C̿12_new fitted parameters for both laboratory results after normalization

C̅11_new fitted parameters for all three specimens after normalization

C̅12_new fitted parameters for all three specimens after normalization

Ci pseudosecant modulus at the current time step 

Ci – 1 pseudosecant modulus at the previous time step

C* cyclic pseudosecant modulus

C*n cyclic pseudosecant modulus at the current analysis cycle 

C*n – 1 cyclic pseudosecant modulus at the previous analysis cycle

C̅(S)i the fitted function for ith laboratory after normalization

C̿(S) the fitted function of two laboratories being compared after normalization
viii

C(Snew)i the fitted function for ith replicate after normalization for vnorm calculation

C̅(Snew) the fitted function of all the data after normalization for vnorm calculation

COVrpt% repeatability COV for Sapp

COVrpd% reproducibility COV for Sapp

COV(x,y) COV between x and y

d2srpt(%) allowable difference between two test results from one operator expressed as a 

COV

d2srpd(%) allowable difference between test results reported by two laboratories expressed 

as a COV

DR pseudostrain energy-based fatigue failure criterion

 DR value for ith specimen

average DR value for all the specimens

DMR dynamic modulus ratio

%D̅ average %Damage

%DP predicted damage at the reliability level P

%Damage percent of damage predicted using FlexPAVETM 

E′fit_i,j fitted storage modulus from the fitted curve for the ith specimen and jth 

temperature and frequency

E′fit_avg,j storage modulus from the fitted curve using all of the specimens at the jth 

temperature and frequency

δ 2S2P1D fitting coefficient

E′2S2P1D storage modulus from 2S2P1D model 

E0 maximum storage modulus value (kPa or psi) 

E00 minimum storage modulus value (kPa or psi) 

|E*| dynamic modulus

|E*|fingerprint dynamic modulus determined from fingerprint test 

|E*|LVE dynamic modulus from 2S2P1D model 

εR pseudostrain

 peak-to-peak pseudostrain 

tension amplitude of the pseudostrain 

fV density distribution function for vnorm

γ 2S2P1D fitting coefficient

Γ gamma function 

h 2S2P1D fitting coefficient

η between-laboratory consistency statistic

θ within-laboratory consistency statistic 

ILVE_test linear viscoelastic (dynamic modulus model) variation index from test result

DR variation index from test results

k 1×105, multiplier to change the magnitude for vnorm calculation

K the shape parameter in gamma distribution

K1 form adjustment factor

κ 2S2P1D fitting coefficient 

λ the scale parameter in gamma distribution

MAR maximum allowable range 

max E′ maximum storage modulus 

n total number of cyclic fatigue test specimens 
ix

nLVE total number of dynamic modulus test specimens 

nS number of simulations in the 95-percent confidence interval, 4,750 

N number of cycles

Nf number of cycles to failure

ω angular frequency (rad/s)

ωR reduced angular frequency (rad/s)

p number of laboratories in ILS

S damage parameter

Sall all the S values from n damage characteristic curves

Sapp apparent damage capacity

Sapp-i Sapp value for the ith simulation

S̅app average Sapp value across all simulations

Sapp-1

2.5 the 2.5 percentile of Sapp for laboratory 1 

Sapp-1

97.5 the 97.5 percentile of Sapp for laboratory 1 

Sapp-2

2.5 the 2.5 percentile of Sapp for laboratory 2 

Sapp-2

97.5 the 97.5 percentile of Sapp for laboratory 2 

Sfactor factor used to normalize damage value in vnorm calculation

Smin the minimum of the corresponding failure value of each dataset

Smean the average of the corresponding failure value of each dataset 

Smax the maximum of the corresponding failure value of each dataset

Snew damage value after normalization

SE standard error of actuator strain

si within-laboratory standard deviation 

sr% repeatability standard deviation (COV) 

sx̅ standard deviation of laboratory averages 

sx̅% reproducibility standard deviation (COV) 

stdTotal standard deviation of %Damage. 

ΔS increment of damage 

ΔSi damage growth between the current and previous time step

ΔSn damage growth between the current and previous analysis cycle

ΔtR reduced time step

σ stress (kPa or psi)

σpp peak-to-peak stress (kPa or psi) 

t time measured from the experiment (s)

tR reduced time (s)

T temperature (°C)

τE 2S2P1D fitting coefficient

x̅i laboratory average for laboratory i 

x̿ average of all laboratory averages

vnorm variation index for damage characteristic curve results

ZP standard normal deviate corresponding to reliability level P

1 

CHAPTER 1. INTRODUCTION

The American Association of State Highway and Transportation Officials (AASHTO) testing 

protocols AASHTO TP 107-22 and AASHTO TP 133-19—collectively referred to as procedures 

for asphalt mixture performance tester (AMPT) cyclic fatigue testing—enable the practical, 

mechanical performance characterization of asphalt mixtures using cyclic fatigue testing in the 

AMPT.(1,2) They are the only testing protocols developed specifically for the AMPT to assess 

fatigue cracking. 

AASHTO TP 107-22 is used for testing cylindrical asphalt mixture specimens that are 100-mm 

in diameter by 130-mm tall (hereafter referred to as large specimens).(1) In contrast, AASHTO 

TP 133-21 applies to cylindrical asphalt mixture specimens that are 38-mm in diameter by 

110-mm tall (hereafter referred to as small specimens).(2) AASHTO TP 107-22 preceded 

AASHTO TP 133-21. The small-specimen geometry was established to improve testing 

efficiency and enable field core testing for mixtures with nominal maximum aggregate size 

(NMAS) values up to 19 mm. 

The results of AMPT cyclic fatigue testing, coupled with the results of dynamic modulus (|E*|) 

testing, are used to obtain two key material functions of the simplified viscoelastic continuum 

damage (S-VECD) model: the damage-characteristic curve (i.e., pseudostiffness (C) versus the 

internal state parameter representing damage (S)) and the pseudostrain energy-based fatigue 

failure criterion (DR). The damage-characteristic curve and failure criteria are independent of the 

loading mode and loading history. Consequently, prediction of the damage response to any given 

loading history of interest is possible using limited test results, thus making the fatigue-cracking 

characterization of asphalt mixtures efficient compared to empirical methods.

The standards for AMPT cyclic fatigue testing recently underwent a ruggedness evaluation to 

define tolerance ranges for the experimental factors that contribute to variability in the test 

results.(3) The standards for AMPT cyclic fatigue testing were revised based on the ruggedness 

evaluation findings. The AMPT cyclic fatigue standards lack precision statements that define the 

repeatability (i.e., with laboratory variation) and reproducibility (i.e., between laboratory 

variation) of the test results when executed properly. Therefore, this study aims to improve the 

AMPT cyclic fatigue standards by developing precision statements. To achieve this objective, an 

interlaboratory study (ILS) was executed wherein seven laboratories conducted AMPT cyclic 

fatigue testing according to the AASHTO TP 107-22 and AASHTO TP 133-21 protocols 

established based on the ruggedness evaluation results.(1,2) Statistical analysis of the ILS yielded 

recommended precision statements for AASHTO TP 107-22 and AASHTO TP 133-21. The ILS

was designed and analyzed according to ASTM E691-20 with integration of ASTM C670-15 

recommendations specifically for construction materials.(4,5)

This report contains four chapters and an appendix. Chapter 1 introduces the study’s motivation, 

objective, and scope. Chapter 2 provides the overall methodology of the study, including the 

materials, test and analysis methods, and experimental plan. Chapter 3 presents the results of the 

experiments and statistical analyses of the results, along with the proposed precision statements 

for AASHTO TP 107-22 and AASHTO TP 133-21. Chapter 4 presents the conclusions and 

recommendations drawn from the results of this study.

(1,2)
2 

This report uses certain naming conventions throughout as follows: 

• “Samples” or “loose mixture samples” are plant-mixed asphalt mixture samples in a loose 

state. 

• “Gyratory-compacted samples” are plant-mixed, laboratory-compacted asphalt mixture

samples that have been compacted in a Superpave gyratory compactor. 

• “Test specimens” or “specimens” refer to plant-mixed materials that have been 

compacted, cut, and cored in the laboratory. 

• “Prefabricated specimens” are specimens prepared by a single laboratory and distributed 

to ILS participants.
3 

CHAPTER 2. METHODOLOGY

ASTM E691-20 and ASTM C670-15 describe the basic methodology for designing, carrying out, 

and analyzing the results of an ILS to define the repeatability and reproducibility of a test 

procedure.(4,5)

 Repeatability defines the variability in test results obtained by a single operator in 

a single laboratory on the same material. Reproducibility defines the variability between test 

results obtained from two different laboratories on the same material. The cyclic fatigue test 

method’s repeatability is represented as the acceptable variation among the three test replicates. 

The reproducibility of the cyclic fatigue test methods characterizes the acceptable variation 

between the average test results of two laboratories. This chapter describes the methodology 

followed in conducting the AMPT cyclic fatigue ILS. 

PARTICIPATING LABORATORIES

The seven laboratories that participated in the ILS, the corresponding AMPT testing equipment, 

and the gyratory compactor type that they used are listed in table 1. The laboratories are codified 

within the results to preserve anonymity. 

Table 1. ILS participants. 

Participant AMPT Equipment* Gyratory Compactor* 

Federal Highway Administration

(FHWA)

Controls AMPT Pine G2

Maine Department of Transportation Controls AMPT Pine G1

National Center for Asphalt Technology Controls SPT (upgraded) Pine G2

North Carolina State University Controls AMPT Pro® Pine G2

University of Massachusetts-Dartmouth Controls AMPT Pine G2

University of Nevada-Reno Controls AMPT Pine G2

University of New Hampshire Controls SPT Controls Servopac®

*The U.S. Government does not endorse products or manufacturers. Trademarks or manufacturers’ names appear in 

this report only because they are considered essential to the objective of the document.

SPT = simple performance tester. 

ILS PROTOCOL

An ILS protocol was developed and distributed to the participating laboratories that provided 

detailed specimen fabrication and testing instructions. The researchers provided the laboratories 

with links to data-logging forms; AMPT cyclic fatigue test standards; and videos demonstrating 

sample fabrication, test specimen preparation, and testing procedures. The data-logging forms 

were used to collect information on specimen fabrication procedures, specimen bulk-specific 

gravity measurements, and testing details (e.g., thermal equilibration time and ambient 

laboratory conditions). In addition, participants were provided detailed instruction sheets that 

specified the specimen and sample identifications provided by the research team. They were 

given each specimen’s and sample’s purpose, mixture compaction temperatures, theoretical 

maximum specific gravity values, target air void contents, and test temperature and frequency 

requirements. The researchers provided participants with shared folders for uploading their datalogging forms and AMPT test results. 
4 

Before beginning the ILS, the research team distributed a prebriefing survey to ask participants 

about specific aspects of the testing equipment, calibration, and additional information to help 

address data inconsistencies and/or ensure compliance with the current requirements of the 

AMPT cyclic fatigue test standards. The research team also held a webinar before beginning the 

ILS to present the cyclic fatigue test standards that resulted from the ruggedness evaluation and 

ILS testing requirements. The participants were allowed to ask questions during the webinar. 

MATERIALS

Four plant-mixed, laboratory-compacted mixtures were used for the ILS: dense-graded 9.5-mm 

NMAS surface mixture from Wisconsin, stone matrix asphalt (SMA) surface mixture with 

12.5-mm NMAS from Maryland, dense-graded 19-mm NMAS surface mixture from Arizona, 

and dense-graded 25-mm NMAS base mixture from North Carolina. Table 2 provides details 

about the selected mixtures. Figure 1 shows the gradations for the selected mixtures. 

Table 2. Selected mixtures for the ILS.

Source

NMAS 

(mm)

Virgin 

Binder 

Grade

Binder 

Content 

(%)

RAP 

Content 

(%)

VMA 

(%)

VFA 

(%)

Number of

Design 

Gyrations

Wisconsin 9.5 PG 58-28 6.1 25 16.2 75.3 75

Maryland 12.5 PG 64E22 6.5 15 18.5 81.0 100 

Arizona 19.0 PG 70-10 4.8 0 14.2 71.5 75

North 

Carolina 25.0 PG 64-22 4.2 30 13.6 71.5 65 

PG = performance grade; RAP = reclaimed asphalt pavement; VMA = voids in mineral aggregate; 

VFA = voids filled with asphalt.
5 

Source: FHWA.

Figure 1. Graph. Sieve size 0.45-mm power chart for gradations.

Plant-mixed, laboratory-compacted samples were used based on the results of a survey 

distributed to potential ILS participants that suggests that most laboratories use plant-mixed 

samples for test specimen fabrication in practice. The mixtures were sourced from different 

climatic regions that are expected to yield different test results. Also, the SMA mixture contained 

polymer-modified virgin asphalt, whereas the other mixtures did not include polymer 

modification. The research team included differences in NMAS and gradation type to cover a 

representative range of test variability. For both small- and large-specimen geometries, the 

researchers used both the 9.5-mm dense-graded surface mixture with an unmodified binder and 

the 12.5-mm SMA mixture with the polymer-modified binder. The 19-mm mixture was only 

used in the small-specimen geometry ILS, whereas the 25-mm mixture was only used in the 

large-specimen geometry ILS. The maximum NMAS permitted in AASHTO TP 133-21 is 

19 mm.

(2) 

To prepare the loose mixture samples for distribution, the participants separated and randomized 

the plant-mixed material into individual samples following the researchers’ loose mixture 

separation procedure.(6) Plant-mixed specimen testing using this procedure has yielded AMPT 

|E*| and cyclic fatigue test results with comparable variability to lab-mixed specimen testing in 

the research team’s laboratory when this separation procedure is followed. Each separated 

sample was placed in a quick-release, heavy-duty, wax-lined box, consistent with those used by 

AASHTO re:source. Each box contained the material required to produce approximately 

one-half a gyratory-compacted sample. All mixture test specimens were prepared to achieve an 

air void content of 5 ± 0.5 percent. 
6 

PILOT TESTING

ASTM E691-20 specifies pilot testing before beginning an ILS to familiarize participants with 

the test procedures and ILS protocol.(4) ILS participants were supplied with prefabricated 

small-test specimens for protocol familiarization. The test specimens were fabricated according 

to AASHTO PP 99-19 by a single operator, with the research team using the 9.5-mm ILS 

mixture.(6) The fabricated specimens were randomized and distributed to the participating 

laboratories. Each laboratory received eight pilot test specimens: three for |E*| testing conducted 

according to AASHTO TP 132-19, three for cyclic fatigue testing following AASHTO 

TP 133-21, and two extra specimens.(2,8) Participants were instructed to conduct a single cyclic 

fatigue test and |E*| test for initial feedback. The researchers reviewed the initial test results to 

identify any procedural issues. If they identified issues, they met with the participating laboratory 

to resolve the problem prior to completing the additional tests. 

To assess pilot |E*| test results, the differences between a participating laboratory and the 

research team’s results were compared against the reproducibility limits included in AASHTO 

T 378-17.(9) Note that these limits were established based on a large specimen fabricated by 

individual laboratories, so the conditions under which they were developed do not coincide with 

the pilot testing conducted in this study. However, they provided a reference to assess the general 

agreement of the results obtained by different laboratories. Cyclic fatigue test results were 

assessed using the proportional, integral, derivative (PID) data quality indicator established in 

Phase I of this project (i.e., maximum limit of 10 percent on the actuator strain standard error in 

cycles two through five of the cyclic fatigue test) and the research team’s experience evaluating 

the reproducibility of cyclic fatigue test results from Phase I of this project.

(3)

 Procedure 

discrepancies identified among the participating laboratories that contributed to biases in initial 

pilot results included the following:

• Thermal equilibration time differences.

• Temperature calibration errors. 

• |E*| tests using compensating springs. 

• Deviations from strain selection guidance in AASHTO TP 133-21.(2)

• PID tuning issues. 

The research team resolved these discrepancies before the full-scale ILS testing. The appendix 

presents a detailed discussion of the steps taken to identify and resolve procedural discrepancies, 

as well as the final pilot test results obtained after addressing procedural issues. Pilot test results 

were not used to develop the precision statements. 

FULL-SCALE TESTING

The participating laboratories prepared full-scale ILS samples and test specimens using separated 

loose mixture samples. Full-scale ILS testing used specimens fabricated by the participant 

laboratories because precision statements for reproducibility are contingent upon variability in 

both the specimen fabrication and the test procedures. In addition, the repeatability of specimens 

fabricated by a single laboratory and within individual laboratories is not expected to differ 

significantly based on the ILS results of AASHTO T 378-17.(10) 
7 

For each ILS mixture, the research team shipped six boxes of separated loose mixture samples 

(i.e., sufficient material to produce three gyratory-compacted samples) to the participating 

laboratories to perform a compaction study. The compaction study would determine the sample 

mass required to achieve the specified air void content in extracted test specimens. The research 

team provided 16 additional boxes of loose mixture for each mixture included in the 

large-specimen ILS and 8 additional boxes of loose mixture for each mixture included in the 

small-specimen ILS. These allocations allowed for the fabrication of two extra gyratory samples 

for both large- and small-specimen fabrication purposes, as needed. Laboratories were required 

to measure and report the test specimen air void content to ensure they were within the specified 

range. 

Laboratories were required to carry out three |E*| and three cyclic fatigue tests on each mixture 

and specimen geometry combination included in the ILS. Large-specimen |E*| testing adhered to 

AASHTO R 83-17 and AASHTO T 378-17.(9,11) Dynamic modulus tests of the small specimens 

followed AASHTO PP 99-19 and AASHTO TP 132-19.(6,8) In both large- and small-specimen 

|E*| tests, asphalt specimens are subjected to frequency-sweep testing at 10, 1, and 0.1 Hz at 

three temperatures. All specimens were tested at 4 and 20 °C. The third test temperature for the 

9.5-mm mixture was 35 °C. For the other mixtures, the third test temperature was 40 °C. The test 

temperatures were selected on the basis of the binder performance grade in accordance with 

AASHTO R 84-17.(11) 

Large-specimen cyclic fatigue testing followed AASHTO TP 107-22, whereas small-specimen 

cyclic fatigue testing adhered to AASHTO PP 99-19 and AASHTO TP 133-21.(1,2,7) The 

researchers selected the test temperature based on the guidance given in AASHTO TP 133-21, 

which specifies test temperatures based on the expected blended binder grade for the mixtures 

containing reclaimed or recycled materials. When a test resulted in a specimen failure outside the 

instrumented gauge points (i.e., end failure), as depicted in figure 2 of AASHTO TP 133-21, the 

test was considered invalid, and an additional specimen was tested.

TEST ANALYSIS PROCEDURES

The results of both AASHTO TP 133-21 and TP 107-22 are the S-VECD damage characteristic 

curve and failure criterion.

(1,2) The linear viscoelastic properties of the asphalt mixture are 

required to implement the S-VECD model, which is obtained from the |E*| test. FlexMATTM 2.0 

was used to analyze all AMPT |E*| and cyclic fatigue test data.

(13)

 The following sections 

provide information about the analysis of the ILS test results conducted within FlexMAT 2.0 to 

obtain the damage characteristic curve and failure criterion results. 

Dynamic Modulus Testing

The calculation of |E*|, phase angle, and data-quality indicators using AMPT dynamic modulus 

test results followed the method outlined in each respective |E*| standard. Consistent with 

AASHTO TP 133-21 and TP 107-22, the researchers used the two springs, two parabolic 

elements, and one dashpot (2S2P1D) model to characterize the |E*| as a function of temperature 

and frequency.

(1,2,14)

 Equations 1 through 3 show the 2S2P1D model for storage modulus. 
8 

(1) 

(2) 

(3) 

Where: 

E0 = maximum storage modulus value (kPa or psi). 

E00 = minimum storage modulus value (kPa or psi).

κ, δ, γ, h, β, τE = fitting coefficients. 

ωR = reduced angular frequency (rad/s). 

Eʹ2S2P1D = storage modulus from 2S2P1D model. 

To best reflect the effect of the differences in cyclic fatigue tests alone on the damage 

characteristic curve and failure criterion, the research team analyzed the |E*| test results of all 

laboratories for a given mixture and specimen geometry together and coupled the averaged result 

with all of the cyclic fatigue testing obtained from each lab. One way in which the 2S2P1D 

model is used within the S-VECD model is the calculation of the dynamic modulus ratio (DMR), 

defined in equation 4. The |E*| from the 2S2P1D model yields |E*|LVE in this equation, 

calculated at the test temperature and frequency corresponding to cyclic fatigue testing. In 

equation 4, |E*|fingerprint is defined as the |E*| determined from the fingerprint portion of the 

cyclic fatigue test. DMR is used in AASHTO TP 107-22 and AASHTO TP 133-21 to 

characterize the potential difference between the specimens tested in the |E*| test and the 

specimens used in the cyclic fatigue test.

(1,2) This value is also used to calculate parameters that 

define the damage characteristic curve. 

The AMPT cyclic fatigue test standards stipulate that the DMR must fall between 0.85 and 1.15 

to ensure sufficient consistency in the test results based on the ruggedness evaluation.(3)

 In some 

cases, the use of the average |E*| test results from all laboratories to calibrate the 2S2P1D model 

yielded DMR values that fell outside of the specified limits; in these cases, a subset of the |E*|

test results of the collective laboratories was identified for calibrating the 2S2P1D model that 

yielded DMR values within the specified range for all laboratories. The appendix includes the 

full-scale ILS dynamic modulus results and a summary of individual laboratories’ |E*| results 

that were used collectively to generate |E*| inputs for cyclic fatigue analysis of each mixture and 

specimen geometry combination. 

    

  

 

  '

 

 

 

              

 
9 

(4) 

Cyclic Fatigue Testing 

The damage characteristic curve constitutes the relationship between the normalized 

pseudosecant modulus C, which is a material integrity indicator, and damage S. The research 

team followed the standard equations for calculating these quantities, as presented in the 

AASHTO TP 107-22 and TP 133-21 test protocols.

(1,2) The pseudosecant modulus and damage 

are defined in equations 5 through 8; equations 5 and 6 are used to calculate the pseudosecant 

modulus and damage, respectively, for the tensile portion of the initial loading cycle, whereas 

equations 7 and 8 are used in later cycles.(15)

(5) 

Where:

σ = stress (kPa or psi). 

εR = pseudostrain. 

(6) 

Where:

ΔSi = damage growth between the current and previous time step. 

Ci = pseudosecant modulus at the current time step. 

Ci – 1 = pseudosecant modulus at the previous time step. 

ΔtR = reduced time step.

α = continuum damage power term, equal to 1+1/m where m is the maximum slope of the 

relaxation modulus master curve in log space. 

 

 

 
10 

(7) 

Where:

C* = cyclic pseudosecant modulus.

σpp = peak-to-peak stress (kPa or psi). 

= peak-to-peak pseudostrain. 

(8) 

Where: 

ΔSn = damage growth between the current and previous analysis cycle. 

= tension amplitude of the pseudostrain. 

K1 = form adjustment factor.

C*n = cyclic pseudosecant modulus at the current analysis cycle.

C*n – 1 = cyclic pseudosecant modulus at the previous analysis cycle.

Equation 9 is used to model the relationship between C and S, constituting the damage 

characteristic curve using the collective results of three test specimens.

(16) The fitting coefficients 

C11 and C12 are reported as one of the primary test results from the AMPT cyclic fatigue test. 

S represents the microstructural damage state variable.

(9) 

The failure criterion (DR) is another cyclic fatigue test result, defined in equation 10 and shown 

in figure 2. The DR is the average reduction in pseudostiffness up to failure.(17) The DR of a given 

mixture is determined as the average DR value of the collective test specimens. Thus, the 

deviation of the individual data points from the best fit line is a measure of within-laboratory 

specimen-to-specimen variability in the failure criterion results. The Nf is defined as the cycle 

where the product of peak-to-peak stress and cycle number reaches a peak value in AASHTO 

TP 133-21 and TP 107-22.(1,2)

(10) 

 

 

 
11 

Source: FHWA.

Cum = cumulative. 

Figure 2. Graph. DR failure criterion. 

STATISTICAL ANALYSIS

The researchers conducted repeatability and reproducibility analysis according to ASTM 

E691-20 and ASTM C670-15.(4,5) However, an inherent limitation of these standards is that they 

are “concerned exclusively with test methods which yield a single numerical figure as the test 

result.”(4) The damage characteristic curve is a cyclic fatigue test result; this functional 

relationship does not lend directly to the analysis procedure prescribed in ASTM E691-20 or 

C670-15. Therefore, the researchers also investigated an alternative means to define the 

repeatability and reproducibility of the damage characteristic curve using a recently developed 

functional data metric.

(18)

Single-Point Measure Analysis Conducted Using ASTM E691 and ASTM C670 

There are two underlying assumptions regarding the interpretation of ILS test results according 

to ASTM E691-20.(4) First, all laboratories are assumed to have the same variability level under 

the repeatability (within-laboratory) conditions for a given mixture. Second, mean laboratory test 

results are assumed to vary according to a normal distribution in each mixture. Following the 

terminology in the standard, a test result represents the value of a characteristic obtained by 

carrying out a specified test method. A test determination represents the value of a characteristic 

or dimension of a single test specimen derived from one or more observed values. An example of 

a test determination is a single test specimen’s damage characteristic curve.

ASTM E691-20 provides guidance for preparing a precision statement for test results.(4) ASTM 

C670-15 provides additional guidance for preparing a precision statement using the results of test

determinations.

(5)

 Based on the definition of test result and test determination, this study includes 

three test determinations for each laboratory per material. The test results are the best-fit damage 

characteristic curve across three specimens collectively and the average DR value from three 
12 

specimens. A test determination is the damage characteristic curve and DR value of one test 

specimen. Given that a laboratory would typically execute three test determinations to obtain a 

test result and not obtain two separate test results, repeatability of the cyclic fatigue test standards 

should reflect variability among test determinations within a lab. Therefore, the repeatability and 

reproducibility precision state development herein incorporate guidance from ASTM C670-15.(5)

The ASTM E691-20 and C670-15 standards strive to define limits for the repeatability and 

reproducibility of a test that will only be exceeded with a probability of 5 percent if the test is 

executed properly.(4,5)

As noted, ASTM E691-20 is written for tests that yield a single numerical figure as the test 

result, which is directly applicable to the failure criterion result, DR.

(4) However, the damage 

characteristic curve results are functional data rather than a single numerical figure. To comply 

with the standard, the research team selected the following six single-point values to represent 

the damage characteristic curve for initial analyses: C value at Smin, Smean, and Smax and shape 

factor at Smin, Smean, and Smax, Smin, Smean, and Smax represent the minimum, average, and maximum 

of all the S at failure for each material/specimen geometry. The shape factor represents the area 

between C = 1 and the C versus S curve at predefined levels of S. In this analysis, the shape 

factor was computed when S equals Smin, Smean, or Smax. Figure 3 shows an example of the 

C value and shape factor at Smax. 

Source: FHWA.

C = C value; S = shape factor.

Figure 3. Graph. An example of C value and shape factor. 

The analysis framework prescribed in ASTM E691-20 can be divided into three steps:(4)
13 

• Step 1: Evaluate test consistency. Test statistics are used to determine whether the 

collected data are adequately consistent to form the basis for a test method precision 

statement.

• Step 2: Investigate inconsistent data. The data that do not meet the consistency 

requirement are investigated to identify any possible errors, such as clerical or procedural 

errors. Then, the analysts should decide whether to include or delete the inconsistent data. 

• Step 3: Obtain the precision statistics. The repeatability and reproducibility standard 

deviations are calculated to establish the repeatability and reproducibility statements.

The researchers executed these three analysis steps as described in the following sections. 

Test Consistency

The statistical analysis for the estimates of precision statistics is simply a one-way analysis of 

variance (within and between laboratories) for each material. Therefore, severe outliers can 

significantly affect the analysis and invalidate the results. The consistency of the data should be 

examined before developing precision statistics. Equations 11 through 14 present the statistics 

used in this analysis according to ASTM E691-20.(4) Statistics h and k were used to evaluate the 

between-laboratory consistency and within-laboratory consistency, respectively. 

(11) 

Where: 

sx̅ = standard deviation of laboratory averages. 

x̅i = laboratory average for laboratory i. 

x̿= average of all laboratory averages. 

p = number of laboratories in ILS.

(12) 

Where: 

sr = repeatability standard deviation. 

si = within-laboratory standard deviation. 

(13) 
14 

(14) 

Where: 

η = between-laboratory consistency statistic. 

θ = within-laboratory consistency statistic.

The consistency statistics η and θ were calculated for each laboratory, geometry, and material 

combination to evaluate the consistency of the data collected in the ILS. The critical value of 

each statistic is based on the number of laboratories and number of tests each laboratory 

conducts. This study was designed to include seven laboratories and three cyclic fatigue tests for 

each laboratory. The corresponding critical values for η and θ for these conditions are ±2.05 and 

2.03, respectively (ASTM E691-20).(4) Consistency statistics were calculated for both the DR and 

damage characteristic curve results described in the previous section. 

Investigate Inconsistent Data

The inconsistent data flagged in the consistency test were investigated to make sure that there 

were no procedural or reporting errors. It should be noted that the critical values of the h and k

consistency statistics are calculated based on the 0.5 percent significance level rather than the 

5 percent often used in statistical analyses of materials (ASTM E691-20).(4) There are 42 h (or k) 

values for one single-point measure representing seven laboratories and six mixtures at a 

particular testing geometry. Theoretically, 0.21 out of 42 values are expected to stand out as 

inconsistent according to the 0.5-percent significance level. Therefore, any data flagged as 

inconsistent require careful investigation for rejection from precision statement development. 

Data ultimately deemed inconsistent were removed from the dataset before the repeatability and 

reproducibility statistics were computed. 

Obtain the Precision Statistics

Each laboratory conducted three test determinations in the AMPT cyclic fatigue ILS rather than 

obtaining three test results. Therefore, the repeatability and reproducibility statistics developed 

follow the guidance in ASTM C670-15, which applies to the development of repeatability 

statements based on individual test determinations and results. ASTM E691-20 only provides 

guidance for developing precision statements based on test results.(4,5)

The repeatability standard deviation, sr, represents the standard deviation of test determinations 

obtained under repeatability conditions. In this study, the reproducibility standard deviation was 

taken to be sx̅, which represents the standard deviation of test results obtained under 

reproducibility conditions (i.e., quantifies the difference between the test results of two 

laboratories). For the C at a specific S value, single-point measures of damage characteristic 

curve variability, 1−the average C was used as the basis for the calculation of the average value 

across laboratories because the standard deviation in the C value increases as the C value 

decreases (and correspondingly the S value increases). 

The corresponding repeatability and reproducibility coefficients of variation (COVs) were 

calculated using equation 15 and equation 16, respectively. 
15 

(15) 

(16) 

Where: 

sx̅% = reproducibility COV. 

x̿= average of all laboratory averages for shape factor, 1−average of all laboratory averages 

for C value. 

Due to the nature of the damage characteristic curve, the C or S value selected to carry out a 

single-point measure is highly likely to affect the repeatability and reproducibility results. Plots 

of repeatability and reproducibility COVs at different 1−C values were evaluated to identify the 

sensitivity of the repeatability and reproducibility to the chosen point within the damage 

characteristic curve. 

When the test determinations are the basis for repeatability, the single-operator precision 

statement is defined as the maximum allowable difference between multiple test determinations 

or the maximum acceptable difference between test results obtained as the average of multiple 

determinations (ASTM C670-15).(5)

For the repeatability of one test result with n test determinations, the maximum allowable range

(MAR) (i.e., the difference between the highest and lowest test determinations) expressed as a

COV is calculated using equation 17. 

(17) 

Where: 

an = multiplier affected by the number of test determinations, n, listed in table 3. 

sr% = repeatability COV.
16 

Table 3. Multiplier for repeatability statistic. 

Number of Test Determinations Multiplier

2 2.8 

3 3.3 

4 3.6 

5 3.9 

6 4.0 

7 4.2 

8 4.3 

9 4.4 

10 4.5 

ASTM C670-15 does not give very prescriptive instructions for defining the reproducibility of 

test results. It only specifies that “If the test method calls for reporting the average of more than 

one test determination, multilaboratory precision is expressed as a maximum allowable 

difference between averages of such groups obtained by two laboratories.”(5)

 Based on this 

guidance, the reproducibility precision of the AMPT cyclic fatigue tests was defined using sx̄, 

which constitutes the standard deviation of the test results between two laboratories. 

Correspondingly, equation 18 was used to define the reproducibility limit in terms of the 

allowable difference in test results reported by two laboratories, expressed as a COV. 

(18) 

Where: 

d2srpd(%) = allowable difference between test results reported by two laboratories expressed

as a COV. 

Advanced Analysis of Damage Characteristic Curve Repeatability and Reproducibility 

Using a Functional Data Metric

To capture the variation of the entire damage characteristic curve among test replicates better 

than the single-point measures, Ding et al. developed a variance index (v) (18)

 Herein, the 

definition of the variance index has been modified by first normalizing the damage characteristic 

curve and then performing the same calculation as shown in Ding et al. Because the curve is first 

normalized, the name of the index used for this project is “normalized variance index,” vnorm, and 

distinct definitions of vnorm for repeatability and reproducibility are given. However, using vnorm

to define the repeatability and reproducibility of damage characteristic curve results requires 

some deviation from the ASTM standards because the vnorm follows a gamma, rather than a 

normal, distribution. 

The vnorm is defined in equations 19 through 21. Equations 20 and 21 show that when the vnorm is 

calculated, the S values are first normalized to avoid bias based on the damage characteristic 

curve length (i.e., S at failure). As shown by the integral in equation 19, vnorm encompasses the 

total area between individual C versus S curves and the curve that resulted from optimization of 
17 

all individual curves with normalization based on the number of curves considered and a scaling 

factor, k. The multiplier k was applied to change the magnitude of vnorm from approximately 10−5

to 1 for convenience. Further details pertaining to the development of v, and thus vnorm, are 

provided in Ding et al.(18)

(19) 

(20) 

(21) 

Where: 

k = 1×105

, multiplier to change the magnitude.

Sall = all the S values from n replicates.

C(Snew)i = the fitted function for i

th replicate after normalization. 

C̅(Snew) = the fitted function of all the data after normalization. 

n = the total number of cyclic fatigue test specimens. 

Equations 22 through 24 are used to obtain the fitted functions after normalization (i.e., C(Snew)i 

and C̅(Snew)) in equation 19, using the originally defined damage characteristic curve model 

coefficients in equation 9 and Sfactor (defined in equation 21). 

(22) 

(23) 

(24) 

Where: 

C11_new, C12_new = the fitted parameters for damage-characteristic curve model after 

normalization. 

C11, C12 = the fitted parameters before normalization.

Using vnorm to define repeatability. 

Figure 4 illustrates the calculation of vnorm to evaluate damage characteristic curve repeatability. 

Damage characteristic curve results for three specimens are shown in figure 4-A. The Smean value 
18 

is calculated by averaging all S at failure for three specimens, and the corresponding Sfactor is

calculated using equation 21. All S values are normalized using equation 20, and the 

corresponding rescaled damage characteristic curve results are shown in figure 4-B. Fitted 

functions for individual specimens (i.e., fit_individual) and the overall fit from the aggregated 

results of all specimens (i.e., fit_all) are shown in figure 4-B as well. The fitting coefficients in 

these cases are determined using equations 23 and 24. The individual specimen curves are used 

to calculate C(Snew)i, and the overall fit is used to calculate C̅(Snew) and determine vnorm using 

equation 25 (obtained by substituting equation 22 and the equivalent for the overall fit into 

equation 19 and solving the integral). 

(25) 

Where: 

C11-i_new, C12-i_new = the fitted parameters for specimen i after normalization. 

C̅11_new, C̅12_new = the fitted parameters for all three specimens after normalization.

Source: FHWA.

A. Finding Smean for three specimens.

 
19 

Source: FHWA.

B. Normalizing S values and fitting individual and overall damage characteristic curves. 

Figure 4. Graphs. An example for calculating repeatability vnorm. 

Using vnorm to Define Reproducibility

Equation 26 provides the vnorm to define damage characteristic curve reproducibility. The 

denominator is 1 in equation 19 when vnorm is calculated to evaluate the reproducibility of the 

results of two laboratories because n = 2 in this case. 

(26) 

Where: 

C̅(S)i = the fitted function for ith laboratory after normalization. 

C̿(S) = the fitted function of two laboratories being compared after normalization.

Figure 5 illustrates the vnorm approach for calculating the reproducibility of damage characteristic 

curves using equation 26. Figure 5-A shows the selected data points of fitted curves for the 

individual laboratories obtained from the collective results of three test specimens. The damage 

at failure for each fitted curve is the Smax across all the specimens of each laboratory. The 

selected data points for each fitted curve are based on predefined damage values of 0, 100, 300, 

500, and 1,000, and damage values with an interval of 0.05×Smax from 0.05×Smax to Smax. The

Smean value is calculated by averaging Smax of two laboratories, and the Sfactor for reproducibility is 

calculated using equation 21. Figure 5-B presents the fitted curve across two laboratory results

with the selected data points. The fitted curves for each laboratory and the fitted curve across two 

laboratory results are then rescaled, and C̿(Snew) is determined for each of the three fitted curves. 

Corresponding rescaled model fitting coefficients are determined using equations 23 and 24. The 

reproducibility vnorm captures the reproducibility of two test results and is calculated using 

equation 27. Recall that a test result is the damage characteristic curve that best fits the collective 

C(S) data from three test specimens 

 
20 

(27) 

Where: 

C̅11-i_new, C̅12-i_new = the fitted parameters for laboratory i after normalization. 

C̿11_new, C̿12_new = the fitted parameters for both laboratory results after normalization.

Source: FHWA.

A. Finding Smean for two laboratory results.

Source: FHWA.

B. Normalizing S values and fitting overall damage characteristic curves.

Figure 5. Graphs. An example for calculating reproducibility vnorm. 

Precision Statements

Ding et al. showed that vnorm values obtained from different operators for a given material follow 

a gamma distribution, defined in equation 28.(18)

 The analysis procedures in ASTM E691–20 and 

C670-15 are only directly applicable to test results that follow a normal distribution. Therefore,

 

 
21 

vnorm is not directly amenable to the guidance for precision statement development given in the 

ASTM standards.(4,5)

 However, the research team used the underlying principle behind the 

precision statement developed in the ASTM standards as the basis for developing a precision 

statement using vnorm herein. A gamma distribution was fit using the ILS results and was 

subsequently used to define the maximum limit for vnorm that will be exceeded with a probability 

of 5 percent if the test is executed properly. 

(28) 

Where: 

fV = probability density function for vnorm.

K = the shape parameter in gamma distribution.

Γ = the gamma function 

λ = the scale parameter in gamma distribution.

ANALYSIS OF THE PRECISION STATEMENT IMPLICATIONS ON UNCERTAINTY 

IN THE TEST RESULTS’ PRACTICAL APPLICATIONS

Uncertainty in FlexPAVETM %Damage Predictions 

One of the primary uses of the AMPT cyclic fatigue test results is the mechanistic prediction of 

fatigue damage evolution using FlexPAVE.(19) FlexPAVE version 1.1, developed under the 

sponsorship of FHWA, is a pavement-performance analysis software package that uses |E*| and 

S-VECD models to integrate the effects of loading rate and temperature into a structural model 

that then computes the pavement’s responses and damage evolution. The FlexPAVE program 

calculates the long-term fatigue damage and rut depth of asphalt pavements under changing 

environmental and loading conditions. The output from FlexPAVE fatigue cracking simulations 

is the percentage of damage (%Damage), which is computed as the ratio of the damaged 

cross-sectional area to the total effective cross-sectional area; additional details are provided 

elsewhere.(20) Ghanbari recently established a method to estimate how the uncertainties from 

material model inputs propagate to uncertainty in %Damage predictions obtained from 

FlexPAVE.

(21)

 This study adopted this framework to evaluate the implications of the combined 

uncertainty in the damage characteristic curve and failure criterion results on the uncertainty 

in %Damage predictions. 

Ghanbari used Bayesian inference-based Markov Chain Monte Carlo (MCMC) methods to 

estimate how the uncertainties from the material model’s input parameters propagate to 

uncertainty in the FlexPAVE Version 1.1 predicted pavement performance.(21) First, predictive 

envelopes at predefined statistical levels were generated independently for each material model 

(i.e., |E*|, damage characteristic curve, and failure criterion) using the MCMC method. The 

predictive envelope defines the range that a new observation will fall into with a defined level of 

probability. Based on predictive envelope, 1,000 possible results for each of the three models 

were generated. Next, 1,000 possible sets of FlexPAVE inputs were generated by randomly 

sampling from the distributions of material model inputs without replacement. 
22 

FlexPAVE simulations were then conducted using these inputs to generate 1,000 sets 

of %Damage results that reflect the uncertainty in pavement performance simulation as a 

consequence of the material model uncertainty. This analysis framework was carried out using 

different materials, pavement structures, traffic conditions, climate conditions, and base and 

subgrade moduli. Ghanbari’s analysis demonstrated that the uncertainty in %Damage, defined by 

the standard deviation in the predicted %Damage, depends on the average %Damage and 

material model uncertainty, irrespective of the pavement simulation conditions (i.e., structure, 

traffic, climate), as shown in equation 29. Correspondingly, equation 30 was defined to relate 

uncertainty in %Damage predictions to uncertainty in the |E*| and cyclic fatigue model 

uncertainties. The aTotal in equation 30 is defined by summing the contributions from uncertainty 

in the |E*| model, damage characteristic model, and failure criterion, as shown in

equations 31-36. 

(29) 

(30) 

(31) 

(32) 

(33) 

(34) 

(35) 

(36) 

Where: 

%DP = predicted damage at the reliability level P. 

ZP = standard normal deviation corresponding to reliability level P. 
23 

stdTotal = standard deviation of %Damage. 

%D̅= average %Damage. 

ILVE_test = linear viscoelastic (|E*| model) variation index from test results.

= DR variation index from test results.

n = number of cyclic fatigue test specimens. 

nLVE = number of dynamic modulus test specimens.

 = DR value for ith specimen. 

= average DR value for all the n specimens.

E′fit_i,j = fitted storage modulus from the fitted curve for the ith specimen and jth temperature 

and frequency. 

E′fit_avg,j = storage modulus from the fitted curve using all of nLVE specimens at the jth 

temperature and frequency.

According to Ghanbari’s framework, all the ILS results that meet the proposed repeatability and 

reproducibility precision limits were used to identify the maximum uncertainty in the %Damage

predictions from FlexPAVE.

(21) Figure 6 presents the procedure used to assess the repeatability 

of %Damage predictions, which defines the uncertainty in %Damage in one test result (i.e., three 

test specimens) obtained in a single laboratory on a single material. As shown in this figure, 

ultimately, vnorm is selected for defining damage characteristic curve repeatability and, thus, used 

herein. 

The research team calculated the repeatability vnorm and maximum DR difference across three 

specimens for each laboratory and mixture. If both vnorm and maximum DR results met the

repeatability precision limits—indicating the repeatability was acceptable—then the repeatability 

vnorm and were used to calculate aTotal in equation 31. Since this analysis focused on 

identifying uncertainty induced from the cyclic fatigue test alone, a1—which reflects the 

contribution of variation in the |E*| curve results to %Damage uncertainty—was set to zero. The 

researchers then used the calculated aTotal to calculate the standard deviation in %Damage using 

equation 30. Finally, the standard deviation was used to calculate the 95 percent confidence 

interval of %Damage using equation 29. A %Damage of 15 percent was used in all analyses 

conducted because it reflects a critical point in the transfer function that relates %Damage to 

field observations of cracking.(22) At less than 15 %Damage, little fatigue cracking is observed 

on pavement surfaces, but after this limit, small increments in %Damage correspond to notable 

increases in field cracking.
24 

Source: FHWA. 

Figure 6. Flowchart. The procedure to assess uncertainty in %Damage predictions under 

repeatability conditions. 

Figure 7 presents the procedure used to assess uncertainty in %Damage predictions for any two 

laboratory results obtained using the same material under reproducibility conditions. First, the 

research team evaluated the individual results of the two laboratories under consideration. If the 

results exceeded the repeatability limits for vnorm or DR, the data were not used for reproducibility 

analysis. If the differences in the test results between the two laboratories exceeded 

reproducibility limits for these parameters, the data were also rejected from the analysis. If the 

pair of laboratories’ results met both repeatability and reproducibility limits, then the 

reproducibility vnorm and were used to calculate aTotal in equation 31. Consistent with 

repeatability analyses, a1—which reflects the contribution of variation in the |E*| curve results 

to %Damage uncertainty—was set to zero. It should be noted that reproducibility uses 

the average DR for each laboratory as . The calculated aTotal was then used to calculate the 

standard deviation in %Damage using equation 30. Finally, the standard deviation was used to 

calculate the 95 percent confidence interval of %Damage using equation 29. An 

average %Damage of 15 percent was used in all analyses, consistent with the repeatability 

analyses.

The repeatability uncertainty was computed for each of the seven laboratory’s results for each 

mixture and specimen geometry combination. The reproducibility uncertainty was calculated for 

each pair of the seven laboratory’s results for each mixture and geometry combination. In total, 
25 

there were 42 results for the repeatability analysis and 126 (i.e., 21×6) results for the

reproducibility analysis before results that did not meet the proposed precision limits were 

rejected. 

Ultimately, the results of the FlexPAVE analysis defined the maximum and median repeatability 

and reproducibility standard deviations of %Damage for each mixture and specimen geometry 

combination. These maximum and median standard deviations were used to perform assessments 

of the maximum and common uncertainty expected for test results that meet the precision 

statements established in this study. 

Source: FHWA. 

Figure 7. Flowchart. The procedure to assess uncertainty in %Damage predictions under 

reproducibility conditions. 

Uncertainty in Apparent Damage Capacity 

AMPT cyclic fatigue test results can also be used to calculate an index parameter to indicate 

asphalt mixture fatigue resistance, termed the apparent damage capacity (Sapp) and defined in 

equation 37.(23) The Sapp value incorporates the effects of the material’s modulus and toughness 
26 

on its fatigue resistance and measures the amount of fatigue damage the material can tolerate

under loading. The temperature for the Sapp calculation is equal to the average of high- and 

low-temperature grades at the standard 98-percent reliability level minus 3 °C for the climate 

where the mixture is to be placed. Higher Sapp values indicate increased fatigue resistance. The 

Sapp value is sensitive to mixture factors (e.g., aggregate gradation, asphalt binder content, 

reclaimed asphalt pavement content, and asphalt binder grade), compaction, and aging and meets 

general expectations regarding the effects of these parameters on fatigue-cracking 

performance.(23) Similar to the %Damage analysis, the implications of the combined variation in 

the damage characteristic curve and failure criterion results permitted under the proposed 

precision limits on the uncertainty in Sapp were evaluated. 

(37) 

Where:

α = damage growth rate. 

aT = time–temperature shift factor at a given temperature.

To evaluate the uncertainty in Sapp, the Sapp values were calculated using the results of individual 

test determinations. The researchers calculated the Sapp value for each laboratory, termed the 

“measured” Sapp values herein. The measured Sapp values were used to assess the variation of Sapp

values among test determinations within a given laboratory and the reproducibility of Sapp results 

among laboratories. In addition, the researchers developed a framework to further evaluate Sapp

uncertainty due to the combined uncertainty that is analogous to the framework that Ghanbari 

used to quantify %Damage uncertainty.(21) The repeatability of Sapp values among test 

determinations could be directly assessed from the measured Sapp values. 

However, the repeatability of Sapp results could not be directly measured from the ILS results 

since each laboratory only generated one test result on a given mixture and specimen geometry 

combination. The researchers used supplementary analysis to assess the repeatability of Sapp

results. Figure 8 presents the flowchart for assessing the uncertainty in Sapp in one test result (i.e., 

three test specimens) obtained in a single laboratory on a single material. Consistent with 

the %Damage analysis, data were only incorporated into the analysis if the repeatability limits 

for vnorm or DR were met when the uncertainty in Sapp was analyzed. The researchers used results 

meeting the repeatability limits to generate independent predictive intervals for damage 

characteristic curves and DR using the MCMC method. The predictive intervals represent the 
27 

range within which a new test determination (either damage characteristic curve or DR value) 

will fall within with 95 probability. 

Source: FHWA. 

Figure 8. Flowchart. The procedure to assess uncertainty in Sapp values under repeatability 

conditions.

Then, 5,000 possible damage characteristic curves and DR determinations were simulated based 

on the prediction intervals. Next, three sets of damage characteristic curves and DR

determinations were randomly selected from the 5,000 sets of simulated parameters to reflect 

three test determinations that comprise a possible test result. The collective results of the three 

simulated test determinations were used to calculate an average DR value and damage 

characteristic curve fitting coefficients in equation 9, which constitute an AMPT cyclic fatigue 

test result. 
28 

The process of randomly sampling three test determinations to constitute a possible test result 

was conducted 5,000 times with replacement and used to calculate 5,000 corresponding 

simulated Sapp values from a given lab. Consistent with the other analysis within this study, 

consistent |E*| were used in all Sapp calculations for a given specimen geometry and mixture 

combination, irrespective of the lab. 

The 95 percent confidence interval was defined using the 5,000 simulated Sapp values. The values 

falling within the bounds, corresponding to 2.5 and 97.5 percentiles, were used to calculate the 

Sapp COV according to equation 38. The analysis was conducted for each mixture, laboratory, 

and specimen geometry combination. 

(38) 

Where:

COVrpt% = repeatability COV for Sapp.

nS = number of simulations in the 95-percent confidence interval, 4,750. 

Sapp-i = Sapp value for the ith simulation. 

S̅app = average Sapp value across nS simulations.

Supplementary analysis was also conducted to provide a measure of the maximum expected 

variation in Sapp among the laboratories, given the proposed reproducibility precision limits. 

Figure 9 presents the framework used to assess the uncertainty in Sapp obtained using the same 

material and specimen geometry under reproducibility conditions. Consistent with the %Damage

analysis, only data that met both repeatability and reproducibility precision limits were included 

in the analysis of the uncertainty in Sapp under reproducibility conditions. If a pair of laboratories’ 

results met both repeatability and reproducibility limits, then the 2.5 percentile and 

97.5 percentile Sapp values from each of the two laboratories—obtained from the repeatability 

analysis from the process depicted in figure 8—were used to calculate the reproducibility COV% 

for Sapp, according to equations 39 and 40. That is, the reported reproducibility COV% reflects 
29 

the maximum possible between-laboratory variations in Sapp results based on the 95 percent 

confidence intervals for Sapp results within a given lab. 

Source: FHWA. 

Figure 9. Flowchart. The procedure to assess uncertainty in Sapp values under 

reproducibility conditions.

(39) 
30 

(40) 

Where:

COVrpd% = reproducibility COV for Sapp. 

Sapp-1

2.5 = the 2.5 percentile of Sapp for laboratory 1. 

Sapp-1

97.5 = the 97.5 percentile of Sapp for laboratory 1. 

Sapp-2

2.5 = the 2.5 percentile of Sapp for laboratory 2. 

Sapp-2

97.5 = the 97.5 percentile of Sapp for laboratory 2. 

COV(x,y) = COV between x and y. 

Consistent with the %Damage analysis, the researchers computed the Sapp repeatability 

uncertainty for the results for each mixture and specimen geometry combination from the seven 

laboratories. The researchers computed the reproducibility uncertainty for each pair of the seven 

laboratory’s results for each mixture and geometry combination. In total, there were 42 results 

for the repeatability analysis and 126 (i.e., 21×6) results for the reproducibility analysis before 

the results that did not meet the proposed precision limits were rejected. The maximum and 

median repeatability and reproducibility Sapp COV% of Sapp for each mixture and specimen 

geometry combination were extracted to provide measures of the typical and maximum possible 

uncertainty that can be expected for laboratory results meeting the proposed precision limits. 
31 

CHAPTER 3. RESULTS

ILS RESULTS OVERVIEW

Figure 10 through figure 33 show the ILS test results for each mixture and test geometry. Graphs 

of individual specimen damage characteristic curves convey the results of individual test 

determinations. In contrast, the fitted damage characteristic curve graphs convey the test result

(the curve best fit to the results of the three test determinations). The bar graphs of the DR results 

show the average DR values from each laboratory, with error bars denoting the range of 

individual specimen values from each laboratory. The cumulative (1−C) values versus fatigue 

life graphs provide another means to visualize failure criterion repeatability and reproducibility.

Based on visual observation of the results, the 9.5-mm mixture (small- and large-specimen 

geometries) has the least variation among and within laboratories, and the 19-mm small and 

25-mm large materials have the most variation among and within the laboratories. Trends among 

the repeatability and reproducibility of the two specimen geometries for the 9.5-mm and 

12.5-mm mixtures are difficult to ascertain visually.

Source: FHWA.

Figure 10. Graph. Individual specimen damage characteristic curves for 9.5-mm small 

specimens. 
32 

Source: FHWA.

Figure 11. Graph. Fitted damage characteristic curves for 9.5-mm small specimens. 

Source: FHWA.

Figure 12. Graph. Cumulative (1−C) results from individual tests for 9.5-mm small 

specimens. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50,000 100,000 150,000

C

S

Lab A

Lab B

Lab C

Lab D

Lab E

Lab F

Lab G

0

5,000

10,000

15,000

20,000

25,000

30,000

35,000

40,000

45,000

0 20,000 40,000 60,000 80,000 100,000

Cum (1 – C)

Nf

Lab A: y = 0.5108x 

Lab B: y = 0.5202x 

Lab C: y = 0.4979x 

Lab D: y = 0.5579x 

Lab E: y = 0.4317x 

Lab F: y = 0.5091x 

Lab G: y = 0.5389x 
33 

Source: FHWA.

Figure 13. Graph. DR results for 9.5-mm small specimens. 

Source: FHWA.

Figure 14. Graph. Individual specimen damage characteristic curves for 9.5-mm large 

specimens. 
34 

Source: FHWA.

Figure 15. Graph. Fitted damage characteristic curves for 9.5-mm large specimens. 

Source: FHWA.

Figure 16. Graph. Cumulative (1−C) results from individual tests for 9.5-mm large 

specimens. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50,000 100,000 150,000

C

S

Lab A

Lab B

Lab C

Lab D

Lab E

Lab F

Lab G

Lab A: y = 0.4528x 

Lab B: y = 0.5118x 

Lab C: y = 0.3955x 

Lab D: y = 0.4764x 

Lab E: y = 0.4418x 

Lab F: y = 0.5041x 

Lab G: y = 0.4911x 
35 

Source: FHWA.

Figure 17. Graph. DR results for 9.5-mm large specimens. 

Source: FHWA.

Figure 18. Graph. Individual specimen damage characteristic curves for 12.5-mm small 

specimens. 
36 

Source: FHWA.

Figure 19. Graph. Fitted damage characteristic curves for 12.5-mm small specimens. 

Source: FHWA.

Figure 20. Graph. Cumulative (1−C) results from individual tests for 12.5-mm small 

specimens. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100,000 200,000 300,000 400,000 500,000

C

S

Lab A

Lab B

Lab C

Lab D

Lab E

Lab F

Lab G

Lab A: y = 0.6074x 

Lab B: y = 0.6416x 

Lab C: y = 0.6162x 

Lab D: y = 0.6602x 

Lab E: y = 0.5624x 

Lab F: y = 0.6017x 

Lab G: y = 0.6107x 
37 

Source: FHWA.

Figure 21. Graph. DR results for 12.5-mm small specimens. 

Source: FHWA.

Figure 22. Graph. Individual specimen damage characteristic curves for 12.5-mm large 

specimens. 
38 

Source: FHWA.

Figure 23. Graph. Fitted damage characteristic curves for 12.5-mm large specimens. 

Source: FHWA.

Figure 24. Graph. Cumulative (1−C) results from individual tests for 12.5-mm large 

specimens. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100,000 200,000 300,000 400,000 500,000

C

S

Lab A

Lab B

Lab C

Lab D

Lab E

Lab F

Lab G

Lab A: y = 0.5689x

Lab B: y = 0.614x

Lab C: y = 0.5502x

Lab D: y = 0.5207x

Lab E: y = 0.5331x

Lab F: y = 0.602x

Lab G: y = 0.5652x
39 

Source: FHWA.

Figure 25. Graph. DR results for 12.5-mm large specimens. 

Source: FHWA.

Figure 26. Graph. Individual specimen damage characteristic curves for 19-mm small 

specimens. 
40 

Source: FHWA.

Figure 27. Graph. Fitted damage characteristic curves for 19-mm small specimens. 

Source: FHWA.

Figure 28. Graph. Cumulative (1−C) results from individual tests for 19-mm small 

specimens. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50,000 100,000 150,000 200,000 250,000 300,000

C

S

Lab A

Lab B

Lab C

Lab D

Lab E

Lab F

Lab G

Lab A: y = 0.5066x 

Lab B: y = 0.4969x 

Lab C: y = 0.4753x 

Lab D: y = 0.5098x 

Lab E: y = 0.3997x 

Lab F: y = 0.5087x 

Lab G: y = 0.5138x 
41 

Source: FHWA.

Figure 29. Graph. DR results for 19-mm small specimens. 

Source: FHWA.

Figure 30. Graph. Individual specimen damage characteristic curves for 25-mm large 

specimens. 
42 

Source: FHWA.

Figure 31. Graph. Fitted damage characteristic curves for 25-mm large specimens. 

Source: FHWA.

Figure 32. Graph. Cumulative (1−C) results from individual tests for 25-mm large 

specimens. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50,000 100,000 150,000 200,000

C

S

Lab A

Lab B

Lab C

Lab D

Lab E

Lab F

Lab G

Lab A: y = 0.5268x 

Lab B: y = 0.5054x 

Lab C: y = 0.4648x 

Lab D: y = 0.4876x 

Lab E: y = 0.5085x 

Lab F: y = 0.5077x 

Lab G: y = 0.4692x 
43 

Source: FHWA.

Figure 33. Graph. DR results for 25-mm large specimens. 

ILS RESULTS STATISTICAL ANALYSIS

Consistency Statistics

The researchers evaluated the consistency of the test results using the seven selected single-point 

parameters to assess the damage characteristic curve and failure criterion per ASTM E691.(4) The 

analysis yielded 84 consistency statistics (42 for η and 42 for θ) for each single-point parameter. 

Table 4 summarizes the percentage of the consistency statistics that exceed the critical value for 

each parameter. The analysis indicated a high degree of consistency within and between 

laboratories. All of the flagged data failed the within-laboratory consistency test. All of the 

inconsistent results belong to laboratory E’s 9.5-mm large-mixture results corresponding to the 

C values at Smin, Smean, and Smax and shape factors at Smean and Smax. 

The researchers reviewed laboratory E’s 9.5-mm large-specimen results and identified no 

procedural or data quality issues. They further investigated the impact of laboratory E’s 

inconsistent results by conducting the repeatability and reproducibility analysis for the 9.5-mm 

mixture large-specimen results, both including laboratory E’s results and excluding them. The 

comparison indicated that excluding the laboratory E 9.5-mm mixture large-specimen data 

provides more consistent trends with respect to NMAS than when the results are included. 

Therefore, laboratory E’s 9.5-mm mixture large-specimen data were ultimately excluded from 

the statistical analysis used to develop the precision statement. The following results do not 

include laboratory E’s 9.5-mm large-specimen data.
44 

Table 4. Percentage of data exceeding critical consistency value. 

Percent

C at

Smin

C at

Smean

C at

Smax

Shape

at Smin

Shape

at Smean

Shape

at Smax DR All

η

(between

-lab) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

θ

(withinlab) 2.38 2.38 2.38 0.00 2.38 2.38 0.00 2.04

Single-Point Measure Repeatability and Reproducibility Analysis Conducted Following

ASTM E691

The repeatability (sr%) and reproducibility (sx̅%) COV results for damage characteristic curve

are presented in table 5. The column labeled “9.5-mm large-E” excludes laboratory E data; the 

remaining tables presented in this report follow this naming structure. The results show that the 

COVs change with the chosen S value for analysis in terms of both the C value and shape factor. 

However, there is no clear trend between the chosen S value and the corresponding COV that 

suits all the mixtures with different geometry. The repeatability and reproducibility COVs for the 

damage characteristic curve parameters are generally higher for the 19-mm and 25-mm mixtures 

than the smaller NMAS mixture results for a given specimen geometry, which matches 

expectations. However, trends for NMAS are less clear in the DR results. The DR COV results 

show that the 19-mm small mixture has the highest repeatability COV, and 9.5-mm large mixture 

has the highest reproducibility COV, even when laboratory E’s data are excluded.
45 

Table 5. Repeatability and reproducibility COV. 

Parameters

9.5-mm 

Small

9.5-mm 

Small

9.5-mm 

Large-E 

9.5-mm 

Large-E 

12.5-

mm 

Small

12.5-

mm 

Small

12.5-

mm 

Large

12.5-

mm 

Large

19-mm 

Small

19-mm 

Small

25-mm 

Large

25-mm 

Large

- sr% sx̅% sr% sx̅% sr% sx̅% sr% sx̅% sr% sx̅% sr% sx̅%

C at Smin 1.79 2.30 1.88 2.74 2.57 2.62 2.58 2.91 6.24 7.40 6.10 5.82

C at Smean 1.77 2.24 1.71 3.17 3.42 3.21 2.82 3.65 6.54 7.69 6.74 5.12

C at Smax 1.82 2.25 1.69 3.32 4.47 3.88 3.15 4.56 6.80 7.92 7.39 4.86

Shape at

Smin

2.17 2.64 2.41 2.52 2.76 2.46 2.19 1.58 6.07 7.15 6.02 6.80

Shape at

Smean

1.91 2.41 2.08 2.54 2.43 2.46 2.35 2.15 6.19 7.34 6.08 5.83

Shape at

Smax

1.82 2.33 2.00 2.59 2.85 2.80 2.60 2.98 6.35 7.50 6.42 5.35

DR 5.24 6.19 4.14 9.27 6.41 4.85 4.83 5.82 12.85 7.07 7.68 6.23
46 

Figure 34 and figure 35 present the variation in the repeatability and reproducibility COVs as a 

function of the average C value for C values at a specific S value and shape factors, respectively. 

The trend for each mixture in terms of either C value or shape factor was captured by using a 

finely spaced increment with ∆S, where ∆S is the increment of damage (1,000 units until a 

maximum value of 105

). Therefore, the trends conveyed by the lines in figure 34 and figure 35

reflect the true trend since interpolation to form the lines was minimal. The average C values 

corresponding to Smin, Smean, and Smax are identified as points (chosen S) to provide context to the 

results shown in table 5. Both C values and shape factor show similar trends in their COVs with 

respect to the average C value at which the parameter is evaluated. These results indicate that 

shape factor, although expressed as area of damage characteristic curve results, provides similar 

information to the C value alone. The results also suggest that defining the test precision based 

on C (or shape factor) at the average S corresponding to (1−C̿) = 0.7 may help avoid bias to the 

chosen S value when a single point measure of the damage characteristic curve is used. The 

results clearly indicate higher variation within and across laboratories in the 19-mm and 25-mm 

mixtures compared to the smaller NMAS mixtures.

Source: FHWA.

L = large; L-E = large with exclusion of laboratory E; S = small.

A. Sensitivity analysis of repeatability COV (sr%) on single-point C value selection. 

 

 
47 

Source: FHWA.

B. Sensitivity analysis of reproducibility COV (sx̅%) on single-point C value selection. 

Figure 34. Graphs. Sensitivity analysis of repeatability and reproducibility COV on 

single-point C value selection. 

Source: FHWA.

A. Sensitivity analysis of repeatability COV (sr%) on single-point shape factor selection. 

 

 

 
48 

Source: FHWA.

B. Sensitivity analysis of reproducibility COV (sx̅%) on single-point shape factor selection. 

Figure 35. Graphs. Sensitivity analysis of repeatability and reproducibility COV on 

single-point shape factor selection. 

The researchers developed repeatability and reproducibility limits for the single-point measures 

(C value and shape factor) using the repeatability and reproducibility COVs with (1−C̿)=0.7 (i.e., 

an average C value of 0.3) for damage characteristic curve results based on the results of the 

sensitivity analysis. Limits for DR were defined using the repeatability and reproducibility 

standard deviation because DR is a single-point measure that is independent of the selection of S. 

Following the ASTM E1169-20 and ASTM C670-15 standards, the repeatability and 

reproducibility limits define the variation in the result that will not be exceeded with 95 percent

confidence.(4,5) The results are shown in table 6 through table 8. The limits for the repeatability of 

three specimens are conveyed as the maximum allowable difference among three specimens 

(COV%), according to equation 17. The reproducibility limits convey the maximum allowable 

difference between two test results obtained in different laboratories (COV%), defined according 

to equation 18. In general, the repeatability and reproducibility limits rise with increasing NMAS 

for a given specimen geometry. The only exception is the DR reproducibility limits, which do not 

reflect a clear trend with respect to NMAS. Note that the repeatability limits for |E*| and phase 

angle in AASHTO T 378-17 depend on modulus and NMAS but the reproducibility limits are 

independent of NMAS.(9) Thus, the trends in repeatability and reproducibility regarding NMAS 

observed in this study align with the findings of AASHTO T 378 ILS. 

Table 6. Repeatability and reproducibility limits (COV) for C value at (1−C̿)=0.7. 

Mixtures

Repeatability

(MAR%) 

Reproducibility

(d2srpd%)

9.5 S 5.85 6.28 

9.5 L 5.75 8.49 

12.5 S 9.38 7.90 

12.5 L 8.52 8.18 

19 S 21.45 21.44 

25 L 21.48 14.78 

 

 
49 

Table 7. Repeatability and reproducibility limits (COV) for shape factor at (1−C̿)=0.7. 

Mixtures

Repeatability

(MAR%)

Reproducibility

(d2srpd%)

9.5 S 6.26 6.72

9.5 L 7.03 7.04

12.5 S 8.33 6.72

12.5 L 7.25 4.46

19 S 20.37 20.48

25 L 19.81 17.03

Table 8. Repeatability and reproducibility precision statements (standard deviation) of DR. 

Mixture

Average 

DR

Repeatability

(MAR)

Reproducibility

(d2srpt)

9.5 S 0.507 0.088 0.088

9.5 L-E 0.479 0.065 0.124

12.5 S 0.608 0.129 0.083

12.5 L 0.559 0.089 0.091

19 S 0.505 0.214 0.100

25 L 0.488 0.124 0.085

Advanced Analysis of Damage Characteristic Curve Repeatability and Reproducibility 

Using a Functional Data Metric

Table 9 and table 10 show the vnorm values calculated using the ILS test results in terms of 

repeatability and reproducibility, respectively. The corresponding precision limit results are 

shown in table 11. These precision limits were defined by separately fitting the gamma function 

given in equation 28 to the distribution of vnorm repeatability and reproducibility results and using 

the resultant gamma functions to define the vnorm limits that would not be exceeded with 

95 percent confidence. Analysis for 9.5-mm large mixture was conducted without laboratory E’s

data. Precision limit analysis for repeatability in table 11 identified the repeatability of one test 

result (i.e., repeatability among three test determinations). 

Generally, increasing the NMAS yields higher repeatability and reproducibility limits for vnorm

(and hence, indicates higher variance) for a given specimen geometry. Trends with respect to 

specimen geometry vary, indicating separate precision limits may be warranted for the two 

specimen geometries. In two cases, 12.5-mm small-specimen geometry and 25-mm mixture 

large-specimen geometry, the reproducibility vnorm limits are smaller than the corresponding 

repeatability limits. Because vnorm for assessing repeatability evaluates variation among test 

determinations (i.e., among three individual test specimens), and vnorm for assessing 

reproducibility evaluates variation among test results (i.e., results of three test specimens each in 

two laboratories), they are not directly comparable. The variation in three individual specimens 

is expected to exceed the variation in the average results of two sets of three specimens, which 

may explain the observed trends for the 25-mm mixture and 12.5-mm small-specimen results. 
50 

Table 9. Repeatability vnorm for different laboratories and different materials.

Laboratory 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L

A 5.5 14.6 0.7 98.3 6.4 527.9

B 12.1 1.5 38.3 21.7 274.8 142.7

C 2.3 6.3 38.7 0.3 38.2 11.2

D 14.7 0.4 88.4 3.7 19.9 66.0

E 20.6 107.0 21.4 16.6 163.0 68.8

F 7.3 4.2 49.9 25.9 117.7 68.6

G 2.9 39.8 7.5 14.9 1.9 38.4

Table 10. Reproducibility vnorm for different laboratories and different materials. 

Laboratory Comparison 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L

A-B 6 9 2 58 170 89

A-C 15 46 49 82 105 19

A-D 23 8 8 131 1 62

A-E 64 65 26 99 32 174

A-F 43 11 20 1 113 81

A-G 27 6 4 55 174 76

B-C 2 40 37 8 7 42

B-D 6 9 10 23 188 15

B-E 33 38 21 13 54 15

B-F 18 6 11 65 8 1

B-G 8 20 13 6 680 280

C-D 2 18 84 4 117 9

C-E 18 7 19 0 22 83

C-F 7 88 9 97 3 51

C-G 3 87 104 4 516 102

D-E 14 32 47 3 39 46

D-F 6 29 35 154 130 24

D-G 0 28 16 16 144 184

E-F 2 77 10 119 22 11

E-G 12 104 80 6 332 400

F-G 5 9 54 69 592 290

Table 11. Repeatability and reproducibility precision statements with vnorm. 

Mix/Geometry

Repeatability Limit 95 Percent

Confidence

Reproducibility Limit 95 Percent

Confidence

9.5 S 22 46

9.5 L-E 38 82

12.5 S 108 86

12.5 L 88 170

19 S 314 580

25 L 401 313
51 

Evaluation of Repeatability and Reproducibility Trends for NMAS

Figure 36 and figure 37 present the evaluation of repeatability and reproducibility trends for 

NMAS and specimen geometry for damage characteristic curve (C value, shape factor, vnorm) and 

DR, respectively. For C value and shape factor, the repeatability statistic, sr%, and reproducibility 

statistic, sx̅%, were extracted for an average (1−C̿) = 0.7. The analysis for 9.5-mm large mixture 

does not include laboratory E’s results.

Figure 36 shows that within-laboratory variation increases as the NMAS increases for a given 

specimen geometry, irrespective of the measure in terms of C value, shape factor, vnorm, and DR. 

The damage characteristic curve measures (C value, shape factor, vnorm) generally indicate

similar repeatability between the large- and small-specimen geometry for the 9.5-mm and 

12.5-mm mixtures, regardless of the parameter of evaluation. The 25-mm mixture largespecimen results show similar repeatability to 19-mm mixture small-specimen results in terms of 

C value and shape factor but a higher repeatability limit based on the vnorm approach. Although it 

shows the same increasing trend with NMAS for a given specimen geometry, the trend with DR

is somewhat different than the damage characteristic curve trends. The small-specimen geometry 

DR results of the 9.5-mm and 12.5-mm mixtures have higher within-laboratory variation than 

large-specimen geometry. Furthermore, the 25-mm mixture large-specimen results indicated less 

within-laboratory variation in DR compared to the 19-mm mixture small-specimen geometry 

results.

Source: FHWA.

A. Repeatability analysis using single-point C value. 
52 

Source: FHWA.

B. Repeatability analysis using single-point shape factor. 

Source: FHWA.

C. Repeatability analysis using vnorm. 
53 

Source: FHWA.

D. Repeatability analysis using DR. 

Figure 36. Graphs. Repeatability analysis comparison between single-point measure and 

vnorm approach. 

Figure 37 shows similar trends in vnorm, with increasing NMAS for a given specimen geometry. 

The C value and shape factor measures indicate higher between-laboratory variation in the 

19-mm and 25-mm mixtures compared to the 9.5-mm and 12.5-mm mixtures; however, trends 

among 9.5-mm and 12.5-mm mixtures are somewhat inconsistent. The 25-mm mixture has lower 

between-laboratory variation than the 19-mm mixture in all damage characteristic curve 

measures, regardless of the parameters. All damage characteristic curve measures indicate the 

highest between-laboratory variation in the 19-mm mixture small-specimen results. Since the 

25-mm mixture was used with the large-specimen geometry and the 19-mm mixture was used 

with the small-specimen geometry, these results match expectations. However, DR measures 

again present different trends with NMAS compared to damage characteristic curve measures. 

All mixtures have similar between-laboratory variances for DR, irrespective of NMAS and 

geometries, although the 9.5-mm large mixture has somewhat higher reproducibility variation

than any other mixtures. 
54 

Source: FHWA.

A. Reproducibility analysis using single-point C value. 

Source: FHWA.

B. Reproducibility analysis using single-point shape factor. 

Source: FHWA.

C. Reproducibility analysis using vnorm. 
55 

Source: FHWA.

D. Reproducibility analysis using DR. 

Figure 37. Graphs. Reproducibility analysis comparison between single-point measure and 

vnorm approach. 

Establishment of the Recommended Precision Statements

Damage Characteristic Curve

Given the clear NMAS dependence observed in the vnorm results and its advantage of considering 

the entire damage characteristic curve rather than a single point within the curve, vnorm is 

recommended for quantifying the repeatability and reproducibility of damage characteristic 

curve results. The researchers propose the vnorm limits given in table 11 for defining the 

repeatability and reproducibility of damage characteristic curve results of the AMPT cyclic 

fatigue tests; these limits are a function of NMAS and specimen geometry. 

Failure Criterion

Given the clear NMAS dependence of DR repeatability shown in figure 36, is the researchers

recommend that the NMAS and specimen geometry-specific DR repeatability limits provided in 

table 8 be incorporated into the AMPT cyclic fatigue test standard precision statement. However, 

the DR reproducibility results in figure 37 do not indicate NMAS dependence. In fact, the 

reproducibility precision for different mixtures within a given specimen geometry are close to 

each other except for the 9.5-mm large-mixture specimen results, which exhibit the highest 

between-laboratory variation (for unknown reasons). Given the lack of a clear trend in the 

reproducibility of DR with respect to NMAS, the collective ILS results of each specimen 

geometry were aggregated together and used to define specimen geometry-specific limits for the 

difference in DR test results between two laboratories that are not expected to be exceeded with 

95 percent confidence; this approach follows the underlying premise behind the precision 

statement development guidance given in ASTM E691-20 and C670-15.(4,5)

 Correspondingly, the 

researchers took the following steps to define the precision statement for DR reproducibility:
56 

1. Calculate the average DR value of three specimens for each lab, mixture, and geometry 

combination. 

2. Compute the difference between the average DR of each pair of laboratories for each 

mixture with different NMAS and geometry. Each pair of laboratories yielded two results 

(i.e., laboratory A minus laboratory B and laboratory B minus laboratory A for 9.5-mm 

small mix).

3. Divide the data computed in step 2 into two groups according to the mixture geometry 

(small versus large).

4. Test whether the data are normally distributed using the Shapiro-Wilk test for each 

group.(24)

5. Fit a normal distribution to the data in step 3 in each group with mean and standard 

deviation using the maximum-likelihood fitting method.

(25)

6. Calculate the 95-percent confidence interval to find the maximum allowable difference in 

DR test results between two laboratories. 

A summary of the DR reproducibility analysis results, aggregating all mixture results for a given

geometry, is presented in table 12. The number of data points for the large-specimen geometry 

mixture is smaller than for small-specimen geometry because laboratory E data were excluded 

from the 9.5-mm large-specimen data, as previously discussed. Both small- and large-specimen 

geometry results yielded p-values higher than 0.05 in the Shapiro-Wilk test, which indicates the 

hypothesis that the results are normally distributed was not rejected.(24)

Table 12. DR reproducibility analysis results aggregating NMAS for different geometry.

Geometry

Number of 

Data Points 

Shapiro-Wilk 

Test p-value(24)

Normal 

Distribution

Maximum 

Allowable 

Difference

Small 126 0.075 Yes (0, 0.0462) 0.090 

Large 114 0.38 Yes (0, 0.0502) 0.098 

Figure 38 presents the corresponding histogram of DR differences between laboratory pairs, with 

the corresponding best fit normal distribution. The results show that the maximum allowable 

difference between two laboratory results for DR is 0.090 for mixtures with the small geometry 

and 0.098 for mixtures with the large geometry using a confidence level of 95 percent; it is 

recommended that these limits be incorporated into the reproducibility precision statement in the 

AMPT cyclic fatigue test standards. Table 13 presents, the recommended precision statements 

for the repeatability and reproducibility of DR. 
57 

Source: FHWA. 

A. Histograms of DR difference with the fitted normal distribution for mixtures with small 

geometry. 

Source: FHWA. 

B. Histograms of DR difference with the fitted normal distribution for mixtures with large 

geometry. 

Figure 38. Graphs. Histograms of DR difference with the fitted normal distribution. 

 

 
58 

Table 13. Recommended repeatability and reproducibility precision statements (maximum 

allowable difference) of DR.

Mixture

Repeatability

(MAR) 

Reproducibility

(d2srpd) 

9.5 S 0.088 0.090 

9.5 L-E 0.065 0.098 

12.5 S 0.129 0.090 

12.5 L 0.089 0.098 

19 S 0.214 0.090 

25 L 0.124 0.098 

PROPOSED PRECISION STATEMENTS

The following provides a summary of the proposed precision statements for the AMPT cyclic 

fatigue test standards:

• Precision—Criteria for judging the acceptability of cyclic fatigue test results, including 

damage characteristic curve and DR obtained by this method, are given in table 14 and 

table 15. The vnorm is calculated by an analytical integration of the sum of the squared 

difference between the fitted curve for each specimen and the fitted curve across all the 

specimens divided by the number of specimens. The vnorm precision limits given in table 

14 and table 15 are based on the test results from seven laboratories. 

• Single operator precision (repeatability)—The figures in columns 3 and 4 of table 14 are 

the acceptable limits that are appropriate for the mixtures described in column 2. Results 

obtained in the same laboratory by the same operator using the same equipment in the 

shortest practical period should be accepted unless the vnorm or DR range exceeds that 

given in table 14. 

• Multilaboratory precision (reproducibility)—The figures in columns 3 and 4 of table 15 

are the acceptable limits that are appropriate for the mixtures described in column 2. 

Results obtained by two different operators testing the same material in two different 

laboratories should be accepted unless the vnorm or DR difference exceeds that given in

table 15.

The precision limits given in table 14 and table 15 are based on six mixtures with different 

NMAS and geometries tested in seven laboratories.
59 

Table 14. Single-operator precision for damage characteristic curve and DR results.

Applicable 

Standard NMAS (mm)

vnorm

Limit

Acceptable Limit for DR

(MAR) 

AAHTO TP 107(1) 9.5 38 0.065 

AAHTO TP 107 12.5 88 0.089 

AAHTO TP 107 19.0 401 0.124 

AAHTO TP 107 25.0 401 0.124 

AAHTO TP 133(2) 9.5 22 0.088 

AAHTO TP 133 12.5 108 0.129 

AAHTO TP 133 19.0 314 0.214 

Table 15. Multilaboratory precision for damage characteristic curve and DR results.

Applicable 

Standard NMAS (mm)

vnorm

Limit

Acceptable Limit for DR

(d2srpd) 

AAHTO TP 107(1) 9.5 82 0.098 

AAHTO TP 107 12.5 170 0.098 

AAHTO TP 107 19.0 313 0.098 

AAHTO TP 107 25.0 313 0.098 

AAHTO TP 133(2) 9.5 46 0.090 

AAHTO TP 133 12.5 86 0.090 

AAHTO TP 133 19.0 580 0.090 

ANALYSIS OF THE PRECISION STATEMENT IMPLICATIONS ON UNCERTAINTY 

IN PRACTICAL APPLICATIONS OF THE TEST RESULTS 

Data Excluded From the Analysis 

Table 16 and table 17 present the individual laboratory results and laboratory pair results that did 

not meet the proposed precision limits for repeatability and reproducibility, respectively. Given 

the proposed precision limits, these results were excluded from the analysis of the uncertainty in 

FlexPAVE %Damage and Sapp. Of the 42 total results, 6 laboratories (14.3 percent) did not meet 

the repeatability limits, and 44 out of 126 laboratory pairs (34.9 percent) did not meet the 

reproducibility limits. Recall that the vnorm and DR precision limits were both established based 

on a 95-percent confidence interval. Because each was established using a 95-percent confidence 

level, it is expected that roughly 5 percent of laboratories would exceed each of the limits. A test 

will not meet the repeatability limit if either of the vnorm or DR exceeds the repeatability precision 

limits. Therefore, the probability of a test not meeting repeatability limits is approximately 

10 percent (1–95 percent×95 percent), higher than 5 percent. Within the six laboratories that did 

not meet the repeatability limits, two (4.8 percent) were excluded due to failure to meet the DR

precision limit, and four (9.5 percent) were excluded due to failure to meet vnorm limit. The 

number of laboratory pairs that do not meet reproducibility results reflects a much higher 

percentage than 10 percent because data were discarded for consideration in the reproducibility 

analysis if repeatability (within a laboratory) or reproducibility (between-laboratory) criteria

were exceeded. 
60 

Table 16. Data failing repeatability limits. 

Laboratory 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L 

A — — — X — X 

B X — — — — — 

C — — — — — — 

D — — — — — — 

E — X — — — — 

F — — X — — — 

G — X — — — — 

—No data.

X = excluded due to not meeting the precision limits.

Table 17. Data excluded from reproducibility analysis. 

Laboratory 

Pair 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L 

A-B X — — X — X 

A-C — — — X — X 

A-D — — — X — X 

A-E X X — X — X 

A-F — — X X — X 

A-G — X — X — X 

B-C X X — — — — 

B-D X — — — — — 

B-E X X — — — — 

B-F X — X — — — 

B-G X X — — X — 

C-D — — — — — — 

C-E — X — — — — 

C-F — X X — X — 

C-G — X X — — — 

D-E X X — — — — 

D-F — — X — — — 

D-G — X — — — — 

E-F — X X — — — 

E-G — X — — — X 

F-G — X X — X — 

—No data.
61 

Uncertainty in FlexPAVE %Damage Predictions

The uncertainty in %Damage, conveyed by the standard deviation calculated using Ghanbari’s 

framework for each mixture and specimen geometry combination in terms of repeatability and 

reproducibility, is shown in table 18 and table 19, respectively.

(19) The X’s correspond to those 

excluded due to not meeting the precision limits. Table 20 and table 21 summarize the maximum 

%Damage standard deviation result for each mixture and specimen geometry combination of all 

of the ILS results that met the proposed precision limits. Table 22 and table 23 summarize the 

median %Damage standard deviation for each mixture and specimen geometry combination. 

These maximum and median standard deviations were used to calculate the 95-percent 

confidence intervals for %Damage. The maximum repeatability uncertainty of %Damage

increases with NMAS for a given geometry, which matches expectations given that the 

repeatability precision limits for vnorm and DR both increase as a function of NMAS. The 19-mm 

small-specimen results show that this NMAS and geometry combination yielded the highest 

uncertainty in repeatability %Damage of all conditions evaluated. The small-specimen 19-mm 

mixtures results have the second highest repeatability vnorm limit of all conditions evaluated, 

including both small- and large-specimen geometries, while the large-specimen 25-mm mixture 

results have the highest. However, the 19-mm small-specimen results have a higher repeatability 

DR limit than the 25-mm large-specimen results; this result suggests that the influence of DR

uncertainty had a more pronounced effect on the uncertainty of %Damage than the damage 

characteristic curve uncertainty.

Table 18. Repeatability standard deviation of %Damage for all the laboratories and 

mixtures.

Laboratory 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L

A 0.37 0.43 0.16 X 1.05 X

B X 0.26 0.53 0.77 2.48 1.26

C 0.40 0.36 0.70 0.25 0.88 0.42

D 0.51 0.53 1.50 0.49 0.57 1.30

E 0.90 X 0.30 0.87 1.73 0.80

F 0.33 0.50 X 0.69 2.21 1.47

G 0.22 X 0.56 0.61 0.27 0.76

Median 0.39 0.43 0.54 0.65 1.05 1.03

Maximum 0.90 0.53 1.50 0.87 2.48 1.47
62 

Table 19. Reproducibility standard deviation of %Damage for all the laboratories and 

mixtures.

Laboratory 

Pair 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L

A-B X 0.78 0.34 X 1.10 X

A-C 0.25 0.79 0.44 X 1.14 X

A-D 0.69 0.42 0.54 X 0.11 X

A-E X X 0.75 X 0.70 X

A-F 0.44 0.95 X X 1.05 X

A-G 0.63 X 0.26 X 0.89 X

B-C X X 0.64 0.60 0.99 0.95

B-D X 0.54 0.29 0.96 1.19 0.89

B-E X X 0.98 0.78 1.11 0.51

B-F X 0.29 X 0.60 0.27 0.49

B-G X X 0.62 0.52 X 1.47

C-D 0.47 0.88 0.93 0.36 1.12 0.27

C-E 0.76 X 0.70 0.16 0.44 0.84

C-F 0.19 X X 1.15 X 0.55

C-G 0.41 X X 0.18 1.82 0.72

D-E X X 1.22 0.22 0.69 0.79

D-F 0.51 0.82 X 1.54 1.15 0.50

D-G 0.10 X 0.74 0.55 0.77 1.03

E-F 0.62 X X 1.34 1.04 0.37

E-G 1.02 X 0.84 0.33 1.38 X

F-G 0.41 X X 0.99 X 1.05

Median 0.47 0.79 0.67 0.6 1.05 0.75

Maximum 1.02 0.95 1.22 1.54 1.82 1.47

Table 20. Maximum repeatability variation for each mixture.

Mixture stdTotal

Average 

%Damage±1.96 std

9.5 S 0.90 15±1.76

9.5 L 0.53 15±1.05

12.5 S 1.50 15±2.94

12.5 L 0.87 15±1.70

19 S 2.48 15±4.86

25 L 1.47 15±2.88

std = standard deviation. 
63 

Table 21. Maximum reproducibility variation for each mixture.

Mixture stdTotal

Average 

%Damage±1.96 std

9.5 S 1.02 15±2.00

9.5 L 0.95 15±1.85

12.5 S 1.22 15±2.39

12.5 L 1.54 15±3.03

19 S 1.82 15±3.56

25 L 1.47 15±2.89

Table 22. Median repeatability variation for each mixture.

Mixture stdTotal

Average 

%Damage±1.96 std

9.5 S 0.39 15±0.76

9.5 L 0.43 15±0.83

12.5 S 0.54 15±1.07

12.5 L 0.65 15±1.28

19 S 1.05 15±2.05

25 L 1.03 15±2.01

Table 23. Median reproducibility variation for each mixture.

Mixture stdTotal

Average 

%Damage±1.96 std

9.5 S 0.47 15±0.91

9.5 L 0.79 15±1.54

12.5 S 0.67 15±1.31

12.5 L 0.60 15±1.18

19 S 1.05 15±2.05

25 L 0.75 15±1.48

An increase in the maximum %Damage uncertainty with increasing NMAS was expected under 

reproducibility conditions, given that the reproducibility limits for vnorm increase with NMAS and 

the reproducibility limits for DR are independent of the NMAS. The maximum uncertainty in 

%Damage under reproducibility conditions increases with NMAS for the small-specimen 

geometry but not for the large-specimen geometry; the reason for this trend is unknown. The 

19-mm small-specimen geometry results encompass the maximum uncertainty in %Damage

under reproducibility conditions for all mixtures and geometries evaluated. Because repeatability 

assesses variation among three individual test specimens and reproducibility assesses variation 

among test results (i.e., results of three test specimens each in two laboratories), they are not 

directly comparable. Thus, trends among repeatability and reproducibility %Damage uncertainty 

for a given mixture are varied.

Comparison between median and maximum confidence intervals in percent damage predictions 

provides context to the typical versus extreme uncertainty ranges expected in practice. For 
64 

example, even though the 95-percent confidence interval of predicted %Damage at an average 

%Damage of 15 percent under repeatability conditions for the 19-mm mixture using the smallspecimen geometry is from 10.14 percent to 19.86 percent (approximately ±30 percent error), the 

median 95-percent confidence interval is 12.95 percent to 17.05 percent (corresponding to 

roughly ±14 percent error). For the 9.5-mm mixture with the small-specimen geometry, the 

median condition yields a 95-percent confidence interval for %Damage of 14.24–15.76 percent 

(corresponding to roughly ±5 percent error). 

Similar trends are observed for reproducibility. The maximum 95-percent confidence interval of 

all conditions under reproducibility conditions coincides with the 19-mm mixture with the smallspecimen geometry, yielding a %Damage span from 11.44–18.56 percent (roughly ±23 percent 

error), but the median confidence interval for the same condition is notably lower at 

12.95-17.05 percent (roughly ±14 percent error). The practical implications of these errors merit 

further investigation in future work. Efforts to improve the consistency of specimen fabrication 

procedures may also enable a reduction in the uncertainty of test outcomes, particularly for the 

larger NMAS mixtures

Uncertainty in Apparent Damage Capacity 

Figure 39 through figure 44 present the measured Sapp values for the different mixture and 

specimen geometry combinations. The height of the bars indicates the representative Sapp values 

determined from the damage characteristic curve and failure criterion results from a given 

laboratory. The error bars convey the maximum and minimum values determined from the 

individual test determinations. Solid bars convey that the representative laboratory’s results met 

the proposed repeatability limits for vnorm and DR, whereas the hollow bars indicate that the 

laboratory’s results failed to meet the proposed repeatability limits for vnorm and/or DR. The 

horizontal lines convey the proposed Sapp thresholds for standard (S), heavy (H), very heavy (V), 

and extreme (E) traffic levels.(23) Note that consistent dynamic modulus results were used for all 

laboratories when calculating the Sapp value for a given mixture and specimen geometry 

combination, consistent with the statistical analysis to develop the precision limits. 

Figure 39 through figure 42 demonstrate that the Sapp values obtained from the large-specimen 

geometry were generally lower than those for the small-specimen geometry for both the 9.5-mm 

and 12.5-mm mixtures. Differences in the dynamic moduli and α values among the two 

specimen geometries could have contributed to these trends. In addition, it is apparent that 

different test determinations within a given laboratory and the representative Sapp results from 

different laboratories can yield different traffic designations according to the proposed Sapp

thresholds. These findings suggest that the limits may merit reconsideration given the measured 

uncertainty in the test results. While figure 43 and figure 44 indicate all laboratories’ 

representative Sapp results for the 19-mm and 25-mm mixtures have the same standard traffic 

level designation according to the proposed Sapp thresholds, the results do show considerable 

variation within and across laboratories with respect to the mean values. 
65 

Source: FHWA.

The hollow bar indicates test results that failed to meet the proposed 

repeatability precision limits.

Figure 39. Graph. Measured Sapp values for the 9.5-mm mixture small-specimen geometry.

Source: FHWA.

The hollow bars indicate test results that failed to meet the proposed 

repeatability precision limits.

Figure 40. Graph. Measured Sapp values for the 9.5-mm mixture large-specimen geometry.
66 

Source: FHWA.

The hollow bar indicates test results that failed to meet the proposed 

repeatability precision limits.

Figure 41. Graph. Measured Sapp values for the 12.5-mm mixture small-specimen geometry.

Source: FHWA.

The hollow bar indicates test results that failed to meet the proposed 

repeatability precision limits.

Figure 42. Graph. Measured Sapp values for the 12.5-mm mixture large-specimen geometry.
67 

Source: FHWA.

Figure 43. Graph. Measured Sapp values for the 19-mm mixture small-specimen geometry.

Source: FHWA.

The hollow bar indicates test results that failed to meet the proposed 

repeatability precision limits.

Figure 44. Graph. Measured Sapp values for the 25-mm mixture large-specimen geometry.

Table 24 and table 25 show the uncertainty in Sapp under repeatability conditions measured 

among test determinations and predicted among test results from the analysis in terms of the 

COV% for each mixture and specimen geometry combination. The X’s represent the

laboratory’s results that were excluded due to not meeting the repeatability precision limits. As 

described in chapter 2, the repeatability COV% values for test results were calculated using 

simulated Sapp results that fell within a 95-percent confidence interval. The Sapp repeatability 
68 

COV% values among test determinations were calculated directly from the ILS test results. The 

simulated Sapp values were generated based on the observed within-laboratory variability in both 

damage characteristic curve and failure criterion results. Figure 45 shows the maximum Sapp

COV% values for each mixture and specimen geometry combination reported in table 24 and 

table 25. 

The results demonstrate that the uncertainty in Sapp under repeatability conditions generally 

increases as the NMAS increases for a given specimen geometry, which matches expectations 

since the precision limits increase with respect to NMAS. Furthermore, the results demonstrate 

that the variation in Sapp values among three test determinations is considerably higher than that 

among two test results (i.e., two sets of three determinations). The maximum Sapp COV% values 

for test results span from approximately 7 percent to 18 percent. The maximum COV% value for 

the repeatability of Sapp results of 9.5-mm and 12.5-mm mixtures falls below 11 percent. The 

repeatability COV% values are somewhat higher for the 19-mm and 25-mm mixtures, with a 

maximum median value of 11 percent and a maximum value of 17.8 percent. 

Table 24. Measured repeatability COV% of Sapp among test determinations for all the 

laboratories and mixtures.

Laboratory 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L 

A 6.38 10.21 2.64 16.75 19.14 X 

B X 4.63 7.95 5.93 20.80 6.24 

C 10.38 7.72 11.60 11.35 17.83 3.85 

D 8.80 13.12 23.93 15.26 10.07 26.55 

E 15.64 X 5.01 14.87 31.14 3.31 

F 1.71 11.46 X 12.51 23.71 28.27 

G 3.62 X 8.36 16.75 3.06 7.81 

Table 25. Repeatability COV% of Sapp results for all the laboratories and mixtures from 

the uncertainty analysis. 

Laboratory 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L 

A 3.24 3.32 1.50 X 10.90 X 

B X 2.75 2.92 7.14 17.77 8.40 

C 4.19 3.24 4.34 3.15 6.83 3.22 

D 3.87 6.84 10.91 5.26 3.83 12.04 

E 9.79 X 1.70 8.78 12.21 4.78 

F 2.51 5.33 X 5.76 16.90 14.26 

G 1.79 X 5.54 5.11 2.41 5.31 
69 

Source: FHWA.

Max = maximum.

Figure 45. Graph. Comparison of the maximum Sapp COV% values measured among test 

determinations and calculated among test results from the uncertainty analysis. 

Table 26 and table 27 present the measured Sapp COV% values and the Sapp COV% values from 

the uncertainty analysis, respectively. The X’s represent the laboratories’ or laboratory pairs’ 

results that were excluded due to not meeting the precision limits. The reproducibility COV% 

values from the uncertainty analysis reflect the maximum possible value from a pair of 

laboratories’ results based on the 95-percent confidence intervals for the simulated Sapp values in 

each laboratory and, thus, constitute an extreme limit possible given the precision limits. 

 
70 

Table 26. Measured reproducibility COV% of Sapp for all the laboratories and mixtures.

Laboratory 

Pair 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L

A-B X 14.28 5.71 X 5.58 X

A-C 4.66 20.15 8.42 X 24.32 X

A-D 5.82 4.76 11.55 X 0.18 X

A-E X X 15.39 X 14.52 X

A-F 7.37 19.84 X X 0.30 X

A-G 3.02 X 2.26 X 11.20 X

B-C X X 14.10 12.04 18.87 8.36

B-D X 9.56 5.86 20.51 5.40 13.43

B-E X X 21.01 15.50 8.98 9.78

B-F X 5.64 X 10.33 5.28 14.86

B-G X X 7.96 8.08 X 7.03

C-D 10.46 24.79 19.88 8.57 24.15 5.09

C-E 22.24 X 7.02 3.49 9.97 1.42

C-F 2.72 X X 22.24 X 6.53

C-G 7.67 X X 3.99 35.04 15.35

D-E X X 26.71 5.09 14.35 3.67

D-F 13.16 15.15 X 30.52 0.13 1.44

D-G 2.81 X 13.79 12.53 11.37 20.36

E-F 19.58 X X 25.63 14.23 5.12

E-G 29.66 X 13.16 7.47 25.51 X

F-G 10.37 X X 18.33 X 21.77
71 

Table 27. Reproducibility COV% of Sapp for all the laboratories and mixtures from the 

uncertainty analysis. 

Laboratory 

Pair 9.5 S 9.5 L 12.5 S 12.5 L 19 S 25 L

A-B X 23.70 13.21 X 50.34 X

A-C 16.47 29.61 17.97 X 49.58 X

A-D 17.09 20.32 30.35 X 24.20 X

A-E X X 20.22 X 49.39 X

A-F 16.54 33.16 X X 44.97 X

A-G 11.22 X 13.97 X 33.08 X

B-C X X 26.11 27.45 53.07 24.78

B-D X 24.64 26.67 38.73 41.12 45.33

B-E X X 28.31 39.76 52.88 28.70

B-F X 18.67 X 30.48 57.48 48.66

B-G X X 22.17 27.18 X 29.48

C-D 23.23 39.08 42.63 21.44 40.32 31.67

C-E 43.67 X 16.26 22.52 37.30 14.28

C-F 13.74 X X 35.07 X 35.19

C-G 17.43 X X 17.27 48.67 28.21

D-E X X 44.70 25.73 40.13 32.02

D-F 23.30 34.07 X 46.02 35.51 41.20

D-G 11.66 X 38.92 28.88 21.29 48.50

E-F 38.48 X X 47.02 56.57 35.54

E-G 47.09 X 24.06 29.95 48.48 X

F-G 17.50 X X 34.80 X 51.76

Figure 46 shows the maximum Sapp COV% values for each mixture and specimen geometry 

combination reported in table 26 and table 27. Trends in the reproducibility COV% values with 

respect to NMAS are less clear than for repeatability. The lack of clear trends in DR variability 

among laboratories for NMAS may explain this result. The measured maximum reproducibility 

Sapp COV% values are considerably lower than those calculated from the uncertainty, 

highlighting those calculated through the uncertainty analysis constitute an extreme scenario that 

is possible, but unlikely, given the proposed precision limits. The maximum measured Sapp

reproducibility COV% values spanned from roughly 25 to 35 percent for the different mixture 

and specimen geometry combinations, whereas the maximum possible values from the 

uncertainty analysis spanned from roughly 39 to 58 percent. 
72 

Source: FHWA.

Figure 46. Graph. Comparison of the maximum Sapp COV% values measured among test 

determinations and calculated among test results from the uncertainty analysis.

The reproducibility COV% values are considerably larger than the repeatability COV% values. 

The researchers attributed this result to differences in how the repeatability and reproducibility 

COV% values were calculated. Increased uncertainty in test results between laboratories 

compared to within a given laboratory also contributed to this result. The implications of the 

relatively high Sapp COV% values observed under reproducibility conditions merit further 

investigation in future research. 

Table 28 summarizes the maximum and median COV% values from the uncertainty analysis

regarding repeatability and reproducibility for each mixture and specimen geometry 

combination. 

 
73 

Table 28. Maximum and median COV% of Sapp for each mixture from the uncertainty 

analysis. 

Mixture

Maximum 

Repeatability 

COV% 

Maximum 

Reproducibility 

COV% 

Median 

Repeatability 

COV% 

Median 

Reproducibility 

COV% 

9.5 S 9.79 47.09 3.55 17.43 

9.5 L 6.84 39.08 3.32 27.13 

12.5 S 10.91 44.70 3.63 25.08 

12.5 L 8.78 47.02 5.51 29.95 

19 S 17.77 57.48 10.90 46.72 

25 L 14.26 51.76 6.86 33.61 

Comparisons between the uncertainty measures for %Damage and Sapp under repeatability 

conditions and reproducibility conditions are shown in figure 47. The comparisons demonstrate 

that the measures of uncertainty in %Damage and Sapp are correlated. The uncertainty in 

%Damage and Sapp, given the uncertainty in AMPT cyclic fatigue test results, merits 

consideration when practical inferences from these parameters are made. 

Source: FHWA. 

stdtot = standard deviation of %Damage.

A. Comparison between uncertainty of %Damage and uncertainty of Sapp in terms of 

repeatability.
74 

Source: FHWA

B. Comparison between uncertainty of %Damage and uncertainty of Sapp in terms of 

reproducibility. 

Figure 47. Graphs. Comparison between uncertainty of %Damage and uncertainty of Sapp. 
75 

CHAPTER 4. CONCLUSIONS AND RECOMMENDED CHANGES TO THE 

STANDARDS FOR AMPT CYCLIC FATIGUE TESTING 

The following summarizes the primary findings and recommended changes to the standards for 

AMPT cyclic fatigue testing developed as a result of this ILS. Herein, repeatability quantifies the 

acceptable variation among three test determinations (specimens) obtained within a laboratory on 

a single material. The reproducibility quantifies the acceptable variation between the average test 

results of two laboratories conducted on the same material. The research team’s findings and 

recommendations are as follows:

• Statistical tests of the consistency of the ILS results acquired from the seven participating 

laboratories indicate that the results obtained are highly consistent according to ASTM 

E691-20.(4)

 Only a single laboratory’s results of a single mixture tested using the largespecimen geometry failed the consistency tests and were omitted from the statistical 

analysis used to develop the proposed precision limits for the AMPT cyclic fatigue test 

standards.

• Statistical analysis of the damage characteristic curve repeatability and reproducibility 

conducted using single-point measures of the damage characteristic curve to directly 

align with the guidance in ASTM E691-20 and ASTM C670-15 demonstrates that the 

inferred uncertainty in test results and trends among different mixtures and specimen 

geometries can vary with the chosen point of reference (e.g., S value at which C is 

evaluated).(4,5)

• Modified forms of the variance index originally proposed by Ding et al., termed vnorm and 

developed herein to quantify repeatability and reproducibility of damage characteristic 

curves, better capture variation in the curve than the single-point measures.

(18) The vnorm

results demonstrated clear increases in within- and between-laboratory variability in 

damage characteristic curve with increasing NMAS values for both small- and 

large-specimen geometries. 

• The vnorm results followed a gamma rather than a normal distribution, which precluded 

the direct use in precision-limit development according to ASTM E691-20 and ASTM 

C670-15.(4,5)

 However, fitted gamma distributions of the test results were used to define 

vnorm limits for repeatability and reproducibility conditions that will be exceeded with a 

probability of 5 percent, which aligns with the guiding principles for precision-limit 

development, according to the ASTM standards. The established repeatability and 

reproducibility limits are a function of NMAS and specimen geometry. 

• The DR constitutes a single-point measure and, therefore, is directly amenable to 

precision-statement development, according to ASTM E691-20 and ASTM C670-15.(4,5)

The DR results demonstrate that within-laboratory variation expands with increasing 

NMAS for a given specimen geometry. However, the reproducibility of DR does not 

depend on the NMAS. Correspondingly, repeatability precision limits for DR were 

established as a function of NMAS and specimen geometry, whereas reproducibility 

limits were established as a function of specimen geometry only. 
76 

• The ILS results show that cyclic fatigue test results for a given mixture and specimen 

geometry, which meet the proposed precision limits, can yield different traffic 

designations according to the proposed Sapp thresholds.(23)

 This finding suggests that the 

spacing of Sapp thresholds for different traffic designations may merit reconsideration. 

Under the proposed precision limits, the Sapp COV under reproducibility conditions can 

be as high as 57 percent for 19-mm mixtures with the small-specimen geometry. An 

analysis of the uncertainty in FlexPAVER version 1.1 predictions yielded results that 

were correlated to those from Sapp. The implications of the observed uncertainty in Sapp

and FlexPAVE, given the proposed precision limits, merits further consideration in future 

work. Possible measures in specimen fabrication procedures to reduce test variability also 

merit investigation to reduce uncertainty in the test results. 

• The researchers recommend that the proposed precision statements presented in chapter 3 

be incorporated into the AMPT cyclic fatigue test procedures.
77 

APPENDIX. INITIAL PILOT TEST RESULTS EVALUATION AND RESOLUTION OF 

PROCEDURAL DISCREPANCIES

The research team evaluated the initial pilot dynamic modulus test results from participating 

laboratories according to the data quality statistics specified in AASHTO TP 132-19 and the 

agreement to the reference results.(8)

 Agreement to the reference results was assessed using the 

precision limits for |E*| and phase angle specified in AASHTO T 378-17 since precision limits 

were not available for AASHTO TP 132-19.(8,9) In general, laboratories met the data quality 

statistics requirements. However, the research team identified cases where the differences 

between the participating laboratory’s results and those of the research team exceeded the 

reproducibility precision limits specified in AASHTO T 378-17.(9) In the majority of these cases, 

the participating laboratory did not provide adequate thermal equilibration time after placing the 

test specimen in the AMPT chamber. AASHTO TP 132-19 specifies that a specimen can be first 

preconditioned to the test temperature in an external chamber and then transferred to the AMPT 

chamber.(8) If the transfer, linear variable differential transformer (LVDT) bracket, and LVDT 

are attached, and the AMPT chamber returned to the testing temperature in 5 min, then the 

standard allows the user to begin the test as soon as the AMPT chamber returns to the testing 

chamber. 

The pilot test results suggest that this procedure did not allow adequate thermal equilibration 

time and led to discrepancies with the reference results where thermal equilibration was 

achieved. In cases identified as having insufficient thermal equilibration times, the participants 

were asked to retest following the temperature conditioning guidance established for AASHTO 

TP 133-21 in Phase I of this contract using a heat transfer model.(2) The temperature conditioning 

guidance in AASHTO TP 133-21 specifies the required thermal equilibration time for a small 

test specimen in the AMPT chamber as a function of the difference between the test temperature 

and room temperature, whether the specimen is conditioned directly in the AMPT or first 

preconditioned in an external chamber, and the test setup time for cases where the specimen is 

preconditioned in an external chamber.(2) Increasing the thermal equilibrations to this guidance 

was found to resolve the majority of cases that failed to meet the reproducibility limits. This 

result suggests that the temperature conditioning guidance in AASHTO TP 132-19 should be 

revised.(8) The research team recommends revisions that are, in part, informed by the experiences 

gained during these pilot studies. 

In two cases, a discrepancy between the reference results and the participating laboratories’ 

results still existed after thermal equilibration time differences were resolved. At this stage, these 

two laboratories were asked to check the temperature of their AMPT chamber using an external 

temperature probe and compare the reading to the AMPT software’s reported value. In one case, 

a difference between the temperature probe and AMPT software readings was identified, 

indicating a temperature calibration error. The laboratory recalibrated its AMPT temperature 

chamber, which was found to resolve the discrepancy with the reference results. In the other 

case, no temperature calibration error was found. Furthermore, in this case, the discrepancy 

between the reference results and the participating laboratory’s results was only evident at 35 °C. 

The researchers found that the laboratory was using compensation springs at 35 °C, whereas the 

other laboratories (including the research team’s laboratory) were not. Compensation springs are 

allowed in AASHTO TP 132-19 to avoid positive strain drift during testing.(8) Repeating the tests 
78 

without the compensation springs yielded results that agreed with the reference results, according 

to the AASHTO T 378-17 precision limits.(9) This finding suggests that the allowance of 

compensation springs merits further evaluation in future work. Compensation springs were not 

used in any subsequent ILS testing to avoid potential bias. Again, the research team recommends 

revisions to the allowance (or disallowance) of compensating springs within AASHTO TP 132 

that are, in part, informed by the experiences gained while evaluating troubleshooting with this 

laboratory. 

The researchers evaluated the initial AMPT cyclic fatigue pilot test results based on adherence to 

the strain selection guide in AASHTO TP 133-21 and the average actuator strain standard error 

in cycles two through five of the cyclic fatigue test.(2) AASHTO TP 133-21 specifies that the 

average actuator strain standard error in cycles two through five of the cyclic fatigue test shall 

not exceed 10 percent. AASHTO TP 133-21 specifies initial strain selection based on the linear 

viscoelastic |E*| measured from either AASHTO TP 132-19 or AASHTO T 378 testing or the 

fingerprint portion of the cyclic fatigue test. Thus, a comparison between the strains specified in 

AASHTO TP 133-21, according to the reported |E*| values, was used to assess adherence to the 

standard. Identified deviations from the standard were discussed with the participants and 

resolved in subsequent tests. 

The researchers reviewed the actuator strain waveform in cases where the maximum limit on 

actuator strain standard error was exceeded. If the actuator strain curve appeared smooth, such as 

in the case shown in figure 48, the laboratory was instructed to perform PID tuning and then 

repeat the cyclic fatigue test. In these cases, the repeated test results were in good agreement with 

those of the research team in terms of the damage characteristic curve and failure criterion. In 

one case, the actuator strain curve was not smooth, as shown in figure 49. The “hiccup” in the 

actuator strain signal indicates a loose connection in the cyclic fatigue test setup. In this case, the 

research team worked with the participating laboratory to identify the source and resolve the 

loose connection. After resolving this issue, the test was repeated, and the actuator strain signal 

appeared smooth. Also, after resolving the loose connection, the damage characteristic curve and 

failure criterion results agreed with the research team’s results.

The final pilot test results obtained after resolving all procedural issues are presented in the 

subsequent sections.
79 

Source: FHWA.

Figure 48. Graph. Example of actuator strain results without a loose connection. 

Source: FHWA.

Figure 49. Graph. Example of actuator strain standard error results indicating a loose 

connection in the cyclic fatigue setup.

 

 
80 

FINAL PILOT TEST RESULTS

Figure 50 and figure 51 show the |E*| pilot test results using log-log and semilog scales, 

respectively. Figure 52 and figure 53 show the damage characteristic curve and failure criterion 

pilot cyclic fatigue test results, respectively. 

Source: FHWA.

Figure 50. Graph. Dynamic modulus master curve for pilot test results (log-log scale). 

Source: FHWA.

Figure 51. Graph. Dynamic modulus master curve for pilot test results (semilog scale).
81 

Source: FHWA.

Figure 52. Graph. Damage characteristic curves for pilot test results.

Source: FHWA.

Figure 53. Graph. Failure criterion for pilot test results.

FULL-SCALE ILS DYNAMIC MODULUS TEST RESULTS

Figure 54 through figure 59 show the full-scale |E*| test results for the 9.5-mm large specimen, 

9.5-mm small specimen, 12.5-mm large specimen, 12.5-mm small specimen, 19-mm small 

specimen, and 25-mm large specimens, respectively. Table 29 summarizes the |E*| test results 

used for cyclic fatigue test analysis.
82 

Source: FHWA.

Figure 54. Graph. Dynamic modulus test results for 9.5-mm mixture large specimen.

Source: FHWA.

Figure 55. Graph. Dynamic modulus test results for 9.5-mm mixture small specimen.
83 

Source: FHWA.

Figure 56. Graph. Dynamic modulus test results for 12.5-mm mixture large specimen.

Source: FHWA.

Figure 57. Graph. Dynamic modulus test results for 12.5-mm mixture small specimen.
84 

Source: FHWA.

Figure 58. Graph. Dynamic modulus test results for 19-mm mixture small specimen.

Source: FHWA.

Figure 59. Graph. Dynamic modulus test results for 25-mm mixture large specimen.
85 

Table 29. Dynamic modulus data used for ILS cyclic fatigue analysis. 

Specimen Lab A Lab B Lab C Lab D Lab E Lab F Lab G 

9.5-mm 

Large 

Yes Yes Yes No Yes Yes Yes

9.5-mm 

Small

Yes Yes Yes No Yes Yes Yes

12.5-mm 

Large 

Yes Yes Yes No Yes Yes Yes

12.5-mm 

Small

Yes Yes Yes No Yes Yes Yes

19-mm Small Yes Yes Yes No Yes Yes Yes

25-mm Large Yes Yes Yes No Yes Yes Yes

87 

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HRDI-10/06-23(WEB)E Recycled

Recyclable

Recommended citation: Federal Highway Administration, 

Ruggedness and Interlaboratory Studies for Asphalt Mixture 

Performance Tester (AMPT) Cyclic Fatigue Test: Phase II Report

(Washington, DC: 2023) https://doi.org/10.21949/1521943