Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

LTPP Computed Parameter: 

Dynamic Modulus

Publication No. FHWA-HRT-10-035 september 2011
FOREWORD

The dynamic modulus, |E*|, is a fundamental property that defines the strain response 

characteristics of asphalt concrete mixtures as a function of loading rate and temperature. The 

significance of this material property is threefold. First, it is one of the primary material property 

inputs in the Mechanistic Empirical Pavement Design Guide (MEPDG) and software developed 

by National Cooperative Highway Research Program Project 1-37A.(1,2) MEPDG uses a master 

curve and time-temperature shift factors in its internal modulus computation.(1) In MEPDG, the 

master curve is constructed using a hierarchical structure of inputs ranging from estimates 

based on mixture volumetrics and binder tests to full-scale mixture |E*| testing. |E*| is one 

of the primary properties measured in the Asphalt Mixture Performance Test protocol that 

complements the volumetric mix design.(3,4) Additionally, it is one of the fundamental linear 

viscoelastic material properties that can be used in advanced pavement response models based 

on viscoelasticity. 

Given the significance of |E*|, this study evaluated existing prediction models, developed new 

models, and populated the Long-Term Pavement Performance database to provide a valuable 

data source for the pavement community. Supplementing the full suite of material properties, 

performance history, traffic, and climate with |E*| estimates will be advantageous in conducting 

MEPDG calibration, validation, and implementation. 

Jorge E. Pagán-Ortiz

Director, Office of Infrastructure

Research and Development

Notice

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

in the interest of information exchange. The U.S. Government assumes no liability for the use of 

the information contained in this document. This report does not constitute a standard, 

specification, or regulation. 

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-10-035

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

4. Title and Subtitle

LTPP Computed Parameter: Dynamic Modulus

5. Report Date

September 2011 

6. Performing Organization Code: 

7. Author(s)

Y. Richard Kim, B. Underwood, M. Sakhaei Far, 

N. Jackson, and J. Puccinelli

8. Performing Organization Report No. 

1240.10

9. Performing Organization Name and Address

Nichols Consulting Engineers, Chtd.

1885 S. Arlington Avenue, Suite 111

Reno, NV 89509

10. Work Unit No.

11. Contract or Grant No.

DTFH61-02-D-00139

12. Sponsoring Agency Name and Address

Turner-Fairbank Highway Research Center

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101-2296

13. Type of Report and Period Covered

Final Report

14. Sponsoring Agency Code

15. Supplementary Notes

Prepared in cooperation with the U.S. Department of Transportation, the Contracting Officer’s Technical 

Representative (COTR) was Larry Wiser, Long-Term Pavement Performance Team.

16. Abstract

The dynamic modulus, |E*|, is a fundamental property that defines the stiffness characteristics of hot mix asphalt 

(HMA) mixtures as a function of loading rate and temperature. In spite of the demonstrated significance of |E*|, it 

is not included in the current Long-Term Pavement Performance (LTPP) materials tables because the database 

structure was established before |E*| was identified as the main HMA property in the Mechanistic Empirical 

Pavement Design Guide (MEPDG). The objective of this study was to use readily available binder, volumetric, 

and resilient material properties in the LTPP database to develop |E*| estimates. This report provides a thorough 

review of existing prediction models. In addition, several models have been developed using artificial neural 

networks for use in this project. This report includes assessments of each model, quality control checks applied to 

the data, and the final structure and format of the dynamic modulus data added to the LTPP database. A program 

was also developed to assist in populating the LTPP database, and the details of the program are provided in 

this report.

17. Key Words

Dynamic modulus, MEPDG, LTPP, Hot mix 

asphalt, Artificial neural network (ANN)

18. Distribution Statement

Document is available to the U.S. public through the 

National Technical Information Service, Springfield, VA, 

22161

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

263 

22. Price

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

SI* (MODERN METRIC) CONVERSION FACTORS 

APPROXIMATE CONVERSIONS TO SI UNITS

Symbol When You Know Multiply By To Find Symbol 

LENGTH 

in inches 25.4 millimeters mm 

ft feet 0.305 meters m 

yd yards 0.914 meters m 

mi miles 1.61 kilometers km 

AREA 

in2 square inches 645.2 square millimeters mm2

ft2 square feet 0.093 square meters m2

yd2 square yard 0.836 square meters m2

ac acres 0.405 hectares ha 

mi2 square miles 2.59 square kilometers km2

VOLUME 

fl oz fluid ounces 29.57 milliliters mL 

gal gallons 3.785 liters L 

ft3 cubic feet 0.028 cubic meters m3 

yd3 cubic yards 0.765 cubic meters m3 

NOTE: volumes greater than 1000 L shall be shown in m3

MASS oz ounces 28.35 grams g

lb pounds 0.454 kilograms kg

T short tons (2000 lb) 0.907 megagrams (or "metric ton") Mg (or "t") 

TEMPERATURE (exact degrees) o

F Fahrenheit 5 (F-32)/9 Celsius o

C 

or (F-32)/1.8 

ILLUMINATION 

fc foot-candles 10.76 lux lx 

fl foot-Lamberts 3.426 candela/m2 cd/m2

FORCE and PRESSURE or STRESS 

lbf poundforce 4.45 newtons N 

lbf/in2 poundforce per square inch 6.89 kilopascals kPa 

APPROXIMATE CONVERSIONS FROM SI UNITS 

Symbol When You Know Multiply By To Find Symbol 

LENGTH mm millimeters 0.039 inches in 

m meters 3.28 feet ft 

m meters 1.09 yards yd 

km kilometers 0.621 miles mi 

AREA mm2 square millimeters 0.0016 square inches in2 

m2 square meters 10.764 square feet ft2 

m2 square meters 1.195 square yards yd2 

ha hectares 2.47 acres ac 

km2 square kilometers 0.386 square miles mi2 

VOLUME 

mL milliliters 0.034 fluid ounces fl oz 

L liters 0.264 gallons gal 

m3 cubic meters 35.314 cubic feet ft3 

m3 cubic meters 1.307 cubic yards yd3 

MASS 

g grams 0.035 ounces oz

kg kilograms 2.202 pounds lb

Mg (or "t") megagrams (or "metric ton") 1.103 short tons (2000 lb) T 

TEMPERATURE (exact degrees) o

C Celsius 1.8C+32 Fahrenheit o

F 

ILLUMINATION 

lx lux 0.0929 foot-candles fc 

cd/m2 candela/m2 0.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS 

N newtons 0.225 poundforce lbf 

kPa kilopascals 0.145 poundforce per square inch lbf/in2

*SI is the symbol for the International System of Units. Appropriate rounding should be made to comply with Section 4 of ASTM E380. 

(Revised March 2003)
iii

TABLE OF CONTENTS

EXECUTIVE SUMMARY .......................................................................................................... 1

1.0 INTRODUCTION................................................................................................................... 5

2.0 PREDICTIVE MODELS ....................................................................................................... 7

2.1 ORIGINAL WITCZAK EQUATION (NCHRP 1-37A) ............................................... 7

2.2 MODIFIED WITCZAK EQUATION BASED ON |G*| (NCHRP 1-40D) .................. 8

2.3 HIRSCH MODEL ............................................................................................................. 9

2.4 LAW OF MIXTURES PARALLEL MODEL (AL-KHATEEB MODEL) .............. 10

2.5 ANN MODELS................................................................................................................ 10

2.6 SUMMARY OF INPUT VARIABLES ......................................................................... 11

3.0 DATABASES......................................................................................................................... 13

3.1 STATEMENT OF NEED ............................................................................................... 13

3.1.1 Inconsistent Definition of Frequency....................................................................... 13

3.1.2 Use of Estimated |G*| Values .................................................................................. 14

3.1.3 Low-Temperature |G*| Values ................................................................................. 18

3.1.4 Aging Effects ........................................................................................................... 18

3.2 BINDER DATABASES .................................................................................................. 19

3.2.1 Witczak Binder Database ......................................................................................... 19

3.2.2 FHWA Mobile Trailer Binder Database .................................................................. 20

3.2.3 FHWA TPF-5(019) Binder Database ...................................................................... 21

3.2.4 NCDOT Binder Database ........................................................................................ 22

3.2.5 Citgo Binder Database ............................................................................................. 22

3.2.6 WRI Binder Database .............................................................................................. 22

3.3 MIXTURE DATABASES .............................................................................................. 23

3.3.1 Witczak Mixture Database ....................................................................................... 23

3.3.2 FHWA Mobile Trailer Mixture Database ................................................................ 23

3.3.3 FHWA TPF-5(019) Mixture Database .................................................................... 24

3.3.4 NCDOT Mixture Database ...................................................................................... 29

3.3.4 NCDOT Mixture Database ...................................................................................... 29

3.3.5 WRI Mixture Database ............................................................................................ 29

3.3.6 Citgo Mixture Database ........................................................................................... 30

3.3.7 FHWA MR Database ................................................................................................ 31

3.3.8 LTPP Data ................................................................................................................ 32

3.3.9 Input Source Hierarchy ............................................................................................ 32

3.3.10 Data Structure and Detail ....................................................................................... 33

4.0 EVALUATION OF |E*| PREDICTIVE MODELS ........................................................... 37

4.1 ANN STRUCTURE USED IN THIS STUDY .............................................................. 37

4.2 EVALUATION OF BINDER |G*|-BASED MODELS ................................................ 38

4.3 VISCOSITY-BASED ANN ............................................................................................ 52

5.0 ANN MODELS FOR POPULATING THE LTPP DATABASE ..................................... 61

5.1 MR ANN MODEL ........................................................................................................... 63

5.1.1 Forward Modeling ................................................................................................... 64

5.1.2 ANN Model |E*| Backcalculation ............................................................................ 72
iv

5.2 VV ANN MODEL ........................................................................................................... 75

5.3 GV ANN MODEL ........................................................................................................... 77

5.4 OTHER ANN MODELS ................................................................................................ 80

5.5 ERROR ASSESSMENT ................................................................................................. 80

6.0 PREDICTION OF |E*| FOR LTPP LAYERS ................................................................... 83

6.1 MODEL PRIORITIZATION AND DECISION TREE DEVELOPMENT ............. 83

6.2 DATA QC ........................................................................................................................ 84

6.2.1 QC #1 ....................................................................................................................... 84

6.2.2 QC #2 ....................................................................................................................... 88

6.2.3 QC #3 ....................................................................................................................... 89

6.2.4 QC #4 ....................................................................................................................... 91

6.2.5 QC #5 ....................................................................................................................... 93

6.2.6 QC #6 ....................................................................................................................... 95

6.2.7 QC #7 ....................................................................................................................... 95

6.3 CONTRACTOR JUDGMENT ...................................................................................... 96

6.4 CONTRACTOR MASTERCURVE JUDGMENT ..................................................... 96

6.5 PREDICTION STATISTICS ........................................................................................ 96

6.6 LTPP DATABASE ADDITIONS .................................................................................. 97

7.0 SUMMARY AND FUTURE WORK ................................................................................ 103

APPENDIX A: PROCESSING ASPHALT BINDER VISCOSITY DATA ........................ 105

A.1 INTRODUCTION ........................................................................................................ 105

A.2 TEMPERATURE SUSCEPTIBILITY RELATIONSHIP ...................................... 105

A.3 R&BT TEMPERATURE ............................................................................................ 106

A.4 PENETRATION .......................................................................................................... 106

A.5 ABSOLUTE VISCOSITY ........................................................................................... 106

A.6 KINEMATIC VISCOSITY ......................................................................................... 107

A.7 EXAMPLE PROBLEM .............................................................................................. 107

A.8 TYPICAL VALUES FOR PURCHASE SPECIFICATION GRADE .................... 107

APPENDIX B: PROCESSING ASPHALT BINDER SHEAR MODULUS DATA ........... 109

B.1 INTRODUCTION ........................................................................................................ 109

B.2 USE OF BBR DATA IN |G*| MASTERCURVE GENERATION .......................... 109

B.3 ANALYSIS USING DSR MEASUREMENTS .......................................................... 109

B.4 ANALYSIS WITH BBR MEASUREMENTS ........................................................... 112

B.5 COMBINING DSR AND BBR ANALYSIS RESULTS ........................................... 115

B.6 COMPARISON OF RESULTS WITH AND WITHOUT BBR 

MEASUREMENTS ............................................................................................................ 117

B.7 EXTRACTING FULL BINDER |G*| DATA FROM LIMITED 

ISOCHRONAL MEASUREMENTS ................................................................................ 125

B.8 t-T SHIFT FACTOR SURROGATE MODEL DEVELOPMENT ......................... 127

B.8.1 Phenomenological t-T Shift Factor Function Model............................................. 128

B.8.2 Use of Average t-T Shift Factor Function ............................................................. 130

B.9 INCLUSION OF BBR DATA IN MASTERCURVE GENERATION ................... 131

B.10 PAV- TO RTFO-AGING MODEL (PAR MODEL) .............................................. 133

B.11 AGING EFFECT ON M-VALUE ............................................................................ 141

B.12 VERIFICATION OF LIMITED DATA DEVELOPMENT .................................. 142
v 

B.12.1 Level 1a Verification ........................................................................................... 142

B.12.2 Level 1b Verification .......................................................................................... 146

B.12.3 Level 2 Verification ............................................................................................ 150

B.12.4 Level 3 Verification ............................................................................................ 153

APPENDIX C: AMPT VERSUS TP-62 .................................................................................. 157

C.1 EXPERIMENTAL VERIFICATION OF AMPT AND TP-62 

DIFFERENCES .................................................................................................................. 157

C.2 COMPARISON OF AMPT AND TP-62 PROTOCOLS WITH THE 

AVAILABLE DATABASE ................................................................................................ 160

C.3 EVALUATION OF AMPT VERSUS TP-62 PROTOCOLS USING ANN 

MODEL ............................................................................................................................... 175

APPENDIX D: ANN MODEL FACTORS ............................................................................. 201

D.1 INTRODUCTION ........................................................................................................ 201

D.2 MR ANN ......................................................................................................................... 203

D.3 VV ANN ........................................................................................................................ 208

D.4 |G*|-BASED ANN (GV ANN) ..................................................................................... 212

APPENDIX E: ANNACAP SOFTWARE MANUAL ........................................................... 217

E.1 INTRODUCTION ........................................................................................................ 217

E.2 DISCLAIMER .............................................................................................................. 217

E.3 INSTALLATION INSTRUCTIONS .......................................................................... 218

E.4 USING ANNACAP ...................................................................................................... 218

E.4.1 Program Main Screen ............................................................................................ 218

E.4.2 Input Data .............................................................................................................. 219

E.4.3 Layer ID................................................................................................................. 220

E.4.4 MR-Based ANN ..................................................................................................... 221

E.4.5 |G*|-Based ANN .................................................................................................... 221

E.4.6 Viscosity-Based ANN ........................................................................................... 222

E.5 BATCH MODE ............................................................................................................ 223

E.6 OUTPUT DIRECTORY .............................................................................................. 228

E.7 RUN ANALYSIS .......................................................................................................... 229

E.8 FORM OF SUPPLEMENTARY FUNCTIONS ....................................................... 233

E.8.1 CAM Model Function ........................................................................................... 233

E.9 TIME-TEMPERATURE SHIFT FACTOR FUNCTION FOR |G*| ...................... 233

E.9.1 Level 1 (WLF Function) ........................................................................................ 233

E.9.2 Levels 2 and 3 ........................................................................................................ 233

E.10 SIGMOIDAL FUNCTION ........................................................................................ 234

E.11 TIME-TEMPERATURE SHIFT FACTOR FUNCTION FOR |E*| ..................... 234

REFERENCES .......................................................................................................................... 235
vi 

LIST OF FIGURES

Figure 1. Illustration. Modulus prediction model decision tree ...................................................... 3

Figure 2. Graph. Comparison between the Witczak predictive model and measured |G*|b 

values in arithmetic scale .............................................................................................................. 15

Figure 3. Graph. Comparison between the Witczak predictive model and measured |G*|b 

values in logarithmic scale ............................................................................................................ 16

Figure 4. Graph. Comparison between the Witczak predictive model and measured δb to 

develop the δb model .................................................................................................................... 16

Figure 5. Graph. Comparison between the Witczak predictive model and measured |G*|b 

values using Citgo binders in the North Carolina Department of Transportation (NCDOT) 

database in arithmetic scale .......................................................................................................... 17

Figure 6. Graph. Comparison between the Witczak predictive model and measured |G*|b 

values using Citgo binders in the NCDOT database in logarithmic scale .................................... 17

Figure 7. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the modified Witczak in arithmetic scale .............. 40

Figure 8. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the modified Witczak in logarithmic scale ........... 40

Figure 9. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the Hirsch model in arithmetic scale ..................... 41

Figure 10. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the Hirsch model in logarithmic scale ................... 41

Figure 11. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the Al-Khateeb model in arithmetic scale ............. 42

Figure 12. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the Al-Khateeb model in logarithmic scale........... 42

Figure 13. Graph. Prediction of training data containing processed Witczak, FHWA I, 

NCDOT I, and WRI databases using |G*| binder and gradation-based pilot ANN model 

(G-GR pANN) in arithmetic scale ................................................................................................ 43

Figure 14. Graph. Prediction of training data containing processed Witczak, FHWA I, 

NCDOT I, and WRI databases using G-GR pANN in logarithmic scale ..................................... 43

Figure 15. Graph. Prediction of training data containing processed Witczak, FHWA I, 

NCDOT I, and WRI databases using preliminary |G*|-based models used in phase I of 

the study (G-V) pANN in arithmetic scale ................................................................................... 44

Figure 16. Graph. Prediction of training data containing processed Witczak, FHWA I, 

NCDOT I, and WRI databases using G-V pANN in logarithmic scale........................................ 44

Figure 17. Graph. Predicted moduli using G-GR pANN and modified Witczak models 

for the FHWA II database in arithmetic scale .............................................................................. 45

Figure 18. Graph. Predicted moduli using G-GR pANN and modified Witczak models 

for the FHWA II database in logarithmic scale ............................................................................ 45

Figure 19. Graph. Predicted moduli using G-V pANN and Hirsch models for the FHWA II 

database in arithmetic scale .......................................................................................................... 46

Figure 20. Graph. Predicted moduli using G-V pANN and Hirsch models for the FHWA II 

database in logarithmic scale ........................................................................................................ 46

Figure 21. Graph. Predicted moduli using G-GR pANN and modified Witczak models 

for the NCDOT II database in arithmetic scale ............................................................................ 47
vii 

Figure 22. Graph. Predicted moduli using G-GR pANN and modified Witczak models 

for the NCDOT II database in logarithmic scale .......................................................................... 47

Figure 23. Graph. Predicted moduli using G-V pANN and Hirsch models for the NCDOT II 

database in arithmetic scale .......................................................................................................... 48

Figure 24. Graph. Predicted moduli using G-V pANN and Hirsch models for the NCDOT II 

database in logarithmic scale ........................................................................................................ 48

Figure 25. Graph. Predicted moduli using G-GR pANN and modified Witczak models 

for the Citgo database in arithmetic scale ..................................................................................... 49

Figure 26. Graph. Predicted moduli using G-GR pANN and modified Witczak models 

for the Citgo database in logarithmic scale ................................................................................... 49

Figure 27. Graph. Predicted moduli using G-V pANN and Hirsch models for the Citgo 

database in arithmetic scale .......................................................................................................... 50

Figure 28. Graph. Predicted moduli using G-V pANN and Hirsch models for the Citgo 

database in logarithmic scale ........................................................................................................ 50

Figure 29. Graph. Prediction of training data containing Witczak, FHWA I, and NCDOT I 

databases using Visc-GR pANN in arithmetic scale .................................................................... 53

Figure 30. Graph. Prediction of training data containing Witczak, FHWA I, and NCDOT I 

databases using Visc-GR pANN in logarithmic scale .................................................................. 54

Figure 31. Graph. Prediction of training data containing Witczak, FHWA I, and NCDOT I 

databases using Visc-V pANN in arithmetic scale ....................................................................... 54

Figure 32. Graph. Prediction of training data containing Witczak, FHWA I, and NCDOT I 

databases using Visc-V pANN in logarithmic scale ..................................................................... 55

Figure 33. Graph. Predicted moduli using Visc-GR pANN model for the FHWA II database 

in arithmetic scale ......................................................................................................................... 55

Figure 34. Graph. Predicted moduli using Visc-GR pANN model for the FHWA II database 

in logarithmic scale ....................................................................................................................... 56

Figure 35. Graph. Predicted moduli using Visc-V pANN model for the FHWA II database 

in arithmetic scale ......................................................................................................................... 56

Figure 36. Graph. Predicted moduli using Visc-V pANN model for the FHWA II database 

in logarithmic scale ....................................................................................................................... 57

Figure 37. Graph. Predicted moduli using Visc-GR pANN model for the NCDOT II database 

in arithmetic scale ......................................................................................................................... 57

Figure 38. Graph. Predicted moduli using Visc-GR pANN model for the NCDOT II database 

in logarithmic scale ....................................................................................................................... 58

Figure 39. Graph. Predicted moduli using Visc-V pANN model for the NCDOT II database 

in arithmetic scale ......................................................................................................................... 58

Figure 40. Graph. Predicted moduli using Visc-V pANN model for the NCDOT II database 

in logarithmic scale ....................................................................................................................... 59

Figure 41. Illustration. Network structure used for training the ANN models ............................. 63

Figure 42. Illustration. Stress distribution in the IDT specimen subjected to a strip load ............ 65

Figure 43. Graph. Poisson’s ratio versus reduced time for S12.5C .............................................. 69

Figure 44. Graph. Comparison of predicted and measured MR values for S12.5C mixture ......... 71

Figure 45. Graph. Comparison of predicted and measured MR values for S12.5CM mixture ..... 71

Figure 46. Graph. Comparison of predicted and measured MR values for S12.5FE mixture ....... 72

Figure 47. Graph. Comparison of predicted and measured MR values for B25.0C mixture ......... 72
viii

Figure 48. Graph. MR ANN model using 90 percent of randomly selected data as a training 

set in arithmetic scale .................................................................................................................... 73

Figure 49. Graph. MR ANN model using 90 percent of randomly selected data as a training 

set in logarithmic scale.................................................................................................................. 73

Figure 50. Graph. MR ANN model using 10 percent of randomly selected data as a 

verification set in arithmetic scale ................................................................................................ 74

Figure 51. Graph. MR ANN model using 10 percent of randomly selected data as a 

verification set in logarithmic scale .............................................................................................. 74

Figure 52. Graph. VV ANN model using 90 percent of randomly selected data as a training 

set in arithmetic scale .................................................................................................................... 75

Figure 53. Graph. VV ANN model using 90 percent of randomly selected data as a training 

set in logarithmic scale.................................................................................................................. 76

Figure 54. Graph. VV ANN model using 10 percent of randomly selected data as a 

verification set in arithmetic scale ................................................................................................ 76

Figure 55. Graph. VV ANN model using 10 percent of randomly selected data as a 

verification set in logarithmic scale .............................................................................................. 77

Figure 56. Graph. GV ANN model using 90 percent of randomly selected data as a training 

set in arithmetic scale .................................................................................................................... 78

Figure 57. Graph. GV ANN model using 90 percent of randomly selected data as a training 

set in logarithmic scale.................................................................................................................. 78

Figure 58. Graph. GV ANN model using 10 percent of randomly selected data as a 

verification set in arithmetic scale ................................................................................................ 79

Figure 59. Graph. GV ANN model using 10 percent of randomly selected data as a 

verification set in logarithmic scale .............................................................................................. 79

Figure 60. Illustration. Decision tree applied to population of LTPP database based on 

ranking of ANN models ................................................................................................................ 83

Figure 61. Graph. Example of the effect of a violation of QC #1 for MR ANN model in 

semi-log scale................................................................................................................................ 85

Figure 62. Graph. Example of the effect of a violation of QC #1 for MR ANN model in 

log-log scale .................................................................................................................................. 86

Figure 63. Graph. Example of the effect of a violation of QC #1 for VV ANN model in 

semi-log scale................................................................................................................................ 86

Figure 64. Graph. Example of the effect of a violation of QC #1 for VV ANN model in 

log-log scale .................................................................................................................................. 87

Figure 65. Graph. Example of the effect of a violation of QC #1 for VV ANN model 

unshifted data ................................................................................................................................ 87

Figure 66. Graph. Example of the effect of a violation of QC #2 in semi-log scale .................... 88

Figure 67. Graph. Example of the effect of a violation of QC #2 in log-log scale ....................... 89

Figure 68. Graph. Example of the effect of a violation of QC #2 unshifted data ......................... 89

Figure 69. Graph. Example of the effect of a violation of QC #3 in semi-log scale .................... 90

Figure 70. Graph. Example of the effect of a violation of QC #3 in log-log scale ....................... 91

Figure 71. Graph. Example of the effect of a violation of QC #3 unshifted data ......................... 91

Figure 72. Graph. Example of the effect of a violation of QC #4 in semi-log scale .................... 92

Figure 73. Graph. Example of the effect of a violation of QC #4 in log-log scale ....................... 92

Figure 74. Graph. Example of the effect of a violation of QC #4 unshifted data ......................... 93

Figure 75. Graph. Example of the effect of a violation of QC #5 in semi-log scale .................... 94
ix

Figure 76. Graph. Example of the effect of a violation of QC #5 in log-log scale ....................... 94

Figure 77. Graph. Example of the effect of a violation of QC #5 unshifted data ......................... 95

Figure 78. Illustration. ANN models and their appropriate input and output tables .................. 102

Figure 79. Graph. A-VTS relationship ....................................................................................... 105

Figure 80. Graph. |G*| versus frequency curves at different temperatures ................................. 110

Figure 81. Graph. |G*| versus reduced frequency mastercurve at 59 oF (15 °C) ........................ 111

Figure 82. Graph. t-T shift factor function at 59 oF (15 °C) ....................................................... 111

Figure 83. Graph. D(t) at different temperatures for ALF AC5 binder ...................................... 113

Figure 84. Graph. Comparison of optimization and regression GPL characterization results ... 114

Figure 85. Graph. |G*| mastercurve for the WesTrack binder at 59 oF (15 °C) .......................... 116

Figure 86. Graph. Optimized t-T shift function for WesTrack binder at 59 oF (15 °C) ............. 116

Figure 87. Graph. Comparison of t-T shift factor functions calibrated with and without 

BBR data ..................................................................................................................................... 118

Figure 88. Graph. |G*| mastercurve using t-T shift factor function calibrated without

BBR data ..................................................................................................................................... 119

Figure 89. Graph. |G*| mastercurves characterized with and without BBR data for ALF 

AC5 binder .................................................................................................................................. 121

Figure 90. Graph. |G*| mastercurves characterized with and without BBR data for the 

MnRoad Pen 120-150 binder ...................................................................................................... 122

Figure 91. Graph. |G*| mastercurves characterized with and without BBR data for the 

WesTrack binder ......................................................................................................................... 122

Figure 92. Graph. t-T shift factor function characterized with and without BBR data for 

the ALF AC5 binder ................................................................................................................... 123

Figure 93. Graph. t-T shift factor function characterized with and without BBR data for the 

MnRoad Pen 120-150 binder ...................................................................................................... 123

Figure 94. Graph. t-T shift factor function characterized with and without BBR data for the 

WesTrack binder ......................................................................................................................... 124

Figure 95. Graph. Phenomenological t-T shift factor function ................................................... 129

Figure 96. Graph. Effect of change in time from DSR to BBR on the t-T shift factor 

function model ............................................................................................................................ 129

Figure 97. Graph. t-T shift factor model verification with phenomenological model ................ 130

Figure 98. Graph. Average t-T shift factor function ................................................................... 131

Figure 99. Graph. t-T shift factor model verification with average function model .................. 131

Figure 100. Graph. Effect of temperature on aging ratio of asphalt binder ................................ 134

Figure 101. Graph. Effect of angular frequency on aging ratio of asphalt binder ...................... 135

Figure 102. Graph. Calibrated PAR model in arithmetic scale .................................................. 135

Figure 103. Graph. Calibrated PAR model in logarithmic scale ................................................ 136

Figure 104. Graph. Strength of PAR model in arithmetic scale ................................................. 136

Figure 105. Graph. Strength of PAR model in logarithmic scale ............................................... 137

Figure 106. Graph. PAR model verification in arithmetic logarithmic scale ............................. 137

Figure 107. Graph. PAR model verification in logarithmic scale .............................................. 138

Figure 108. Graph. BBR-calibrated κT relationship ................................................................... 139

Figure 109. Graph. Comparison of |G*| mastercurve analysis under PAV and RTFO 

conditions for ALF AC10 binder ................................................................................................ 139

Figure 110. Graph. Calibrated PAR model for BBR conditions in arithmetic scale .................. 140

Figure 111. Graph. Calibrated PAR model for BBR conditions in logarithmic scale ................ 140
x 

Figure 112. Graph. Verification of m-value relationship ............................................................ 142

Figure 113. Graph. Comparison of typical |G*| mastercurves characterized using full 

database and SuperpaveTM only database ................................................................................... 143

Figure 114. Graph. Comparison of typical t-T shift factors characterized using full 

database and SuperpaveTM only database ................................................................................... 144

Figure 115. Graph. Level 1a verification of limited data analysis procedure in arithmetic 

scale using phenomenological shift factor function model......................................................... 144

Figure 116. Graph. Level 1a verification of limited data analysis procedure in logarithmic 

scale using phenomenological shift factor function model......................................................... 145

Figure 117. Graph. Level 1a verification of limited data analysis procedure in arithmetic 

scale using average shift factor function model .......................................................................... 145

Figure 118. Graph. Level 1a verification of limited data analysis procedure in logarithmic 

scale using average shift factor function model .......................................................................... 146

Figure 119. Graph. Comparison of typical |G*| mastercurve characterized using full 

database and SuperpaveTM only database plus PAR model ........................................................ 147

Figure 120. Graph. Comparison of typical t-T shift factors characterized using full 

database and SuperpaveTM only database plus PAR model ........................................................ 148

Figure 121. Graph. Level 1b verification of limited data analysis procedure in arithmetic 

scale using phenomenological shift factor function model......................................................... 148

Figure 122. Graph. Level 1b verification of limited data analysis procedure in logarithmic 

scale using phenomenological shift factor function model......................................................... 149

Figure 123. Graph. Level 1b verification of limited data analysis procedure in arithmetic 

scale using average shift factor function model .......................................................................... 149

Figure 124. Graph. Level 1b verification of limited data analysis procedure in logarithmic 

scale using average shift factor function model .......................................................................... 150

Figure 125. Graph. Level 2 verification of limited data analysis procedure in arithmetic 

scale using phenomenological shift factor function model......................................................... 151

Figure 126. Graph. Level 2 verification of limited data analysis procedure in logarithmic 

scale using phenomenological shift factor function model......................................................... 152

Figure 127. Graph. Level 2 verification of limited data analysis procedure in arithmetic 

scale using average shift factor function model .......................................................................... 152

Figure 128. Graph. Level 2 verification of limited data analysis procedure in logarithmic 

scale using average shift factor function model .......................................................................... 153

Figure 129. Graph. Level 3 verification of limited data analysis procedure in arithmetic 

scale using phenomenological shift factor function model......................................................... 154

Figure 130. Graph. Level 3 verification of limited data analysis procedure in logarithmic 

scale using phenomenological shift factor function model......................................................... 155

Figure 131. Graph. Level 3 verification of limited data analysis procedure in arithmetic 

scale using average shift factor function model .......................................................................... 155

Figure 132. Graph. Level 3 verification of limited data analysis procedure in logarithmic 

scale using average shift factor function model .......................................................................... 156

Figure 133. Graph. Test mixture gradation ................................................................................. 158

Figure 134. Graph. Comparison of |E*| measured via AMPT and TP-62 protocols in 

arithmetic scale ........................................................................................................................... 159

Figure 135. Graph. Comparison of |E*| measured via AMPT and TP-62 protocols in 

logarithmic scale ......................................................................................................................... 159
xi

Figure 136. Graph. Frequency distribution of temperature in AMPT versus TP-62 

databases ..................................................................................................................................... 161

Figure 137. Graph. Range of temperature in AMPT versus TP-62 databases ............................ 162

Figure 138. Graph. Frequency distribution of frequency in AMPT versus TP-62 databases ..... 162

Figure 139. Graph. Range of loading frequency in AMPT versus TP-62 databases .................. 163

Figure 140. Graph. Frequency distribution of percentage retained on 3/4-inch (19.05-mm) 

sieve (ρ 34) in AMPT versus TP-62 databases ............................................................................ 163

Figure 141. Graph. Range of percentage retained on 3/4-inch (19.05-mm) sieve (ρ 34) in 

AMPT versus TP-62 databases ................................................................................................... 164

Figure 142. Graph. Frequency distribution of percentage retained on 3/8-inch (9.56-mm) 

sieve (ρ 38) in AMPT versus TP-62 databases ............................................................................ 164

Figure 143. Graph. Range of percentage retained on 3/8-inch (9.56-mm) sieve (ρ 38) in 

AMPT versus TP-62 databases ................................................................................................... 165

Figure 144. Graph. Frequency distribution of percentage retained on #4 sieve (ρ 4) in 

AMPT versus TP-62 databases ................................................................................................... 165

Figure 145. Graph. Range of percentage retained on #4 sieve (ρ 4) in AMPT versus 

TP-62 databases .......................................................................................................................... 166

Figure 146. Graph. Frequency distribution of percentage passing #200 sieve (ρ 200) in 

AMPT versus TP-62 databases ................................................................................................... 166

Figure 147. Graph. Range of percentage passing #200 sieve (ρ 200) in AMPT versus 

TP-62 databases .......................................................................................................................... 167

Figure 148. Graph. Frequency distribution of specimen air voids in AMPT versus 

TP-62 databases .......................................................................................................................... 167

Figure 149. Graph. Range of specimen air voids in AMPT versus TP-62 databases ................. 168

Figure 150. Graph. Frequency distribution of effective binder volume in AMPT versus 

TP-62 databases .......................................................................................................................... 168

Figure 151. Graph. Range of effective binder volume in AMPT versus TP-62 databases ......... 169

Figure 152. Graph. Frequency distribution of VMA in AMPT versus TP-62 databases ........... 169

Figure 153. Graph. Range of VMA in AMPT versus TP-62 databases ..................................... 170

Figure 154. Graph. Frequency distribution of VFA in AMPT versus TP-62 databases ............. 170

Figure 155. Graph. Range of VFA in AMPT versus TP-62 databases ...................................... 171

Figure 156. Graph. Frequency distribution of |G*| in AMPT versus TP-62 databases .............. 171

Figure 157. Range of |G*| in AMPT versus TP-62 databases .................................................... 172

Figure 158. Graph. Percentage of difference between AMPT versus TP-62 databases 

based on similar ranges of different variables at 39.9 oF (4.4 °C) .............................................. 172

Figure 159. Graph. Percentage of difference between AMPT versus TP-62 databases 

based on similar ranges of different variables at 69.9 oF (21.1 oC) ............................................ 173

Figure 160. Graph. Percentage of difference between AMPT versus TP-62 databases 

based on similar ranges of different variables at 100 oF (37.8 oC) ............................................. 173

Figure 161. Graph. Percentage of difference between AMPT versus TP-62 databases 

based on similar ranges of different variables at 129.2 oF (54.0 oC) .......................................... 174

Figure 162. Graph. Percentage of difference between AMPT versus TP-62 databases 

based on similar ranges of different variables at 129.9 oF (54.4 oC) .......................................... 174

Figure 163. Graph. Prediction of the combination of AMPT and TP-62 data using the 

modified Witczak and G-GR pANN models in arithmetic scale ................................................ 179
xii

Figure 164. Graph. Prediction of the combination of AMPT and TP-62 data using the 

modified Witczak and G-GR pANN models in logarithmic scale ............................................. 179

Figure 165. Graph. Prediction of the combination of AMPT and TP-62 data using the 

Hirsch model in arithmetic scale................................................................................................. 180

Figure 166. Graph. Prediction of the combination of AMPT and TP-62 data using the 

Hirsch model in logarithmic scale .............................................................................................. 180

Figure 167. Graph. Prediction of the combination of AMPT and TP-62 data using the 

Al-Khateeb model in arithmetic scale......................................................................................... 181

Figure 168. Graph. Prediction of the combination of AMPT and TP-62 data using the 

Al-Khateeb model in logarithmic scale ...................................................................................... 181

Figure 169. Graph. Prediction of the AMPT data using the modified Witczak and AMPT 

pANN models in arithmetic scale ............................................................................................... 182

Figure 170. Graph. Prediction of the AMPT data using the modified Witczak and AMPT 

pANN models in logarithmic scale ............................................................................................. 182

Figure 171. Graph. Prediction of the AMPT data using the Hirsch model in arithmetic 

scale............................................................................................................................................. 183

Figure 172. Graph. Prediction of the AMPT data using the Hirsch model in logarithmic 

scale............................................................................................................................................. 183

Figure 173. Graph. Prediction of the AMPT data using the Al-Khateeb model in arithmetic 

scale............................................................................................................................................. 184

Figure 174. Graph. Prediction of the AMPT data using the Al-Khateeb model in logarithmic 

scale............................................................................................................................................. 184

Figure 175. Graph. Prediction of the TP-62 data using the modified Witczak and TP-62 

pANN models in arithmetic scale ............................................................................................... 185

Figure 176. Graph. Prediction of the TP-62 data using the modified Witczak and TP-62 

pANN models in logarithmic scale ............................................................................................. 185

Figure 177. Graph. Prediction of the TP-62 data using the Hirsch model in arithmetic scale ... 186

Figure 178. Graph. Prediction of the TP-62 data using the Hirsch model in logarithmic 

scale............................................................................................................................................. 186

Figure 179. Graph. Prediction of the TP-62 data using the Al-Khateeb model in arithmetic 

scale............................................................................................................................................. 187

Figure 180. Graph. Prediction of the TP-62 data using the Al-Khateeb model in logarithmic 

scale............................................................................................................................................. 187

Figure 181. Graph. Prediction of the FHWA II data using the modified Witczak and G-GR 

pANN models in arithmetic scale ............................................................................................... 188

Figure 182. Graph. Prediction of the FHWA II data using the modified Witczak and G-GR 

pANN models in logarithmic scale ............................................................................................. 188

Figure 183. Graph. Prediction of the FHWA II data using the AMPT pANN and TP-62 

pANN models in arithmetic scale ............................................................................................... 189

Figure 184. Graph. Prediction of the FHWA II data using the AMPT pANN and TP-62 

pANN models in logarithmic scale ............................................................................................. 189

Figure 185. Graph. Prediction of the FHWA II data using the Hirsch model in arithmetic 

scale............................................................................................................................................. 190

Figure 186. Graph. Prediction of the FHWA II data using the Hirsch model in logarithmic 

scale............................................................................................................................................. 190
xiii

Figure 187. Graph. Prediction of the FHWA II data using the Al-Khateeb model in 

arithmetic scale ........................................................................................................................... 191

Figure 188. Graph. Prediction of the FHWA II data using the Al-Khateeb model in 

logarithmic scale ......................................................................................................................... 191

Figure 189. Graph. Prediction of the NCDOT II data using the modified Witczak and G-GR 

pANN models in arithmetic scale ............................................................................................... 192

Figure 190. Graph. Prediction of the NCDOT II data using the modified Witczak and G-GR 

pANN models in logarithmic scale ............................................................................................. 192

Figure 191. Graph. Prediction of the NCDOT II data using the AMPT pANN and TP-62 

pANN models in arithmetic scale ............................................................................................... 193

Figure 192. Graph. Prediction of the NCDOT II data using the AMPT pANN and TP-62 

pANN models in logarithmic scale ............................................................................................. 193

Figure 193. Graph. Prediction of the NCDOT II data using the Hirsch model in arithmetic 

scale............................................................................................................................................. 194

Figure 194. Graph. Prediction of the NCDOT II data using the Hirsch model in logarithmic 

scale............................................................................................................................................. 194

Figure 195. Graph. Prediction of the NCDOT II data using the Al-Khateeb model in 

arithmetic scale ........................................................................................................................... 195

Figure 196. Graph. Prediction of the NCDOT II data using the Al-Khateeb model in 

logarithmic scale ......................................................................................................................... 195

Figure 197. Graph. Prediction of the Citgo data using the modified Witczak and G-GR 

pANN models in arithmetic scale ............................................................................................... 196

Figure 198. Graph. Prediction of the Citgo data using the modified Witczak and G-GR 

pANN models in logarithmic scale ............................................................................................. 196

Figure 199. Graph. Prediction of the Citgo data using the AMPT pANN and TP-62 pANN 

models in arithmetic scale ........................................................................................................... 197

Figure 200. Graph. Prediction of the Citgo data using the AMPT pANN and TP-62 pANN 

models in logarithmic scale ........................................................................................................ 197

Figure 201. Graph. Prediction of the Citgo data using the Hirsch model in arithmetic scale .... 198

Figure 202. Graph. Prediction of the Citgo data using the Hirsch model in logarithmic 

scale............................................................................................................................................. 198

Figure 203. Graph. Prediction of the Citgo data using the Al-Khateeb model in arithmetic 

scale............................................................................................................................................. 199

Figure 204. Graph. Prediction of the Citgo data using the Al-Khateeb model in logarithmic 

scale............................................................................................................................................. 199

Figure 205. Screenshot. ANNACAP main screen ...................................................................... 219

Figure 206. Illustration. ANNACAP menu diagram .................................................................. 219

Figure 207. Screenshot. ANNACAP input screen ...................................................................... 220

Figure 208. Screenshot. Output directory dialog ........................................................................ 229

Figure 209. Screenshot. Choosing output directory.................................................................... 229
xiv

LIST OF TABLES

Table 1. Predictive relationships for |E*| ........................................................................................ 7

Table 2. Model variables............................................................................................................... 11

Table 3. Example of summarized mixture and binder properties in the original Witczak 

database ......................................................................................................................................... 14

Table 4. Summary of |G*| data available in the Witczak binder database .................................... 19

Table 5. Summary of |G*| data available in the FHWA mobile trailer database .......................... 20

Table 6. Summary of |G*| data available in the FHWA TPF-5(019) binder database ................. 21

Table 7. Summary of |G*| data available in the NCDOT binder database ................................... 22

Table 8. Summary of |G*| data available in the Citgo binder database ........................................ 22

Table 9. Summary of |G*| data available in the WRI database .................................................... 23

Table 10. Summary of |E*| data available in the processed Witczak mixture database ............... 25

Table 11. Summary of |E*| data available in the FHWA I mobile trailer mixture database ........ 26

Table 12. Summary of |E*| data available in the FHWA II mobile trailer mixture database ....... 27

Table 13. Summary of |E*| data available in the FHWA TPF-5(019) mixture database.............. 28

Table 14. Summary of |E*| data available in the NCDOT mixture database ............................... 29

Table 15. Summary of |E*| data available in the WRI database .................................................. 30

Table 16. Summary of |E*| data available in the Citgo mixture database .................................... 31

Table 17. Summary of |E*| data available in the MR database ...................................................... 31

Table 18. Summary of extracted LTPP data ................................................................................. 34

Table 19. Description of |G*|-based ANN models ....................................................................... 51

Table 20. Description of viscosity-based ANN models................................................................ 53

Table 21. Parameters and number of nodes used in developed ANN models .............................. 62

Table 22. Geometry coefficients ................................................................................................... 65

Table 23. Probability distribution of error functions of each ANN model ................................... 81

Table 24. Input ranges of different ANN models ......................................................................... 85

Table 25. Limiting |E*| values used for QC #6 ............................................................................. 95

Table 26. Statistics of LTPP data populated with |E*| .................................................................. 97

Table 27. Structure and format of the TST_ESTAR_MASTER table ......................................... 99

Table 28. Structure and format of the TST_ESTAR_MODULUS table ...................................... 99

Table 29. Structure and format of the TST_ESTAR_MODULUS_COEFF table ..................... 100

Table 30. Structure and format of the TST_ESTAR_GSTAR_INPUT table ............................. 100

Table 31. Structure and format of the TST_ESTAR_GSTAR_CAM_COEFF table ................. 100

Table 32. Structure and format of the TST_ESTAR_VOLUM_INPUT table ........................... 101

Table 33. Structure and format of the TST_ESTAR_VISC_INPUT table ................................. 101

Table 34. Structure and format of the TST_ESTAR_VISC_MODEL_COEFF table ................ 101

Table 35. Structure and format of the TST_ESTAR_MR_INPUT table.................................... 101

Table 36. Relationship between asphalt binder grade and viscosity parameters ........................ 108

Table 37. Calibrated CAM model and t-T shift factor function coefficients using BBR 

data .............................................................................................................................................. 117

Table 38. Calibrated CAM model and t-T shift factor function coefficients omitting 

BBR data ..................................................................................................................................... 118
xv

Table 39. Percentage of difference between BBR-calibrated and non-BBR-calibrated |G*| 

at TP-62 temperatures and frequencies ....................................................................................... 120

Table 40. Percentage of log difference between BBR-calibrated and non-BBR-calibrated 

|G*| at TP-62 temperatures and frequencies ............................................................................... 125

Table 41. PAR model coefficient summary for SuperpaveTM high temperature grades ............ 134

Table 42. PAR model verification dataset summary .................................................................. 138

Table 43. TP summary ................................................................................................................ 157

Table 44. Test mixture volumetric properties ............................................................................. 158

Table 45. Statistical summary of AMPT and TP-62 test results................................................. 160

Table 46. Percentage of difference between AMPT versus TP-62 database based on similar 

ranges of different variables ........................................................................................................ 175

Table 47. Summary of database used for training ANN models ................................................ 175

Table 48. Summary of the database used for verification of ANN models ................................ 176

Table 49. Description of the developed ANN Models and their validation statistics................. 177

Table 50. Wik

1 matrix elements for MR ANN (part 1) ................................................................. 204

Table 51. Wik

1 matrix elements for MR ANN (part 2) ................................................................. 204

Table 52. Bk

1 vector elements for MR ANN (transposed for convenience)................................. 205

Table 53. Wkj

2 matrix elements for MR ANN (part 1) ................................................................. 205

Table 54. Wkj

2 matrix elements for MR ANN (part 2) ................................................................. 206

Table 55. Wkj

2 matrix elements for MR ANN (part 3) ................................................................. 206

Table 56. Bj

2 vector elements for MR ANN (transposed for convenience) ................................. 207

Table 57. Wjl

3 vector elements for MR ANN (part 1) .................................................................. 207

Table 58. Wjl

3 vector elements for MR ANN (part 2) .................................................................. 207

Table 59. Wjl

3 vector elements for MR ANN (part 3) .................................................................. 207

Table 60. Normalization parameters for MR ANN ..................................................................... 208

Table 61. Wik

1 matrix elements for VV ANN ............................................................................. 208

Table 62. Bk

1 vector elements for VV ANN (transposed for convenience) ................................ 209

Table 63. Wkj

2 matrix elements for VV ANN (part 1) ................................................................ 209

Table 64. Wkj

2 matrix elements for VV ANN (part 2) ................................................................ 210

Table 65. Wkj

2 matrix elements for VV ANN (part 3) ................................................................ 210

Table 66. Wkj

2 matrix elements for VV ANN (part 4) ................................................................ 211

Table 67. Bj

2 vector elements for VV ANN (transposed for convenience) ................................ 211

Table 68. Wj

3

 vector elements for VV ANN ............................................................................... 212

Table 69. Normalization parameters for VV ANN ..................................................................... 212

Table 70. Wik

1 matrix elements for GV ANN ............................................................................. 213

Table 71. Bk

1 vector elements for GV ANN (transposed for convenience) ................................ 213

Table 72. Wkj

2 matrix elements for GV ANN (part 1) ................................................................ 214

Table 73. Wkj

2 matrix elements for GV ANN (part 2) ................................................................ 214

Table 74. Wkj

2 matrix elements for GV ANN (part 3) ................................................................ 215

Table 75. Bj

2 vector elements for GV ANN (transposed for convenience) ................................ 215

Table 76. Wj

3

 vector elements for GV ANN ............................................................................... 216

Table 77. Normalization parameters for GV ANN ..................................................................... 216

Table 78. Relationship between binder purchase specification grade and A and VTS 

parameters ................................................................................................................................... 223

Table 79. File format for lab-aged files ...................................................................................... 225

Table 80. File format for field-aged files .................................................................................... 227
xvi 

Table 81. Output data format for single ANNACAP use ........................................................... 230

Table 82. Summary file format from batch mode in ANNACAP .............................................. 231

Table 83. Output data format for batch mode ANNACAP use. ................................................. 231
xvii 

LIST OF ABBREVIATIONS AND SYMBOLS

Abbreviations 

A Regression intercept

AASHTO American Association of State Highway and Transportation Officials

AC Asphalt concrete

ALF Accelerated loading facility

AMPT Asphalt mixture performance tester

ANN Artificial neural network 

ANNACAP Artificial Neural Networks for Asphalt Concrete Dynamic Modulus 

Prediction

ASTM American Society for Testing and Materials 

A-VTS Regression intercept-regression slope of viscosity temperature 

susceptibility

BBR Bending beam rheometer 

CAM Christensen Anderson Marasteanu 

DSR Dynamic shear rheometer 

FHWA Federal Highway Administration 

G-GR |G*| binder and gradation-based

GPL Generalized power law

GPS General Pavement Study 

GR Gradation

G-V Preliminary |G*|-based model used in phase 1 of the study 

GV |G*|-based model using consistent aged binder data 

GV-PAR |G*|-based model using inconsistent aged binder data of pressure-aging 

vessel and rolling thin film oven aging conditions 

HMA Hot mix asphalt

IDT Indirect tensile

LOE Line of equality

LTOA Long-term oven aging

LTPP Long-Term Pavement Performance

LVE Linear viscoelastic

MEPDG Mechanistic Empirical Pavement Design Guide

NCDOT North Carolina Department of Transportation 
xviii

NCHRP National Cooperative Highway Research Program 

NCSU North Carolina State University 

NMSA Nominal maximum-sized aggregate

pANN Pilot artificial neural network

PAV Pressure-aging vessel 

PG Performance grade

QC Quality control 

R&BT Ring and ball temperature 

RTFO Rolling thin film oven 

SHRP Strategic Highway Research Program 

SPS Specific Pavement Study 

STOA Short-term oven aging

TFHRC Turner-Fairbank Highway Research Center

TP Test protocol 

VFA Percentage of voids filled with asphalt 

Visc-GR Viscosity-gradation model 

Visc-V Viscosity-volumetric model

VMA Percentage of voids in mineral aggregate 

VTS Slope of temperature susceptibility relationship

VV Viscosity-based model 

VV-grade Viscosity-based model using specification grade of the asphalt binder as 

recommended in the Mechanistic Empirical Pavement Design Guide 

WLF Williams-Landel-Ferry Model

WRI Western Research Institute 

Symbols 

|E*| Dynamic modulus 

|G*| Dynamic shear modulus 

MR Resilient modulus

t-T Time-temperature shift factors

ρ 200 Percentage of aggregate passing #200 sieve 

ρ 4 Percentage of aggregate retained #4 sieve

ρ3/8 Percentage of aggregate retained 3

/8-inch sieve
xix

ρ 3/4 Percentage of aggregate retained 3

/4-inch sieve

Va Percentage of air void 

Vbeff Percentage of effective asphalt content (by volume of mix) 

f Loading frequency 

η Binder viscosity 

| *| G b Dynamic shear modulus of asphalt binder 

b δ Binder phase angle associated with |G*|b

sf Dynamic shear frequency

T Temperature 

TR Temperature in Rankine scale

|E*|m Dynamic modulus of hot mix asphalt (pounds per square inch) 

Pc Aggregate contact volume 

φ Phase angle of hot mix asphalt 

Dynamic shear modulus of asphalt binder at the glassy state 

Vmix Air voids by volume of the mix

ω Angular frequency in radians per second 

t Time 

Rf Reduced frequency

me Fitting coefficient

k Fitting coefficient 

δ Binder phase angle (degree)

Gmb Bulk specific gravity 

Gmm Maximum specific gravity

Sum of squared error 

Standard deviation 

Standard error

σ a Applied stress 

r ε Recoverable strain

υ Poisson’s ratio 
xx 

P Applied load 

U Recoverable horizontal displacement 

V Recoverable vertical displacement

1 k Constant value in NCHRP 1-28 elastic solution for Poisson’s ratio 

2 k Constant value in NCHRP 1-28 elastic solution for Poisson’s ratio 

3 k Constant value in NCHRP 1-28 elastic solution for Poisson’s ratio 

4 k Constant value in NCHRP 1-28 elastic solution for Poisson’s ratio 

E Young’s modulus 

D Compliance 

R Radius of specimen 

x Horizontal distance from center of specimen 

E ' Storage modulus 

E '' Loss modulus 

E∞ Equilibrium modulus 

ωr Angular reduced frequency 

Ei Modulus of the ith Maxwell element

ρi Relaxation time of the ith Maxwell element 

Akj Matrix element in the kth row and jth column of matrix A 

Bk Vector element in the kth row of vector B 

j τ Retardation time of the jth Voigt element determined a priori 

kt Time of interest 

α1 Shift factor coefficient 1 

α2 Shift factor coefficient 2

α3 Shift factor coefficient 3

(log|E*|ANN)i Logarithm of the modulus determined from the ANN models at a 

 particular temperature frequency combination 

(log|E*|fit)i Logarithm of the modulus determined from the optimized sigmoidal fit

(log|E*|avg)i Logarithm of the average modulus determined from the ANN models for a 

 given layer 

A Intercept of temperature susceptibility relationship 
xxi 

Tcritical Temperature in Rankine at which the viscosity is exactly equal to 

2.7 x 1012 cP (0.0027 x 1012 Pas)

PEN Penetration number at given test temperature 

ν Kinematic viscosity (cSt)

ρ Density (oz/in3 (g/cm3))

T a Time-temperature shift factor 

ωT Frequency at the physical temperature 

ωTR Reduced frequency of ωT at the reference temperature 

ωc Fitting coefficient

C1 Fitting coefficient

C2 Fitting coefficient

C3 Fitting coefficient

S t( ) Beam stiffness 

D0 Regression coefficient 

D1 Regression coefficient

| *| D Dynamic axial creep compliance

∂|G*|/∂T Derivative |G*| with respect to time

E(t) Uniaxial relaxation modulus

E ́(ω) Storage modulus 

D t( ) Creep compliance 

κ T Temperature factors

κ ω Frequency factors 

β1 Second order polynomial coefficient 

β2 Second order polynomial coefficient 

β3 Second order polynomial coefficient 

f T( ) Transfer function

1 Hk Value of the nodes at the first hidden layer 

1 ˆHk Transferred value of the nodes at the first hidden layer 
xxii 

2 H j Value of the nodes at the second hidden layer 

2 ˆH j Transferred value of the nodes at the second hidden layer 

3 Hl Value of output node l (MR ANN)

3 H Value of the output node (GV ANN and VV ANN) 

3 ˆHl Transferred value of the output node l, normalized output (MR ANN)

3 Hˆ Transferred value of the output node, normalized output (GV ANN and 

VV ANN)

Pi Input variables 

1 Wik Weight factors for the first hidden layer 

2 Wkj Weight factors for the second hidden layer 

1 Bk Bias factors for the first layer

2 Bj Bias factors for the second layer

0 Bl Bias factors for the outer layer (MR ANN) 

0 B Bias factor for the outer layer (GV ANN and VV ANN) 

( )l max Sig Maximum value for sigmoidal coefficient l in the trained data (MR ANN)

( )l min Sig Minimum value for sigmoidal coefficient l in the trained data (MR ANN)

max log | *| E Maximum log |E*| of the trained data (GV ANN and VV ANN)

min log | *| E Minimum log |E*| of the trained data (GV ANN and VV ANN)

ˆ

Pi Normalized value for given input variable, i

MINi Minimum value of variable, i, in calibration dataset

MAXi Maximum value of variable, i, in calibration dataset

1 Sig First sigmoidal function coefficient

2 Sig Second sigmoidal function coefficient

3 Sig Third sigmoidal function coefficient 

4 Sig Fourth sigmoidal function coefficient 

5 MR Resilient modulus at 41° F (5 °C), MPa 
xxiii

25 MR Resilient modulus at 77° F (25 °C), MPa 

40 MR Resilient modulus at 104° F (40 °C), MPa 

R Coefficient in WLF function 

Ea Coefficient in WLF function 

Rt Inverse of reduced frequency of loading (Hz)

δ Sigmoidal function fitting coefficient

β Sigmoidal function fitting coefficient

D' First vector component of |D*| 

D" Second vector component of |D*| 

Vbe The effective volume of the binder 

1 

EXECUTIVE SUMMARY

The dynamic modulus, |E*|, is a fundamental property that defines the stiffness characteristics of 

hot mix asphalt (HMA) mixtures as a function of loading rate and temperature. Given the 

significance of |E*| in pavement engineering, this project was undertaken to provide the LongTerm Pavement Performance (LTPP) database with |E*| estimates using material properties 

currently available for LTPP test sections. In this report, existing models used to estimate |E*| 

values and additional models that have been developed based on the use of artificial neural 

networks (ANNs) were evaluated. Using the results of the model evaluation, the research team 

developed a model selection hierarchy and populated the LTPP database with |E*| estimates at 

five temperatures and six frequencies. It also developed shift factors and sigmodial functions that 

can be used to construct mastercurves.

The seven models identified at the outset of this project as potentially suitable for the task at

hand include the following: 

1. Original Witczak equation (National Cooperative Highway Research Program 

(NCHRP) 1-37A).(2)

2. Modified Witczack dynamic shear (|G*|) equation (NCHRP 1-40D).(5)

3. Hirsch model.(6)

4. Law of mixtures parallel model.(7)

5. Resilient modulus (MR)-based ANN model. 

6. Viscosity-based ANN model. 

7. Binding shear modulus (|G*|)-based ANN model. 

The existing predictive models (1–4 above) are collectively referred to in this report as 

“closed-form models.” Specific comparisons are drawn regarding their forms and required 

input parameters. 

An extensive independent database was required to develop the ANN models and to fairly assess 

the predictive capabilities of each model in the list of possible models. At the outset of the 

project, the most comprehensive material database available was compiled through the efforts of 

Dr. Matthew Witczak at Arizona State University. Witczak’s database consists of 7,400 data 

points from 346 mixtures, all of which were used in the calibration of the NCHRP 1-40D 

predictive models.(5) A smaller subset of the data (2,750 data points from 205 mixtures) was also 

used in developing the NCHRP 1-37A predictive model.(2) In addition, the database contains |G*| 

data obtained from different materials and aging conditions. Through this research project, the 

Witczak database was combined with mixtures from other national projects and efforts 

undertaken at North Carolina State University (NCSU). The expanded mixture database

currently includes 22,505 data points. 
2 

In addition to a mixture database, binder properties were compiled into a similarly expansive 

database. Substantial efforts have been expended to develop the appropriate binder data 

processing techniques. The required processing varied depending on the type of data available 

(i.e., |G*|, viscosity, or binder grade). Only the critical points are presented in this report, and the 

details are provided in the appendices. 

Closed-form models were compared using datasets that were not used in the calibration of the 

respective models. It was found that the law of mixtures parallel model shows a significant bias, 

but the Hirsch model shows reasonable predictions, except for insensitivity under extreme 

conditions.(7,6) For the verification database, the Hirsch model shows slightly better statistical 

predictions than either of the Witczak models.(6) This finding, along with other statistical 

analyses, led the research team to adopt the Hirsch model input parameters into the viscositybased (VV) and |G*|-based ANN models. 

Comparisons between the ANN models and the closed-form models were made. Overall, the 

ANN models provide better predictability than any of the closed-form solutions. Additionally, 

the ANN models are more sensitive to the input parameters. Based on these findings, the ANN 

models were chosen to populate the LTPP database moduli values. The primary advantage of 

using ANN modeling over statistical regression techniques is that the functional form of the 

relationship is not needed a priori. Considering that many variables affect |E*| values and their 

interactions, the ANN technique may capture complicated nonlinear relationships between |E*| 

and other mixture variables better than regression analysis. 

Early in the project, concerns arose because the database combined moduli that had been 

measured using two different methods, the American Association of State Highway and 

Transportation Officials (AASHTO) test protocol (TP)-62 and the asphalt mixture performance 

tester (AMPT) protocol.(8,9,4) A study of the available databases revealed that the mixtures that 

are tested according to the AASHTO TP-62 protocol tend to yield higher moduli values than 

similar mixtures tested using the AMPT protocol. Statistical analysis to assess the significance of 

the difference was not performed, but the two data ranges tend to overlap, suggesting a lack of 

statistical significance in their differences. A limited experimental study wherein the modulus of 

a single mixture was measured using the two protocols is also discussed. The study shows a 

statistically significant difference of about 12 percent in the measured moduli across all studied 

temperatures and frequencies. However, in light of the fact that both protocols are readily 

available and that neither of the available protocols can be discounted without a more 

comprehensive and controlled experimental program, the decision was made to include all 

available data from both the AMPT protocol and AASHTO TP-62 in the calibration process. 

Details of the three ANN models, including the required input parameters, model structure, and 

input range, are presented in this report. The models are prioritized based on engineering 

judgment and statistical analysis. From this prioritization, a decision tree was developed for 

populating |E*| of the LTPP layers (see figure 1). A user may follow this decision structure and 

determine the best model to use for the available input parameters. 
3 

Resilient Modulus Resilient Modulus

Viscosity and Volumetric 

Properties

Viscosity and Volumetric 

Properties

No Yes

Y

N Y

Prediction Models Prediction Models

Available Information and Temperature 

and Frequency of Interest

Available Information and Temperature 

and Frequency of Interest

|G*| and Volumetric Properties 

at Consistent Age

|G*| and Volumetric Properties 

at Consistent Age

Data Processing and 

Surrogate Models

Data Processing and 

Surrogate Models

t-T Shift 

Factor

VV MR

A-VTS

GV-PAR No Prediction 

Possible

t-T Shift 

Factor

PAR 

Model

Isochronal 

Analysis

Binder Grade and Volumetric 

Properties

Binder Grade and Volumetric 

Properties N Y

M-E PDG 

Values

VVGrade

|G*| and Volumetric Properties 

at Inconsistent Age

|G*| and Volumetric Properties 

at Inconsistent Age

N Y

GV

t-T Shift 

Factor

Isochronal 

Analysis

N

Resilient Modulus Resilient Modulus

Viscosity and Volumetric 

Properties

Viscosity and Volumetric 

Properties

No Yes

Y

N Y

Prediction Models Prediction Models

Available Information and Temperature 

and Frequency of Interest

Available Information and Temperature 

and Frequency of Interest

|G*| and Volumetric Properties 

at Consistent Age

|G*| and Volumetric Properties 

at Consistent Age

Data Processing and 

Surrogate Models

Data Processing and 

Surrogate Models

t-T Shift 

Factor

VV MR

A-VTS

GV-PAR No Prediction 

Possible

t-T Shift 

Factor

PAR 

Model

Isochronal 

Analysis

t-T Shift 

Factor

PAR 

Model

Isochronal 

Analysis

Binder Grade and Volumetric 

Properties

Binder Grade and Volumetric 

Properties N Y

M-E PDG 

Values

VVGrade

|G*| and Volumetric Properties 

at Inconsistent Age

|G*| and Volumetric Properties 

at Inconsistent Age

N Y

GV

t-T Shift 

Factor

Isochronal 

Analysis

N

Figure 1. Illustration. Modulus prediction model decision tree.

At the end of phase I of this study, it was discovered that some State agencies report effective 

binder content by mass instead of by volume (i.e., gravimetric instead of volumetric). As a result, 

there were concerns about the use of a predictive model based on the volumetric properties of the 

asphalt mixtures. After reviewing the database and carrying out some volumetric computations, 

volumetric-based properties could still be calculated when gravimetric quantities were reported. 

Details of how these volumetric-based quantities were computed are provided in this report. 

A key component to the prediction of moduli values is ensuring that the predicted values are 

rational and acceptable. To meet this criterion for the finalized ANN predictions, a set of quality 

control (QC) checks on both the input parameters and model predictions were performed. In 

total, seven QC checks were developed, one for the inputs and six for the modulus predictions, 

and are described in detail in section 6.2 of this report. Executable software, Artificial Neural 

Networks for Asphalt Concrete Dynamic Modulus Prediction (ANNACAP), was developed as 

part of this project for this purpose. The software can be run for individual layers (manual mode) 

or all layers simultaneously (batch mode). An unpublished manual for the software is provided in 

appendix E of this report. 

Statistics for the population effort are also presented in this report. The LTPP database contains 

information for 1,806 layers that meet the criteria established for this project. These layers have 

binder data available at a combination of different aging conditions, including unaged or 

original-aged, rolling thin film oven (RTFO)-aged, pressure-aging vessel (PAV)-aged, and fieldaged. For the field-aged data, 2,223 records are available because some layer properties have 
4 

been measured at different dates. The total resulting number of records is 7,641. Using 

the combined ANN models and requisite internal QC checks, modulus values are predicted 

for 363 records/layers in the original-aged level, 469 records/layers in the RTFO-aged level, 

1 record/layer in the PAV-aged level, and 503 records in the field-aged level. Combined, these 

numbers translate to predictions for 17.5 percent of the total number of records available. 

However, these records are distributed in such a way that a higher percentage of the layers has 

some sort of valid prediction. Of the 1,806 layers in the database, 1,010 layers, or 56 percent, 

have a modulus prediction for some aging condition. Of these 1,010 layers, 615 layers, or 

34 percent of the total 1,806 layers, have reasonable predictions (i.e., an “A” grade), and 

89 layers, or 4.9 percent of the total 1,806 layers, have unreasonable predictions (i.e., an 

“F” grade). The remaining 306 layers, representing 17 percent of the 1,806 layers, have 

questionable predictions (i.e., a “C” grade). Thus, the total percentage of layers with a 

completely valid or questionable prediction is 51 percent. The quality grading system referenced 

is different from the standard record status definition used in the LTPP database.(10) The research 

team established strict QC checks to ensure that only the highest quality data were assigned an 

“A” grade. The data that did not achieve an “A” grade were not considered unusable data. All 

predictions are included in the database so that users can determine the data that are suitable for 

their needs. In addition, the Federal Highway Administration (FHWA) can revise the criteria 

used for the quality checks as deemed appropriate based on the opinions of its experts.
5 

1.0 INTRODUCTION

|E*| is a fundamental property that defines the stiffness characteristics of HMA mixtures as a 

function of loading rate and temperature. The significance of this material property is threefold. 

First, it is one of the primary material property inputs in the Mechanistic Empirical Pavement 

Design Guide (MEPDG) and software developed by NCHRP Project 1-37A.(2) MEPDG uses 

a mastercurve and time-temperature (t-T) shift factors in its internal computations. The 

mastercurve is constructed using a hierarchical structure of inputs ranging from laboratory tests 

on HMA mixtures and binders to estimates based on properties of the HMA mixtures. Second, 

|E*| is one of the primary HMA properties measured in the Superior PERforming Asphalt 

PAVEment (SuperpaveTM) simple performance TP that complements the volumetric mix design. 

Third, |E*| is one of the fundamental linear viscoelastic (LVE) material properties that can be 

used in advanced HMA and pavement models that are based on viscoelasticity. 

Despite the demonstrated significance of |E*|, it is not included in the current LTPP materials 

tables because the database structure was established long before |E*| was identified as the 

main HMA property in the MEPDG. It is not practical to perform MEPDG level 1 laboratory 

|E*| tests on material samples from LTPP test sections at this time due to a lack of materials, 

budget limitations, and the absence of a suitable test method that is applicable to field samples 

obtained from relatively thin pavement structures. However, the LTPP database does contain 

other data that can be used to estimate the |E*| mastercurve and associated shift factors, estimate 

|E*| at specific load durations and temperatures, or develop inputs to the models contained 

in the MEPDG. 

The primary objective of this project, as stated in the task order proposal request, was to 

“...develop estimates of the dynamic modulus of HMA layers on LTPP test sections following 

the models used in the MEPDG....” The team evaluated existing models used to estimate |E*| 

values and additional models that are developed based on the use of ANNs. 

7 

2.0 PREDICTIVE MODELS

Several alternative predictive relationships have been developed to estimate |E*| from simpler 

material properties and volumetrics. These predictive relationships can be used to populate the 

LTPP database with estimated |E*| values. Table 1 lists the predictive relationships identified by 

the research team. These relationships are described briefly in the following subsections. 

Table 1. Predictive relationships for |E*|. 

Model 

Number Model

1 Original Witczak equation (NCHRP 1-37A)(2,11)

2 Modified Witczak |G*| equation (NCHRP 1-40D)(5)

3 Hirsch model(6)

4 Law of mixtures parallel model(7)

5 ANN model

6 MR-|E*| model

2.1 ORIGINAL WITCZAK EQUATION (NCHRP 1-37A)

Andrei et al. revised the original Witczak |E*| predictive equation based on data from 

205 mixtures with 2,750 data points.(11) The revised equation is as follows: 

2

10 4

2

/ 3/8 /

log | *| 1.249937 0.02923 0.001767( ) 0.002841 0.05809

3.871977 0.0021 0.003958 0.000017( ) 0.00547 0.082208

1 exp( 0.603313 0.313351log 0.393532log

a

beff

beff a

E p p pV

V pp p p

V V f η

200 200

4 3 8 3 4

=− + − − −

−+ − +

− +

+ +− − − ) (1)

Where:

p200 = Percentage of aggregate passing #200 sieve. 

p4 = Percentage of aggregate retained in #4 sieve. 

p3/8 = Percentage of aggregate retained in 3/8-inch (9.56-mm) sieve. 

p3/4 = Percentage of aggregate retained in 3

/4-inch (19.01-mm) sieve. 

Va = Percentage of air voids (by volume of mix). 

Vbeff = Percentage of effective asphalt content (by volume of mix). 

f = Loading frequency (hertz). 

η = Binder viscosity at temperature of interest (106 P (105 Pas).

Witczak’s equation is based on a nonlinear regression analysis using the generalized reduced 

gradient optimization approach in Microsoft Excel’s Solver. This model incorporates mixture 

volumetrics and aggregate gradation and is currently one of two options for level 3 analysis 

using the NCHRP 1-37A MEPDG program.(2) For the viscosity term in equation 1, the program 

converts all level 2 and 3 inputs into regression intercept-regression slope of viscosity 

temperature susceptibility (A-VTS) values for the formulation of the |E*| mastercurve. 

Furthermore, Witczak’s model has an equation (not listed) to convert A-VTS coefficients from 

virgin or tank binders to RTFO- and PAV-aged binder values. 
8 

The limitations of Witczak’s equation, acknowledged by Bari, include relying on other models to 

translate the currently used |G*| measurement into binder viscosity.(5) Because the original 

Witzcak predictive equation is based on regression analysis, extrapolation beyond the calibration 

database should be restricted. Bari also mentions that there is limited volumetric influence 

(precision) when the model is compared to the Shell Oil model.(5) Other researchers have also 

noted the need for improved sensitivity to volumetrics, such as the percentage of voids in mineral 

aggregate (VMA), the percentage of voids filled with asphalt (VFA), asphalt concrete percentage 

(AC), and Va.

(12)

2.2 MODIFIED WITCZAK EQUATION BASED ON |G*| (NCHRP 1-40D)(5)

To include binder |G*| in the predictive model, Witczak reformulated the model to include the 

binder variable directly. The updated model is as follows: 

(2)

Where: 

|G*|b = Dynamic shear modulus of asphalt binder (pounds per square inch). 

δb = Binder phase angle associated with |G*|b (degrees).

As with the NCHRP 1-37A model, equation 2 is based on a nonlinear regression analysis using 

346 mixtures with 7,400 data points. The measured results of the unmodified binders have a 

better correlation with the model (R2 = 0.87) compared to those of the modified binders (R2= 

0.79) in arithmetic scale. In logarithmic scale, both binder types have R2

 = 0.99. The binder 

phase angle is predicted using an empirical equation (R2 = 0.83). This equation is one of two 

options for level 3 analysis in the most current MEPDG program. 

Because some of the mixtures in this database do not contain |G*|b data, the Cox-Mertz rule, 

using correction factors for the non-Newtonian behaviors (see equations 3–5), is used to 

calculate |G*|b from A-VTS values as follows: 

2 7.1542 0.4929 0.0211

, | * | 0.0051 (sin ) s s

s

f f G f b s fT b η δ − + = (3)

,

2

,

90 ( 7.3146 2.6162* ')*log( * )

(0.1124 0.2029* ')*log( * )

s

s

b s fT

s fT

VTS f

VTS f

δ η

η

= +− −

+ + (4)

0.0527 0.0575

, log log 0.9699 * 0.9668 * log sf T s s R η f A f VTS T − − = + (5)
9 

Where:

fs = Dynamic shear frequency.

δb = Binder phase angle predicted from equation 4 (degrees). 

ηfs,T = Viscosity of asphalt binder at a particular loading frequency (fs) and temperature (T) 

determined from equation 5 (centipoise). 

TR = Temperature in Rankine scale.

2.3 HIRSCH MODEL

Christensen et al. examined four different models based on the law of mixtures parallel model 

and chose the model that incorporates the binder modulus, VMA, and VFA because it provides 

accurate results in the simplest form.

(6,7) The other more complicated forms attempt to 

incorporate the modulus of the mastic or the film thickness, which are difficult parameters to 

measure. The suggested model for |E*| estimation is provided in equations 6–8 as follows: 

( )

( )

( )

* 1

| * | 4, 200,000 1 3 | * | 100 10,000 1 100

4, 200,000 3 | * |

c

m c b

b

VMA VFA VMA P

E P G

VMA VMA

G VFA

      − = −+     +         −

+

(6)

2 φ =− ( − 21 log ) 55log Pc Pc (7)

( ( ) ( ))

( ( ) ( ))

0.58

0.58

20 3 | * |

650 3 | * |

b

c

b

G VFA VMA

P

G VFA VMA

+ =

+ (8)

Where:

|E*|m = Dynamic modulus of HMA (pounds per square inch). 

Pc = Aggregate contact volume. 

φ = Phase angle of HMA.

A strength of this model is the empirical phase angle equation, which is important for the 

interconversion of |E*| to the relaxation modulus or creep compliance. Weaknesses of the model 

include a lack of a strong dependence on volumetric parameters, particularly at low Va and VFA 

conditions. Also, questions arise regarding the ability of the |G*|b parameter to account for the 

possible beneficial effects of modifiers.(7) It must be noted that only 206 data points were used to 

determine the coefficients in the Hirsch model compared to 2,750 data points for the original 

Witczak model and 7,400 data points for the modified Witczak model. 
10 

2.4 LAW OF MIXTURES PARALLEL MODEL (AL-KHATEEB MODEL)

Based on their findings from the Hirsch model, Al-Khateeb et al. suggest the following model:(7)

0.66

0.66

| *| 90 10,000 100 | *| 3 | *| 100 | *| 1,100 900

b

m g

b

G

VMA VMA

E G

G

VMA

         +     −     =           +          (9)

Where: 

|G*|g = Dynamic shear modulus of asphalt binder at the glassy state (assumed to be 

145,000 psi (999,050 kPa).

Like the Hirsch model, this formulation is based on the law of mixtures for composite materials. 

In this model, the different material phases (aggregate, asphalt binder, and air) are considered to 

exist in parallel. Therefore, this model is a simpler interpretation of the Hirsch model. The 

researchers note that their model addresses one of the primary shortcomings of the Hirsch model 

(i.e., the Hirsch model’s inability to accurately predict |E*| of the mixture at low frequencies and 

high temperatures). 

Strengths of this model include the improved prediction of high-temperature and low-frequency 

|E*| data for mixtures used in the FHWA accelerated loading facility (ALF) test strips. 

Weaknesses include a lack of model verification and the fact that the researchers who developed 

this model did so based on |E*| values obtained from tests at higher than recommended strain 

amplitudes (200 με versus the recommended maximum of 75–150 με). 

2.5 ANN MODELS

The NCSU research team employed the ANN technique to develop new |E*| predictive models. 

The primary advantage of this approach over statistical regression is that the functional form of 

the relationship is not needed a priori. Considering that so many variables affect |E*| values and 

their interactions, the ANN technique captures complicated nonlinear relationships between |E*| 

and other mixture variables better than regression analysis. 

The ANN technique was used in this research to develop several different models. The first 

model is the ANN model that predicts |E*| values using the input variables employed in the 

modified Witczak equation (i.e., binder dynamic modulus and phase angle, aggregate gradation, 

and volumetrics of the HMA mixture). The effort to develop this ANN model is described later 

in this report. The ANN technique was also applied to backcalculate |E*| values from MR. During 

the FHWA DTFH61-05-RA-00108 project, the NCSU research team developed a mechanistic 

approach to compute MR from |E*| of HMA.

(13,14)

 This approach was verified successfully using 

measured data from mixtures with varying gradations and binder characteristics. The verified 

solutions were then applied to an available |E*| database to estimate the MR values corresponding 

to the |E*| values. This database was used to develop an inverse algorithm based on the ANN 

technique that can predict |E*| from MR. The development and verification of MR-|E*| ANN are 
11 

presented later in this report. Finally, the ANN technique was used to develop an |E*| predictive 

model based on binder viscosity information. This ANN model is also described in this report. 

2.6 SUMMARY OF INPUT VARIABLES

Table 2 presents the necessary input variables for each predictive relationship discussed 

previously. For models that utilize |G*|b, predictions are only possible at the temperatures and 

frequencies where |G*|b values are available. Having |G*|b only at the conditions used in 

SuperpaveTM testing is not sufficient for generating |E*| values over the range of conditions 

typically needed for mechanistic analysis (14–129.2 °F (-10–54 °C)). For example, if the user 

has |G*|b at only 147.2 °F (64 °C) and 10 radians per second (rad/s), then it is possible to predict 

the |E*| value only at 147.2 °F (64 °C) and 10 rad/s. To predict |E*| at 77 °F (25 °C) and 25 rad/s, 

the user must measure |G*|b at 77 °F (25 °C) and 25 rad/s. 

Table 2. Model variables. 

Variable Description

Model Number

LTPP Data 

Availability

1 2 3 4 5 6

Specific 

Pavement 

Study 

(SPS)

1

General 

Pavement 

Study 

(GPS)

1

Mixture MR X Yes Yes

|G*|b X X X X Yes2 No

δb X X Yes No

VMA (percent) X X Yes Yes4

VFA (percent) X Yes Yes4

Aggregate passing #200 sieve (percent) X X X Yes Yes

Aggregate passing #4 sieve (percent) X X X Yes Yes

Aggregate passing 3/8-inch sieve (percent) X X X Yes Yes

Aggregate passing 3/4-inch sieve (percent) X X X Yes Yes

Va (by volume of the mix (Vmix)) (percent) X X X Yes Yes

Effective asphalt content (by total volume of 

the mixture) (percent) X X X Yes Yes4

Loading frequency (Hz) X Yes No

A-VTS X Yes3 Yes5

1 inch = 25.4 mm

1

The in-service pavement sections are classified in the LTPP program as GPS and SPS. 2

|G*|b was tested for SPS-9 sections only.

3

Data for penetration at 77 and 115 °F (25 and 46 °C), cone and plate viscometer at 77 °F (25 °C), absolute viscosity at

140 °F (46 °C), and kinematic viscosity at 275 °F (135 °C ) are available. 4

Indicates information was reported by an agency. 5

Data for ring and ball softening point, penetration at 39.2 and 77 °F (4 and 25 °C), absolute viscosity at 140 °F 

(46 °C), and kinematic viscosity at 275 °F (135 °C) are available (reported by agency).

Note: Blank cells indicate that the input parameter is not required in the model.
12 

A preliminary review of available information in the LTPP materials database revealed that 

measured |G*|b data are only available for most of the SPS-9 projects and only at 10 rad/s at 

multiple temperatures. Even the available |G*|b data are measured from binders aged at different 

levels (i.e., RTFO-aged versus PAV-aged binders). The lack of complete |G*|b data at multiple 

temperatures and multiple frequencies is a serious problem because all of the models except the

original Witczak equation require this information. (Note that the MR-|E*| ANN model needs the 

binder t-T shift factor as well.) During the course of this project, the NCSU research team 

developed empirical models that allow the estimation of the RTFO-aged |G*|b values at multiple 

temperatures and frequencies from the |G*|b values obtained at a single frequency and multiple 

temperatures and aging levels.
13 

3.0 DATABASES

To accomplish the goals of the project, modulus values from multiple mixtures and binders were

necessary. These databases were assembled from existing national efforts and from data obtained 

at NCSU. In the following sections, the databases are summarized by material type (i.e., binder 

or mixture). First, the specific reasons for such databases are given.

3.1 STATEMENT OF NEED

The most comprehensive database for both binder moduli and viscosity and mixture moduli that 

is currently available is the one used to develop the Witczak and modified Witczak models. This 

database is extensive and covers a range of material characteristics. At the outset of this project, 

such a database was sufficient to meet the project goals. However, an indepth evaluation of the 

database identified the following three problems in using it for model development: 

1. Different definitions of frequency for the binder |G*| and mixture |E*| data are used. 

2. Estimated rather than measured |G*| and b δ data are used to populate the database. 

3. The estimated |G*| data at temperatures lower than or equal to 39.9 °F (4.4 °C) are estimated 

in a manner that is inconsistent with other temperatures, and the method of estimation is not 

satisfactorily explained. 

3.1.1 Inconsistent Definition of Frequency

In developing the Witczak database, researchers used two definitions for frequency. The decision 

to use two definitions was motivated by widespread confusion between frequency and time for 

LVE materials and the differences in the binder and mixture fields. Researchers studying asphalt 

binders have been influenced by rheology and often cite the relationship between time and 

frequency as follows: 

1 1

2 s

t

ω π f = = (10)

Where:

ω = Angular frequency in radians per second. 

t = Time. 

fs = Shear frequency. 

Conversely, researchers who focus on AC tend to use the argument that time and frequency are 

related as follows: 

1 1

c

t

f f = = (11)
14 

Where:

f = Frequency (hertz).

fc = Frequency under axial compression. 

This latter interpretation is more accurate if the time under consideration is the pulse time (i.e., it 

equates the modulus at a given frequency to the appropriate modulus when the material is 

subjected to a load pulse of a given duration). The former definition is more accurate when it is 

necessary to equate the modulus determined at a given frequency to the material modulus under 

a fixed load after a given duration. 

The use of different definitions for frequency appears to be an effort to appease both mixture and 

binder branches and their techniques for data interpretation. To coordinate these two datasets, the 

Witczak data match time from equations 10 and 11. Such an approach allows the prediction of 

the modulus of AC at 10 Hz and 77 °F (25 °C). As a result, the appropriate binder modulus that 

can substitute into equation 2 is the one at 10 rad/s (1.59 Hz) and 77 °F (25 °C). This is referred 

to as the inconsistent definition of frequency in this report because the frequency at which |E*| 

and |G*| are measured is not consistent. A small sample of the database that clearly shows this 

inconsistent definition is shown in table 3. 

Table 3. Example of summarized mixture and binder properties in the original Witczak 

database. 

Mix 

Number

Temperature

(°C)

Mixture |E*| Test Data Binder Stiffness Data

fc (Hz) |E*| (psi) fs (Hz) |G*| (psi) δb (degrees) 

1 4.4 1 1.35E+06 0.16 9.81E+02 54.7

°C = (°F-32)/1.8

1 psi = 6.89 kPa

A more logical approach, which was used in all of the other |G*|-based models and is presented 

in table 2, is to use a consistent definition for frequency. To predict the mixture modulus at 

10 Hz and 77 °F (25 °C), the logical binder modulus that should be used is the one at 10 Hz and 

77 °F (25 °C). Such an effort required the complete repopulation of the Witczak database, which 

was part of the effort in this project. 

Because part of this research effort was to assess the sensitivity of existing models, independent 

databases were needed. These databases are summarized in the following sections; however, it 

should be noted that for fairness, care was taken to ensure that the inconsistent definition was 

used to make predictions using the modified Witczak model. The consistent definition was used 

for development of the ANN-based model.

3.1.2 Use of Estimated |G*| Values

The |G*| data in the Witczak database are not measured values and were estimated using the 

predictive equations given in equations 3–5. Such an approach was deemed necessary for 

consistency purposes. Even though the |G*| binders were measured for several of the mixtures, 

no such data were available for approximately half of the mixtures in the database. By using |G*|

from the predictive equations, the entire database could be used to develop the modified Witczak 

model. Therefore, the effects of the errors in the binder modulus model were integrated into the 
15 

predictive capabilities of the model. To evaluate the predictive power of these models and to 

assess the need to return to the measured data, the measured and predicted |G*| and b δ values for 

the 8,940 data points from 41 binders (including 9 modified binders) used in developing 

equations 3–5 were compared. Figure 3 illustrates that the model shows little bias, but the errors 

in the predictive models are significant. In some cases, the errors exceeded 100 percent, which 

could translate directly to errors in the predicted mixture moduli. The predicted b δ is shown in 

figure 4, and the errors were even more significant than they were for the predicted |G*|. Again, 

such errors can have a significant effect on the predictive model’s capabilities. An independent 

dataset was also used to assess the errors, and the results are shown in figure 5 (arithmetic scale) 

and figure 6 (logarithmic scale). This analysis shows that any given binder may have significant 

bias, the effects of which will directly translate to |E*| predictions. 

To correct this problem, the binder data in the Witczak database were completely ignored, and 

only mixtures that had independently measured |G*| values available were used. This approach 

reduced the number of usable mixtures to approximately half the number that was initially

thought to be available. 

0.0E+00

4.0E+07

8.0E+07

1.2E+08

1.6E+08

0.0E+00 4.0E+07 8.0E+07 1.2E+08 1.6E+08

Measured |G*|b (Pa)

Predicted |G*|b (Pa)

1 psi = 6.86 kPa

Figure 2. Graph. Comparison between the Witczak predictive model and measured |G*|b

values in arithmetic scale. 
16 

1.E-01

1.E+01

1.E+03

1.E+05

1.E+07

1.E+09

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07 1.E+09

Measured |G*|b (Pa)

Predicted |G*|b (Pa)

1 psi = 6.86 kPa

Figure 3. Graph. Comparison between the Witczak predictive model and measured |G*|b

values in logarithmic scale. 

0

20

40

60

80

100

0 20 40 60 80 100

Measured δ b (deg)

Predicted 

δ b (deg)

Figure 4. Graph. Comparison between the Witczak predictive model and measured δb to 

develop the δb model. 
17 

0.0E+00

5.0E+06

1.0E+07

1.5E+07

2.0E+07

0.0E+00 5.0E+06 1.0E+07 1.5E+07 2.0E+07

Measured |G*|b (Pa)

Predicted |G*|b (Pa)

Citgo-Wil-PG64-22

Citgo-Wil-PG 70-22

LOE

1 psi = 6.86 kPa

Figure 5. Graph. Comparison between the Witczak predictive model and measured |G*|b

values using Citgo binders in the North Carolina Department of Transportation (NCDOT) 

database in arithmetic scale. 

1.E+01

1.E+03

1.E+05

1.E+07

1.E+09

1.E+01 1.E+03 1.E+05 1.E+07 1.E+09

Measured |G*|b (Pa)

Predicted |G*|b (Pa)

Citgo-Wil-PG64-22

Citgo-Wil-PG 70-22

LOE

1 psi = 6.86 kPa

Figure 6. Graph. Comparison between the Witczak predictive model and measured |G*|b

values using Citgo binders in the NCDOT database in logarithmic scale. 
18 

3.1.3 Low-Temperature |G*| Values

Through the process of determining that the binder data in the Witczak database consist of 

predicted values, it was also found that the low-temperature binder moduli were not determined 

in the same way as the intermediate and high temperatures. It was not possible to discern how 

these values were determined for the database because they do not agree with the predictions 

made from a surrogate model. Additionally, they are approximately two orders of magnitude 

smaller than typical values for these temperatures. Measured moduli for the available binders 

were provided only for temperatures as low as 59 °F (15 °C). 

To address this problem, the measured binder data that were available had to be processed in a 

way that would allow extrapolation. According to the Christensen Anderson Marasteanu (CAM) 

model, shown in equations 12 and 13, the following is assumed for all binders in the database: 

*

1

e

g

m k k

c

R

G

G

f

f

=

      +         (12) 

90

1( / )

e

k

R c

m

f f

δ = + (13)

Where:

Gg = Maximum shear modulus or glassy modulus (pascal). 

δ = Binder phase angle (degree). 

fR = Reduced frequency (hertz). 

fc, me, and k = Fitting coefficients.

Equation 12 has been characterized for each of the binders using least square optimization 

techniques. The fit was verified, and the low-temperature binder stiffness was determined. Note 

that for consistency, this functional form was also used to populate the binder data at the same 

temperatures as the |E*| measurements. Unfortunately, the Witczak binders do not provide 

available data to assess potential errors using this extrapolation methodology. To fully assess the 

errors, additional binder data, which include bending beam rheometer (BBR) measurements or 

shear modulus measurements taken at extremely low temperatures, and dynamic shear rheometer 

(DSR) measurements at intermediate and high temperatures are required.

3.1.4 Aging Effects

After accounting for these binder data issues, it was found that the lack of binder data at different 

aging levels is a serious limitation of the Witczak database. This limitation is important because 

the LTPP database contains binder modulus values at original-, RTFO-, PAV-aged, and even 

recovered conditions. Accounting for these inconsistencies in the analysis process required an 

understanding of their effects on the binder shear modulus. Because the data in the Witczak 

database are insufficient for this purpose, additional binder databases were necessary. 
19 

3.2 BINDER DATABASES

There are six binder datasets: (1) Witczak, (2) FHWA mobile trailer, (3) FHWA TPF-5(019), 

(4) NCDOT, (5) Western Research Institute (WRI), and (6) Citgo.(15) The following sections 

describe these binder datasets. 

3.2.1 Witczak Binder Database 

Table 4 summarizes the binders that make up the Witczak binder database. Each of these binders 

corresponds to a particular mixture or mixtures in the Witczak |E*| database. The binders in this 

database have measured |G*| values available at a range of temperatures and frequency 

combinations as well as various aging conditions (original, RTFO, and PAV). Additionally, in

table 4, a subset of these binders has measured BBR results. Note that the Citgo and WRI binders 

are the only ones with BBR data. Each binder has BBR stiffness and slope values at -11.2, -0.4, 

and 10.4 °F (-24, -18, and -12 °C) (WesTrack) or -22, -11.2, and -0.4 °F (-30, -24, and -18 °C) 

(all others) and times of 8, 15, 30, 60, 120, and 240 s, respectively. All binders in this database 

have viscosity data in the form of temperature susceptibility function parameters A and VTS. 

Table 4. Summary of |G*| data available in the Witczak binder database. 

Binder Code

Temperature

(°C)

Frequency

(rad/s)

Performance 

Grade (PG) Original RTFO PAV

BBR 

Available

ALF AC-5

15, 25, 35, 45, 

60, 70, 80, 95, 

105, and 115 

1, 1.59, 

2.51, 3.98, 

6.31, 10, 

15.9, 25.1, 

39.8, 63.1, 

and 100 

58 

   

ALF AC-10    

Paramount 

PG 58-22   

ALF AC-20

64 

   

WesTrack    

Chevron PG 64-22   

Paramount 

PG 64-16   

Navajo PG 70-10 70   

ALF-Novophalt

76 

   

Chevron PG 76-16   

Navajo PG 76-16   

ALF-Styrelf 82    

MNRD120P 15, 25, 35, 45, 

60, 70, 80, 95, 

and 105 

58    

MNRDAC20 64    

MPA Citgo 70-22

25, 35, 45, 60, 

70, 80, 95, 

105, and 115 

70   

MPA-Elvaloy 76 

  

MPA-TA   

MPA-Novophalt 82 

  

MPA-Stylink   

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.
20 

3.2.2 FHWA Mobile Trailer Binder Database

The FHWA mobile trailer database contains original-, RTFO-, and PAV-aged |G*| values for a 

range of binders throughout the United States (see table 5). Note that the temperature and 

frequency combinations are not as consistent or as broad as those in the Witczak database. All of 

the asphalt binders in this database have viscosity data available.

Table 5. Summary of |G*| data available in the FHWA mobile trailer database. 

Binder Code

Temperature

(°C)

Frequency

(rad/s) PG Original RTFO PAV

BBR 

Available

IA_0358

5, 17, 23, 25, 40, 

and 45

0.1, 0.63, 1, 

3.14, 6.3, 10, 

31.45, 62.9, 

100, 157.199 

58    

WI_0357 

5, 15.6, 19.6, 22, 

23.6, 25, 31.2, 45 

64 

   

ME_0359 

5, 17.9, 23.9, 25, 

37.5, and 45    

LA_0462 5, 25, 45, and 54    

WA_0463 5, 15, 25, 38, and 45    

KS_0464 5, 31, and 45    

NY_0466 

4.4, 5, 13, 21.1, 25, 

37.8, 45, and 54.4    

MA_0467 

4.4, 5, 13, 15, 22, 

25, 37.8, 45, and 

54.4    

NC_0360 5, 20.8, 25, and 45

70 

   

MN_0465

5, 13, 25, 45, and 

54.4    

AZ_0356 5, 22, 25, and 45 76    

CO_0777-BA

-10, 4.4, 21.1, 37.8, 

and 54.5 

58 

 

CO_0777-

BA+ADVERA  

CO_0777-

BA+SASOBIT 64  

OK_0673  

SD_0674  

MO_0672 70  

AL_0675 76  

NJ_0671  

ME_0570

4.4, 21.1, 37.8, and 

54.10 

64  

NE_0569  

KS_0568 70  

KS_0568 (2)  

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.
21 

3.2.3 FHWA TPF-5(019) Binder Database 

A portion of the current FHWA ALF binder data is available to the research team and is

summarized in table 6. For these materials, DSR values are available at a wider range of 

frequencies than in the other databases. At the time this report was written, BBR results were not 

available; however, the possibility of obtaining such values along with data from other binders in 

the TPF-5(019) study is under investigation. Also, these binders have |G*| values measured 

under original-, RTFO-, and PAV-aged conditions. Viscosity values are not available for 

these materials.

Table 6. Summary of |G*| data available in the FHWA TPF-5(019) binder database. 

Binder Code

Temperature

(°C)

Frequency

(rad/s) PG Original RTFO PAV

BBR 

Available

Binder (AB)-B6261

7, 19, 25, 31, 

40, 46, 58, 64, 

70, and 82 

0.100, 0.126, 

0.158, 0.200, 

0.251, 0.316, 

0.398, 0.501, 

0.631, 0.794, 

1.000, 1.259, 

1.585, 1.995, 

2.512, 3.162, 

3.981, 5.012, 

6.309, 7.942, 

10.000, 12.589, 

15.849, 19.952, 

2.118, 31.622, 

39.809, 50.115, 

62.091, 79.426, 

and 100.000

52 

SBS LG baseB-6275 58 

PG 70-22 1-B6267

70 



PG 70-22 2-B6272 

PG 70-22 3-B6298 

SBS LG-B6295 

Terpoly-B-6289 76 



CRTB-B-6286 

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.
22 

3.2.4 NCDOT Binder Database

The binders corresponding to NCDOT mixture data are available in the NCDOT binder database, 

which is summarized in table 7. Data for these binders, both |G*| and viscosity, are available 

only for the RTFO-aged condition. 

Table 7. Summary of |G*| data available in the NCDOT binder database.

Binder Code

Temperature 

(°C)

Frequency 

(rad/s) PG Original RTFO PAV

BBR 

Available

AA-Inman

16, 22, 28, 

and 40 

0.06, 0.31, 

0.63, 3.14, 

6.28, 31.42, 

62.83, and 

94.25 

64 



El Paso-Apex 

El PasoCharlotte 

Citgo-Wil-64 

AA-Sali-70 70 



Citgo-Wil-70 

AA-Sali-76 76 

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.

3.2.5 Citgo Binder Database 

Table 8 summarizes the binders that are used in the Citgo database. This database contains 

original-, RTFO-, and PAV-aged |G*| and BBR results for two different binders used in 

NCHRP 9-25 and 9-31.(16,17) No viscosity measures are available for these binders.

Table 8. Summary of |G*| data available in the Citgo binder database. 

Binder 

Code

Temperature 

(°C)

Frequency 

(rad/s) PG Original RTFO PAV

BBR 

Available

Citgo_A 15, 25, 35, 45, 

60, and 75 

0.1, 0.159, 0.251, 

0.398, 0.631, 1, 

1.59, 2.51, 3.98, 

6.31, 10, 15.9, 25.1, 

39.8, 63.1, and 100

70    

Citgo_B    

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.

3.2.6 WRI Binder Database 

The WRI binder database consists of two sets of binders from the Kansas and Nevada test sites. 

The four binders used at the Kansas site are from different crude sources, but all are PG 64-22. 

Additionally, the four binders used at the Nevada site are from different crude sources, each of 

which meets the AC-30 requirements of AASHTO M226-80, “Standard Specification for 

Viscosity Graded Asphalt Cement.”(18) Table 9 summarizes the relevant information for the 

binders in this database. Viscosity values for these binders are not available.
23 

Table 9. Summary of |G*| data available in the WRI database. 

Binder Code

Temperature

(°C)

Frequency

(rad/s) PG Original RTFO PAV

BBR 

Available

KS-1-Sinclair Tulsa 0.0, 10.0, 

20.0, 30.0, 

40.0, 50.0, 

60.0, 70.0, 

and 80.0 

0.10, 0.13, 

0.16, 0.20, 

0.25, 0.32, 

0.40, 0.50, 

0.63, 0.79, 1.0, 

1.26, 1.58, 2.0, 

2.51, 3.16, 3.9, 

5.0, 6.3, 7.9, 

10.0, 12.59, 

15.6, 19.9, 

25.11, 31.6, 

39.8, 50.1, 

63.1, 79.4, 

and 100.0 

64 

  

KS-2-Coastal 

Eldorado   

KS-3-Koch Muskogee   

KS-4-Royal Trading   

NV-1-Sinclair WY

0.0, 10.0, 

20.0, 30.0, 

40.0, 50.0, 

60.0, and 

70.0 

AC-30 

  

NV-2-Crown Nevada   

NV-3-Crown 

Venezuela   

NV-4-Crown 

Canadian   

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.

3.3 MIXTURE DATABASES

The mixture databases for this project are similar in content to the binder databases; however, the 

temperatures and frequencies are different. These mixture databases contain the volumetric 

information listed in table 2. There are seven mixture datasets: (1) Witczak, (2) FHWA mobile 

trailer I, (3) FHWA mobile trailer II, (4) FHWA TPF-5(019), (5) NCDOT, (6) WRI, and 

(7) Citgo. The following sections describe the mixture databases.

3.3.1 Witczak Mixture Database 

Over the years, differences in TPs used to develop the database have led to limitations of the use 

of some mixtures in the Witczak database. The complete database contains 7,400 data points; 

however, the database used for this project has been limited to 3,180 data points for 

106 mixtures. To avoid confusion, this limited database is referred to as the “processed 

Witczak database.” These mixtures are summarized in table 10 along with the spread of 

volumetric properties. Note that for all of the mixtures in this database, the true measured |G*| 

data are available (see table 4). 

3.3.2 FHWA Mobile Trailer Mixture Database

A primary difference between the FHWA mobile trailer database and the processed Witczak 

database is TP. The mobile trailer database is populated with |E*| data obtained from the 

procedure and equipment suggested for AMPTs, whereas the processed Witczak database is 

populated using data obtained from the AASHTO TP-62 protocol.(8) For modeling purposes, 

the mobile trailer mixture database is separated into two groups: FHWA I and FHWA II, which

are summarized in table 11 and table 12, respectively. Note that a comparison of table 10 with 

table 11 and table 12 shows that the processed Witczak database covers an overall broader range 
24 

of conditions than the mobile trailer mixture database. However, the latter database has smaller 

values of some parameters, particularly those related to the mixture gradation (i.e., ρ3/4, ρ3/8, 

and ρ4). This indicates that smaller nominal maximum-sized aggregate (NMSA) mixtures 

are represented more strongly in the mobile trailer database than they are in the processed 

Witczak database. 

3.3.3 FHWA TPF-5(019) Mixture Database 

The FHWA ALF study mixtures are available for the TP-62 test conditions. These mixtures 

consist of the same aggregate type and gradation, air void content, and asphalt content. The only 

differences between these mixtures are the asphalt type and a slight difference in the effective 

asphalt content. Table 13 presents a summary of the FHWA TPF-5(019) mixtures.(15)
25 

Table 10. Summary of |E*| data available in the processed Witczak mixture database. 

Binder Code

Number 

of Mix 

Variations

Total 

Number

of Tests

Temperature

(°C)

Frequency

(Hz)

Aging Condition

Plant or 

Unaged

Short-Term 

Oven Aging 

(STOA)

Long-Term 

Oven Aging 

(LTOA) or 

Field

MPA Citgo 

70-22 4 4 

-17.8, 4.4, 

21.1, 37.8, 

and 54.4 

25, 10, 5, 1, 

0.5, and 0.1 

4 

MPA-Elvaloy 4 4 4

MPA-Novophalt 4 4 4

MPA-Stylink 4 4 4

MPA-TA 4 4 4

ALF AC-5 6 6

-10, 4.4, 21.1, 

37.8, and 54.4 

4 2

ALF AC-10 2 2 1 1

ALF AC-20 6 6 4 2

WesTrack 34 34 22 12

Chevron 64-22 17 17 17

Paramount 

64-16 1 1 1

Navajo70-10 2 2 2

ALF-Novophalt 2 2 1 1

Chevron 76-16 2 2 2

ALF-Styrelf 2 2 1 1

MNRD120P 8 8 6 2

MNRDAC20 4 4 1 3

ρ3/4

Maximum 26.1

Minimum 0

ρ3/8

Maximum 41

Minimum 15

ρ4

Maximum 73

Minimum 30

ρ200

Maximum 6.6

Minimum 2.6

Va

Maximum 12.5

Minimum 0.7

Vbeff

Maximum 13.5

Minimum 6.1

VMA Maximum 22.2

Minimum 11.2

VFA Maximum 95.1

Minimum 32.8

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.
26 

Table 11. Summary of |E*| data available in the FHWA I mobile trailer mixture database. 

Binder Code

Number of 

Mix

Variations

Total 

Number

of Tests Temperature (°C)

Frequency

(Hz)

Aging Condition

Plant or 

Unaged STOA

LTOA 

or 

Field

WI_0357 9 36 15.6,19.6, 23.6, and 31.3

25, 10, 5, 1, 

0.5, and 0.1 

24 12

ME_0359 9 36 17.9, 23.9, and 37.5 24 12

LA_0462 9 28 25, 45, and 54 15 13

NY_0466 12 32 13, 25, and 45 20 12

MA_0467 5 16 15, 25, and 45 16

NC_0360 13 52 21, 25, and 45 40 12

MN_0465 11 39 13, 25, and 45 27 12

AZ_0356 9 26 22 and 44 14 12

CO_0777 1 4

4.4, 21.1, 37.8, and 54.4 

4

CO_0777+ ADV 1 4 4

CO_0777+ SAS 1 4 4

OK_0673 13 34 22 12

SD_06741 6 24 12 12 

MO_06722 6 24 24 

AL_06753 24 96 72 24 

NJ_06711 6 24 12 12 

ME_0570 11 44

21.1, 37.8, and 54.4 

32 12

NE_0569 16 54 46 8

KS_05681 6 30

KS_0568(2)1 4 20

ρ3/4

Maximum 11.6

Minimum 0

ρ3/8

Maximum 45.1

Minimum 2.3

ρ4

Maximum 67.4

Minimum 22.3

ρ200

Maximum 6.6

Minimum 2.7

Va

Maximum 8.7

Minimum 4.5

Vbeff

Maximum 12.7

Minimum 4.8

VMA Maximum 20

Minimum 9.5

VFA Maximum 70.3

Minimum 43

°C = (°F-32)/1.8 1

Vbeff is not given, 2inconsistent ID numbers for correlating volumetric and |E*| data, and 3measured |E*| under confining pressure. 

Note: Blank cells indicate data are unavailable in the database.
27 

Table 12. Summary of |E*| data available in the FHWA II mobile trailer mixture database. 

Binder Code

Number

of Mix 

Variations

Total 

Number

of Tests

Temperature

(°C)

Frequency

(Hz)

Aging Condition

Plant or 

Unaged STOA

LTOA 

or 

Field

IA_0358 7 28 17, 23, and 40 25, 10, 5, 1, 

0.5,and 0.1 

16 12

WA_0463 6 24 15, 25, 38, and 45 12 12

KS_0464 8 32 25, 31, and 45 20 12

ρ3/4

Maximum 2.1

Minimum 0

ρ3/8

Maximum 17

Minimum 14.7

ρ4

Maximum 45.9

Minimum 34.1

ρ200

Maximum 6.5

Minimum 3.8

Va

Maximum 8.5

Minimum 5.7

Vbeff

Maximum 11.5

Minimum 8.2

VMA Maximum 19.9

Minimum 15.1

VFA Maximum 65.5

Minimum 52.1

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.
28 

Table 13. Summary of |E*| data available in the FHWA TPF-5(019) mixture database. 

Binder Code

Number

of Mix 

Variations

Total 

Number

of Tests

Temperature 

(°C)

Frequency

(Hz)

Aging Condition

Plant or 

Unaged STOA

LTOA 

or 

Field

PG 70-22 3-B6298 1 3 -10, 5, 20, 40, 

and 54 

25, 10, 5, 

1, 0.5, and 

0.1 

3 

SBS LG-B-6295 1 3 3

Terpoly-B-6289 1 3 3

CRTB-B-6286 1 3 3

ρ3/4 0.0

ρ3/8 14.8

ρ4 44

ρ200 6.7

Va 4

Vbeff

Maximum 12.4

Minimum 12.3

VMA

Maximum 16.4

Minimum 16.3

VFA

Maximum 75.6

Minimum 75.5

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.
29 

3.3.4 NCDOT Mixture Database

The NCDOT mixture database consists of 36 AC mixtures covering a range of materials. All of 

the mixtures were tested at NCSU. The test conditions for the mixtures in this database are most 

similar to the TP-62 protocol. The database is summarized in table 14.

Table 14. Summary of |E*| data available in the NCDOT mixture database. 

Binder Code

No. of Mix 

Variations

Total 

No. of 

Tests

Temperature

(°C)

Frequency

(Hz)

Aging Condition

Plant 

or 

Unaged STOA

LTOA 

or 

Field

El Paso-Charlotte-64 2 2

-10, 10, 35, 

and 54 

25, 10, 5, 1, 

0.5, 0.1, 

0.05, and 

0.01 

2

Citgo-Wil-64 24 24 24

AA-Sali-70 1 1 1

Citgo-Wil-70 8 8 8

AA-Sali-76 1 1 1

ρ3/4

Maximum 31

Minimum 0

ρ3/8

Maximum 55

Minimum 1

ρ4

Maximum 67

Minimum 14

ρ200

Maximum 6.6

Minimum 3

Va

Maximum 4.9

Minimum 3

Vbeff

maximum 14.2

Minimum 7.3

VMA Maximum 18.4

Minimum 10.8

VFA Maximum 79.1

Minimum 67.7

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.

3.3.5 WRI Mixture Database 

The results of laboratory testing and analysis of plant-produced AC mixtures from two 

sets of four test sections constructed for the WRI on US-77 in Kansas and on I-15 in 

Clark County, NV, are documented in this database. In each of these two sites, the four test 

sections were constructed using the same asphalt mixture and four binders from different crude 

sources. The laboratory testing was conducted on plant-produced samples that were compacted 

to a target air void of 7 percent using a gyratory compactor. The tests were conducted for the asreceived plant-aged condition and after LTOA in accordance with AASHTO R30-02, “Standard 

Specification for Mixture Conditioning of Hot Mix Asphalt (HMA).”(19) A summary of the 

relevant information in this database is shown in table 15. 
30 

Table 15. Summary of |E*| data available in the WRI database. 

Binder Code

No. of Mix 

Variations

Total 

No. of 

Tests

Temperature

(°C)

Frequency

(Hz)

Aging Condition

Plant or 

Unaged STOA

LTOA 

or 

Field

KS-1-Sinclair 4 4

4.4, 21.1, and 

40 

10, 1, 0.1, 

and 0.01 

2 2

KS-2-Coastal 4 4 2 2

KS-3-Koch 4 4 2 2

KS-4-Royal 4 4 2 2

NV-1-Sinclair WY 2 2

4, 20, and 40 

10, 5, 1, 0.5, 

0.1, 0.05, 

and 0.01 

1 1

NV-2-Crown 2 2 1 1

NV-3-Crown 2 2 1 1

NV-4-Crown 2 2 1 1

ρ3/4

Maximum 9

Minimum 1

ρ3/8

Maximum 40

Minimum 14

ρ4

Maximum 50

Minimum 32

ρ200

Maximum 6.6

Minimum 4.2

Va

Maximum 7.4

Minimum 6.6

Vbeff

Maximum 9.8

Minimum 6.4

VMA Maximum 17.2

Minimum 13.4

VFA Maximum 57.4

Minimum 47.1

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.

3.3.6 Citgo Mixture Database 

The Citgo database consists of two mixtures fabricated with the same aggregate structure 

(0.4-inch (9.5-mm) SuperpaveTM mixture) but with different asphalt binders. Both STOA and 

LTOA were conducted on these mixtures. A summary of the mixture properties is presented in 

table 16. These two mixtures are included in NCHRP projects 9-25 and 9-31.(16,17) For this 

database, the modulus values were measured in accordance with the AASHTO TP-62 protocol, 

and the aging was conducted in accordance with AASHTO R30-02.(8,19)
31 

Table 16. Summary of |E*| data available in the Citgo mixture database. 

Binder Code

No. of Mix 

Variations

Total 

No. of 

Tests

Temperature

(°C)

Frequency

(Hz)

Aging Condition

Plant or 

Unaged STOA

LTOA 

or 

Field

Citgo-A 2 4 4, 20, and 40 

10, 5, 1, 

0.5, 0.1, 

0.05, and

0.01

2 2

Citgo-B 2 4 2 2

ρ3/4 0

ρ3/8 6

ρ4 50

ρ200 5.1

Va 4.3

Vbeff 11.5

VMA 15.8

VFA 72.6

°C = (°F-32)/1.8

Note: Blank cells indicate data are unavailable in the database.

3.3.7 FHWA MR Database 

Under the FHWA DTFH61-05-RA-00108 project, NCSU researchers tested a set of mixtures for 

both |E*| and MR.

(14)

 The importance of this database is that it can be used to verify the predictive 

capabilities of the MR ANN model. In total, seven different mixtures are included with two 

different asphalt binder types. The asphalt binders used for these mixtures are not part of the 

binder database. Because these mixtures are only used for the MR ANN verification, the relevant 

information is different (see table 17). The designation for these mixes follows the NCDOT 

convention where the first letter represents either a surface mix (S) or a base mix (B). The second 

number represents the NMSA, and the final letter represents the gradation of fine (F), coarse (C), 

or flat and elongated (FE). The “M” on the last mix denotes that the binder was modified. Two of 

the study mixtures are derivations of the S12.5C mix in that only the asphalt content was 

changed by either ±1 percent. 

Table 17. Summary of |E*| data available in the MR database. 

Mix

Percent 

Va

Percent

AC

Asphalt 

Grade

S12.5C 4 5.5 

PG 64-22 

S12.5C-AC+1 4 6.5

S12.5C-AC-1 4 4.5

S12.5FE 4 5.7

S12.5F 4 4.8

B25.0C 4 4.9

S12.5CM 4 5.5 PG 76-22
32 

3.3.8 LTPP Data

Mixture and binder properties of AC layers on LTPP test sections at different aging conditions 

(i.e., original-, TFO/RTFO-, PAV-aged, and field-aged) were extracted from the LTPP database 

for use as inputs for the models developed as part of this study. The following list provides the 

criteria for selecting the layers to be considered in dynamic modulus estimates:(20)

• Layer thickness of 1 inch (25.4 mm) or greater as reported in the TST_LO5B table. 

• Virgin or recycled hot mix, hot laid, and dense graded AC (i.e., MATL_CODE 1 or 13 in 

the TST_L05B table).

• Placed as an original layer, overlay layer, or AC layer below the surface 

(i.e., DESCRIPTION 1, 3, or 4 in the TST_L05B table).

3.3.9 Input Source Hierarchy

Many properties of interest for this project are available in multiple locations within the LTPP 

database. Because of this, the team established a hierarchy of data sources to extract information

from the LTPP database. The following lists provide the priority used in extracting data (lower 

numbers denote higher priority) for each of the aging conditions. Included are the LTPP module 

and the specific table name for each data source. 

Binder shear modulus and phase angle are as follows:

(20)

Original properties: 

1. TST module (TST_AE07).

2. SPS module (SPS9_SP_PMA_AC_PROPERTIES).

TFO/RTFO- and PAV-aged properties: 

1. TST module (TST_AE07). 

2. SPS module (SPS9_SP_PMA_AC_PROPERTIES).

Field-aged properties: 

1. TST module (TST_AE07). 

Viscosity ring/ball, penetration, absolute viscosity, and kinematic viscosity are as follows:

(20)

Original properties: 

1. TST module (TST_AE02/AE04/AE05). 

2. SPS module (SPS?_PMA_AC_PROPERTIES). 
33 

3. RHB module (RHB_ACO_PROP/HMRAP_COMBINE_AC). 

4. INV module (INV_PMA_ASPHALT). 

TFO/RTFO and PAV-aged properties: 

1. RHB module (RHB_ACO_LAB_AGED_AC/HMRAP_LAB_AGED_AC). 

2. INV module (INV_PMA_ASPHALT). 

Field-aged properties: 

1. TST module (TST_AE02/AE04/AE05). 

Volumetrics-VMA, VFA, Va, theoretical maximum specific gravity (Gmb), and bulk specific 

gravity (Gmm) are as follows:

(20)

Original-, TFO/RTFO-, and PAV-aged properties: 

1. SPS module (SPS?_PMA_MIXTURE_PROP) (as-placed).

2. RHB module (RHB_ACO_MIX_PROP/HMRAP_MIX_PROP) (as-placed).

3. INV module (INV_PMA_ORG_MIX). 

Field-aged properties: 

1. TST module (TST_AC02/AC03). 

MR is as follows:(20)

Field-aged: 

1. TST module (TST_AC07). 

3.3.10 Data Structure and Detail

Table 18 provides the location within the LTPP database where the information was extracted 

and a summary of the data utilized in this project. 
34 

Table 18. Summary of extracted LTPP data. 

Field Name Description

STATE_CODE Code representing State or province

PROJECT_ID SHRP_ID for GPS sections and PROJECT_ID for SPS sections

PROJECT_LAYER

Project layer code as established in TST_LO5B for SPS sections; layer 

code as established in TST_LO5B for GPS sections(20)

CONSTRUCTION_DATE Date layer was constructed

SAMPLE_TYPE

1—original binder, 2—lab-aged (TFO/RTFO), 3—lab-aged (PAV), 

and 4—field-aged

GSTAR_SAMPLE_DATE Date of |G*| sampling (if field-aged sample)

GSTAR_1 Binder complex modulus at temperature 1

PHASE_ANGLE_1 Phase angle at temperature 1

GSTAR_TEMP_1 Binder complex modulus at temperature 1

GSTAR_2 Binder complex modulus at temperature 2

PHASE_ANGLE_2 Phase angle at temperature 2

GSTAR_TEMP_2 Binder complex modulus at temperature 2

GSTAR_3 Binder complex modulus at temperature 3

PHASE_ANGLE_3 Phase angle at temperature 3

GSTAR_TEMP_3 Binder complex modulus at temperature 3

GSTAR_4 Binder complex modulus at temperature 4

PHASE_ANGLE_4 Phase angle at temperature 4

GSTAR_TEMP_4 Binder complex modulus at temperature 4

GSTAR_5 Binder complex modulus at temperature 5

PHASE_ANGLE_5 Phase angle at temperature 5

GSTAR_TEMP_5 Binder complex modulus at temperature 5

GSTAR_6 Binder complex modulus at temperature 6

PHASE_ANGLE_6 Phase angle at temperature 6

GSTAR_TEMP_6 Binder complex modulus at temperature 6

GSTAR_SOURCE

LTPP module from which binder complex modulus was extracted 

(i.e., TST = testing, RHB = rehabilitation, SPS = SPS construction, 

INV = inventory, and CALC = calculated)

BINDER_SAMPLE_DATE Date of binder sampling (if field-aged sample)

RING_BALL Ring/ball (°F) results

RING_BALL_SOURCE

LTPP module from which ring and ball was extracted (i.e., TST =

testing, RHB = rehabilitation, SPS = SPS construction, INV = 

inventory, and CALC = calculated)

PENETRATION_39.2F Penetration at 39.2 °F

PENETRATION_39.2F

_SOURCE

LTPP module from which penetration at 39.2 °F was extracted 

(i.e., TST = testing, RHB = rehabilitation, SPS = SPS construction, 

INV = inventory, and CALC = calculated)

PENETRATION_77F Penetration at 77 °F

PENETRATION_77F

_SOURCE

LTPP module from which penetration at 77 °F was extracted 

(i.e., TST = testing, RHB = rehabilitation, SPS = SPS construction, 

INV = inventory, and CALC = calculated)

PENETRATION_115F Penetration at 115 °F
35 

PENETRATION_115F

_SOURCE

LTPP module from which penetration at 115 °F was extracted (i.e., 

TST = testing, RHB = rehabilitation, SPS = SPS construction, INV = 

inventory, and CALC = calculated)

ABSOLUTE_VISCOSITY Absolute viscosity at 140 °F (poises)

ABSOLUTE_VISCOSITY

_SOURCE

LTPP module from which absolute viscosity was extracted (i.e., TST =

testing, RHB = rehabilitation, SPS = SPS construction, INV = 

inventory, and CALC = calculated)

KINEMATIC_VISCOSITY Kinematic viscosity at 275 °F (centistokes)

KINEMATIC_VISCOSITY

_SOURCE

LTPP module from which kinematic viscosity was extracted (i.e., 

TST = testing, RHB = rehabilitation, SPS = SPS construction, INV = 

inventory, and CALC = calculated)

VMA Voids in mineral aggregate

VMA_SOURCE

LTPP module from which VMA was extracted (i.e., TST = testing, 

RHB = rehabilitation, SPS = SPS construction, INV = inventory, and 

CALC = calculated)

VFA Voids filled with asphalt

VFA_SOURCE

LTPP module from which VFA was extracted (i.e., TST = testing, 

RHB = rehabilitation, SPS = SPS construction, INV = inventory, and 

CALC = calculated)

AIR_VOID 

SAMPLE_DATE Date of air void sampling (if field-aged sample)

AIR_VOIDS Percent air voids

AIR_VOIDS_SOURCE

LTPP module from which air voids was extracted (i.e., TST = testing, 

RHB = rehabilitation, SPS = SPS construction, INV = inventory, and 

CALC = calculated)

GMB Bulk specific gravity of the mix

GMB_SOURCE

LTPP module from which bulk specific gravity was extracted ((i.e., 

TST = testing, RHB = rehabilitation, SPS = SPS construction, INV = 

inventory, and CALC = calculated) 

GMM Maximum specific gravity

GMM_SOURCE

LTPP module from which maximum specific gravity was extracted 

(i.e., TST = testing, RHB = rehabilitation, SPS = SPS construction, 

INV = inventory, and CALC = calculated)

MR_SAMPLE_DATE Date of resilient modulus sampling (if field-aged sample)

MR_5 Resilient modulus at 5 °C

MR_25 Resilient modulus at 25 °C

MR_40 Resilient modulus at 40 °C

MR_SOURCE

LTPP module from which resilient modulus was extracted 

(i.e., TST = testing, RHB = rehabilitation, SPS = SPS construction, 

INV = inventory, and CALC = calculated)

BINDER_GRADE Binder grade information (only populated for RTFO data)

°C = (°F-32)/1.8
36 

For GPS projects, each layer is specific to one test section. As such, the data utilized from the 

LTPP database are section-specific. However, for SPS projects, the same materials were used in 

the construction of many test sections on one SPS project. Inputs for these SPS projects were 

computed as the average of the layer over the entire SPS project. Therefore, |E*| estimates were 

developed for each PROJECT_LAYER_CODE (as reported in the TST_LO5B table) and 

reported as project-level data.(20) For SPS projects that were linked to GPS test sections, priority 

was given to data from the SPS project. If those data were not available, the SPS records were 

populated with data from the corresponding GPS test section. 

In some cases, the AC layers were sampled and tested multiple times and at varying ages. If test 

results were available from multiple sample dates, |E*| was computed for each sample date. The 

corresponding age was calculated and included with the data. 

Va for field-aged samples are not directly available in the LTPP database; therefore, the 

following equation was used to calculate air voids from bulk- and maximum-specific gravities 

measured from field samples:

%100*1 













 −=

mm

mb

a G

G V

(14)

Where:

Va = Air voids (percent). 

Gmb = Bulk specific gravity.

Gmm = Maximum specific gravity.

In addition, because VFA information is not available directly in the LTPP database, the 

following equation was used to compute VFA:

 %100*1 

 



 −=

VMA

V VFA a

(15)
37 

4.0 EVALUATION OF |E*| PREDICTIVE MODELS 

ANNs are used to solve problems in the asphalt pavement field with the primary application 

being the backcalculation of pavement layer moduli from falling weight deflectometer 

measurements.

(21–23) This method has also been applied successfully to assess the roughness 

progression in flexible pavements and to analyze the surface wave data obtained from the 

nondestructive testing of asphalt pavements.(24,25) An often cited drawback of these approaches is 

their inability to extrapolate when a situation arises that is beyond that used to train the ANNs. 

To overcome this shortcoming, it is important that the dataset used to train the ANNs covers the 

entire range of conditions expected to be encountered during use. The objective of this study is to 

show that ANN modeling techniques can be applied to predict |E*| values of AC to a higher 

degree of accuracy than currently available models using the same input parameters.

4.1 ANN STRUCTURE USED IN THIS STUDY

A preliminary study was conducted to determine the feasibility and predictability of the ANN 

modeling technique relative to the existing models. This feasibility study was first conducted 

based on |G*| because more closed-form models exist that use this parameter as their primary 

input parameter. The ANN models used in this preliminary study are not the final models 

suggested by the research team, but they are similar in form and validation. To ensure full 

coverage of the expected conditions, the most recent Witczak database with available measured 

|G*| data and a portion of the dataset obtained at NCSU with support from the NCDOT were 

utilized as the TP-62 training database. Also, appropriate portions of the FHWA mobile trailer 

database and the WRI database (from the Kansas and Nevada sites) were considered the AMPT 

training database. A combination of the AMPT and TP-62 databases has been used to train the 

network after investigations, which are summarized in appendix C of this report. 

The ANN model developed herein contains a mapping ANN architecture and is based on 

supervised learning. In the developed network, the learning method is a feed forward back 

propagation, which is one of the best known types of ANN. The sigmoidal function was chosen 

as the transfer function. The three-layer network was selected as the best network configuration. 

The first two layers consist of 12–14 nodes based on the different cases studied for the 

development of the models. 

To evaluate the goodness of fit in arithmetic scale, |E*| is considered to be the dependent 

variable, and the error is given as follows: 

2

Sum of Squared Error, * *

normal predicted measured SSE E E =   − ∑  (16)

Sy is defined as the standard deviation of the measured |E*| values. To evaluate the goodness of 

fit in logarithmic scale, the dependent variable is the log (|E*|), and the error is as follows: 

( ) ( ) 2

Sum of Squared Error, log log * log * predicted measured SSE =   E − E ∑  (17)
38 

Sy is defined as the standard deviation of the measured log (|E*|) values. The standard error is 

defined as follows: 

Standard Error, 1

SSE Se

n = − (18)

Where:

n = Number of observations. 

4.2 EVALUATION OF BINDER |G*|-BASED MODELS 

A literature review of current versions of the modified Witczak and Hirsch models showed that 

these two models have high goodness-of-fit statistics for their original training databases of 

7,400 and 206 data points, respectively. (See references 11, 5, 12, 6, 7, and 13.) The literature 

purports high correlation coefficients and small errors of R2 = 0.80 and Se/Sy = 0.45 in arithmetic 

scale, R2 = 0.90 and Se/Sy = 0.32 in logarithmic scale for the modified Witczak model, and R2 

= 

0.98 in logarithmic scale for the Hirsch model. The developers of the Hirsch model report only 

the logarithmic-based R2 value.

These results demonstrate that the goodness-of-fit for each model is dependent on the number of 

observations or the width of the range of the different variables considered in the development of 

each model. The deterioration of the statistical parameters when these models are applied to 

some expanded and independent databases is not entirely unexpected given the nature of 

regression models. Additional insight is gained by examining line-of-equality (LOE) plots for 

each of the models in both arithmetic and logarithmic scales.

In this section, the existing models (including the modified Witczak, Hirsch, and Al-Khateeb 

models) are evaluated along with the two ANNs using the verification databases shown in 

figure 7 through figure 16. Also, figure 17 through figure 28 show predictions of the |E*| values 

from the verification databases using the different models. The first observation is that the 

Al-Khateeb model exhibits a significant bias (i.e., a power trend between the predicted |E*| and 

the measured |E*| values) in all the predictions. This finding is not entirely unexpected given the 

limited database used in calibrating the model. However, due to the bias relative to the other two 

existing models, it was decided that this model would be eliminated from consideration in any 

future analysis. It should be noted that the scales in figure 7 through figure 28 vary to provide the 

greatest clarity in the data.

The Hirsch model behaves in a reasonable fashion, although it exhibits undesirable behavior at 

low |E*| values in the LOE graphs shown in figure 9 and figure 10. When the prediction is good, 

the expectation is that a group of data points following the LOE with an oval shape in the LOE 

graph would be seen. However, the Hirsch predictions shown in figure 9 and figure 10 exhibit a 

horizontal pattern amplified in the bottom left side of the log-log LOE graph. This undesirable 

pattern in the Hirsch model predictions is related to the insensitivity of the model and its inability 

to distinguish the performance differences among different mixtures for a given set of 

environmental and other conditions.(26) This issue is discussed later in this section. 
39 

The developers of the Hirsch model note that during initial development, substantial errors 

occurred when predictions were made at extremely high and low modulus values. The authors of 

the model attempted to correct this issue in subsequent efforts by expanding the calibration 

database to include 15.8 and 129.2 °F (-9 and 54 °C) data.

(6) However, it is unclear if such efforts 

succeeded in reducing potential prediction errors because the authors did not have access to

and/or show a large enough verification database. Regardless of this potential model 

shortcoming, predictions were made for the complete range of temperatures that a typical user is 

likely to apply to the model (i.e., 14 to 129.2 °F (-10 to 54 °C)). For an independent and 

expanded database, it appears that the Hirsch model developers, with their available dataset, 

were not able to completely address the original shortcomings of their model because plateau 

areas appear at the high and low modulus values, as shown in figure 9, figure 10, figure 19, 

figure 20, figure 23, figure 24, figure 27, and figure 28. The undesirable pattern in the Hirsch 

model predictions at low modulus values (high temperatures) is related to the insensitivity of the 

model to the changes in volumetric parameters. Errors at the low temperature of 14 °F (-10 °C) 

(high modulus values) are caused by the model having a limited modulus of approximately 

3,500,000 psi (24,115,000 kPa) even though values higher than this are often measured in the 

laboratory at 14 °F (-10 °C). These drawbacks raise concerns over the use of the Hirsch model 

for the prediction of modulus values over the complete range needed for the MEPDG input. 

However, it will be shown later that the parameters on which this model is based seem to be 

promising and adequate for consideration in predicting the |E*| values using the ANN model. 

The modified Witczak model shows a larger scatter than the ANN models. The performance of 

the modified Witczak model in all the databases presented in figure 7 and figure 8 indicates that 

this model tends to overestimate the measured |E*| values over the entire range, particularly at 

the extreme modulus values. It is believed that this effect is due to the aforementioned use of 

inappropriate |G*| values at temperatures less than or equal to 39.9 °F (4.4 °C) (overestimation at 

high modulus values) and the use of |E*| values at higher than recommended strain levels 

(overestimation at low modulus values). The modified Witczak predictions also tend to have a 

bias at high temperatures relating to insensitivity to volumetric changes and the inability of this 

model to clearly capture these differences.

(26)

The prediction of the Citgo |E*| data is presented in figure 25 through figure 28. It should be 

noted that the binder data at 39.2 oF (4 °C) are extrapolated from the CAM model more so than 

other binders in the database. For predictions of both aging conditions, the model inputs are the 

same, which causes the slight horizontal pattern. For this dataset, figure 25 through figure 28

show that the ANN-based |E*| predictions are more erratic compared to the modified Witczak 

and Hirsch models. It is believed that this dataset was used in the calibration of the Hirsch 

model. As a result, it is somewhat unfair to compare the predictions using this model. After a 

careful exploration of the training dataset, it was found that mixtures that have similar |G*| 

values at 39.2 oF (4 °C) and have similar volumetric properties also have higher measured |E*| 

values than the Citgo mixtures. It was also found that these similar mixtures are coarse 

gradations, whereas the Citgo mixtures are finely graded. For this reason, the ANN models are 

capable of finding differences between these mixtures. However, including only the modified 

Witczak volumetric parameters causes model confusion at the low modulus values and, hence, 

the observed variability. The reason this variability shows so clearly with the Citgo mixture is 

likely due to the close similarity of it with mixtures in the training database. To address this issue 

thoroughly, new parameters that better represent the relative effects of the gradation and 
40 

volumetric properties may need to be identified. This effort is beyond the scope of this current 

project. 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

LOE

Modified Witczak

R2

=0.85, Se/Sy=0.75

1 psi = 6.86 kPa

Figure 7. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the modified Witczak in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

LOE

Modified Witczak

R2

=0.92, Se/Sy=0.41

1 psi = 6.86 kPa

Figure 8. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the modified Witczak in logarithmic scale. 
41 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.81, Se/Sy=0.48

1 psi = 6.86 kPa

Figure 9. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the Hirsch model in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2=0.92, Se/Sy=0.31 

1 psi = 6.86 kPa

Figure 10. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the Hirsch model in logarithmic scale. 
42 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.71, Se/Sy=0.65 

1 psi = 6.86 kPa

Figure 11. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the Al-Khateeb model in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.89, Se/Sy=0.34 

1 psi = 6.86 kPa

Figure 12. Graph. Prediction of the processed Witczak, FHWA I, FHWA II, NCDOT I, 

NCDOT II, WRI, and Citgo databases using the Al-Khateeb model in logarithmic scale. 
43 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

G-GR pANN

LOE

G-GR pANN

R2

=0.92, Se/Sy=0.29

1 psi = 6.86 kPa

Figure 13. Graph. Prediction of training data containing processed Witczak, FHWA I, 

NCDOT I, and WRI databases using |G*| binder and gradation-based pilot ANN model 

(G-GR pANN) in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

G-GR pANN

LOE

G-GR pANN

R2

=0.98, Se/Sy=0.15

1 psi = 6.86 kPa

Figure 14. Graph. Prediction of training data containing processed Witczak, FHWA I, 

NCDOT I, and WRI databases using G-GR pANN in logarithmic scale. 
44 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

G-V pANN

LOE

G-V pANN

R2

=0.89, Se/Sy=0.34

1 psi = 6.86 kPa

Figure 15. Graph. Prediction of training data containing processed Witczak, FHWA I, 

NCDOT I, and WRI databases using preliminary |G*|-based models used in phase I of the 

study (G-V) pANN in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

G-V pANN

LOE

G-V pANN

R2

=0.96, Se/Sy=0.21

1 psi = 6.86 kPa

Figure 16. Graph. Prediction of training data containing processed Witczak, FHWA I, 

NCDOT I, and WRI databases using G-V pANN in logarithmic scale. 
45 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.91, Se/Sy=0.92

G-GR pANN

R2

=0.86, Se/Sy=0.38

1 psi = 6.86 kPa

Figure 17. Graph. Predicted moduli using G-GR pANN and modified Witczak models for 

the FHWA II database in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.92, Se/Sy=0.58

G-GR pANN

R2

=0.91, Se/Sy=0.35

1 psi = 6.86 kPa

Figure 18. Graph. Predicted moduli using G-GR pANN and modified Witczak models for 

the FHWA II database in logarithmic scale. 
46 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

G-V pANN

LOE

Hirsch

R2

=0.92, Se/Sy=0.30

G-V pANN

R2

=0.90, Se/Sy=0.33

1 psi = 6.86 kPa

Figure 19. Graph. Predicted moduli using G-V pANN and Hirsch models for the FHWA II 

database in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

G-V pANN

LOE

Hirsch

R2

=0.92, Se/Sy=0.39

G-V pANN

R2

=0.94, Se/Sy=0.28

1 psi = 6.86 kPa

Figure 20. Graph. Predicted moduli using G-V pANN and Hirsch models for the FHWA II 

database in logarithmic scale. 
47 

0.0E+00

2.5E+06

5.0E+06

7.5E+06

1.0E+07

0.0E+00 2.5E+06 5.0E+06 7.5E+06 1.0E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.91, Se/Sy=0.71

G-GR pANN

R2

=0.97, Se/Sy=0.33

1 psi = 6.86 kPa

Figure 21. Graph. Predicted moduli using G-GR pANN and modified Witczak models for 

the NCDOT II database in arithmetic scale. 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.98, Se/Sy=0.19

G-GR pANN

R2

=0.96, Se/Sy=0.27

1 psi = 6.86 kPa

Figure 22. Graph. Predicted moduli using G-GR pANN and modified Witczak models for 

the NCDOT II database in logarithmic scale. 
48 

0.0E+00

2.5E+06

5.0E+06

7.5E+06

1.0E+07

0.0E+00 2.5E+06 5.0E+06 7.5E+06 1.0E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

G-V pANN

LOE

Hirsch

R2

=0.94, Se/Sy=0.70

G-V pANN

R2

=0.91, Se/Sy=0.31

1 psi = 6.86 kPa

Figure 23. Graph. Predicted moduli using G-V pANN and Hirsch models for the 

NCDOT II database in arithmetic scale. 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

G-V pANN

LOE

Hirsch

R2

=0.98, Se/Sy=0.31

G-V pANN

R2

=0.96, Se/Sy=0.23

1 psi = 6.86 kPa

Figure 24. Graph. Predicted moduli using G-V pANN and Hirsch models for the 

NCDOT II database in logarithmic scale. 
49 

0.E+00

2.E+06

4.E+06

6.E+06

0.E+00 2.E+06 4.E+06 6.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.98, Se/Sy=0.64

G-GR pANN

R2

=0.94, Se/Sy=0.52

1 psi = 6.86 kPa

Figure 25. Graph. Predicted moduli using G-GR pANN and modified Witczak models for 

the Citgo database in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.99, Se/Sy=0.26

G-GR pANN

R2

=0.96, Se/Sy=0.59

1 psi = 6.86 kPa

Figure 26. Graph. Predicted moduli using G-GR pANN and modified Witczak models for 

the Citgo database in logarithmic scale. 
50 

0.E+00

2.E+06

4.E+06

6.E+06

0.E+00 2.E+06 4.E+06 6.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

G-V pANN

LOE

Hirsch

R2

=0.99, Se/Sy=0.11

G-V pANN

R2

=0.97, Se/Sy=0.88

1 psi = 6.86 kPa

Figure 27. Graph. Predicted moduli using G-V pANN and Hirsch models for the Citgo 

database in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

G-V pANN

LOE

Hirsch

R2

=0.99, Se/Sy=0.09

G-V pANN

R2

=0.99, Se/Sy=0.24

1 psi = 6.86 kPa

Figure 28. Graph. Predicted moduli using G-V pANN and Hirsch models for the Citgo 

database in logarithmic scale. 
51 

The feasibility of calibrating an ANN using the Hirsch parameters and the modified Witczak 

model parameters has been investigated. Table 19 shows that the key difference in input 

requirements for these two models is the lack of gradation parameters for the Hirsch model. Note 

the naming convention for the ANN models followed in this portion of the report. The first letter 

group represents the major binder property used (“G” for |G*| and “Visc” for viscosity). The 

second letter group signifies whether gradation parameters were used (“GR” when they are 

included and “V” when only volumetric properties are used). Finally, the designation pANN 

denotes that the model is an ANN model but that it is used only for the pilot studies. The 

finalized ANN models developed in this project have no prefix before ANN. Due to the issues 

discussed in appendix C regarding AMPT and TP-62 measured moduli, both types of data were 

used in the calibration of the ANN model. The calibration results for this model are shown in 

figure 13 to figure 16, and the verification data are shown in figure 17 to figure 28. 

Table 19. Description of |G*|-based ANN models.

ANN Model

Parameters 

Used to Train Training Database

Verification 

Database

G-GR pANN

|G*|

FHWA I

Processed Witczak1

NCDOT I

FHWA II

NCDOT II

Va

Vbeff

ρ34

ρ38

ρ4

ρ200

G-V pANN

|G*|

VMA (percent)

VFA (percent) 1

Portions of the Witczak database (mixtures 1–135) and also some mixtures from the 

remaining portion that do not have reliable measurements (very high |E*| measurements) 

were omitted. The portions of the Witczak database used for developing |G*|-based models 

are the ones that have measured |G*| values.

For the FHWA II dataset shown in figure 17 and figure 18, the G-GR pANN model shows 

more scatter than the predictions made from the G-V pANN model and the Hirsch model. 

Because none of the FHWA II mixtures contain data at 14 oF (-10 °C) (|E*| measurement based 

on AMPT protocol), there is no observed bias in the Hirsch model shown in figure 19 and figure 

20. For the NCDOT II database that contains low temperature data (figure 21 through figure 24), 

the bias in the Hirsch model is clear. Also, the G-GR pANN and G-V pANN models are 

similar, with the G-GR pANN model showing a slight improvement both visually and from 

the statistical measurements. 

With the exception of the Citgo dataset, which was discussed previously with regard to the 

Witczak-based ANNs, the |G*|-based ANN model appears to yield better predictions than the 

Hirsch and modified Witczak models. The overall performance of this |G*|-based ANN model 

shows that considering the VMA and VFA parameters together with the t-T dependent binder 

rheological parameter provides more promising predictions for both the training and verification 

databases than the parameters used in the modified Witczak model.
52 

The findings from figure 7 to figure 28 are as follows:

• The Al-Khateeb model displays a significant bias and, therefore, is removed from 

future analysis.

• The Hirsch model performs reasonably well, although the horizontal pattern shown in 

the LOE graph indicates problems associated with the insensitivity of the model to the 

input variables. 

• The pANN models show the least scatter with the least bias overall.

• The pANN models trained with the Hirsch model parameters should yield predictions as 

good as if not better than the models trained with the modified Witczak parameters. The 

reason for this was not explored in detail. 

• The primary advantage of ANN modeling over statistical regression techniques is that 

the functional form of the relationship is not needed a priori. Considering that many 

variables affect |E*| values and their interaction, the ANN technique may capture 

complicated nonlinear relationships between |E*| and other mixture variables better than 

regression analysis.

4.3 VISCOSITY-BASED ANN

Two viscosity-based ANN models with two different sets of parameters were developed (see 

table 20). The performance of each model is shown in figure 29 through figure 40 for both 

training and verification databases. Like the |G*|-based ANN models, the difference in these two 

viscosity-based ANN models are related to the input parameters used for training. In the first 

ANN, viscosity-gradation (Visc-GR) pANN, the parameters suggested by the original Witczak 

model are adopted, whereas in the second ANN (viscosity-volumetric (Visc-V) pANN), the 

Hirsch model parameters are chosen, with the exception that frequency and viscosity are chosen 

instead of |G*|. Results of the training for the two models are shown in figure 29 to figure 32. 

Figure 33 to figure 40 show the verification dataset. Note that some of the points used in the 

training data have also been used in calibrating the original Witczak model. These points 

represent only a very small portion of the total data shown in figure 29 to figure 32. Results of 

the independent model verification process are shown for different databases in figure 33 to 

figure 40. Through these figures, it appears that the ANN models perform better than the original 

Witczak model. It is also evident that although the Visc-GR pANN model performs better in 

training, the Visc-V pANN model is better for model verification. Based on these findings, it 

appears that, like the |G*|-based models, removing gradation parameters from the necessary 

inputs yielded improved and more stable modulus predictions. 
53 

Table 20. Description of viscosity-based ANN models. 

ANN Model

Parameters 

Used to Train

Training 

Database

Verification 

Database

Visc-GR pANN

f (hertz)

FHWA I

Witczak1

NCDOT I

FHWA II

NCDOT II

η (109 P)

Va

Vbeff

ρ34

ρ38

ρ4

ρ200

Visc-V pANN

f (hertz)

η (109 P)

VMA (percent)

VFA (percent)

1 Pas = 10 P 1

Portions of the Witczak database that do not have reliable measurements 

(high |E*| measurements) were neglected.

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-GR pANN

LOE

Visc-GR pANN

R2

=0.90, Se/Sy=0.32

1 psi = 6.86 kPa

Figure 29. Graph. Prediction of training data containing Witczak, FHWA I, and NCDOT I 

databases using Visc-GR pANN in arithmetic scale. 
54 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-GR pANN

LOE

Visc-GR pANN

R2

=0.97, Se/Sy=0.18

1 psi = 6.86 kPa

Figure 30. Graph. Prediction of training data containing Witczak, FHWA I, and NCDOT I 

databases using Visc-GR pANN in logarithmic scale. 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-V pANN

LOE

Visc-V pANN

R2

=0.87, Se/Sy=0.36

1 psi = 6.86 kPa

Figure 31. Graph. Prediction of training data containing Witczak, FHWA I, and NCDOT I 

databases using Visc-V pANN in arithmetic scale. 
55 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-V pANN

LOE

Visc-V pANN

R2

=0.94, Se/Sy=0.25

1 psi = 6.86 kPa

Figure 32. Graph. Prediction of training data containing Witczak, FHWA I, and NCDOT I 

databases using Visc-V pANN in logarithmic scale. 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-GR pANN

LOE

Visc-GR pANN

R2

=0.66, Se/Sy=0.67

1 psi = 6.86 kPa

Figure 33. Graph. Predicted moduli using Visc-GR pANN model for the FHWA II 

database in arithmetic scale. 
56 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-GR pANN

LOE

Visc-GR pANN

R2=0.71, Se/Sy=0.56

1 psi = 6.86 kPa

Figure 34. Graph. Predicted moduli using Visc-GR pANN model for the FHWA II 

database in logarithmic scale. 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-V pANN

LOE

Visc-V pANN

R2

=0.87, Se/Sy=0.36

1 psi = 6.86 kPa

Figure 35. Graph. Predicted moduli using Visc-V pANN model for the FHWA II database 

in arithmetic scale. 
57 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-V pANN

LOE

Visc-V pANN

R2

=0.93, Se/Sy=0.28

1 psi = 6.86 kPa

Figure 36. Graph. Predicted moduli using Visc-V pANN model for the FHWA II database 

in logarithmic scale. 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-GR pANN

LOE

Visc-GR pANN

R2=0.94, Se/Sy=0.27

1 psi = 6.86 kPa

Figure 37. Graph. Predicted moduli using Visc-GR pANN model for the NCDOT II 

database in arithmetic scale. 
58 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-GR pANN

LOE

Visc-GR pANN

R2

=0.91, Se/Sy=0.36

1 psi = 6.86 kPa

Figure 38. Graph. Predicted moduli using Visc-GR pANN model for the NCDOT II

database in logarithmic scale. 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-V pANN

LOE

Visc-V pANN

R2=0.95, Se/Sy=0.21

1 psi = 6.86 kPa

Figure 39. Graph. Predicted moduli using Visc-V pANN model for the NCDOT II database 

in arithmetic scale. 
59 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Visc-V pANN

LOE

Visc-V pANN

R2

=0.97, Se/Sy=0.17

1 psi = 6.86 kPa

Figure 40. Graph. Predicted moduli using Visc-V pANN model for the NCDOT II database 

in logarithmic scale. 

61 

5.0 ANN MODELS FOR POPULATING THE LTPP DATABASE

Three ANN models have been developed for populating the LTPP database sections. These 

models are differentiated by the following primary input parameters: (1) the MR model uses the 

resilient modulus, (2) the VV model uses the binder viscosity, and (3) the |G*|-based (GV) model 

uses the binder shear modulus. The subsequent sections provide a description of each model 

along with the final verification plots and statistics. The models in this chapter use normalized 

inputs, whereas those discussed in the previous section use nonnormalized inputs. 

The term normalizing a vector often refers to changing the magnitude of an element by dividing 

it by a norm of the vector. For neural networks, this definition often means changing the scale of 

a vector by the minimum value and the range of the vector so that all the components are

between 0 and 1 or 1 and -1. Linear and uniform normalizations are the common methods used 

for this purpose. In some cases, such as lab- and field-measured data used to develop the ANN 

models in this study, linear normalization seems to be more meaningful when no specific 

distributions of the data are known or present. In addition to normal distribution, a Gaussian 

normalization (based on the mean and standard deviation of the data for each parameter) could 

be performed. In some cases, these statistics could be used to get rid of outliers (e.g., data points 

outside of the three standard deviations). Although such types of nonlinear normalization and 

related procedures may be useful in some applications (e.g., when the measurement range is 

meant to be normally distributed or distributed with some statistical uniformity), for the type of 

engineering applications where the data represent different conditions, it is not possible to find a 

statistical distribution of those input values. For example, if temperature is an input and 

different datasets represent different discrete values/ranges, then it is not useful to find a mean 

temperature value and its standard deviation to normalize the data. In this study, the linear model 

was adopted because of its simplicity. After applying this normalization (scaling) scheme to the 

data and developing the ANN models, it was found that the predictions were acceptable, and

there was no further investigation into the need of nonlinear normalization. This issue is one that 

possibly warrants future study. 

The decision to utilize normalized input-based ANN models was not finalized at the time 

of the study presented in the previous chapter. After deciding to use normalized input-based 

ANN models, the pilot analysis was not redone because the normalization only improved 

the predictability and thus did not change the final conclusions of the pilot study. Based on 

these fitting statistics and on engineering judgment, the models were ranked to develop a 

decision tree so that a user can determine which ANN model is best suited to a specific set of 

input parameters.

Due to differences in the required inputs for each model, different subsets of the entire database 

were used in training and verification. The total number of points used for each model, along 

with a summary of the required input parameters, is provided in table 21. Of the total available 

points for each model, 90 percent were randomly selected for the purpose of training the 

networks, and 10 percent were used for verification. The Witczak database was used for 

calibrating the VV ANN model, except that the data at the temperatures equal to or less than 

32 ° F (0 °C) were not used because of unacceptably high |E*| values. It should be mentioned 

that the GV and |G*|-based model using inconsistent aged binder data of PAV- and RTFO-aging 
62 

conditions (GV-PAR) models, like the VV and viscosity-based model using specification grade 

of the asphalt binder (VV-grade) models, represent the same trained network, but they 

are identified by different terms due to the binder values used in each case. In the GV-PAR 

model, the binder values are based on two aging conditions, PAV and RTFO. In the VV-grade 

model, the values of A and VTS are chosen, as recommended in MEPDG, based on the 

specification grade of the asphalt binder. Sections using this model have less than two viscosity 

measurements, but the VMA, VFA, and binder grade values are available. Descriptions of the 

binder analysis necessary for using these models are given in appendices A and B. 

All of the ANN models developed herein contain a mapping ANN architecture and are based on 

supervised learning. In the developed network, the learning method used is a feed forward back 

propagation, and the sigmoidal function is the transfer function. The three-layer network with 

two hidden layers was selected as the best configuration. The number of nodes in each layer 

differs according to the selected model (see table 21). These node numbers were determined after 

a systematic study of each model. In each case, the network follows the same basic structure that 

is schematically illustrated in figure 41. A more formal mathematical representation for each of 

these models is given in appendix D. 

Table 21. Parameters and number of nodes used in developed ANN models. 

ANN 

Model

Parameters Used to Train ANN Models

Number

of Data 

Points

Number

of Nodes

MR at 5, 25,

and 40 °C 

(MPa)

Shift 

Factor 

(α1, α2, α3)

fR

(Hz)

Viscosity 

(109 P)

VMA 

(Percent)

VFA 

(Percent)

|G*| 

(psi)

MR   11,730 12

VV     14,682 14

GV    12,907 12

GV-PAR    12,907 12

VV-grade     14,682 14

°C = (°F-32)/1.8

1 Pas = 10 P

1 psi = 6.86 kPa

Note: Blank cells indicate that the parameter was not used to train the model.
63 

1

Input 

Layer

Input 

Layer

Hidden 

Layer #1

Hidden 

Layer #1

Output 

Layer

Output 

Layer

Hidden 

Layer #2

Hidden 

Layer #2

Wik, Bk

1

Log |E*|

2

3

n

1

2

3

n

1

1

i

n

|P1|

|Pn|

Normalized 

Input 1 Predicted

Log |E*|

Measured

Target

Wkj, Bj

2

Wj, B0

Error

i k j

Backpropagation of Error to 

update weights and Biases

P1

Pn

Input n Normalized 

Input n

Input 1

Normalization

1

Input 

Layer

Input 

Layer

Hidden 

Layer #1

Hidden 

Layer #1

Output 

Layer

Output 

Layer

Hidden 

Layer #2

Hidden 

Layer #2

Wik, Bk

1

Log |E*|

2

3

n

1

2

3

n

1

1

i

n

|P1|

|Pn|

Normalized 

Input 1 Predicted

Log |E*|

Measured

Target

Wkj, Bj

2

Wj, B0

Error

i k j

Backpropagation of Error to 

update weights and Biases

P1

Pn

Input n Normalized 

Input n

Input 1

Normalization

Figure 41. Illustration. Network structure used for training the ANN models. 

5.1 MR ANN MODEL

The 1993 AASHTO Guide for Design of Pavement Structures employs MR as the material 

property representing the stiffness characteristics of layer materials.(27) MR is defined as the ratio 

between applied stress (σa) and recoverable strain (εr) as follows: 

a

R

r

M σ

ε = (19)

Several testing standards have been developed for the determination of MR of AC using the 

indirect tensile (IDT) test method (American Society for Testing and Materials (ASTM) D4123-

82, NCHRP 1-28, State Highway Research Program (SHRP) P-07, and NCHRP 1-28A). 

(See references 28–31.) In light of these facts, the LTPP database has stored MR as the primary 

measured mixture stiffness term for many of the layers. Due to the industry’s transition from MR

to |E*|, a significant amount of MR data that have been collected in State highway agencies may 

become obsolete unless the MR values can be converted to |E*| values. The NCSU research team 

successfully developed a method to make this conversion using an ANN-based methodology to 

predict the |E*| values at multiple temperatures and frequencies when only the MR values at three 

temperatures are available.

The difficulty in performing this conversion stems from the fact that MR provides a snapshot of 

the material behavior under one loading history (i.e., 0.1-s haversine loading followed by a 0.9-s 

rest period) at different testing temperatures (normally three temperatures at 41, 77, and 104 °F 

(5, 25, and 40 °C)). Zhang explored the possibility of characterizing the viscoelastic properties 

obtained from MR tests using Fourier analysis.(32) It was concluded that this type of analysis is 

impractical due to the difficulty in solving a large number of variables and the limited range of 
64 

results, which provided little information other than MR data. Another challenge is that Fourier 

analysis requires a loading and deformation history in order to fit, which most databases do not 

contain. Because an analytical method is not feasible, any attempt to predict |E*| from MR is, at 

best, empirical in nature.

One difficulty in any empirical model is acquiring a database large enough to represent the 

range of possible inputs. Because a large database containing both MR and |E*| does not exist, 

a theoretical approach has been developed based on LVE principles. The resulting forward 

model is used along with the mixtures in the Witczak database to develop the ANN model 

training database. 

5.1.1 Forward Modeling

Developing a database with |E*| and MR values can be a time-consuming task if the properties 

are measured in the laboratory, especially for a database comprehensive enough to encompass a 

large range of mixture variables such as binders, gradations, NMSA, VMA, VFA, air voids, etc. 

The proposed method was to use a comprehensive |E*| database and populate the database with 

MR values by using LVE principles. 

MR can be calculated by several different methods, including ASTM D4123-82, NCHRP 1-28 

method, the SHRP P07 protocol, AASHTO TP31-96 standard, or the Roque and Buttlar 

equation 17, which accounts for the bulging effects of the specimen. (See references 28, 29, 30, 

33, and 34.) The NCHRP 1-28 elastic solutions are used in this report.

(29) The equations for 

calculating MR and Poisson’s ratio are as follows: 

1 2 ( R

P M kk

Ud

= −) ν

(20)

( )

( )

3 1

4 2

k k VU

k k VU

ν

+ = + (21)

Where:

MR = Resilient modulus (MPa).

ν = Poisson’s ratio.

P = Applied load (N).

U = Recoverable horizontal displacement (m).

V = Recoverable vertical displacement (m). 

k1, k2, k3, k4 = Constants. 

Equations 20 and 21 can be combined to yield the following relationship: 

( )

( )

3 1

1 2

4 2

R

P k k VU

M kk

Ud k k V U

  + = −  

+   (22)

These equations are based on the linear elastic solutions developed by Hondros after accounting 

for the nonuniform stress and strain distributions in the IDT specimen (see figure 42).(35) The 

constants in equations 20 and 21 are listed in table 22. Note that these constants are different 
65 

from the constants for |E*| because the coefficients for MR are derived using linear elastic theory, 

whereas those for |E*| are derived using LVE theory. 

Table 1. Geometry coefficients.

IDT |E*| β 1 β 2 γ 1 γ 2 

-0.0134 -0.0042 0.0037 0.0116

MR k3 k4 k1 k2

-0.00067 0.000209 -0.00018 0.000578

Figure 1. Illustration. Stress distribution in the IDT specimen subjected to a strip load.

For linear elastic materials, the stress-strain relationship is represented by the generalized 

Hooke’s law. It is assumed that the rectangular coordinate of x1 is in the horizontal direction, x2

is in the vertical direction, and x3 is in the depth direction of the IDT specimen. Two strains that 

are of interest are ε11 and ε22. According to the generalized Hooke’s law, these two strains are 

related to stresses as follows:

11 11 22 11 22

1 ( )( ) D

E

ε σ νσ σ νσ = −= − (23)

22 22 11 22 11

1 ( )( ) D

E

ε σ νσ σ νσ = −= − (24)

Where:

E = Young’s modulus. 

D = Compliance. 
66 

Application of the elastic-viscoelastic correspondence principle to these linear elastic solutions 

results in the following LVE stress-strain relationships: 

11 11

0

( )[ ( )]

t

ε τ σ ν τσ τ D t t d

τ

22

∂ = − −− ∂ ∫ (25)

22 22 11

0

( ) [ ( )]

t

ε τ σ ν τσ τ D t t d

τ

∂ = − −− ∂ ∫ (26)

The Hondros equations used to calculate the resulting stresses in the horizontal and vertical 

directions are as follows: 

11

2 [ ( ) ( )] P f x gx ad

σ

π = − (27)

22

2 [ ( ) ( )] P f x gx ad

σ

π

=− +

(28) 

2 2

2 2 4 4

(1 / )sin 2 ( ) 1 (2 / )cos 2 /x R f x xR xRα

α

− = + + (29)

2 2

1

2 2

1 / ( ) tan tan

1 /

x R

g x x R

α −   − =     + (30) 

Where:

R = Radius of specimen (m).

x = Horizontal distance from center of specimen (m). 

α = Radial angle (radians).

Combining equation 27 through equation 30 results in the following expressions for the 

horizontal and vertical strains: 

11 1

0

2 ( , ) ( ) {[ ( ) ( )] ( )[ ( ) ( )]}

t P x t Dt f x g x t f x g x d ad

ε τ ν τ τ

π τ

∂

= − − +− +

∂ ∫ (31)

22 2

0

2 ( , ) ( ) {[ ( ) ( )] ( )[ ( ) ( )]}

t P x t Dt f x g x t f x g x d ad

ε τ ν τ τ

π τ

∂

=− − + + − −

∂ ∫ (32) 
67 

The displacements at the gauge length can be calculated by integrating the nonuniform strains in 

equations 31 and 32 along the gauge length (i.e., -l to +l) as follows: 

11 1 ( ,)

l

l

U x t dx ε

−

= ∫ (33)

22 2 ( ,)

l

l

V x t dx ε

−

= ∫ (34)

or 

0

2 ( ) [ ( ) ( )] ( ) [ ( ) ( )]

t l l

l l

P U D t f x g x dx t f x g x dx d ad

τ ν τ τ

π τ − −

∂  

= − − +− +   ∂   ∫∫ ∫ (35)

0

2 ( ) [ ( ) ( )] ( ) [ ( ) ( )]

t l l

l l

P V D t f x g x dx t g x f x dx d ad

τ ν τ τ

π τ − −

∂  

= − − + +− −   ∂   ∫∫ ∫ (36)

For the gauge length of 2.0 inches (50.8 mm), equations 35 and 36 reduce to the following: 

1 2

0

2 ( ) { ( )}

t P U Dt t d

ad τ γ ν τγ τ

π τ

∂

= − +−

∂ ∫ (37)

1 2

0

2 ( ) { ( )}

t P V Dt t d

ad τ β ν τβ τ

π τ

∂

= − +−

∂ ∫ (38)

Where the values of β 1, β 2, γ 1, and γ 2 are shown in table 22. Equations 37 and 38 require two 

time-dependent material properties (i.e., creep compliance and Poisson’s ratio). It has been 

proven that the creep compliance of AC can be predicted from |E*| using theoretical 

relationships.(36) In this study, the following approach is used to convert the |E*| mastercurve 

to the creep compliance. 

The complex modulus, E*, is represented in equation 39 as follows: 

E E iE *'" = + (39) 

Where: 

E' = Storage modulus = |E*| cosine φ . 

E" = Loss modulus = |E*| sine φ . 

φ = Phase angle.
68 

The storage modulus, E’, can be represented in terms of a Prony series in equation 40 as follows: 

2 2

2 2

1

'( ) 1

n

r ii

r

i r i

E E E ω ρ ω

ω ρ ∞

=

= +

+ ∑ (40) 

Where: 

E∞ = Elastic modulus (MPa). 

ωr = Angular reduced frequency. 

Ei = Modulus of the ith Maxwell element. 

ρi = Relaxation time of the ith Maxwell element.

An exact conversion to D(t) can be obtained by solving the following equations: 

[ ] AD B { } { } = , or AkjDj = Bk (41) 

( ) ( ) (/ ) (/ ) (/ )

1

1 k j ki ki

m t i i t t

kj i j

i i j

E AE e e e τ ρ ρ ρ ρ τ

ρ τ

− − −

∞

=

=− + − ≠ − ∑ (42)

(/ )

1 1

1 k i

m m t

ki i

i i

B E Ee E E − ρ

∞ ∞

= =

  

=− + +       ∑ ∑ (43)

Where:

Akj = Matrix element in the kth row and jth column of matrix A.

Bk = Vector element in the kth row of vector B. 

E∞ = Equilibrium modulus (MPa).

Ei = Modulus of the ith Maxwell element. 

ρi = Relaxation time of the ith Maxwell element determined a priori. 

τj = Retardation time of the jth Voigt element determined a priori. 

tk = Time of interest. 

m = Number of Prony coefficients.

By solving for compliance of column j of matrix A (Dj), the Prony series representation for D(t) 

can be determined. Equation 42 does not show a solution for ρi = τ j because the error increases 

when such a case exists.

(36,37)

The second time-dependent material property in equation 37 is Poisson’s ratio. The timedependent nature of Poisson’s ratio has been reported.

(37) Poisson’s ratio values for the S12.5C 

mixture are shown in figure 43 against the reduced time to show a typical trend. 
69 

0.0

0.2

0.4

0.6

0.8

0.0000001 0.0001 0.1 100 100000

Reduced Time (s)

Poission's Ratio (

ν)

Replicate 1

Replicate 2

Replicate 4

Replicate 5

Figure 43. Graph. Poisson’s ratio versus reduced time for S12.5C. 

The Poisson’s ratio values are calculated from equation 21 using the data from the IDT |E*| test. 

As can be seen in figure 43, a large sample-to-sample variability is evident in Poisson’s ratio 

values. Another problem with Poisson’s ratio is that equation 25 is difficult to solve because two 

time-dependent properties are inherent in the equation. If Poisson’s ratio in equations 25 and 26 

is assumed to be constant, the two equations reduce to the following: 

1 2

0

2( ) ( )

t P U Dt d

ad

γ νγ τ τ

π τ

+ ∂ = − ∂ ∫ (44)

1 2

0

2( ) ( )

t P V Dt d

ad

β νβ τ τ

π τ

+ ∂ = − ∂ ∫ (45)

In figure 43, the measured Poisson’s ratio exceeds the theoretical limits of zero to 0.5. A high 

Poisson’s ratio occurs at large reduced times when the temperature is high and/or the loading 

time is long. Some Poisson’s ratios are higher than 0.5, indicating that damage has occurred in 

the specimen.

(28,38) The biggest challenge in determining Poisson’s ratio in the IDT test is to 

induce a large enough horizontal displacement to overcome the electronic noise in the testing 

system without causing damage in the specimen. Mirza et al. provide a method to evaluate the 

quality of the data using a deflection ratio.(39)

The effect on MR when a constant Poisson’s ratio is assumed is evaluated by using three 

Poisson’s ratios at each temperature of each mixture and calculating the percentage of difference 

in the predicted MR values. The Poisson’s ratio values used in this comparison are 0.15, 0.2, and 

0.25 for 41 oF (5 °C); 0.25, 0.3, and 0.35 for 77 oF (25 °C); and 0.4, 0.45, and 0.5 for 104 oF 

(40 °C). It was found that the difference in MR values is negligible when Poisson’s ratio, which 

is used to predict the displacements, is used to calculate MR. A more indepth analysis of 

equations 20 and 37 along with the geometry coefficients in these equations, which are shown in 
70 

table 22 (i.e., k1, k2, γ 1, and γ 2), reveals that the effect of change in Poisson’s ratio on the term 

(k1-k2ν) in the numerator of equation 20 is about the same as that on the horizontal displacement 

calculated from equation 37, which appears in the denominator of equation 20. Therefore, it 

is concluded that the effect of Poisson’s ratio on MR calculation is minimal. Vinson draws a 

similar conclusion that a Poisson’s ratio of 0.15 to 0.45 does not affect MR much based on a 

theoretical finite-element analysis.

(40,28) For the remainder of this report, equation 44 is used, 

with constant Poisson’s ratios of 0.2, 0.35, and 0.45 for 41 oF (5 °C), 77 oF (25 °C), and 104 oF 

(40 °C), respectively.

Because the numerical integration of equation 44 requires calculating all the previous time steps 

to arrive at the current time step, calculation times can grow exponentially.

(41) To reduce the 

calculation time, the state variable approach described in equation 46 through equation 48 is used 

in this study. 

1

1 1 1 1

0 1

( )

n

t n n n n n

o i

i

d Dt d

dt

σ

ε τ τη η

+

+ + + +

=

= − =− ∫ ∑

(46)

Where:

1 1

1

( )( ) n o m n t t

ooi

i

η D D σ σ + +

=

=+ − ∑ (47)

1 1 2 [ ] n n

t t

n n tt

i i i ee D ο ο τ τ η η σσ +

∆ ∆ − − + =+ − (48) 

Figure 44 through figure 47 show the results of predicted versus measured values using a LOE

graph. For both mixtures, the predicted and measured MR values are in good agreement, 

suggesting that the proposed approach based on the theory of linear viscoelasticity using the IDT 

|E*| can provide a reasonable estimate of MR of AC. 
71 

0

10000

20000

30000

0 10000 20000 30000

Measured MR (MPa)

Predicted M

R (MPa)

1 psi = 6.86 kPa

Figure 44. Graph. Comparison of predicted and measured MR values for S12.5C mixture. 

0

10000

20000

30000

0 10000 20000 30000

Measured MR (MPa)

Predicted M

R (MPa)

1 psi = 6.86 kPa

Figure 45. Graph. Comparison of predicted and measured MR values for S12.5CM mixture. 
72 

0

10000

20000

30000

0 10000 20000 30000

Measured MR (MPa)

Predicted M

R (MPa)

1 psi = 6.86 kPa

Figure 46. Graph. Comparison of predicted and measured MR values for S12.5FE mixture. 

0

10000

20000

30000

0 10000 20000 30000

Measured MR (MPa)

Predicted M

R (MPa)

1 psi = 6.86 kPa

Figure 47. Graph. Comparison of predicted and measured MR values for B25.0C mixture. 

5.1.2 ANN Model |E*| Backcalculation 

The characterization and verification results of the MR ANN model are shown in figure 48

through figure 51. In figure 49, the results of characterization are shown in logarithmic space, 

and the model displays excellent fitting statistics with a high R2 = 0.98 (0.90 in arithmetic space) 

and a low Se/Sy = 0.15 (0.32 in arithmetic space). The verification plots are shown in figure 50
73 

and figure 51, and the model is found to predict the moduli values with statistics similar to those 

from the calibration dataset. 

0.E+00

2.E+04

4.E+04

6.E+04

0.E+00 2.E+04 4.E+04 6.E+04

Measured |E*| (MPa)

Predicted |E*| (MPa)

MR Model

LOE

R2=0.90 , Se/Sy=0.32

1 psi = 6.86 kPa

Figure 48. Graph. MR ANN model using 90 percent of randomly selected data as a training 

set in arithmetic scale. 

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Measured |E*| (MPa)

Predicted |E*| (MPa)

MR Model

LOE

R2=0.98 , Se/Sy=0.15

1 psi = 6.86 kPa

Figure 49. Graph. MR ANN model using 90 percent of randomly selected data as a training 

set in logarithmic scale. 
74 

0.E+00

2.E+04

4.E+04

6.E+04

0.E+00 2.E+04 4.E+04 6.E+04

Measured |E*| (MPa)

Predicted |E*| (MPa)

MR Model

LOE

R2=0.91 , Se/Sy=0.31

1 psi = 6.86 kPa

Figure 50. Graph. MR ANN model using 10 percent of randomly selected data as a 

verification set in arithmetic scale. 

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Measured |E*| (MPa)

Predicted |E*| (MPa)

MR Model

LOE

R2=0.98 , Se/Sy=0.16

1 psi = 6.86 kPa

Figure 51. Graph. MR ANN model using 10 percent of randomly selected data as a 

verification set in logarithmic scale. 
75 

5.2 VV ANN MODEL

The characterization and verification results of the VV ANN model are shown in figure 52

through figure 55 in a manner similar to that shown for the MR ANN model. Like the MR ANN 

model, the VV model shows good fitting statistics with a high R2 = 0.95 (0.93 in arithmetic 

space) and a low Se/Sy = 0.23 (0.26 in arithmetic space). The verification plots are shown in 

figure 54 and figure 55. Although the verification statistics are not as favorable as those 

found for the MR ANN model, they are still better overall than those of the existing closedform solutions. 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

VV Model

LOE

R2=0.93, Se/Sy=0.26

1 psi = 6.86 kPa

Figure 52. Graph. VV ANN model using 90 percent of randomly selected data as a training 

set in arithmetic scale. 
76 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

VV Model

LOE

R2=0.95, Se/Sy=0.23

1 psi = 6.86 kPa

Figure 53. Graph. VV ANN model using 90 percent of randomly selected data as a training 

set in logarithmic scale. 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

VV Model

LOE

R2=0.89, Se/Sy=0.34

1 psi = 6.86 kPa

Figure 54. Graph. VV ANN model using 10 percent of randomly selected data as a 

verification set in arithmetic scale. 
77 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

VV Model

LOE

R2

=0.94, Se/Sy=0.24

1 psi = 6.86 kPa

Figure 55. Graph. VV ANN model using 10 percent of randomly selected data as a 

verification set in logarithmic scale. 

5.3 GV ANN MODEL

The characterization and verification results of the GV ANN model are shown in figure 56

through figure 59. The calibration dataset for this model shows similar or slightly better fitting 

statistics than the VV ANN model, with an R2

 = 0.96 (0.90 in arithmetic space) and a Se/Sy = 

0.19 (0.32 in arithmetic space). The verification plots are shown in figure 58 and figure 59, and 

the model predictions agree favorably with the measured moduli. Care must be taken when 

visually comparing the predictability of the VV ANN and GV ANN models because each uses a 

different number of datapoints for calibration and verification (see table 21). 
78 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.0E+00 2.0E+06 4.0E+06 6.0E+06 8.0E+06

Measured |E*| (psi)

Predicted |E*| (psi)

GV Model

LOE

R2

=0.90, Se/Sy=0.32

1 psi = 6.86 kPa

Figure 56. Graph. GV ANN model using 90 percent of randomly selected data as a training 

set in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

GV Model

LOE

R2=0.96, Se/Sy=0.19

1 psi = 6.86 kPa

Figure 57. Graph. GV ANN model using 90 percent of randomly selected data as a training 

set in logarithmic scale. 
79 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

GV Model

LOE

R2

=0.89, Se/Sy=0.33

1 psi = 6.86 kPa

Figure 58. Graph. GV ANN model using 10 percent of randomly selected data as a 

verification set in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

GV Model

LOE

R2=0.96, Se/Sy=0.19

1 psi = 6.86 kPa

Figure 59. Graph. GV ANN model using 10 percent of randomly selected data as a 

verification set in logarithmic scale. 
80 

5.4 OTHER ANN MODELS

Through the course of this study, some effort was made to develop ANN models with 

gravimetric (as opposed to volumetric) input variables. Ultimately, these models were not fully 

developed nor were they used in the modeling efforts because, after a preliminary review, it was 

determined that such models would not be useful. Although it is true that some layers contained

gravimetric-based effective asphalt contents, these sections also contained enough information to 

allow the volumetric properties VMA and VFA to be computed. The most common relationship 

for the three key volumetric properties is shown in equation 49 as follows: 

1

100 a

VFA V VMA  = −     (49)

5.5 ERROR ASSESSMENT

For statistical validity, the expected error for each of the three main ANN models has been 

assessed using the following error function: 

* *

% *100

*

Measured Predicted

Measured

E E

Error

E

− = (50)

A probability distribution function for the error was computed for each model by using the 

verification dataset for the respective model. These datasets are shown in figure 50 and 

figure 51 for MR ANN, figure 54 and figure 55 for VV ANN, and figure 58 and figure 59 for 

GV ANN. The distribution function is shown for each model, including the previously discussed 

closed-form models presented in table 23. The distribution of each error function is 

approximately normal and, as such, the error values that encompass a 95 percent reliability 

interval for the percentage of error in each model can be readily computed. 

Because the number of data points used to verify each model is so large (see the last row in 

table 23), the distribution is assumed normal, and a z-score of 1.96 is used to compute the 

reliability ranges. Sy is computed after compiling the percent error values for each prediction of 

the respective model. After rounding slightly for convenience, the suggested reliability range for 

each ANN model is as follows:

• MR ANN: ±55 percent.

• VV ANN: ±75 percent.

• GV ANN: ±60 percent.

It is also found from table 23 that when the closed-form solutions are applied to a completely 

independent dataset, each solution yields a substantial number of predictions with errors 

exceeding -105 percent. The dataset used to compute these statistics is completely independent 

of the data used in calibrating any of these closed-form solutions. It also includes a mixture of 

AMPT and TP-62 measured moduli.(8) The high number of large negative error observations 
81 

suggests a bias towards over-prediction of the |E*| values with each model. The bias in 

predictions can be clearly observed by examining figure 8 for the modified Witczak model and 

figure 10 for the Hirsch model. A comparison of figure 7 and figure 9 shows a tendency of the 

Hirsch model to underpredict |E*| at higher modulus values more so than the modified Witczak 

model. This trend is also captured in table 23, where the error distribution for the Hirsch model is 

skewed towards positive values. As a result of the existence of these prediction errors, the 

reliability ranges for the closed-form solutions are much larger than the ranges observed for the 

ANN models as follows: 

• Original Witczak: ±200 percent.

• Modified Witczak: ±175 percent.

• Hirsch: ±125 percent.

Table 23. Probability distribution of error functions of each ANN model. 

Percentage 

Error Range

ANN Model (Percentage Within Range)

MR VV GV

Original 

Witczak

Modified 

Witczak Hirsch

< -105 0.75 2.31 0.49 22.04 31.12 10.25

-105 to -90 0.32 0.86 0.77 2.86 5.11 2.65

-90 to -75 0.43 1.91 1.05 3.15 8.43 3.14

-75 to -60 0.21 2.44 2.02 3.72 8.97 4.46

-60 to -45 0.53 4.75 3.42 7.06 10.49 5.86

-45 to -30 5.98 7.59 8.02 8.30 11.39 8.58

-30 to -15 15.06 14.32 16.11 9.83 9.24 12.34

-15 to 0 24.89 17.56 21.34 10.11 7.89 14.57

0 to 15 28.53 19.14 21.48 12.02 4.48 13.81

15 to 30 19.02 15.12 15.69 13.65 2.06 10.11

30 to 45 3.74 8.65 6.28 6.20 0.54 7.67

45 to 60 0.32 4.16 2.58 1.05 0.27 4.81

60 to 75 0.21 1.06 0.63 0.00 0.00 1.39

75 to 90 0.00 0.13 0.14 0.00 0.00 0.35

> 105 0.00 0.00 0.00 0.00 0.00 0.00

Number of data 

points used 936 1,515 1,434 1,048 1,115 1,434

83 

6.0 PREDICTION OF |E*| FOR LTPP LAYERS

6.1 MODEL PRIORITIZATION AND DECISION TREE DEVELOPMENT

The developed ANN models were ranked based on the performance of each model for an 

independent dataset that contains all of the required input parameters to be used in the different 

models. The decision tree in figure 60 is the result of this prioritization and was used to populate 

the LTPP database. As suspected at the outset of this project, the MR ANN model yields the best 

prediction possible and is the preferred model. This finding is not surprising given that the 

MR ANN model takes actual mixture measured quantities as input. The intuition is also supported 

in the fitting statistics shown in the figures presented in section 5.0. If MR data are not available, 

the next preferred model is the VV ANN model. Although the VV ANN shows poorer fitting 

statistics than the GV ANN model in terms of logarithmic space, it shows better statistics in 

terms of arithmetic space. The arithmetic statistics were used to rank these models because the

moduli values are expressed and utilized in terms of their arithmetic value. Of the three ANN 

models, GV ANN received the lowest ranking. However, GV-PAR outranked the VV-grade 

because the model GV-PAR uses measurements taken directly from the material of interest. The 

VV-grade model uses viscosity values representative of the binder grade. 

Resilient Modulus Resilient Modulus

Viscosity and Volumetric 

Properties

Viscosity and Volumetric 

Properties

No Yes

Y

N Y

Prediction Models Prediction Models

Available Information and Temperature 

and Frequency of Interest

Available Information and Temperature 

and Frequency of Interest

|G*| and Volumetric Properties 

at Consistent Age

|G*| and Volumetric Properties 

at Consistent Age

Data Processing and 

Surrogate Models

Data Processing and 

Surrogate Models

t-T Shift 

Factor

VV MR

A-VTS

GV-PAR No Prediction 

Possible

t-T Shift 

Factor

PAR 

Model

Isochronal 

Analysis

Binder Grade and Volumetric 

Properties

Binder Grade and Volumetric 

Properties N Y

M-E PDG 

Values

VVGrade

|G*| and Volumetric Properties 

at Inconsistent Age

|G*| and Volumetric Properties 

at Inconsistent Age

N Y

GV

t-T Shift 

Factor

Isochronal 

Analysis

N

Resilient Modulus Resilient Modulus

Viscosity and Volumetric 

Properties

Viscosity and Volumetric 

Properties

No Yes

Y

N Y

Prediction Models Prediction Models

Available Information and Temperature 

and Frequency of Interest

Available Information and Temperature 

and Frequency of Interest

|G*| and Volumetric Properties 

at Consistent Age

|G*| and Volumetric Properties 

at Consistent Age

Data Processing and 

Surrogate Models

Data Processing and 

Surrogate Models

t-T Shift 

Factor

VV MR

A-VTS

GV-PAR No Prediction 

Possible

t-T Shift 

Factor

PAR 

Model

Isochronal 

Analysis

t-T Shift 

Factor

PAR 

Model

Isochronal 

Analysis

Binder Grade and Volumetric 

Properties

Binder Grade and Volumetric 

Properties N Y

M-E PDG 

Values

VVGrade

|G*| and Volumetric Properties 

at Inconsistent Age

|G*| and Volumetric Properties 

at Inconsistent Age

N Y

GV

t-T Shift 

Factor

Isochronal 

Analysis

N

Figure 60. Illustration. Decision tree applied to population of LTPP database based on 

ranking of ANN models.
84 

The input sets available for each section are different. Consequently, the models that can be 

utilized for a given section may differ. All the models that can be used for a specific section have 

been identified. If any of the inputs are outside the range of the calibration data, then the model 

for that section is considered to be a violated model, and predictions with it may or may not be 

reasonable. Any available model that does not violate the input range criteria is used as the 

predictive model for that section. In cases where only two violated models are available, the 

model for that section must be decided manually by the user. 

6.2 DATA QC 

Two sets of internal QC checks were applied to the |E*| data produced for submission to the 

LTPP database. The first QC check was used on the input values. All the other QCs were used to 

check the output |E*| predictions. In the following sections, each of these QC criteria is 

described. The work criterion is implied but not explicitly stated for each of these checks. The 

quality grading system referenced is different than the standard record status definition used in 

the LTPP database. For example, data assigned an “A” by the research team represent the highest 

quality data, whereas the LTPP convention assigns an “E” to the highest quality data. The 

research team established strict QC checks for these data to ensure that only the highest quality 

data were assigned an “A” grade. The data that did not achieve an “A” grade should be used with 

caution, and users should be fully aware that the data did not pass the QC check. All predictions 

are included in the database so users can determine whether or not the data are suitable for their 

needs. In addition, FHWA can revise the criteria used in the QC checks as deemed appropriate 

based on the opinions of their experts. 

6.2.1 QC #1 

QC #1 checks for a violation of the input range used in each model based on the input ranges of 

the calibration dataset, as seen in (see table 24). Figure 61 through figure 65 show example 

predictions of |E*| in two sections where violations of the input criteria have been detected. This 

check is performed on a line-by-line basis, meaning that each time a modulus is predicted, the 

input parameters are checked. Lines that pass the QC check receive a grade of “A,” whereas

those that do not pass the QC check receive a grade of “F.” 

For the layers where |E*| predictions are estimated using the MR ANN model but QC #1 is 

violated, the mastercurve created appears visually continuous (see figure 61 and figure 62). The 

reason for this appearance is that the output of this model is the mastercurve itself. However, the 

output of the other two ANN models is the estimation of |E*| under a specific condition. In this 

case, a violation of QC #1 may create a clear error in the mastercurve, as demonstrated in 

figure 63 through figure 65. The acceptable range of input parameters is shown for the three 

ANN models in table 24. These models do not estimate the information needed to create 

the mastercurve. 
85 

Table 24. Input ranges of different ANN models. 

Model Range

Parameter

MR

5 °C 

(GPa)

MR

25 °C 

(GPa)

MR

40 °C 

(GPa)

fR

(Hz)

Viscosity 

(109 P)

VMA 

(percent)

VFA 

(percent)

|G*| 

(psi)

Log(|E*|) 

(psi)

MR Min 4.8003 1.081 0.3789

Max 34.053 15.411 6.8637

VVgrade1

Min 0.01 1.99E-06 9.51 32.82 3.52

Max 25 2.70E+01 34.64 95.07 6.82

GVPAR1

Min 9.51 32.82 2.93E-02 3.52

Max 22.21 95.07 6.76E+05 6.81

°C = (°F-32)/1.8

1 Pas = 10 P

1 psi = 6.86 kPa

1

Indicates that VV-grade and GV-PAR have the same range of input.

Note: Blank cells indicate that the input parameter is not required in the model.

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

1.E-08 1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 61. Graph. Example of the effect of a violation of QC #1 for MR ANN model in 

semi-log scale. 
86 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E-08 1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 62. Graph. Example of the effect of a violation of QC #1 for MR ANN model in 

log-log scale. 

0.E+00

1.E+06

2.E+06

3.E+06

1.E-05 1.E-02 1.E+01 1.E+04 1.E+07 1.E+10

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 63. Graph. Example of the effect of a violation of QC #1 for VV ANN model in 

semi-log scale. 
87 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E-05 1.E-02 1.E+01 1.E+04 1.E+07 1.E+10

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 64. Graph. Example of the effect of a violation of QC #1 for VV ANN model in 

log-log scale. 

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

0.01 0.1 1 10 100

Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 65. Graph. Example of the effect of a violation of QC #1 for VV ANN model 

unshifted data. 
88 

6.2.2 QC #2 

QC #2 checks the trends of |E*| as a function of temperature and frequency. It is expected that 

|E*| decreases as the temperature increases and the loading frequency decreases. Figure 66

through figure 68 show sample cases of a QC #2 violation. In these examples, the predicted |E*| 

value at 40 °F (4.4 °C) and 25 Hz is smaller than the predicted value at 40 °F (4.4 °C) and 10 Hz. 

Similar to QC #1, this QC check is performed line-by-line; the lines that pass receive a grade of 

“A,” and those that fail receive a grade of “F.”

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

5.E+06

1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 66. Graph. Example of the effect of a violation of QC #2 in semi-log scale. 
89 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 67. Graph. Example of the effect of a violation of QC #2 in log-log scale. 

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

0.01 0.1 1 10 100

Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 68. Graph. Example of the effect of a violation of QC #2 unshifted data. 

6.2.3 QC #3

The typical percentage of difference between 0.1 Hz at one temperature and 25 Hz at the next 

warmest temperature is checked using QC #3. Equation 51 states the percentage of difference 

and a typical value of this term based on the available data in this study. Figure 69 through 
90 

figure 71 show the violation of QC #3 in one of the sections. For this particular section, a 

violation occurred between 40 and 70 °F (4.4 and 21.1 oC). For this sample case, at 0.1 Hz 

and 40 °F (4.4 oC), the modulus is equal to 1.29 x 106

 psi (8.85 x 106 kPa). At 25 Hz and 

70 °F (21.1 oC), the modulus is equal to 8.28 x 105 psi (56.8 x 105 kPa), which is a percentage 

difference of 55.4 percent. 

HigherTemperature Hz

LowerTemperature Hz HigherTemperature Hz % Difference

25@

1.0@ − 25@ = 

(51)

• From 14 to 40 °F (-10 to 4.4 oC): +25 percent, -25 percent.

• From 40 to 70 °F (4.4 to 21.1 oC): +50 percent, -75 percent.

• From 70 to 100 °F (21.1 to 37.7 oC): +50 percent, -75 percent.

• From 100 to 130 °F (37.7 to 54.4 oC): +50 percent, -75 percent.

Unlike the previous QC checks, QC #3 is performed on the basis of temperature. For example, if 

the percentage of difference between 14 and 40 °F (-10 and 4.4 °C) exceeds 25 percent, then all 

of the predictions at 14 °F (-10 oC) would receive a grade of “F.” 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 69. Graph. Example of the effect of a violation of QC #3 in semi-log scale. 
91 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 70. Graph. Example of the effect of a violation of QC #3 in log-log scale. 

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

0.01 0.1 1 10 100

Frequency (Hz)

| E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 71. Graph. Example of the effect of a violation of QC #3 unshifted data. 

6.2.4 QC #4 

in QC #4, the difference in |E*| values predicted between 0.1 Hz at one temperature and 0.1 Hz 

at the next warmest temperature is checked to see if appropriate trends with regard to 

temperature and modulus hold. Sample cases of QC #4 violation are shown in figure 72 through 
92 

figure 74. For the data in these figures, |E*| at 0.1 Hz and 40 °F (4.4 oC) is smaller than |E*| at 

0.1 Hz and 70 °F (21.1 oC). In the case shown in these figures, this situation has led to a 

discontinuous mastercurve because the optimization algorithm becomes confused when the 

modulus does not decrease as the temperature increases. Similar to QC #3, QC #4 is applied on a 

temperature basis.

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

5.E+06

1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 72. Graph. Example of the effect of a violation of QC #4 in semi-log scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 73. Graph. Example of the effect of a violation of QC #4 in log-log scale. 
93 

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

0.01 0.1 1 10 100

Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 74. Graph. Example of the effect of a violation of QC #4 unshifted data. 

6.2.5 QC #5 

QC #5 checks the range of shift factor values in the output predictions against typical values. 

This check is used to indicate potential problems with the mastercurve generation process. The 

limits identified and subsequently used are as follows: 

• At 14 °F (-10 oC): 3 < log(aT) < 7 

• At 130 °F (54.4 oC): -5 < log(aT) < -2 

The output parameters of interest from the ANNACAP software are SHIFT_FACTOR_

COEFFICIENT 1 (α1), SHIFT_FACTOR_COEFFICIENT 2 (α2), and SHIFT_FACTOR_ 

COEFFICIENT 3 (α3). The shift factor is computed using equation 52 as follows: 

( ) 2

1 23 log T a TT = ++ α αα (52) 

Because the goal of this QC criterion is to judge the mastercurve generation process, only shift 

factor values at extreme temperatures need to be examined. Note that because the MR ANN 

model predicts the mastercurve directly, any predictions made with this model will automatically 

pass QC #5 without the need for calculation. When this QC is not passed, the entire section (at 

all temperatures) receives a grade of “F.” 

Figure 75 through figure 77 show violations of QC #5. In these cases, the shift factor at 14 °F 

(-10 oC) is equal to 7.03. This effect results in a visual discontinuity in the mastercurve between 

the 14- and 40-°F (-10- and 4.4-oC) datasets. 
94 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

1.E-06 1.E-03 1.E+00 1.E+03 1.E+06 1.E+09

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 75. Graph. Example of the effect of a violation of QC #5 in semi-log scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E-06 1.E-03 1.E+00 1.E+03 1.E+06 1.E+09

Reduced Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 76. Graph. Example of the effect of a violation of QC #5 in log-log scale. 
95 

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

0.01 0.1 1 10 100

Frequency (Hz)

|E*| (psi)

T = 14F

T = 40F

T = 70F

T = 100F

T = 130F

1 psi = 6.86 kPa

Figure 77. Graph. Example of the effect of a violation of QC #5 unshifted data. 

6.2.6 QC #6 

In an earlier version of the ANN models developed for this study, some cases had inputs within 

the calibration input range, but the |E*| predictions obtained values of the limits of the calibration 

data. In some other cases, |E*| did not change with varying temperature and frequency. QC #6 

was expected to be the last QC check, but after improving the ANN models, the problem no 

longer exists. Nevertheless, the check is performed on the data. The limiting |E*| values used to 

judge whether a given prediction passes this QC check are given in table 25. Like QC #1 and 

QC #2, this QC check is applied line-by-line.

Table 25. Limiting |E*| values used for QC #6. 

Model

Upper 

Limit (psi)

Lower 

Limit (psi)

MR ANN N/A N/A

VV ANN 5,888,437 3,311.311

GV ANN 6,456,542 3,311.311

1 psi = 6.86 kPa

N/A indicates that data are not available.

6.2.7 QC #7 

Similar to QC #5, QC #7 is designed to judge the mastercurve generation process. Each time a 

mastercurve is generated, fitting statistics are computed. These statistics include the Se/Sy (of 

combined predictions), and R2

. According to the draft standard for mastercurve generation using 

AASHTO TP-62 data, the explained variance should be greater than 0.99, and the ratio of Se/Sy

should be less than 0.05.(42) When these limits are exceeded, QC #7 is triggered, and the layer 
96 

fails. Because the MR ANN model predicts the mastercurve directly, any predictions made with 

this model will automatically pass QC #7 without the need for calculation. Equations 53–55 are 

used to compute the necessary statistical parameters. Whenever this QC is violated, the entire 

section receives a grade of “F.” 

( ) ( ) 2

1

1 log * log * 23

n

e ANN fit i i

S EE   = − ∑    (53)

Where:

23 = Number of temperature/frequency combinations used minus the number of 

fitting parameters minus 1. 

(log|E*|ANN)i = Logarithm of the modulus determined from the ANN models at a particular 

temperature frequency combination. 

(log|E*|fit)i = Logarithm of the modulus determined from the optimized sigmoidal fit.

( ) ( ) 2

1

1 log * log * 29

n

y ANN i avg S EE =   − ∑  (54) 

Where:

29 = Number of temperature/frequency combinations used minus 1. 

(log|E*|avg)i = Logarithm of the average modulus determined from the ANN models for a 

given layer. 

2

2

2

(23) 1

(29)

e

y

S

R

S = − (55)

6.3 CONTRACTOR JUDGMENT

The NCSU researchers used a combination of the above QC factors to judge the quality of the 

modulus prediction for an individual modulus prediction. If QCs #1, #3, #4, and #6 all pass, then 

the section receives a grade of “A.” If QC #1 fails but QCs #3, #4, and #6 all pass, then the 

modulus receives a grade of “C,” which means the predicted moduli are questionable. Otherwise, 

the modulus receives a grade of “F.” 

6.4 CONTRACTOR MASTERCURVE JUDGMENT

QCs #5 and #7 are used to judge the quality of the calibrated mastercurve and shift factor 

function. When both QCs pass, the section receives a grade of “A,” but when both fail, the 

section receives a grade of “F.” If one passes and one fails, then the mastercurve and shift factor 

function values are questionable, and the section receives a grade of “C.” 

6.5 PREDICTION STATISTICS 

The LTPP database contains information for a total of 1,806 layers that meet the criteria 

described in section 3.4 of this report. These layers have binder data available at a combination 
97 

of different aging conditions including unaged or original-, RTFO-, PAV-, or field-aged. In the 

field-aged data, 2,223 records are available because some layers’ properties may have been 

measured at different dates. The total resulting number of records is 7,641. Using the combined 

ANN models and requisite QC checks, modulus values were predicted for 363 records/layers in 

the original-aged level, 469 records/layers in the RTFO-aged level, 1 record/layer in the 

PAV level, and 503 records in the field-aged level. These numbers translate to predictions for 

17.5 percent of the total number of records available. However, these records are distributed in 

such a way that a higher percentage of the layers have some sort of valid prediction. Of the 

1,806 layers in the database, 1,010, or 56 percent, have a modulus prediction at some aging 

condition. Of these 1,010 layers, 615, or 34 percent of the total 1,806 layers, have completely 

reasonable predictions (i.e., an “A” grade), and 89, or 4.9 percent of the total 1,806 layers, have 

unreasonable predictions (i.e., an “F” grade). The remaining 306 layers (17 percent of the 

1,806 layers) have questionable predictions (i.e., a “C” grade). Thus, the total percentage of 

layers with a completely valid or questionable prediction is 51 percent. Table 26 shows the 

summarized statistics of the population effort. Although it cannot be interpreted directly from 

this table, the majority of the valid records were populated using the MR ANN model followed 

next by the VV-grade ANN model and the VV ANN model. 

Table 26. Statistics of LTPP data populated with |E*|. 

Populated LTPP data

Aging Condition

Original RTFO PAV Field Total

Number of records 1,806 1,806 1,806 2,223 7,641

Number of populated records 363 469 1 503 1,336

Number of records by contractor 

individual judgment grade

Grade A 147 252 0 465 864

Grade C 0 1 1 38 40

Grade F 216 216 0 0 432

Number of records by contractor 

mastercurve judgment grade

Grade A 44 142 0 0 186

Grade C 211 237 0 503 951

Grade F 108 90 1 0 199

Populated records using MR 5 0 0 503 508

Populated records using VV 358 59 0 0 417

Populated records using GV 0 0 1 0 1

Populated records using GV-PAR 0 2 0 0 2

Populated records using VV-grade 0 408 0 0 408

Number of populated layers 1,010

Number of layers with valid and questionable predictions 921

Number of layers with fully valid predictions 615

Number of layers with failed predictions 89

6.6 LTPP DATABASE ADDITIONS

As a result of this project, nine tables were included in the LTPP database to document the inputs 

used in the dynamic modulus models as well as resultant predictions.(20) The tables were added to 

the materials testing module (i.e., TST) of the LTPP database. Each table is described below. It 

should be noted that inputs from the LTPP database as well as outputs from the models are in 
98 

units that are consistent with popular convention, but both SI and English units are used in 

the tables. 

• TST_ESTAR_MASTER: Identification information for all records in the ESTAR tables 

(see table 27 for format).

• TST_ESTAR_MODULUS: |E*| predictions from the models described in the report 

(see table 28 for format).

• TST_ESTAR_MODULUS_COEFF: Sigmodial and shift factor information generated 

from the |E*| models (see table 29 for format).

• TST_ESTAR_GSTAR_INPUT: Binder shear modulus information used as inputs into 

the GV ANN model (see table 30 for format).

• TST_ESTAR_GSTAR_CAM_COEFF: CAM coefficients used as inputs into the 

GV ANN model (see table 31 for format).

• TST_ESTAR_VOLUM_INPUT: Volumetric properties used as inputs into the 

VV ANN model (see table 32 for format).

• TST_ESTAR_VISC_INPUT: Binder viscosity information used as inputs into the 

VV ANN model (see table 33 for format).

• TST_ESTAR_VISC_MODEL_COEFF: A and VTS data generated from viscosity 

inputs for use in the VV ANN model (see table 34 for format).

• TST_ESTAR_MR_INPUT: MR data used as inputs into the MR ANN model (see table 

35 for format).
99 

Table 27. Structure and format of the TST_ESTAR_MASTER table. 

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data

STATE_CODE

Numerical code for State or province. U.S. codes are 

consistent with Federal Information Processing 

Standards

SHRP_ID

Test section identification number assigned by LTPP 

program. Must be combined with STATE_CODE to 

be unique

LAYER_NO

Unique sequential number assigned to pavement 

layers starting with layer 1 as the deepest layer 

(subgrade)

PROJECT_ID

The four-character identifier for the SPS project used 

to identify elements of information which are 

common to all sections in that project

PROJECT_LAYER_CODE

Sequential alphabetic code assigned to identify 

group project-wide layers

PREDICTIVE_MODEL

Code indicating the predictive model used to 

generate ESTAR estimates

CONSTRUCTION_DATE Construction date of the layer

SAMPLE_TYPE_ESTAR

Code indicating the aging condition of the samples 

used for inputs into the ESTAR predictive models

SAMPLE_DATE Sampling date if field-aged

SAMPLE_AGE Sample age if field-aged

RECORD_STATUS

Code indicating the general quality of the data as 

outlined, based on the level of QC checks described 

in the Database User Reference Guide(10)

Table 28. Structure and format of the TST_ESTAR_MODULUS table. 

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data

TEMPERATURE Temperature of modulus prediction

FREQUENCY Frequency of modulus prediction

ESTAR Predicted dynamic modulus

RECORD_STATUS

Code indicating the general quality of the data as 

outlined, based on the level of QC checks described 

in the Database User Reference Guide(10)
100 

Table 29. Structure and format of the TST_ESTAR_MODULUS_COEFF table. 

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data

SIGMOIDAL_COEFF_1 Sigmoidal fitting function coefficient delta

SIGMOIDAL_COEFF_2 Sigmoidal fitting function coefficient alpha

SIGMOIDAL_COEFF_3 Sigmoidal fitting function coefficient beta

SIGMOIDAL_COEFF_4 Sigmoidal fitting function coefficient gamma

SHIFT_FACTOR_COEFF_1 Shift factor fitting function coefficient alpha 1

SHIFT_FACTOR_COEFF_2 Shift factor fitting function coefficient alpha 2

SHIFT_FACTOR_COEFF_3 Shift factor fitting function coefficient alpha 3

MASTERCURVE_QUALITY

Code indicating the general quality of the 

mastercurve generation process. Pass if explained 

variance is greater than 0.99 and ratio of standard 

error to standard deviation is less than 0.05

RECORD_STATUS

Code indicating the general quality of the data as 

outlined based on the level of QC checks described 

in the Database User Reference Guide(10)

Table 30. Structure and format of the TST_ESTAR_GSTAR_INPUT table. 

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data

TEMPERATURE Temperature of modulus prediction

FREQUENCY Frequency of modulus prediction

GSTAR Binder shear modulus used for G* ANN model

RECORD_STATUS

Code indicating the general quality of the data as 

outlined based on the level of QC checks described 

in the Database User Reference Guide(10)

 

Table 31. Structure and format of the TST_ESTAR_GSTAR_CAM_COEFF table. 

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data.

CAM_COEFF_1 CAM fitting function coefficient Gg

CAM_COEFF_2 CAM fitting function coefficient wc

CAM_COEFF_3 CAM fitting function coefficient k

CAM_COEFF_4 CAM fitting function coefficient me

RECORD_STATUS

Code indicating the general quality of the data as 

outlined based on the level of QC checks described 

in the Database User Reference Guide(10)
101 

Table 32. Structure and format of the TST_ESTAR_VOLUM_INPUT table. 

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data

VMA

Voids in mineral aggregate as a percent of total 

volume

VFA Voids filled with asphalt as a percent of VMA

RECORD_STATUS

Code indicating the general quality of the data as 

outlined, based on the level of QC checks described 

in the Database User Reference Guide(10)

Table 33. Structure and format of the TST_ESTAR_VISC_INPUT table.

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data

TEMPERATURE Temperature of modulus prediction

VISCOSITY Voids filled with asphalt as a percent of VMA

RECORD_STATUS

Code indicating the general quality of the data as 

outlined, based on the level of QC checks described 

in the Database User Reference Guide(10)

Table 34. Structure and format of the TST_ESTAR_VISC_MODEL_COEFF table.

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data

VISC_A Viscosity model intercept

VISC_VTS Viscosity model slope

RECORD_STATUS

Code indicating the general quality of the data as 

outlined, based on the level of QC checks described 

in the Database User Reference Guide(10)

Table 35. Structure and format of the TST_ESTAR_MR_INPUT table.

Field Name Field Description

ESTAR_LINK Generic key for linking ESTAR data

MR_5C Resilient modulus at 5 °C

MR_25C Resilient modulus at 25 °C

MR_40C Resilient modulus at 40 °C

RECORD_STATUS

Code indicating the general quality of the data as 

outlined, based on the level of QC checks described 

in the Database User Reference Guide(10)

°C = (°F-32)/1.8

QC checks were developed to be applied to the LTPP database. These checks include many 

of the internal checks developed as part of this study as well as additional checks deemed 

appropriate. As such, the data available in the LTPP database have been subjected to both the 

research team’s internal QC checks and the LTPP database QC checks. Figure 78 shows the 

ANN models and their appropriate input and output tables. 
102 

Figure 78. Illustration. ANN models and their appropriate input and output tables.

 
103 

7.0 SUMMARY AND FUTURE WORK

Existing predictive equations, including the original Witczak equation (see equation 1), the 

modified Witczak equation (NCHRP 1-40D), the Hirsch model, and the law of mixtures parallel 

model were evaluated for accuracy and potential bias. (See references 2, 5, 6, and 7.) For 

fairness to each model, a database consisting of hundreds of mixtures and binders and thousands 

of single data points from projects across the United States was compiled and used in the 

evaluation. This effort showed that although each model has certain benefits, no single model is 

capable of highly accurate predictions over the complete range of necessary conditions. 

Furthermore, none of these predictive models could predict the |E*| values when only the MR was 

available. This final criterion is crucial because more layers had available MR values than the 

information needed for the existing predictive models. Multiple models are necessary because 

the layers in the LTPP database are not uniform in their population of material properties. As a 

result, the research team developed ANN models that, through a pilot study, were shown to yield 

reasonable and accurate predictions for the complete range of conditions needed. This 

hierarchical approach thus allows a more complete population of |E*| values. 

In the end, three ANN models were developed. Each model differed in the required input 

parameters. The most accurate ANN model was found to utilize MR as its primary input 

parameter. The other two models use mixture volumetric properties as well as a binder property 

as input variables. The VV ANN model uses the binder viscosity and input frequency, whereas 

the GV ANN model uses the binder |G*| property. These models were extended to include 

conditions where perfect input values were not available, such as when |G*| had been measured 

at warm temperatures for the RTFO-aged binder and measured at intermediate temperatures for 

the PAV binder or when only the binder grade was available. Statistical analysis and engineering 

judgment were utilized to rank the predictive models, with the MR ANN model being the best, 

the VV ANN model being the second best, and the GV ANN model being the third best. 

Imperfect input conditions were also ranked below these three models. 

The individual ANN models developed for this project have practical implications beyond the 

current study. The most direct use of these ANN models is the prediction of |E*| values for

MEPDG or other structural/performance analysis of AC pavements. They may be used in the 

same way that existing closed-form solutions are used. The advantage of using ANN models for 

this purpose is their improved accuracy when compared to existing closed-form solutions. The 

MR ANN model developed in this project is the only available method for predicting |E*| over 

the range of temperatures and frequencies needed for complete analysis. Agencies that have 

managed to compile large databases of MR values may find such a tool useful in local 

calibration efforts. 

The LTPP database was populated with |E*| values at five temperatures and six frequencies by 

using the prioritized ANN models. This database contains information for a total of 1,806 layers. 

These layers have binder data available at a combination of different aging conditions, including 

unaged or original-, RTFO-, PAV-, or field-aged. In the field-aged data, 2,223 records are 

available because, for some layers, properties may have been measured at different dates. The 

total resulting number of records is 7,641. Using the combined ANN models and requisite QC 

checks, modulus values were predicted for 363 records/layers in the original-aged level, 
104 

469 records/layers in the RTFO-aged level, 1 record/layer in the PAV level, and 503 records in 

the field-aged level. These numbers translate to predictions for 17.5 percent of the total number 

of records available. However, these records are distributed in such a way that a higher 

percentage of the layers have some sort of valid prediction. Of the 1,806 layers in the database, 

1,010, or 56 percent, have a modulus prediction at some aging condition. Of these 1,010 layers, 

615, or 34 percent of the total 1,806 layers, have completely reasonable predictions (i.e., an 

“A” grade), and 89, or 4.9 percent of the total 1,806 layers, have unreasonable predictions 

(i.e., an “F” grade). The remaining 306 layers, 17 percent of the 1,806 layers, have questionable 

predictions (i.e., a “C” grade). Thus, the total percentage of layers with a completely valid or 

questionable prediction is 51 percent. 

These populated values will allow users to develop a mastercurve for independent analysis or 

directly into MEPDG. In addition, mastercurve sigmoidal parameters and temperature shift 

factors were also computed and included in the population effort. The computed parameters are 

included in the computed parameter data submitted to FHWA.

The following tasks constitute suggested future research efforts: 

• Development and/or refinement of a closed-form |E*| predictive model to estimate the 

dynamic modulus values as a function of temperature and frequency. The insight gained 

with the current ANN models should aid in the development of such models. 

• Development of individual temperature-based ANN or closed-form |E*| models. By 

developing such models or the comprehensive closed-form model suggested in the future 

task listed above, a maximum of 306 of the layers that currently have a questionable 

modulus prediction could be populated with more accurate values. 

• Performance of an indepth and comprehensive experimental study to gain better 

understanding of the differences between the moduli measured via the AMPT and 

TP-62 protocols.(8) Understanding the factors that have led to the clear differences found 

in the datasets compiled by each of these techniques could lead to more robust 

predictive models. 
105 

APPENDIX A: PROCESSING ASPHALT BINDER VISCOSITY DATA

A.1 INTRODUCTION

Until the advent of the SuperpaveTM mix design system in the early 1990s, the viscosity of 

asphalt binders was the key measure used in purchase specifications. As a result, many sections 

that are now reaching their design lives and, as a result, are included in the LTPP database, 

contain various measures of viscosity. Over the years, four primary types of viscosity measures 

have been adopted: (1) ring and ball temperature (R&BT), (2) penetration (at various 

temperatures), (3) absolute viscosity at 140 °F (60 °C), and (4) kinematic viscosity at 275 °F 

(135 °C). These measures have been used together or separately in various grading schemes. 

With the exception of penetration, these viscosity measures are beyond the range of temperature 

conditions typically needed for modulus prediction. As a result, a certain amount of processing 

the available data is necessary. In this appendix, the steps necessary to properly analyze available 

viscosity data are presented.

A.2 TEMPERATURE SUSCEPTIBILITY RELATIONSHIP

Although the relationship between viscosity and temperature is highly nonlinear, it has been 

found that when proper transformations are made to temperature and viscosity, a linear 

relationship exists. This relationship is commonly referred to as the “A-VTS relationship.”(43) 

This relationship is shown schematically in figure 79 and mathematically in equation 56. The 

plateau region in figure 79 is based on arguments that are, in turn, based on the chemical 

structure of asphalt binder and suggest that the maximum viscosity for asphalt binder is 

2.7 x 1012 cP (0.0027 x 1012 Pas).

(44) For the purposes of this report, this same limiting criterion 

was also adopted. 

0

0.2

0.4

0.6

0.8

1

1.2

2.55 2.6 2.65 2.7 2.75 2.8 2.85 2.9

log(TR)

log(log(

η))

Measured Viscosity

A-VTS Relationship

Figure 79. Graph. A-VTS relationship. 
106 

( ) ( )

12

log log log 2.7 10

R R critical

R critical

A VTS T T T

x TT η

 + >  =     ≤ (56)

Where:

η = Viscosity (cP). 

A = Intercept of temperature susceptibility relationship. 

VTS = Slope of temperature susceptibility relationship. 

TR = Temperature in Rankine. 

Tcritical = Temperature in Rankine at which the viscosity is equal to 2.7 x 1012 cP 

 (0.0027 x 1012 Pas). 

Because the A-VTS relationship is linear, only two of the four viscosity measures are needed to 

completely characterize the relationship. The following sections provide the equations necessary 

to convert the four measures to actual viscosity.

A.3 R&BT TEMPERATURE

The softening point of asphalt binder as measured by AASHTO T53-08 is also known as 

R&BT.

(45) According to literature on the topic, this temperature, measured in Fahrenheit, 

corresponds to the temperature at which asphalt binder has a viscosity of 13,000 P 

(1,300 Pas).(44)

A.4 PENETRATION

The penetration number for asphalt binder is determined via AASHTO T49-07.(46) In this test, a 

3.5-oz (100-g) needle is used to penetrate an asphalt sample for 5 s. The amount of penetration, 

measured in tenths of a millimeter, is the penetration number for the asphalt binder at the 

particular test temperature. The measurement temperature typically used for specification 

purposes is 77 °F (25 °C); however, other temperatures, including 39.2 and 115 °F (4 and 46 °C), 

may also be measured. Penetration values are converted to viscosity using the relationship 

suggested by Mirza and Witczak as seen in equation 57:(44)

( ) ( ) ( ) 2

log 10.5012 2.2601log 0.00389 log η =− + PEN PEN (57) 

Where:

η = Viscosity (P). 

PEN = Penetration number at a given test temperature.

A.5 ABSOLUTE VISCOSITY

The absolute viscosity is the viscosity of asphalt binder measured at 140 °F (60 °C) by AASHTO 

T202-03.(47) Because this quantity is typically reported in poise instead of centipoise, the only 

conversion needed is to multiply the given quantity by 100. 
107 

A.6 KINEMATIC VISCOSITY

The kinematic viscosity of asphalt binder is determined at 275 °F (135 °C) via AASHTO 

T201-03.(48) Kinematic and absolute viscosities are related by the density of the medium under 

investigation. The relationship between these two quantities is shown in equation 58. For 

LTPP purposes, it is assumed that the density for all binders is equal to 0.6 oz/in3 (1.03 g/cm3).

ηνρ = * (58) 

Where:

η = Absolute viscosity (cP). 

ν = Kinematic viscosity (cSt). ρ = Density (oz/in3

 ((g/cm3)). 

A.7 EXAMPLE PROBLEM

For a given asphalt binder, the following properties are measured: 

R&BT temperature = 104 °F (40 oC) 

PEN at 39.2 °F (4 oC) = 19 

PEN at 77.0 °F (25 oC) = 156 

η = 774 P (77.4 Pas) 

ν = 266.1 centistokes (cSt) 

 

Using these relationships, the viscosity is computed for different temperatures as follows: 

39.2 °F (4 oC) = 4.14 x 109 cP (0.00414 x 109 Pas)

77.0 °F (25 oC) = 3.66 x 107 cP (0.00366 x 107 Pas)

104 °F (40 oC) = 1.30 x 106 cP (0.0013 x 106 Pas)

140 °F (60 oC) = 7.74 x 105 cP (0.00774 x 105 Pas)

275 °F (135 oC) = 2.74 x 102 cP (0.00274 x 102 Pas)

After performing linear regression on these quantities, it is found that A = 10.599, 

VTS = -3.5646, and Tcritical = 464.8 °R. 

A.8 TYPICAL VALUES FOR PURCHASE SPECIFICATION GRADE

As part of the NCHRP 1-37A effort, researchers compiled typical A and VTS values for different 

purchase specification grades.

(2) These typical values include SuperpaveTM PG binders, AC 

viscosity-graded binders, and penetration-graded binders. Data were not compiled for all grades 

in use across the United States, but most grades were included. Because some of the LTPP layers 

fell in the category where the only binder information known was the grade, these relationships 

are considered important. In table 36, the A and VTS parameters are presented for the different 

grades reported in the MEPDG documentation.(2)
108 

Table 36. Relationship between asphalt binder grade and viscosity parameters. 

Asphalt 

Binder Grade A VTS

Asphalt 

Binder Grade A VTS

PG 46-34 11.5040 -3.9010 PG 70-28 9.7150 -3.2170

PG 46-40 10.1010 -3.3930 PG 70-34 8.9650 -2.9480

PG 46-46 8.7550 -2.9050 PG 70-40 8.1290 -2.6480

PG 52-10 13.3860 -4.5700 PG 76-10 10.0590 -3.3310

PG 52-16 13.3050 -4.5410 PG 76-16 10.0150 -3.3150

PG 52-22 12.7550 -4.3420 PG 76-22 9.7150 -3.2080

PG 52-28 11.8400 -4.0120 PG 76-28 9.2000 -3.0240

PG 52-34 10.7070 -3.6020 PG 76-34 8.5320 -2.7850

PG 52-40 9.4960 -3.1640 PG 82-10 9.5140 -3.1280

PG 52-46 8.3100 -2.7360 PG 82-16 9.4750 -3.1140

PG 58-10 12.3160 -4.1720 PG 82-22 9.2090 -3.0190

PG 58-16 12.2480 -4.1470 PG 82-28 8.7500 -2.8560

PG 58-22 11.7870 -3.9810 PG 82-34 8.1510 -2.6420

PG 58-28 11.0100 -3.7010 AC-2.5 11.5167 -3.8900

PG 58-34 10.0350 -3.3500 AC-5 11.2614 -3.7914

PG 58-40 8.9760 -2.9680 AC-10 11.0134 -3.6954

PG 64-10 11.4320 -3.8420 AC-20 10.7709 -3.6017

PG 64-16 11.3750 -3.8220 AC-3 10.6316 -3.5480

PG 64-22 10.9800 -3.6800 AC-40 10.5338 -3.5104

PG 64-28 10.3120 -3.4400 PEN 40-50 10.5254 -3.5047

PG 64-34 9.4610 -3.1340 PEN 60-70 10.6508 -3.5537

PG 64-40 8.5240 -2.7980 PEN 85-100 11.8232 -3.6210

PG 70-10 10.6900 -3.5660 PEN 120-150 11.0897 -3.7252

PG 70-16 10.6410 -3.5480 PEN 200-300 11.8107 -4.0068

PG 70-22 10.2990 -3.4260 — — —

— Indicate that no additional relationships exist.
109 

APPENDIX B: PROCESSING ASPHALT BINDER SHEAR MODULUS DATA

B.1 INTRODUCTION

A potential shortcoming of the predictive models that rely on the binder |G*| is the need for the 

moduli to fall within the complete range of conditions under which the material mixture modulus 

can be predicted. This range of conditions typically includes temperatures from 14 to 129.2 oF 

(-10 to 54 °C) and frequencies from 25 to 0.1 Hz. All of the possible combinations of these 

temperatures and frequencies should be known in order to effectively use the predictive 

equations for pavement design purposes. The specification data cover the necessary range, but 

four main problems arise in using the specification results to determine the material properties: 

(1) the availability of data at only a limited number of temperatures, (2) the evaluation of LVE 

properties at only a single frequency or time, (3) the mixture of time domain and frequency 

domain measurements, and (4) a mixture of characteristic behaviors under different aging 

conditions. It is possible to measure the material properties at a sufficient number of 

frequencies/times and temperatures to perform the appropriate analysis. However, this process is 

not part of standard agency practice and, therefore, is not included in the current LTPP database. 

In this appendix, a combined phenomenological and mechanical approach is developed and 

presented. This approach, when coupled with a standard optimization technique, can be used 

with the existing specification test results to determine |G*| over the necessary range. This 

approach provides sufficient information for |E*| predictions without increasing the testing 

requirements. This analytical methodology, although more complicated than that typically used 

in agency offices, may be coded into a software package or a spreadsheet to allow easy, direct, 

and rapid characterization. 

B.2 USE OF BBR DATA IN |G*| MASTERCURVE GENERATION

In asphalt binder testing, two TPs are used to extract the LVE properties of the material. At high 

and intermediate temperatures, DSR is used, and at low temperatures, BBR is used. To determine 

the properties of a material over an entire range of in-service conditions, results from these two 

TPs must be combined. The purpose of this section is to present a method that combines these 

two outcomes into a single relationship. 

A comparison of the mastercurve and the t-T superposition shift factor function development 

with and without BBR data is also presented. The methodology used for mastercurve and t-T 

shift factor function development using DSR and BBR data generally follows the method 

presented by Kim et al.(49)

 The method used herein differs from the Kim et al. method in that it 

makes more extensive use of optimization techniques. 

B.3 ANALYSIS USING DSR MEASUREMENTS

Significant literature exists on the development of mastercurves from DSR measurements. 

Stiffness values at multiple combinations of temperature and frequency, such as those shown in 

figure 80, are first determined via experimentation. The data are then horizontally shifted by 

temperature along the logarithmic frequency axis to form a continuous function (see figure 81). 

The amount of horizontal shift is the t-T shift factor for that particular temperature and the 
110 

shifted frequency is referred to as the “reduced frequency.” This operation is expressed 

mathematically in equation 59, and an example function is shown in figure 82. 

TR

T

T

a

ω

ω =

(59)

Where: 

aT = Time-temperature shift factor. 

ωT = Frequency at the physical temperature. 

ωTR = Reduced frequency of ωT at the reference temperature.

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

0.1 1 10 100 1000

Frequency (rad/s)

|G*| (Pa) °C = (°F-32)/1.8

Figure 80. Graph. |G*| versus frequency curves at different temperatures. 

15C

25C

35C

45C

60C

70C

80C 

95C 

105C

115C 
111 

1.E+00

1.E+02

1.E+04

1.E+06

1.E+08

1E-07 1E-05 1E-03 1E-01 1E+01 1E+03

Reduced Frequency (rad/s)

|G*| (Pa)

1 psi = 6.86 kPa

Figure 81. Graph. |G*| versus reduced frequency mastercurve at 59 oF (15 °C). 

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

0 20 40 60 80 100 120 140

Temperature (C)

log aT °C = (°F-32)/1.8

Figure 82. Graph. t-T shift factor function at 59 oF (15 °C). 

Prior to desktop computers and optimization spreadsheet functions, the determination of the shift 

factors and the functional form comprised distinct steps. However, a current common technique 

is to assume some functional form for the mastercurve a priori and then optimize the coefficients 

of this function along with either the t-T shift factors directly or a function that relates these 

factors. Equation 60, which is an extension of the CAM model developed through the SHRP 
112 

program, is assumed for the mastercurve, and the Williams-Landel-Ferry (WLF) model is 

assumed for the t-T shift factor function (see equation 61). For all cases, 59 oF (15 °C) is taken as 

the reference temperature (i.e., aT = 1 at 59 oF (15 °C)). 

*

1

e

g

m k k

c

R

G

G

ω

ω

=

      +         (60)

Where:

Gg = Maximum shear modulus or glassy modulus (pascals). 

ωR = Reduced frequency (rad/s). 

ωc, me, and k = Fitting coefficients.

1 ( )

2

log R

T

R

CT T

a

C TT

− = + −

(61) 

Where: 

aT = t-T shift factor at temperature T (°C). 

TR = Reference temperature (°C). 

C1 and C2 = Fitting coefficients.

B.4 ANALYSIS WITH BBR MEASUREMENTS

To utilize the BBR data in |G*| mastercurve development, the data must be further processed 

because BBR measurements are taken in the time domain whereas DSR measurements are taken 

in the frequency domain. Also, BBR measurements are taken in the bending mode whereas DSR 

measurements are taken in the shear mode. 

The PG specification defines beam stiffness, S(t), as the inverse of creep compliance, D(t) (see 

equation 62). The creep compliance values of ALF AC5 at -22, -11.2, and -0.4 oF (-30, -24, and 

-18 °C) are calculated from the BBR stiffness values and plotted in figure 83. In a manner similar 

to the mastercurve development method mentioned previously, these curves are horizontally 

shifted to form a continuous curve. Note that because these measurements are made in the time 

domain, the reduced time is defined by equation 63. 

( ) ( )

1 S t

D t = (62)

R

T

T

T

t

t

a = (63)
113 

1.E-09

1.E-08

1.E-07

1 10 100 1000

Time (s)

D(t) (1/Pa)

-18 C

-24 C

-30 C

1 psi = 6.86 kPa

Figure 83. Graph. D(t) at different temperatures for ALF AC5 binder. 

For D(t), a generalized power law (GPL) is assumed as seen in equation 64 as follows: 

( ) 0 1

m D t D Dt = + R (64) 

Where D0, D1, and m are regression coefficients. In Kim et al., the determination of the 

coefficients in equation 64 is shown through a regression technique.(49) For the work presented 

here, the coefficients are determined via optimization using the Microsoft Excel Solver 

function. The two techniques are compared in figure 84 and found to yield indistinguishable 

results. Note that for the BBR measurements, the t-T shift factors form a linear relationship with 

temperature as seen in equation 65 as follows: 

log T BBR , 3 ( ) R a CT T = − (65) 
114 

1E-09

1E-08

1E-07

1E-07 1E-06 1E-05 1E-04 1E-03 1E-02

Reduced Time (s)

D(t) (1/Pa)

-30

-24

-18

Fit by optimization

Fit by regression

1 psi = 6.86 kPa

Figure 84. Graph. Comparison of optimization and regression GPL 

characterization results. 

To use the BBR results with DSR measurements, D(t) must be converted to |G*|. To make 

this conversion, the mathematical consequences of linear viscoelasticity and equation 64 along 

with equation 66 are utilized. Note that in this process, Poisson’s ratio is assumed to be timeindependent and has a value of 0.50. 

( )

1 *

21 *

G

ν D = + (66)

In this equation, ν is Poisson’s ratio and |D*| is the dynamic axial creep compliance. From linear 

viscoelasticity, this compliance can be determined by the following relationship: 

( ) ( ) 2 2 D DD * '" = + (67)

Where: 

D' = First vector component of |D*|. 

D" = Second vector component of |D*|. 

Both are defined for the GPL in equations 68 and 69 as follows: 

' 0 1 ( ) 1 cos

2

m

R

m DDDm π

ω −   = +Γ +     (68)
115 

" 1 sin 1 ( ) 2

m

R

m D Dm π

ω −   =Γ +     (69)

Although it is not necessary for this methodology, the frequencies at which the shear modulus is 

determined are consistent with LVE time-frequency equivalency principles. The equivalency is 

given in equation 70 as follows: 

2

R

Rt

ω

π =

(70)

B.5 COMBINING DSR AND BBR ANALYSIS RESULTS

To utilize the BBR measurements for CAM model development and t-T shift factor 

determination, the Microsoft Excel Solver optimization package is used. In this technique, the 

coefficients of the CAM model (see equation 60), the GPL model (see equation 64), the WLF 

equation for t-T shift factors above 32 oF (0 °C), and the t-T shift factor slope for temperatures 

below 32 oF (0 °C) are simultaneously optimized such that the objective function in equation 71 

is minimized. The results from this optimization are shown in figure 85 and figure 86 for the 

WesTrack binder. Results of the characterization for all binders for a reference temperature of 

59 oF (15 °C) are shown in table 37. 

( ( ) ( ))

( ( ) ( ) ( ) ( ) )

( ( ) ( ))

2

1

2

1

2

1

log | * | log | * |

log log

log | * | log | * |

M

DSR CAM i

i

N

BBR GPL j j

N

BBR CAM j j

Objective Function G G

Dt Dt

G G

=

=

=

= −+

− +

−

∑

∑

∑ (71)
116 

1E+00

1E+03

1E+06

1E+09

1E-08 1E-05 1E-02 1E+01 1E+04 1E+07

Reduced Angular Frequency (rad/s)

|G*| (Pa)

DSR

BBR

CAM Fit

1 psi = 6.86 kPa

Figure 85. Graph. |G*| mastercurve for the WesTrack binder at 59 oF (15 °C). 

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

-50 0 50 100

Temperature

log aT °C = (°F-32)/1.8

Figure 86. Graph. Optimized t-T shift function for WesTrack binder at 59 oF (15 °C). 

( )

( )

3

1

2

15

0 log 15

0

15

T

C T

T C

a C T

T C

C T

 −  < =  − ≥ 

 + −




117 

Table 37. Calibrated CAM model and t-T shift factor function coefficients using BBR data. 

Binder

CAM t-T

Gg (Pa) fc (rad/s) k me C1 C2 C3

ALF AC5 3.60E+10 0.06 0.0777 1.4245 -13.4747 99.8914 -0.1447

ALF AC10 2.49E+10 0.09 0.0851 1.4299 -12.9868 90.7373 -0.1481

ALF AC20 1.81E+09 1.60 0.1278 1.1693 -13.6667 94.2261 -0.1520

ALF Novophalt 8.38E+08 0.60 0.1265 1.0477 -12.6598 80.6345 -0.1624

ALF Styrelf 8.35E+10 0.00 0.0622 1.2543 -15.5337 121.0536 -0.1525

MnRd. AC20 1.44E+09 0.54 0.1288 1.2208 -12.6909 77.2684 -0.1638

MnRd. Pen120-150 4.04E+10 0.11 0.0842 1.4556 -12.1122 78.8723 -0.1453

WesTrack 1.17E+09 0.60 0.1411 1.1828 -12.9616 73.4526 -0.1547

1 psi = 6.86 kPa

B.6 COMPARISON OF RESULTS WITH AND WITHOUT BBR MEASUREMENTS

The methodology presented in the previous section has also been performed using only the DSR 

measurements. In this analysis, only the CAM model coefficients and WLF model coefficients 

are allowed to change, and the objective function is modified to that (see equation 72). The 

purpose of this analysis is to determine the effect of omitting BBR measurements when

determining |G*| values outside the DSR measured range. With this goal in mind, it is 

assumed that the analysis using BBR data represents the most accurate determination of the 

|G*| mastercurve. 

( ( ) ( )) 2

1

log | * | log | * |

M

DSR CAM i

i

Objective Function G G =

= ∑ − (72) 

The calibrated CAM model coefficients and the t-T shift factor function coefficients for each of 

the binders are shown in table 38. A comparison of the values presented in table 37 and table 38

reveals no consistent trend. In some cases, the values from the calibrated model are higher than 

those from the BBR-calibrated models (higher Gg and k values and lower me), whereas the 

opposite occurs in other models. In addition, the t-T shift factor function does not extrapolate 

well to lower temperatures when only DSR values are used in the calibration. An example of this 

behavior can be seen in figure 87. However, the figure shows that consistent results between the 

two characterization schemes occur at the DSR temperatures. To examine the effect of the 

extrapolation error, the GPL model is fitted to the BBR measurements, and the resulting overlap 

between the two datasets can be observed. When using the DSR-calibrated t-T shift factors, the 

BBR data are not continuous with the DSR data (see figure 88). This result suggests that the 

effect of extrapolating the t-T shift factor function is significant and may lead to errors. 
118 

Table 38. Calibrated CAM model and t-T shift factor function coefficients omitting 

BBR data. 

Binder

CAM t-T

Gg (Pa) fc (rad/s) k me C1 C2

ALF AC5 5.24E+09 0.66 0.0976 1.2887 -13.2075 98.3537

ALF AC10 3.60E+10 0.62 0.0872 1.3514 -13.0002 92.7463

ALF AC20 1.25E+10 0.34 0.0968 1.2981 -14.0171 98.1479

ALF Novophalt 3.35E+07 1.74 0.3069 0.8552 -12.3904 79.8044

ALF Styrelft 4.52E+09 0.04 0.0824 1.1214 -14.8831 108.9959

MnRd. AC20 3.72E+10 0.57 0.0924 1.3303 -13.1226 86.8517

MnRd. Pen120-150 7.42E+10 0.90 0.0851 1.3752 -12.1104 80.9397

WesTrack 1.43E+08 4.33 0.2623 1.0068 -13.1142 81.7971

1 psi = 6.86 kPa

-8.0

-4.0

0.0

4.0

8.0

12.0

16.0

-50 0 50 100

Temperature

log aT

With BBR

Without BBR

Figure 87. Graph. Comparison of t-T shift factor functions calibrated with and without 

BBR data. 
119 

1E+00

1E+03

1E+06

1E+09

1E-08 1E-05 1E-02 1E+01 1E+04 1E+07

Reduced Angular Frequency (rad/s)

|G*| (Pa)

DSR

BBR

1 psi = 6.86 kPa

Figure 88. Graph. |G*| mastercurve using t-T shift factor function calibrated without

BBR data. 

For the purposes of this report, the most critical information is found in the differences between 

these two techniques at TP-62 temperatures (14, 41, 68, 104, and 129.2 oF (-10, 5, 20, 40, and 

54 °C)) and frequencies (25, 10, 5, 1, 0.5, and 0.1 Hz). To explore these conditions, the 

difference formula shown in equation 73 is utilized for |G*| values that are predicted using the 

calibrated CAM models at the TP-62 temperatures and frequencies. The results are summarized 

in table 39 for each of the binders. 

* *

% *100

*

with BBR without BBR

with BBR

G G

Diff G

− = (73) 
120 

Table 39. Percentage of difference between BBR-calibrated and non-BBR-calibrated |G*| 

at TP-62 temperatures and frequencies.

Temperature 

(°C)

Frequency 

(Hz)

ALF 

AC5

ALF 

AC10

ALF 

AC20

ALF 

Novochip

ALF 

Styrene

MnRoad 

AC20

Pen 

120-150

Wes 

Track

-10 25 -11.09 -153.94 -211.16 87.30 29.32 -629.40 -334.59 67.37

-10 10 -20.53 -162.36 -203.86 85.55 23.47 -589.44 -358.74 63.23

-10 5 -28.23 -169.15 -198.98 83.95 18.73 -561.10 -378.72 59.42

-10 1 -48.12 -186.48 -189.99 78.88 6.61 -501.81 -431.57 47.49

-10 0.5 -57.59 -194.67 -187.24 75.91 0.90 -479.17 -457.47 40.55

-10 0.1 -81.72 -215.54 -183.75 66.24 -13.55 -433.54 -526.17 18.18

5 25 16.83 -11.35 -29.59 72.25 20.04 -77.81 -13.86 54.92

5 10 14.07 -9.28 -22.56 67.17 15.44 -58.57 -10.94 50.21

5 5 12.09 -7.70 -17.75 62.64 11.89 -45.57 -8.76 46.44

5 1 7.96 -4.02 -8.19 49.60 3.57 -20.03 -3.78 37.54

5 0.5 6.42 -2.44 -4.73 42.92 -0.02 -10.83 -1.69 33.93

5 0.1 3.50 1.16 1.84 25.70 -8.25 7.00 2.98 27.26

20 25 3.60 -6.13 -3.81 21.12 9.08 -18.13 -5.92 1.89

20 10 2.11 -4.30 -0.90 10.85 5.61 -9.73 -3.60 -2.43

20 5 1.20 -2.94 0.90 3.71 3.09 -4.23 -1.91 -4.39

20 1 -0.15 0.06 3.86 -8.63 -2.32 6.09 1.75 -4.19

20 0.5 -0.40 1.28 4.63 -11.52 -4.42 9.61 3.21 -2.22

20 0.1 -0.24 3.92 5.42 -12.25 -8.72 15.95 6.27 5.47

40 25 -3.12 -2.17 3.73 -17.56 3.18 -1.44 -1.13 -20.55

40 10 -2.98 -0.87 3.76 -15.91 1.48 1.13 0.35 -15.26

40 5 -2.66 0.05 3.52 -13.42 0.38 2.60 1.37 -10.88

40 1 -1.30 1.90 2.23 -5.35 -1.53 4.65 3.36 -0.80

40 0.5 -0.49 2.56 1.41 -1.57 -2.06 5.01 4.04 3.10

40 0.1 1.78 3.78 -0.92 6.39 -2.69 4.89 5.22 10.41

54 25 -4.30 -0.72 3.71 -12.03 1.64 0.44 0.31 -11.80

54 10 -3.44 0.20 2.81 -7.08 0.87 1.07 1.28 -6.28

54 5 -2.64 0.81 1.97 -3.32 0.49 1.21 1.90 -2.54

54 1 -0.39 1.87 -0.37 4.40 0.19 0.58 2.90 4.19

54 0.5 0.70 2.18 -1.46 7.01 0.31 -0.02 3.14 6.13

54 0.1 3.38 2.48 -4.01 10.96 1.07 -1.99 3.22 8.28

°C = (°F-32)/1.8

The results in table 39 show that extreme errors may result when using only DSR measurements 

in the |G*| mastercurve characterization. However, these errors are contained entirely within the 

low temperature range. At temperatures included in the DSR testing (greater than 59 oF (15 °C)), 

the errors are small. To place perspective on the differences shown in table 39, the mastercurves 

for the ALF AC5 and MnRoad Pen 120-150 binders are shown in figure 89 and figure 90, 

respectively. Note that the mastercurves themselves are calibrated well and that the observed 

errors are almost entirely related to the t-T shift factor errors (see figure 87). It is interesting to 

note that the two mastercurves that seem to have the poorest match (i.e., the WesTrack binder in 

figure 91) do not have the highest error. This result is due to the nature of the error definition and 
121 

also due to differences in the t-T shift factors obtained from the two characterization methods. 

This latter effect is seen in figure 89 through figure 91 where all the mastercurves change from 

an equivalent reduced frequency of 14 oF (-10 °C) and 25 Hz to 129.2 oF (54 °C) and 0.1 Hz. 

This effect is seen also in the t-T shift factor functions in figure 92 through figure 94. 

Furthermore, equation 73 indicates that the error is defined in normal space, and figure 91

indicates that the magnitude of |G*| is smaller for the Westrack binder than for the other binders. 

If the error is defined in logarithmic space, as seen in equation 74, the error is found to follow the 

graphical results more closely, as shown in table 40. 

The results in figure 89 through figure 94, table 39, and table 40 affect the LTPP database 

population effort because BBR data are not currently available for any of the LTPP sections. In 

fact, the only data available in the LTPP database are DSR results at 10 rad/s and limited 

temperatures. A technique for considering these limited data is presented in the next section. 

However, these figures and tables reveal that without the BBR test results, the strength of the 

|E*| predictive models, which rely on the |G*|, would be severely limited at low temperatures. 

1E+00

1E+03

1E+06

1E+09

1E+12

1E-08 1E-05 1E-02 1E+01 1E+04 1E+07

Reduced Angular Frequency (rad/s)

|G*| (Pa)

CAM Fit with BBR

CAM Fit without BBR

1 psi = 6.86 kPa

Figure 89. Graph. |G*| mastercurves characterized with and without BBR data for 

ALF AC5 binder. 
122 

1E+00

1E+03

1E+06

1E+09

1E+12

1E-08 1E-05 1E-02 1E+01 1E+04 1E+07

Reduced Angular Frequency (rad/s)

|G*| (Pa)

CAM Fit with BBR

CAM Fit without BBR

1 psi = 6.86 kPa

Figure 90. Graph. |G*| mastercurves characterized with and without BBR data for the 

MnRoad Pen 120-150 binder. 

1E+00

1E+03

1E+06

1E+09

1E+12

1E-08 1E-05 1E-02 1E+01 1E+04 1E+07

Reduced Angular Frequency (rad/s)

|G*| (Pa)

CAM Fit with BBR

CAM Fit without BBR

1 psi = 6.86 kPa

Figure 91. Graph. |G*| mastercurves characterized with and without BBR data for the 

WesTrack binder. 
123 

-8.0

-4.0

0.0

4.0

8.0

12.0

16.0

-50 0 50 100

Temperature

log aT

With BBR

Without BBR

Figure 92. Graph. t-T shift factor function characterized with and without BBR data for 

the ALF AC5 binder. 

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

-50 0 50 100

Temperature

log aT

With BBR data

Without BBR data

Figure 93. Graph. t-T shift factor function characterized with and without BBR data for 

the MnRoad Pen 120-150 binder. 
124 

-10.0

-5.0

0.0

5.0

10.0

15.0

-50 0 50 100

Temperature (C)

log aT

With BBR data

Without BBR data

°C = (°F-32)/1.8

Figure 94. Graph. t-T shift factor function characterized with and without BBR data for 

the WesTrack binder. 

( ) ( )

( )

log * log *

% *100

log *

with BBR without BBR

with BBR

G G

Diff

G

−

=

(74)
125 

Table 40. Percentage of log difference between BBR-calibrated and non-BBR-calibrated 

|G*| at TP-62 temperatures and frequencies. 

Temperature

(°C)

Frequency

(Hz)

ALF 

AC5

ALF 

AC10

ALF 

AC20

ALF 

Novochip

ALF 

Styrine

MnRoad 

AC20

PEN 

120-150

Wes 

Track

-10 25 -0.54 -4.06 -5.16 10.71 1.75 -8.35 -4.65 5.97

-10 10 -0.97 -4.24 -5.04 10.14 1.37 -8.01 -4.81 5.49

-10 5 -1.30 -4.39 -4.96 9.67 1.07 -7.75 -4.95 5.09

-10 1 -2.12 -4.76 -4.80 8.43 0.36 -7.14 -5.31 4.09

-10 0.5 -2.49 -4.94 -4.76 7.83 0.05 -6.89 -5.48 3.62

-10 0.1 -3.39 -5.40 -4.71 6.29 -0.70 -6.32 -5.92 2.44

5 25 1.03 -0.58 -1.39 6.96 1.24 -3.06 -0.69 4.17

5 10 0.86 -0.49 -1.11 6.12 0.95 -2.49 -0.56 3.69

5 5 0.75 -0.42 -0.90 5.47 0.73 -2.05 -0.46 3.34

5 1 0.50 -0.23 -0.45 3.91 0.22 -1.03 -0.21 2.58

5 0.5 0.41 -0.14 -0.27 3.24 0.00 -0.59 -0.10 2.30

5 0.1 0.23 0.07 0.11 1.78 -0.50 0.43 0.19 1.83

20 25 0.24 -0.37 -0.23 1.43 0.59 -1.00 -0.36 0.11

20 10 0.14 -0.27 -0.06 0.71 0.37 -0.57 -0.23 -0.15

20 5 0.08 -0.19 0.06 0.24 0.20 -0.26 -0.13 -0.27

20 1 -0.01 0.00 0.26 -0.55 -0.16 0.42 0.13 -0.27

20 0.5 -0.03 0.09 0.33 -0.74 -0.30 0.70 0.24 -0.15

20 0.1 -0.02 0.32 0.42 -0.83 -0.63 1.29 0.52 0.40

40 25 -0.24 -0.17 0.28 -1.15 0.23 -0.11 -0.09 -1.35

40 10 -0.24 -0.07 0.29 -1.09 0.11 0.09 0.03 -1.08

40 5 -0.23 0.00 0.28 -0.96 0.03 0.21 0.12 -0.82

40 1 -0.12 0.18 0.20 -0.43 -0.13 0.42 0.33 -0.07

40 0.5 -0.05 0.26 0.13 -0.13 -0.18 0.48 0.42 0.29

40 0.1 0.21 0.44 -0.10 0.63 -0.25 0.54 0.64 1.16

54 25 -0.38 -0.06 0.32 -0.91 0.13 0.04 0.03 -0.95

54 10 -0.33 0.02 0.26 -0.58 0.07 0.10 0.13 -0.55

54 5 -0.27 0.08 0.19 -0.29 0.04 0.12 0.20 -0.24

54 1 -0.05 0.22 -0.04 0.44 0.02 0.07 0.36 0.47

54 0.5 0.09 0.28 -0.17 0.75 0.03 0.00 0.42 0.75

54 0.1 0.53 0.39 -0.54 1.37 0.12 -0.29 0.53 1.25

°C = (°F-32)/1.8

B.7 EXTRACTING FULL BINDER |G*| DATA FROM LIMITED ISOCHRONAL 

MEASUREMENTS

To fully utilize |G*| in mixture |E*| predictive models, a complete dataset is needed. This dataset 

must be complete because it must cover the temperature and frequency ranges over which the 

mixture modulus should be predicted. The full suite of SuperpaveTM binder specification tests 

covers this range; however, the data are inconsistent (i.e., |G*| from DSR, and S(t) and m from 

BBR) and require further processing for complete utilization. This section of the report presents 
126 

a method for analyzing standard PG binder characterization tests so that the results can provide a 

more complete dataset. 

This method is currently under development by the research team and will be refined as 

a more complete picture of the available data is obtained. It is assumed that DSR (i.e., |G*|) 

measurements taken at the standard frequency of 10 rad/s, or 1.67 Hz, at multiple temperatures 

are available. Results from Superpave TM BBR testing are not necessary in this methodology, 

but a theoretically justified technique for including such data in the process is shown and 

evaluated. Currently, such data are not available from the LTPP database; however, the 

methodology is evaluated nonetheless in case such characterization is included in future 

research efforts. 

To extract a more complete dataset, an analytical expression for |G*| as a function of frequency 

and temperature must be found. Because asphalt binder is thermorheologically simple, these two 

factors can be combined into a single parameter, known as “reduced frequency” and shown in 

equations 75 and 76. Note that although the nomenclature in this report represents frequency 

using ω, which implies the unit is radians per second, the methodology is equally applicable to 

frequency in hertz. 

( ) ( ) * * , GT G ω ω = R (75) 

R T ω ω= × a (76) 

Where:

ωR = Reduced frequency. 

aT = t-T shift factor, which is a function of temperature. 

This factor can be determined experimentally using results from a temperature and frequency 

sweep test, such as the test found in AASHTO TP-62.(8) From these tests, the t-T shift factor is 

determined by horizontally shifting the data at different temperatures until a smooth varying 

function results. This process is shown previously in this report and is only possible when 

measurements are taken at multiple frequencies and temperatures. 

Measurements from a single-frequency temperature sweep test, such as SuperpaveTM DSR 

testing, cannot be used directly to extract these t-T shift factors by simple horizontal translations. 

This complication arises because the slope of the modulus-temperature relationship, a value that 

can be determined from single-frequency temperature sweep tests, is dependent on both the 

frequency and the temperature. Mathematically, this discrepancy can be expressed as follows:

( )

* * log( )

log

R

R

G G

T T

ω

ω

∂ ∂ ∂ = ∂∂ ∂ (77)
127 

Where:

T = Temperature. 

ωR = Reduced frequency. 

∂|G*|/∂T = Derivative of |G*| with respect to time, known from Superpave TM DSR testing. 

Direct determination is not possible using these measurements, but equation 77 can be solved 

analytically if functional forms for the t-T shift factor function and for |G*| as a function of 

reduced frequency are known. In this methodology, the CAM model is assumed sufficient for 

the latter, and a secondary surrogate model along with the WLF equation is assumed for the 

logarithm of the former (note that two surrogate models are discussed in the following section). 

These functions are shown in equations 78 and 79, respectively. 

*

1

e

g

m k k

c

R

G

G

ω

ω

=

      +         (78)

1 ( )

2

log R

T

R

CT T

a

C TT

− = + − (79)

Where Gg, ωc, k, C1, C2, and C3 are fitting coefficients. Combining equations 76 and 79 yields 

the following analytical expression for reduced frequency: 

1( )

2 *10

R

R

CTT

C TT ω ω R

−

+ − = (80) 

From equation 77, DSR measurements can be used directly to determine the left-hand side of the 

equation, and the right-hand side can be solved analytically. However, initial trials using this 

approach proved unsuccessful due to the large spacing between typical SuperpaveTM DSR 

measurements. Instead, a more direct approach that uses equation 78 was successful; however, 

it is necessary to first discuss the model used to predict the t-T shift factors. 

B.8 T-T SHIFT FACTOR SURROGATE MODEL DEVELOPMENT

Because no clear theoretical link was found between typical DSR measurements and the t-T shift 

factors, a phenomenological approach and a simple averaging approach were taken. As shown in 

the following sections, both approaches provide approximately the same accuracy; however, the 

final recommended procedure will be the subject of further investigation. Because it is important 

that this t-T shift factor function covers the range over which the mixture modulus would be 

predicted, only binders with BBR test data have been used. A list of these binders can be found 

in table 4. For each of these binders, the t-T shift factors were found from optimization fitting 

using the CAM and WLF equations. 
128 

B.8.1 Phenomenological t-T Shift Factor Function Model

A similarity exists between the normalized shear modulus at 10 rad/s and the log shift factor 

function. Additionally, a similar relationship exists in the BBR test results at 60 s. These 

relationships are shown in figure 95 for the binders with BBR test data. For simplicity, both 

relationships are normalized to |G*| at 10 rad/s at the chosen reference temperature. Because this 

value is taken as the reference condition and because BBR data are essentially DSR results at 

0.01 rad/s (ω=2/π t), an apparent discontinuity is found in the BBR curve at the reference 

condition. To understand this discontinuity, it must be recognized that to obtain a given ratio 

larger than one from DSR test results, the test must be performed at a higher temperature than for 

a BBR test. This temperature effect is integral to the relationship shown in figure 95 for the 

DSR results. Because all of the BBR test results are obtained at the same time at 60 s, each is 

affected similarly, resulting in the discontinuity. However, the slope of this relationship stays 

much the same as seen in figure 96, where the offset discontinuity in the BBR data has been 

artificially removed. 

It should be noted that the relationships shown in figure 95 have been developed at a reference 

temperature of 59 oF (15 °C). If DSR measurements are taken at this temperature, then the 

relationship may be used directly. Alternatively, |G*| at 59 oF (15 °C) may be interpolated if DSR 

measurements are taken near this temperature. Also, the relationship can be used even if the 

reference condition is not 59 oF (15 °C), although some error would be inevitable. However, 

the approach taken here recognizes that at around 59 oF (15 °C) (between 41 and 77 oF (5 and 

25 °C)), the t-T shift factors are not significantly different. This relationship follows a second 

order polynomial, and its implications are shown in the calibrated t-T shift factor relationship 

shown in equations 81 and 82. The verification of these relationships is shown in figure 97 

for FHWA mobile trailer and FHWA ALF binders, which were not included in the 

calibration process.

( ) ( ) ( ) 2 2

, log .1639 2.0182 * 0.000638* 0.1469 * 2.06 T DSR R R a A A TT =+ − − + (81) 

( ) ( ) 2

, log 2.1859 3.1685 0.000638* 0.1469 * 2.06 T BBR R R a A TT = +− − + (82) 

Where: 

( ) 2

*

log 6.16 3.05 0.000418* 0.10 * 5.42 * R

R R

T

G

A T T

G

 

= −+   − +    

. (83)
129 

y = 0.1639x2

 + 2.0182x

R2

 = 0.978

y = 2.1859x + 3.1685

R2

 = 0.6752

-10

-8

-6

-4

-2

0

2

4

6

8

10

-6 -4 -2 0 2 4

log(|G*|/|G*|TR)

log a

T

BBR Relationship

DSR Relationship

Figure 95. Graph. Phenomenological t-T shift factor function. 

-10

-8

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4

log(|G*|/|G*|TR)

log a

T

Shifted BBR Relationship

DSR Relationship

Figure 96. Graph. Effect of change in time from DSR to BBR on the t-T shift factor 

function model. 
130 

-8

-6

-4

-2

0

-8 -6 -4 -2 0

log aT, actual 

log aT, predicted

R2

=0.9721

Figure 97. Graph. t-T shift factor model verification with phenomenological model. 

B.8.2 Use of Average t-T Shift Factor Function 

A second, more simplistic approach has also shown promise. From the available binder 

databases, it was found that the t-T shift factor functions are somewhat similar across binder 

types. In this approach, the same binders used to develop the phenomenological model are 

processed to find an average representative shift factor function (see figure 98). The advantage 

of this approach is its simplicity as well as the fact that BBR test results are not needed to 

obtain a reasonable estimate of the t-T shift factor at low temperatures. Verification of this 

model is shown in figure 99. Comparing figure 99 to figure 97, it is observed that the average 

technique has a slightly improved R2

 value. In addition, the average technique does not tend to 

show systematic underestimation of the phenomenological approach in the intermediate 

temperature ranges. However, in general, the effectiveness of the model does not appear to be 

significantly superior or inferior to the phenomenological approach. Both are evaluated in the 

subsequent sections. 
131 

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

-20 0 20 40 60

Temperature (C)

log aT

Characterization Data

Average Model Function

°C = (°F-32)/1.8

Figure 98. Graph. Average t-T shift factor function. 

-8

-6

-4

-2

0

-8 -6 -4 -2 0

log aT, actual 

log aT, predicted

R2

=0.987

Figure 99. Graph. t-T shift factor model verification with average function model. 

B.9 INCLUSION OF BBR DATA IN MASTERCURVE GENERATION

Including SuperpaveTM-level BBR data in the optimization process is not as straightforward as 

the process that combines full DSR and BBR characterization. Complications arise due to the 

limited dataset that is available (typically only the beam stiffness term, S(t), and m at 60 s for up 

to three temperatures). Therefore, to use the data as accurately as possible, a few approximate 

conversion and typical values need to be assumed. The process begins by observing from the 
132 

theory of linear viscoelasticity that S is actually the inverse of the uniaxial extensional D(t). In 

order to convert to |G*|, the uniaxial relaxation modulus, E(t), must be found. Because it is 

possible that only a single BBR test result is available, the conversion technique must be capable 

of single-point conversion. For this purpose, the approximate relationship given in equation 84 is 

utilized as follows: 

( ) ( )

( )

( ) ( ) sin n n 1 sin

E t S t

n Dt n

π π

π π = = (84)

Where n is equivalent to the m-value determined from the BBR test. After determining E(t), the 

approximate interrelationship in equation 85 is used to convert from time-domain to frequencydomain functions as follows: 

( ) ( ) 2

t

Et E ω πω = ≈ ′ (85)

Where E ́(ω) is the storage modulus. E ́(ω) is then converted to the dynamic uniaxial modulus 

by using equation 86 as follows: 

( ) ( )

( )

*

cos

E

E

ω

ω

φ

′ = (86)

Where φ is the phase angle in the material. Experience has found that this value is approximately 

24 degrees under BBR test conditions. Therefore, the actual relationship used in this 

methodology is given by equation 87 as follows: 

( )

* 1.095*

cos 24

E E E ′ = = ′ (87)

Finally, to determine |G*|, the mechanistic relationship between uniaxial and shear deformation 

shown in equation 88 is used. For this relationship, Poisson’s ratio is assumed to be timeindependent and to have a value of 0.5. 

( ) ( )

* * 1 * *

2 1 2 1 0.5 3

E E

G E

ν = = = + + (88)

Combining equations 84 through 87, the relationship between |G*| and S(t) is given by 

equation 89, as follows: 

( ) ( ) ( )

2

0.365*sin

*

t t

n

G S t

n π

π

ω

π =

=

(89)

In addition to calculating the shear stiffness values from beam stiffness and m-values, it is

worthwhile to consider fitting the m-values. Recognizing that m-values are the derivative 
133 

of the logarithm of stiffness with respect to the logarithm of time and that simple multiplicative 

relationships are used to convert from frequency to time, the analytical expression for the 

m-value using the CAM model can be given by equation 90, as follows: 

log *

log

1

k

c

e

R

calculated k

R c

R

m

G

m

ω

ω

ω ω

ω

    ∂   = = ∂       +         (90) 

B.10 PAV- TO RTFO-AGING MODEL (PAR MODEL)

To develop a complete mastercurve from limited SuperpaveTM data, the conversion between 

PAV- and RTFO-aging conditions is necessary. This conversion is also needed in the LTPP 

database where |G*| values are given in terms of RTFO- and PAV-aging conditions. It is noted 

that original |G*| values are also available in the database. A similar approach could be used to 

convert from original (or field-aged) to RTFO. Currently, the more important conversion is from 

PAV to RTFO. To develop this relationship, the Witczak database binders listed in section 3.0 

are used. In general, the relationship between |G*|PAV and |G*|RTFO is found to be dependent on 

temperature, frequency, and the chemical properties of the asphalt binder. Because none of the 

SuperpaveTM tests directly measure the chemical properties of the asphalt binder, the two 

primary factors in the PAR model are temperature and frequency. Although not a major factor, 

the SuperpaveTM high temperature grade is found to be a secondary factor and may indirectly 

capture some of the chemical composition effects. 

The general form of the PAR model is shown in equation 91 where κT and κω are the 

temperature and frequency factors, respectively. 

** *

RTFO PAV T PAV GG G = = κ κκ ω (91) 

In general, κT follows a second order polynomial relationship, with the minimum occurring near 

the high temperature PG (see figure 100). After calibrating κT, κω is characterized. This 

relationship follows a power law relationship, with the parameters depending on the high 

temperature grade of the asphalt binder, as seen in figure 101. With these relationships in mind, 

equation 91 can be rewritten as equation 92 as follows: 

( )( ) 2

1 23

* *

RTFO PAV G TT G τ = ++ β β β γω (92) 

Where T is temperature. Each of the parameters in equation 92 follow a first or second order 

polynomial according to the high temperature PG of the binder. These parameters are 

summarized in table 41. The calibration dataset along with the calibrated model are shown 

according to high temperature PG in arithmetic and logarithmic spaces in figure 102 through 

figure 105. For model verification, a database of binders obtained from the FHWA mobile trailer 

and the binders used in the current FHWA ALF study are summarized in table 42. The 
134 

verification is shown in figure 106 and figure 107, and agreement is found between the model 

and actual data at all high temperature PG levels. 

Table 41. PAR model coefficient summary for SuperpaveTM high temperature grades. 

PG β 1 β 2 β 3 γ τ λ1 λ2 

Number of 

Observations R2

46 4.82E-05 -0.002 0.169 0.858 0.082 -0.039 0.062 Extrapolated 52 6.08E-05 -0.006 0.359 0.842 0.082 -0.043 0.180

58 7.35E-05 -0.010 0.514 0.819 0.084 -0.047 0.281 429 0.983

64 8.61E-05 -0.012 0.633 0.790 0.086 -0.050 0.363 1712 0.986

70 9.88E-05 -0.014 0.716 0.754 0.090 -0.053 0.428 182 0.937

76 1.11E-04 -0.015 0.763 0.712 0.095 -0.056 0.474 537 0.964

82 1.24E-04 -0.015 0.774 0.663 0.101 -0.058 0.503 326 0.935

Unknown 8.21E-05 -0.011 0.641 0.787 0.082 -0.033 0.689 2013 0.945

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140

Temperature (C)

κ

Τ

PG 46 PG 52 PG 58

PG 64 PG 70 PG 76

PG 82

°C = (°F-32)/1.8

Figure 100. Graph. Effect of temperature on aging ratio of asphalt binder. 
135 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 10 100

Frequency (rad/s)

κ

ω

PG 46 PG 52 PG 58

PG 64 PG 70 PG 76

PG 82

Figure 101. Graph. Effect of angular frequency on aging ratio of asphalt binder. 

0.0E+00

2.0E+07

4.0E+07

6.0E+07

8.0E+07

1.0E+08

0.0.E+00 5.0.E+07 1.0.E+08 1.5.E+08 2.0.E+08

|G*|PAV (Pa)

|G*|RTFO (Pa)

58-model 58-meas

64-model 64-meas

70-model 70-meas

76-model 76-meas

82-model 82-meas

1 psi = 6.86 kPa

Figure 102. Graph. Calibrated PAR model in arithmetic scale. 
136 

1.0E+00

1.0E+03

1.0E+06

1.0E+09

1.0.E+00 1.0.E+02 1.0.E+04 1.0.E+06 1.0.E+08

|G*|PAV (Pa)

|G*|RTFO (Pa)

1 psi = 6.86 kPa

Figure 103. Graph. Calibrated PAR model in logarithmic scale. 

0.0E+00

2.0E+07

4.0E+07

6.0E+07

8.0E+07

1.0E+08

0.0.E+00 2.0.E+07 4.0.E+07 6.0.E+07 8.0.E+07 1.0.E+08

|G*|RTFO Predicted (Pa)

|G*|RTFO Measured (Pa)

58-meas 64-meas

70-meas 76-meas

82-meas LOE

1 psi = 6.86 kPa

Figure 104. Graph. Strength of PAR model in arithmetic scale. 
137 

1.0E+00

1.0E+03

1.0E+06

1.0E+09

1.0.E+00 1.0.E+03 1.0.E+06 1.0.E+09

|G*|RTFO Predicted (Pa)

|G*|RTFO Measured (Pa)

1 psi = 6.86 kPa

Figure 105. Graph. Strength of PAR model in logarithmic scale. 

0.0E+00

3.0E+07

6.0E+07

9.0E+07

1.2E+08

0.0.E+00 3.0.E+07 6.0.E+07 9.0.E+07 1.2.E+08

|G*|RTFO Predicted (Pa)

|G*|RTFO Measurement (Pa)

PG 58 PG 64

PG 70 PG 76

LOE

1 psi = 6.86 kPa

Figure 106. Graph. PAR model verification in arithmetic logarithmic scale. 
138 

1.0E+00

1.0E+02

1.0E+04

1.0E+06

1.0E+08

1.0E+10

1.0.E+00 1.0.E+02 1.0.E+04 1.0.E+06 1.0.E+08 1.0.E+10

|G*|RTFO Predicted (Pa)

|G*|RTFO Measurement (Pa)

1 psi = 6.86 kPa

Figure 107. Graph. PAR model verification in logarithmic scale. 

Table 42. PAR model verification dataset summary. 

PG

Number of 

Observations R2

52 297 0.702

58 605 0.981

64 2289 0.970

70 769 0.988

76 700 0.985

Because the DSR data extend only to 59 °F (15 °C), a second relationship must be calibrated for 

κT at temperatures consistent with the BBR test results. Such a relationship is calibrated by 

assuming κω is accurate for BBR conditions and then backcalculating κT from equation 91 and 

BBR test results under PAV and RTFO conditions. The results suggest that κT changes linearly 

with temperature under the BBR test conditions and that aging softens the material at the 

extremely high reduced-frequency region, as seen in figure 108. This observation is confirmed 

from |G*| mastercurve analysis of the aforementioned binders and is believed to be the first of 

such an observation. An example of this analysis is shown for the ALF AC 10 binder in 

figure 109. Due to the limited database, this observation cannot be confirmed with certainty; 

rather, κT is treated as a fitting parameter, which, along with κω, approximates the effect of 

RTFO aging to PAV aging. The complete relationship between RTFO and PAV conditions for 

BBR tests is given in equation 93, and the parameters are summarized along with the other 

parameters in table 41. Finally, the calibrated BBR data model is shown in figure 110 and 

figure 111. 
139 

0

0.5

1

1.5

2

2.5

-35 -30 -25 -20 -15 -10 -5 0

Temperature (C)

κ

Τ

PG 46 PG 52 PG 58

PG 64 PG 70 PG 76

PG 82

°C = (°F-32)/1.8

Figure 108. Graph. BBR-calibrated κT relationship. 

1E+00

1E+03

1E+06

1E+09

1E-08 1E-05 1E-02 1E+01 1E+04 1E+07

Reduced Angular Frequency (rad/s)

|G*| (Pa)

PAV

RTFO

1 psi = 6.86 kPa

Figure 109. Graph. Comparison of |G*| mastercurve analysis under PAV and RTFO 

conditions for ALF AC10 binder. 

* * ( ) 1 2 ( ) RTFO PAV GT G τ = + λ λ γω (93) 
140 

0.0E+00

1.0E+08

2.0E+08

3.0E+08

4.0E+08

0.0.E+00 1.0.E+08 2.0.E+08 3.0.E+08 4.0.E+08

|G*|PAV (Pa)

|G*|RTFO (Pa)

Measurement

Model

1 psi = 6.86 kPa

Figure 110. Graph. Calibrated PAR model for BBR conditions in arithmetic scale. 

1.0E+07

1.0E+08

1.0E+09

1.0.E+07 1.0.E+08 1.0.E+09

|G*|PAV (Pa)

|G*|RTFO (Pa)

1 psi = 6.86 kPa

Figure 111. Graph. Calibrated PAR model for BBR conditions in logarithmic scale. 
141 

B.11 AGING EFFECT ON M-VALUE

To address the issue of aging effects on the m-value, the PAR model is used to derive the 

proper relationship. Consider that m, n, the log-log slope of S(t), and |G*|(ω) are close in 

absolute value at time-frequency equivalency points. In this case, the following equation 

is considered: 

2

log log *

log log t

S G

m

t ω π

ω =

∂ ∂ =− ≈ ∂ ∂ (94)

Which leads to the following equation: 

1 2

1 2

log * log *

log log

G G

m

ω ω

− ≈ − (95)

Casting equation 95 for mRTFO as follows: 

( ) ,1 ( ) ,2

1 2

log * log *

log log

RTFO RTFO

RTFO

G G

m

ω ω

− ≈ − (96)

Applying equation 91 to equation 96 as follows: 

( ,1 1 2 ) ( ,2 )

1 2

log * * log * *

log log

PAV PAV

RTFO

G G

m

κ κ

ω ω

− ≈ − (97)

Simplification leads to the following:

( ) ( ) 1 2

1 2

log log

log log

m m RTFO PAV

κ κ

ω ω

− ≈ + − (98) 

The second term in equation 98 can be solved analytically given the mathematical relationship 

for κ provided in equation 92. Performing the differentiation leads to the following: 

ln 10 ( ) m m RTFO PAV

τ

≈ +

(99)

Equation 99 is verified using the m-values for the binder data in the Witczak database, as shown 

in figure 112. 
142 

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

m-value, PAV

m-value, RTFO

Measurement

Model

Figure 112. Graph. Verification of m-value relationship. 

B.12 VERIFICATION OF LIMITED DATA DEVELOPMENT

Verification of the mastercurve construction using only isochronal measurements is divided 

into three steps. The first verification level (divided into levels 1a and 1b) uses the Witczak 

binder data, which are also used in the shift factor function model development (i.e., the 

binder data in table 4 using available BBR measurements). However, this level is not considered 

true verification because the data were used in the calibration process. Instead, this level assesses 

the impact of model fitting errors. Levels 1a and 1b differ only in the use of the PAR model. 

Also, level 1 verification includes BBR data in the analysis process, which is not representative 

of the currently available LTPP database. However, the second two verification levels are 

representative of data in the current LTPP database. For sections that have the complete 

SuperpaveTM DSR suite, level 2 is the most representative. Sections without the complete 

high temperature characterization data are better represented by the level 3 verification. Note 

that for all sections, the phenomenological and average shift factor function models are used 

and presented. 

B.12.1 Level 1a Verification

The first check of these shifting principles consists of a circular check whereby the binders used 

in the previous section help determine the WLF and CAM model coefficients using only data 

that would typically be available for SuperpaveTM testing. For level 1a verification purposes, 

only RTFO data are used in the analysis; level 1b verification adds an aging model to account for 

the fact that SuperpaveTM testing consists of results under original-, RTFO-, and PAV-aged

conditions. The steps taken in level 1a verification are as follows: 

1. Find DSR results at 10 rad/s at 59, 77, 95, 140, 158, and 176 oF (15, 25, 35, 60, 70, 

and 80 °C). 
143 

2. Find BBR results at 60 s at either -22, -11.2, and -0.4 oF (-30, -24, and -18 °C) or at -11.2, 

-0.4, and 10.4 oF (-24, -18, and -12 °C). 

3. Convert BBR results to |G*| values using equation 89. 

4. Determine the t-T shift factors via equations 81 and 82 or by using the average shift 

factor function. 

5. Fit the t-T shift factors to the WLF model using equation 79. 

6. Optimize the CAM model coefficients to minimize the error between the DSR and 

BBR results.

Typical analysis results are shown in figure 113 and figure 114. In general, the SuperpaveTM

only data analysis results show good agreement with the full analysis results across all of the 

selected binders. The complete results of level 1a verification are summarized in figure 115

through figure 118 in both normal and logarithmic scales. Overall, the calibrated model fits the 

data well, with an overall average error of approximately 3.5 percent and a higher average 

absolute error at 14 percent. These error percentages are comparable for both of the t-T shift 

factor function models. The major difference between these two models appears to be the 

tendency of the average shift factor function model to underpredict the modulus values in the 

high region. Conversely, the average shift factor function tends to show a smaller spread in the 

prediction error. 

1E+00

1E+03

1E+06

1E+09

1E-08 1E-05 1E-02 1E+01 1E+04 1E+07

Reduced Angular Frequency (rad/s)

|G*| (Pa)

Full CAM Fit

Limited CAM Fit

DSR

BBR

1 psi = 6.86 kPa

Figure 113. Graph. Comparison of typical |G*| mastercurves characterized using full 

database and SuperpaveTM only database. 
144 

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

-50 0 50 100 150

Temperature (C)

log aT

Full Data

Limited Data

°C = (°F-32)/1.8

Figure 114. Graph. Comparison of typical t-T shift factors characterized using full

database and SuperpaveTM only database. 

0.E+00

1.E+08

2.E+08

3.E+08

4.E+08

5.E+08

6.E+08

7.E+08

0.E+00 1.E+08 2.E+08 3.E+08 4.E+08 5.E+08 6.E+08 7.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2

 = 0.865

1 psi = 6.86 kPa

Figure 115. Graph. Level 1a verification of limited data analysis procedure in arithmetic 

scale using phenomenological shift factor function model. 
145 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.997

1 psi = 6.86 kPa

Figure 116. Graph. Level 1a verification of limited data analysis procedure in logarithmic 

scale using phenomenological shift factor function model. 

0.E+00

1.E+08

2.E+08

3.E+08

4.E+08

5.E+08

6.E+08

7.E+08

0.E+00 1.E+08 2.E+08 3.E+08 4.E+08 5.E+08 6.E+08 7.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.894

1 psi = 6.86 kPa

Figure 117. Graph. Level 1a verification of limited data analysis procedure in arithmetic 

scale using average shift factor function model. 
146 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.997

1 psi = 6.86 kPa

Figure 118. Graph. Level 1a verification of limited data analysis procedure in logarithmic 

scale using average shift factor function model. 

B.12.2 Level 1b Verification

After verifying the limited data shifting procedure, attention is focused on SuperpaveTM testing 

on original-, RTFO-, and PAV-aged binders. Because the shear modulus data are desired under 

the RTFO conditions for the mixture |E*| predictions and because most tests performed on the 

original-aged binder are also performed on the RTFO-aged binder, the primary model of interest 

is the relationship between the PAV and RTFO binders. Such a model would then be applied to 

the DSR tests at the intermediate temperatures and the BBR test results. The steps taken in 

level 1b verification are as follows: 

1. Find DSR results under RTFO-aging conditions at 10 rad/s at 140, 158, and 176 oF 

(60, 70, and 80 °C). 

2. Find DSR results under PAV-aging conditions at 10 rad/s at 59, 77, and 95 oF (15, 25, 

and 35 °C). 

3. Find BBR results under PAV-aging conditions at 60 s at either -22, -11.2, and -0.4 oF 

(-30, -24, and -18 °C) or at -11.2, -0.4, and 10.4 oF (-24, -18, and -12 °C). 

4. Apply the PAR model from equations 92 or 93 to predict RTFO values for intermediate 

temperature DSR results and BBR results. 

5. Convert the BBR results to |G*| values using equation 89. 

6. Determine the t-T shift factors using equations 81 and 82 or via the average shift 

factor function. 
147 

7. Fit the t-T shift factors to the WLF model using equation 79. 

8. Optimize the CAM model coefficients to minimize the error between the DSR and 

BBR results.

Typical analysis results for this method are shown in figure 119 and figure 120. A slight decrease 

in the model strength is observed when compared to the level 1a verification, but in general, the 

SuperpaveTM only data analysis results show good agreement with the full analysis results across 

all of the selected binders. The complete results of level 1b verification are summarized in 

figure 121 and figure 122 for the phenomenological shift factor function model and in figure 123

and figure 124 for the average shift factor function in both normal and logarithmic scales. 

Overall, the calibrated model shows an approximately 13 percent error, with the average absolute 

error higher at 19 percent. As with level 1a verification, these numbers are comparable for both 

of the t-T shift factor function models. Also, these results tend to show a smaller variability with 

the average shift factor function model. 

1E+00

1E+03

1E+06

1E+09

1E+12

1E-08 1E-05 1E-02 1E+01 1E+04 1E+07

Reduced Angular Frequency (rad/s)

|G*| (Pa)

Full CAM Fit

Limited + PAR Fit

DSR

BBR

1 psi = 6.86 kPa

Figure 119. Graph. Comparison of typical |G*| mastercurve characterized using full 

database and SuperpaveTM only database plus PAR model. 
148 

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

-50 0 50 100 150

Temperature (C)

log aT

Full Data

Limited Data + PAR

°C = (°F-32)/1.8

Figure 120. Graph. Comparison of typical t-T shift factors characterized using full 

database and SuperpaveTM only database plus PAR model. 

0.E+00

1.E+08

2.E+08

3.E+08

4.E+08

5.E+08

6.E+08

7.E+08

0.E+00 1.E+08 2.E+08 3.E+08 4.E+08 5.E+08 6.E+08 7.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.813

1 psi = 6.86 kPa

Figure 121. Graph. Level 1b verification of limited data analysis procedure in arithmetic 

scale using phenomenological shift factor function model. 
149 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.995

1 psi = 6.86 kPa

Figure 122. Graph. Level 1b verification of limited data analysis procedure in logarithmic 

scale using phenomenological shift factor function model. 

0.E+00

1.E+08

2.E+08

3.E+08

4.E+08

5.E+08

6.E+08

7.E+08

0.E+00 1.E+08 2.E+08 3.E+08 4.E+08 5.E+08 6.E+08 7.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.880

1 psi = 6.86 kPa

Figure 123. Graph. Level 1b verification of limited data analysis procedure in arithmetic 

scale using average shift factor function model. 
150 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2

 = 0.996

1 psi = 6.86 kPa

Figure 124. Graph. Level 1b verification of limited data analysis procedure in logarithmic 

scale using average shift factor function model. 

B.12.3 Level 2 Verification

Level 2 verification consists of data that have not been used in the development of either the 

PAR model or the t-T shift factor relationship. In addition, it consists of only DSR data, but 

the data cover the complete range expected in normal SuperpaveTM DSR characterization. A 

third level of verification that does not have the complete DSR data available is shown 

subsequently. The FHWA ALF study binders used previously to verify the PAR model are 

used for level 2 verification. 

The steps taken in level 2 verification are as follows: 

1. Find DSR results under RTFO-aging conditions at 10 rad/s at the two standard temperatures 

available that are closest to the high temperature PG of the binder—136.4 and 147.2 oF 

(58 and 64 °C), 147.2 and 168.8 oF (64 and 76 °C), or 147.2 and 179.6 oF (76 and 82 °C). 

2. Find DSR results under PAV-aging conditions at 10 rad/s at 59, 77, and 95 oF (15, 25, 

and 35 °C). 

3. Apply the PAR model from equation 92 to predict RTFO values for intermediate temperature 

DSR results. 

4. Determine the t-T shift factors using equation 81 or by using the average shift factor 

function. 

5. Fit the t-T shift factors to the WLF model using equation 79. 
151 

6. Optimize the CAM model coefficients to minimize the error in the DSR results.

The major difference between level 1 and 2 verification is the lack of BBR data in the process. 

Because it is shown earlier in this section that differences in |G*| mastercurves can be large 

at temperatures below 41 °F (5 °C) when BBR data are not used for the calibration, only 41, 

68, 105, and 129.2 °F (5, 20, 40, and 54 °C) data are used to analyze the differences in 

level 2 verification. 

The results of level 2 verification are shown in figure 125 through figure 128. The model 

strength is good with an excellent coefficient of correlation. Both the average shift factor 

function and the phenomenological model show good results. The average function performs 

better in arithmetic space, and the phenomenological function appears to perform better in 

logarithmic space. 

0.E+00

1.E+08

2.E+08

0.E+00 1.E+08 2.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.942

1 psi = 6.86 kPa

Figure 125. Graph. Level 2 verification of limited data analysis procedure in arithmetic 

scale using phenomenological shift factor function model. 
152 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2

 = 0.997

1 psi = 6.86 kPa

Figure 126. Graph. Level 2 verification of limited data analysis procedure in logarithmic 

scale using phenomenological shift factor function model. 

0.E+00

1.E+08

2.E+08

0.E+00 1.E+08 2.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.953

1 psi = 6.86 kPa

Figure 127. Graph. Level 2 verification of limited data analysis procedure in arithmetic 

scale using average shift factor function model. 
153 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2

 = 0.996

1 psi = 6.86 kPa

Figure 128. Graph. Level 2 verification of limited data analysis procedure in logarithmic 

scale using average shift factor function model. 

B.12.4 Level 3 Verification

The final verification level considered in this report is the verification of the analysis procedure 

for DSR data that are not complete. Specifically, the data at high temperatures are assumed to be 

unavailable. For this purpose, the FHWA mobile trailer database outlined in table 42 is used. As 

with level 2 verification, BBR data are unavailable for calibration purposes, and 14 oF (-10 °C) 

data are not included in the verification process. Without available high temperature data, it is 

assumed that each binder receives its high temperature grade based on the RTFO-aged binder 

(i.e., |G*|/sin(δ ) = 0.32 psi (2.2 kPa) exactly at the high PG). In addition, it is known that asphalt 

binder has a phase angle of approximately 80 degrees at these high temperatures. These 

assumptions imply that |G*| at the high temperature PG is 0.31 psi (2,166 Pa), which is used in 

the fitting process. 

The steps taken in level 3 verification are as follows: 

1. Find DSR results under PAV-aging conditions at 10 rad/s at the three coolest temperatures 

available for a given binder (see table 42). 

2. Apply the PAR model using equation 92 to predict RTFO values for intermediate 

temperature DSR results. 

3. Assume |G*| under RTFO conditions at the high temperature PG to be 0.31 psi (2,166 Pa). 

4. Determine the t-T shift factors using equation 81 or via the average shift factor function. 

5. Fit the t-T shift factors to the WLF model using equation 79. 
154 

6. Optimize the CAM model coefficients to minimize the error in the DSR results.

The results of level 3 verification are shown in figure 129 through figure 132. From these 

figures, it is observed that this dataset shows a reduction in predictability compared to the level 2 

analysis but improved predictability as measured by R2

. However, it should be noted that level 3 

analysis does not include model data at 14 °F (-10 °C), while the two level 1 verification 

analyses do include these 14 °F (-10 °C) data.

0.E+00

1.E+08

2.E+08

0.0.E+00 1.0.E+08 2.0.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.927

1 psi = 6.86 kPa

Figure 129. Graph. Level 3 verification of limited data analysis procedure in arithmetic 

scale using phenomenological shift factor function model. 
155 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.988

1 psi = 6.86 kPa

Figure 130. Graph. Level 3 verification of limited data analysis procedure in logarithmic 

scale using phenomenological shift factor function model. 

0.E+00

1.E+08

2.E+08

0.0.E+00 1.0.E+08 2.0.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2 = 0.894

1 psi = 6.86 kPa

Figure 131. Graph. Level 3 verification of limited data analysis procedure in arithmetic 

scale using average shift factor function model. 
156 

1.E+02

1.E+04

1.E+06

1.E+08

1.E+02 1.E+04 1.E+06 1.E+08

|G*| Predicted (Pa)

|G*| from Measurement (Pa)

R2

 = 0.992

1 psi = 6.86 kPa

Figure 132. Graph. Level 3 verification of limited data analysis procedure in logarithmic 

scale using average shift factor function model.
157 

APPENDIX C: AMPT VERSUS TP-62 

C.1 EXPERIMENTAL VERIFICATION OF AMPT AND TP-62 DIFFERENCES

To assess differences in the measured moduli determined from the AMPT and TP-62 protocols, a 

joint study was carried out between researchers at the Turner-Fairbank Highway Research Center 

(TFHRC) and NCSU. For this study, TFHRC performed dynamic modulus testing on a mixture 

following the AMPT TP, and NCSU performed testing on the same mixture using the TP-62 

protocol.(8) In both cases, three replicates have been tested. To reduce any variability not related 

to the equipment and protocols, all specimens were fabricated at NCSU and randomly sampled 

for either AMPT testing or TP-62 testing. The details of each testing protocol are summarized 

in table 43. 

Table 43. TP summary. 

Factor AMPT TP-62

Temperature (°F) 40, 70, 100, and 130 14, 40, 70, 100, and 130 

Frequency (Hz) 20, 10, 5, 1, 0.5, and 0.1 25, 10, 5, 1, 0.5, and 0.1

Microstrain target 75–125 50–75

LVDT gauge length 

(mm) 70 100

Load direction Bottom loading Top loading

End treatment Teflon®

Greased double latex 

membranes

Conditioning 

External temperature 

chamber, then equalize 

in AMPT for 3 min

Equalize for 2.5–3.0 h in 

test machine

Rest period between 

frequencies (s) 0 300

Calculations

NCHRP 09-29 

final 10 cycles(50)

NCHRP 09-29 

final five cycles(50)

°C = (°F-32)/1.8

1 inch = 25.4 mm

The mixture used for this purpose is a 0.371-inch (9.5-mm) SuperpaveTM mixture typically used 

in North Carolina for surface courses. The gradation of this mixture is given in figure 133, and 

the relevant volumetric properties are summarized in table 44. All tests were conducted at 

5.9 percent ±0.1 percent air void levels.
158 

0

20

40

60

80

100

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Sieve.45

% Passing

Maximum Density

Control Points

Gradation

Figure 133. Graph. Test mixture gradation. 

Table 44. Test mixture volumetric properties. 

Volumetric Property

Mix 

Design

Test 

Samples

Va (percent) 3.8 5.9

VMA (percent) 15.6 17.5

VFA (percent) 75.7 66.2

Asphalt content (percent) 5.2 5.2

Percent effective binder content 4.9 4.9

Dust percentage 1.2 1.2

Gmm 2.616 2.616

Bulk specific gravity of the aggregate 2.828 2.828

Effective specific gravity of the aggregate 2.855 2.855

Gb 1.035 1.035

Results from the experimental study are summarized in figure 134 and figure 135, where the 

average dynamic moduli from the TP-62 protocol are plotted against the average moduli from 

the AMPT protocol. Error bars in these figures represent a single standard deviation from the 

mean. From these figures, it is observed that the AMPT test results are systematically lower 

than those from the TP-62 protocol; the difference between the two datasets is approximately 

13 percent. Statistical analysis of these values using the step-down bootstrap method has also 

been performed. This method is used in lieu of multiple paired t-tests due to the effect of 

experimentwise error rates, which results in statistical errors when making multiple comparisons. 

Specifically, failing to account for this error rate increases the probability of finding significance 

when none is present. The statistical analysis results are shown by temperature and frequency in 

table 45. Note that in this table, the conditions under which the means are statistically similar 

are bold. 
159 

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

0.0E+00 5.0E+03 1.0E+04 1.5E+04 2.0E+04 2.5E+04

|E*| via TP-62 (MPa)

|E*| via AMPT (MPa)

1 psi = 6.86 kPa

Figure 134. Graph. Comparison of |E*| measured via AMPT and TP-62 protocols in 

arithmetic scale. 

1.E+02

1.E+03

1.E+04

1.E+05

1.E+02 1.E+03 1.E+04 1.E+05

|E*| via TP-62 (MPa)

|E*| via AMPT (MPa)

1 psi = 6.86 kPa

Figure 135. Graph. Comparison of |E*| measured via AMPT and TP-62 protocols in 

logarithmic scale. 
160 

Table 45. Statistical summary of AMPT and TP-62 test results. 

Temperature 

(°C)

Frequency 

(Hz)

|E*| 

AMPT 

(psi)

|E*| TP-62 

(psi) p-Value

4 25.00 2,145,226 2,420,540 0.032

4 10.00 1,989,606 2,284,746 0.020

4 5.00 1,838,144 2,111,129 0.030

4 1.00 1,503,747 1,726,774 0.026

4 0.50 1,359,729 1,601,117 0.019

4 0.10 1,050,375 1,234,431 0.023

21 25.00 1,030,409 1,237,696 0.020

21 10.00 899,831 1,022,446 0.023

21 5.00 785,545 881,347 0.025

21 1.00 550,882 628,569 0.033

21 0.50 468,842 524,847 0.068

21 0.10 306,841 358,120 0.057

37 25.00 385,448 464,233 0.008

37 10.00 318,540 384,219 0.010

37 5.00 263,476 330,282 0.002

37 1.00 160,938 198,110 0.008

37 0.50 130,346 167,580 0.005

37 0.10 75,190 96,587 0.011

54 25.00 153,735 177,050 0.003

54 10.00 127,039 128,097 0.801

54 5.00 102,669 101,164 0.672

54 1.00 58,086 59,737 0.377

54 0.50 42,997 48,863 0.022

54 0.10 23,863 33,547 0.005

°C = (°F-32)/1.8

1 psi = 6.86 kPa

Note: Bold text indicates conditions where means are statistically similar.

C.2 COMPARISON OF AMPT AND TP-62 PROTOCOLS WITH THE AVAILABLE 

DATABASE 

To assess the differences observed between the two |E*| measurement protocols, a more 

comprehensive analysis was performed using the databases available in this study. The two 

AMPT and TP-62 databases were segregated based on the temperatures at which the |E*| values 

were measured. Because these two databases cover different ranges of parameters, it is useful to 

examine the distribution of the relevant parameters for the two databases. Figure 136 through 

figure 157 present the distribution and range of each parameter in the two databases. In 

figure 158 through figure 162, the measured |E*| data points available for some specific 

temperatures for each type of database are shown by frequency. Based on observations from 

these figures and the difference equation shown in equation 100, differences between the 

databases containing AMPT and TP-62 measurements are evident, as can be seen in table 46. 
161 

62

62

* *

% *100

*

TP AMPT

TP

E E

Difference E

−

−

− =

(100)

Based on this description, the following differences are observed at each temperature:

• 40 °F (4.4 oC): 43.63 percent.

• 70 °F (21.1 oC): 57.84 percent.

• 100 °F (37.8 oC): 61.95 percent.

• 129 °F (53.9 oC): 46.26 percent.

• 130 °F (54.4 oC): 57.83 percent.

0

500

1000

1500

2000

2500

3000

3500

4000

14 32 50 68 86 104 122 140

Temperature (F)

Number of Datapoints

TP62 AMPT

°C = (°F-32)/1.8

Figure 136. Graph. Frequency distribution of temperature in AMPT versus 

TP-62 databases. 
162 

14.0

69.7

130.0

39.2

86.7

129.9

0

20

40

60

80

100

120

140

Min Avg Max

Temperature (F)

TP62 AMPT

°C = (°F-32)/1.8

Figure 137. Graph. Range of temperature in AMPT versus TP-62 databases. 

0

500

1000

1500

2000

2500

3000

3500

0 0.2 0.4 1 3 5 7 9 11 13 15 17 19 21 23 25

Frequency (Hz)

Number of Datapoints

TP62 AMPT

Figure 138. Graph. Frequency distribution of frequency in AMPT versus TP-62 databases. 
163 

0.01

6.5

25.0

0.01

6.7

25.0

0

5

10

15

20

25

30

Min Avg Max

Frequency (Hz)

TP62 AMPT

Figure 139. Graph. Range of loading frequency in AMPT versus TP-62 databases. 

0

1000

2000

3000

4000

5000

6000

7000

0 5 10 15 20 25 30 35

%Retained on 3/4" Sieve (ρ34), %

Number of Datapoints

TP62 AMPT

Figure 140. Graph. Frequency distribution of percentage retained on 3/4-inch (19.05-mm) 

sieve (ρ 

34) in AMPT versus TP-62 databases. 
164 

0.0

5.1

31.0

0.0

1.4

11.6

0

5

10

15

20

25

30

35

Min Avg Max

ρ34 (%)

TP62 AMPT

Figure 141. Graph. Range of percentage retained on 3

/4-inch (19.05-mm) sieve (ρ 

34) in 

AMPT versus TP-62 databases. 

0

500

1000

1500

2000

2500

3000

0 5 10 15 20 25 30 35 40 45 50 55

%Retained on 3/8" Sieve (ρ38), %

Number of Datapoints

TP62 AMPT

Figure 142. Graph. Frequency distribution of percentage retained on 3/8-inch (9.56-mm)

sieve (ρ 

38) in AMPT versus TP-62 databases. 
165 

1.0 2.3

26.5

55.0

15.2

45.1

0

10

20

30

40

50

60

Min Avg Max

ρ38 (%)

TP62 AMPT

Figure 143. Graph. Range of percentage retained on 3

/8-inch (9.56-mm) sieve (ρ 

38) in 

AMPT versus TP-62 databases. 

0

1000

2000

3000

4000

5000

6000

7000

20 30 40 50 60 70 80

%Retained on #4 Sieve (ρ4), %

Number of Datapoints

TP62 AMPT

Figure 144. Graph. Frequency distribution of percentage retained on #4 sieve (ρ 

4) in 

AMPT versus TP-62 databases. 
166 

49.5

22.3

37.6

67.4

14.0

73.0

0

10

20

30

40

50

60

70

80

Min Avg Max

ρ4 (%)

TP62 AMPT

Figure 145. Graph. Range of percentage retained on #4 sieve (ρ 

4) in AMPT versus 

TP-62 databases. 

0

500

1000

1500

2000

2500

3000

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5

%Passing #200 Sieve (ρ200), %

Number of Datapoints

TP62 AMPT

Figure 146. Graph. Frequency distribution of percentage passing #200 sieve (ρ 

200) in 

AMPT versus TP-62 databases. 
167 

4.9

2.6

6.6

4.9

2.7

6.6

0

1

2

3

4

5

6

7

8

Min Avg Max

ρ200 (%)

TP62 AMPT

Figure 147. Graph. Range of percentage passing #200 sieve (ρ 

200) in AMPT versus 

TP-62 databases. 

0

500

1000

1500

2000

2500

3000

3500

4000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Air Voids (Va), %

Number of Datapoints

TP62 AMPT

Figure 148. Graph. Frequency distribution of specimen air voids in AMPT versus TP-62 

databases. 
168 

6.4

12.5

4.5

6.9

8.7

0.7

0

2

4

6

8

10

12

14

Min Avg Max

Va, %

TP62 AMPT

Figure 149. Graph. Range of specimen air voids in AMPT versus TP-62 databases. 

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

5 8 11 14 17

Effective Binder Volume (Vbeff), %

Number of Datapoints

TP62 AMPT

Figure 150. Graph. Frequency distribution of effective binder volume in AMPT versus 

TP-62 databases. 
169 

6.1

10.0

14.2

9.5

12.7

4.8

0

2

4

6

8

10

12

14

16

Min Avg Max

Vbeff, 

%

TP62 AMPT

Figure 151. Graph. Range of effective binder volume in AMPT versus TP-62 databases. 

0

1000

2000

3000

4000

5000

6000

8 12 16 20 24

Voids in Mineral Aggregates (VMA), %

Number of Datapoints

TP62 AMPT

Figure 152. Graph. Frequency distribution of VMA in AMPT versus TP-62 databases. 
170 

10.8

16.4

22.2

9.5

16.4

19.9

0

5

10

15

20

25

Min Avg Max

VMA, %

TP62 AMPT

Figure 153. Graph. Range of VMA in AMPT versus TP-62 databases. 

0

1000

2000

3000

4000

5000

6000

7000

30 40 50 60 70 80 90 100

Voids Filled with Asphalt (VFA), %

Number of Datapoints

TP62 AMPT

Figure 154. Graph. Frequency distribution of VFA in AMPT versus TP-62 databases. 
171 

58.1

32.8

61.9

95.1

42.9

70.3

0

10

20

30

40

50

60

70

80

90

100

Min Avg Max

VFA, %

TP62 AMPT

Figure 155. Graph. Range of VFA in AMPT versus TP-62 databases. 

0

500

1000

1500

2000

2500

3000

3500

0.01 0.1 10 100 1000 10000 100000 1E+06

|G*| (psi)

Number of Datapoints

TP62 AMPT

Figure 156. Graph. Frequency distribution of |G*| in AMPT versus TP-62 databases. 
172 

6.8E+05

2.9E-02 1.4E+04 4.8E-02 5.4E+02 1.8E+04

2.E-02

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

6.E+05

7.E+05

Min Avg Max

|G*| (psi)

TP62 AMPT

1 psi = 6.86 kPa

Figure 157. Range of |G*| in AMPT versus TP-62 databases. 

T = 40F

30

35

40

45

50

55

60

65

ρ34 ρ38 ρ4 ρ200 Va Vbeff VMA VFA |G*|

Variable

% Difference

Figure 158. Graph. Percentage of difference between AMPT versus TP-62 databases based 

on similar ranges of different variables at 39.9 oF (4.4 °C). 
173 

T =70F

30

35

40

45

50

55

60

65

ρ34 ρ38 ρ4 ρ200 Va Vbeff VMA VFA |G*|

Variable

% Difference

Figure 159. Graph. Percentage of difference between AMPT versus TP-62 databases based 

on similar ranges of different variables at 69.9 oF (21.1 oC). 

T = 100F

30

35

40

45

50

55

60

65

ρ34 ρ38 ρ4 ρ200 Va Vbeff VMA VFA |G*|

Variable

% Difference

Figure 160. Graph. Percentage of difference between AMPT versus TP-62 databases based 

on similar ranges of different variables at 100 oF (37.8 oC). 
174 

T = 129.2F

30

35

40

45

50

55

60

65

ρ34 ρ38 ρ4 ρ200 Va Vbeff VMA VFA |G*|

Variable

% Difference

Figure 161. Graph. Percentage of difference between AMPT versus TP-62 databases based 

on similar ranges of different variables at 129.2 oF (54.0 oC). 

T = 130F

30

35

40

45

50

55

60

65

ρ34 ρ38 ρ4 ρ200 Va Vbeff VMA VFA |G*|

Variable

% Difference

Figure 162. Graph. Percentage of difference between AMPT versus TP-62 databases based 

on similar ranges of different variables at 129.9 oF (54.4 oC). 
175 

Table 46. Percentage of difference between AMPT versus TP-62 database based on similar 

ranges of different variables.

Temp 

(°F)

0 ≤ ρ 

34 

≤ 15

5 ≤ ρ 

38 

≤ 50

30 ≤ ρ 

4 

≤ 70

3 ≤ ρ 

200 

≤ 7

5 ≤ Va

≤ 9

8 ≤ Vbeff 

≤ 14

12 ≤ VMA 

≤ 20

50 ≤ VFA 

≤ 80

1e-2 ≤ 

|G*| 

≤ 1e5 

40 46.08 39.80 41.14 43.54 42.75 45.65 44.81 44.29 43.60

70 59.39 47.54 51.74 57.66 57.67 60.23 59.91 58.32 57.84

100 62.63 49.53 51.35 61.61 63.38 64.49 64.36 62.66 61.95

129 45.60 51.02 49.65 46.26 N.A 63.01 46.16 52.09 46.26

130 57.55 40.76 44.14 57.46 60.50 59.99 60.54 57.93 57.83

°C = (°F-32)/1.8

Similar ranges of each variable have been considered for each temperature, and the percentage 

of error has been calculated based on the difference of average TP-62 versus AMPT |E*| 

measurements for the corresponding temperature.

C.3 EVALUATION OF AMPT VERSUS TP-62 PROTOCOLS USING ANN MODEL

A preliminary study was conducted to determine the feasibility and predictability of the ANN 

modeling technique relative to the existing models. This feasibility study was first conducted 

based on |G*| because more existing closed-form models use this parameter as their primary 

input parameter. The ANN models used in this preliminary study are not the final models 

suggested by the research team, but they are similar in form and validation. To ensure full 

coverage of the expected conditions, the most recent Witczak database with available measured 

|G*| data and a portion of the dataset obtained at NCSU with support from the NCDOT were

utilized as the TP-62 training database. Also, appropriate portions of the FHWA mobile trailer 

database and the WRI database (from Kansas and Nevada sites) were considered as the AMPT 

training database (see table 47).(51,52) New parameters were not identified through this study. 

Instead, only those that have been used in the modified Witczak model are incorporated. For 

verification purposes, three different sets of independent databases were used (see table 48). As a 

corollary to this study, an additional ANN model was trained that uses the Hirsch model input 

parameters. The results from this model are given in this section, as well.

Table 47. Summary of database used for training ANN models. 

Type of Database

AMPT TP-62

FHWA I WRI Witczak NCDOT I Total

Number of mixtures 409 24 106 24 563

Number of data points 7,827 500 3,180 644 12,151

Number of binders 13 8 17 5 43

Number of gradation 

variations 13 12 13 19 57 

Number of volumetric 

variations 256 13 98 24 391

Note: FHWA I consists of the mixtures from 12 States in the FHWA mobile trailer database.
176 

Table 48. Summary of the database used for verification of ANN models. 

Type of Database

AMPT TP-62

FHWA II Citgo NCDOT II Total

Number of mixtures 84 8 12 104

Number of data points 1,652 168 338 2,158

Number of binders 3 2 3 8

Number of gradation 

variations 3 1 12 16

number of volumetric 

variations 75 1 12 88

Note: FHWA II consists of the mixtures from three States in the FHWA mobile trailer database 

with the following site IDs: 1-IA0358, 2-WA0463, and 3-KS464.

It should be noted that the two TPs, AMPT and TP-62, were used to measure the |E*| values in 

the various databases. To illustrate any possible differences between the two protocols, three 

different ANNs were developed using the Witczak-based input parameters, as shown in table 49. 

G-GR pANN was trained using data from both the AMPT and TP-62 protocols, whereas AMPT 

pANN and TP-62 pANN models were trained using the data from AMPT only and TP-62 only. 

Table 49 summarizes the databases used to train and verify the ANNs. 
Table 49. Description of the developed ANN Models and their validation statistics. 

Model

Data Used in ANN 

Training

Description

Reference 

Scale

Statistical 

Parameters for 

Training Data

Statistical Parameters for 

Verification Data

AMPT TP-62 FHWA II NCDOT II Citgo

G-GR pANN FHWA I Witczak

ANNs trained 

with modified 

Witczak 

parameters

Arithmetic

Se/Sy = 0.29

R2= 0.92 

Se/Sy = 0.38

R2 = 0.86 

Se/Sy = 0.33

R2= 0.97 

Se/Sy = 0.52

R2= 0.94 

WRI NCDOT I Log

Se/Sy = 0.15

R2= 0.98

Se/Sy = 0.35

R2= 0.91

Se/Sy = 0.27

R2= 0.96

Se/Sy = 0.59

R2= 0.96

AMPT pANN FHWA I Arithmetic

Se/Sy = 0.24

R2= 0.94

Se/Sy = 0.36

R2= 0.91

Se/Sy = 0.63

R2= 0.87

Se/Sy = 0.37

R2= 0.88

WRI Log

Se/Sy = 0.16

R2= 0.97

Se/Sy = 0.38

R2= 0.90

Se/Sy = 0.60

R2= 0.89

Se/Sy = 0.48

R2= 0.91

TP-62 pANN Witczak Arithmetic

Se/Sy = 0.34

R2= 0.88

Se/Sy = 2.08

R2= 0.77

Se/Sy = 0.24

R2= 0.95

Se/Sy = 1.20

R2= 0.97

NCDOT I Log

Se/Sy = 0.18

R2= 0.97

Se/Sy = 0.99

R2= 0.82

Se/Sy = 0.27

R2= 0.93

Se/Sy = 0.53

R2= 0.99

Modified Witczak 

Model 

Arithmetic

Se/Sy = 0.92

R2= 0.91 

Se/Sy = 0.71

R2= 0.91 

Se/Sy = 0.64

R2= 0.98 

Log

Se/Sy = 0.58

R2= 0.92 

Se/Sy = 0.19

R2= 0.98 

Se/Sy = 0.26

R2= 0.99 

Hirsch Model Arithmetic

Se/Sy = 0.30

R2= 0.92

Se/Sy = 0.47

R2= 0.97

Se/Sy = 0.11

R2= 0.99

Log

Se/Sy = 0.39

R2= 0.92

Se/Sy = 0.26

R2 = 0.97

Se/Sy = 0.09

R2= 0.99

Al-Khateeb 

Model 

Arithmetic

Se/Sy = 0.48

R2 = 0.89

Se/Sy = 0.55

R2= 0.93

Se/Sy = 0.36

R2= 0.93

Log

Se/Sy = 0.43

R2 = 0.92 

Se/Sy = 0.40

R2= 0.93 

Se/Sy = 0.17

R2= 0.97 

Note: Blank cells indicate information is not applicable.

177 
178 

The ANN models perform well, as shown in figure 163 to figure 180, which display the 

prediction accuracies of the different models for the combined AMPT and TP-62 data 

(figure 163 to figure 168), TP-62 data only (figure 169 to figure 174), and AMPT data only 

(figure 175 to figure 180). Also, these three groups of figures show the prediction accuracies of 

the ANNs separately. In these three figures, the type of data (i.e., AMPT versus TP-62) used in 

the ANN training matches the type of data used in the verification (e.g., figure 163 shows the 

prediction accuracy of the G-GR pANN model trained with the combined AMPT and TP-62 data 

on the combined AMPT and TP-62 data, etc.). It is noted that the data used in these figures were 

not included in the ANN training. 

Figure 181 through figure 204 further demonstrate the differences between the AMPT and the 

TP-62 data and their effect on the prediction accuracies of the different ANNs. FHWA II data 

used in figure 181 through figure 188 are obtained using the AMPT protocol. The TP-62 pANN 

model trained with the TP-62 data and the modified Witczak model overpredict the measured 

|E*| values. Figure 189 through figure 196 present the prediction results for the NCDOT II data, 

which were measured using the TP-62 protocol. These figures illustrate the opposite effect on the 

prediction bias, that is, the effect of using the TP-62 data in the ANN training and predicting the 

AMPT data. In this case, the AMPT pANN model, trained using the AMPT data, underpredicts 

the |E*| values. The G-GR pANN model provides a promising ANN-based |E*| model, and the 

TP-62 pANN model shows good predictions without any significant bias. With the exception of 

the Citgo dataset, the G-GR pANN model provides high goodness of fit and correlation, as seen 

in table 49. The promising feature of the G-GR pANN model is that it improves the bias of |E*| 

predictions, particularly at high and low temperatures. This new ANN model is more sensitive 

to, and thus more likely to capture, the changes in volumetric parameters than all the other 

existing predictive models.

The findings from figure 163 to figure 204 are summarized as follows:

• The |E*| values measured by the AMPT protocol seem to be slightly different from those 

measured by the TP-62 protocol. The |E*| predictive models developed using the |E*|

values measured by the TP-62 overpredict the |E*| values measured by the AMPT. 

• Overall, the G-GR pANN model, trained with the combination of the AMPT and TP-62 

data, shows excellent statistics in terms of high accuracy and low bias, especially at 

extremely high and low temperatures.
179 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.84, Se/Sy=0.76

G-GR pANN

R2

=0.92, Se/Sy=0.29

1 psi = 6.86 kPa

Figure 163. Graph. Prediction of the combination of AMPT and TP-62 data using the

modified Witczak and G-GR pANN models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2=0.92, Se/Sy=0.40

G-GR pANN

R2

=0.98, Se/Sy=0.15

1 psi = 6.86 kPa

Figure 164. Graph. Prediction of the combination of AMPT and TP-62 data using the 

modified Witczak and G-GR pANN models in logarithmic scale. 
180 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.81, Se/Sy=0.48

1 psi = 6.86 kPa

Figure 165. Graph. Prediction of the combination of AMPT and TP-62 data using the 

Hirsch model in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.92, Se/Sy=0.31

1 psi = 6.86 kPa

Figure 166. Graph. Prediction of the combination of AMPT and TP-62 data using the 

Hirsch model in logarithmic scale. 
181 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.71, Se/Sy=0.65

1 psi = 6.86 kPa

Figure 167. Graph. Prediction of the combination of AMPT and TP-62 data using the 

Al-Khateeb model in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.89, Se/Sy=0.34

1 psi = 6.86 kPa

Figure 168. Graph. Prediction of the combination of AMPT and TP-62 data using the 

Al-Khateeb model in logarithmic scale. 
182 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

AMPT pANN

LOE

Modified Witczak

R2

=0.86, Se/Sy=1.04

AMPT pANN

R2

=0.94, Se/Sy=0.24

1 psi = 6.86 kPa

Figure 169. Graph. Prediction of the AMPT data using the modified Witczak and AMPT

pANN models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

AMPT pANN

LOE

Modified Witczak

R2=0.93, Se/Sy=0.48

AMPT pANN

R2

=0.97, Se/Sy=0.16

1 psi = 6.86 kPa

Figure 170. Graph. Prediction of the AMPT data using the modified Witczak and AMPT 

pANN models in logarithmic scale. 
183 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.91, Se/Sy=0.31

1 psi = 6.86 kPa

Figure 171. Graph. Prediction of the AMPT data using the Hirsch model in 

arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.93, Se/Sy=0.34

1 psi = 6.86 kPa

Figure 172. Graph. Prediction of the AMPT data using the Hirsch model in 

logarithmic scale. 
184 

0.E+00

2.E+06

4.E+06

6.E+06

8.E+06

0.E+00 2.E+06 4.E+06 6.E+06 8.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2=0.88, Se/Sy=0.39

1 psi = 6.86 kPa

Figure 173. Graph. Prediction of the AMPT data using the Al-Khateeb model in 

arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.94, Se/Sy=0.33

1 psi = 6.86 kPa

Figure 174. Graph. Prediction of the AMPT data using the Al-Khateeb model in 

logarithmic scale. 
185 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

TP62 pANN

LOE

Modified Witczak

R2

=0.79, Se/Sy=0.83

TP62 pANN

R2

=0.88, Se/Sy=0.34

1 psi = 6.86 kPa

Figure 175. Graph. Prediction of the TP-62 data using the modified Witczak and TP-62 

pANN models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

TP62 pANN

LOE

Modified Witczak

R2

=0.92, Se/Sy=0.33

TP62 pANN

R2

=0.97, Se/Sy=0.18

1 psi = 6.86 kPa

Figure 176. Graph. Prediction of the TP-62 data using the modified Witczak and TP-62 

pANN models in logarithmic scale. 
186 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.79, Se/Sy=0.53

1 psi = 6.86 kPa

Figure 177. Graph. Prediction of the TP-62 data using the Hirsch model in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.92, Se/Sy=0.30

1 psi = 6.86 kPa

Figure 178. Graph. Prediction of the TP-62 data using the Hirsch model in 

logarithmic scale. 
187 

0.0E+00

4.0E+06

8.0E+06

1.2E+07

0.0E+00 4.0E+06 8.0E+06 1.2E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.75, Se/Sy=0.75

1 psi = 6.86 kPa

Figure 179. Graph. Prediction of the TP-62 data using the Al-Khateeb model in 

arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.88, Se/Sy=0.40

1 psi = 6.86 kPa

Figure 180. Graph. Prediction of the TP-62 data using the Al-Khateeb model in logarithmic

scale. 
188 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.91, Se/Sy=0.92

G-GR pANN

R2

=0.86, Se/Sy=0.38

1 psi = 6.86 kPa

Figure 181. Graph. Prediction of the FHWA II data using the modified Witczak and G-GR 

pANN models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.92, Se/Sy=0.58

G-GR pANN

R2

=0.91, Se/Sy=0.35

1 psi = 6.86 kPa

Figure 182. Graph. Prediction of the FHWA II data using the modified Witczak and G-GR 

pANN models in logarithmic scale. 
189 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

AMPT pANN

TP62 pANN

LOE

AMPT pANN

R2=0.91, Se/Sy=0.36

TP62 pANN

R2

=0.77, Se/Sy=2.08

1 psi = 6.86 kPa

Figure 183. Graph. Prediction of the FHWA II data using the AMPT pANN and TP-62 

pANN models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

AMPT pANN

TP62 pANN

LOE

AMPT pANN

R2

=0.90, Se/Sy=0.38

TP62 pANN

R2

=0.82, Se/Sy=0.99

1 psi = 6.86 kPa

Figure 184. Graph. Prediction of the FHWA II data using the AMPT pANN and TP-62 

pANN models in logarithmic scale. 
190 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.92, Se/Sy=0.30

1 psi = 6.86 kPa

Figure 185. Graph. Prediction of the FHWA II data using the Hirsch model in 

arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.92, Se/Sy=0.39

1 psi = 6.86 kPa

Figure 186. Graph. Prediction of the FHWA II data using the Hirsch model in 

logarithmic scale. 
191 

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

0.E+00 1.E+06 2.E+06 3.E+06 4.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.89, Se/Sy=0.48

1 psi = 6.86 kPa

Figure 187. Graph. Prediction of the FHWA II data using the Al-Khateeb model in 

arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.92, Se/Sy=0.43

1 psi = 6.86 kPa

Figure 188. Graph. Prediction of the FHWA II data using the Al-Khateeb model in 

logarithmic scale. 
192 

0.0E+00

2.5E+06

5.0E+06

7.5E+06

1.0E+07

0.0E+00 2.5E+06 5.0E+06 7.5E+06 1.0E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2=0.91, Se/Sy=0.71

G-GR pANN

R2

=0.97, Se/Sy=0.33

1 psi = 6.86 kPa

Figure 189. Graph. Prediction of the NCDOT II data using the modified Witczak and 

G-GR pANN models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.98, Se/Sy=0.19

G-GR pANN

R2

=0.96, Se/Sy=0.27

1 psi = 6.86 kPa

Figure 190. Graph. Prediction of the NCDOT II data using the modified Witczak and 

G-GR pANN models in logarithmic scale. 
193 

0.0E+00

2.5E+06

5.0E+06

7.5E+06

1.0E+07

0.0E+00 2.5E+06 5.0E+06 7.5E+06 1.0E+07

Measured |E*| (psi)

Predicted |E*| (psi)

AMPT pANN

TP62 pANN

LOE

AMPT pANN

R2

=0.87, Se/Sy=0.63

TP62 pANN

R2

=0.95, Se/Sy=0.24

1 psi = 6.86 kPa

Figure 191. Graph. Prediction of the NCDOT II data using the AMPT pANN and TP-62 

pANN models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

AMPT pANN

TP62 pANN

LOE

AMPT pANN

R2

=0.89, Se/Sy=0.60

TP62 pANN

R2

=0.93, Se/Sy=0.27

1 psi = 6.86 kPa

Figure 192. Graph. Prediction of the NCDOT II data using the AMPT pANN and TP-62 

pANN models in logarithmic scale. 
194 

0.0E+00

2.5E+06

5.0E+06

7.5E+06

1.0E+07

0.0E+00 2.5E+06 5.0E+06 7.5E+06 1.0E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2=0.97, Se/Sy=0.47

1 psi = 6.86 kPa

Figure 193. Graph. Prediction of the NCDOT II data using the Hirsch model in 

arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.97, Se/Sy=0.26

1 psi = 6.86 kPa

Figure 194. Graph. Prediction of the NCDOT II data using the Hirsch model in 

logarithmic scale. 
195 

0.0E+00

2.5E+06

5.0E+06

7.5E+06

1.0E+07

0.0E+00 2.5E+06 5.0E+06 7.5E+06 1.0E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2=0.93, Se/Sy=0.55

1 psi = 6.86 kPa

Figure 195. Graph. Prediction of the NCDOT II data using the Al-Khateeb model in 

arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.93, Se/Sy=0.40

1 psi = 6.86 kPa

Figure 196. Graph. Prediction of the NCDOT II data using the Al-Khateeb model in 

logarithmic scale. 
196 

0.E+00

2.E+06

4.E+06

6.E+06

0.E+00 2.E+06 4.E+06 6.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2=0.98, Se/Sy=0.64

G-GR pANN

R2

=0.94, Se/Sy=0.52

1 psi = 6.86 kPa

Figure 197. Graph. Prediction of the Citgo data using the modified Witczak and G-GR 

pANN models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Modified Witczak

G-GR pANN

LOE

Modified Witczak

R2

=0.99, Se/Sy=0.26

G-GR pANN

R2

=0.96, Se/Sy=0.59

1 psi = 6.86 kPa

Figure 198. Graph. Prediction of the Citgo data using the modified Witczak and G-GR 

pANN models in logarithmic scale. 
197 

0.E+00

2.E+06

4.E+06

6.E+06

0.E+00 2.E+06 4.E+06 6.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

AMPT pANN

TP62 pANN

LOE

AMPT pANN

R2=0.88, Se/Sy=0.37

TP62 pANN

R2

=0.97, Se/Sy=1.20

1 psi = 6.86 kPa

Figure 199. Graph. Prediction of the Citgo data using the AMPT pANN and TP-62 pANN 

models in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

AMPT pANN

TP62 pANN

LOE

AMPT pANN

R2

=0.91, Se/Sy=0.48

TP62 pANN

R2

=0.99, Se/Sy=0.53

1 psi = 6.86 kPa

Figure 200. Graph. Prediction of the Citgo data using the AMPT pANN and TP-62 pANN 

models in logarithmic scale. 
198 

0.E+00

2.E+06

4.E+06

6.E+06

0.E+00 2.E+06 4.E+06 6.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2=0.99, Se/Sy=0.11

1 psi = 6.86 kPa

Figure 201. Graph. Prediction of the Citgo data using the Hirsch model in arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Hirsch

LOE

Hirsch

R2

=0.99, Se/Sy=0.09

1 psi = 6.86 kPa

Figure 202. Graph. Prediction of the Citgo data using the Hirsch model in 

logarithmic scale. 
199 

0.E+00

2.E+06

4.E+06

6.E+06

0.E+00 2.E+06 4.E+06 6.E+06

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.93, Se/Sy=0.36

1 psi = 6.86 kPa

Figure 203. Graph. Prediction of the Citgo data using the Al-Khateeb model in 

arithmetic scale. 

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Measured |E*| (psi)

Predicted |E*| (psi)

Al-Khateeb

LOE

Al-Khateeb

R2

=0.97, Se/Sy=0.17

1 psi = 6.86 kPa

Figure 204. Graph. Prediction of the Citgo data using the Al-Khateeb model in 

logarithmic scale. 

201 

APPENDIX D: ANN MODEL FACTORS

D.1 INTRODUCTION

The ANN model developed herein contains a mapping ANN architecture and is based on 

supervised learning. In the developed network, the learning method used is a feed forward back 

propagation, which is one of the best known types of ANN models. The sigmoidal function, 

which is shown in equation 101, was chosen as the transfer function. After an indepth 

investigation of network configurations, it was found that the three-layer network with equal 

nodes in the first two layers is the most appropriate configuration. All three ANN models share 

some basic functions, which are shown in equations 101–105. Equations specific to the 

MR ANN are equations 106–108, and equations specific to the GV ANN and VV ANN models 

are equations 109–111. For equations 101–111, it should be understood that when a single index 

is used, it indicates an array. When dual indices are used, they represent a matrix with the first 

letter indicating the values in the row and the second letter indicating the values in the column. 

Index i represents the number of input parameters, index k represents the number of nodes in the 

first hidden layer, and index j represents the number of nodes in the second hidden layer. For the 

MR ANN, l represents the number of output values. All inputs are scaled to have a value between 

-1 and 1. The normalization equation is shown in equation 112. Also, note that the MR model 

produces coefficients for the sigmoidal function and is provided in equation 113. The units of the 

prediction are in megapascals as compared to pounds per square inch, which has been used 

elsewhere in this report.

( ) 2

2 1

1 T f T

e− = − + (101)

11 1

1

ˆ m

k k ik i

i

H B WP == +∑ (102)

( ) 1 1 ˆH fH k k = (103) 

2 2 12

1

ˆ n

j j k kj

k

H B HW =

= +∑ (104)

( ) 2 2 ˆH fH j j = (105) 

3 0 23

1

ˆ n

l l j jl

j

H B HW =

= +∑ (106)

( ) 3 3 ˆH fH l l = (107) 
202 

( ) (( ) ( ) ) ( ) 3 max min

min

ˆ 1

2

l l

l l l

Sig Sig

Sig H Sig − = + +

(108)

3 0 23

1

ˆ n

j j

j

H B HW == +∑ (109)

( ) 3 3 H fH ˆ = (110) 

( ) ( max min ) 3

min

log | *| log | *| ˆ log * 1 log | *| 2E E

E H E − = + +

(111)

Where:

T = The placeholder variable. 

f T( ) = The transfer function. 

1 Hk = The value of the nodes at the first hidden layer. 

1 ˆHk = The transferred value of the nodes at the first hidden layer. 

2 H j = The value of the nodes at the second hidden layer.

2 ˆH j = The transferred value of the nodes at the second hidden layer. 

3 Hl = The value of output node l (MR ANN).

3 H = The value of the output node (GV ANN and VV ANN).

3 ˆHl = The transferred value of the output node l, normalized output (MR ANN).

3 Hˆ = The transferred value of the output node, normalized output (GV ANN and 

VV ANN).

Pi = The input variables. 

1 Wik = The weight factors for the first hidden layer. 

2 Wkj = The weight factors for the second hidden layer. 

1 Bk = The bias factors for the first layer. 

2 Bj = The bias factors for the second layer. 

0 Bl = The bias factors for the outer layer (MR ANN). 

0 B = The bias factor for the outer layer (GV ANN and VV ANN). 

( )l max Sig = Maximum value for sigmoidal coefficient l in the trained data (MR ANN).

( )l min Sig = Minimum value for sigmoidal coefficient l in the trained data (MR ANN).

max log | *| E = Maximum log|E*| of the trained data (GV ANN and VV ANN).

min log | *| E = Minimum log|E*| of the trained data (GV ANN and VV ANN).

m = The number of nodes in the first hidden layer (see table 21). 

n = The number of nodes in the second hidden layer (see table 21).
203 

( )

( )

2 ˆ 1 i i

i

i i

P MIN

P

MAX MIN

− = − − (112)

Where:

Pi = Given input variable, i. 

ˆ

Pi = Normalized value for given input variable, i.

MINi = Minimum value of i in calibration dataset. 

MAXi = Maximum value of i in the calibration dataset. 

3 4 ( )

2

1

*log

log * 1 1 R Sig Sig f

Sig E Sig

e +

= +

+

(113) 

Where:

fR = Reduced frequency (hertz). 

Sig1 = First sigmoidal function coefficient. 

Sig2 = Second sigmoidal function coefficient. 

Sig3 = Third sigmoidal function coefficient. 

Sig4 = Fourth sigmoidal function coefficient.

Three ANN models have been developed from this architecture: (1) MR ANN, (2) VV ANN, and 

(3) |G*| ANN. In the following sections, the value of the weight factors, bias factors, input 

parameters, and normalization parameters are given for each of these models. 

D.2 MR ANN

5 25 40 Pi RR R MM M ααα 123 =     (114)

Where: 

MR

5

 = Resilient modulus at 41 oF (5 °C) (megapascals). 

MR

25

 = Resilient modulus at 77 oF (25 °C) (megapascals).

MR

40

 = Resilient modulus at 104 oF (40 °C) (megapascals).

α 

1 = Shift factor coefficient 1 (0.0007). 

α 

2 = Shift factor coefficient 2 (-0.1646). 

α 

3 = Shift factor coefficient 3 (0.806). 
204 

Table 50. Wik

1 matrix elements for MR ANN (part 1). 

Element 

k

Element i

1 2 3

1 -0.00200156589300513 0.00128692125170315 0.00284438217881075

2 0.28516256606761700 0.49118675435971600 -0.01820933290109810

3 -0.13012155717600300 0.07853842949381770 0.17680064334424700

4 -0.29231959611729100 0.17617282979184900 0.39775504273252700

5 -1.41445013310154000 6.44297995264973000 -1.77482643703439000

6 -0.21241167223734400 0.12821712282974400 0.28884600944520300

7 0.03400370099657090 -0.02101635307328680 -0.04672053052579720

8 0.09548185985586570 -0.05752189953709870 -0.13026426280596500

9 0.32507208708348800 -0.19505459789439700 -0.44164369167303300

10 0.28899439562768500 1.14183643175563000 1.48347522466600000

11 -0.98864001747551200 -2.86851244653187000 -3.23450809016547000

12 1.62092571912098000 -4.58381702398853000 2.21293392742914000

Table 51. Wik

1

 matrix elements for MR ANN (part 2). 

Element 

k

Element i

4 5 6

1 -712.8628402976380000 9.83219882009042000 -1.94319401017490000

2 753.53741515309100000 7.67671366971338000 -0.84350062295273200

3 -1093.068754047720000 9.23892947309088000 0.14382032104724800

4 -1129.557247424060000 -1.50590848559491000 -0.66955349262730300

5 -1.83158558608035000 3.97456523446851000 7.12208818273337000

6 -839.5292060420890000 8.73889517406220000 0.67603083268390200

7 755.74883976821200000 -12.71270279414850000 -1.34117207352107000

8 847.61568238744000000 -12.59290188069720000 -0.47503813284310300

9 410.33781806362100000 -7.80175950866107000 -2.36148957427579000

10 -805.5026910683050000 -4.21610143211805000 4.07885029040178000

11 471.04240973832900000 13.45559381618210000 -3.61642908665572000

12 28.37591023338010000 12.28550446563640000 0.70805040715617700
205 

Table 52. Bk

1

 vector elements for MR ANN (transposed for convenience). 

Element 

k B1

1 6.17091399170251000

2 7.71811183199239000

3 5.04220257285617000

4 1.36211474573909000

5 1.27586815658043000

6 1.24112853958781000

7 -0.55920953360997600

8 -1.54793233282465000

9 -2.38955526082692000

10 -5.31749985483251000

11 6.04452696422904000

12 2.43088472177158000

Table 53. Wkj

2

 matrix elements for MR ANN (part 1). 

Element 

j

Element k

1 2 3 4

1 -1.27212265453448000 0.98941325757367200 -1.90058367559899000 -1.69308092227318000

2 2.24917988742036000 -1.47046066978322000 2.81430927085362000 2.35680695833274000

3 0.90938132410845800 -1.87010703846175000 1.72113608938929000 0.53365095186003000

4 0.71420875730581000 -2.02463155580309000 1.24092642037176000 -0.75971225627055200

5 1.02573805084948000 -5.05227027803185000 -0.50811201125450400 0.12398342947210100

6 2.87344198982080000 1.1825109019882500 1.83686226283920000 1.07343492432610000

7 -6.84673965801500000 8.75898275822440000 -6.07092164029969000 -5.61770119661654000

8 -0.27215636118304200 -0.71265020644285800 0.10337231975487200 -0.50732389147556800

9 -1.24280778319817000 -0.42688724228155600 -1.49197352443203000 -1.1916709449455000

10 -0.87186156614646800 -0.55253243107747700 0.01014336151285740 -1.38109064819317000

11 5.05262125180162000 -3.58899791830666000 5.32477118646701000 6.08771475391962000

12 1.94315627250074000 -11.72058718677100000 1.02780825311488000 -0.07727872946434780
206 

Table 54. Wkj

2

 matrix elements for MR ANN (part 2). 

Element 

j

Element k

5 6 7 8

1 1.72171850082868000 -1.22361303255407000 0.26380146379060500 1.59052924999161000

2 -0.69983955029987400 2.15750218152474000 -2.72092291591448000 -2.56417332772541000

3 5.68303374707809000 -0.22567435819557400 -0.38683461689596400 -1.33743701004047000

4 -5.29792907293055000 -0.79868120409271800 0.56639895589096100 -0.15145370730044200

5 -6.9600412768491900 -0.25943738604012600 -0.07507398274261910 -0.60082153969042900

6 -6.90406457489009000 0.66048638607472900 -0.74734595052120800 -1.77757048725853000

7 14.28996210073420000 -4.93956480500064000 5.47678679301124000 5.04474379279507000

8 5.33401793234324000 -0.67814558115517500 1.35435682469839000 0.67517616517050400

9 6.13017525412482000 -0.25895030883091000 0.76688050726348400 0.59567627875874100

10 -4.52656667316437000 -0.61198000964802400 0.66990865283891800 1.79648574970781000

11 61.00473788943800000 5.94808205391293000 -6.49782178526526000 -5.63585227166289000

12 -0.74313876146962200 -0.49502882731012100 -0.04343884290084600 0.43382323116577400

Table 55. Wkj

2 matrix elements for MR ANN (part 3). 

Element 

j

Element k

9 10 11 12

1 1.28982526693395000 -1.36139827942535000 0.81242090814931100 0.75421177520166000

2 -1.98856516178736000 2.88665638759158000 -2.22345527946482000 47.32908703710720000

3 -1.46019991113224000 0.83288304533622300 -0.18737329755176000 7.94189735755131000

4 -0.62488286109702500 0.99358597783727000 -1.80707863873754000 -1.51697406306899000

5 -0.53491093234660500 0.03718852912964090 0.50412006696695100 -2.27324267108375000

6 -1.75495744905123000 0.04848011387017500 0.65137190325020200 -19.45130367879080000

7 6.34621810767813000 -4.33945264323816000 8.48179242696090000 -55.79280738270100000

8 -0.02703184429355360 -1.38007437316169000 -2.03641139073515000 7.15173377918400000

9 0.18219945071653500 -0.43950447308134000 0.88864139595275200 17.00276947159220000

10 0.18331101506990700 -1.29082311881848000 0.17907218849290100 0.00442335303199802

11 -4.70890802094776000 5.58102070625218000 -4.42512396747533000 34.44356640107110000

12 -0.58348656053904200 11.76004227042600000 -18.62573141905100000 40.36522874277380000
207 

Table 56. Bj

2 vector elements for MR ANN (transposed for convenience). 

Element 

j B2

1 0.69029100571690900

2 24.78083328475230000

3 -14.21641765871200000

4 0.59454387354854500

5 0.83441149973420900

6 2.00770642875969000

7 -6.80461833224959000

8 -3.04506035154618000

9 -2.81699338543501000

10 -0.01908957707522740

11 -23.82659200096020000

12 -2.84950882850698000

Table 57. Wjl

3

 vector elements for MR ANN (part 1). 

Element 

l

Element j

1 2 3 4

1 1.55631817035049000 -6.71717964044816000 5.52866041884021000 -23.13533591138010000

2 -2.10815560445854000 5.53651802240772000 -4.20046449739353000 13.84767577640070000

3 10.42186407217690000 1.07658778061795000 -1.00952254919933000 1.35728376824998000

4 12.53270054290200000 -2.82666156182142000 0.36140292673336200 11.63883002692080000

Table 58. Wjl

3 vector elements for MR ANN (part 2). 

Element 

l

Element j

5 6 7 8

1 7.89233240566186000 3.41508482981355000 0.23067944090008000 -6.35835335516061000

2 -4.35253958227055000 -4.33105623723765000 -0.23450370767118300 4.90555584894330000

3 -0.53624851772844700 4.26302489876063000 0.18477243720173400 0.95437352582270800

4 -5.27378018192719000 12.58074218873790000 0.30883922900382400 -0.81566323458280500

Table 59. Wjl

3 vector elements for MR ANN (part 3). 

Element 

l

Element j

9 10 11 12

1 3.12448098517872000 7.76943031096589000 0.16915092735543200 17.81482352191000000

2 -4.10144987129725000 -8.41207948330685000 -0.13394241862122600 -14.80599826102690000

3 4.33224573662362000 -12.80943716195050000 0.05065733872668660 -2.21601216037733000

4 12.62825381229430000 2.74268434100584000 0.08908525498240080 8.18232817352028000
208 

Table 60. Normalization parameters for MR ANN. 

Parameter Maximum Minimum

MR

5 (MPa) 34053.0 4800.3

MR

25 (MPa) 15411.0 1081

MR

40 (MPa) 6863.7 378.9 

α 1 (1/ °C2

) 0.002400 -0.000194

α 2 (1/ °C) -0.098 -0.300

α 3 1.430 0.490

Sig1 (MPa) 2.660 -0.043

Sig2 4.700 1.500

Sig3 4.100 0.650

Sig4 0.850 0.260

[ ] 0 12.88483251257 -10.23705702365 -1.957618903369 3.273305236416 Bl = (115) 

D.3 VV ANN

[ ] P f VMA VFA i = η (116) 

Where: 

f = Frequency, Hz. 

η = Viscosity, 109

 P (108 Pas). 

Table 61. Wik

1 matrix elements for VV ANN. 

Element 

k

Element i

1 2 3 4

1 0.00275807935276415 9.65414347487492000 2.03545176382323000 9.60483606710040000

2 0.00699289293419363 23.57848634613580000 -14.68700301452580000 -7.85111160151172000

3 -29.85518987292190000 -0.01748605711868700 -0.05226790767927880 0.10630299509262800

4 0.00272712204182532 0.22756591526942600 -0.44128805965281300 0.51123020782505100

5 0.00885326169582901 2.81905088512526000 -16.74274657827420000 -11.51343150553350000

6 -0.00604496464330366 -3.83323067510033000 -3.76213295881730000 -4.29370215749809000

7 0.00001271343965432 178.92786633284100000 -0.00411385898011919 0.00960133688419267

8 0.01990111297026090 -1.16149781971796000 3.28168963629711000 7.53282726564379000

9 0.01005974523187870 -0.39351960943073500 -2.51237935759967000 -4.29116433585095000

10 2.20552340599479000 -0.04056518032735070 0.20605359978971700 0.15739161949478100

11 -0.02404084443474640 -6.58972054945535000 -0.71981980149770700 1.52555084062215000

12 -0.00480809655510197 -0.27885745601162500 0.20260632406742000 -0.77108867937747000

13 0.00078471111967578 28.44415144259920000 -0.87571840373706600 1.95186725224564000

14 0.00771761131363787 5.33676984624904000 -11.97622550219350000 7.71358914882133000
209 

Table 62. Bk

1 vector elements for VV ANN (transposed for convenience). 

Element 

k B1

1 15.74835436269630000

2 12.95405722517280000

3 -32.89896917533040000

4 -1.10052248959346000

5 -10.78091133520380000

6 -6.05302142493261000

7 179.14709718593800000

8 0.65710599534106200

9 0.97891317883012100

10 4.72223785338294000

11 -6.55165918673815000

12 0.16731428319782600

13 27.93047751952220000

14 -1.37408939580495000

Table 63. Wkj

2 matrix elements for VV ANN (part 1). 

Element 

j

Element k

1 2 3 4

1 0.10511744512798600 0.10087889704665500 -6.08319321387257000 -6.59951922338895000

2 0.48485201906871100 -0.98061150567901500 5.28213726518926000 -0.99307488071221600

3 0.38666686767509600 0.14407696249404200 -0.64821237624430900 -0.68553604874521800

4 -0.29103025921087400 -0.12300943951262000 0.20660407620275500 0.84431006573009900

5 7.20145641626984000 151.58382852395500000 -35.04225274891410000 50.31324421340920000

6 57.80869276380200000 45.53819717827600000 34.31952056143690000 24.37102641478090000

7 0.04766296493976920 -0.00042062272941119 -1.64780015541921000 -0.14767680975504800

8 1.33157778260189000 -0.00864139577441407 -10.53694387134640000 -28.77316964591470000

9 0.95432357054777500 -1.02389021831262000 46.82499405300610000 61.27640626961720000

10 0.09476608746471290 -0.04013797559617900 15.08416115539120000 1.26248388850460000

11 0.65673641456535000 25.30163159128400000 6.86855242012598000 52.55865424871390000

12 -57.90977231272230000 57.64668837199080000 -36.95301005956680000 -51.19423424496710000

13 2.27010107107278000 -2.76393198240337000 14.26953262603660000 20.57219630346000000

14 -0.81442427169913900 -1.58075985235425000 20.32130385637230000 -21.37864285760230000
210 

Table 64. Wkj

2 matrix elements for VV ANN (part 2). 

Element 

j

Element k

5 6 7 8

1 0.10958125840908900 0.17103742786951300 7.53738934561000000 0.32157150848996400

2 1.10538840213556000 -0.06005243494707140 -0.46816179656466200 -0.11362528740921000

3 -0.03668587237246070 -0.20195215673052800 0.50583875586658800 0.04117830356601680

4 0.04165310223654260 0.12412276818401500 -0.71560911337291800 -0.06643717805416860

5 -135.8256006242760000 4.39896372970834000 2.12048232621756000 1.61468697764029000

6 12.18509505034930000 4.51309256843370000 7.19546156774681000 11.77248078904480000

7 0.00470501377909039 -0.00810109614328989 -0.20468271977533900 -0.00831305598572143

8 0.14429797312686300 0.32443645525838900 -18.31175213933740000 -0.17988695146765000

9 0.48668670982841300 -1.06169701219564000 -0.47416257725070700 2.01336853261287000

10 0.02124957257916830 -0.01491154156235320 214.20333131224300000 0.05344273444888970

11 -16.55295761437330000 31.36367340855210000 0.14069168225602600 20.67506639536700000

12 105.32395565518000000 -68.59510275333000000 -0.57180295407110300 24.60816018017130000

13 3.20913352965092000 -1.46922615656989000 -0.45618908696155800 -0.74587534667358800

14 2.14653394183588000 2.86656902371875000 -1.38951291360800000 1.51498306935364000

Table 65. Wkj

2 matrix elements for VV ANN (part 3). 

Element 

j

Element k

9 10 11 12

1 0.67436953255572400 1.52993707048353000 -1.56378559364968000 -5.97316328323617000

2 0.83884786099458000 0.72056843262828500 0.33823343591634200 0.32995739963357400

3 0.13129904885190400 0.03393257336078320 -0.33495422276929500 1.21059369021165000

4 -0.08374413143914640 -0.14011290475212000 0.40330307484415800 -0.92068329846481200

5 -4.74183346589286000 -7.18458646900432000 -8.20175738069755000 21.46322182793860000

6 -2.22050947171933000 -3.19929780229875000 43.60123067158340000 -38.15447962643450000

7 0.04970822735474000 0.38186826752702800 0.02975701201504030 -0.07187684022207080

8 18.96102127708150000 0.34577779423701500 0.10112166202643200 -4.34637124684574000

9 136.10585968855900000 3.75489444791017000 5.73368672918881000 35.37279289616110000

10 -0.36747370936357000 1.90705035404089000 -1.42366525777415000 6.11736464999887000

11 -20.86114381473210000 4.50957253702840000 0.73758107479932900 -3.21435115847231000

12 -19.00772620196210000 44.26713771722670000 38.62379852233980000 -1.87378784809568000

13 0.17584310006536200 3.18528344468959000 0.62107582578786300 8.04752096583409000

14 2.97468553092343000 -4.21086927843771000 2.73490619946452000 -3.67293829703328000
211 

Table 66. Wkj

2 matrix elements for VV ANN (part 4). 

Element 

j

Element k

13 14

1 -1.12245731810529000 0.67867713986347200

2 0.41929338131065000 -1.37665914977400000

3 0.58160889599262400 0.36777315706927400

4 -0.51666057998473900 -0.35135512626691600

5 3.27948738968729000 1.94926465685885000

6 -45.89506402313440000 70.49683570639200000

7 0.01612131389179630 -0.01152122049407600

8 3.77564707007114000 -4.91896963383303000

9 3.89920658003942000 -49.40155143998830000

10 2.24644797717875000 -0.11299018277251500

11 -9.63227261518900000 -119.1073499783990000

12 -41.84530118313840000 -35.82982088240670000

13 0.56754251531620400 -3.01209559093347000

14 2.43256581696669000 -1.47837832821775000

Table 67. Bj

2 vector elements for VV ANN (transposed for convenience). 

Element 

j B2

1 -12.23193846569030000

2 0.61294509854138200

3 -2.26121589380006000

4 1.99708243932415000

5 22.07400990717020000

6 -31.86954422722210000

7 -2.41399982513437000

8 -55.27590960198670000

9 -106.0101552356340000

10 -30.89403826224180000

11 -76.24782342311330000

12 69.28157710468700000

13 20.21945409941280000

14 6.12083493792229000
212 

Table 68. Wj

3 vector elements for VV ANN. 

Element 

j W3

1 1.15373219918479000

2 -6.43559921490721000

3 -27.68360314597900000

4 -29.84574981371640000

5 -0.19407309170597600

6 0.11995708036519800

7 37.96893504107600000

8 34.00719477253400000

9 0.43014683810727000

10 58.47886817741680000

11 11.94908594315590000

12 -0.07969689689272150

13 1.28313108040340000

14 0.53296310828566400

Table 69. Normalization parameters for VV ANN. 

Parameter Maximum Minimum

Frequency (Hz) 25 0.01

Viscosity (109 P) 1.99 x 10-6 27.00

VMA (percent) 34.64 9.51

VFA (percent) 95.07 32.82

|E*| (psi) 6.77 3.52

0 B = -3.484481025467 (117) 

D.4 |G*|-BASED ANN (GV ANN) 

* P VMA VFA i =     G (118) 
213 

Table 70. Wik

1 matrix elements for GV ANN. 

Element 

k

Element i

1 2 3

1 -0.00024607061578008 0.39573112252617300 0.03137522820868100

2 0.06369415599503260 1.20396485280366000 -0.41366999743116300

3 0.12560166488947000 12.23092512080170000 -15.40428508265840000

4 144.78479285235900000 0.03602175009356860 -0.01486430375847190

5 -0.88590000706402800 -6.64620868848810000 6.76784019433308000

6 -0.24480066614146600 -175.126570586870000 110.38396928012700000

7 0.35658487276453800 -1.13633711409192000 -2.69698652853612000

8 0.05160380299493010 0.96914775793747200 -0.45058047297787600

9 -0.11763211836169300 -2.07308576925671000 0.85862809150608600

10 -0.10260986412141000 -1.89647330970902000 0.63576557888985500

11 0.83832125440623900 5.01864835388400000 -7.07830843630593000

12 -22.27016740128570000 2.82680472067850000 -1.10402495920697000

Table 71. Bk

1 vector elements for GV ANN (transposed for convenience). 

Element 

k B1

1 -13.23020275429390000

2 -1.15912963101547000

3 4.49668100338684000

4 144.90818089626700000

5 0.92374996199841500

6 187.12386013330800000

7 -0.64852041864182200

8 0.79297555520979200

9 1.27505357580263000

10 1.56260322810567000

11 -1.80964836750601000

12 -24.23180989095970000
214 

Table 72. Wkj

2 matrix elements for GV ANN (part 1). 

Element 

j

Element k

1 2 3 4

1 1.68662477353906000 0.81911931325403900 -0.03005180291517210 0.00569345005879186

2 -4.86360723468991000 105.69622449695400000 -2.47849339660309000 0.18455777935409300

3 91.15855920616590000 92.66954611901590000 -40.91260113059330000 -1.67503899696897000

4 106.61660556715000000 22.90152433894880000 229.56133863086000000 -0.21741320127254800

5 0.73120568649091500 -2.04683544732935000 0.02907899740128110 -0.00680956721358145

6 20.01334655907270000 -124.1136850495880000 -12.93483076443820000 0.16320420801843400

7 11.03261781901980000 -55.35942281688430000 -0.25840813426995300 2.36249059396197000

8 6.30522135450740000 -72.37646707852670000 -2.13173103347156000 23.41004001398310000

9 -34.10766755925970000 26.04890333595410000 0.13622620891216500 -534.4781944799400000

10 -18.68321667137760000 103.90356571837300000 -2.46159833726818000 0.18433171253622200

11 19.14302423015890000 3.59998964741180000 0.25399603119775500 -44.54673896061580000

12 -0.33713658604817700 12.87184727785630000 0.13134300289504600 6.54659549746008000

Table 73. Wkj

2 matrix elements for GV ANN (part 2). 

Element 

j

Element k

5 6 7 8

1 0.01859496903873790 7.79141326044631000 0.01459371336584290 -4.01406752777277000

2 30.71090633980400000 5.37671972945662000 3.18886351743859000 -2.85108340883822000

3 87.71359895456880000 -77.08967734352910000 22.84350940748950000 335.75204711704400000

4 68.74692689690920000 -45.87194063996940000 12.79313464511960000 -37.77625414085690000

5 -0.0094809359102436 -1.94014227142969000 -0.01256576806807960 4.16910051124176000

6 -37.35043967525970000 5.27365300784224000 1.31038988899583000 43.16167413535480000

7 91.10288269578740000 -29.48793850026970000 -0.04849305692638800 3.23868144074583000

8 3.53501265914675000 0.83322882269153700 1.62246929641914000 6.69105720723216000

9 -0.09814064912135180 -0.02099590067358970 -0.19796523923434900 17.65358100702660000

10 55.61599363732450000 -5.35164182127019000 3.15862176137102000 -2.82977620499617000

11 -0.30334853272738200 43.07269450273440000 -0.22059949976226000 1.05780537005262000

12 0.11377535318423100 0.11441693675638400 0.15289220845030100 -1.13970260089778000
215 

Table 74. Wkj

2 matrix elements for GV ANN (part 3). 

Element 

j

Element k

9 10 11 12

1 3.47596025162077000 -3.82827454649034000 0.01169888530415500 0.12954405222771200

2 94.55869709663890000 -7.34809568152935000 10.41344790169180000 21.32866301200200000

3 205.23211172111800000 -34.98627953103440000 76.29312010940660000 -30.59094661558560000

4 64.60376447639590000 -21.68091976305070000 66.41011966659030000 -26.22110679142560000

5 0.08966925765449130 -0.90205636358616400 -0.00585495017589258 0.00011267457317091

6 12.38797404837280000 -96.26501641581090000 -37.17032464251540000 0.12957951502862800

7 -50.14112337006290000 -8.38137267965061000 40.93959435883270000 -0.32474898284932000

8 -55.82517744922910000 -57.75954730123160000 2.17269724264550000 -54.42459642341290000

9 -1.13861767294711000 6.61476050123734000 0.25450956423993600 -1.38236982388117000

10 93.41332136138760000 -9.48311646261710000 49.01504550800940000 20.79054924536240000

11 2.55740586147654000 2.61695086136455000 -0.09940520779407240 3.00291187179692000

12 4.67064477635346000 8.20560279728423000 0.10724854309492400 3.33502620583327000

Table 75. Bj

2 vector elements for GV ANN (transposed for convenience). 

Element 

j B2

1 -3.02263601315923000

2 5.97704828771739000

3 -91.56648216379910000

4 -106.5989770883490000

5 -0.36740268358330700

6 -20.04954785032860000

7 -10.42449435680280000

8 -5.32258083806398000

9 32.81032420706510000

10 17.56993360741350000

11 -20.65675932973890000

12 2.35956011260892000
216 

Table 76. Wj

3 vector elements for GV ANN. 

Element 

j W3

1 -134.7651195030360000

2 92.96416099463190000

3 0.07847134028534860

4 0.58725325729413100

5 -146.9138139427070000

6 -12.15444670634670000

7 34.34304769221520000

8 -63.09541362586340000

9 -76.62381419958460000

10 -93.21865080014820000

11 -29.95120632059920000

12 0.71757242259101300

Table 77. Normalization parameters for GV ANN. 

Parameter Maximum Minimum

|G*| (psi) 676,000 0.0293

VMA (percent) 22.21 9.51

VFA (percent) 95.07 32.82

|E*| (psi) 6.81 3.52

0 B = -8.469734576039 (119) 
217 

APPENDIX E: ANNACAP SOFTWARE MANUAL

E.1 INTRODUCTION

The Artificial Neural Networks for Asphalt Concrete Dynamic Modulus Prediction 

(ANNACAP) program has been developed under Contract No. DTFH61-02-00139 Task Order 

#10 LTPP Computed Parameter Dynamic Modulus to aid in populating the LTPP database with 

dynamic modulus data. Technical details concerning the analysis used in this program can be 

found in the final report for the referenced project. The purpose of this document is to provide a 

manual of operation for the ANNACAP program. 

The dynamic modulus, |E*|, is a fundamental property that defines the stiffness characteristic of 

hot mix asphalt (HMA) mixtures as a function of loading rate and temperature. The significance 

of this material property is threefold. First, it is one of the primary material property inputs in the 

Mechanistic-Empirical Pavement Design Guide (MEPDG) and software developed by NCHRP 

Project 1-37A. The MEPDG uses mastercurves and time-temperature shift factors in its internal 

computations. The mastercurve is constructed using a hierarchical structure of inputs ranging 

from laboratory tests on HMA mixtures and binders to estimates based on properties of the HMA 

mixtures. Second, the |E*| is one of the primary HMA properties measured in the SuperpaveTM

simple performance test protocol that complements the volumetric mix design. Third, |E*| is one 

of the fundamental linear viscoelastic (LVE) material properties that can be used in advanced 

HMA and pavement models that are based on viscoelasticity.

In spite of the demonstrated significance of |E*|, it is not included in the current LTPP materials 

tables because the database structure was established long before |E*| was identified as the main 

HMA property in the MEPDG. It is not practical to perform MEPDG level 1 laboratory |E*| tests 

on material samples from LTPP test sections at this time due to a lack of materials, budget 

limitations, and the absence of a suitable test method applicable to field samples obtained from 

relatively thin pavement structures. However, the LTPP database does contain other data that can 

be used to estimate the |E*| mastercurve and associated shift factors, estimate the |E*| at specific 

load durations and temperatures, or develop inputs to the models contained in MEPDG. 

E.2 DISCLAIMER

This software is provided “as is,” and any express or implied warranties, including but not 

limited to the implied warranties of merchantability and fitness for a particular purpose, are 

disclaimed. In no event shall the authors be liable for any direct, indirect, incidental, special, 

exemplary, or consequential damages (including, but not limited to, procurement of substitute 

goods or services; loss of use, data, or profits; or business interruption) however caused and in 

any way theory of liability, whether in contract, strict liability, or tort (including negligence or 

otherwise) arising in any way out of the use of this software, even if advised of the possibility of 

such damage.
218 

E.3 INSTALLATION INSTRUCTIONS

1. Insert ANNACAP Installation CD into the CD ROM or download zipped installer file from 

Web site. Note: If downloading the file, remember the location where you have unzipped the 

files. 

2. If the CD does not autorun, press the Windows Start button. 

3. Click on Run. 

4. Change the directory to the location of the installer file (either the CD ROM drive or the 

unzipped file locations. 

5. Select the Setup program. 

6. Press OK.

7. Follow the onscreen instructions. 

8. ANNACAP and all necessary support files will be installed.

9. A shortcut will be placed in the start menu under the Programs  ANNACAP path. To place 

a shortcut onto the desktop, users must manually perform the operation. 

E.4 USING ANNACAP

E.4.1 Program Main Screen

The main ANNACAP screen is shown in figure 205. This screen appears when ANNACAP is 

launched. All analysis is menu-driven, and the menu schematic is shown in figure 206. To 

perform modulus predictions, users must provide the necessary inputs by following the File 

Input path (described in detail below). Users must also provide a directory for output to be 

written by following the File  Output Directory path. Once both have been properly input, 

users may perform data analysis by following the File  Run Analysis path. Users may access 

this document by following the Help  Manual path or find basic program information by 

choosing the Help  About path. 
219 

Figure 205. Screenshot. ANNACAP main screen. 

File

Input Data

Run Analysis

Output Folder

Exit

Help

Manual

About

-

-

File

Input Data

Run Analysis

Output Folder

Exit

Help

Manual

About

-

-

Figure 206. Illustration. ANNACAP menu diagram. 

E.4.2 Input Data

Selecting the File  Input path will automatically launch the input utility. The initial screen shot 

is shown in figure 207. Users have four modes to choose from: (1) MR-based ANN, (2) |G*|-

based ANN, (3) viscosity-based ANN, and (4) batch mode. The screen is divided into two 

regions referred to as the left side and right side and separated by bordered regions. On the left 

side are areas for inputting basic information for the nonbatch mode runs, choosing the mode to 

use, and choosing the input parameter complexity to use. On the right side are inputs specific to 
220 

the chosen analysis mode. Necessary inputs will appear on both the left and right sides of the 

screen as users make selections. After selecting all of the appropriate factors and entering all 

necessary inputs, pressing “Done” on the menu bar will return users to the main ANNACAP 

screen. Note that all modes will produce |E*| predictions at 14, 40, 70, 100, and 130 °F (-10, 4.4, 

21.1, 37.8, 54.4 oC) and 25, 10, 5, 1, 0.5, and 0.1 Hz. 

Figure 207. Screenshot. ANNACAP input screen. 

E.4.3 Layer ID

Under the Layer ID section, users should enter the following items: 

• State code.

• Project ID. 

• Project layer.

• Construction date.

• Aging level (only RTFO aging is available for certain models). 

• Test date (only if “Other” aging level is chosen). 
221 

The naming convention for these items is the same as that followed in the LTPP database. Users 

may choose to enter the dates directly into the appropriate boxes and, if they do so, the format 

should be Day-Abbreviated Month-Year (i.e., 20-Mar-1985). Users wishing to use the calendar 

utility should press the Choose button beside the date box and navigate to the appropriate date. 

When the appropriate date is chosen, users must press the OK button. 

E.4.4 MR-Based ANN

Selecting MR-based ANN from the model drop-down menu will allow users to develop |E*| 

predictions based on MR inputs. On the right side of the screen, users must enter MR values in 

gigapascals at three specific temperatures (41, 77, and 104 oF (5, 25, and 40 °C)) into the 

provided table. The appropriate input ranges for these values are as follows: 

• 41o F (5 °C) = 4,938,970.10 to 696,224.65 psi (34.053 to 4.8003 GPa). 

• 77 oF (25 °C) = 2,235,176.58 to 156,785.79 psi (15.411 to 1.0810 GPa). 

• 104 oF (40 °C) = 995,495.52 to 54,954.80 psi (6.8637 to 0.3789 GPa). 

Inputting values outside this range will cause ANNACAP to display a warning indicating that 

input values exceed the calibration range, but predictions will still be made. Users should also 

select the aging condition for the measured MR values. 

E.4.5 |G*|-Based ANN

Selecting |G*|-based ANN from the model drop-down menu will allow users to develop |E*| 

predictions based on |G*| inputs. There are three different levels of input complexity: level 1, 

level 2, and level 3. For each level, users should input the percentage of voids in mineral 

aggregate (VMA) and percentage of voids filled with asphalt (VFA) in addition to |G*| values. 

The appropriate ranges for these input variables are shown beside the input controls. For all input 

levels, ANNACAP will use the CAM model to generate a mastercurve of the data. Users may 

choose to force the CAM model to fit the data with a certain glassy modulus value by choosing 

Input from the Find Gg by drop-down menu. If Input is chosen, the default value is 

145,037.74 psi (1 GPa), but users may change it to any desired value. If users choose Fitting

from the Find Gg by drop-down menu, then the glassy modulus is treated as any other 

optimization parameter. If no data are available at extremely low temperatures (below 32 oF 

(0 °C)), users are recommended to choose Input and select a value between 145,037.74 and 

1,450,377.38 psi (1 and 10 GPa). If for some reason a fitting error occurs with the provided 

input, the program will display a fitting error dialog and not allow users to predict |E*|. If the 

calculated |G*| is greater than 675,875.86 psi (4.66 x 109 Pa) or less than 0.029 psi (202 Pa), a 

warning dialog will appear, but |E*| predictions will be performed. 

E.4.5.1 Level 1

For level 1, users have access to complete |G*| values at multiple temperatures and frequencies. 

These values should be entered directly into the table that appears on the right side of the screen. 

Users may choose to create this data file in another application and load it into the table by using 
222 

the Load button. The file should be a tab delimited text file with the same column format as the 

input table. The file should have column labels. 

E.4.5.2 Level 2

For level 2, users have access to |G*| values and possibly BBR stiffness values at multiple 

temperatures (at least two above 114.8 oF (46 °C) and two below 114.8 oF (46°C)) but at the 

fixed frequency of 10 rad/s and load time of 60 s. All of the measured moduli values should 

be at a consistent aging level. The |G*| values are entered in units of kilopascals, whereas the 

S(t) values are entered in units of megapascals following convention. The temperature is 

always entered in degrees Celsius. Users should enter the values into the tables on the right 

side of the screen.

E.4.5.3 Level 3

For level 3, users have access to |G*| values and possibly BBR stiffness values at multiple 

temperatures (at least two above 114.8 oF (46 °C) and two below 114.8 oF (46 °C)) at a fixed 

frequency of 10 rad/s and load time of 60 s. The aging conditions are a mixture of RTFO and 

PAV. The units are the same as those used in level 2 input. In addition to entering these values, 

users should choose the high temperature SuperpaveTM PG for the binder from the High Temp 

PG drop-down menu. If this information is not known or cannot be determined, users may select 

Unknown from the drop-down menu. In Level 3 analysis, only the RTFO-aging conditions may 

be predicted.

E.4.6 Viscosity-Based ANN

Selecting Viscosity-based ANN from the model drop-down menu will allow users to develop 

|E*| predictions based on viscosity inputs. There are three different levels of input complexity: 

level 1, level 2, and level 3. For each level, users should input the VMA and VFA. The 

appropriate ranges for these input variables are shown beside the input controls. If the calculated 

viscosity is less than 199,000 cP (199 Pas), a warning dialog will appear, but |E*| predictions will 

be performed. 

E.4.6.1 Level 1

In level 1, users enter A and VTS values directly into the right side of the screen. 

E.4.6.2 Level 2

In level 2, users choose the types of viscosity measures available by selecting or deselecting the 

radio buttons on the left side of the screen. The available measures include: R&BT temperature, 

penetration, absolute viscosity, and kinematic viscosity. Selecting or deselecting these measures 

will make appropriate input tables or controls appear on the right side of the screen. Following 

standard convention, the R&BT temperature is given in degrees Fahrenheit, the penetration is 

given by the PEN number, the absolute viscosity is input in poise, and the kinematic viscosity is 

input in centistokes. If users select kinematic viscosity, then they must also enter the binderspecific gravity in the Gb control. By default, ANNACAP inputs 1.03 for Gb. Users must input at 

least two measures of viscosity so that A and VTS can be computed. 
223 

E.4.6.3 Level 3

In level 3, users choose the binder grade. Typical values compiled during the NCHRP 1-37A 

project are available, and the binder grades can be either SuperpaveTM -based, viscosity-based 

(AC system only), or penetration-based. The binder grade-to-viscosity relationship is 

summarized in table 78. 

Table 78. Relationship between binder purchase specification grade and 

A and VTS parameters. 

Asphalt 

Binder 

Grade A VTS

Asphalt 

Binder Grade A VTS

PG 46-34 11.5040 -3.9010 PG 70-28 9.7150 -3.2170

PG 46-40 10.1010 -3.3930 PG 70-34 8.9650 -2.9480

PG 46-46 8.7550 -2.9050 PG 70-40 8.1290 -2.6480

PG 52-10 13.3860 -4.5700 PG 76-10 10.0590 -3.3310

PG 52-16 13.3050 -4.5410 PG 76-16 10.0150 -3.3150

PG 52-22 12.7550 -4.3420 PG 76-22 9.7150 -3.2080

PG 52-28 11.8400 -4.0120 PG 76-28 9.2000 -3.0240

PG 52-34 10.7070 -3.6020 PG 76-34 8.5320 -2.7850

PG 52-40 9.4960 -3.1640 PG 82-10 9.5140 -3.1280

PG 52-46 8.3100 -2.7360 PG 82-16 9.4750 -3.1140

PG 58-10 12.3160 -4.1720 PG 82-22 9.2090 -3.0190

PG 58-16 12.2480 -4.1470 PG 82-28 8.7500 -2.8560

PG 58-22 11.7870 -3.9810 PG 82-34 8.1510 -2.6420

PG 58-28 11.0100 -3.7010 AC-2.5 11.5167 -3.8900

PG 58-34 10.0350 -3.3500 AC-5 11.2614 -3.7914

PG 58-40 8.9760 -2.9680 AC-10 11.0134 -3.6954

PG 64-10 11.4320 -3.8420 AC-20 10.7709 -3.6017

PG 64-16 11.3750 -3.8220 AC-3 10.6316 -3.5480

PG 64-22 10.9800 -3.6800 AC-40 10.5338 -3.5104

PG 64-28 10.3120 -3.4400 PEN 40-50 10.5254 -3.5047

PG 64-34 9.4610 -3.1340 PEN 60-70 10.6508 -3.5537

PG 64-40 8.5240 -2.7980 PEN 85-100 11.8232 -3.6210

PG 70-10 10.6900 -3.5660 PEN 120-150 11.0897 -3.7252

PG 70-16 10.6410 -3.5480 PEN 200-300 11.8107 -4.0068

PG 70-22 10.2990 -3.4260 — — —

— Indicates that no additional relationships exist.

E.5 BATCH MODE

In batch mode, users enter four different files for the four different aging levels: (1) unaged or 

original binder data file, (2) RTFO-aged binder file, (3) PAV-aged binder file, and (4) field-aged 

binder file. Each file must be a tab delimited text file in order for ANNACAP to read the file. 

The file should have a header. Even if no data are available for some aging conditions, a file 
224 

name must be entered into the directory path on the right side. The formatting for the original-, 

RTFO-, and PAV-aged conditions, collectively referred to as the lab-aged files, is different than 

the formatting for the field-aged binder file. The formats for the two files are presented in 

table 79 and table 80. Users may also view the format by selecting either the Format for LabAged File or Format for Field-Aged File buttons. 
225 

Table 79. File format for lab-aged files. 

Item Description

STATE_CODE Code representing State or province

PROJECT_ID Project ID code

PROJECT_LAYER Project layer code as established in TST_LO5B

CONSTRUCTION_DATE Date layer was constructed

SAMPLE_TYPE 1–original binder, 2–RTFO/TFO binder, 3–PAV binder

SAMPLE_DATE Date of sampling

TEST_DATE Date of testing

GSTAR_1 Binder |G*| at temperature 1 (kPa)

GSTAR_PHASE_ANGLE_1 Phase angle at temperature 1 (degree)

GSTAR_TEMP_1 |G*| temperature 1 (°C)

GSTAR_2 Binder |G*| at temperature 2 (kPa)

GSTAR_PHASE_ANGLE_2 Phase angle at temperature 2 (degree)

GSTAR_TEMP_2 |G*| temperature 2 (°C)

GSTAR_3 Binder |G*| at temperature 3 (kPa)

GSTAR_PHASE_ANGLE_3 Phase angle at temperature 3 (degree)

GSTAR_TEMP_3 |G*| temperature 3 (°C).

GSTAR_4 Binder |G*| at temperature 4 (kPa)

GSTAR_PHASE_ANGLE_4 Phase angle at temperature 4 (degree)

GSTAR_TEMP_4 |G*| temperature 4 (°C)

GSTAR_5 Binder |G*| at temperature 5 (kPa)

GSTAR_PHASE_ANGLE_5 Phase angle at temperature 5 (degree)

GSTAR_TEMP_5 |G*| temperature 5 (°C)

GSTAR_6 Binder |G*| at temperature 6 (kPa)

GSTAR_PHASE_ANGLE_6 Phase angle at temperature 6 (degree)

GSTAR_TEMP_6 |G*| temperature 6 (°C)

GSTAR_SOURCE LTPP module from which |G*| was extracted (i.e., TST)

RING_BALL Ring/ball temperature in Fahrenheit

RING_BALL_SOURCE

LTPP module from which TR&B was extracted (i.e., 

TST)

PENETRATION_39.2F Penetration at 39.2 °F (PEN)

PENETRATION_39.2F_SOURCE

LTPP module from which PEN at 39.2 °F was extracted

(i.e., TST)

PENETRATION_77F Penetration at 77 °F (PEN)

PENETRATION_77F_SOURCE

LTPP module from which PEN at 77 °F was extracted 

(i.e., TST)

PENETRATION_115F Penetration at 115 °F (PEN)

PENETRATION_115F_SOURCE

LTPP module from which PEN at 115 °F was extracted 

(i.e., TST)

ABSOLUTE_VISCOSITY Absolute viscosity at 140 °F (poise)

ABSOLUTE_VISCOSITY_SOURCE

LTPP module from which absolute viscosity was 

extracted (i.e., TST)
226 

KINEMATIC_VISCOSITY Kinematic viscosity at 275 °F (centistokes)

KINEMATIC_VISCOSITY_SOURCE

LTPP module from which kinematic viscosity was 

extracted (i.e., TST)

VMA Voids in mineral aggregate as percent total volume

VMA_SOURCE

LTPP module from which VMA was extracted 

(i.e., TST)

VFA Voids filled with asphalt as percent VMA

VFA_SOURCE

LTPP module from which VFA was extracted 

(i.e., TST)

AIR_VOIDS Air voids as percent total volume

AIR_VOIDS_SOURCE

LTPP module from which air voids was extracted 

(i.e., TST)

GMB Bulk-specific gravity of the mix.

GMB_SOURCE LTPP module from which Gmb was extracted (i.e., TST)

GMM Maximum specific gravity of the mix

GMM_SOURCE LTPP module from which Gmm was extracted (i.e., TST)

EFFECTIVE_AC

Effective asphalt content as percentage of total mix 

volume

EFFECTIVE_AC_SOURCE

LTPP module from which the effective volume of the 

binder (Vbe) was extracted (i.e., TST)

MR_5 Resilient modulus at 5 °C (GPa)

MR_25 Resilient modulus at 25 °C (GPa)

MR_40 Resilient modulus at 40 °C (GPa)

MR_SOURCE LTPP module from which MR was extracted (i.e., TST)

BINDER_GRADE

Purchase specification grade of binder (RTFO aging 

only)

°C = (°F-32)/1.8

1 P = 10 Pas

1 psi = 6.86 kPa
227 

Table 80. File format for field-aged files. 

Item Description

STATE_CODE Code representing State or province.

PROJECT_ID Project ID code.

PROJECT_LAYER Project layer code as established in TST_LO5B.

CONSTRUCTION_DATE Date layer was constructed.

SAMPLE_TYPE Four-field-aged binder.

GSTAR_SAMPLE_DATE Date of sampling for |G*|.

GSTAR_1 Binder |G*| at temperature 1 (kPa).

GSTAR_PHASE_ANGLE_1 Phase angle at temperature 1 (degree).

GSTAR_TEMP_1 |G*| temperature 1 (°C).

GSTAR_2 Binder |G*| at temperature 2 (kPa).

GSTAR_PHASE_ANGLE_2 Phase angle at temperature 2 (degree).

GSTAR_TEMP_2 |G*| temperature 2 (°C).

GSTAR_3 Binder |G*| at temperature 3 (kPa).

GSTAR_PHASE_ANGLE_3 Phase angle at temperature 3 (degree).

GSTAR_TEMP_3 |G*| temperature 3 (°C).

GSTAR_4 Binder |G*| at temperature 4 (kPa).

GSTAR_PHASE_ANGLE_4 Phase angle at temperature 4 (degree).

GSTAR_TEMP_4 |G*| temperature 4 (°C).

GSTAR_5 Binder |G*| at temperature 5 (kPa).

GSTAR_PHASE_ANGLE_5 Phase angle at temperature 5 (degree).

GSTAR_TEMP_5 |G*| temperature 5 (°C).

GSTAR_6 Binder |G*| at temperature 6 (kPa).

GSTAR_PHASE_ANGLE_6 Phase angle at temperature 6 (degree).

GSTAR_TEMP_6 |G*| temperature 6 (°C).

GSTAR_SOURCE

LTPP module from which |G*| was extracted 

(i.e., TST).

BINDER_SAMPLE_DATE Date of sampling for viscosity.

RING_BALL Ring/ball temperature in Fahrenheit.

RING_BALL_SOURCE

LTPP module from which TR&B was extracted 

(i.e., TST).

PENETRATION_39.2F Penetration at 39.2 °F (PEN).

PENETRATION_39.2F_SOURCE

LTPP module from which PEN at 39.2 °F was 

extracted (i.e., TST).

PENETRATION_77F Penetration at 77 °F (PEN).

PENETRATION_77F_SOURCE

LTPP module from which PEN at 77 °F was 

extracted (i.e., TST).

PENETRATION_115F Penetration at 115 °F (PEN).

PENETRATION_115F_SOURCE

LTPP module from which PEN at 115 °F was 

extracted (i.e., TST).

ABSOLUTE_VISCOSITY Absolute viscosity at 140 °F (poises).
228 

ABSOLUTE_VISCOSITY_SOURCE

LTPP module from which absolute viscosity was 

extracted (i.e., TST).

KINEMATIC_VISCOSITY Kinematic viscosity at 275 °F (centistokes).

KINEMATIC_VISCOSITY_SOURCE

LTPP module from which kinematic viscosity was 

extracted (i.e., TST).

VMA Voids in mineral aggregate as percent total volume.

VMA_SOURCE

LTPP module from which VMA was extracted 

(i.e., TST).

VFA Voids filled with asphalt as percent VMA.

VFA_SOURCE

LTPP module from which VFA was extracted 

(i.e., TST).

AIR_VOID_SAMPLE_DATE Date of sampling for air voids.

AIR_VOIDS Air voids as percent total volume.

AIR_VOIDS_SOURCE

LTPP module from which air voids was extracted 

(i.e., TST). 

GMB Bulk specific gravity of the mix.

GMB_SOURCE

LTPP module from which Gmb was extracted 

(i.e., TST). 

GMM Maximum specific gravity of the mix.

GMM_SOURCE

LTPP module from which Gmm was extracted 

(i.e., TST). 

EFFECTIVE_AC

Effective asphalt content as percent of total mix 

volume.

EFFECTIVE_AC_SOURCE

LTPP module from which Vbe was extracted 

(i.e., TST).

MR_SAMPLE_DATE Date of sampling for MR.

MR_5 Resilient modulus at 5 °C (GPa).

MR_25 Resilient modulus at 25 °C (GPa).

MR_40 Resilient modulus at 40 °C (GPa).

MR_SOURCE

LTPP module from which MR was extracted 

(i.e., TST).

°C = (°F-32)/1.8

1 psi = 6.86 kPa

E.6 OUTPUT DIRECTORY

Following the File  Output Directory path will launch the output directory dialog, as seen in 

figure 208. If no output directory is chosen or if users would like to change the current output 

directory, they should press the browse folder button to the right of the directory path (circled in 

black in figure 208). When this button is pressed, the folder selection dialog will appear (see 

figure 209). Users should then navigate to the desired output folder and select the Current 

Folder button (circled in black in figure 209). When selected, users return to the output directory 

dialog screen. To keep the chosen directory, users should press OK to return to the main screen. 

If users do not choose to keep the directory, press Cancel. 
229 

Figure 208. Screenshot. Output directory dialog. 

Figure 209. Screenshot. Choosing output directory. 

E.7 RUN ANALYSIS

To perform the dynamic modulus calculation, users should choose the File  Run Analysis path. 

The run analysis feature will become active only after ANNACAP has received valid input 

values. If an output directory has not been selected, an error message will appear, and the 

analysis will not be performed. Users must select a valid output directory and follow the File 

Run Analysis path. 

If users have chosen to follow either the MR-based ANN, |G*|-based ANN, or viscosity-based 

ANN, a single output file with the quantities shown in table 81 will be generated and located in 

the output directory. The file name for this file will be Project_ID-Project_Layer-Aging CodeModel Type-Predicted Modulus.out. If users have chosen to follow the batch mode analysis 

technique, two different files will be generated: (1) a summary file with one row per layer and 

(2) a detailed output data file including 30 rows per layer (one row for each temperature and 

frequency combination). The summary analysis file for the batch mode will be formatted as 

shown in table 82. The file name for this file will be Models_Summary_Batch_Mode.out. The 

format for the main output file will be similar to that of the individual layer analysis and is 

shown in table 83. This file will be titled Predicted_Modulus_Batch_Mode.out. Both the 

summary and detailed output files will be located in the user-selected output directory. When 
230 

running batch mode, users must rename the previous runs that are in the output directory because

ANNACAP will overwrite existing file names without any warning to the user. 

Table 81. Output data format for single ANNACAP use. 

°C = (°F-32)/1.8

1 psi = 6.86 kPa

Item Description

STATE_CODE Code representing State or province as input by user

PROJECT_ID Project ID code as input by user

PROJECT_LAYER Project layer code as input by user

CONSTRUCTION_DATE Date layer was constructed as input by user

SAMPLE_TYPE

1–original binder, 2–RTFO/TFO binder, 3–PAV binder, and 

4–field binder

PREDICTIVE_MODEL MR ANN, VV- ANN, GV-|G*| ANN

TEMPERATURE Temperature of modulus prediction (°F)

FREQUENCY Frequency of modulus prediction (Hz)

|E*|_PREDICTION Predicted dynamic modulus (psi)

VMA Voids in mineral aggregate as percent total volume

VFA Voids filled with asphalt as percent VMA

VISCOSITY Viscosity input (109 P) only for viscosity ANN model

A Viscosity model intercept A (only for viscosity ANN model)

VTS Viscosity model slope (only for viscosity ANN model)

MR_5C Resilient modulus at 5 °C (only for MR model) (MPa)

MR_25C Resilient modulus at 25 °C (only for MR model) (MPa)

MR_40C Resilient modulus at 40 °C (only for MR model) (MPa)

|G*| Binder shear modulus (psi) only for |G*| ANN model

WLF_COEFFICIENT_1

WLF shifting function coefficient C1 (only for level 1 input |G*| 

ANN model) 

WLF_COEFFICIENT_2

WLF shifting function coefficient C2 (only for level 1 input |G*| 

ANN model)

CAM_COEFFICIENT_1 CAM fitting coefficient Gg (only for |G*| ANN model) (Pa)

CAM_COEFFICIENT_2 CAM fitting coefficient ωc (only for |G*| ANN model) (Pa)

CAM_COEFFICIENT_3 CAM fitting coefficient k (only for |G*| ANN model) (Pa)

CAM_COEFFICIENT_4 CAM fitting coefficient “me” (only for |G*| ANN model) (Pa)

SIGMOIDAL_COEFFICIENT_1 Sigmoidal fitting function coefficient δ (psi)

SIGMOIDAL_COEFFICIENT_2 Sigmoidal fitting function coefficient α 

SIGMOIDAL_COEFFICIENT_3 Sigmoidal fitting function coefficient β

SIGMOIDAL_COEFFICIENT_4 Sigmoidal fitting function coefficient γ

SHIFT_FACTOR_COEFFICIENT 1 Shift factor fitting function coefficient α 1 (°C)

SHIFT_FACTOR_COEFFICIENT 2 Shift factor fitting function coefficient α 2 (°C)

SHIFT_FACTOR_COEFFICIENT 3 Shift factor fitting function coefficient α 3 (°C)

SAMPLE_DATE Date that binder was sampled (only for field-aged binder)

SAMPLE_AGE

Age of test sample relative to construction (days) (only for fieldaged binder)
231 

Table 82. Summary file format from batch mode in ANNACAP. 

Table 83. Output data format for batch mode ANNACAP use. 

Item Description

STATE_CODE Code representing State or province as input by user

PROJECT_ID Project ID code as input by user

PROJECT_LAYER Project layer code as input by user

CONSTRUCTION_DATE Date layer was constructed as input by user

SAMPLE_TYPE

1–original binder, 2–RTFO/TFO binder, 3–PAV binder, and

4–field binder

AVAILABLE_MODELS

Listing of models that can be used in modulus prediction with 

the available input data 

VIOLATED_MODELS

Listing of available models for which the input data violates the 

calibration range

CHOSEN_MODEL

MR-resilient modulus ANN, VV-viscosity ANN, GV-|G*| ANN, 

GV-PAR-|G*| ANN with inconsistent aging conditions, VVgrade, viscosity ANN with viscosity coming from binder grade, 

*-V, *ANN model with inputs violating the input range

Item Description

SECTION_ID

Unique ID combing State code, project ID and layer, sample type, and 

model name

STATE_CODE Code representing State or province as input by user

PROJECT_ID Project ID code as input by user

PROJECT_LAYER Project layer code as input by user

CONSTRUCTION_DATE Date layer was constructed as input by user

SAMPLE_TYPE 1–original binder, 2–RTFO/TFO binder, 3–PAV binder, and 4–field binder

PREDICTIVE_MODEL

MR-resilient modulus ANN, VV-viscosity ANN, GV-|G*| ANN, GV-PAR-

|G*| ANN with inconsistent aging conditions, VV-grade, viscosity ANN with 

viscosity coming from binder grade, *-V, *ANN model with inputs violating 

the input range

TEMPERATURE Temperature of modulus prediction (°F)

FREQUENCY Frequency of modulus prediction (Hz)

|E*|_PREDICTION Predicted dynamic modulus (psi).

VMA

Voids in mineral aggregate as percent total volume (blank for 

MR ANN)

VFA Voids filled with asphalt as percent VMA (only for viscosity and |G*| ANN)

VISCOSITY Viscosity input (109 P) (only for viscosity ANN model)

A Viscosity model intercept A (only for viscosity ANN model)

VTS Viscosity model slope (only for viscosity ANN model)

MR_5C Resilient modulus at 5 °C (only for MR model) (MPa)

MR_25C Resilient modulus at 25 °C (only for MR model) (MPa)

MR_40C Resilient modulus at 40 °C (only for MR model) (MPa)

|G*| Binder shear modulus (psi) only for |G*| ANN model

WLF_COEFFICIENT_1 WLF shifting function coefficient C1 (not used in batch mode)
232 

°C = (°F-32)/1.8

1 psi = 6.86 kPa

WLF_COEFFICIENT_2 WLF shifting function Coefficient C2 (Not used in batch mode)

CAM_COEFFICIENT_1 CAM fitting coefficient Gg (only for |G*| ANN model) (Pa)

CAM_COEFFICIENT_2 CAM fitting coefficient ωc (only for |G*| ANN model) (Pa)

CAM_COEFFICIENT_3 CAM fitting coefficient k (only for |G*| ANN model) (Pa)

CAM_COEFFICIENT_4 CAM fitting coefficient me (only for |G*| ANN model) (Pa)

SIGMOIDAL_COEFFICIENT_1 Sigmoidal fitting function coefficient δ (psi)

SIGMOIDAL_COEFFICIENT_2 Sigmoidal fitting function coefficient α 

SIGMOIDAL_COEFFICIENT_3 Sigmoidal fitting function coefficient β

SIGMOIDAL_COEFFICIENT_4 Sigmoidal fitting function coefficient γ

SHIFT_FACTOR_COEFFICIENT 1 Shift factor fitting function coefficient α 1 (°C)

SHIFT_FACTOR_COEFFICIENT 2 Shift factor fitting function coefficient α 2 (°C)

SHIFT_FACTOR_COEFFICIENT 3 Shift factor fitting function coefficient α 3 (°C)

QUALITY_CONTROL_#1 A—the output data passed QC #1; F—the input data did not pass QC #1

QUALITY_CONTROL_#2 A—the output data passed QC check 2; F—the input data did not pass QC #2

QUALITY_CONTROL_#3

A—the output data passed QC check 3 or QC #3 did not apply; F—the input 

data did not pass QC #3

QUALITY_CONTROL_#4

A—the output data passed QC #4 or QC #4 did not apply; F—the input data 

did not pass QC #4

QUALITY_CONTROL_#5 A—the output data passed QC #5; F—the input data did not pass QC #5

QUALITY_CONTROL_#6

A—the output data passed QC #6 or QC #6 did not apply; F—the input data 

did not pass QC #6 

QUALITY_CONTROL_#7

A—the output data passed QC #7 or QC #7 did not apply; F—the input data 

did not pass QC #7 

AVAILABLE_MODELS

Listing of models that can be used in modulus prediction with the available 

input data

VIOLATED_MODELS

Listing of available models for which the input data violates the calibration 

range

CHOSEN_MODEL Listing of model chosen for predicting the modulus

SAMPLE_DATE Date that binder was sampled (only for field-aged binder)

SAMPLE_AGE

Age of test sample relative to construction (days) (only for field-aged binder, 

blank means either SAMPLE_DATE or CONSTRUCTION_DATE were 

not given)

INDIVIDUAL_DATA_GRADE

NCSU grade for modulus prediction; “A”—the predicted modulus is 

acceptable; “C”—the predicted modulus is questionable; and “F”—the 

predicted modulus may have severe problems 

MASTERCURVE_GRADE

NCSU grade for mastercurve prediction; “A”—the predicted curve is 

acceptable; “C”—the predicted curve is questionable; and “F”—the 

predicted curve may have severe problems
233 

E.8 FORM OF SUPPLEMENTARY FUNCTIONS

E.8.1 CAM Model Function

*

1

e

g

m k k

c

R

G

G

ω

ω

=

      +        

(120) 

Where:

ω R = Reduced angular frequency. 

Gg, ω c, k, and me = Fitting coefficients. 

*

R T ω ω= a (121) 

Where:

ω = Physical angular frequency of load. 

aT = Time-temperature shift factor.

E.9 TIME-TEMPERATURE SHIFT FACTOR FUNCTION FOR |G*| 

E.9.1 Level 1 (WLF Function) 

1 ( )

2

log R

T

R

CT T

a

C TT

− = + −

(122) 

Where:

T = Test temperature of interest. 

TR = Reference temperature (chosen as 59 °F (15 °C) for ANNACAP). 

C1 and C2 = WLF fitting coefficients.

E.9.2 Levels 2 and 3 

For the GV ANN models not using level 1 input, the shift factor is given by the following: 

( )

1

2

1

2

1 1 0

2.303* 273 273

log

0

+ a R

R

T

R

R

E C T T

R T CT

a

C T T

T

C TT

    −   − ≤     + − =   −   >

  + − (123)
234 

Where:

R = 8.314 x 10-3 kJ·K-1·mol-1. 

Ea = 189.879 kJ/mol. 

C1 = -13.227. 

C2 = 90.349. 

E.10 SIGMOIDAL FUNCTION

*log( ) log * 1 Rt E

e

β γ

α δ + = +

+

(124) 

Where:

tR = The inverse of reduced frequency of loading, which is defined in the same 

way as reduced angular frequency in equation 121 but with frequency in hertz 

 instead of radians per second. 

δ , α , β , and γ = Fitting coefficients.

E.11 TIME-TEMPERATURE SHIFT FACTOR FUNCTION FOR |E*| 

2

1 23 log T aTT = ++ α αα (125) 

Where: 

aT = Mixture time-temperature shift factor. 

T = Temperature of interest. 

α 1, α 2, and α 3 = Fitting coefficients. 
235 

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HRDI-30/09-11(600)E Recycled

Recyclable