Superpave Office and Advanced Pavement Laboratory at Arizona State University (ASU) for all of ...... 2.9 Log Shift Factor versus Temperature for Two-Guns Mix.
DEVELOPMENT OF A NEW REVISED VERSION OF THE WITCZAK E* PREDICTIVE MODELS FOR HOT MIX ASPHALT MIXTURES by Javed Bari
A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
ARIZONA STATE UNIVERSITY December 2005
DEVELOPMENT OF A NEW REVISED VERSION OF THE WITCZAK E* PREDICTIVE MODELS FOR HOT MIX ASPHALT MIXTURES by Javed Bari
has been approved November 2005
APPROVED: , Chair r r Supervisory Committee
ACCEPTED:
____________________________________ Department Chair
____________________________________ Dean, Division of Graduate Studies
ABSTRACT The main purpose of this dissertation is to present the development of a new set of predictive models for stiffness of asphalt cement binders and a new revised version of the Witczak dynamic modulus (E*) predictive model of hot mix asphalt (HMA) mixtures. Master databases of binder stiffness and HMA E* were compiled and revised. The binder stiffness database contained lab shear modulus (|G b*|) and phase angle (δ b) data with ASTM (American Society for Testing Materials) viscosity (Ai-VTS i) data. The database has 8,940 data points from 41 different binders. The majority of the HMA E* database has been developed from the test results obtained during the National Cooperative Highway Research Program (NCHRP) 9-19 Project (also conducted by Arizona State University). This database contained lab HMA E* data with aggregate gradation, mixture volumetric and binder Ai-VTS i data. The database has 7400 data points from 346 different mixtures. It is an extension of the Witczak E* predictive model database that has been periodically updated since 1969. The binder stiffness master database was used to develop a new predictive model for binder stiffness. The first set of model equations in this model is a fully revised version of the widely known “ASTM Ai-VTS i Viscosity Model”. The second and third sets of model equations in the model predict |Gb*| and associated δ b, respectively. The binder stiffness predictive model has been found to be rational, unbiased, accurate, and statistically sound. The E* master database was used to develop a new revised version of the Witczak E* predictive model for HMA mixtures. This model was aimed at improving the limitations associated with the current version that has been used in the present draft of iii
the new mechanistic-empirical pavement design guide (M-E PDG) developed under the NCHRP Project 1-37A. The new E* model has been found to be rational, unbiased, accurate, and statistically sound. The new models developed under this Ph.D. research have similar mathematical structures as the ones used in the new M-E PDG. Therefore, it is hypothesized that the new models can be easily incorporated in a future revision of this pavement design guide.
iv
DEDICATION To my beloved wife Rozina and daughter Faria, respected parents, brothers, and friends. Their loving encouragement and support have always helped me in staying focused and keeping things in perspective in life.
v
ACKNOWLEDGMENTS I wish to express sincerest appreciation and gratitude to Professor Matthew W. Witczak, my Ph.D. Committee Chair, for his all-round support throughout my graduate study, for his guidance, encouragement, and constructive criticism offered in various phases of this research, and for his critical review of the manuscript. Special appreciation and gratitude is expressed to my wife (Rozina) and daughter (Faria) for their encouragement, patience and support during my academic program. Sincere gratitude is also extended to my parents for getting me ready to pursue such a challenging endeavor in my life. Special thanks and gratitude are due to Professors Michael S. Mamlouk and Kamil E. Kaloush for their invaluable insight and advice regarding this research and their review of the manuscript. I would like to state my sincerest appreciation and thanks to my colleagues in the Superpave Office and Advanced Pavement Laboratory at Arizona State University (ASU) for all of the supporting work they contributed for this research. Many of their contributions are included and acknowledged in this dissertation. Finally, appreciation is also expressed to the Department of Civil and Environmental Engineering at ASU for making its facilities available for this research work.
vi
TABLE OF CONTENTS Page LIST OF TABLES ..…..…….……………………….…………………….……..…..………….…... xvi LIST OF FIGURES ……..…….…………………...…………………..………………….…...…… xviii CHAPTER 1
INTRODUCTION .................................................................................................. 1 1.1
Background ..........................................................................................................1
1.2
Mechanistic-Empirical (M-E) Design Efforts .....................................................2
1.3
Dynamic Modulus as HMA Stiffness ..................................................................3
1.4
Features of Dynamic Modulus (E*) ....................................................................5
1.5
Role of a Predictive Model for Dynamic Modulus .............................................8
1.6
Expansion of E* Database ...................................................................................9
1.7
E* Master Database ...........................................................................................11
1.8
Binder Database .................................................................................................26
1.9
Problems with the Current E* Model................................................................27
1.10
Assessing the Reliability of the Current Model.................................................30
1.11
Objectives ..........................................................................................................31
1.12
Scope of Research..............................................................................................34
1.13
Dissertation Organization..................................................................................35
vii
CHAPTER 2
Page
LITERATURE REVIEW ..................................................................................... 37 2.1
Asphalt Binder Characterization........................................................................37
2.1.1
Behavior of Asphalt Binder .......................................................................... 37
2.1.2
Properties of Asphalt Binders ....................................................................... 38
2.1.2.1
Ductility ................................................................................................ 39
2.1.2.2
Penetration ............................................................................................ 40
2.1.2.3
Softening Point...................................................................................... 40
2.1.2.4
Viscosity................................................................................................ 41
2.1.2.5
Stiffness................................................................................................. 42
2.1.2.6
Age Hardening ...................................................................................... 42
2.1.2.7
Temperature Susceptibility ................................................................... 46
2.1.2.8
Shear Susceptibility............................................................................... 46
2.1.3
Pavement Distresses Re lated to Rheological Properties of Binder .............. 47
2.1.4
Asphalt Binder Characterization Tests ......................................................... 47
2.2
Stiffness of Asphalt Binders ..............................................................................48
2.2.1
Common Stiffness Parameters ...................................................................... 48
2.2.2
Determination of |Gb*| and δ b by DSR Test.................................................. 49
2.2.3
Predictive Models for Stiffness of Asphalt Binders ..................................... 52 viii
CHAPTER
Page
2.2.3.1
Van der Poel (Shell Oil’s Early Version) Model.................................. 52
2.2.3.2
ASTM Ai-VTS i Model.......................................................................... 54
2.2.3.3
Witczak et al. Models ............................................................................ 56
2.3
Asphalt Concrete Mixture Characterization......................................................61
2.3.1
Behavior of Asphalt Concrete Mixture ......................................................... 61
2.3.2
Volumetric Properties of Asphalt Concrete Mixture .................................... 61
2.3.3
Pavement Distresses Related to Asphalt Concrete Mixture Properties ........ 63
2.3.4
Asphalt Concrete Mixture Characterization Tests........................................ 63
2.4
Stiffness of Asphalt Concrete Mixtures.............................................................64
2.4.1
E* Stiffness of Asphalt Concrete Mixtures .................................................. 64
2.4.2
Determination of Stiffness by E* Test .......................................................... 65
2.4.3
Time-Temperature Superposition of Dynamic Modulus .............................. 67
2.4.4
Use of E* in the 2002 Design Guide ............................................................ 71
2.4.5
Predictive Models for E* Stiffness ............................................................... 73
2.4.5.1
Van der Poel (Shell Oil’s Early Version) Model.................................. 73
2.4.5.2
Bonnaure (Shell Oil’s Later Version) Model........................................ 75
2.4.5.3
Shook and Kallas’ Models .................................................................... 76
2.4.5.4
Witczak’s Early Model ......................................................................... 78 ix
CHAPTER 2.4.5.5
Witczak and Shook’s Model................................................................. 79
2.4.5.6
Witczak’s 1981 Model.......................................................................... 80
2.4.5.7
Witczak, Miller and Uzan’s Model....................................................... 80
2.4.5.8
Witczak and Akhter’s Models ............................................................... 81
2.4.5.9
Witczak, Leahy, Caves and Uzan’s Models ......................................... 83
2.4.5.10
Witczak and Fonseca’s Model.......................................................... 84
2.4.5.11
Andrei, Witczak and Mirza’s Revised Model................................... 88
2.4.5.12
Hirsch Model of Christensen, Pellinen and Bonaquist ..................... 90
2.5
Statistical Background .......................................................................................91
2.5.1
Statistical Analysis ........................................................................................ 92
2.5.1.1
Model Comparison................................................................................ 92
2.5.1.2
Goodness of Fit ..................................................................................... 93
2.5.2 3
Page
Model Optimization...................................................................................... 94
ASPHALT BINDER STIFFNESS DATABASE................................................. 96 3.1
Introduction........................................................................................................96
3.2
Original η-Gb* Database ...................................................................................96
3.3
Expanded η-Gb* Database.................................................................................97
3.4
Test Methodologies Related to the Gb*-η Database..........................................99 x
CHAPTER 3.5
Page
Conversion of Test Data ..................................................................................101 DEVELOPMENT OF A COMPREHENSIVE η-|Gb*|-δ b MODEL .................. 102
4 4.1
Introduction......................................................................................................102
4.2
Problems with the Current Gb* Models ...........................................................104
4.3
Conceptual Model Development .....................................................................106
4.3.1
Modified ASTM Ai-VTSi Relationship Development ............................... 106
4.3.2
|Gb*| Model Development ........................................................................... 109
4.3.3
δ b Model Development ............................................................................... 116
4.4
Optimization Technique for the Comprehensive η-|Gb*|-δ b Model ................120
4.5
Final Comprehensive η-|Gb*|-δ b Model ..........................................................121
4.5.1
Final Modified Ai-VTSi Relationship (η-Model) ...................................... 121
4.5.2
Final δ b Model............................................................................................. 122
4.5.3
Final |Gb*| Model........................................................................................ 124 Performance of the Final Comprehensive η-|Gb*|-δ b Model...........................125
4.6 4.6.1
Performance of the Final Comprehensive η-|Gb*| Model........................... 125
4.6.1.1
Accuracy of η-|Gb*| Model................................................................. 125
4.6.1.2
Comparison with Previous η-|Gb*| Models ........................................ 128
xi
CHAPTER 4.6.2
5
Page Performance of the Phase Angle (δ b) Model .............................................. 134
4.6.2.1
Accuracy of δ b Model......................................................................... 134
4.6.2.2
Comparison with Previous δ b Models ................................................. 137
ASPHALT MIXTURE E* STIFFNESS DATABASE ...................................... 142 5.1
Introduction......................................................................................................142
5.2
Original E* Database .......................................................................................142
5.3
Expanded E* Database ....................................................................................143
5.4
Revised E* Database .......................................................................................144
5.5
Test Methodologies Related to the E* Database .............................................146
5.6
Master Curve Parameters.................................................................................150
6
VARIABLES OF E* MODEL ........................................................................... 151 6.1
List of |E*| Models ...........................................................................................151
6.2
Variables in E* Models ...................................................................................151
6.3
Important Predictor Variables..........................................................................152
6.4
Sigmoidal Model Form....................................................................................154
6.5
Characteristics of Predictor Variables .............................................................155
6.6
Correlation of |E*| with Predictor Variables....................................................156
xii
CHAPTER 7
Page
NEW E* MODEL DEVELOPMENT ................................................................ 162 7.1
Introduction......................................................................................................162
7.2
Basic Hypotheses .............................................................................................162
7.3
Modeling Approach.........................................................................................163
7.3.1
Sigmoidal Form .......................................................................................... 163
7.3.2
Sub-Models ................................................................................................. 164
7.3.3
Model Optimization Technique .................................................................. 164
7.3.4
Calculation of Goodness of Fit ................................................................... 165
7.3.5
Sequential Optimization of Sub-Models and the Main Model ................... 166
7.3.5.1
Optimization of the δ-Model............................................................... 167
7.3.5.2
Optimization of the α-Model.............................................................. 168
7.3.5.3
Optimization of the Exponent Model ................................................. 168
7.3.6 7.4 8
Model Rationality ....................................................................................... 169 Model Selection...............................................................................................170
PERFORMANCE OF THE NEW E* MODEL ................................................. 172 8.1
Introduction......................................................................................................172
8.2
Statistics of the New E* Model .......................................................................172
8.3
Comparison between Predicted and Observed |E*| Values .............................173 xiii
CHAPTER
Page
8.4
Predicting Capability of the New E* Model....................................................175
8.5
Frequency Distribution of Residuals ...............................................................177
8.5.1
Frequency Distribution for Full Data Range .............................................. 177
8.5.2
Frequency Distribution at Ranges of Temperatures ................................... 180
8.5.3
Frequency Distribution at Ranges of E* Stiffness Values .......................... 183
8.6
Sensitivity Analysis of the New E* Model......................................................185
8.6.1
Ranges of Predictor Variables..................................................................... 186
8.6.2
Model Sensitivity to Gradation................................................................... 188
8.6.2.1
Model Sensitivity to ρ200..................................................................... 188
8.6.2.2
Model Sensitivity to ρ4 ....................................................................... 191
8.6.2.3
Model Sensitivity to ρ38 ...................................................................... 194
8.6.2.4
Model Sensitivity to ρ34 ...................................................................... 194
8.6.3
Model Sensitivity to Mix Volumetric Properties ........................................ 200
8.6.3.1
Model Sensitivity to Mix Air Voids (Va)............................................ 200
8.6.3.2
Model Sensitivity to Mix Effective Binder Content ........................... 203
8.6.4
Model Sensitivity to Binder Stiffness ......................................................... 205
8.6.4.1
Model Sensitivity to |Gb*| ................................................................... 206
8.6.4.2
Model Sensitivity to δ b........................................................................ 209 xiv
CHAPTER 8.6.5
Page Model Sensitivity to Temperature .............................................................. 209
8.7
Response of Predicted E* to Mix Volumetrics................................................216
8.8
Comparison with Previous Models..................................................................223
8.8.1
Comparison of Statistical Goodness of Fit ................................................. 223
8.8.2
Comparison of Plots for Predicted versus Observed Data .......................... 225
8.8.3
Comparison with Hirsch Model.................................................................. 229
8.9 9
Final E* Model ................................................................................................231 CONCLUSIONS AND RECOMMENDATIONS ............................................. 232
9.1
Introduction......................................................................................................232
9.2
Conclusions ......................................................................................................232
9.3
Recommendations ............................................................................................240
REFERENCES ….………………………………………..………………...…………………………243 APPENDIX A
MIXTURE E* MASTER DATABASE.…...…………………….…………..…………250 A.1
HMA Mixture I.D. …………..………..………..…………..……………………………251
A.2
HMA Mixture Data.…………………….…………………..……………………………266
A.3
E* Data of HMA Mixtures .………………………………..……………………………280
B
BINDER η-GB* MASTER DATABASE.……………….……..…………………....…556
C
MIXTURE E* MASTER CURVE PARAMETERS .……………………..…………828
D
FINAL CANDIDATE E* PREDICTIVE MODELS …...……………..…….………843 xv
LIST OF TABLES Table
Page
1.1 Summary of Data Points of the E* Database ............................................................ 10 1.2 Statistics of Witczak E* Model for the UMD and ASU Database ........................... 30 3.1 Summary of Asphalt Binders of the η-Gb* Database ............................................... 98 4.1 Statistics of Predictive Models for η-|Gb*| of Asphalt Binders .............................. 133 4.2 Statistics of Predictive Models for Binder Phase Angle (δ b) .................................. 141 5.1 Summary of Mixtures in ASU E* Database ........................................................... 145 5.2 Test Conditions of the ASU E* Tests ..................................................................... 147 6.1 List of E* Models.................................................................................................... 151 6.2 Variables in Historic E* Models ............................................................................. 152 6.3 Correlation of E* with Predictor Variables at Different Temperature Zones......... 161 8.1 Statistics of the New E* Model .............................................................................. 173 8.2 Range of E* Model Variables at T = 14°F and fc = 10 Hz..................................... 187 8.3 Range of E* Model Variables at T = 70°F and f = 10 Hz ...................................... 187 8.4 Range of E* Model Variables at T = 130°F and f = 10 Hz .................................... 188 8.5 Input Data wit h Observed and Predicted E* Data for ρ200 ..................................... 189 8.6 Input Data with Observed and Predicted E* Data for ρ4 ........................................ 192 8.7 Input Data with Observed and Predicted E* Data for ρ38 ....................................... 195 8.8 Input Data with Observed and Predicted E* Data for ρ34 ....................................... 198 8.9 Input Data with Observed and Predicted E* Data for Va........................................ 201 8.10 Input Data with Observed and Predicted E* Data for Vbeff .................................. 204 xvi
Table
Page
8.11 Input Data with Observed and Predicted E* Data for |Gb*| .................................. 207 8.12 Input Data with Observed and Predicted E* Data for δ b ...................................... 210 8.13 Input Data for Model Response to Mix Volumetrics............................................ 217 8.14 Goodness of Fit Statistics of E* Models ............................................................... 225 8.15 Range of Variables Used in the Development of Hirsch and New Model ........... 230 A.1 HMA Mixture I.D ...…………………………………..…………………………252 A.2 HMA Mixture Data …...…………………………………………………………267 A.3
E* Master Database ……………………...………….….…………………….…281
B.1 Binder η-Gb* Master Database ………………...…………..……………………557 C.1 Master Curve Parameters of HMA Mixtures ….…………………………………829 D.1
Summary of Final Candidate E* Models ……..…………………………………844
D.2
Goodness of Fit Statistics of Final Candidate E* Models …....……….…….……845
xvii
LIST OF FIGURES Figure
Page
1.1 Storage and Loss Modulus .......................................................................................... 7 1.2 Frequency Distribution and Range of Temperature.................................................. 12 1.3 Frequency Distribution and Range of Loading Frequency....................................... 13 1.4 Frequency Distribution and Range of Effective Binder Volume.............................. 14 1.5 Frequency Distribution and Range of Specimen Air Voids ..................................... 15 1.6 Frequency Distribution and Range of Voids in Mineral Aggregates ....................... 16 1.7 Frequency Distribution and Range of Voids Filled with Asphalt............................. 17 1.8 Frequency Distribution and Range of Percent Retained on ¾″ Sieve (ρ34 ).............. 18 1.9 Frequency Distribution and Range of Percent Retained on 3/8″ Sieve (ρ38 )............ 19 1.10 Frequency Distribution and Range of Percent Retained on #4 Sieve (ρ4 ).............. 20 1.11 Frequency Distribution and Range of Percent Passing #200 Sieve (ρ200 ).............. 21 1.12 Frequency Distribution Test Temperatures for E* Master Database...................... 22 1.13 Frequency Distribution of Loading Frequency for E* Master Database ................ 22 1.14 Frequency Distribution Effective Binder Volume for E* Master Database ........... 23 1.15 Frequency Distribution of Specimen Air Voids for E* Master Database .............. 23 1.16 Frequency Distribution of VMA for E* Master Database ...................................... 24 1.17 Frequency Distribution of VFA for E* Master Database ....................................... 24 1.18 Frequency Distribution of Aggregate Sizes for E* Master Database ..................... 25 1.19 Predicted versus Measured E* for Current Witczak E* Model.............................. 32 2.1 Time of Loading Dependence of Binder Stiffness for Axial Tension...................... 43 2.2 Viscosity versus Depth as a Function of Time at a Given Temperature .................. 45 xviii
Figure .
Page
2.3 Dynamic Shear Rheometer Operation ...................................................................... 50 2.4 Nomograph for Stiffness of Asphalt Binders (2) ...................................................... 53 2.5 Binder and Mix Behavior in Black Space................................................................. 57 2.6 Specimen Instrumentation of E* Testing.................................................................. 66 2.7 Laboratory E* versus Loading Time for Two-Guns Mix ......................................... 70 2.8 Master Curve with Shifted E* Data for Two-Guns Mix........................................... 70 2.9 Log Shift Factor versus Temperature for Two-Guns Mix ........................................ 71 2.10 Nomograph for Stiffness Modulus of AC Mixtures (32)........................................ 77 2.11 Concepts of Precision and Bias................................................................................ 92 4.1 CF as a Function of Loading Frequency (fs) for PG 58-22 Binder at 25°F ............ 113 4.2 CF as a Function of Phase Angle for Paramount PG 58-22 Binder at 25°F........... 114 4.3 CF as a Function of sinδ b for Paramount PG 58-22 Binder at 25°F ....................... 114 f
4.4 CF as a Function of (sinδ b) s for Paramount PG 58-22 Binder at 25°F .................. 115 4.5 Relationship between A and VTS for All Binders.................................................. 117 4.6 Relationship between log(fs x ηfs ,T ) and Phase Angle (δ b) for All Binders ............ 118 4.7 Log(fs x ηfs ,T ) versus δ b for Paramount PG 58-22 Binder at Different Ages .......... 119 4.8 Observed versus Predicted |G b*| (Using the New η-|Gb*| Model) ......................... 127 4.9 Distribution of Residuals for the η-|Gb*| Model..................................................... 128 4.10 Comparison of η-|Gb*| Models (Based on Original Data) .................................... 130 4.11 Comparison of η-|Gb*| Models (Based on the Expanded Database) .................... 132 4.12 Observed versus Predicted Phase Angle Using the New δ b Model ...................... 136 xix
Figure .
Page
4.13 Distribution of δ b Residuals for the New δ b Model.............................................. 137 4.14 Comparison of δ b Models (Based on Original Model Data)................................. 138 4.15 Comparison of δ b Models (Based on the Expanded Database) ............................ 139 5.1 Vertical Stress versus Loading Time in Actual E* Test of an ADOT Mix ............ 149 6.1 |E*| Master Curve of a Sigmoidal Form for Two-Guns Mix .................................. 155 6.2 Relationship of |E*| with Gradation Parameters (ρ200 , ρ4 , ρ38 and ρ34 ) .................. 158 6.3 Relationship of |E*| with Volumetric Parameters (Va, Vbeff, VMA, VFA)............. 159 6.4 Relationship of |E*| with Binder Stiffness Parameters (|Gb*| and δ b)..................... 160 8.1 Comparison between Predicted and Observed Log E* Values .............................. 174 8.2 Comparison between Predicted and Observed E* Values ...................................... 174 8.3 Predicted versus Observed |E*| for MnRoad Mixtures ........................................... 176 8.4 Predicted versus Observed |E*| for Salt River ¾ inch Mixtures ............................. 177 8.5 Frequency Distribution of E* Residuals ................................................................. 178 8.6 Frequency Distribution of LogE* Residuals........................................................... 179 8.7 Frequency Distribution of E* Ratio ........................................................................ 180 8.8 Frequency Distribution of E* Residuals at Temperature Ranges ........................... 181 8.9 Frequency Distribution of Log E* Residuals at Temperature Ranges ................... 181 8.10 Frequency Distribution of E* Ratio at Temperature Ranges ................................ 182 8.11 Frequency Distribution of E* Residuals at E* Stiffness Ranges .......................... 184 8.12 Frequency Distribution of Log E* Residuals at E* Stiffness Ranges .................. 184 8.13 Frequency Distribution of E* Ratio at E* Stiffness Ranges ................................. 185 8.14 E* Model Sensitivity to ρ200 ................................................................................. 190 xx
Figure .
Page
8.15 E* Model Sensitivity to ρ4 .................................................................................... 193 8.16 E* Model Sensitivity to ρ38................................................................................... 196 8.17 E* Model Sensitivity to ρ34................................................................................... 199 8.18 E* Model Sensitivity to Va ................................................................................... 202 8.19 E* Model Sensitivity to Vbeff ................................................................................ 205 8.20 E* Model Sensitivity to |G b*|............................................................................... 208 8.21 E* Model Sensitivity to δ b .................................................................................... 211 8.22 ∆LogE* Versus ρ200 at fc = 10 Hz and T = 14, 70 and 130°F .................................. 212 8.23 ∆LogE* Versus ρ4 at fc = 10 Hz and T = 14, 70 and 130°F..................................... 213 8.24 ∆LogE* Versus ρ38 at fc = 10 Hz and T = 14, 70 and 130°F.................................... 213 8.25 ∆LogE* Versus ρ34 at fc = 10 Hz and T = 14, 70 and 130°F.................................... 214 8.26 ∆LogE* Versus Va at fc = 10 Hz and T = 14, 70 and 130°F .................................... 214 8.27 ∆LogE* Versus Vbeff at fc = 10 Hz and T = 14, 70 and 130°F ................................. 215 8.28 ∆LogE* Versus |Gb*| at fc = 10 Hz and T = 14, 70 and 130°F................................. 215 8.29 ∆LogE* Versus δ b at fc = 10 Hz and T = 14, 70 and 130°F ..................................... 216 8.30 Response of Predicted E* to Mix Volumetrics at T = 14°F and f = 1 Hz ............ 218 8.31 Response of Predicted E* to Mix Volumetrics at T = 40°F and f = 1 Hz ............ 219 8.32 Response of Predicted E* to Mix Volumetrics at T = 70°F and f = 1 Hz ............ 220 8.33 Response of Predicted E* to Mix Volumetrics at T = 100°F and f = 1 Hz .......... 221 8.34 Response of Predicted E* to Mix Volumetrics at T = 130°F and f = 1 Hz .......... 222
xxi
Figure
Page
8.35 Predicted Versus Observed E* for Shell Oil (2nd Version) Model (Expanded Data) ................................................................................................................................. 226 8.36 Predicted Versus Observed E* for Shell Oil (1977) Model (Expanded Data) ..... 227 8.37 Predicted Versus Observed E* for Current Witczak (1999) Model (Original 2750 Data)........................................................................................................................ 227 8.38 Predicted Versus Observed E* for Current Witczak (1999) Model (Expanded Data) ................................................................................................................................. 228 8.39 Predicted Versus Observed E* for Hirsch (2003) Model (Expanded Data)......... 228 8.40 Predicted Versus Observed E* for New Witczak (2005) Model (Expanded Data) ................................................................................................................................. 229
xxii
1
INTRODUCTION
1.1
Background The approximately 4 million mile- long nationally interconnected road network of
the U.S. provides the basis for America’s economic prosperity. The demands and reliance upon the American pavement system for mobility and commerce have increased substantially over the past few decades. The total length of the roads in the U.S., surfaced with either asphalt or cement concrete, is about 2.3 million miles, out of which approximately 96% have hot mix asphalt (HMA) surface (1). HMA mixtures in these pavements are subjected to a wide range of load and environmental conditions. The response to these condit ions is complex and involves the elastic, viscoelastic and plastic characteristics of the material used in the pavement. The stiffness of a HMA mix is a specific material response parameter that determines the strains and displacements pavement structure as it is loaded or unloaded. In the early 1950’s, Van der Poel of the Shell Oil Company introduced the term “stiffness” (or stiffness modulus) (2). The stiffness of a HMA mix is a modulus that is dependent upon the loading time and temperature of the mix. Due to the immense importance of stiffness in the analysis, design and performance evaluation of HMA mixture and flexible pavement structures; researchers have been trying to develop accurate stiffness (modulus) laboratory test protocols as well as to deve lop accurate predictive models and equations. Over the last fifty years, numerous models and regression equations have been developed to predict the stiffness of a HMA mix. Historically, the stiffness predictive models and equations were developed based on the conventional multivariate linear regression or non- linear
2 regression analysis of laboratory test data and the established or anticipated basic engineering behavior and/or properties of the HMA mixture and/or its components.
1.2
Mechanistic-Empirical (M-E) Design Efforts In the worldwide pavement community, potential research efforts are currently
being focused on the Mechanistic-Empirical (M-E) design of pavements. Since potential M-E designs are mechanics based, they can adapt to varied and changing dis tress modes, load limits and load configurations. They also allow for rational materials tests and characterization and allow for the direct interaction between structural and materials design. There are several computer programs available to mechanistic-empirically evaluate stresses and strains in a pavement for a given set of loading, environmental and pavement cross section conditions. Most of them, however, use some form of linear elastic layer analysis. The basic material properties required for multi- layer linear elastic analysis are the material modulus (E) and Poisson’s ratio (ν). Unfortunately, most, if not all, pavement materials are not purely elastic. For AC layers, the modulus varies considerably with temperature and rate (or time) of loading (i.e. true viscoelastic response). That is why, in recent years, pavement material researchers have concentrated their research efforts on the time-temperature dependant stiffness properties of hot mix asphalt (HMA) pavement mixtures. Recently, Dr. Mathew W. Witczak, leader of the asphalt team for the National Cooperative Highway Research Program (NCHRP) Project 1-37A Project and Professor at Arizona State University (ASU), completed a multi- year comprehensive research under NCHRP 1-37A Project. The primary goal of this research effort was to develop the
3 flexible pavement analysis and design part of the draft AASHTO Pavement Design Guide titled “Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures”. While this product was initially known as the “2002 Design Guide”, it is now referred to as the M-E PDG (Mechanistic-Empirical Pavement Design Guide) (3). In the remaining portion of this dissertation, this design guide will be mentioned as either the “2002 Design Guide” or “M-E PDG” for convenience. The ASU research team also developed a comprehensive pavement analysis and design computer software named “APADS” (Asphalt Pavement Analysis and Design System) for the M-E analysis and design of asphalt pavements. This software is an integral part of the 2002 Design Guide. It has the capability of designing and analyzing new and rehabilitated asphalt pavements.
1.3
Dynamic Modulus as HMA Stiffness Historically, various types of material parameters have been used for presenting
the stiffness characteristics of asphalt mixtures that include flexural stiffness, creep compliance, relaxation modulus, resilient modulus, dynamic modulus etc. At present, one of the most universally used methodologies to characterize the modulus of asphalt mixtures is the dynamic (complex) modulus (E*). Research, led by Dr. M. W. Witczak, conducted at ASU, under the NCHRP 9-19 project demonstrated that the complex (dynamic) modulus (E*) can be used as a good performance indicator for the HMA design stage (4). Witczak and other colleagues working on the NCHRP 9-19 project have summarized several advantages of the use of E* in the HMA pavement analysis and design over other stiffness parameters (such as the Resilient Modulus, Mr) as follows:
4 •
E* data allows a hierarchical HMA mixture characterization approach to be used,
•
aging can be taken into account,
•
vehicle speed (time of load) can be taken into account,
•
E* can be linked to the SHRP Performance Graded binder specifications, and
•
E* is more fundamentally and theoretically compared to the Falling Weight Deflectometer (FWD) back-calculated modulus of HMA mixtures.
The NCHRP Report 465 ranked the |E*|/sinφ parameter, which can be obtained from complex (dynamic) modulus (E*) test, as one of the most preferred candidate parameters for the SPT (Simple Performance Test) for use in the Superpave Mix Design procedure (5). In this final Task C analysis of the NCHRP 9-19 study, Witczak et al identified the advantages of using the dynamic modulus (|E*|) in pavement analysis and design as follows: •
It provides necessary input for structural analysis for a considerably wide range of temperature (very low to moderately high) and loading rate (low to high).
•
It provides a rational way to establish mix criteria for rutting, cracking etc.
•
It can be linked to a predictive model.
The 2002 Design Guide uses the dynamic modulus (|E*|) as the primary stiffness property of interest for asphalt materials for all three levels of hierarchical inputs for the HMA mix characterization and the E* test as the primary HMA mix characterization mode. In the 2002 Design Guide, the stiffness of any HMA, at all analysis levels of
5 temperature and time rates of load, is determined from a master curve constructed at a reference temperature (usually 70ºF). The E* master curve for the Level 1 analysis is developed using numerical optimization to shift the laboratory mixture E* test data into a smooth master curve of a sigmoidal form. Before shifting the mixing data, the relationship between binder viscosity and temperature is established by the use of specific asphalt cement (AC) binder test data. The master curve for the Level 2 analysis is developed using the current version of the Witczak E* Predictive Equation from specific laboratory test data (6). The Level 3 analysis requires no laboratory test data for the asphalt binder but requires those mixture properties for the Witczak E* Predictive Equation.
1.4
Features of Dynamic Modulus (E*) For linear viscoelastic materials such as HMA mixes and asphalt binders, the
stress-to-strain relationship under a continuous sinusoidal loading in the frequency domain is defined by the complex modulus. This parameter is the ratio of the amplitude of the sinusoidal stress and the amplitude of the sinusoidal strain, at the same time and frequency, which results in a steady state material response. The complex modulus can be obtained from a standard laboratory testing. The laboratory testing can be done using either a normal or shear stress mode. When the applied stress is normal, the complex modulus is denoted by E*; whereas when a shear stress is applied, the complex modulus is denoted by G*. The sinusoidal stress (σ) can be represented as follows: σ = σ 0 eiω t = σ 0 cos (ω t ) + iσ 0 sin (ω t )
where,
(1.1)
6 σ = sinusoidal stress magnitude at time, t σ0 = maximum stress amplitude ω = angular velocity, radians/s t = loading time, seconds The angular velocity (ω) is related to the loading frequency (f) by: ω = 2π f
(1.2)
As a result of the sinusoidal stress (σ), the viscoelastic material experiences a sinusoidal strain (ε), which generally (but not always) lags the stress by a phase angle. In the case when the applied stress is normal, the phase angle is denoted by φ; whereas when a shear stress is applied, the phase angle is denoted by δ. For a normally applied sinusoidal stress, the lagging sinusoidal strain is expressed as: ε = ε 0 e i( ω t −φ ) = ε 0 cos (ω t − φ ) + iε 0 sin (ω t − φ )
(1.3)
in which ε 0 is the maximum strain amplitude. The ratio of the sinusoidal stress to the sinusoidal strain defines the complex modulus, denoted by either E* or G* depending on the stress type. For a normal stress application, E* can be mathematically expressed as follows: E* =
σ σ eiω t = 0i (ω t −φ ) ε ε 0e
(1.4)
The complex modulus has two distinct parts; a real part and an imaginary part. The real is called the storage (or elastic) modulus or elastic stiffness, and is denoted by E1 or E′ (G1 or G′ in case of shear stress). On the other hand, the imaginary part is called the loss (or viscous) modulus or viscous damping, and is denoted by E2 or E′′ (G2 or G′′ in
7 case of shear stress). For a normally applied stress, the storage and loss modulus, as shown in Figure 1.1, are mathematically related as follo ws:
Imaginary
E* = E1 + iE2
(1.5)
E* |E*| E2
φ E1
Real
FIGURE 1.1 Storage and Loss Modulus The ratio of loss to storage modulus is called the loss tangent (tan φ), which is the ratio of the energy lost to the energy stored in a cyclic deformation, and can be expressed as follows: tanφ = E1 / E2
(1.6)
Finally, the ratio of stress to strain amplitude defines the norms (vector lengths) of dynamic modulus (|E*|) and shear modulus (|G*|). Thus the dynamic modulus is: | E* |=
σ0 ε0
(1.7)
In case of a shear type of loading; the stress, stress amplitude, strain and strain amplitude are denoted by τ, τ0 , γ and γ0 , respectively. So, the shear modulus can be expressed similarly as follows: | G * |=
τ0 γ0
(1.8)
8 It is important to note that the dynamic modulus (|E*|) and the phase angle (φ) jointly describe the complex (dynamic) modulus. Similarly, the shear modulus (|G*|) and the phase angle (δ) jointly describe the complex (shear) modulus. In the remaining part of this dissertation, to be consistent with the conventional practice, the dynamic modulus is denoted as E* (not |E*|) and the shear modulus is denoted as G* (not |G*|) unless otherwise stated.
1.5
Role of a Predictive Model for Dynamic Modulus From the previous discussions, it is clear that the use of E* as both a principal
design tool for the new and rehabilitated paveme nts and a performance indicator for the mix design stage could unify and largely simplify the HMA mixture testing needed by agencies. One may intuitively understand that E* will play a very dominant role in the material characterization behavior of all HMA mixtures in any future technological methodologies. Because of this, a good predictive model (equation) will then further simplify the HMA mix design and performance prediction process in terms of resources such as money, time and labor. In fact during the last 50 years, a multitude of predictive equations and methods of modeling E* stiffness behavior have been developed and the E* test method itself has been enhanced with time. The historic development of the HMA mix stiffness predictive models and equations is discussed in the following chapter. Of all existing E* predictive models (equations), the one developed by Witczak and his colleagues at the University of Maryland and Arizona State University over the last 35 years is considered one of the most, if not the most, rational and comprehensive. It is commonly known as the Witczak E* Predictive Equation. The most recent version of
9 this equation is based upon 2750 test points and 205 different asphalt mixtures (34 of which are modified). Due to its mathematical structure (sigmoidal curve) and wide data basis, it can be used to estimate the stiffness of a HMA mix at a wide temperature and frequency range for load-associated distresses. In contrast to the 1986 and 1993 AASHTO Design Guides, the 2002 Design Guide utilizes dynamic modulus (E*) for all HMA layers of the pavement structure, in the hierarchical level scheme. As noted, the 2002 Design Guide methodology uses a particular form of model (a sigmoidal form) in all three levels of analysis and a particular predictive equation (Witczak E* Predictive Equation) in levels 2 and 3.
1.6
Expansion of E* Database The original Witczak E* Predictive Model was developed and calibrated by
Witczak and his colleagues at the University of Maryland (UMD) based on 1430 test data points from 149 un-aged laboratory blended HMA mixtures that contained only conventional non-modified binders (7). By 1999, Witczak further expanded the database. The revised (current) version of the model equation was based on an expanded database (known as the “UMD E* Database”) that contained 2750 test data points from 205 HMA mixtures (6). As before, all of the new 56 mixtures were un-aged laboratory blended mixes. It should be recognized that this expanded 1999 database consisted of certain types of aggregate and binder, narrow ranges of air void and binder content, few aggregate gradations and no mixture aging. As a result of several research projects and studies performed at Arizona State University (ASU) under the direction of Professor M. W. Witczak, the author (along with
10 his colleagues) completed E* testing on 176 additional new HMA mixtures that provided 5820 more E* test data points (8). The new database, known as “ASU E* Database”, is in a state of continual expansion. The size of this database is now more than double the UMD E* database. Compared with the previous database, the ASU E* database has a much wider range of aggregate and binder types, aggregate gradations, binder and air void contents, mixture aging, test temperature and loading time (or frequency). Table 1.1 shows a tabular comparison of the UMD (current) and ASU (new) E* data points.
TABLE 1.1 Summary of Data Points of the E* Database Database
UMD
ASU
Total
Mix
STOA1 Lab Typical Lab Plant Field Total Total Data
Type
Blend Mix Blend 2 Mix Mix Core Mix
Points
Conventional
0
171
0
0
171
1980
Modified Binder
0
34
0
0
34
770
Subtotal
0
205
0
0
205
2750
Conventional
66
0
76
25
167
5070
Modified Binder
2
0
0
2
4
120
AR3 -Dense Grade
0
0
2
0
2
60
AR-Gap Grade
0
0
4
0
4
120
AR-Open Grade
0
0
4
0
4
120
Lime Modified
11
0
0
0
11
330
Subtotal
79
0
86
27
192
5820
Total
79
205
86
27
397
8570
Note: 1. STOA = Short-Term Oven Aging for 4 hours at 275ºF 2. Typical lab blend mixes that are not short-term aged at lab 3. AR = Asphalt rubber mix For both the UMD and ASU database, Figures 1.2 through 1.11 show the frequency distribution (part “a” of the figure) and ranges (part “b” of the figure) of test
11 temperature (T), loading frequency (f), effective binder volume (%Vbeff), specimen air voids (%Va), voids in mineral aggregates (%VMA), voids filled with asphalt (%VFA), percent retained on ¾″ sieve (ρ34 ), percent retained on 3/8″ sieve (ρ38 ), percent retained on #4 sieve (ρ4 ) and percent passing #200 sieve (ρ200 ), respectively. It is evident that if the UMD and ASU E* database are combined, the expanded database provides a very large and diversified E* and related mixture data set, which is the most important prerequisite of developing an accurate and robust predictive model equation for dynamic modulus. When combined, the two databases create a single comprehensive “E* master database”.
1.7
E* Master Database The final version of the E* master database is presented in Append ix A (A.1 and
A.2 combined). As noted, this database was obtained from combining the UMD and ASU E* database. Based on the currently available E* master database, Figures 1.12 to 1.18 show the frequency distribution of test temperature (T), loading frequency (f), effective binder volume (%Vbeff), specimen air voids (%Va), voids in mineral aggregates (%VMA), voids filled with asphalt (%VFA), and aggregate sizes respectively. It is evident that the E* master database provides a very large and diversified E* predictor variable data.
12 2000
Numbers
UMD
ASU
1500 1000 500 0 0
15
25
35
45
55 65 75 85 o Temperature, F
95 105 115 125
a. Frequency Distribution 160
o
Temperature, F
130
130
120 71
70
80 40
14 0 0 Min
Max
Avg
Min
UMD
Max ASU
b. Range
FIGURE 1.2 Frequency Distribution and Range of Temperature
Avg
13 2000 UMD
ASU
Numbers
1500 1000 500 0 0
0.2 0.4
1
3
5 7 9 11 13 15 17 19 21 23 25 Loading Frequency (f), Hz a. Frequency Distribution
30 25.0
25.0
f, Hz
20
10
7.0 0.1
6.9 0.1
0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.3 Frequency Distribution and Range of Loading Frequency
14 2000 UMD
ASU
Numbers
1500 1000 500 0 5
8
11
14 17 20 23 Effective Binder Volume (Vbeff), %
26
29
a. Frequency Distribution 30 25.1 19.0
Vbeff, %
20
11.0 10
6.2
10.6 6.1
0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.4 Frequency Distribution and Range of Effective Binder Volume
15 1500
Numbers
UMD
ASU
1000
500
0 0
2
4
6
8 10 12 Air Voids (Va), %
14
16
18
a. Frequency Distribution
Specimen Air Voids (Va), %
20
18.1 15.9
15
10 7.1
6.3 5 0.7
0.1 0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.5 Frequency Distribution and Range of Specimen Air Voids
20
16 1500
Numbers
UMD
ASU
1000
500
0 0
4
8
12 16 20 24 28 32 Voids in Mineral Aggregates (VMA), %
36
40
a. Frequency Distribution 40
34.6
31.9 VMA, %
30 17.7
17.3
20
11.2
10.3 10 0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.6 Frequency Distribution and Range of Voids in Mineral Aggregates
17 2000 UMD
ASU
Numbers
1500 1000 500 0 30
35 40 45
50 55 60 65 70 75 80 85 Voids Filled with Asphalt (VFA), %
90
95 100
a. Frequency Distribution 100
99.4
95.1
80 VFA, %
64.2
60.1
60 40
36.2
32.8
20 0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.7 Frequency Distribution and Range of Voids Filled with Asphalt
18 4000 Numbers
UMD
ASU
3000 2000 1000 0 0
2.5
5
7.5
10 12.5 15 17.5 20 22.5 25 27.5 30 %Retained on 3/4" Sieve (ρ34 ), % a. Frequency Distribution
40 29.3
ρ34 , %
30
26.1
20
10
6.5 3.1 0.0
0.0
0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.8 Frequency Distribution and Range of Percent Retained on ¾″ Sieve (ρ34 )
19 1500
Numbers
UMD
ASU
1000
500
0 0
5
10
15 20 25 30 35 40 %Retained on 3/8" Sieve (ρ38), %
45
50
55
a. Frequency Distribution 80 56.0
ρ38 , %
60
43.0 40 28.0 19.8 20 0.0
0.0
0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.9 Frequency Distribution and Range of Percent Retained on 3/8″ Sieve (ρ38 )
20 1500
Numbers
UMD
ASU
1000
500
0 0
10
20
30 40 50 %Retained on #4 Sieve ( ρ4), %
60
70
a. Frequency Distribution 100
ρ4 , %
74.0
73.0
80 60
50.4
44.4 40
30.0
20 3.0 0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.10 Frequency Distribution and Range of Percent Retained on #4 Sieve (ρ4 )
21 5000 UMD
ASU
Numbers
4000 3000 2000 1000 0 0
2.5
5 7.5 10 %Passing #200 Sieve ( ρ200 ), %
12.5
a. Frequency Distribution 15 11.8 10.6 ρ200 , %
10
5.0
5
5.0 1.8
0.4 0 Min
Max
Avg
Min
UMD
Max
Avg
ASU b. Range
FIGURE 1.11 Frequency Distribution and Range of Percent Passing #200 Sieve (ρ200 )
22
Numbers
2000 1500 1000 500 0 0
20
40
60 80 Temperature, o F
100
120
140
FIGURE 1.12 Frequency Distribution Test Temperatures for E* Master Database
Numbers
2000 1500 1000 500 0 0
0.1
0.2
0.3
0.4 0.5 1 5 10 Loading Frequency (f), Hz
15
20
25
FIGURE 1.13 Frequency Distribution of Loading Frequency for E* Master Database
23 3000
Numbers
2500 2000 1500 1000 500 0 6
8
10
12 14 16 18 20 22 24 Effective Binder Volume (Vbeff), %
26
28
FIGURE 1.14 Frequency Distribution Effective Binder Volume for E* Master Database
2000
Numbers
1500 1000 500 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Air Voids (Va), % FIGURE 1.15 Frequency Distribution of Specimen Air Voids for E* Master Database
24
Numbers
3000
2000
1000
0 10
14
18 22 26 30 34 Voids in Mineral Aggregates (VMA), %
38
FIGURE 1.16 Frequency Distribution of VMA for E* Master Database
2000
Numbers
1500 1000 500 0 30
40
50 60 70 80 Voids Filled with Asphalt (VFA), %
90
100
FIGURE 1.17 Frequency Distribution of VFA for E* Master Database
25
Numbers
4000 3000 2000 1000 0 0
5
10 15 20 25 Retained on 3/4" Sieve (ρ34), %
30
a.
Numbers
1500 1000 500 0 0
5
10
15
20 25 30 35 40 45 50 Retained on 3/8" Sieve (ρ38), %
55
60
65
b.
Numbers
2000 1500 1000 500 0 0
10
20
30 40 50 60 Retained on #4 Sieve (ρ4), %
70
80
c.
Numbers
2400 1800 1200 600 0 0
1
2
3
4 5 6 7 8 9 10 Passing #200 Sieve (ρ200), %
11
12
13
d.
FIGURE 1.18 Frequency Distribution of Aggregate Sizes for E* Master Database
26 1.8
Binder Database The viscosity of the asphalt binder at the temperature of interest may be
determined from the ASTM viscosity temperature relationship, which is commonly known as the ASTM Ai-VTSi relationship (9, 10). Similar to the mix E* database, a large database exists for the Ai-VTS i data as obtained from the Penetration, Ring and Ball, Absolute Viscosity, Kinematic Viscosity, Brookfield Viscosity tests and the binder complex shear modulus (Gb*) data as obtained from the Dynamic Shear Rheometer (DSR) test. Previously, Bonaquist et al. (11) summarized such data for 33 different conventional and modified asphalt binders with a partial factorial of the following three aging conditions: (a) Original or Tank condition; (b) Construction phase aging of asphalt binder using the Rolling Thin Film Oven (RTFO); and (c) Accelerated in-service aging of asphalt binder using the Pressure Aging Vessel (PAV) at 100°C. The data were obtained from research studies conducted at the University of Maryland under the NCHRP 9-19 project and hence the database has been identified in this dissertation as the “UMD Binder Database”. This database contains 5,640 Gb* data points and associated Ai-VTSi data. Later, under the direction of Dr. M. W. Witczak, the author conducted binder testing on eight more binders at ASU, which provided 3,300 new Gb* data points and associated Ai-VTSi data (12-14). The database has been identified in this dissertation as the “ASU Binder Database”. Thus the binder database obtained from combining the UMD and ASU binder databases contains a total of 8,940
27 Gb* test data points and associated Ai-VTS i data. Thus when combined, the two databases created a single comprehensive “G b*-viscosity (η) master database”, presented in Appendix B.
1.9
Problems with the Current E* Model A detailed discussion of the historic development of E* predictive models and
equations are presented in the following chapter. Historically, the first generation E* predictive models were obtained mostly from linear multivariate regression analysis. While important information was obtained on the general importance of mixture variables as well their interactive effects, theoretical considerations on the variation of dynamic modulus with the most significant variables eventually led the researchers to pursue a non- linear regression analysis. Witczak and Fonseca developed an E* predictive model in a simple mathematical sigmoidal form based on non- linear regression analysis (7). The enhanced version of this model, which is the current “Witczak E* Predictive Equation”, is the equation that has been used to obtain mix stiffness in the level 2 and 3 analyses of the 2002 Design Guide (6). While this model is considered very reliable, it has several limitations that must be understood by the user. The most critical limitations are discussed in the following paragraphs. The model relies on other models or techniques to determine viscosity of the asphalt binder at the temperature of interest. The ASTM Ai-VTS i relationship for the asphalt binders was used in the development of the model. The inherent problem with the ASTM Ai-VTS i relationship is that it does not consider the effect of loading frequency (or time) on the stiffness of the binder itself used in the mix. In fact, the complex shear
28 modulus (Gb*) of the binder is a much more representative stiffness of the binder than the stiffness obtained from a typical Ai-VTSi relationship. Furthermo re, with the adoption of the Performance Grading (PG) System and the associated binder testing, the Gb* data will be more available than the typical Ai-VTS i data. So it is logical to develop a future E* predictive model equation that would directly link the dynamic modulus of HMA mix with the complex shear modulus of the binder used in the mix. As with any predictive equation, the classical statistical principles regarding model extrapolation outside the range of variables used to develop the model apply. The recent version of the Witczak E* Predictive Model was initially based upon a sigmoidal form (Witczak-Fonseca model) and calibrated based on 1430 test data points from 149 un-aged laboratory blended HMA mixtures made with only conventional un-modified binders. The current (1999) version of the Witczak E* predictive model also utilized a sigmoidal form. This version was based on 2750 test data points from 205 HMA mixtures, which included 34 mixtures with modified binders. Unfortunately, the new mixes were, again, only un-aged laboratory blended mixes. As a result, the accuracy of the E* stiffness prediction for plant mix, field core and short-term aged laboratory blend mixes is based upon the initial assumption that the representative viscosity of the mix (at the time of testing), whether it is from plant mix, field aged cores, laboratory short/long term aging, will lead to accurate predictions of E* as long as the proper viscosity is input into the equation. One other frequently used E* (stiffness) predictive methodology is the Shell Oil model (15) for predicting the stiffness of HMA and asphalt binder. In 2001, Pellinen conducted a comparison between the Witczak E* model and Shell Oil model (16). The
29 data used actual measured E* stiffness of MnRoad, ALF and WesTrack sites for a temperature range of 40 to 130°F. The findings showed that the Witczak model had good correlation to the measured values over the entire data range. The Shell Oil model, on the other hand, under predicted mixture stiffness, particularly at high temperatures. A very significant bias in the measured and Shell Oil predicted E* values were found. It was found that at very high temperatures and low loading rates, the Shell Oil model leads to E* values that are off by a factor of 6 to 10 from lab measured E* values. Despite the large bias found in the Shell Oil model for the full temperature range, the precision of that model was better than the Witczak model while predicting the relative E* stiffness trends (caused by mixture volumetric properties). It was also concluded that the magnitude of the prediction was much closer to the measured values using the Witczak model. In other words, while the Witczak model is acceptable in terms of accuracy and bias, it may need further refinement to improve its precision, especially by improving its sensitivity to volumetric properties of the HMA mixtures. In 2005, Dongré and his co-researchers investigated the E* prediction capability of the Witczak model (17). Their research was based on very limited actual E* test data obtained from only 5 HMA mixtures. Nonetheless, in the course of comparison; the research team showed that the Witczak Model was found to converge beyond an E* value of approximately 125,000 psi. The model seemed to overestimate E* stiffness below that point. The accuracy of the Witczak model also reduced as the volumetric properties and binder content deviated from the mix design levels, which is common in HMA production. Dongré et al recommended that the model constants of the Witczak model may require future refinement to accommodate the new Superpave shear modulus (|Gb*|)
30 and phase angle (δ b) of asphalt binder data directly. They further recommended that the model should be improved to account for changes in volumetric properties, asphalt content and air voids.
1.10
Assessing the Reliability of the Current Model Table 1.2 shows the comparative statistics of the Witczak E* Model for the UMD,
ASU and UMD+ASU (expanded) database. Applied to the UMD E* database, the current version of the Witczak E* predictive model had an excellent goodness of fit statistics in logarithmic scale: R2 = 0.94 and Se/Sy = 0.25. In the same scale, the model exhibited good but slightly lower statistics for the ASU E* database: R2 = 0.85 and Se/Sy = 0.39. When the model was applied to the combined UMD+ASU E* database, the Witczak E* predictive model had R2 of 0.88 and Se/Sy of 0.35 in log scale.
TABLE 1.2 Statistics of Witczak E* Model for the UMD and ASU Database Parameters Total Mix Modified Mix Data Points Se/Sy R2 Se/Sy R2
E* Database UMD ASU UMD+ASU 191 155 346 0 17 17 2750 4650 7400 Logarithmic Scale 0.25 0.39 0.35 0.94 0.85 0.88 Arithmetic Scale 0.34 0.65 0.60 0.89 0.57 0.65
31 The plots of the predicted versus measured dynamic modulus for the UMD database and ASU database using the current Witczak E* model are presented in Figure 1.19. It is obvious that a poorer agreement for the ASU (new) database is present while compared with the UMD database, upon which the current Witczak E* model is based. It is postulated that the major reasons for this discrepancy are associated with: •
a much broader range of initial input variables with the ASU generated database (i.e. problems associated with model input extrapolation); and
•
a wider range of binder stiffness (viscosity) with the ASU generated database due to the testing on lab-aged specimens, plant mix and field aged cores.
1.11
Objectives With this background in mind, the main goal of the research study presented in
this Ph.D. dissertation was to develop an enhanced version of the Witczak dynamic modulus predictive model for HMA mixtures capable of estimating changes in modulus as a function of changes in mixture volumetrics, material properties, temperature and loading frequency (or time) for the combined database of E* lab results. Of equal importance was the goal of incorporating the Gb* parameter as the primary source of quantifying the asphalt binder stiffness, in lieu of using asphalt binder viscosity. In order to achieve these goals, the research was aimed at fulfilling the following three specific objectives:
32 1000
5
Predicted E*, 10 psi
100
UMD Database: Ndata = 2750 Nmix = 191
10 1 0.1 Arithmetic scale: R2 = 0.89, Se/Sy = 0.34 Log scale : R2 = 0.94, Se/Sy = 0.25
0.01 0.001 0.001
0.01
0.1
1
10
100
1000
5
Obdesrved E*, 10 psi a. Predicted versus Observed E* Plot Based on UMD Database 1000
5
Predicted E*, 10 psi
100
ASU Database: Ndata = 4650 Nmix = 155
10 1 0.1 Arithmetic scale: R2 = 0.57, Se/Sy = 0.65 Log scale : R2 = 0.85, Se/Sy = 0.39
0.01 0.001 0.001
0.01
0.1
1
10
100
1000
5
Obdesrved E*, 10 psi b. Predicted versus Observed E* Plot Based on ASU Database
FIGURE 1.19 Predicted versus Measured E* for Current Witczak E* Model
33 The first objective of this research study was to create a master database of mix dynamic modulus (E*) test data, binder complex (shear) modulus (Gb*) test data, binder viscosity and Ai-VTSi data, and all relevant material and mixture data. This master database would be obtained by combining the UMD and ASU E*, Gb* and Ai-VTS i databases available to the author. The next objective was to develop a predictive model for complex shear modulus (Gb*) capable of accurately predicting both the dynamic shear modulus |G b*| and phase angle (δ b) of an asphalt binder from given “A” and “VTS” values as obtained from the conventional ASTM Ai-VTSi relationship. This would be done with the Gb* and binder viscosity (η) database available in the study. In the course of the Gb* model development, the conventional ASTM Ai-VTS i relationship to use in the predictive models for |Gb*| and δ b would also be revised so that the modified relationship would be capable of incorporating the effect of loading frequency on the asphalt binder viscosity. The final objective of the study was to develop an enhanced Witczak predictive model for the dynamic modulus (E*) of HMA mixtures. The revised η-|Gb*|-δ b model would be incorporated in the new E* predictive model to overcome the current problem with using proper binder viscosity. The final E* model to be pursued should be relatively simple, statistically accurate, and implementable within the general framework already established in the 2002 Design Guide. Hence, it was considered that the sigmoidal model form would be a mandatory consideration of the new model.
34 1.12
Scope of Research The scope of this research included the following work tasks: Conduct Literature Research - The goal of the literature review was to document
previous and existing studies on the HMA dynamic modulus and binder complex shear modulus needed to accomplish the objectives of this study. The completion of the literature review was done to ensure that all the essential information needed to accomplish the objectives of this study was obtained. Completion of Binder Database - The UMD and ASU binder testing databases were completed in terms of the complex shear modulus (Gb*) data as obtained from the DSR test and the Ai-VTSi viscosity (η) data as obtained from the Penetration, Ring and Ball, Absolute Viscosity, Kinematic Viscosity and Brookfield Viscosity tests. Both databases were combined into a single comprehensive “G b*-η master database”. Revise ASTM Ai-VTSi Relationship - The Gb*-η master database was analyzed in order to develop a revised ASTM Ai-VTSi relationship capable of taking care of the effect of loading frequency on the asphalt binder viscosity. Develop Predictive Model for |G b*| - The Gb*-η master database and the revised ASTM Ai-VTS i relationship were analyzed to develop a new predictive model for the binder shear modulus (Gb*). Develop Predictive Model for δ b - In course of developing the |Gb*| model, a new predictive model for the phase angle (δ b) associated with the binder complex (shear) modulus was also developed.
35 Completion of E* Database - The UMD and ASU E*-φ database were consolidated in terms of E* testing data (temperature, loading frequency, dynamic modulus and corresponding phase angle), mixture data (specific gravities, air voids, binder content, effective binder volume, VMA and VFA), material data (binder grade, aggregate gradation, specific gravities, modifier, source etc.), specimen data (geometry, compaction, replication and aging), and binder testing data (temperature, loading frequency, ASTM A and VTS values, dynamic shear modulus and corresponding phase angle). The new comprehensive η-|G b*|-δ b predictive model was used to calculate the |Gb*| and δ b values from the corresponding A and VTS values of the binders used in the HMA mixes. Finally, both the UMD and ASU databases were combined into a single comprehensive “E* Master Database”. New E* Model Development – Finally, the E* master database was analyzed to develop the latest enhanced version of the Witczak et al predictive model for the dynamic modulus of asphalt mixtures and hence, fulfill the associated objectives of this proposed research.
1.13
Dissertation Organization This Ph. D. dissertation is organized as follows. First, the purpose, objective and
scope of this research are discussed. After the introductory discussio n within the first chapter, Chapter 2 presents a comprehensive literature review related to the research. Chapter 3 describes the details of the asphalt binder stiffness database. The development of a new comprehensive model for binder stiffness, based upon Gb* is described in detail in Chapter 4. Chapter 5 describes the asphalt mixture E* stiffness database. The variables
36 used in the new enhanced E* model are described and analyzed in Chapter 6. Then the E* model development process is discussed in Chapter 7, while the final E* model evaluation process is described in Chapter 8. Finally, Chapter 9 is dedicated to summarize the conclusions and recommendations obtained from this research. Appendices A and B contain the master E* database and η-Gb* database, respectively. Other appendices are referred at appropriate sections of the dissertation.
2
LITERATURE REVIEW
2.1
Asphalt Binder Characterization
2.1.1
Behavior of Asphalt Binder In modern days, the main source of asphalt binder is the distillation of crude
petroleum using different refining processes. At ambient temperatures, asphalt binder usually is a black, semisolid and highly viscous material. It is considered a valuable construction and maintenance material because of its strong, adhesive, waterproofing and durability characteristics. It provides flexibility to mixtures of mineral aggregates with which they are usually combined. It is also highly resistant to reaction with most acids, alkalis and salts (1). Although it generally exists in a solid or semi-solid state at ordinary atmospheric temperatures, it may be liquefied by applying heat, dissolving in petroleum solvents or emulsifying in water. Asphalt binder is a viscoelastic material. Its behavior depends on both temperature and rate of loading. In hot conditions such as in the desert climate in Arizona, or under sustained loads (slow moving or parked vehicles), the asphalt binder behaves more like a viscous liquid. As a Newtonian liquid, hot asphalt has a linear relationship between the resisting force and relative velocity often at temperatures greater than 140°F (60°C). Viscous liquids like hot asphalt are sometimes called plastic because once they start flowing, they do not return to their original position. In hot climates, less stable asphalt pavements flow under repeated wheel loads and form ruts. On the other hand, in cold climate or under rapidly applied loads asphalt binders behave more like elastic solids. Here, the asphalt cement may become brittle and crack when excessively loaded.
38 Most environmental conditions associated with pavement construction and performance lie between the extreme hot and cold temperatures previously discussed. In these climates, asphalt binders may exhibit the characteristics of both viscous liquid and elastic solid. Because of this range of behavior, asphalt is an excellent adhesive material to use in paving, but also becomes an extremely complicated material to characterize.
2.1.2
Properties of Asphalt Binders Due to the complex chemical nature and variability involved in available asphalt
binders, their properties are usually evaluated in two broad categories: physical and rheological properties. Historically, some characteristics have been placed in both of these categories. The physical properties of asphalt binders are usually determined by the following tests: •
Consistency tests: Ductility, Penetration, Softening Point, Absolute Viscosity, Kinematic Viscosity, and BrookfieldT M Viscosity test.
•
Stiffness test: Dynamic Shear Rheometer (DSR) and Bending Beam Rheometer (BBR).
•
Durability tests: Rolling Thin Film Oven (RTFO) and Pressure Aging Vessel (PAV) test.
•
Purity test: Solubility test.
•
Safety test: Flash Point test.
•
Other tests: Specific Gravity and Spot test.
39 The rheological properties of asphalt binders are very important to characterize them. These properties include: •
Ductility,
•
Penetration,
•
Viscosity,
•
Stiffness,
•
Age hardening,
•
Temperature susceptibility, and
•
Shear susceptibility.
Rheological properties of the asphalt binder highly affect pavement performance. These properties change during hot mix asphalt (HMA) production and continue to change subsequently in service. These properties are briefly discussed in the following paragraphs. Many binder grading, HMA mix design and quality assurance/control protocols have been deve loped based on some combinations of the physical and rheological properties of asphalt binders. Important binder properties are briefly discussed in the following paragraphs.
2.1.2.1 Ductility The ductility of a paving binder is measured by the distance it will elongate before breaking when two ends of a briquette specimen are pulled apart at a specified speed and temperature. It is a very empirical measure of the fracture characteristics of a binder. The
40 significance of the ductility test as a means of asphalt binder quality control has been debated because of its empirical nature and poor reproducibility (1).
2.1.2.2 Penetration Penetration is an empirical measure of asphalt binder consistency, which is measured by a standard penetration test. It is one of the original ways by which consistency of asphalt binder was measured. Penetration at 77°F has widely been used in asphalt cement specifications because no simple method of viscosity determination at 77°F or lower temperature is currently available. Penetration at 77°F generally gives the consistency of asphalt binder near the average yearly service temperature. Thus, it has some influence on the overall performance of HMA pavements and produces a standardized way of ranking bitumen hardness (stiffness).
2.1.2.3 Softening Point Softening point is measured by the ring and ball test. It is defined as the temperature at which as asphalt binder cannot support the weight of a steel ball and starts flowing. Its purpose is to determine the temperature at which the binder undergoes a phase change. The softening point is indicative of the tendency of the binder to flow at elevated temperatures encountered in service. The softening point is used in the classification of asphalt binders and as one of the elements in establishing the uniformity of shipments or sources of supply. For most unmodified asphalt binders, the ring and ball softening point corresponds to a viscosity of 13,000 poise (18). For this reason, this
41 temperature (TRB) is often referred as an equiviscous point; i.e. a point where all binders have the same viscosity.
2.1.2.4 Viscosity The viscosity (η) of a liquid is a measure of that fluid’s resistance to flow when acted upon by an external force. At any given temperature and shear rate, viscosity is the ratio of shear stress to shear strain rate. Some liquids, such as jelly, asphalt, and syrup are very viscous. Other liquids, such as water, lighter hydrocarbon, and gas are not as viscous. Most viscous liquids will flow more easily when their temperatures are raised. Ductility, penetration and softening point are common empirical tests for measuring the consistency of asphalt binders. Unlike these empirical tests, viscosity is a fundamental consistency measurement of asphalt binders that is generally not affected by changes in test configurations or geometry of the sample (19). However, it is affected by temperature and rate of loading. At high temperature such as 275°F, asphalt binders tend to behave as a Newtonian liquid; that is, the ratio of shear stress to shear strain rate is constant. At low temperatures, the ratio is usually no longer a constant, and the binder behaves like a nonNewtonian liquid. Asphalt binder viscosity at 140°F has some influence on the performance of the HMA pavements during hot summer days when the pavement surface temperatures are near 140°F. A low viscosity at 140°F can induce flushing and/or rutting if other factors are identical. It has also been observed that aging of the HMA pavement results in progressively higher viscosity with time.
42 2.1.2.5 Stiffness Stiffness modulus or simply stiffness (S) of the asphalt binder is the relationship between stress and strain as a function of time of loading and temperature. This definition was originally introduced by Huekelom of the Royal Dutch Shell Oil lab. This relationship is also referred to as the rheological behavior of the binder. In many applications of HMA, its stiffness characteristics must be known not only to assess the behavior of the mix itself, but also to evaluate the performance of the pavement. Ideally, for a highway pavement surface course, increased binder stiffness is desirable at high service temperatures (near 140°F) to avoid rutting, and decreased binder stiffness is desirable at low service temperature to avoid low temperature shrinkage cracking (1). According to Finn, asphalt stiffness is practically time independent at very short loading time (i.e. very high loading frequency) (20). In this case, as shown in Figure 2.1, the stiffness approaches the elastic modulus, E. For an intermediate range on the time scale, the stiffness decreases with an increase in the time of loading. At very long loading times (i.e. very small loading frequencies), the stiffness may still decrease, but at a uniform rate, and the behavior of the asphalt binder may be cons idered as purely viscous.
2.1.2.6 Age Hardening The first significant hardening of asphalt binder takes place in the pug mill or drum mixer where the heated aggregate is mixed with hot asphalt cement. During the short mixing period, the very thin films of the asphalt binder are usually exposed to air at temperatures that range from 275°F to 325°F (135°C to 163°C) or more. Substantial
43 rheological changes then occur. Some of these changes are a decrease in penetration or an increase in viscosity of the binder. The se changes take place mostly from air oxidation and loss of more volatile components. This age hardening continues, although at a much slower rate, while the HMA is processed through surge or storage silo, transported to the paving site, laid and compacted. This part of the aging is frequently called “short term aging”.
Stiffness, S, psi (in Log Scale) or
Elastic Behavior
Delayed or Retarded Elastic Behavior
(1) At Short Loading Times, S = E (2) At Intermediate Loading times, S = E(t) (3) At Long Loading Times, S = λ/t where, λ = Viscous Traction = 3η ( a Measure of Viscous Deformation)
Viscous Behavior
Time, T (in Log Scale) FIGURE 2.1 Time of Loading Dependence of Binder Stiffness for Axial Tension
44 When the pavement is opened to traffic, the age hardening process continues for its service life, though at much slower rates. This is generally called “long term aging”. The following factors have been reported to contribute to the age hardening of asphalt binders: oxidation, volatilization, polymerization, thixo tropy, syneresis, and separation (20, 21). Because asphalt cements are composed of organic molecules, they react with oxygen from the environment. As a result, the structure and composition of the asphalt molecules are changed. This causes oxidative or age hardening resulting in a more brittle asphalt cement. Oxidative hardening occurs at a slow rate, but is accelerated in warmer climates. Improperly compacted asphalt pavements that usually have higher levels of air voids, will allow more oxidative hardening. In practice, a considerable amount of oxidative hardening occurs before the asphalt is placed, especially in a hot mix facility. Volatilization is the evaporation of the lighter constituents from asphalt cement and is primarily a function of temperature. However, this is usually not a significant factor contributing to the long-term in-service aging. Beside polymerization, thixotropy, syneresis and separation; Traxler suggested some additional factors such as effect of light and water, chemical reaction with aggregates, microbiological deterioration and adsorption of heavy asphalt components on the surface of the aggregates (22). The well-known Global Aging System, developed by Mirza and Witczak, presents a systematic understanding and analysis procedure of short and long-term aging of asphalt binders (18). The system describes the change in asphalt binder properties with depth and time at a specific pavement temperature as shown in Figure 2.2.
45
FIGURE 2.2 Viscosity versus Depth as a Function of Time at a Given Temperature
The three lines define the three major conditions of asphalt binder that occurs during the life of the pavement system. The binder properties at these stages can be categorized as follows: •
Original (tank) properties (shown by Line #1 in Figure 2.2),
•
Mix/laydown (short term) properties at time of construction, where time t = 0 (shown by Line #2), and
•
Field aged (long term) properties, where time t > 0 (shown by Line #3).
For the original (tank) and mix/laydown conditions, the viscosity is practically constant with pavement depth. The shift from original to mix/laydown viscosity is the result of hardening that occurs during the mixing and laydown operation. This increase in viscosity represents the short-term aging phenomena. Line #3 represents the field aging
46 properties of asphalt binder at any time after the mix/laydown process. This change in viscosity is not uniform with depth below the pavement surface because the long-term aging of the binder is mainly due to the oxidation process. The higher oxidation near the surface, as shown in Figure 2.2, is due to the binder in direct contact with the circulating air and higher surface temperature from direct solar radiation. The effect decreases sharply with depth and become almost negligible at a few inches below the pavement surface.
2.1.2.7 Temperature Susceptibility As a thermoplastic material, the consistency of asphalt binder changes with temperature. The rate of this change is known as temperature susceptibility. Asphalt cement with high temperature susceptibility is not usually desirable because at the compaction temperature of HMA, the binder viscosity may be very low causing tender mix and compaction problems. On the other hand, the viscosity at the lowest service temperature may be too high that it results in low temperature cracking. Penetration Index (PI), Pen-Vis Number (PVN) and Viscosity -Temperature Susceptibility (VTS) are three common measurements of this behavior of asphalt binders.
2.1.2.8 Shear Susceptibility At low temperatures, most asphalt binders exhibit non-Newtonian or viscoelastic flow. As such their viscosity is dependent on the shear rate; the viscosity increases as the rate of shear increases and vice versa. The rate of change of viscosity with the shear rate is known as shear susceptibility, which is considered to be an intrinsic property of the
47 asphalt cement. The rate of gain in shear susceptibility relative to increases in viscosity at 77°F seems to be one of the major factors affecting HMA pavement performance. Relatively lower gain in shear susceptibility with the corresponding increase in viscosity has been reported to be associated with better pavement performance (23).
2.1.3
Pavement Distresses Related to Rheological Properties of Binder There are some specific types of HMA pavement distresses affecting the
pavement performance that are related to the rheological properties of the binder. Among them, rutting and cracking (both load-associated and non-load associated) are most common.
2.1.4
Asphalt Binder Characterization Tests HMA pavement distresses related to the rheological properties of the binder affect
the pavement performance in many ways and degrees. As a result, it is extremely important to properly characterize asphalt binders to ensure proper pavement performances. Many tests are available to characterize the binders. Some tests are commonly used by the highway agencies, while others are used for research. Since the properties of the asphalt are highly sensitive to temperature and time of loading, all asphalt binder tests must be conducted at specified temperatures and/or time of loading within very tight tolerances. Common asphalt binder characterization tests are as follows: •
Consistency tests: Penetration, Softening Point, Absolute Viscosity, Kinematic Viscosity, and BrookfieldT M Viscosity test.
48 •
Stiffness test: Dynamic Shear Rheometer (DSR) and Bending Beam Rheometer (BBR).
•
Durability tests: Rolling Thin Film Oven (RTFO) and Pressure Aging Vessel (PAV) test.
•
Purity test: Solubility test.
•
Safety test: Flash Point test.
•
Physical property test: Specific Gravity test.
2.2
Stiffness of Asphalt Binders
2.2.1
Common Stiffness Parameters Over last few decades, researchers have developed a handful of empirical and
mechanistic stiffness parameters. They have also developed empirical equations to convert common consistency parameters such as Penetration and Softening Point to viscosity. Models are also available to convert viscosity data to modulus data. The most important and widely used binder modulus is the complex (shear) modulus (Gb*). The theory behind Gb* has already been discussed in Chapter 1. Note that this modulus is usually expressed with the absolute value (|G b*|) and the associated phase angle (δ b). The Dynamic Shear Rheometer (DSR) test is widely used for determining |Gb*| and δ b at a wide range of temperature and loading frequency. The AASHTO T315-02 protocol is followed for this purpose.
49 2.2.2
Determination of |Gb*| and δ b by DSR Test The Dynamic Shear Rheometer (DSR) test covers the determination of the
dynamic shear modulus (Gb*) and the phase angle (δ b) of asphalt binder when tested in dynamic (oscillatory) shear using parallel plate test geometry. The AASHTO T315-02 protocol is followed in this test. The test is applicable to asphalt binders having dynamic shear modulus values in the range from 100 Pa to 10 MPa. This range in modulus is typically obtained between 5°C and 85°C for most conventional binders. As discussed in Chapter 1, Complex Shear Modulus (G b*) is the ratio of absolute value of the peak-to-peak shear stress (τ) to the absolute value of the peak-to-peak shear strain (γ). The Phase Angle (δ b) is the angle in degrees between a sinusoidally applied strain and the resultant sinusoidal stress in a controlled-strain testing mode or between the applied stress and the resultant strain in a controlled-stress testing mode. There is two types of dynamic shear rheometers: controlled stress and controlled strain. Superpave binder tests are conducted in the controlled stress mode. In the DSR operation, as shown in Figure 2.3, the asphalt binder is “sandwiched” between two parallel plates: one that is fixed and one that oscillates. As the plate oscillates, the centerline of the plate at point A (indicated by the dark vertical line) moves to point B. From point B, the plate centerline moves back and passes point A to point C. From point C the plate centerline moves back to point A. This oscillation is one cycle and is continuously repeated during the DSR operation. The speed of oscillation is frequency. All Superpave DSR binder tests are performed at an angular frequency of 10 radians per second, which is equal to approximately 1.59 Hz (cycle per second).
50
FIGURE 2.3 Dynamic Shear Rheometer Operation The DSR test is used to characterize both viscous and elastic behavior by measuring the shear modulus (|G b*|) and phase angle (δ b) of asphalt binders. Gb* is a measure of the total resistance of a material to deformation when exposed to repeated pulses of shear stress. It consists of two components: elastic (recoverable) and viscous (non-recoverable). The phase angle (δ b) is an indicator of the relative amounts of recoverable and non-recoverable deformation. The value of |Gb*| and δ b for asphalt binders are highly dependent on the temperature and frequency of loading. At high temperatures, asphalt binders behave like a viscous fluid with no capacity for recovering or rebounding. At very low temperatures, asphalt binders behave like elastic solids that rebound from deformation completely. Under normal pavement temperature and traffic loading, asphalt binders act with the characteristics of both viscous liquid and elastic solid. Thus, by measuring |Gb*| and δ b, the DSR provides a more complete picture of the behavior of asphalt binders at pavement service temperatures.
51 The Dynamic Shear Rheometer (DSR) System consists of test plates, environmental chamber, loading device, control and data acquisition system, specimen mold (optional), specimen trimmer and a calibrated temperature detector. Test specimens 1 mm thick by 25 mm in diameter or 2 mm thick by 8 mm in diameter are formed between parallel metal plates. During testing, one of the parallel plates is oscillated with respect to the other at pre-selected frequencies and rotational deformation amplitudes (or torque amplitudes). The required amplitude depends upon the value of the complex shear modulus of the asphalt binder being tested. These amplitudes are selected to ensure that measurements are within the region of linear behavior. The test specimen is maintained at the test temperature to within + 0.1°C by positive heating and cooling of the upper and lower plates. Oscillatory (angular) loading frequencies using this standard can range from 1 to 100 rad/s using a sinusoidal wave form. Specification testing is performed at a test frequency of 10 radians per second, which is equal to approximately 1.59 Hz (cycles per second). The complex shear modulus and the phase angle are calculated automatically as part of the rheometer using proprietary computer software supplied by the equipment manufacturer. Original (i.e. tank) binders and RTFO aged binders are tested at strain values of about 10% to 12%. PAV-aged binders are tested at strain values of about 1%. In all cases, strain values must be small enough that the response of the binders (i.e. |Gb*| value) remains in the linear viscoelastic range. In this range, |Gb*| is virtually unaffected by changes in strain level. The DSR measures the rheological properties (complex shear modulus and phase angle) at intermediate to high temperatures experienced by the pavement in the geographical area for which the asphalt binder is intended. The DSR test provides
52 stiffness behavior of asphalt binders over a wide range of temperatures. Two forms of Gb* and δ b are used in the binder specification. Permanent deformation is governed by limiting the (Gb*)/sinδ b at the test temperatures to values greater than 1.00 kPa for original binder and 2.20 kPa after RTFO aging. Fatigue cracking is governed by limiting (Gb*)sinδ b of PAV aged material to values less than 5000 kPa at the test temperature. However, there is not a full universal agreement for the Fatigue DSR specifications.
2.2.3
Predictive Models for Stiffness of Asphalt Binders Very often, there are situations when it becomes impractical to conduct a full
range DSR test to obtain the stiffness properties of an asphalt binder. Therefore, over time, researchers have come up with different predictive models for the indirect measurement of stiffness of asphalt binders. Available prediction models are chronologically briefly described in the next few paragraphs.
2.2.3.1 Van der Poel (Shell Oil’s Early Version) Model One of the earliest, but most well-known asphalt binder viscosity (stiffness) predictive models, was developed by Van der Poel of the Shell Oil Company based upon over 20 years of laboratory work at “Koninklijke/Shell- Laboratorium, Amsterdam (KSLA)” (2). This predictive method is a part of the well-known “KSLA Method”. It uses a nomographic solution to obtain the asphalt binder stiffness (Sb), which is assumed to be a function of the temperature, time of loading and the characteristics of the bitumen in a mix expressed in terms of the penetration index (PI). The Shell Oil nomograph is shown in Figure 2.4.
53
FIGURE 2.4 Nomograph for Stiffness of Asphalt Binders (2) The binder properties needed in the nomograph are the viscosity at T800pen and PI. T800pen is the temperature at which the penetration would be 800. It was found that all bitumens have an equiviscous magnitude or penetration of 800 at their Ring and Ball Softening Point (TR&B). Thus, T800pen can be set equal to TR&B. Thus Tdif = TRB − T
(2.1)
defines a normalized temperature under which differences in viscous behavior types disappear. For the intermediate (viscoelastic) behavior, use of the penetration index (PI) is made as follows: 20 − PI ∂ log Pen = 50 10 + PI ∂T
(2.2)
54 In other words: 20 − PI log 800 − log Pen = 50 10 + PI TRB − T
(2.3)
The binder PI is calculated as follows: 20 − 500 A 1 + 50 A
(2.4)
log( Penetration at T ) − log 800 T − TR & B
(2.5)
PI =
A=
where, Pen
= penetration at the temperature of interest as obtained from the standard Penetration Test, 0.1 mm
A
= temperature susceptibility (see equation 2.5)
T
= temperature of interest, ºC
In lab, the binder stiffness can be determined by either a creep test with a loading time t or dynamic shear test under an angular loading frequency of ω (or a sinusoidal loading frequency of fs). Van der Poel defined t as follows: t=
1 1 = ω 2πf s
(2.6)
2.2.3.2 ASTM Ai-VTS i Model For any aging condition of an asphalt binder, the viscosity of the asphalt binder at the temperature of interest may be determined from the ASTM viscosity temperature relationship defined in equation 2.7. This approach is commonly known as the ASTM AiVTSi relationship (9, 10).
55 log log η = A + VTS log TR
(2.7)
where, η = viscosity, cP T R= temperature, degree Rankine (°R = °F + 459.7) A = regression intercept VTS = regression slope (viscosity temperature susceptibility parameter) This relationship is applicable not only to virgin asphalt cement binders but also to a wide variety of modified binders, provided that the modification percentages are not excessively high (less than 2%-3%). Although the ASTM Ai-VTSi relationship is usually used with data from viscosity measurements at 140o F and 275o F to develop mixing and compaction temperatures, it can be extended to lower temperatures using ring and ball softening point and penetration data. Research by Shell Oil, which was later confirmed by Mirza and Witczak, indicates that for most unmodified asphalt binders, the ring and ball softening point corresponds to a viscosity of 13,000 poise (18). Penetrations at a range of test temperatures, from tests using 100 g, 5 sec loading can be converted to viscosity using the following relationship (18): log η = 10.5012 − 2.2601 log( Pen ) + 0.00389(log( Pen)) 2
(2.8)
where, η = viscosity, P Pen = measured penetration for 100g, 5 sec loading, 0.10 mm Thus, from a combination of penetration, ring and ball softening point, and kinematic, absolute and BrookfieldT M viscosity measurements; the viscosity of the binder
56 over a wide range of temperatures can be predicted. Mirza and Witczak also developed equations, which shift the viscosity temperature relationship of the original binder for short-term aging that occurs during mixing and compaction, and for long-term in-situ aging (18). These equations take into account the aging potential of the binder, the temperature in the pavement, the time in service and the particular depth within an AC layer.
2.2.3.3 Witczak et al. Models The prediction of the complex modulus (E*) and related phase angle (φ) of an asphalt mixture from properties of asphalt binder and volumetric composition of the mixture plays a very important role in the rational pavement design procedure. The phase angle of asphalt mixtures is partially related to the phase angle of the asphalt binder (δ b). At low temperatures, phase angles of the binder tend to mirror the phase angle behavior of the mixture. However, at moderate to high temperatures, a large divergence between the two appears. As the temperature gets very large, the phase angle of the binder approaches that of a pure viscous material and will approach 90 degrees. In contrast, the mixture phase angle will start decreasing towards a value of zero and will denote a pure elastic behavior of the mixture. Figure 2.5 shows similar behavior of asphalt binder and mix in a phase angle (δ b or φ) versus dynamic stiffness (|Gb*| or |E*|) plot (16).
57
Phase Angle, degrees
90 75 60
High Temperature Region
45
Intermediate Temperature Region
Low Temperature Region
30 15 0 -3
-2
-1
0
1 2 Log |G b*|, psi
3
4
5
a. Binder Behavior in Black Space
Phase Angle, degrees
40
High Temperature Region
30
Low Temperature Region Intermediate Temperature Region
20
10
0 3
3.5
4
4.5
5 5.5 Log |E*|, psi
6
6.5
7
b. Mixture Behavior in Black Space FIGURE 2.5 Binder and Mix Behavior in Black Space According to Christensen, at cold temperatures, less than 32°F (0°C), asphalt concrete behavior can be approximated as being linear viscoelastic, because strains remain small under loading (24). At higher temperatures, the HMA mix behaves more like a granular
58 non- linear elastic material. According to Goodrich, HMA mixtures largely reflect the binder rheology at low temperatures (122°F (50°C)), the HMA mix rheology is predominantly influenced by the aggregates. It is well known with the adoption of the Superpave Performance Grading system and its associated testing, the binder characterization parameters required to establish the ASTM Ai-VTS i relationship may no longer be routinely collected. One of the major limitations associated with the use of the ASTM Ai-VTSi viscosity relationship is that it does not tell anything about the change of binder viscosity under dynamic loading with changing loading time (or frequency). This phenomenon is well known to be observed in the low to intermediate temperature ranges (14). Under the current Superpave binder characterization methodologies, the use of the binder complex shear modulus (Gb*), which is obtained from the Dynamic Shear Rheometer (DSR) test according to the AASHTO T 315-02 protocol, is considered very important. In contrast to the ASTM AiVTSi relationship, the Gb* data describes the change of binder stiffness under dynamic (shear) loading with changing loading time (or frequency). It is obvious that this parameter is the most fundamental and rational variable that can be used in prediction models for HMA mixture dynamic modulus. Witczak and his colleagues at the University of Maryland and Arizona State University developed a set of provisional models that relate binder complex shear modulus with binder viscosity (η). The 2002 Design Guide uses a model deve loped by
59 Witczak and his colleagues at the University of Maryland termed as the “WitczakBonaquist η-Gb* Model” in this dissertation (11). The model equations were expressed by: | G * | 1 ? = b ? sin db
a o + a 1? + a 2 ?
2
(2.9)
where, η
= binder viscosity (from ASTM Ai-VTS i equation), cP
|Gb*|
= binder shear modulus (measured), Pa
δb
= binder phase angle (measured), degree
ω
= angular frequency used to measure G* and δ b, radians/s
a0 , a1 and a2
= fitting parameter for all type of binders respectively 3.639216, 0.131373 and -0.000901
and, δ b = 90 + (b1 + b2 VTS)log(|G b*|) + (b3 + b4 VTS) {log(|G b*|)} 2
(2.10)
where, b1 , b2 , b3 and b4
= fitting parameter of values respectively -24.878,-7.2734, 1.8118 and 0.77054
VTS
= regression slope of the log- log viscosity (cP) versus log temperature (°R) plot; a viscosity temperature susceptibility parameter
For ω = 10 rad/s (fs = 1.59 Hz), which is the specified test frequency in the Superpave Performance Grading system, Equation 2.6 becomes:
60 | G *| 1 η = b 10 sin δ b
4 .8628
(2.11)
Witczak, Sybilski, Bonaquist and Pellinen further suggested revised model equations, termed as the “Witczak-Sybilski η-Gb* Model” in this dissertation, which are as follows (26): | G *| 1 ? = b ? sin db
a o + a1? +a 2 ? 2
(2.12)
where, a0 , a1 and a2
= fitting parameter for all type of binders respectively 4.9461, 0.0537 and - 0.0004 | Gg * | − | Gb * | = a(tan δ ) b | Gb * |
(2.13)
where, |Gg*|
= glassy shear modulus of binders, which was assumed equal for all asphalt binders, typically 1 GPa
|Gb*|
= shear modulus, Pa
δb
= phase angle, degree
a
= -1.5112 + 0.4159 . A
b
= 22.848 + 5.144 . VTS
A
= regression intercept of the log- log viscosity (cP) versus log temperature (°R) plot
61 2.3
Asphalt Concrete Mixture Characterization
2.3.1
Behavior of Asphalt Concrete Mixture It is well known that the mechanical behavior of a HMA mix, which is a
composite material, is primarily governed by the viscoelastic nature of the asphalt binder contained in the mixture. One should recall that in elastic materials, all work done by the external stresses during an increment of deformation is stored and fully recovered on unloading, i.e. the material does not dissipate or lose energy. Additionally, elastic deformations depend solely upon the stress magnitude and not upon the straining or loading history. On the other hand, in purely viscous fluids, strain increases continuously and linearly with time as energy is dissipated in flow, yielding permanent deformation. Material behavior that incorporates both elastic and viscous behavior is referred to as viscoelastic behavior. The elastic solid and viscous fluid represents opposite endpoints of a wide spectrum of viscoelastic beha vior.
2.3.2
Volumetric Properties of Asphalt Concrete Mixture The volumetric properties of a HMA mix are defined as follows in accordance
with the representative volume of a mix (27). Air Voids, Va : the percent volume of air between the coated aggregates in the compacted asphalt mixture relative to the total bulk volume of the compacted mix. Voids in Mineral Aggregates, VMA: the percent volume of compacted paving mix not occupied by the aggregate relative to the total bulk volume of the compacted mix (when the volume of the aggregate is calculated from its bulk specific gravity).
62 Absorbed Asphalt Volume, Vba : percent by volume of asphalt binder absorbed into the aggregates relative to the total bulk volume of the compacted mix. Asphalt Content, P b or AC : the percent by weight of asphalt binder in the mix relative to the total weight of the compacted mix. Effective Asphalt (or Binder) Volume (or Content), Vbeff: the percent volume of asphalt binder, which is not absorbed into the aggregates, relative to the total bulk volume of the compacted mix. Voids Filled with Asphalt, VFA: the percentage of VMA filled with asphalt binder. Air voids in the compacted HMA specimen as a percent of the total volume can be calculated as follows: Va = 100 ×
Gmm − Gmb G mm
(2.14)
where, Gmm = theoretical maximum specific gravity of the loose HMA mixture Gmb = bulk specific gravity of the compacted HMA mixture Voids in mineral aggregates (VMA) and voids filled with asphalt (VFA) of the compacted HMA specimen can be calculated as follows: G mb Ps G sb
(2.15)
VMA − Va VMA
(2.16)
VMA = 100 −
VFA = 100 ×
where,
63 Ps = amount of aggregates as percentage of total weight of the mixture Gsb = bulk specific gravity of aggregates
2.3.3
Pavement Distresses Related to Asphalt Concrete Mixture Properties Pavements may be subjected to a wide variety of distresses, leading to a decrease
in long-term pavement performance. Broadly speaking, these distresses can be cracking (fatigue, thermal, longitudinal, block, reflective, slippage etc.), distortion (rutting, shoving, corrugation etc.) or disintegration (raveling, wear loss, stripping etc.). Among them, rutting and fatigue cracking are considered very critical in terms of the pavement longevity and its life cycle cost and are distresses that are definitely influenced by the external design inputs such as pavement structure, layer quality, traffic loading, site environmental factors and foundation conditions.
2.3.4
Asphalt Concrete Mixture Characterization Tests Asphalt concrete mixture characterization is the measurement and analysis of the
response of HMA mixes to load, deformation, and/or the environment at a wide range of temperatures and loading conditions. Tests historically used for this purpose, such as Marshall Stability and Flow, and Hveem Stability tests, are empirical in nature. These tests are not reliable when conditions are outside those in which the tests were developed (1). For example, they are not reliable for predicting pavement performance as axle loads and tire pressure continue to increase since the tests were developed. Recent research done under the NCHRP 9-19 project showed that the Dynamic Modulus (E*) test is a leading candidate as a simple performance test to use a
64 mechanistic performance test. As a result, it has already been used in the new 2002 Pavement Design Guide as the principal HMA stiffness characterization test. The most current E* test procedures have been described in “Standard Method of Test for Determining Dynamic Modulus of Hot-Mix Asphalt Concrete Mixtures” with AASHTO designation TP 62-03 (28).
2.4
Stiffness of Asphalt Concrete Mixtures
2.4.1
E* Stiffness of Asphalt Concrete Mixtures For more than 50 years, it is well known in the pavement community that the
loading time and temperature have pronounced effects upon the rheological responses of asphalt binder and mixtures. In 1954, Van der Poel introduced the term “stiffness” (St,T ) to distinguish this parameter from the elastic modulus response. He expressed stiffness as follows: σ St ,T = ε t ,T
(2.17)
where, σ = stress ε = strain t = time of loading T = temperature The stiffness of a mix under dynamic sinusoidal loading is more commonly known as its complex modulus (E*). The performance of an asphalt pavement structure is significantly influenced by the modulus of the AC layers. The theory behind the complex
65 (dynamic) modulus has already been discussed in the previous chapter. Different forms (types) of dynamic modulus can be evaluated by several existing direct testing procedures. Among them, several common tests used in practice are: (a) Complex (Dynamic) Modulus, (b) Flexural Stiffness, and (c) Diametral (Resilient or Indirect) Modulus. In addition to the direct testing approaches, several widely utilized indirect predictive techniques are also available for HMA mix modulus evaluation, which are discussed in the later part of this chapter.
2.4.2
Determination of Stiffness by E* Test AASHTO test protocol TP 62-03 (Standard Method of Test for Determining
Dynamic Modulus of Hot-Mix Asphalt Concrete Mixtures) is the most current dynamic modulus (E*) test protocol (29). This method covers procedures for preparing and testing asphalt concrete mixtures to determining the dynamic modulus (|E*|) and phase angle (φ) over a range of temperatures and loading frequencies. A sinusoidal (haversine) axial compressive stress is applied to a specimen of asphalt concrete at a given temperature and loading frequency. The laboratory prepared mixtures are conditioned in accordance with the 4-hour short-term oven conditioning procedure described in AASHTO R30. Field mixtures are not conditioned. Any lab blended or plant obtained mixture is compacted in a gyratory compactor to 6- in diameter x 6.69- in high according to AASHTO T 312 Protocol. The test specimen ends are sawed and cored to obtain the final 4- in diameter x 6- in high E* test specimen. AASHTO T269 is then conducted to measure the air voids of the final test specimens.
66 For full characterization of the mix, E* tests are generally conducted at 14, 40, 70, 100 and 130°F under 25, 10, 5, 1, 0.5 and 0.1 Hz loading frequencies. A 60 second rest period is used between each frequency to allow some specimen recovery before applying the new loading at a lower frequency. The E* tests are done using a controlled stress mode, which produces strains between 50 and 150 micro-strain. This ensures, to the best possible degree, that the response of the material is linear across the temperature range used. Figure 2.6 shows the schematic presentation of the instrumentation of a E* Test sample. The axial deformations of the specimens are measured through at least two spring- loaded Linear Variable Differential Transducers (LVDTs) placed vertically on diametrically opposite sides of the specimen. Parallel brass studs are used to secure the LVDTs in place. Two pairs of studs are glued on the two opposite cylindrical surfaces of a specimen; each stud in a pair, being 100- mm (4 inch) apart and located at approximately the same distance from the top and bottom of the specimen.
Frictionless Bushing Guiding Rod
LVDT Mounting Stud Holding Bracket a. Sample Assembly
b. Lateral View
FIGURE 2.6 Specimen Instrumentation of E* Testing
67 Top and bottom surface friction is a very practical problem for compressive type testing. In order to eliminate the possibility of having shear stresses on the specimen ends during testing, pairs of rubber membranes, with vacuum grease within the pairs, are placed on the top and bottom of each specimen during testing. All E* tests are conducted in a temperature-controlled chamber.
2.4.3
Time-Temperature Superposition of Dynamic Modulus Dynamic modulus of HMA mix is dependant on both time of loading and
temperature. This special characteristic makes the characterization of AC a three dimensional problem. This problem can be reduced to a two-dimensional problem by the use of time-temperature superposition. The 2002 Design Guide also uses this technique at all analysis levels as the primary way of characterizing HMA mixtures. The latest procedure for the time-temperature superposition is developed at ASU. In this method, a master curve for the dynamic modulus (E*) of a specific HMA mix is constructed at a reference temperature (generally taken as 70°F). The E* data at other temperatures are shifted with respect to time until the curves merge into single smooth sigmoidal function. The master curve of the modulus, as a function of time, formed in this manner describes the time dependency of the material. The amount of shifting at each temperature required to form the master curve describes the temperature dependency of the material. In general, the master modulus curve can be mathematically modeled by a sigmoidal function described as:
68 LogE* = δ +
α 1+ e
β + γ (logt r )
(2.18)
where, tr
= reduced time of loading at reference temperature
δ
= minimum value of E*
δ + α = maximum value of E* β, γ
= parameters describing the shape of the sigmoidal function
The shift factor can be shown in the following form: a(T) =
t tr
(2.19)
where, a(T)
= shift factor as a function of temperature
t
= time of loading at desired temperature
tr
= reduced time of loading at reference temperature
T
= temperature of interest
For the greatest degree of precision, a second order polynomial relationship between the logarithm of the shift factor i.e. log a(Ti) and the temperature in degrees Fahrenheit is used. The relationship can be expressed as follows: Log a(Ti) = aTi2 + bTi + c where, a(Ti )
= shift factor as a function of temperature Ti
Ti
= temperature of interest, °F
a, b and c
= coefficients of the second order polynomial
(2.20)
69 The second order polynomial has been found to be the most accurate methodology to develop the time-temperature superposition relationship for HMA mixtures. While theory recognizes linear viscoelastic superposition, this would be equivalent to regressing a coefficient value of “a” to be equal to a = 0. Nonetheless, many mixtures have been found not to conform to the linear model form. This is why a more universal second order polynomial is recommended with real lab test results. Master curves and the corresponding shift factors can be developed experimentally by shifting laboratory frequency sweep data from dynamic modulus test (AASHTO designation TP 62-03) (29). Once the master curve for a specific HMA mix is obtained, the time of loading at the reference temperature can be calculated using Equation 2.19 for any given time of loading at any given temperature. Then the appropriate E* value can be calculated from Equation 2.18 using the time of loading at the reference temperature. As an example, construction of master curve for the ADOT Two-Guns mixture with 1% lime is shown in Figures 2.7 through 2.10. Figure 2.7 is a plot of E* (in psi) versus loading time (in seconds). In Figures 2.8 and 2.9, the E* data are shifted using a non- linear optimization by simultaneously solving seven master curve and shift parameters (δ, α, β, γ, a, b and c). These seven parameters are then used in the equations 2.17, 2.18 and 2.19 to calculate the E* of the particular mix at any temperature and loading frequency within the range used in the E* testing.
70 1.E+08 14 deg F
40 deg F
100 deg F
130 deg F
70 deg F
E*, psi
1.E+07
1.E+06
1.E+05
1.E+04 0.01
0.1
1 Log Loading Time, s
10
100
FIGURE 2.7 Laboratory E* versus Loading Time for Two-Guns Mix
1.E+08 δ =3.9289, α = 2.9309, β = −0.7793, and γ = 0.4214
14 deg F 40 deg F 70 deg F 100 deg F 130 deg F Master Curve
|E*|, psi
1.E+07
1.E+06
1.E+05
1.E+04 -8
-6
-4
-2 0 Log Reduced Time, s
2
4
FIGURE 2.8 Master Curve with Shifted E* Data for Two-Guns Mix
6
71
10
6 log aT
14 deg F 40 deg F 70 deg F 100 deg F 130 deg F Regression Line
y = 0.0003x2 - 0.1298x + 7.5344, R2 = 1 a = 0.0003, b = -0.1298 and c = 7.5344
2
-2
-6 0
20
40
60 80 Temperature, °F
100
120
140
FIGURE 2.9 Log Shift Factor versus Temperature for Two-Guns Mix
2.4.4
Use of E* in the 2002 Design Guide At all analysis levels of the 2002 Design Guide, any stiffness or viscosity data
used for new flexible pavement studies is obtained or predicted for HMA mix short-term aged according to AASHTO Test Method AASHTO PP2 and binder short-term aged by RTFO test according to AASHTO T240 (3). Once the E* data is obtained or predicted, the stiffness of HMA is determined from a E* master curve constructed at a reference temperature of 70°F using Equations 2.18 and 2.19. In the Level 1 analysis, specific laboratory test data are used in Equation 2.4 (and Equation 2.8 in case Gb* data is available) to establish the viscosity-temperature relationship of the asphalt binder. Then the laboratory frequency sweep data obtained
72 from the dynamic modulus test are shifted into a smooth master curve through computer aided numerical optimization using the following equation: LogE* = δ +
α 1+ e
β +γ [log(t )−c{log(η ) −log(ηTr )}]
(2.21)
where, t
= time of loading
η
= viscosity at temperature of interest
ηT r
= viscosity at reference temperature
δ, α, β, γ and c = model fitting parameters obtained by the optimization. The logarithm of the reduced time tr can be is expressed as follows: log(t) – c{(log(η) – log(ηT r)} = log (tr)
(2.22)
Thus, Equation 2.21 is actually another form of Equation 2.18. Once the master curve and associated parameters are obtained, the mix E* stiffness at any specific temperature and loading time can easily be calculated from Equation 2.21. In the Level 2 analysis of the 2002 Design Guide, the 1999 version of the Witczak E* predictive equation (6) is combined with the same laboratory binder test data needed in Level 1. No E* test data is required at this level. The Witczak E* predictive equation is described in the later part of this chapter. In the Level 3 analysis, the current version of the Witczak E* predictive equation is combined with the typical temperature-viscosity relationship established for a specific binder grade specified in AASHTO MP1. In both Level 2 and Level 3 analyses, some mixture and binder data are needed for use in the E* predictive equation. Once the E* data at different temperatures and loading times are
73 predicted, the E* master curve is constructed and mix stiffness at specific temperature and loading time is obtained similarly as in the Level 1 analysis.
2.4.5
Predictive Models for E* Stiffness Numerous E* predictive models and related equations have been developed over
the last 50 years. Historically, the E* predictive models and equations were developed on the basis of the conventional multivariate linear regression or non- linear regression analysis of laboratory test data and the established or anticipated basic engineering behavior and/or properties of the HMA mixture and/or its components. These models can be broadly classified as the following categories: •
Linear polynomial for logarithm (10-based) of |E*| with related nomograph for bitumen stiffness modulus such as the Shell Oil method.
•
Linear polynomial for logarithm (10-based) of |E*| such as the Shook and Kallas model and all early Witczak et al. models.
•
Non-linear polynomial for logarithm (10-based) of |E*| using a sigmoidal function such as the Witczak et al. model.
•
E* models primarily based the law of mixtures such as the Hirsch model (30).
•
The available |E*| predictive models are chronologically briefly described in the next few paragraphs.
2.4.5.1 Van der Poel (Shell Oil’s Early Version) Model One of the early but well-known asphalt mix stiffness predictive model was developed by Van der Poel of the Shell Oil Company based upon over 20 years of
74 laboratory work at “Koninklijke/Shell-Laboratorium, Amsterdam (KSLA)” (2). This predictive method is also known as the “KSLA Method”. A primary assumption of the Shell Oil method is that the HMA mix stiffness is a function of the asphalt binder stiffness (Sb). It uses a nomographic solution to obtain the Sb as outlined earlier in the Section 2.2.3.1 of this dissertation. Once the binder stiffness (Sb ) is determined from the nomograph, the mix stiffness (Sm) is calculated from the following equation: 2.5 Cv S m = Sb 1 + n 1 − Cv
n
(2.23)
where, Sm = mix stiffness, kg/cm2 N = 0.83 log10 [4 x 105 )/Sb ] Cv =
Vg Vg + Vb
Vg = volume of aggregates Vb = volume of bitumen (asphalt binder) The mix stiffness predictive equation was originally considered applicable for air voids ≈ 3% and Cv = 0.7 to 0.9. Further research at Shell Oil concluded that for mixtures having air voids > 3%, Cv ´ should be used in lieu of Cv using the following relationship: Cv ' =
where, H = (Actual air voids – 3)/100
Cv 1+ H
(2.24)
75 This correction is applicable only to mixtures having an asphalt volume concentration factor (C B) satisfying the following relationship: CB ≥
2 (1 − Cv ' ) 3
CB =
Vb Vg + Vb
(2.25)
where, (2.25a)
2.4.5.2 Bonnaure (Shell Oil’s Later Version) Model In 1977, a revised version of the Shell Oil model was developed by Bonnaure and his co-researchers (15). Their mix stiffness equations were developed using 9 different HMA mixtures tested in a two-point bending apparatus developed by Shell Oil. Trapezoidal specimens fixed at the bottom were subjected to a sinusoidal load at the free end. The stiffness modulus was obtained by evaluating the stress and strain measured at the free end. Bonnaure and his team developed the following equations for predicting the HMA mix stiffness (Sm) based on the volume of the binder (Vb), volume of the aggregate (Vg) and the binder stiffness (Sb) (31): β1 = 10.82 −
1.342(100 −V g )
(2.26)
Vg + Vb
β 2 = 8.0 + 0.00568Vg + 0.0002135V g
2
(2.27)
1.37Vb 2 − 1 β3 = 0.6 log 1 . 33 V − 1 b
(2.28)
β 4 = 0.7582( β1 − β 2 )
(2.29)
76 For 5 x 106 N/m2 < Sb < 109 N/m2 , log S m =
β4 + β3 β − β3 (log S b − 8) + 4 log S b − 8 + β 2 2 2
(2.30)
For 109 N/m2 < Sb < 3 x 109 N/m2 , log S m = β 2 + β 4 + 2.0959( β1 − β 2 − β 4 )(log S b − 9)
(2.31)
The previous two equations are based on SI units with Sb and Sm in N/m2 . The following two equations were formulated when Sb and Sm are expressed in psi: For 725 psi < Sb < 145,000 psi, log S m =
β4 + β3 β − β3 (log S b − 4.1612) + 4 log S b − 4.1612 + β 2 − 3.8383 (2.32) 2 2
For 145,000 psi < Sb < 435,000 psi, log S m = β 2 + β 4 + 2.0959( β1 − β 2 − β 4 )(log S b − 5.1612) − 3.8388
(2.33)
Figure 2.10 shows the nomograph for determining the stiffness modulus of AC mixtures using the Bonnaure et al. model equations.
2.4.5.3 Shook and Kallas’ Models In the late 1960’s, Shook and Kallas of the Asphalt Institute developed another model to predict dynamic modulus of asphalt mixtures (33). The model was based on 29 HMA mixtures tested at only one loading frequency (f = 4 Hz), which provided a total of 87 data points. The model equation they developed was: log 10 E * = a 0 + a1 ρ 200 + a2Va + a 3η 70;106 + a 4 ρ ac 5 t p a
a6
(2.34)
77
FIGURE 2.10 Nomograph for Stiffness Modulus of AC Mixtures (32)
78 where, |E*|
= dynamic modulus, 105 psi
ρ200
= percentage of aggregates (by weight of the total aggregates) passing the no. 200 sieve, %
Va
= percent air voids (by volume of the mix), %
η70;106
= asphalt binder viscosity at 70°F, 106 poise
ρac
= percent asphalt absorption (by weight of the mix), %
tp
= test temperature, °F
a0 , a1 , a2 , a3 , a4 , a5 and a6 = 1.5436, 0.020108, -0.031861, 0.068142, -0.00127, 0.4 and 1.4, respectively (regression parameters) In 1969, Shook and Kallas came up with a modified version of their model equation, which was as follows: log 10 E * = b0 + b1 ρ 200 + b2Va + b3 ρac 4 (log ηt )b5 b
(2.35)
where, |E*| = dynamic modulus, 105 psi ηt = asphalt binder viscosity at test temperature “t”, poise b0 , b1 , b2 , b3 , b4 and b5 = 3.12197, 0.0248722, -0.034587, -9.02594, 0.19 and -0.9, respectively (regression parameters)
2.4.5.4 Witczak’s Early Model In 1972, Witczak reanalyzed the Shook and Kallas’s data and came up with the following simple equation for the f = 4 Hz data (34):
79
E* = c 0c1
− ( t p c2 )
(2.36)
where, E*
= dynamic modulus, psi
c0 , c1 and c2
= 3.8E+06, 1.0046 and -1.45, respectively (regression parameters)
He used this relationship in his development of the Asphalt Institute MS-II “Design of Full Depth Pavements for Airfields” (35).
2.4.5.5 Witczak and Shook’s Model By the late 1970’s, the E* database had increased to 41 mixtures. Each was tested under a temperature- frequency factorial of three levels each. Using the increased database, Witczak developed a revision to the enlarged database and developed the following relationship:
log10 |E*| = d 0 + d1 ρ 200 f d 2 + d 3Va + d 4η 70;106 + d 5 [t p ( d6 + d7 log f ) ρ ac d8 ]
+ d 9 [t p ( d6 + d7 log f ) ρ ac d8 f d10 ] + d11 f
(2.37)
d12
where, |E*| = dynamic modulus, 105 psi f
= loading frequency, Hz
d0 , d1 , d2 , d3 , d4 , d5 , d6 , d7 , d8 , d9 , d10 , d11 and d12 = 0.553833, 0.028829, -0.17033, -0.03476, 0.070377, 0.000005, 1.3, 0.49825, 0.5, -0.00189, -1.1, 0.931757 and -0.2774, respectively (regression parameters)
80 2.4.5.6 Witczak’s 1981 Model During the development of the Asphalt Institute MS-I Full Depth Design for Highways, Witczak introduced another term, the effective bitumen volume term (V beff), into the previous model (35). This equation was based upon a percent aggregate (by weight) passing sieve no. 200 (ρ200 ) ≈ 5 along with a design frequency (arbitrarily selected to be f = 10 Hz) to simulate highway traffic conditions. This model was not meant to be revised as a new generalized E* model, but rather, one that was “tailored” to typical highway design conditions. The model equation was as follows: log 10 E * = e0 + e1Va + e 2η70;106 + e3t p 4 Vbeff e
e5
(2.38)
where, |E*| = dynamic modulus, 105 psi Vbeff = percent effective asphalt content (by volume of the mix), % e0 , e1 , e2 , e3 , e4 and e5 = 1.52531, -0.03476, 0.070377, -0.0001, 1.79825 and 0.5, respectively (regression parameters)
2.4.5.7 Witczak, Miller and Uzan’s Model During the late 1970’s - early 1980’s, an extensive laboratory test study on the dynamic modulus of asphalt mixtures used by the Maryland State Highway Administration was conducted by Witczak at the University of Maryland. This study involved 90 additional mixtures and 810 additional data points, bringing the total base to nm = 131 mixes and nt = 1179 test points. In contrast to the previous database, which contained data from mixtures made of only dense graded aggregates, the new database
81 had data from mixtures made of gravels, slags and sand asphalts as well. Based on the combined database; Miller, Witczak and Uzan modeled the following E* predictive equation, which had a new term (ρopt ) for the percent optimum asphalt content (by weight of the mix) (36):
log10 | E* | = g 0 + g1ρ200 f g 2 + g 3Va + g 4η70;106 + g 5 f g6 + [ g 7t p( g8 + g9 log f ) + g10 f g11 t p( g 8 + g9 log f ) ][ ρ ac − ρopt + g12 ] g13
(2.39)
where, |E*| = dynamic modulus, 105 psi ρopt = percent optimum asphalt content (by weight of the mix), % g0 , g1 , g2 , g3 , g4 , g5 , g6 , g7 , g8 , g9 , g10 , g11 , g12 and g13 = 0.553833, 0.0288229, -0.17033, -0.03476, 0.070377, 0.931757, -0.02774, 0.000005, 1.3, 0.49825, -0.00189, -1.1, 4 and 0.5, respectively (regression parameters)
2.4.5.8 Witczak and Akhter’s Models During the 1983 to 1984 period, Akhter and Witczak re-analyzed the database and came up with a revised version of the previous model (37), which is as follows:
log 10 E * = h0 + h1Va + h2 ρ3 / 4 + h3η70;106 + h4t p + h5 log f + h6 log( f t p ) 2
+ h7 (Vbeff − Vbeffopt + h8 ) h9 t p 2 + h10 (Vbeff ρ 4 ) + h11 ρ200 ρ abs ) h9 [t p ( d6 +d7 log f ) ρ ac d8 f d10 ] + d11 f d12 (2.40) where, |E*|
= dynamic modulus, 105 psi
ρ3/4
= percentage of aggregates (by weight of the total aggregates) retained
82 on the ¾ inch sieve, % Vbeffopt
= percent effective optimum asphalt content (by volume of the mix), %
ρ4
= percentage of aggregates (by weight of the total aggregates) retained on the No. 4 sieve, %
h0 , h1 , h2 , h3 , h4 , h5 , h6 , h7 , h8 , h9 , h10 and h11 = 1.45, -0.0256272, 0.0127921, 0.0627099, -0.0083735, 0.147306, 0.00001932, -0.00002541, 8, 0.5, -0.0001492 and 0.005918, respectively (regression parameters) This new model had three new terms; Vbeffopt , ρ3/4 and ρ4 , to take into account the effects of effective binder volume and amount of both the coarse and fine aggregates. In the 2nd version of this model, another new term (ρ3/8 ) was introduced. The revised model is as follows: log 10 E * = k 0 + k1Vbeff + k 2Va + k 3 ρ 200 + k 4 ρ 4 + k 5 ρ abs + k 6t p + k 7 f + k 8t p + k 9Vbeff 2
+ k10 ρ 200 + k11ρ 3 / 4 + k12 ρ 3 / 8 + k13 ρ 4 + k14 ρ abs + k15η70;106 + k16 f 2
2
2
2
2
2
2
+ k17 ρ 3 / 8Vbeff + k18 ρ 3 / 4Vbeff + k19 ρ 3 / 4 ρ 4 + k 20 ρ 3/ 8 ρ 4 + k 21 ρ 3 / 8 ρ abs
(2.41) where, |E*|
= dynamic modulus, 105 psi
ρ3/8
= percentage of aggregates (by weight of the total aggregates) retained on the 3/8 inch sieve, %
k0 , k1 , k2 , k3 , k4 , k5 , k6 , k7 , k8 , k9 , k10 , k11 k12 , k13 , k14 , k15 , k16 , k17 , k18 , k19 , k20 and k21 = 2.468, -0.1155, -0.0299, -0.0975, -0.00963, 0.359, -0.00815, 0.066 , -0.0000618, 0.00253, 0.0083, -0.00164, 0.000308, 0.000204, -0.105, 0.0171, -0.00268, 0.00167, 0.000709, 0.000937, -0.00069 and -0.0031, respectively
83 2.4.5.9 Witczak, Leahy, Caves and Uzan’s Models During 1985 to 1989, further E* testing was conducted by Leahy using the dynamic loading equipment at the FHWA research facility under the supervision of Professor Matthew W. Witczak of the University of Maryland. With these E* tests, the E* database was expanded to 149 mixtures and 1429 data points. Based on this expanded database, Witczak and his co-researchers suggested the following predictive model for dynamic modulus (38): log 10 E * = l 0 + l1Vbeff + l 2Va + l3 ρ 200 + l 4 ρ abs + l 5t p + l 6 f + l 7t p + l 8Vbeff + l 9 ρ 200 2
2
2
+ l10 ρ 3 / 4 2 + l 11ρ 3 / 8 2 + l 12 ρ abs 2 + l13 (η 70;106 ) 2 + l14 f 2 + l 15 ρ3 / 8Vbeff + l16 ρ 3 / 4Vbeff + l17 ρ3 / 4 ρ 4 + l18 ρ 3 / 8 ρ 4 + l19 ρ 3 / 8 ρ abs (2.42) where, |E*| = dynamic modulus, 105 psi l0 , l1 , l2 , l3 , l4 , l5 , l6 , l7 , l8 , l9 , l10 , l11 l12 , l13 , l14 , l15 , l16 , l17 , l18 and l19 = 2.250053, -0.091756, -0.027949, -0.096881, 0.250094, -0.006447, 0.060612, -0.00007404, 0.00191539, 0.0082813, -0.0010225, 0.0001909, -0.0801155, 0.0148592, -0.0024159, 0.00094015, 0.00084534, 0.0004965, -0.00034328 and -0.00316297, respectively (regression parameters) Later, the same research team came up with a revised model, which is as follows:
log 10 E * = m0 + m1Va + m2 ρ3 / 4 + m3η 70;106 + m4 t p + m5 log f + m6 log( ft p ) 2
+ m7 (Vbeff − Vbeffopt + m8 ) m9 t p 2 + m10Vbeff ρ 4 + m11ρ 200 ρ abs where, |E*| = dynamic modulus, 105 psi
(2.43)
84 m0 , m1 , m2 , m3 , m4 , m5 , m6 , m7 , m8 , m9 , m10 and m11 = 1.457061, -0.02481, 0.012597, 0.060764, -0.0081771, 0.146439, 0.00001844, -0.00002559, 8, 0.5, -0.0001522 and 0.005006, respectively (regressio n parameters)
2.4.5.10 Witczak and Fonseca’s Model In the mid-1990’s, Fonseca and Witczak identified that while the current (at that time) E* predictive models were highly accurate, they possessed several limitations as well (7). The most important limitations were: The master database was based on dynamic stiffness testing on only lab prepared specimens. All models had been based upon either the penetration at 77°F (25°C) or the viscosity at 70°F (21.1°C) of the original binder, which were intended to reflect the general grade (hardness) of the binder used. As a result, it was completely invalid for one to use field extracted (i.e., field aged) consistency (penetration or viscosity) measurements as input into any of the previous models developed. Thus, those models could not be used to predict the dynamic modulus of long-term field aged mixtures. One of the most significant limitations was associated with the fact that all previous models used a variety of linear polynomial or logarithmic forms and were obtained from test data generated within a temperature range of 41 to 104 °F (5 to 40°C). For all practical purposes, the master curves of such a limited temperature- frequency test matrix fall on the linear sloped portion of the sigmoidal master curve. Hence, extrapolation of any parameter outside the range of variables used to develop the model would follow a log E* trend and would lead to erroneous predictions, especially at extreme temperature-time values beyond those used in the testing.
85 To overcome these two major limitations, research studies were conducted at the University of Maryland to “correct” these deficiencies. One major study was devoted to evaluating the influence of both short and long-term aging upon the original viscositytemperature relationships of conventional asphalt binders. Other significant studies focused on developing a new model for the dynamic modulus using the actual binder viscosity as predictor variable for binder stiffness in lieu of the temperature. Finally, a completely new model form was selected so that very short and very long reduced time of loading (i.e. cold/hot temperatures) would be accurately modeled for the E* prediction. This new model developed was based on a sigmoidal function. In conjunction with this decision, Witczak also abandoned the classical 3 factorial temperature - frequency analysis proposed by ASTM and expanded the laboratory procedure to 5 temperatures and 6 frequencies. The global aging model of Mirza and Witczak “opened the door” to account for any short and long-term aging effect by just using the actual viscosity (regardless of aged state) as direct input into the E* model (18). In fact, most refined asphalt binders, with the exception of heavily air blown or high wax content crudes, exhibit a linear relationship when a log log viscosity (in centipoises, η) versus log temperature (in degrees Rankine, TR) plot is drawn (ASTM D 2493-85). The relationship is commonly known as the ASTM Ai-VTSi relationship, which has previously been shown in Equation 2.7. Once a set of A and VTS is known for a particular aging condition, A and VTS for other aging conditions can be calculated by use of the aging models developed by Mirza and Witczak. They also suggested the typical values of A and VTS for different aging
86 conditio ns of asphalt binders, which may be used in lieu of the lab data in case of insufficient or no lab data. Fonseca and Witczak used the regression coefficients for original conditions as found by Mirza and Witczak in order to modify the viscosity data contained in the E* master database (39). It is known that, at very cold temperatures, there is an upper limit value for the viscosity of all asphalt binders. This value is approximately 2.7 x 1012 centipoises (cP). Thus, the full characterization of the viscosity-temperature relationship of a given binder was viewed as a combined model reflecting the A and VTS parameters and the maximum viscosity limit (≈ 2.7 x 1012 cP) of asphalt binders. It was assumed that the hardening (stiffening) effects produced during conventional laboratory mixing and compaction procedure was practically nil as compared to those taking place during field (plant) mix operations (actually this was proven in a limited lab test study). This procedure was considered quite reasonable according to what is presented in the model validation phase of the study. After this modification, asphalt binder viscosity values became available for each mix at the temperatures at which the dynamic modulus test results were actually obtained. To eliminate the unrealistic estimates of mix stiffness at extreme values of temperatures, aging and loading rate, a sigmoidal growth curve was found as the best functional form. A simple symmetrical sigmoidal function consistent with the limitations of extreme values can be written as follows: y=δ +
α β −γx 1+ e
(2.44)
87 where, y
= the criteria variable
δ
= a location parameter for y (the minimum value for y)
α
= the range of possible values to be added to δ
x
= the predictor variable under consideration
β/γ
= a location parameter for x corresponding to a value of: y = δ + α/2
The final dynamic modulus predictive model, developed by Fonseca and Witczak, was as follows:
log 10 E* = −0.261 + 0.008225ρ 200 − 0.00000101( ρ 200 ) 2 − 0.00196 ρ 4 − 0.03157Va − 0.415
Vbeff Vbeff + Va
+
1.87 + 0.002808 ρ 4 + 0.0000404ρ 38 − 0.0001786( ρ38 ) 2 + 0.0164 ρ 34 1 + e ( −0.716 log f −0.7425 logη ) (2.45)
where, |E*|
= dynamic modulus, 105 psi
ρ200
= percentage of aggregates (by weight of the total aggregates) passing through no. 200 sieve, %
ρ4
= percentage of aggregates (by weight) retained on no. 4 sieve, %
Va
= percent air voids (by volume of the mix), %
Vbeff
= percent effective asphalt content (by volume of the mix), %
ρ38
= percentage of aggregates (by weight) retained on the 3/8 inch sieve, %
ρ34
= percentage of aggregates (by weight) retained on the 3/4 inch sieve, %
f
= loading frequency, Hz
η
= binder viscosity at temperature of interest, 106 poise
88 It should be recognized that the form of the model equation can be easily converted into a general sigmoidal form similar to the one described in Equation 2.44 by defining: δ
= -0.261+0.008225ρ200 -0.00000101(ρ200)2 -0.00196ρ4-0.03157Va -0.415 (
Vbeff Vbeff + Va
)
α
= 1.87+0.002808ρ4 +0.0000404ρ38-0.0001786(ρ38 )2 +0.0164ρ34
β
= -0.7425 log (ηT r)
ηT r
= binder viscosity at the reference temperature, 106 Poise
γ
= 0.716, and
log tr
= log t - 1.037 x (log η - log ηTr), where: tr = reduced time of load at reference temperature (hence, tr = 0 at reference temperature Tr)
2.4.5.11 Andrei, Witczak and Mirza’s Revised Model As time progressed, several more research projects and studies were performed at the University of Maryland. This led to an enhanced E* database becoming available by 1999. The new database represented dynamic modulus test results for 56 additional HMA mixtures (including 34 mixtures with modified binders) that provided 1320 more new data points for analysis. All test samples used in the new database were laboratory prepared by gyratory compaction. Cylindrical 2.75 in diameter by 5.5 in height specimens were cored from each 6 in diameter gyratory plug. Compared to the previous database, the new database covered a much wider range of viscosity values: 20 binders @ 5 test temperatures. It should be noted that the previous database was based on mixtures
89 generally having only conventional binders. However, only 5 aggregate gradations were used throughout the new database. Andrei et al. analyzed the expanded database having 2750 data points obtained from 205 HMA mixtures and came up with a revised E* predictive model using the similar sigmoidal form as developed earlier by Fonseca and Witczak. To the pavement community, this model is presently widely known as the “Witczak E* Predictive Equation” and is as follows:
log 10 E* = −1.249937 + 0.02923ρ 200 − 0.001767 ( ρ 200 ) 2 − 0.002841ρ 4 − 0.058097Va − 0.82208
Vbeff Vbeff + Va
+
3.871977 − 0.0021ρ 4 + 0.003958ρ 38 − 0.000017( ρ 38 ) 2 + 0.00547 ρ 34 1 + e ( −0.603313− 0. 313351log f −0.393532logη ) (2.46)
It is noteworthy that this is the Witczak E* predictive equation that is used in the Level 2 and Level 3 analysis of the 2002 Design Guide. This equation, again, can easily be converted to a sigmoidal form by defining: δ
= -1.249937+0.02923ρ200 -0.001767(ρ200)2 -0.002841ρ4-0.058097Va -0.82208 (
Vbeff Vbeff + Va
)
α
= 3.871977-0.0021ρ4 +0.003958ρ38 -0.000017(ρ38 )2 +0.00547ρ34
β
= - 0.603313 - 0.393532 log (ηTr)
ηT r
= binder viscosity at the reference temperature, 106 Poise
γ
= 0.313351, and
log tr
= log t – 1.255882 x (log η - log ηT r), where tr = reduced time of load at reference temperature (hence, tr = 0 at reference temperature Tr)
90 2.4.5.12 Hirsch Model of Christensen, Pellinen and Bonaquist During 1999 to 2001, Pellinen conducted dynamic modulus testing of 18 HMA mixtures from three project sites: (1) FHWA ALF, (2) MnRoad and (3) WesTrack. All of this testing was conducted at ASU under the guidance of Dr. M. W. Witczak for the NCHRP 9-19 Project (Task C). The test mixtures used 8 conventional and modified binders, 5 aggregate types, while the E* testing used 5 temperatures (15.8, 40, 70, 100 and 130 °F) and 2 loading frequencies (0.1 and 5 Hz). The E* testing provided 206 data points for analyses. Based on this database, Christensen, Pellinen and Bonaquist developed a new E* predictive model based upon an existing version of the law of mixtures, called the Hirsch model, which combines series and parallel elements of phases (30). In applying the Hirsch model to asphalt concrete, the relative portion of material in parallel arrangement, called the contact volume, is not constant but varies with time and temperature. Several versions of the Hirsch model were evaluated. The most effective model was the simplest, in which the modulus of HMA mix is directly estimated from the binder modulus, VMA and VFA. This model is as follows:
VFA × VMA VMA E * mix = Pc 4,200,0001 − + 3 G * binder 100 10,000 VMA 1 − 100 VMA + (1 − Pc) × + 4,200,000 3 × VFA × G * binder where, |E*|mix
= dynamic modulus of the mixture, psi
|G*|binder
= shear modulus of the binder, psi
−1
(2.47)
91 VMA
= voids in the mineral aggregates, %
VFA
= percent of VMA filled with binder, %
Pc
= contact volume estimated from the following equation: 0. 58
VFA × 3 G * binder 20 + VMA Pc = 0. 58 VFA × 3 G * binder 650 + VMA
(2.48)
This model is based on only 206 data points from 18 HMA mixtures. Therefore, it is highly possible that many mixtures will lie outside the range of the original database and the E* prediction for those mix may give erroneous results due to extrapolation of the mathematical model.
2.5
Statistical Background The model development process greatly depends on the statistical analysis and
linear or non- linear optimization process followed. The statistical analysis is aimed at reducing the error from prediction by comparing the predicted values with the observed values for the same values of the input variables in different ways. Model optimization is aimed at finding out the values of the fitting parameters used in a model that typically lead to the lowest prediction ever possible. When these values are used, the model is supposed to provide its best prediction. The following sub-sections discuss the basic statistical concepts needed for a model development.
92 2.5.1
Statistical Analysis
2.5.1.1 Model Comparison While comparing models predictions to known data, there are three important considerations: precision, accuracy and bias. In case of the predictive model, precision refers to how close the predicted and observed data are to each other. The scatter in a plot of observed versus predicted data reflects the precision. Accuracy is the conformity of prediction to the true observed value. Bias is a tendency of predicted data to deviate in one direction from the observed data. In other words, bias is a systematic error between predicted and observed data. Accuracy, precision and bias are influenced by the errors in optimization, factors omitted from the model and wrong function or structure used in the model. The concept of precision and bias is shown pictorially in Figure 2.11 (40).
FIGURE 2.11 Concepts of Precision and Bias
93 2.5.1.2 Goodness of Fit Goodness of fit indicates how well the model input parameters fit into the model. To find this out, the predicted values are calculated using the model and compared with the measured values at the same input conditions. The comparison is obtained by finding the error in the prediction for each data point. The following are the equations that are used to compare the prediction and measure the goodness of fit of the model. ε i = (predicted data)i – (measured data)i
(2.49)
Sum of Error = Σε i
(2.50)
Sum of Squared Error, SSE = Σ(ε i)2
(2.51)
Standard Error, Se =
SSE n −1
(2.52)
Arithmetic Mean, x =
1 n ∑xi n i =1
(2.53)
Standard Deviation, Sy =
(
1 n xi − x n −1∑ i =1
)
2
(2.54)
where, i
= data point number
xi
= value of i-th data point
n
= number of data points
For a model with p number of fitting coefficients, the values of the coefficient of determination (R2 ) can be computed using the following equation. This process provides the adjusted R2 for the model taking into account the degrees of freedom.
94 n − p Se R = 1− ⋅ n − 1 S y 2
2
(2.55)
where,
2.5.2
n
= number of data points
p
= number of regression constants
n– p
= degrees of freedom
Se
= standard error
Sy
= standard deviation of observed data
Model Optimization Once the general mathematical structure of a model is defined, the fitting
coefficients need to be optimized. Through optimization, the regression coefficients or fitting parameters within the model are assigned specific values in such a way that the model equation provides the minimum error when the predicted and observed data are compared. There are two considerations considered during this process; reduction in scatter and elimination of bias. The sum of the squared error (Σε i2 ) should be minimized to reduce the scatter in the data, while the bias is eliminated by setting the sum of errors (Σε i) to zero. Non-linear optimization is almost a mandatory approach for asphalt binder and mixture stiffness prediction models due the complex structure of the models. For this purpose, researchers have found the “solver” function of MicrosoftT M Excel quite convenient and accurate. Solver is based on the Generalized Reduced Gradient (GRG2) algorithm for optimizing non- linear problems. This algorithm was developed by L.S.
95 Lasdon and co-researchers (41). Linear and integer problems use the simplex method. Gradient-based algorithms are useful when trying to determine the absolute best fit of an objective function. Generally, GRG2 technique can find solution with very small tolerance of convergence criteria. However, its disadvantage is that the solution is highly dependent on the initial starting values of the fitting parameters of the model. These values initial are also known as the “seed values”. If the solver function is initiated in an infeasible design space (i.e. with infeasible seed values), it is highly likely that no feasible solution will be found since solver is not capable of exploring the design space very well. To avoid a non-convergent solution, proper caution should be taken to finding out the seed values. To use the solver function, the observed values are first compared with the predicted values. For each set of data, the difference between the predicted and observed value gives the error amount for that data point. The sum of all error squares is first minimized by changing the values of the fitting or regression parameters included in the model under consideration by the use of the built- in “solver” function of MicrosoftT M Excel. This process gives the optimized model with minimal scatter. The arithmetic sum of all errors is then minimized by further changing the values of the fitting parameters by using the solver function again. When proper seed values of the fitting parameters are used, this process gives an unbiased optimized model equation with a minimum Se/Sy and a maximum coefficient of determination (R2 ). This combination (Se/Sy and R2 ) can further be used to compare the statistical goodness of fit of different candidate models.
3
ASPHALT BINDER STIFFNESS DATABASE
3.1
Introduction As noted, a rathe r comprehensive η-Gb* database of asphalt binder stiffness in
terms of ASTM Ai-VTSi viscosity (η), shear modulus (|Gb*|) and phase angle (δ b) has been developed in this research. This database can be effectively used for developing accurate stiffness models for binders. A minor but inconsequential limitation of the database is that the asphalt binder viscosity is reported indirectly in terms of the A and VTS parameters that were originally obtained from the ASTM Ai-VTSi relationship. It may be noted that the ASTM Ai-VTSi relationship is usually calculated from a regression analysis of laboratory obtained viscosity data (i.e. loglog viscosity in centipoises) versus test temperature data (i.e. log temperature in degree Rankine). Suggested values of “A” and “VTS” for a particular performance graded (PG) binder can also be obtained from the NCHRP 1-37A project reports (3).
3.2
Original η-Gb* Database Bonaquist et al. and Witczak et al. developed models for predicting binder
viscosity (η) from laboratory Gb* data in order to use that in the Witczak E* predictive model (11, 26). The Witczak et al. viscosity predictive η-Gb* model was based on the AiVTSi viscosity and Gb* data of 33 different conventional and modified asphalt binders with a partial factorial of the following three aging conditions: (a) Original or Tank condition; (b) Construction phase aging using the Rolling Thin Film Oven (RTFO); and (c) In-service aging using the Pressure Aging Vessel (PAV) at 100°C.
97 3.3
Expanded η-Gb* Database There were 5640 sets of η-Gb* test data points in the Witczak et al. database (26).
Later, this researcher tested complex shear modulus of five conventional and one modified asphalt binder typically used by the Arizona Department of Transportation (ADOT) as part of his Master of Science research at ASU (14). In addition to the 6 ADOT binders, he also tested complex shear modulus of two modified Finnish binders obtained from VTT, Communities and Infrastructure, Finland, for Pellinen’s Ph.D. dissertation (13). In addition to the previously mentioned three aging conditions, one more aging of the binder, PAV at 110°C, was done for these six ADOT and two Finnish binders. Conventional and Superpave consistency tests were conducted to obtain the ASTM Ai-VTSi viscosity (η) data. To obtain the Gb* data (i.e. |Gb*| and δ b data), standard Dynamic Shear Rheometer (DSR) testing was conducted. In the ASU study, for the purpose of elaborate analysis, the DSR tests were conducted at 15, 25, 35, 45, 60, 70, 80, 95, 105 and 115°C under the oscillatory loading frequencies of 1, 10 and 100 rad/s. This ASU testing on the additional 8 binders provided 3,300 new η-Gb* data points. Thus the η-Gb* data of these 6 ADOT and 2 Finnish binders, along with the previously tested 33 binders, comprises a larger database of comp lex shear modulus of 41 different types of asphalt binders (including 9 modified binders) with a wide range of modifications and aging having a total of 8,940 η-Gb* test data points. This database was used in this research to combine the binder characteristics with the HMA mixture characteristics. The complete list of asphalt binders included in the expanded η-Gb* database is summarized in Table 3.1.
98 TABLE 3.1 Summary of Asphalt Binders of the η-Gb* Database Project Maryland Port Administration Study
Mix Study Minnesota Road WesTrack ALF FHWA
MRL
ADOT AC Binder Characterization
Binder Citgo PG 70-22 TLA PG 76-16 [Modified] Elvaloy PG 76-22 [Mod] Novophalt PG 76-22 [Mod] Stylink PG 76-16 [Mod] Citgo PG 64-22 MNRD120P MNRDAC20 WesTrack AC717 ALF AC-5 ALF AC-10 ALF AC-20 ALF Styrelf [Mod] ALF Novophalt [Mod] AAA1 AAA2 AAB1 AAB2 AAC1 AAC2 AAD1 AAD2 AAF1 AAF2 AAG2 AAK1 AAK2 AAM1 AAM2 AAS1 AAS2 AAS3 ABM1 Paramount PG 58-28 Paramount PG 64-16 Chevron PG 64-22 Chevron PG 76-16 Navajo PG 70-10 Navajo PG 76-16 [Mod] Finnish B-80+ [Mod] Finnish PmB [Mod]
Aging Conditions Original, RTFO and PAV at 100ºC Original, RTFO and PAV at 100ºC Original, RTFO and PAV at 100ºC Original, RTFO and PAV at 100ºC Original, RTFO and PAV at 100ºC Original, RTFO, PAV 100ºC & Recovered Original, RTFO, PAV 100ºC & Recovered Original, RTFO, PAV 100ºC & Recovered Original, RTFO & PAV 100ºC Original, RTFO, PAV 100ºC & Recovered Original, RTFO, PAV 100ºC & Recovered Original, RTFO, PAV 100ºC & Recovered Original, RTFO, PAV 100ºC & Recovered Original, RTFO, PAV 100ºC & Recovered Original Original Original Original Original Original Original Original Original Original Original Original Original Original Original Original Original Original Original Original, RTFO, PAV 100ºC & 110ºC Original, RTFO, PAV 100ºC & 110ºC Original, RTFO, PAV 100ºC & 110ºC Original, RTFO, PAV 100ºC & 110ºC Original, RTFO, PAV 100ºC & 110ºC Original, RTFO, PAV 100ºC & 110ºC Original, RTFO, PAV 100ºC & 110ºC Original, RTFO, PAV 100ºC & 110ºC
99 The master η-Gb* database is presented in Appendix-B. It should be noted that a range of nine modified binders are included in the η-Gb* database. In case of the TLA binder, Trinidad Lake Asphalt was added to plain refined asphalt cement to work as a stiffener and filler. The Elvaloy modified binders have a modifier named “Elvaloy” which creates gelation of a binder. The modifier used in the Novophalt binders produces a polymer- modified binder of plastomeric nature. It does not create internal network; rather it works as binder stiffener not producing the elastic recovery characteristics. Stylink (originally Styrelf) is a PmB (polymer modified binder) with a random styrenebutadiene SBR that is cross- linked with sulfur. In this case 4% SBR was applied. This PmB is of elastomeric nature with internal structure. The Navajo PG 76-16 binder is a SBS (styrene-butadiene-styrene) type polymer modified binder. The B-80+ binder is a penetration grade 80 asphalt binder modified with 10% GilsoniteT M. The Finnish PmB binder is a polymer- modified binder with a SBS (styrene-butadiene-styrene) type polymer. In general, it may be concluded that the binder database covers a representative group of modified asphalts as far as the type of modification is concerned, including natural asphalt, gel-type modifier, and plastomeric, elastomeric and SBS type of polymermodified binders. As far as the level of modification (amount of modifier) is concerned, it may be noted that binders in the database represent the typical modified binders, the most widely used in road pavements.
3.4
Test Methodologies Related to the Gb*-η Database The A and VTS values summarized in the master Gb*-η Database were obtained
from the regression analysis of laboratory stiffness data (i.e. loglog viscosity in
100 centipoises) versus test temperature data (i.e. log temperature in degree Rankine) following the ASTM Ai-VTSi relationship, as expressed in Equation 2.9. For the purpose of obtaining the necessary laboratory data to get the A and VTS values of the ASTM AiVTSi relationship, all or most of the following conventional and advanced asphalt binder characterization tests were carried out on each of the binder included in the database: •
Determination of penetration value at 15°C and 25°C with 100 gm load for 5 seconds using standard Penetrometer according to AASHTO T 49-03.
•
Determination of softening point using the Ring and Ball apparatus according to AASHTO T 53-96.
•
Determination of absolute viscosity at 60°C (140°F) using the capillary type vacuum viscometer according to AASHTO T 202-03.
•
Determination of kinematic viscosity at 135°C (275°F) using the Kinematic Viscometer according to AASHTO T 201-03.
•
Determination of rotationa l viscosity at 60°C, 80°C, 100°C, 121.1°C, 135°C and 176.7°C using the Brookfield Viscometer according to AASHTO T 31602.
•
Determination of the complex shear modulus (G*) and phase angle (δ b) at 15, 25, 35, 45, 60, 70, 80, 95, 105 and 115°C under the oscillatory loading rates of 1, 10 and 100 radians per second using the Dynamic Shear Rheometer (DSR) according to AASHTO T 315-02.
101 3.5
Conversion of Test Data The viscosity at ring and ball softening point temperatures corresponded to a fixed
value of 13,000 poise. The penetration values (in one tenth of millimeters) from the Penetration Test were converted to viscosity using the relationship expressed in Equation 2.10 (18). Other tests directly provided binder viscosity values. Finally, the ASTM AiVTSi relations hip (Equation 2.7) provided the pairs of A and VTS values for the binders evaluated.
4
DEVELOPMENT OF A COMPREHENSIVE η-|GB*|-δ B MODEL
4.1
Introduction As noted, the master η-Gb* database completed in this research, as shown in
Appendix-B, contains the laboratory obtained complex shear modulus and viscosity data of 39 different asphalt binders. The database has a total of 8,940 η-Gb* test data points. A minor shortcoming of this database is that the asphalt binder viscosity is reported indirectly in terms of the “A” and “VTS” parameters that were originally obtained from using the ASTM Ai-VTS i relationship (Equation 2.7). The use of the ASTM Ai-VTS i relationship does not tell anything about the change of binder viscosity under dynamic loading with changing loading frequency (or time). This phenomenon is well known to exist in the low to intermediate temperature ranges. With the adoption of the Superpave Performance Grading (PG) system and its associated testing, the data required to establish the ASTM Ai-VTS i relationship, based upon asphalt cement (AC) binder viscosity, will undoubtedly no longer be routinely collected. Under the current Superpave binder characterization methodologies, the use of the binder complex shear modulus (G b*), which is obtained from the Dynamic Shear Rheometer (DSR) test according to the AASHTO T 315-02 protocol, is considered the current (and future) salient binder stiffness parameter. In contrast to the ASTM Ai-VTS i relationship, the Gb* data describes the change of binder stiffness under dynamic (shear) loading with changing loading frequency (or time). It has the potential to be used in conjunction with a predictive model for HMA mixture dynamic modulus. One very important point to note about the loading frequency is that there is a difference in the way loading frequency is defined in asphalt binder and HMA testing.
103 For the case when oscillatory (rotational shear) tests are used to characterize the dynamic characteristics of a material, eg. asphalt binder Gb*; the loading time (t) has been commonly defined by: t=
1 ω
ω = 2πf s
with
(4.1)
(4.2)
where, ω = angular loading frequency, rad/s fs = loading frequency of a dynamic loading in “shear” mode (as used in the complex shear modulus test of asphalt binders), Hz For the case when sinusoidal loading conditions are used to characterize HMA mixtures in the lab, as well as, to characterize the actual in-situ loading time associated with stress pulses associated with a moving wheel load; the true load time is defined by: t=
1 fc
(4.3)
where, fc = loading frequency of a dynamic loading in “compression” mode (as used in the complex dynamic modulus test of AC mixtures), Hz Thus it can be observed that the frequency to be used for E* testing is related to the binder frequency in shear by: f c = 2πf s where,
(4.4)
104 fs = loading frequency of a dynamic loading in “shear” mode (as used in the complex shear modulus test of asphalt binders), Hz fc = loading frequency of a dynamic loading in “compression” mode (as used in the complex dynamic modulus test of AC mixtures), Hz The definition of loading time as shown in Equations 4.1 and 4.2 were suggested in the Shell Oil research and has been used in conjunction with the determination of stiffness modulus of asphalt binders for decades (2, 31). On the other hand, the definition of loading time as shown in Equation 4.3 is the most widely used mathematical relationship between frequency and time. This relationship is used in the determination of stiffness modulus of AC mixtures from the Complex Dynamic Modulus (E*) test (3, 29).
4.2
Problems with the Current Gb* Models It is well established that the viscosity of an asphalt binder is highly dependant on
both the temperature and the loading frequency (i.e. loading time). Van der Poel established that a penetration test represents measurement of stiffness modulus at a loading time 0.4 s where the equivalent angular frequency ω = 2.5 rad/s (2). This was confirmed independently by Gershkoff and Molenaar (42, 43). The Ring and Ball softening point, which corresponds to a viscosity value of 1300 Pas (13000 Poise), is equivalent to the complex shear modulus as obtained from the DSR test, |Gb*| = 1 kPa at ω = 1 rad/s or |G b*| = 10 kPa at ω = 10 rad/s (26, 44). However, for SBS modified binders, the equivalent values of |G b*| are lower, where |Gb*| = 0.3 kPa at ω = 1 rad/s and |Gb*| = 0.5 to 3 kPa at ω = 10 rad/s. The Brookfield rotational viscosity is usually tested at different rotational speeds (i.e. shear rates) ranging from 0.01 rpm to 25 rpm, though
105 20 rpm is the most common, which corresponds to ω = 10 rad/s with a commonly used spindle (SC-27). It is evident that when test data of all these different tests are combined to obtain the regression parameters “A” and “VTS” of the ASTM Ai-VTSi relationship (Equation 2.7), the information of the loading frequency is lost. In fact, the ASTM Ai-VTS i relationship does not contain any variable related to the loading frequency and thus it does not take care of the effect of loading frequency on the viscosity, or stiffness of asphalt binder. As a consequence, enhancement of the accuracy of this relationship can be realized by incorporating the effect of loading frequency on binder viscosity. Both the “Witczak-Bonaquist Gb* Model” and “Witczak-Sybilski Gb* Model” relating binder complex shear modulus with binder viscosity were developed based on the ASTM Ai-VTSi relationship, which itself has an inherent problem of not considering the change of binder viscosity with changing loading frequency (or time). Binder viscosity calculated from the ASTM Ai-VTSi relationship was viewed as the “observed viscosity” (termed as “VTS viscosity”), while the viscosity back-calculated from the proposed model equation, using laboratory Gb* data was viewed as the “predicted viscosity”. The so-called “observed viscosity” values remained constant over different loading frequencies, while the “predicted viscosity” varied as the laboratory obtained Gb* data varied with the same sets of loading frequencies. Finally, the model equations for the phase angle (δ b) were expressed in terms of shear modulus (G b*) and VTS viscosity, which makes it impossible to predict shear modulus (|G b*|) or phase angle solely from the “A” and “VTS” values (i.e. A-VTS viscosity) of an asphalt binder. Intuitively, one can
106 conclude that these Gb*-η models are not accurate enough, especially for further use in comprehensive mechanistic-empirical pavement response models.
4.3
Conceptual Model Development
4.3.1
Modified ASTM Ai-VTSi Relationship Development As noted earlier, the conventional ASTM Ai-VTSi equation does not consider the
effect of loading frequency on viscosity. To overcome this problem, it was hypothesized that the “A” and “VTS” values obtained from a set of conventional binder testing needs to be adjusted for loading frequency in order to use the available “A” and “VTS” values in the predictive models for |Gb*| and δ b. As such, two new frequency adjustment factors, namely “c” and “d” coefficients, for “A” and “VTS”, respectively, were introduced in the current ASTM Ai-VTS i equation. It was initially hypothesized that both variables “c” and “d” would be functions of the loading frequency (fs) in the dynamic shear mode used in the Gb* testing. Furthermore these adjustments should adjust the regression intercept “A” and slope “VTS” in such a way that the resulting viscosity obtained from the modified ASTM Ai-VTS i equation would accurately reflect the effect of loading frequency (fs ) on the binder viscosity. The modified ASTM Ai-VTSi equation was then formulated as follows: log log η fs , T = c × A + d ×VTS log TR
(4.5)
i.e. log log η fs , T = A'+VTS ' log TR
(4.6)
where, ηfs, T
= viscosity of asphalt binder as a function of both loading frequency (fs)
107 and temperature (T), cP fs
= loading frequency in dynamic shear mode as used in the Gb* testing
A
= regression intercept from the conventional ASTM Ai-VTS i equation (Equation 2.7)
VTS
= slope from the conventional ASTM Ai-VTS i equation (Equation 2.7)
c
= frequency adjustment factor for “A”, function of loading frequency (fs)
d
= frequency adjustment factor for “VTS”, function of “fs ”
TR
= temperature in Rankine scale, °R
A'
= adjusted “A” (adjusted for loading frequency)
VTS' = adjusted “VTS” (adjusted for loading frequency) In this study, a number of models were considered for the frequency adjustment factors “c” and “d”. The three most promising candidate models for the “c” parameter considered in this research were as follows: c = c0 + c1 f s
c = c0 + c1 f s + c2 f s c = c0 f s
(4.7) 2
c1
where, fs
= loading frequency in dynamic shear loading mode, Hz
c0 , c1 and c2 = fitting parameters
(4.8) (4.9)
108 Models for both “c” and “d” parameters were considered simultaneously. The three most promising candidate models for the “d” parameter were selected to be: d = d0 + d1 f s
d = d0 + d1 f s + d 2 f s d = d0 f s
(4.10) 2
d1
(4.11) (4.12)
where, fs
= loading frequency in dynamic shear loading mode, Hz
d0 , d1 and d2 = fitting parameters Eventually power models provided the best correlation and goodness of fit for both “c” and “d” parameters. Thus, the final model form used in the optimization process was:
c = c0 f s
c1
(4.13)
d = d0 f s
d1
(4.14)
and,
where, fs
= loading frequency in dynamic shear loading mode, Hz
c0 , c1 , d0 , and d1 = fitting parameters Thus, the final model form of the modified ASTM Ai-VTSi equation contains the following relationships: log log η fs , T = A'+VTS ' log TR
(4.15)
109
A' = c0 f s 1 × A c
(4.16)
VTS ' = d 0 f s 1 × VTS
(4.17)
log log η fs , T = c0 f s 1 × A + d 0 f s 1 ×VTS log TR
(4.18)
d
i.e. c
4.3.2
d
|Gb*| Model Development Depending on the type of applied load, asphalt binders are usually tested in two
different modes. The first type of testing is under near static loading conditions, when the binder is tested in a steady shear state. Penetration, Ring and Ball softening point, capillary (both absolute and kinematic), rotational viscosity (e.g. BrookfieldT M) test etc. represent this type of testing. The second type is testing under dynamic loading. The Dynamic Shear Rheometer (DSR) test represents this type of testing. For the most effective use of testing data, it is very important to possess the tool to combine the data from these two different types of testing. Historically, the most successful attempt for combining them is the Cox-Merz rule (45). The Cox-Merz rule successfully provided a relationship among the steady state viscosity (η), the complex viscosity (η*) and the complex shear modulus Gb* of fluids. According to this rule, the complex viscosity of an asphalt binder is equal to the complex modulus divided by the loading frequency. Mathematically: η* =
Gb * ω
(4.19)
110 where, η*
= complex viscosity
Gb*
= complex shear modulus
ω
= angular frequency, rad/s
Further, for Newtonian fluids where the phase angle approaches 90 degrees, the complex viscosity and the steady state viscosity (η) are equal, which can be expressed as follows: η =| η* |=
| Gb * | ω
(4.20)
Witczak et al. successfully used the Cox-Merz rule in developing the “WitczakBonaquist Gb* Model” that relate binder complex shear modulus with binder viscosity (11). The 2002 Design Guide uses this model to predict viscosity (η) of a certain asphalt binder from laboratory Gb* data. The viscosity predictive equation of this model is as follows: | G * | 1 η = b ω sin δ b
ao + a1ω +a 2ω 2
(4.21)
where, η
= binder viscosity (from ASTM Ai-VTS i equation), cP
|Gb*|
= binder shear modulus (measured), Pa
δb
= binder phase angle (measured), degree
ω
= angular frequency used to measure G* and δ, radians/s
a0 , a1 and a2
= fitting parameter for all type of binders respectively 3.639216,
111 0.131373 and -0.000901 While the “Witczak-Bonaquist Gb* Model” is known to have good performance for prediction of binder viscosity from the Gb* data, the development of an enhanced model for predicting binder Gb* from laboratory viscosity data was aimed in this Ph.D. research to overcome the shortcomings of the existing models. As noted, loading frequency has been given due importance in the development of the new enhanced Gb* predictive model. For a loading frequency of “fs ” Hz used in the dynamic shear loading, ω = 2πfs. Therefore, the relationship obtained from the Cox-Merz rule can be re-written as follows: | Gb * | = 2πf s | η* |
(4.22)
Asphalt binders do not exhibit pure Newtonian behavior over the complete temperature range of interest in pavement applications. So, the Cox-Merz rule cannot be used directly to relate the steady state viscosity (η) with the complex modulus (G b*). Instead, a correction factor for “fs ” may be introduced to account for the non-Newtonian behavior of the asphalt binder at low to intermediate temperature ranges. This “approach” was used by Bonaquist et al. in developing the relationship used in the current version of the M-E PDG. The following equation represents the general form of the relationship between viscosity and complex modulus. | G b * | = 2πη fs , T × CF
(4.23)
where, ηfs,T
= steady state viscosity as a function of loading frequency and temperature
112 (as would be obtained from laboratory obtained A-VTS viscosity data corrected by the new modified ASTM Ai-VTSi model equation form expressed by Equation 4.18) |Gb*|
= shear modulus of binder
δb
= phase angle of binder
ω
= angular frequency used to measure |G*| and δ b
CF
= correction factor as a function of loading (dynamic shear) frequency (fs)
The correction factor (CF) was found to predominantly be a function of the loading frequency (fs) and phase angle (δ b) in the original “Witczak-Bonaquist Gb* Model” (11). For the initial re-analysis conducted in this study, CF was set equal to |Gb*|/(2πη), where η is the viscosity obtained from the ASTM Ai-VTSi relationship (not adjusted for loading frequency). Initially, several model forms were investigated for the CF, which are as follows: CF = kf s
CF = kδ b
(4.24) fs
(4.25)
CF = k sin δ b
(4.26)
CF = k (sin δ b ) a0 fs
(4.27)
CF = k (sin δ b ) a1 + a2 fs +a3 fs
2
where, fs = loading frequency in dynamic shear mode, Hz
(4.28)
113 k, a0 , a1 , a2 , and a3 = fitting parameters The relationships of the CF with loading frequency (fs ), phase angle (δ b), sinδ b, f
and (sinδ b) s were evaluated. For example, Figures 4.1 through 4.4 show these relationships for the ADOT’s tank aged Paramount PG 58-22 tested at 25°F. These plots f
show that the CF is highly correlated with fs, δ b, sinδ b, and (sinδ b) s.
10 CF = -0.0009fs2 + 0.0497fs + 0.0153 R2 = 0.9989
CF
1
0.1
0.01 0
5
10 Loading Frequency, fs (Hz)
15
20
FIGURE 4.1 CF as a Function of Loading Frequency (fs) for PG 58-22 Binder at 25°F
114
0.8 CF = (3E+42)(δb )-23.45 R2 = 0.9892
CF
0.6
0.4
0.2
0 65
70 75 Phase Angle, δb (deg)
80
FIGURE 4.2 CF as a Function of Phase Angle for Paramount PG 58-22 Binder at 25°F
0.8 CF = (0.0049)(sinδb )-55.33 R2 = 0.971
CF
0.6
0.4
0.2
0 0.9
0.92
0.94
0.96
0.98
Sin δb FIGURE 4.3 CF as a Function of sinδ b for Paramount PG 58-22 Binder at 25°F
1
115
0.6 CF = -0.0873{(Sinδb)fs }2 - 0.5918(Sinδb)fs + 0.7061 R2 = 0.9959
CF
0.4
0.2
0 0.2
0.4
0.6
0.8
1
1.2
fs
(Sin δb ) f
FIGURE 4.4 CF as a Function of (sinδ b) s for Paramount PG 58-22 Binder at 25°F Based on a statistical analysis, the general form of the CF selected was: CF = k (sin δ b ) a1 + a2 fs +a3 fs
2
(4.29)
where, fs = loading frequency in dynamic shear mode, Hz k, a1 , a2 , and a3 = fitting parameters Thus the |G b*| model equation had the following general form: a + a2 fs + a3 fs2
| Gb * | = a0 f s η f s ,T (sin δ b ) 1
(4.30)
where, |Gb*|
= dynamic shear modulus, Pa
fs
= dynamic shear loading frequency to be used to predict/measure |G b*| and
116 δ b, Hz
ηfs,T
= viscosity of asphalt binder as a function of both loading frequency (fs) and temperature (T), poise
δb
= phase angle associated with Gb*, deg
a0 , a1 , a2 , and a3 = fitting parameters
4.3.3
δ b Model Development It should be recognized that the phase angle is an essential parameter to describe
the complex shear modulus of asphalt binders. Hence, a new Gb* model is incomplete without a predictive model for the phase angle (i.e. a “δ b model”). As noted, the current Witczak et al. model equations for the phase angle (δ b) are expressed in terms of |Gb*| and “A-VTS Viscosity”, which makes it impossible to predict both shear modulus (|Gb*|) and phase angle solely from the “A” and “VTS” values (i.e. A-VTS viscosity). As a consequence, it is mandatory that the new δ b model for predicting the phase angle be independent of the |Gb*| term. Historic analyses, including this study, have shown that there exists a unique relationship between the “A” and “VTS” values. Figure 4.5 shows that over the range of all “A” and “VTS” values used in this study, the “A” values are about 2.71 times the values of “VTS” plus one. The relationship has an excellent correlation coefficient (R2 ≈ 1). Hence, for further modeling, any use of the “A” parameter was replaced by the use of the “VTS” parameter using the following equation: A = −2.7058 (VTS ) + 1.0371
(4.31)
117
13 A = -2.7058(VTS) + 1.0371
12
R2 = 0.9987
A
11 10 9 8 7 -4.5
-4
-3.5
-3
-2.5
-2
VTS FIGURE 4.5 Relationship between A and VTS for All Binders
From the |Gb*| model analysis, it was recognized that the “fs x ηfs ,T ” factor is highly correlated with the Gb* data. Figure 4.6 shows a plot of log(fs x ηfs ,T ) versus phase angle (δ b) for all the binders evaluated in this study. A polynomial relationship was found to best fit the relationship between log(fs x ηfs ,T ) and δ b with R2 = 0.77. It was further observed that the phase angle (δ b) varies with VTS values as well. Figure 4.7 is an example of this trend where phase angles of the ADOT Paramount PG 58-22 binder show distinct trend lines for different VTS values at different aging conditions.
118 120 Phase Angle = -0.4324[Log(fs x ηfs,T )]2 - 0.7644[Log(fs x ηfs,T)] + 97.533, R2 = 0.8298
Observed Phase Angle, degree
100
80
60
40
20 0
2
4
6
8
10
12
14
Log (fs x ηfs,T )
FIGURE 4.6 Relationship between log(fs x ηfs ,T ) and Phase Angle (δ b) for All Binders
119
Observed Phase Angle, deg
100
80
60
Original (VTS = -3.66) RTFO (VTS = -3.61)
40
PAV100 (VTS = -3.58) PAV110 (VTS = -3.47) 20 0
2
4 log (f x ηf, T )
6
8
FIGURE 4.7 Log(fs x ηfs ,T ) versus δ b for Paramount PG 58-22 Binder at Different Ages Similar to the |Gb*| model development, the conventional Ai-VTS i relationship needed to be modified once again, now based on the given A, VTS and δ b values. The general form of the modified Ai-VTS i relationship was expressed in equation 4.18. The general form of the new δ b model selected for this study contains the following relationships:
δ b = 90 + (b1 + b2VTS ' ) × log ( f s ×η fs , T ) + ( b3 + b4VTS ' ) × log ( f s ×η fs , T ) 2 (4.32) log log η fs , T = A'+VTS ' log TR
(4.33)
A' = c 0 f s 1 × A
(4.34)
c
VTS ' = d 0 f s 1 × VTS d
where,
(4.35)
120 δb
= phase angle, deg
A'
= adjusted A (adjusted for loading frequency)
VTS' = adjusted VTS (adjusted for loading frequency) fs
= loading frequency in dynamic shear mode, Hz
ηfs,T
= viscosity of asphalt binder as a function of both loading frequency (fs) and temperature (T), poise
TR
= temperature in Rankine scale, °R
b1 , b2 , b3 , b4 , c0 , c1 , d0 , and d1 = fitting parameters
4.4
Optimization Technique for the Comprehensive η-|Gb*|-δ b Model The final database used to develop the new η-|Gb*|-δ b model contained 8940 sets
of data points. The adjusted ASTM Ai-VTS i model (Equation 4.18), the new |G b*| model (Equation 4.30) and the new phase angle (δ b) model (Equation 4.32) have a total of 12 (twelve) fitting parameters: c0 , c1 , d0 , d1 , a0 , a1 , a2 , a3 , b1 , b2 , b3 and b4 and 4 direct input parameters: A, VTS, TR and fs. All equations in the model were solved simultaneously by non- linear optimization. For this purpose, the “solver” function of Microsoft T M Excel was used. The fitting parameters were assigned the following initial values: c0 = 1, c1 = 0, d0 = 1, d1 = 0, a0 = 0, a1 = 0, a2 = 0, b1 = 0, b2 = 0, and b3 = 0. First, the A' and VTS' values were calculated using equations 4.16 and 4.17. Then the ηfs ,T values were calculated using equa tion 4.15. These VTS' and ηfs ,T values were used to predict the phase angle (δ b) value by the use of equation 4.32. Then the ηfs ,T and δ b values were substituted into Equation 4.30 to predict |Gb*| values. The observed |Gb*|
121 and δ b values (laboratory Gb* data) were then compared with their respective predicted values. The combined sums of square of errors was minimized by changing values of 12 fitting parameters using the built- in “solver” function of MS Excel. Thus, the values of the fitting parameters were obtained for the best- fit model equations.
4.5
Final Comprehensive η-|Gb*|-δ b Model
4.5.1
Final Modified Ai-VTSi Relationship (η-Model) The final modified ASTM Ai-VTS i model equations obtained from the non- linear
optimization is as follows: log log η fs , T = A'+VTS ' log TR
(4.36)
A' = 0.9699 f s
− 0. 0527
×A
(4.37)
VTS ' = 0.9668 f s
− 0.0575
× VTS
(4.38)
where, ηfs,T
= viscosity of asphalt binder as a function of both loading frequency (fs) and temperature (T), cP
fs
= loading frequency in dynamic shear mode, Hz
A
= regression intercept from the conventional ASTM Ai-VTS i equation (Equation 2.7)
VTS
= slope from the conventional ASTM Ai-VTS i equation (Equation 2.7)
A'
= adjusted “A” (adjusted for loading frequency)
VTS' = adjusted “VTS” (adjusted for loading frequency)
122 = temperature in Rankine scale, °R
TR
The final modified ASTM Ai-VTS i model (in conjunction with the |Gb*| model) is based on 8940 data points from 41 binders (including 9 modified binders). The model has excellent goodness of fit statistics. In arithmetic scale, the R2 = 0.83 and Se/Sy = 0.41; while in logarithmic scale, the R2 = 0.99 and Se/Sy = 0.12.
4.5.2
Final δ b Model The final δ b model obtained from the non- linear optimization technique as
described in the previous section consists of the following equations:
δ b = 90 + (b1 + b2VTS ' ) × log ( f s ×η fs , T ) + ( b3 + b4VTS ' ) × log ( f s ×η fs , T ) 2 (4.39) log log η fs , T = A'+VTS ' log TR
(4.40)
A' = c 0 f s 1 × A
(4.41)
VTS ' = d 0 f s 1 × VTS
(4.42)
c
d
where, δb
= phase angle, deg
A
= regression intercept from the conventional ASTM Ai-VTS i relationship (Equation 2.7)
VTS
= slope from the conventional ASTM Ai-VTS i equation (Equation 2.7)
A'
= adjusted A (adjusted for loading frequency)
VTS' = adjusted VTS (adjusted for loading frequency) fs
= loading frequency in dynamic shear, Hz
123 ηfs,T
= viscosity of asphalt binder as a function of both loading frequency (fs) and temperature (T), cP
TR
= temperature in Rankine scale, °R
b1 , b2 , b3 , b4 , c0 , c1 , d0 , and d1 = fitting parameters = -7.3146, -2.6162, 0.1124, 0.2029, 0.9699, -0.0527, 0.9668, and -0.0575, respectively The final δ b model is also based on 8940 data points from 41 binders (including 9 modified binders). Like η-|Gb*| models, the δ b model has excellent goodness of fit statistics. In arithmetic scale, the R2 = 0.81 and Se/Sy = 0.44; while in logarithmic scale, the R2 = 0.82 and Se/Sy = 0.42. At this point, the reader is reminded to be extremely cautious in using the proper value of the loading frequency (f). In the literature of physical science, loading frequency (f) in Hz is generally defined as the reciprocal of loading time (t). However, it is a common norm in the asphalt binder industry to define f as follows: t=
and
1 ω
ω = 2πf s
(4.43)
(4.44)
where, t = time of a dynamic load applied on a test specimen or a pavement section, s ω = angular loading frequency of a dynamic loading, rad/s fs = loading frequency of a dynamic loading in “shear” mode (as used in the complex shear modulus test of asphalt binders), Hz
124 On the other hand, it is a common norm in the hot mix asphalt (HMA) industry to define f as follows: t=
1 fc
(4.45)
where, t = time of a dynamic load applied on a test specimen or a pavement section, s fc = loading frequency of a dynamic loading in “compression” mode (as used in the complex dynamic modulus test of AC mixtures), Hz Therefore, in case the definitions of loading frequency follow the relationships described from Equations 4.43 through 4.45, one should use the following relationship for interchanging the frequency values: f c = 2πf s
4.5.3
(4.46)
Final |Gb*| Model The new |G b*| model finalized from the non-linear optimization is as follows: | Gb * | = 0.0051 f s η fs , T (sin δ b )
7 .1542 − 0. 4929 f s + 0. 0211 f s 2
(4.47)
where, |Gb*|
= dynamic shear modulus, Pa
fs
= dynamic shear loading frequency to be used with |Gb*| and δ b, Hz
ηfs,T
= viscosity of asphalt binder as a function of both loading frequency (fs) and temperature (T), cP
δb
= phase angle (predicted from the δ b Model), deg
125 The final |Gb*| model (in conjunction with the η-model) is based on 8940 data points from 41 binders (including 9 modified binders). The model has excellent goodness of fit statistics. In arithmetic scale, the R2 = 0.83 and Se/Sy = 0.41; while in logarithmic scale, the R2 = 0.99 and Se/Sy = 0.12.
4.6
Performance of the Final Comprehensive η-|Gb*|-δ b Model The final η-|Gb*|-δ b model consists of three sub- models: •
Revised ASTM Ai-VTSi relationship;
•
Shear modulus prediction model (|Gb*| Model); and
•
Phase angle prediction model (δ b Model).
It is noteworthy that the revised ASTM Ai-VTS i model was used in both δ b and |Gb*| models. Hence, performance of the revised ASTM Ai-VTS i model is embedded in the performance of both the |Gb*| and δ b models.
4.6.1
Performance of the Final Comprehensive η-|Gb*| Model
4.6.1.1 Accuracy of η-|Gb*| Model •
The final η-|Gb*| model statistics are as follows:
•
Total data points
•
Total number of binders = 41
•
Number of modified binders
•
Se/Sy
•
R2 = 0.99 (in logarithmic scale), 0.83 (in arithmetic scale)
= 8940
=9
= 0.12 (in logarithmic scale), 0.41 (in arithmetic scale)
126 The reader should note that the |G b*| model has an excellent correlation coefficient (R2 ≈ 0.99) and a very small Se/Sy (≈ 0.12) in logarithmic scale. It also has an excellent correlation coefficient (R2 ≈ 0.83) and a small Se/Sy (≈ 0.41) in arithmetic scale. It is well known that at very low temperature and/or at very high loading rate, the viscosity of asphalt binders reaches a maximum limit. Researchers have determined that such a threshold value is about 3x1010 poise. For the whole range of database evaluated in this study, the revised ASTM Ai-VTSi model predicted a maximum viscosity of 2.35x1010 poise, which is safely below the threshold (maximum) value of about 3.00x1010 poise. Figure 4.8 shows a plot of observed versus predicted |G b*|. Figure 4.8a is for all data points evaluated in this study. Similar plots are shown in Figures 4.8b and 4.8c for unmodified and modified binders, respectively. The reader should notice that the goodness of fit in log scale is slightly better for the unmodified binders (R2 = 0.99, Se/Sy = 0.09) compared to that of the modified binders (R2 = 0.98, Se/Sy = 0.15). This is most likely due to the fact that the modified binders used in the model development had high variability in stiffness characteristics due to their wide variety of the type and amount of modification. This variability can be graphically observed in Figure 4.8c. It is clear that the variation is practically negligible and the observed versus predicted |G b*| plots are very close to the line of equality for all three scenarios.
127
6
Predicted |Gb*|, 10 Pa
1.E+05 1.E+03
N = 8940, Binder = 41 All (Log), 0.41 (Arith) Se/Sy = 0.12
1.E+01
R = 0.99 (Log), 0.83 (Arith)
2
1.E-01 1.E-03 1.E-05 1.E-07 1.E-09 1.E-09
1.E-07
1.E-05
1.E-03
1.E-01
Observed |G b*|, 106 Pa
1.E+01
1.E+03
1.E+05
1.E+03
1.E+05
1.E+03
1.E+05
a. All Binders (New η-|G b*| Model)
Predicted |Gb*|, 106 Pa
1.E+05 1.E+03
N = 4857, Binder = 33 Se/Sy = 0.09 (Log), 0.36Arith)
1.E+01
R2 = 0.99 (Log), 0.87 (Arith)
1.E-01 1.E-03 1.E-05 1.E-07 1.E-09 1.E-09
1.E-07
1.E-05
1.E-03
1.E-01
Observed |G b*|, 106 Pa
1.E+01
b. Unmodified Binders (New η -|Gb *| Model)
6
Predicted |Gb *|, 10 Pa
1.E+05 1.E+03 1.E+01
N = 4084, Binder = 9 Se/Sy = 0.15 (Log), 0.46 (Arith) 2
R = 0.98 (Log), 0.79 (Arith)
1.E-01 1.E-03 1.E-05 1.E-07 1.E-09 1.E-09
1.E-07
1.E-05
1.E-03
1.E-01
Observed |Gb*|, 106 Pa
1.E+01
c. Modified Binders (New η-|Gb*| Model)
FIGURE 4.8 Observed versus Predicted |G b*| (Using the New η-|Gb*| Model)
128
Frequency, %
50
Mean = -0.19 x 10 6 Pa Std. Dev = 3.14 x 10 6 Pa N = 8940
40 30 20 10 0 -5
-4
-3
-2
-1
0
1
2
3
4
5
Residual |Gb*| (Predicted - Observed), 106 Pa a. Frequency Distribution of |Gb*| Residuals
Frequency, %
50
Mean = 0.00 Std. Dev = 0.22 N = 8940
40 30 20 10 0 -5
-4
-3
-2
-1
0
1
2
3
4
5
Residual Log|Gb*| (Predicted - Observed) b. Frequency Distribution of Log|Gb*| Residuals FIGURE 4.9 Distribution of Residuals for the η-|Gb*| Model Figure 4.9 shows a frequency distribution of residuals, in both arithmetic and log scales, for the new η-|Gb*| model for all the asphalt binders evaluated in this study. The frequency distributions of the |Gb*| and Log|Gb*| residuals show that there is a slight non-
129 zero error variance associated with the model, which was considered acceptable. The distribution of |Gb*| residuals also showed that the mean error was close to zero. In fact, the Log|Gb*| residuals had a mean error of 0.00. These findings allow one to conclude that the model is sufficiently unbiased relative to the calibration data. As a summary finding, it can be concluded that the new |Gb*| model is able to provide accurate predictions of the asphalt binder shear modulus.
4.6.1.2 Comparison with Previous η-|Gb*| Models As noted, the Witczak-Bonaquist η-|G b*| model (Equations 2.9 and 2.10) is used in the current version of the new “NCHRP 1-37A Mechanistic-Empirical Pavement Design Guide” (i.e. the 2002 Design Guide). The Witczak-Sybilski η-|Gb*| model (Equations 2.12 and 2.13) was developed based on a much larger database available at a later time after the Witczak-Bonaquist η-|Gb*| model was already established. It should be clearly understood that both versions of the Witczak et al. η-Gb* model were developed to: (1) predict asphalt binder viscosity (A-VTS viscosity, η) from laboratory |Gb*| and δ b data, and (2) predict phase angle from laboratory |G b*| data, given A and VTS values. On the other hand, the new η-|Gb*| model developed in this study is aimed at independently predicting both |Gb*| and δ b from given A and VTS values. However, the new η-|Gb*| model developed in this study may be visually compared with the Witczak et al. η-|Gb*| models as shown in Figures 4.10 and 4.11.
130 1.E+05 Unmodified Modified
1.E+01
6
Predicted |Gb*|, 10 Pa
1.E+03
1.E-01 1.E-03 1.E-05
N = 3245, Binder = 19, Mod Binder = 0 Se /Sy = 0.08 (Log), 1.30 (Arithmatic)
1.E-07
2
R = 0.99 (Log Scale), -0.69 (Arith) 1.E-09 1.E-09 1.E-07 1.E-05 1.E-03
1.E-01 1.E+01 1.E+03 1.E+05 6
Observed |Gb*|, 10 Pa a. Witczak-Bonaquist η -|G b*| Model 1.E+05 Unmodified Modified
1.E+01
6
Predicted |Gb*|, 10 Pa
1.E+03
1.E-01 1.E-03 1.E-05 N = 5640, Binder = 33, Mod Binder = 6 Se/Sy = 0.16 (Log), 2.07 (Arithmatic)
1.E-07
R2 = 0.97 (Log), -3.26 (Arithmatic) 1.E-09 1.E-09 1.E-07 1.E-05 1.E-03 1.E-016 1.E+01 1.E+03 1.E+05
Observed |Gb*|, 10 Pa b. Witczak-Sybilski η-|G b*| Model
1.E+05
Unmodified Modified
1.E+01
6
Predicted |Gb *|, 10 Pa
1.E+03
1.E-01 1.E-03 1.E-05 N = 8940, Binder = 41, Mod Binder = 9 Se/Sy = 0.12 (Log), 0.41 (Arithmatic)
1.E-07
2
R = 0.99 (Log Scale), 0.83 (Arith)
1.E-09 1.E-09
1.E-07
1.E-05
1.E-03
1.E-01 6 1.E+01
1.E+03
1.E+05
Observed |Gb*|, 10 Pa
c. New η-|Gb*| Model (from this study)
FIGURE 4.10 Comparison of η-|G b*| Models (Based on Original Data)
131 Figure 4.10 shows the plots of predicted versus observed |Gb*| obtained from the Witczak et al. models and the new η-|Gb*| model. Here the specific databases originally used to develop each of these models are used to construct the plots and calculate the goodness of fit statistics. It is quite clear that all three models have excellent goodness of fit statistics in logarithmic scale for their original database. The Witczak-Bonaquist and the new η-|Gb*| model has R2 ≈ 0.99 and Se/Sy ≈ 0.1, and the Witczak-Sybilski model has R2 ≈ 0.97 and Se/Sy ≈ 0.2. It should be noticed that the plots using the WitczakSybilski and new model (Figure 4.10b and 4.10c) showed a very small data scatter that originated mostly from the variability in the type and amount of modification present in the modified binders. Figure 4.11 further shows the plots of predicted versus observed |G b*| obtained from the Witczak et al. models and the new model, where the same expanded database (i.e. the master η-Gb* database) has been used to construct all three plots and calculate the goodness of fit statistics. The very small data scatter seen in Figure 4.11a through 4.11c was mainly contributed by the variability in the type and amount of modification present in the modified binders used in this study. The detailed statistics obtained from the models would, however, provide more meaningful comparison as shown in Table 4.1. As seen in Table 4.1, for the original databases used for developing the models, the Witczak-Bonaquist and Witczak-Sybilski η-|Gb*| models had excellent goodness of fit statistics in the logarithmic scale. The Witczak-Bonaquist model had R2 ≈ 0.99 and Se/Sy ≈ 0.08, while the Witczak-Sybilski model had R2 ≈ 0.97 and Se/Sy ≈ 0.16.
132 1.E+05 Unmodified Modified
1.E+01
6
Predicted |Gb*|, 10 Pa
1.E+03
1.E-01 1.E-03 1.E-05 N = 8940, Binder = 41, Mod Binder = 9 Se /Sy = 0.14 (Log), 2.00 (Arithmatic)
1.E-07
2
R = 0.98 (Log Scale), -3.01 (Arith)
1.E-09 1.E-09 1.E-07
1.E-05
1.E-03
1.E-016 1.E+01 1.E+03
1.E+05
Observed |G b *|, 10 Pa
a. Witczak-Bonaquist η-|G b*| Model 1.E+05 Unmodified Modified
1.E+01
6
Predicted |Gb*|, 10 Pa
1.E+03
1.E-01 1.E-03 1.E-05 N = 8940, Binder = 41, Mod Binder = 9 Se /S y = 0.14 (Log), 1.98 (Arithmatic)
1.E-07
2
R = 0.98 (Log Scale), -2.92 (Arith)
1.E-09 1.E-09 1.E-07
1.E-05
1.E-03
1.E-016 1.E+01 1.E+03
1.E+05
Observed |G b *|, 10 Pa
b. Witczak-Sybilski η-|G b*| Model 1.E+05
Unmodified Modified
1.E+01
6
Predicted |Gb *|, 10 Pa
1.E+03
1.E-01 1.E-03 1.E-05 N = 8940, Binder = 41, Mod Binder = 9 Se /Sy = 0.12 (Log), 0.41 (Arithmatic)
1.E-07
2
R = 0.99 (Log Scale), 0.83 (Arith)
1.E-09 1.E-09
1.E-07
1.E-05
1.E-03
1.E-016 1.E+01
1.E+03 1.E+05
Observed |G b*|, 10 Pa
c. New η -|G b*| Model (from this study)
FIGURE 4.11 Comparison of η-|G b*| Models (Based on the Expanded Database)
133 TABLE 4.1 Statistics of Predictive Models for η-|Gb*| of Asphalt Binders Predictive Models for η-|Gb*| of Asphalt Binders Witczak-Bonaquist Model Witczak-Sybilski Model New Model Original Expanded Original Expanded Expanded Data Data Data Data Data Total Binders 24 41 33 41 41 Modified Binders 4 9 6 9 9 Data Points, N 3245 8940 5640 8940 8940 Goodness of Fit in Logarithmic Scale Se/Sy 0.08 0.14 0.16 0.14 0.12 Parameter
R2
0.99
Se/Sy
1.30
R2
-0.69
0.98 0.97 0.98 Goodness of Fit in Arithmetic Scale 2.00 2.07 1.98 -3.01
-3.26
-2.92
0.99 0.41 0.83
It should be noted here that like previous binder stiffness predictive models, the Witczak-Bonaquist and Witczak-Sybilski η-|Gb*| models were optimized based on minimizing the sum of error squares obtained from the prediction of log(stiffness). As a result, these two models did not provide any realistic goodness of fit statistics when arithmetic scale was used. When the models were applied to the master (expanded) η-Gb* database, both the models still retained their excellent goodness of fit statistics in the logarithmic scale: R2 ≈ 0.98 and Se/Sy ≈ 0.14 for both models. For the same database, the new η-|Gb*| model developed in this study showed excellent and even better goodness of fit statistics in logarithmic scale: R2 = 0.99 and Se/Sy = 0.12. At the same time the model showed very good statistics in the arithmetic scale: R2 = 0.83 and Se/Sy = 0.41 for the entire database, which is much more improved compared with the previous two η-|G b*| models evaluated here.
134 4.6.2
Performance of the Phase Angle (δ b) Model
4.6.2.1 Accuracy of δ b Model The final δ b model statistics are as follows: •
Total data points
•
Total number of binders = 41
•
Number of modified binders
•
Se/Sy = 0.42 (in logarithmic scale), 0.44 (in normal scale)
•
R2 = 0.82 (in logarithmic scale), 0.81 (in normal scale)
= 8940
=9
Except for a few outlier points resulting from the modified binders, the new δ b model showed an excellent correlation coefficient (R2 = 0.82) and a small Se/Sy (= 0.42) in logarithmic scale. It is noteworthy that the new final δ b model developed in this study had also an excellent goodness of fit statistics in arithmetic scale: R2 = 0.81 and Se/Sy = 0.44. In fact, this model has almost identical goodness of fit statistics in arithmetic and logarithmic scales. For all conventional asphalt binders, the practical limit of the phase angle associated with the shear modulus varies from a small angle (at very low temperature and/or very high loading rate) to a maximum of about 90º (at high temperature and/or very small loading rate). The observed phase angle data used in the model calibration ranged from 23º to 100º. For the same conditions, the predicted phase angle values using the new δ b model ranged from 20º to 96º. This is practically almost the same range as found in the observed data. It should be pointed out that a true viscous material has a
135 phase angle equal to 90°. Values in excess of 90° may be associated with either measurement or data analysis (interpolation) errors. Figure 4.12 shows a plot of observed versus predicted phase angle (δ b). Figure 4.12a is for all data points evaluated in this study. Similar plots are shown in Figures 4.12b and 4.12c for unmodified and modified binders, respectively. The reader should notice that the goodness of fit statistics are slightly better for the unmodified binders (R2 = 0.87, Se/Sy = 0.36) compared to that of the modified binders (R2 = 0.71, Se/Sy = 0.54). This is because modified binders usually have higher variability in stiffness characteristics due to their wide variety of modification. This variability can be graphically observed in Figure 4.12c. It is clear that the variation is practically negligible and the observed versus predicted δ b plots are very close to the line of equality for all three scenarios. Figure 4.13 shows the distribution of residuals using the new δ b model for all the asphalt binders evaluated in this study. While a few outliers are present, the observed versus predicted δ b plots are quite close to the line of equality. Figure 4.13 also shows that the standard deviation is not necessarily insignificant (≈ 6.95º) and it indicates that improvements can be made in the model precision (particularly for modified binders). The mean error (≈ 0.06º) was practically zero (≈ 0) allowing one to consider the model unbiased relative to the calibration data. Therefore, it can be concluded that the δ b model is also able to provide quite accurate predictions of phase angle associated with the shear modulus of asphalt binders. However, care should be exercised when using the relationship for modified binders.
Predicted Phase Angle, deg
136 105 90 75 60 45 N = 8940, Binder = 41 Se/Sy = 0.44 (Arith) All 2 R = 0.81 (Arith)
30 15 0 0
15
30
45
60
75
90
105
Observed Phase Angle, deg
Predicted Phase Angle, deg
a. All Binders (New δb Model) 105 90 75 60 45 30
N = 4857, Binder = 33 Se/Sy = 0.37(Arith)
15
R = 0.86 (Arith)
2
0 0
15
30 45 60 75 Observed Phase Angle, deg
90
105
Predicted Phase Angle, deg
b. Unmodified Binders (New δb Model) 105 90 75 60 45 N = 4084, Binder = 9 Se/Sy = 0.54 (Arith)
30 15
2
R = 0.71 (Arith)
0 0
15
30
45 60 75 Observed Phase Angle, deg
90
105
c. Modified Binders (New δb Model)
FIGURE 4.12 Observed versus Predicted Phase Angle Using the New δ b Model
137
Frequency, %
20 Mean = 0.06 ° Std. Dev = 6.95° N = 8940
15
10
5
0 -20
-16
-12
-8
-4
0
4
8
12
16
20
Residual Phase Angle (Predicted - Measured), degree FIGURE 4.13 Distribution of δ b Residuals for the New δ b Model
4.6.2.2 Comparison with Previous δ b Models One of the main achievements made in the new δ b model developed in this study is making it independent of the |G b*| value, provided laboratory or default A and VTS values are available. As noted, the Witczak et al. Phase Angle (δ b) models need |G b*| value as input parameter for predicting the δ b values. The new δ b model developed in this study may be visually compared with the Witczak et al. δ b models as shown in Figures 4.14 and 4.15. Figure 4.14 shows the plots of predicted versus observed δ b obtained from the two Witczak et al. models and the new δ b model, where the specific databases originally used to develop each of these models are used. Unlike the η-|Gb*| models, good goodness of fit statistics of the δ b models were close in arithmetic and log scales.
138
Predicted Ph. Angle, deg ree
100
Unmodified Modified
80 60 40
N = 3245, Binder = 19, Mod Binder = 0 Se/Sy = 0.27 (Arithmetic), 0.31 (Log) R2 = 0.93 (Arithmetic), 0.91 (Log)
20 0 0
20
40
60
80
100
Observed Phase Angle, deg a. Witczak-Bonaquist Phase Angle ( δb) Model Predicted Ph. Angle, deg ree
100 Unmodified Modified
80 60 40
N = 5640, Binder = 33, Mod = 6 Se/Sy = 0.71 (Arith), 0.75 (Log) 2 R = 0.50 (Arith), 0.43 (Log)
20 0 0
20
40 60 80 Observed Phase Angle, deg b. Witczak-Sybilski Phase Angle (δb ) Model
100
Predicted Ph. Angle, deg ree
100 Unmodified Modified
80 60 40
N = 8940, Binder = 41, Mod = 9 Se/Sy = 0.44 (Arith), 0.43 (Log) 2 R = 0.81 (Arith), 0.82 (Log)
20 0 0
20
40 60 80 100 Observed Phase Angle, deg c. New Phase Angle ( δb) Model (from this study) FIGURE 4.14 Comparison of δ b Models (Based on Original Model Data)
139
Predicted Ph. Angle, deg ree
100 Unmodified
80
Modified
60 40 N = 8940, Binder = 41, Mod. Bin. = 9 Se/Sy = 0.42 (Arithmetic), 0.46 (Log) R2 = 0.83 (Arithmetic), 0.79 (Log)
20 0 0
20
40 60 80 Observed Phase Angle, deg a. Witczak-Bonaquist Phase Angle ( δb) Model
100
Predicted Ph. Angle, deg ree
100 Unmodified Modified
80 60 40
N = 8940, Binder = 41, Mod = 9 Se/Sy = 0.75 (Arith), 0.80 (Log) 2 R = 0.44 (Arith), 0.36 (Log)
20 0 0
20
40 60 80 Observed Phase Angle, deg b. Witczak-Sybilski Phase Angle (δb ) Model
100
Predicted Ph. Angle, deg ree
100 Unmodified Modified
80 60 40
N = 8940, Binder = 41, Mod = 9 Se/Sy = 0.44 (Arith), 0.43 (Log) 2 R = 0.81 (Arith), 0.82 (Log)
20 0 0
20
40 60 80 100 Observed Phase Angle, deg c. New Phase Angle ( δb) Model (from this study) FIGURE 4.15 Comparison of δ b Models (Based on the Expanded Database)
140 For the original database, the Witczak-Bonaquist δ b model had excellent goodness of fit statistics: R2 = 0.91 and Se/Sy = 0.31 in log scale, and R2 = 0.93 and Se/Sy = 0.27 in arithmetic scale. But the Witczak-Sybilski δ b model had only fair good goodness of fit statistics: R2 = 0.43 and Se/Sy = 0.75 in log scale, and R2 = 0.50 and Se/Sy = 0.71 in arithmetic scale. The new δ b model has excellent goodness of fit statistics for its entire database in both arithmetic and log scale: R2 = 0.82 and Se/Sy = 0.43 in log scale, and R2 = 0.81 and Se/Sy = 0.44 in arithmetic scale. From Figure 4.14c, it may be noticed that the new model showed a very small data scatter that originated mostly from the high variability in the type and amount of modification present in the modified binders. Figure 4.15 further shows the plots of predicted versus observed δ b obtained from the Witczak et al. models and the new model, using the expanded database (i.e. the master η-Gb* database) to construct all three plots and calculate the goodness of fit statistics. The very small data scatter seen in Figure 4.15a through 4.15c was mainly contributed by the variability in the type and amount of modification present in the modified binders used in this study. The detail statistics of various δ b models provide meaningful comparisons of the model enhancement as shown in Table 4.2. For the original database, the WitczakBonaquist δ b model had excellent good goodness of fit statistics: R2 = 0.91 and Se/Sy = 0.31 in log scale, and R2 = 0.93 and Se/Sy = 0.27 in arithmetic scale. When applied to the expanded (master) η-Gb* database, the model had inferior yet very good statistics: R2 = 0.79 and Se/Sy = 0.46 in log scale, and R2 = 0.83 and Se/Sy = 0.42 in arithmetic scale.
141 TABLE 4.2 Statistics of Predictive Models for Binder Phase Angle (δ b) Parameter
Total Binders Modified Binders Data Points, N Se/Sy
Predictive Models for Phase Angle (δ b) of Asphalt Binders Witczak-Bonaquist Model Witczak-Sybilski Model New Model Original Expanded Original Expanded Expanded Data Data Data Data Data 24 41 33 41 41 4 9 6 9 9 3245 8940 5640 8940 8940 Goodness of Fit in Logarithmic Scale 0.31 0.46 0.82 0.80 0.42
R2
0.91
Se/Sy
0.27
R2
0.93
0.79 0.33 0.36 Goodness of Fit in Arithmetic Scale 0.42 0.77 0.75 0.83
0.42
0.44
0.82 0.44 0.81
For the original database, the Witczak-Sybilski δ b model had only fair statistics: R2 = 0.43 and Se/Sy = 0.75 in log scale, and R2 = 0.50 and Se/Sy = 0.71 in arithmetic scale. When applied to the expanded database, the model further lost its accuracy and had poor goodness of fit statistics: R2 = 0.36 and Se/Sy = 0.80 in log scale, and R2 = 0.44 and Se/Sy = 0.75 in arithmetic scale. On the other hand, the new δ b model has excellent goodness of fit statistics for the entire expanded database in both arithmetic and log scale: R2 = 0.82 and Se/Sy = 0.42 in log scale, and R2 = 0.81 and Se/Sy = 0.44 in arithmetic scale.
5
ASPHALT MIXTURE E* STIFFNESS DATABASE
5.1
Introduction Similar to the master binder Gb*-η database that was developed for this study; a
comprehensive database of dynamic modulus (E*) stiffness of HMA mixtures was synthesized so that accurate E* models could be developed. In this database, the stiffness of the asphalt binder was initially reported indirectly in terms of the “A” and “VTS” parameters that were originally obtained from converting the conventional binder testing data by the use of the ASTM Ai-VTSi relationship. However, as one of the important goals of this research was to insure that any new E* predictive model use the complex shear (Gb*) parameters of the asphalt binder instead of the viscosity parameters. As such, all sets of “A” and “VTS” parameters of the asphalt binders, as reported in the original E* database, were converted to shear modulus |G b*| and phase angle (δ b) values by utilizing the newly developed η-|Gb*|-δ b predictive models from this study (described in Chapter 4 of this report).
5.2
Original E* Database The Witczak-Fonseca E* Predictive Model was developed and calibrated at the
University of Maryland (UMD) based on 1430 test data points from 149 un-aged laboratory blended and HMA mixtures that contained only conventional non- modified binders. All test samples had a cylindrical size of 4″ diameter x 8″ height and were compacted by kneading compaction. As a result of several research projects and studies performed at UMD through 1999, Witczak and his colleagues at the UMD further expanded the E* database. The
143 current version of the Witczak E* Predictive Equation is based on this database (known as the “UMD E* Database”) that contains 2750 test data points from 205 HMA mixtures. Like the initial work, all the new 56 mixtures were un-aged laboratory blended mixes. The new E* test samples, however, had a cylindrical size of 2.75″ diameter x 5.5″ height and were compacted by gyratory compaction.
5.3
Expanded E* Database Since 2000, this researcher, along with others, has been conducting intensive
dynamic modulus testing under the direction of Dr. Witczak at Arizona State University. They have recently completed E* testing on an additional 176 HMA mixtures that provided 5820 more E* test data points (8). This testing was done under 12 tasks of 6 major projects. The projects are as follows: •
NCHRP 9-19 Project: Tasks C4 and C5.
•
ASU-ADOT Project No. 3.
•
ASU-ADOT AR Project: Task-1 (I-40) and Task-2 (I-17).
•
Alberta AR Project.
•
NCHRP 9-23 Project: (a) Calibration of PP2 Protocol, (b) Validation of PP2 Protocol, (c) Verification of PP2 Protocol, (d) Compaction versus noncompaction study, and (e) Aging study.
•
ASU-NLA Project: E* Database of Lime Modified Asphaltic Mixtures.
The new E* database, known as the “ASU E* Database”, is furthermore expanding. The ASU E* test samples had a cylindrical size of 4″ diameter x 6″ height.
144 They were compacted by gyratory compaction. Unlike the UMD database, the ASU E* test samples included a wide variety of sho rt-term oven aged lab blend mix samples, plant mix samples and field cores. The size of the ASU E* database is now more than double the UMD E* database which served as the basis for the current Witczak E* predictive model. Table 5.1 summarizes the different HMA mixtures included in the ASU E* database. Both the UMD and ASU E* database have been combined into one expanded E* database in this research. The expanded database contains an excellent blend of different aggregate gradations, binder type (conventional, polymer modified and rubber modified), mix type (conventional un- modified and lime or rubber modified) and aging (no aging, short-term oven aging, plant aging and field aging) as earlier presented in Table 1.1 of Chapter 1.
5.4
Revised E* Database After a careful examination of the expanded E* database, it was concluded that
some of the data could not be used for the model development and calibration because either their values were felt to be erroneous or some vital mix, aggregate or binder data were missing. Most data obtained from the NCHRP 9-23 mixtures could not be used due to missing mix volumetric data. The final revised database contains 7400 data points from 346 HMA mixtures. This database is termed as the “Master E* Database” and presented in Appendix- A (A.1 through A.3).
145 TABLE 5.1 Summary of Mixtures in ASU E* Database Project
Task
Site ID
NCHRP 9-19 NCHRP 9-19
Task C4 Task C5
ADOT SRB MnRoad NCAT Indiana Nevada I-80 WesTrack FHWA-ALF Arizona I-10
ASU-ADOT
Project #3
Salt River Base Salt River 3/4" Bidahouchi Base Bidahouchi 3/4" ASU-ADOT AR Task-1 Arizona I-40 (I-40) Arizona I-40 ASU-ADOT AR Task-2 Arizona I-17 (I-17) Arizona I-17 Alberta AR Task-1 Alberta Alberta NCHRP 9-23 Calibration MnRoad WesTrack ADOT Validation 11 LTPP Sites Verification Maryland Comp. WesTrack Aging WesTrack ADOT MnRoad ASU-NLA E* Database Two Guns MD DOT Bidahouchi Base Salt River Base WesTrack ALF
Mix Total STOA Plant Field Data Type Mix LB1 Mix Core Points Conv.2 16 16 0 0 480 Conv. 12 5 7 0 360 Conv. 11 0 11 0 330 Conv. 12 0 12 0 360 Conv. 8 0 8 0 240 Conv. 32 10 22 0 960 Conv. 14 9 0 5 420 Poly 3 4 2 0 2 120 Conv. 5 0 5 0 150 AR-A C4 1 0 1 0 30 Conv. 3 3 0 0 90 Conv. 3 3 0 0 90 Conv. 2 2 0 0 60 Conv. 2 2 0 0 60 AR-Gap 1 0 1 0 30 AR-Open 1 0 1 0 30 AR-Gap 3 0 3 0 90 AR-Open 3 0 3 0 90 AR-A C 1 0 1 0 30 Conv. 1 0 1 0 30 Conv. 3 0 0 3 90 Conv. 3 0 0 3 90 Conv. 3 0 0 3 90 Conv. 11 0 0 11 330 Conv. 10 10 0 0 300 Conv. 1 0 1 0 36 Conv. 3 0 3 0 108 Conv. 3 0 3 0 108 Conv. 3 0 3 0 108 Lime 5 3 3 0 0 90 Conv. 1 1 0 0 30 Lime 2 2 0 0 60 Conv. 1 1 0 0 30 Lime 2 2 0 0 60 Conv. 1 1 0 0 30 Lime 2 2 0 0 60 Conv. 1 1 0 0 30 Lime 1 1 0 0 30 Conv. 1 1 0 0 30 Lime 1 1 0 0 30 Conv. 1 1 0 0 30 Table is continued on the next page
146 TABLE 5.1 Summary of Mixtures (Continued) Notes: 1. STOA LB = Short-term oven aged lab blend 2. Conv. = un-modified mix and binder 3. Poly = binder is polymer modified 4. AR = Rubber modified asphalt in mix 5. Lime = Lime modified mix
5.5
Mix Type All Conv. AR Lime Poly
Total STOA Plant Field Data Mix LB1 Mix Core Points 192 79 86 27 5820 167 66 76 25 5070 10 0 10 0 300 11 11 0 0 330 4 2 0 2 120
Test Methodologies Related to the E* Database The Witczak-Fonseca E* database developed at the University of Maryland
(UMD) had 1430 test data points from 149 un-aged laboratory blended and HMA mixtures that contained only conventional non- modified binders. All test samples had a cylindrical size of 4″ diameter x 8″ height and were compacted by kneading compaction. ASTM D3497 protocol was followed for these tests. The later part of the UMD E* database contains 1320 data points from 56 un-aged laboratory blended mixes. The related E* test samples, however, had a cylindrical size of 2.75″ diameter x 5.5″ height and were compacted by gyratory compaction. At ASU, the NCHRP 1-37A Test Method DM-1 titled “Standard Test Method for Dynamic Modulus of Asphalt Concrete Mixtures” and subsequent AASHTO standard method of test for “Determining Dynamic Modulus of Hot-Mix Asphalt Concrete Mixtures” with AASHTO designation TP 62-03 were followed for the laboratory E* test sample preparation and testing (46, 29). The test mixtures contained laboratory blended mixture, plant mixture and field cores. The laboratory blended mixtures were short-term oven aged according to the E* test protocol followed. All final test specimens were cored and sawed to 4-inch diameter and 6 inch height. Table 5.2 presents the E* test conditions.
147 TABLE 5.2 Test Conditions of the ASU E* Tests Test Temp, T Freq, fc
Cycles
Rest Period
Cycles to
(Sec)
Compute E*
Strain Range
(°F)
(Hz)
14, 40, 70,
25
200
-
196 to 200
25 to 150
100, 130
10
100
60
196 to 200
micro-strain
(Generally)
5
50
60
96 to 100
1
20
60
16 to 20
0.5
15
60
11 to 15
0.1
15
60
11 to 15
As noted before, there is a difference in the way loading frequency is defined in asphalt binder testing and HMA testing. The loading time (t), being the same in both cases, is defined as follows: t=
and
ω = 2πf s t=
i.e.,
1 ω
1 fc
f c = 2πf s
(5.1) (5.2) (5.3)
(5.4)
where, t = time of a dynamic load exerted on a test specimen or a pavement section, s ω = angular loading frequency, rad/s fs = loading frequency of a dynamic loading in “shear” mode (as used in the
148 Gb* test of asphalt binders), Hz fc = loading frequency of a dynamic loading in “compression” mode (as used in the E* test of HMA mixtures), Hz The definition of loading time as shown in Equations 5.1 and 5.2 were suggested in the Shell Oil research and has been used in conjunction with the determination of stiffness modulus of asphalt binders for decades (2, 31). On the other hand, the definition of loading time as shown in Equation 5.3 is the most widely used mathematical relationship between frequency and time. This relationship is used in the determination of stiffness modulus of AC mixtures from the Complex Dynamic Modulus (E*) test (3, 29). It is this definition of fc = 1/t that has been used by the Asphalt Institute and Dr. Witczak since the first collection of E* test results that were placed in the database. It is the fc frequency that governs the E* behavior response of mixtures in the lab as well as in the field under dynamic loading. Lab and field response of mixtures is not governed by the fs (shear frequency). As an example, Figure 5.1 shows the output plots of applied vertical compressive dynamic stress (σc) versus loading time (t) based on the actual laboratory E* test data obtained from the E* testing of an ADOT Salt River Base Mix specimen with 4.55% asphalt content and 4% air voids tested at 70°F. Figure 5.1a is the sinusoidal σc versus t plot for a loading frequency (fc) of 10 Hz. It can be observed each cycle length is equal to 0.1 second (i.e. t = 1/10 = 0.1 second). Similarly, Figure 5.1b is a σc versus t plot for fc = 1 Hz, where t = 1/1 = 1 second and Figure 5.1c is a σc versus t plot for fc = 0.1 Hz, where t = 1/0.1 = 10 seconds. From each of these three plots, it is very clear that the dynamic modulus (E*) testing following the current test protocols (as done at ASU and UMD)
Vertical Stress, kPa
149 300 250 200 150 100 50 0 19.3
19.4
19.5
19.6
19.7 Time, s
19.8
19.9
20
20.1
20
21
150
160
Vertical Stress, kPa
a. f c = 10 Hz (loading time = 0.1 second) 300 250 200 150 100 50 0 13
14
15
16
17 Time, s
18
19
Vertical Stress, kPa
b. fc = 1 Hz (loading time = 1 second) 300 250 200 150 100 50 0 80
90
100
110
120 Time, s
130
140
c. fc = 0.1 Hz (loading time = 10 seconds)
FIGURE 5.1 Vertical Stress versus Loading Time in Actual E* Te st of an ADOT Mix
150 gives the E* data with loading frequency (fc) data that follows the mathematical relationships expressed in Equations 5.3 and 5.4. To avoid any confusion, the master E* database, presented in Appendix A (A.1 through A.3), contains both fc and fs values in the unit of hertz (Hz). For simplicity, however, the loading frequency of a dynamic loading in compression mode is expressed as “f” as well as “fc ” in the remaining part of this dissertation.
5.6
Master Curve Parameters The principles of time-temperature superposition using a sigmoidal model
function have been discussed in section 2.4.3. It is important to recognize that the process used by Witczak and his colleagues at ASU is to simultaneously optimize the seven (7) master curve fitting parameters e.g. δ, α, β, γ, a, b and c (previously expressed in equations 2.17 through 2.19) in a single optimization process. If the user performs separate and independent optimizations on both the E* master curve and the timetemperature superposition relationship; different (and probably incorrect) fitting parameters will be obtained. In this study, individual sets of these seven master curve parameters were obtained for each mix evaluated. The parameter values are summarized in Appendix-C.
6
VARIABLES OF E* MODEL
6.1
List of |E*| Models A large number of E* predictive models/techniques have been developed over the
years and the important ones have already been discussed briefly in Chapter 2. Table 6.1 summarizes the important models reviewed, arranged in a chronological order of development. The numbers of mixtures and total data points used in the model development are also shown in this table. It is interesting to note that that model no. 11, the Witczak , Andrei and Mirza’s Revised Model, was based on the largest database (205 mixtures and 2750 data points). This model has already been incorporated in the 2002 Design Guide. TABLE 6.1 List of E* Models Model Model Basic Database No. Mix, Nm Data, Nt 1 Van der Poel (Shell Oil’s Early Version) Model, 1954 NR NR 2 Bonnaure (Shell Oil’s Later Version) Model, 1977 9 93 3 Shook and Kallas’ Models, 1969 29 87 4 Witczak’s Early Model, 1972 29 87 5 Witczak and Shook’s Model, 1978 41 369 6 Witczak’s 1981 Model, 1981 41 369 7 Witczak, Miller and Uzan’s Model, 1983 131 1179 8 Witczak and Akhter’s Models, 1984 131 1179 9 Witczak, Leahy, Caves and Uzan’s Models, 1989 149 1429 10 Witczak and Fonseca’s Model, 1995 149 1429 11 Witczak, Andrei and Mirza’s Revised Model, 1999 205 2750 12 Hirsch Model of Christensen, Pellinen and Bonaquist, 2003 18 206
6.2
Variables in E* Models The literature study also shows that several different variables, related to the
asphalt binder and mix stiffness, have been used in different combinations and in different ways in these models. These variables are briefly summarized in Table 6.2.
152 TABLE 6.2 Variables in Historic E* Models Variable fc fb T T800 PI η η70,106 %Va %AC %AC opt %Vbeff %Vbeffopt Cv Vg Vb VMA VFA ρ200 ρ4 ρ38 ρ34 ρabs Gb* Pc
6.3
Variable Description
Model No. (Variables Checked) 1 2 3 4 5 6 7 8 9 10 11 12 Mixture loading frequency X X X X X X X X X Binder loading frequency X X X Temperature (mix and binder) X X X X X X X X X T at Penetration = 800 X X Binder Penetration Index X X Binder viscosity X X X X o 6 η at 70 F in 10 poise X X X X X X Mix air voids X X X X X X X Asphalt content (by weight) X X X X Optimum %AC X Effective asphalt content (by volume) X X X X X Optimum %V beff X Volume concentration X X Volume of aggregate X Volume of binder X % voids in the mineral aggregates X % of VMA filled with binder X % aggregates passing #200 sieve X X X X X X % aggregates retained on #4 sieve X X X X % aggregates retained on #3/8" sieve X X X X % aggregates retained on #3/4" sieve X X X X % asphalt absorption X X Binder (complex) shear modulus X Contact volume X
Important Predictor Variables The variables used in the models were short- listed to primarily select a set of
predictor variables for HMA mix |E*|. To do so, one should realize that some of the variables can be interchangeably used in developing a new model. For example, the volume concentration Cv , defined as Vg /(Vg + Vb), has been used in a model instead of using Vg and Vb. Another example is using η instead of η70, 10 6 . When one uses both Va and VFA, the effective binder volume (V beff), bulk specific gravities of mix (Gmb) and
153 aggregates (Gsb), maximum theoretical specific gravity (Gmm), binder specific gravity (Gb ) and volume concentration (C v ) are indirectly taken care of. It has already been pointed out that the data required to establish the ASTM AiVTSi relationship may no longer be routinely collected because of the adoption of the Superpave Perfo rmance Grading system and its associated testing. One of the major known limitations associated with the use of the ASTM Ai-VTS i relationship is that it does not incorporate changes of binder viscosity under dynamic loading with changing loading time (or frequency). This is in contrast to the binder complex shear modulus (Gb*) that describes the change of binder stiffness under dynamic (shear) loading with changing loading time (or frequency). The development of a new model for accurately predicting Gb* data from ASTM Ai-VTSi binder viscosity data has already been presented in the previous chapters. Now, the variables |Gb*| and δ b can easily be incorporated in the |E*| predictive model as direct input parameters that can more effectively take care of the binder rheology with changing temperature and loading rate. Thus taking into account of these direct and indirect inter-relationships, the important predictor variables for mix E* were initially selected to be (not in order of importance) as follows: •
Aggrega te gradation (ρ200 , ρ4 , ρ38 and ρ34 )
•
Air voids (Va)
•
Effective binder volume (V beff)
•
Voids in mineral aggregates (VMA)
•
Voids filled with asphalt (VFA)
154
6.4
•
Binder shear modulus (|Gb*|) at specific temperature and loading rate
•
Binder phase angle (δ b) associated with the |G b*|.
Sigmoidal Model Form In the 2002 Design Guide, the stiffness of any HMA mix, at all analysis levels of
temperature and time rates of load, is determined from a sigmoidal shaped master curve constructed at a reference temperature (usually 70ºF). For calculating mix |E*| stiffness, Levels 2 and 3 use the Witczak E* Predictive Model, which itself is mathematically expressed with the equation of a sigmoidal curve. This functional form was found best fitted to eliminate the unrealistic estimates of mix |E*| stiffness at extreme values of temperatures, aging and loading rate (39). At the beginning of this research, it was decided that the sigmoidal form would be kept in any new |E*| predictive model. This would allow the new model to be incorporated into a revised version of the design guide with little to no difficulty. The basic symmetrical sigmoidal function consistent with the limitations of extreme values can be written as follows: y=δ +
α β −γx 1+ e
where, y
= the criteria variable
δ
= a location parameter for y (the minimum value for y)
α
= the range of possible values to be added to δ
x
= the predictor variable under consideration
β/γ
= a location parameter for x corresponding to a value of: y = δ + α/2
(6.1)
155 Figure 6.1 is an example of a sigmoidal function, which is actually the |E*| master curve of the Two-Guns mix constructed from ASU laboratory E* data. Here, |E*| (in 105 psi) = y and log(1/tr) = x in equation 6.1. The concept of reduced time (tr) has been discussed earlier in section 2.4.3 of Chapter 2. From the sigmoidal curve in Figure 6.1, one can obtain the minimum value for |E*| ≈ 9 ksi (= δ) and the maximum value for |E*| ≈ 7,000 ksi (= δ + α). 100
|E*|, 10^5 psi
10
1
0.1
0.01 -15
-10
-5
0 5 Log Reduced Time (tr), s
10
15
20
FIGURE 6.1 |E*| Master Curve of a Sigmoidal Form for Two-Guns Mix
6.5
Characteristics of Predictor Variables It has already been noted that the probable predictor variables for mix |E*|
stiffness were: ρ200 , ρ4 , ρ38 , ρ34 , Va, Vbeff, VMA, VFA, |Gb*|, and δ b. As discussed earlier, at very cold temperatures and/or very small loading times, HMA mix behavior largely
156 reflects the binder rheology, which may be characterized by the viscosity and/or complex shear modulus at the temperature and loading frequency of interest. On the other hand, at very hot temperatures and/or very long loading times, the HMA mix behaves more like a granular non- linear elastic material. In other words, in such conditions, mix behavior can largely be approximated by the smaller gradation (such as ρ200 , ρ4 etc.) and voids characteristics (such as Va, VMA, Vbeff, VFA etc.) of mix/aggregates. At intermediate temperatures, HMA mix rheology is sensitive to the unique binder properties (|Gb*|, δ b, binder viscosity etc.) and voids and larger gradation properties of aggregates (Va, Vbeff, VMA, VFA, ρ38 , ρ34 etc.). However, the noticeable differences observed in binders greatly diminish in mixtures.
6.6
Correlation of |E*| with Predictor Variables At the outset of this research, it was decided that the prospective E* model would
continue to be based on a mathematical sigmoidal function form. As a result, the new E* model should consist of three main parts: δ (the minimum value of Log|E*|), α (the maximum value of that can be added to δ i.e. δ + α = maximum Log|E*|), and the exponent (= β - γx) as described in equation 6.1. The log|E*| value of any HMA mix at very hot temperatures and/or very long loading times usually resembles the δ part of the full sigmoidal master curve of that mix. On the other hand, log|E*| value at very low temperatures and/or very short loading times usually resembles the δ + α part of the sigmoidal curve. The exponent part of the master sigmoidal curve is usually governed by
157 the binder stiffness characteristics (e.g. Gb*) that are functions of both the temperature and loading time. The final master E* database contains laboratory E* test data conducted at temperatures ranging from 0 to 130°F. At this range of temperatures, the correlation of laboratory |E*| value to the predictor variables were evaluated at 1 Hz loading frequency (fc) in compressive dynamic loading mode. Figures 6.2, 6.3 and 6.4 show the plots of the measured Log |E*| versus gradation parameters (ρ200 , ρ4 , ρ38 and ρ34 ), mixture volumetric parameters (Va, Vbeff, VMA and VFA) and binder stiffness parameters (|Gb*| and δ b), respectively. Each plot contains the best- fitted trend lines at three sets of temperature and loading frequency (T = 14, 70 and 130°F each at fc = 0.1, 1 and 10 Hz). It is important to understand that the best fitting trend line may not be indicative of the true correlation of the specific predictor variable with E* because the comparison was not done by fixing all other related variables constant. However, the plots give an indication of the type and trend of the existing correlations. To further evaluate the extent of correlations of the observed E* with all the predictor variables utilized in the study; the ratio of all individual variables’ R2 value with respect to the highest R2 (≈ 0.38) obtained from all the plots shown in Figures 6.2 through 6.4, were calculated. The full range of R2 ratio (0 to 1) was then subdivided and coded for the extent of correlation (none, low, medium, high and very high). Table 6.3 summarizes the coded general correlation of Log |E*| to all of the predictor variables utilized in the study, based on the R2 ratio values thus calculated.
158 100 fc = 10 Hz fc = 1 Hz
100
fc = 10 Hz
T = 14 oF
fc = 0.1 Hz
.
.
fc = 0.1 Hz
fc = 10 Hz
T = 70o F
Observed |E*|, 10 psi
fc = 1 Hz
fc = 0.1 Hz
fc = 10 Hz
1
fc = 10 Hz
10
T = 130o F
fc = 1 Hz fc = 0.1 Hz
1
T = 130o F
fc = 10 Hz
fc = 1 Hz
fc = 1 Hz fc = 0.1 Hz
fc = 0.1 Hz
0.1
0.1 0
3
6
9
0
12
20
40
a. ρ200 100 T = 14o F
T = 14 F fc = 1 Hz
T = 70oF
.
fc = 0.1 Hz fc = 10 Hz
T = 70o F
10
fc = 10 Hz
10
fc = 1 Hz
5
fc = 1 Hz fc = 0.1 Hz
fc = 10 Hz
1
fc = 10 Hz fc = 1 Hz fc = 0.1 Hz
o
Observed |E*|, 10 psi
fc = 10 Hz
.
80
b. ρ 4
100
5
60
ρ4, %
ρ200, %
Observed |E*|, 10 psi
T = 70 oF
5
10
5
Observed |E*|, 10 psi
T = 14 oF
fc = 1 Hz
o
T = 130 F
fc = 0.1 Hz
T = 130o F
fc = 10 Hz
1 fc = 1 Hz
fc = 1 Hz
fc = 0.1 Hz fc = 0.1 Hz
0.1
0.1 0
15
30 ρ38, %
c. ρ38
45
60
0
5
10
15
20
25
30
ρ34, %
d. ρ34
FIGURE 6.2 Relationship of |E*| with Gradation Parameters (ρ200 , ρ4 , ρ38 and ρ34 )
159 100
100 o
T = 14 F
fc = 10 Hz
T = 14oF
fc = 10 Hz fc = 1 Hz
fc = 1 Hz
fc = 0.1 Hz
.
T = 70oF
fc = 10 Hz fc = 1 Hz
10
T = 70o F
5
Observed |E*|, 10 psi
fc = 10 Hz
10
5
Observed |E*|, 10 psi .
fc = 0.1 Hz
fc = 1 Hz fc = 0.1 Hz
T = 130oF fc = 10 Hz
1
fc = 0.1 Hz
fc = 10 Hz
T = 130oF
1
fc = 1 Hz
fc = 1 Hz fc = 0.1 Hz
fc = 0.1 Hz
0.1
0.1 0
5
10
15
20
0
5
10
15
20
Va, %
Vbeff, %
a. Va
b. Vbeff
100 T = 14o F
fc = 10 Hz fc = 1 Hz
T = 14o F
fc = 1 Hz
fc = 0.1 Hz fc = 10 Hz
. fc = 1 Hz fc = 0.1 Hz
T = 70o F fc = 10 Hz
fc = 10 Hz
1
T = 70o F
10
fc = 1 Hz
5
10
Observed |E*|, 10 psi
.
fc = 0.1 Hz
5
30
100 fc = 10 Hz
Observed |E*|, 10 psi
25
T = 130o F
fc = 1 Hz
fc = 0.1 Hz
fc = 10 Hz
1
o
T = 130 F
fc = 1 Hz
fc = 0.1 Hz
fc = 0.1 Hz
0.1
0.1 0
10
20
30
40
30
40
50
60
70
VMA, %
VFA, %
c. VMA
d. VFA
80
90 100
FIGURE 6.3 Relationship of |E*| with Volumetric Parameters (Va, Vbeff, VMA, VFA)
160
5
Observed |E*|, 10 psi .
100
T = 14 o F fc = 10 Hz fc = 1 Hz fc = 10 Hz fc = 1 Hz
10
T = 70o F 1
fc = 0.1 Hz fc = 0.1 Hz
fc = 10 Hz fc = 1 Hz fc = 0.1 Hz
T = 130oF
0.1 0.01
0.1
1
10 |Gb*|, psi
100
1000
10000
a. |Gb|* 100
5
Observed |E*|, 10 psi .
fc = 10 Hz
o
fc = 1 Hz f = 0.1 Hz c
fc = 10 Hz
T = 14 F 10
fc = 1 Hz
T = 70o F
fc = 0.1 Hz
fc = 10 Hz fc = 1 Hz 1
T = 130o F fc = 0.1 Hz
0.1 0
10
20
30
40 50 δb, deg
60
70
80
90
b. δb FIGURE 6.4 Relationship of |E*| with Binder Stiffness Parameters (|Gb*| and δ b)
161 TABLE 6.3 Correlation of E* with Predictor Variables at Different Temperature Zones Variable
Low Temp.
Inter. Temp.
High Temp.
R2 Ratio Code R2 Ratio Code R2 Ratio Code
Note: Correlation R2 Ratio Code
ρ200
0.08
0.00
0.07
None
0.00-0.05
ρ4
0.08
0.39
0.09
Low
0.06-0.15
ρ38
0.03
0.19
0.09
Medium
0.16-0.35
ρ34 Va
0.20 0.06
0.01 0.21
0.00 0.01
Vbeff VMA VFA
0.09 0.16 0.10
0.41 0.57 0.02
0.17 0.07 0.02
Log |Gb*|
0.16
1.00
0.40
Log δ b
0.06
0.72
0.33
High 0.36-0.60 Very High 0.61-1.00
From Figures 6.2 through 6.4 and Table 6.3, the following initial findings were summarized: •
The δ part of the sigmoidal function may be modeled with a partial or full combination of ρ200 , ρ4 , ρ38 , Va, Vbeff, VMA, Gb* and δ b.
•
The α part of the sigmoidal function may be modeled with a partial or full combination of ρ4 , ρ38 , ρ34 , Va, Vbeff, VMA and VFA.
•
The exponent part of the sigmoidal function that controls the change of mix stiffness with time of loading and/or temperature may be modeled with a partial or full combination of |Gb*|, δ b, temperature and loading frequency.
7
NEW E* MODEL DEVELOPMENT
7.1
Introduction The problems associated with the current E* dynamic modulus stiffness models
have been discussed earlier. The greatest possible percentage of the laboratory dynamic modulus database was used to develop and calibrate the new E* model. After a critical investigation of the quality of E* database; 7400 sets of E* data points from 346 asphalt mixtures were eventually selected to use in the final modeling. As previously noted a potential problem was initially apparent in the database in that it did not contain Gb* data (|Gb*| and δ b) of the asphalt binders. Instead, A and VTS viscosity values of the binders had been reported. Hence, to fulfill the objectives of this research, it was critical that a model be developed to convert the A and VTS values of the binders to Gb* data (|G b*| and δ b) in order to eventually allow laboratory Gb* data as a direct input in the new E* model. A comprehensive new η-|Gb*|-δ b model that was successfully able to complete this conversion was developed in this research and previously described in this report. As a consequence, the final E* database was completed by incorporating Gb* data (|Gb*| and δ b) in the database predicted by the AiVTSi viscosity relationships.
7.2
Basic Hypotheses Any empirical model is based on some basic hypotheses. The new E* model is
also based on several basic hypotheses. Without making these assumptions, the model development would have been impossible. The basic hypotheses utilized were as follows:
163 •
The E* complex moduli master curve of asphalt mixtures can be modeled by a mathematical sigmoidal function.
•
Currently available Dynamic Shear Rheometers (DSR) can accurately measure the complex (shear) modulus (G b*) parameters, which comprises of shear modulus (|Gb*|) and phase angle (δ b) of the asphalt binder.
•
The new η-|Gb*|-δ b model can accurately predict |Gb*| and δ b of the asphalt binder from viscosity and the load frequency in oscillatory shear (fs).
•
The angular loading frequency (ω) and loading frequency (fs ) in dynamic shear mode as used in the Gb* test (in a DSR test), the loading frequency (fc) in dynamic compression mode as used in the E* test, and the loading time (t) as used in both Gb* and E* test are related as follows: ω = 2πfs = 1/t, fc = 1/t, and as a consequence fc = 2π fs, where ω, fs, fc and t are expressed with the units of radian/second, hertz (cycles per second), hertz and seconds, respectively.
7.3
Modeling Approach The modeling approach used to establish the new E* model is outlined in the
following paragraphs.
7.3.1
Sigmoidal Form A sigmoidal model form, as expressed by Equation 6.1, has been used for the |E*|
model. This means the model has four distinct parameters; δ, α and the exponent
164 containing the β and γ coefficients. The dependant variable “y” is the logarithm of dynamic modulus in the units of pounds per square inch (psi) i.e. Log|E*| (|E*| in psi). In the sigmoidal function, δ is the asymptotic minimum value of Log|E*|, δ+α is the asymptotic maximum value of Log|E*|, and the exponent (β - γx) in the term 1+e(β - γx) controls the rate of change in E* stiffness with temperature and loading frequency (called reduced time).
7.3.2
Sub-Models Based upon some initial feasibility studies, it appeared that the δ part of the
sigmoidal functio n could be modeled with a partial or full combination of ρ200 , ρ4 , ρ38 , Va, Vbeff, VMA, VFA, |G b*| and δ b. This part of the E* model was termed the δ-model. The α part of the sigmoidal function was modeled with a partial or full combination of ρ4 , ρ38 , ρ34 , Va, Vbeff and VFA. This part of the E* model was termed the α-model. The exponential part of the sigmoidal function was modeled with a partial or full combination of |Gb*|, δ b, temperature and loading frequency. This part of the E* model was termed the exponent model.
7.3.3
Model Optimization Technique Maximum use of available statistical tools was made to determine the goodness of
fit, residual analysis, sensitivity etc. of any model that was investigated. A non- linear optimization technique was used as a mathematical tool for combining the contributions of the predictor variables in the model and thus optimizing the model. For this purpose,
165 the “solver” function of MicrosoftT M Excel was used. As mentioned in Chapter 2, solver is based on the Generalized Reduced Gradient (GRG2) algorithm for optimizing nonlinear problems. Generally, the GRG2 technique can find solutions with very small tolerance of the convergence criteria. However, its solution is highly dependent on the initial starting values or “seed values” of the fitting parameters of the model. To avoid a non-convergent solution, proper caution was taken to finding out the seed values. The observed values were compared with the predicted values. For each set of data, the difference between the predicted and observed value gave the error amount for that data point. The sum of all error squares was first minimized by changing the values of the fitting parameters included in the model under consideration by the use of the solver function. This process gives the optimized model with minimal scatter. The arithmetic sum of all errors was then minimized by further changing the values of the fitting parameters by using the solver function again. This process gives an unbiased optimized model with a maximized coefficient of determination (R2 ).
7.3.4
Calculation of Goodness of Fit For any model under consideration, the values of the coefficient of determination
(R2 ) were computed taking into account the degrees of freedom using the following equation: n − p Se R = 1− ⋅ n − 1 S y 2
where, n
= number of data points
2
(7.1)
166 p
= number of regression constants
n– p
= degrees of freedom
Se
= Standard error of estimate
Sy
= Standard deviation
This process provided the adjusted R2 for a particular model. The optimization using the solver function was targeted at obtaining the minimum Se/Sy and maximum R2 values. This combination was later used to compare the statistical goodness of fit of different previo us models evolved before this research and candidate models evaluated in this research. The goodness of fit calculations were done in both arithmetic and logarithmic scales. For the arithmetic scale, observed and predicted E* values (in 105 psi) were used, while for the log scale, observed and predicted Log E* values (E* in psi) were used. Any optimization was done for minimizing the error in both the arithmetic and log scales. This process provided models optimized in both scales.
7.3.5
Sequential Optimizatio n of Sub-Models and the Main Model As part of developing an accurate enhancement to the E* model; many candidate
models were evaluated. Since all candidate E* models had a mathematical sigmoidal form and contained the distinct major parts, namely δ, α and the exponent; the δ, α and the exponent parts were optimized sequentially. Once all three parts were separately optimized, the full candidate E* model was finally optimized for all E* data used in this study.
167 7.3.5.1 Optimization of the δ-Model Appendix-C contains the final sigmoidal master curve parameters. These parameters were developed based on the procedures described in section 2.3.1. Due to the process followed for obtaining them, δ and α values for any specific mix, as reported in the database, remain cons tant over the range of temperature and loading frequency used in the lab testing. Initial analyses presented in section 6.6, however, showed that the minimum |E*| (i.e. δ) is sensitive to binder stiffness (Gb*). This means that the actual δ value may not be constant for a mix over the same range of temperature and loading frequency due to fact that the binder stiffness itself is highly sensitive to temperature and loading frequency. Hence, it is not justified to optimize the δ-model based on only the δ values of the mixtures reported in Appendix-C. Instead, the candidate δ-models were optimized based on the constant δ values only to find out the most probable seed values of the fitting parameters. It has already been discussed that the δ portion of the sigmoidal E* master curve resembles the |E*| values at very high temperatures and/or very small loading frequencies (fc ). Hence, the candidate δ-models were then optimized based on all the observed Log|E*| values at very high temperature (T = 130°F) and very small loading frequency (fc = 0.1 Hz). A number of candidate δ models were developed and evaluated in the course of this research. The δ model contained in the final candidate E* model contributed an overall R2 = 0.25 with Se/Sy = 0.87 for the full range of 7400 data contained in the master E* database. It should be noted that at this point, the fitting parameter values in the α and exponent part of the full sigmoidal curve of the E* model structure remained zero.
168 7.3.5.2 Optimization of the α-Model Once the δ-model was finalized for a candidate E* model, the next step was to optimize the α-model. As before, the candidate α-models were optimized based on the constant α values reported in Appendix-C only to determine the most probable seed values of the fitting parameters. The δ+α portion of the sigmoidal E* master curve resembles the |E*| values at very low temperatures and/or very high loading frequencies (fc). So, the candidate α-models were then optimized by equating the predicted δ+α values to the observed Log|E*| values at very low temperatures (T = 0 to 14°F) and high loading frequencies (fc = 10 to 25 Hz). These conditions are, theoretically, the most representative of the actual δ+α values of the mix for the range of temperature and loading frequency used in the lab. A number of candidate α models were developed and evaluated. The combined δ and α models contained in the final candidate E* model provided an overall R2 = 0.74 with Se/Sy = 0.51 for the full range of 7400 data contained in the master E* database. It should be noted that at this point, the fitting parameter values in the exponent part of the full sigmoidal curve of the E* model structure remained zero.
7.3.5.3 Optimization of the Exponent Model Once the δ and α models for any candidate E* model were finalized, the next step was to optimize the exponent model. The exponent portion of the sigmoidal E* master curve controls the rate of change in E* of the mix practically from very low to very high temperature and loading frequency regions. Hence, the exponent model was optimized
169 for the full temperature and frequency range of the master database used in this study. A number of candidate exponent models were developed and evaluated in the course of this research. The exponent model along with the δ and α models developed for the final candidate E* model provided an overall R2 = 0.89 with Se/Sy = 0.33 for the full range of 7400 data contained in the master E* database.
7.3.5.4 Final Optimization of the E* Model Once the δ, α and exponent models were developed, the final step of model optimization was to optimize the full E* model. For any candidate E* model, the values of the fitting parameters obtained from the δ, α and exponent models developed, were used as the seed values for the full model. The entire data range (N = 7400) of the master E* database was used to optimize the model by minimizing the error in both the arithmetic and log scale. As an example, after the full optimization was conducted; the final candidate E* model, which would be discussed in the remaining portion of this report, provided an adjusted R2 = 0.90 with Se/Sy = 0.32 in log scale (R2 = 0.80 with Se/Sy = 0.45 in arithmetic scale) for the entire master E* database used in the study.
7.3.6
Model Rationality Rationality constraints were imposed on all regression constants. For the asphalt
binder viscosity (η), the maximum viscosity for all binder types/grades is commonly stated to be: ηmaximum ≈ 3 x 1010 poise. For the HMA mix |E*|, this constraint is believed to be approximately |E*|maximum ≈ 7 x 106 psi (7). However, laboratory E* test data
170 contained in the master E* database used indicated that the |E*|maximum may be as high as 107 psi for highly aged stiff PG grade binders. The sensitivity of the predicted |E*| values to the predictor variables, within their full range, for all candidate E* models were evaluated. The model rationality in terms of aggregate gradation, mixture volumetrics, binder stiffness, and environmental and loading condition were critically evaluated. Finally, each candidate model was compared with previous models. The related analyses for the finally selected E* model are described in the following chapter of this report.
7.4
Model Selection A number of candidate E* models were developed and evaluated in the course of
this research. Final candidates are summarized in Appendix-D. Brief descriptions of characteristics of these candidate models are also provided there. Each model was tested for rationality, accuracy, precision, bias, trend, sensitivity and overall performance. The most promising model obtained in the overall study was found to be:
(
log 10 E* = −0.349 + 0.754 | Gb * | −0 .0052
)
6 .65 − 0 .032 ρ 200 + 0.0027 ρ 200 2 + 0.011 ρ 4 − 0.0001 ρ 4 2 V beff + 0.006 ρ 38 − 0.00014 ρ 38 2 − 0.08 V a − 1.06 Va + V beff
Vbeff + 0 .0124 ρ 38 − 0 .0001 ρ 38 2 − 0.0098 ρ 34 2.558 + 0 .032 Va + 0 .713 V a + Vbeff + 1 + e (−0.7814−0 .5785log|G b *| + 0. 8834 log δ b )
(7.2) where, E*
= dynamic modulus, psi
171 ρ200
= percentage of aggregates (by weight of the total aggregates) passing through no. 200 sieve, %
ρ4
= aggregates (by weight) retained on no. 4 sieve, %
ρ38
= aggregates (by weight) retained on the 3/8 inch sieve, %
ρ34
= aggregates (by weight) retained on the 3/4 inch sieve, %
Va
= air voids (by volume of the mix), %
Vbeff
= effective binder content (by volume of the mix), %
|Gb*|
= dynamic shear modulus of binder, psi
δb
= phase angle of binder associated with |G b*|, degree
The above model, mentioned as the “new E* model” in the remaining portion of this report, was evaluated in the same fashion as all other candidate models. The detail evaluation process of this model is presented in the following chapter.
8
PERFORMANCE OF THE NEW E* MODEL
8.1
Introduction One of the main objectives of this research was to develop a predictive model for
the dynamic modulus (E*) of HMA mixtures. The basic mathematical structure of the model equation is a sigmoidal function. The most accurate model, called the “new E* model”, was presented at the end of chapter 7.
8.2
Statistics of the New E* Model The goodness of fit was evaluated in two ways; in logarithmic scale and in
arithmetic scale. For analyzing the goodness of fit in logarithmic scale, the dependent variable is defined by Log|E*| (|E*| in psi), error is defined by “predicted Log|E*| observed Log|E*|” and Sy is defined by the standard deviation of the observed Log|E*| values. For analyzing the goodness of fit in arithmetic scale, the dependent variable is defined by |E*| (in psi), error is defined by “predicted |E*| - observed |E*|” and Sy is defined by standard deviation of the observed |E*| values. The new E* model was found to possess an excellent correlation coefficient (R2 ≈ 0.90) and a very small Se/Sy (≈ 0.32) in logarithmic scale. When analyzed in a arithmetic scale, the model again shows a very good correlation coefficient (R2 ≈ 0.80) and a small Se/Sy (≈ 0.45). The model statistics are shown in Table 8.1.
8.3
Comparison between Predicted and Observed |E*| Values One major way to visually assess the predictive accuracy of a model is to plot
predicted values against observed values with a line of equality. In general, models
173 TABLE 8.1 Statistics of the New E* Model Parameter Data points, n No. of Mixes, Nm Se Sy Se/Sy R2
Logarithmic Scale 7400 346 0.21 0.66 0.32 0.90
Arithmetic Scale 7400 346 658 ksi 1459 ksi 0.45 0.80
involving the criterion variable in transformed scale should have the predicted values transformed back to the original scale for analysis. Figure 8.1 is a plot of all of the 7400 observed Log E* stiffness data from 346 different HMA mixtures versus the respective Log E* data predicted by the new E* model under the same input conditions. Figure 8.2 is a similar plot where all observed E* stiffness data are plotted against the respective E* values predicted by the new E* model. Figure 8.1 shows that the new E* predictive model has excellent precision and accuracy in log scale. On the other hand, as seen in Figure 8.2, a decrease in the predictive accuracy and increase in model bias, particularly at high E* values, is indicated in the arithmetic mode. Clearly, there is a non-constant variance associated with the E* model in arithmetic scale, which is absent in the logarithmic scale. It may be because like many other asphalt-related characteristics, E* complex modulus is really associated with a non-constant variance and closely follows a logarithmic model. Nonetheless, there is no visible bias in the log scale. It is evident that all 7400 data points are around the line of equality without any trace of significant bias, particularly in the moderate to very high temperature regions. In the very low temperature region (high
174 E*), a very small level of bias, from some special mixes, can be visually observed. This may be associated with the fact that in this region, the actual binder stiffness tends to start reaching a constant value. This phenomenon is hard to fully encompass by any predictive model.
Predicted Log |E*|, |E*| in psi .
7 R2 = 0.90, Se/Sy = 0.32, N = 7400 6
5
4
3 3
4
5 Observed Log |E*|, |E*| in psi
6
7
FIGURE 8.1 Comparison between Predicted and Observed Log E* Values 75 60
5
Predicted |E*|, 10 psi .
R2 = 0.80, Se /Sy = 0.45, N = 7400
45
30 15
0 0
15
30 45 Observed |E*|, 105 psi
60
FIGURE 8.2 Comparison between Predicted and Observed E* Values
75
175 8.4
Predicting Capability of the New E* Model The graphical analysis presented in the previous section showed that except for
very small bias at very cold temperature regions (originating from only a few mixtures), the predicted E* values are in close agreement with the observed E* data. This comparison (i.e. Figure 8.1) was done for all 7400 data points from all 346 mixtures evaluated. It should be understood that the Gb* data (|G b*| and δ b) predicted by the use of the η-|Gb*|-δ b model equations developed in this study were used for the previous graphical analysis. The next step of this analysis would be the comparison of predicted E* values of specific mixtures for which laboratory |Gb*| and δ b data are available. This would validate both the η-|Gb*|-δ b and E* models. Figures 8.3 and 8.4 are plots of predicted versus observed E* values for some MnRoad and Salt River ¾ inch mixtures, respectively, where actual lab |Gb*| and δ b were used for E* prediction. As shown in Figure 8.3, all 4 MnRoad mixtures used the same binder (PG 58-22) and gradation while the air voids and asphalt content were varied from 6.4% to 8.2% and 5.1% to 6%, respectively. In other words, only the mix volumetrics were varied here keeping other input parameters constant. The predictions presented in Figure 8.3 show an extremely high degree of precision and accuracy, without any notable trace of bias. This validates that the new E* model is capable of taking care of even small changes in mix volumetrics while accurately predicting the mix E*. Figure 8.4 shows a similar plot for 3 Salt River ¾ inch mixtures. These mixes used different graded bind ers (e.g. PG 64-22, 70-10 and 76-16), and same volumetrics (Va = 6.85% and AC = 4.25%) and gradation. Figure 8.4 shows that the predictions, as in
176 Figure 8.3, have also very high degree of precision and accuracy, except for a little bias for the PG 64-22 mix at very low temperature (very large E* value) region. This bias was probably originated from the fact that at very low temperatures, the shear modulus (|Gb*|) of binder approaches a maximum limit i.e. the glassy modulus (|Gg*| ≈ 1 GPa). This further validates that the new E* model is capable of taking care of changes in binder types (characterized by rheology, grades etc.). In general, from this brief visual comparisons of the predicted E* values with the observed E* data, where actual lab |Gb*| and δ b were used for E* prediction, it can be said that the new E* model as well as the η-|Gb*|-δ b model possess excellent predictive strength.
Va = 6.4% to 8.2%, AC = 5.1% to 6% PG 58-22 Binder, Same Gradation
5
Predicted |E*|, 10 psi
100
10
MnRoad Cell-3 1
MnRoad Cell-16 MnRoad Cell-17 MnRoad Cell-22
0.1 0.1
1
Observed |E*|, 10 5 psi
10
FIGURE 8.3 Predicted versus Observed |E*| for MnRoad Mixtures
100
177
5
Predicted E*, 10 psi
100
Va = 6.85, AC = 4.25%, Same Gradation PG 64-22, 70-10 and 76-16 Binders 10
1
Salt River 3/4 in PG 64-22 Salt River 3/4 in PG 70-10 Salt River 3/4 in PG 76-16
0.1
0.1
1
10
100
Observed E*, 105 psi
FIGURE 8.4 Predicted versus Observed |E*| for Salt River ¾ inch Mixtures
8.5
Frequency Distribution of Residuals
8.5.1
Frequency Distribution for Full Data Range Like most other engineering parameters, HMA modulus is assumed to follow a
normal distribution. A check of the normality assumption for a model can be made by plotting a histogram of the residuals. The plot should look like a sample from a normal distribution centered at zero (47). Figure 8.5 is a frequency distribution of the residuals of E* (predicted E* observed E*). As expected for a good model, the distribution is very close to a normal distribution with no evident skew. Ideally, for a good model, the average of residuals should be close to zero. It sho uld be noted that the modulus of AC mixtures usually follows a log model and hence, it is almost impossible to get an average residual in arithmetic scale very close to zero. The laboratory E* values evaluated in this study
178 ranged from 0.10 x 105 psi to 86.45 x 105 psi with a standard deviation of 14.59 x 105 psi. For the full data range, the new E* model provided an average of residuals = - 0.56 x 105 psi and a standard deviation = 6.78 x 105 psi. These values are very small compared to the range and scatter of the laboratory data.
Frequency, % .
40 Mean = -0.56 x 10^5 psi Std. Dev = 6.78 x 10^5 psi N = 7400
30
20
10
0 -10
-8
-6
-4
-2
0
2
4
6
8
10
Residual Dynamic Modulus (E*Pred - E* Lab), E* in 10^5 psi FIGURE 8.5 Frequency Distribution of E* Residuals
Figure 8.6 shows the frequency distribution of the residuals of Log E* (= predicted Log E* - observed Log E*). As expected, the distribution is very close to a normal distribution with no evident skew with an average = - 0.05 and a standard deviation = 0.21. Thus, the new E* model was found to be an accurate predictive model that follows a normal distribution both in the arithmetic scale and in the log transformations.
179
25 Mean = -0.05 Std. Dev = 0.21 N = 7400
Frequency, % .
20 15 10 5 0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Residual Log E* (Log E*pred - Log E* lab)
FIGURE 8.6 Frequency Distribution of LogE* Residuals
The frequency distribution of the E* ratio (= E* pred/E*lab), defined as the ratio of predicted E* to observed E*, is presented in Figure 8.7. Due to the nature of the variable (E*pred/E*lab), the plot is not supposed to follow a normal distribution in linear scale. However, as one would expect, the mean of the ratio is exactly 1.00 and the standard deviation is quite small (= 0.47). All three plots allow one to consider the model to be reasonably accurate and unbiased relative to the 7400 calibration data obtained from 346 HMA mixtures used in the model development and calibration process.
180 20 Mean = 1.00 Std. Dev = 0.47 N = 7400
Frequency, % .
15
10
5
0 0.1
0.2
0.3
0.6 1.0 E*pred/E*lab
1.7
3.0
FIGURE 8.7 Frequency Distribution of E* Ratio
8.5.2
Frequency Distribution at Ranges of Temperatures Temperature plays a very important role in characterizing the stiffness of any
HMA mixture. Earlier analysis showed that there is a non-constant variance associated with E* moduli, which changes with temperature. To further evaluate the E* model, the frequency distribution analysis was done in three distinct temperature zones; low (0 to 40°F), intermediate (50 to 80°F) and high (95 to 130°F). Figures 8.8 and 8.9 show frequency distributions of E* residuals and Log E* residuals, respectively, at the three temperature zones investigated. Clearly, the frequency distributions of E* residuals at intermediate to very high temperature range (50 to 130°F) closely follow normality, while the frequency distribution at low temperature range (0 to 40°F) show a slight deviation from normality due to higher scatter associated with the
181 distribution. On the other hand, the frequency distributions of Log E* residuals at all temperature ranges (0 to 130°F) very closely follow normality. 40
Frequency, % .
All Data T = 0 to 40F
30
T = 50 to 80F T = 95 to 130F
20
10
0 -10
-8
-6
-4
-2
0
2
4
6
8
10
Residual Dynamic Modulus (E*Pred - E*Lab), E* in 10^5 psi
FIGURE 8.8 Frequency Distribution of E* Residuals at Temperature Ranges 25 All Data T = 0 to 40F T = 50 to 80F T = 95 to 130F
Frequency, % .
20 15 10 5 0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Residual Log E* (Log E*Pred - Log E*Lab) FIGURE 8.9 Frequency Distribution of Log E* Residuals at Temperature Ranges
182 Figure 8.10 presents the frequency distribution of the E* ratio (= E* pred/E*lab) for the same three temperature zones; low (0 to 40°F), intermediate (50 to 80°F) and high (95 to 130°F). As explained before, due to the nature of this variable (E* pred/E*lab), the plot is not supposed to follow normal distribution in non-normal scale. However, the frequency distribution should ideally show its peak (location of average value) near the value of 1 (one). The frequency distribution plots at the full range of very low to very high temperatures have their peak near the value of 1 (one). While the model rationality and accuracy is further evaluated in the next sections, it is very important to notice that the residuals distribution closely followed normal distribution, as desired, with excellent prediction capability. Overall the performance of the new E* model is quite acceptable for a very wide range of temperature.
Frequency, % .
20 All Data T = 0 to 40F T = 50 to 80F T = 95 to 130F
15
10
5
0 0
0.5
1
1.5
2
2.5
3
3.5
E* Ratio (E*Pred / E*Lab) FIGURE 8.10 Frequency Distribution of E* Ratio at Temperature Ranges
4
183 8.5.3
Frequency Distribution at Ranges of E* Stiffness Values It has already been shown that the E* value is associated with a non-constant
variance. To further evaluate the E* model for this phenomenon, the frequency distribution analysis was done in three distinct E* zones; low (0.1 x 105 psi to 5 x 105 psi), medium (5 x 105 psi to 30 x 105 psi) and high (30 x 105 psi to 100 x 105 psi). Figures 8.11 and 8.12 show frequency distributions of E* residuals and Log E* residuals, respectively, at those the three E* zones. The frequency distributions of E* residuals at low to medium stiffness range very closely follow normality, while the frequency distribution at high E* shows deviation from normality due to higher scatter associated with the distribution. The frequency distributions of Log E* residuals at all stiffness ranges closely follow normality. However, the frequency distributions of Log E* residuals show a small positive bias at the medium E* range and a small negative bias at the high E* range. The frequency distribution of Log E* residuals practically has no bias at the low E* value range, which usually coincides with high temperature ranges. Figure 8.13 presents the frequency distribution of the E* ratio (= E* pred/E*lab) for the same three E* ranges; low, medium and high. As before, the frequency distribution should ideally show its peak (location of average value) near the value of 1 (one). The frequency distribution plot at the low E* range has its peak just at 1 (one). The frequency distribution plot at low and medium E* range show small negative and positive bias, respectively, with their peaks near one. Hence, the model is performing best at low E* stiffness ranges, while its performance is also quite acceptable at low and medium stiffness ranges.
184 40
Frequency, % .
All Data 30
E* = 0.1x10^5 to 5x10^5 psi E* = 5x10^5 to 30x10^5 psi E* =30x10^5 to 100x10^5 psi
20
10
0 -10
-8
-6
-4
-2
0
2
4
6
8
10
Residual Dynamic Modulus (E*Pred - E* Lab), E* in 10^5 psi
FIGURE 8.11 Frequency Distribution of E* Residuals at E* Stiffness Ranges
25 All Data Frequency, % .
20
E* = 0.1x10^5 to 5x10^5 psi E* = 5x10^5 to 30x10^5 psi E* =30x10^5 to 100x10^5 psi
15 10 5 0 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Residual Log E* (Log E*Pred - Log E* Lab) FIGURE 8.12 Frequency Distribution of Log E* Residuals at E* Stiffness Ranges
185 20
Frequency, % .
All Data 15
E* = 0.1x10^5 to 5x10^5 psi E* = 5x10^5 to 30x10^5 psi E* =30x10^5 to 100x10^5 psi
10
5
0 0
0.5
1
1.5
2
2.5
3
3.5
4
E* Ratio (E*Pred / E*Lab)
FIGURE 8.13 Frequency Distribution of E* Ratio at E* Stiffness Ranges
8.6
Sensitivity Analysis of the New E* Model Sensitivity analysis is a very important step in model evaluation and validation. A
model with excellent goodness of fit (high R2 and small Se/Sy) may not pass the sensitivity tests. Models based on a narrow range of input parameters may result in unrealistic predictions. Errors in the model structure can also lead to unrealistic prediction even though the model is based on a very wide range of input parameters. Hence, it is very important to conduct a sensitivity analysis of any new model and evaluate the full range of each predictor variables upon the model rationality. Sensitivity to a specific variable can be accomplished by varying that variable within its full range, while keeping all other input variables constant.
186 As the first step of the sensitivity analysis, the maximum, minimum and average values of each predictor variables at specific combinations of temperature and loading frequency were summarized. Next, the range of a target variable was divided into five to six subdivisions. Then the observed E* values were averaged over each subdivision as the subdivision provided average values of the specific predictor variable. Now, the new E* model was used to predict the E* stiffness of the mix for all those average subdivision values of the target variable by the use of constant average values of other variables for that specific combination of temperature and loading frequency average subdivision. This allowed an avenue for the rational comparison of the observed versus predicted E* values while only one specific predictor variable is varied over its full range. The model sensitivity analysis is presented in the following sub sections.
8.6.1
Ranges of Predictor Variables The first step of the sensitivity analysis was summarizing the full ranges of
predictor variable at specific combinations of temperature and loading frequency. For the purpose of this analysis, the model sensitivity was evaluated for one fixed loading frequency (fc ) of 10 Hz and three test temperatures (T): 14°F, 70°F and 130°F. These three combinations were felt to provide a very good idea on the sensitivity of each variable to the predicted E* value. The predictor variables i.e. input parameters of the new E* stiffness model can be broadly categorized in the following areas: aggregate gradation related (ρ200 , ρ4 , ρ38 and ρ34 ), mix volumetrics related (Va and Vbeff), binder stiffness related (|Gb*| and δ b) and test
187 condition related (temperature and loading frequency). The ranges of these variables within the whole E* database, for the three selected temperatures 14°F, 70°F and 130°F, are shown in Tables 8.2, 8.3 and 8.4, respectively.
TABLE 8.2 Range of E* Model Variables at T = 14°F and fc = 10 Hz Parameters
Maximum Minimum Average
|Gb*| (psi)
3858
1214
1494
δ b (degree)
33
17
18
Va (%)
12.5
0.7
6.9
Vbeff (%)
25.1
6.1
10.7
ρ200 (%)
11.8
1.8
4.9
ρ4 (%)
74.0
30.0
50.9
ρ38 (%)
43.0
5.0
28.5
ρ34 (%)
26.1
0.0
3.5
TABLE 8.3 Range of E* Model Variables at T = 70°F and f = 10 Hz Parameters
Maximum Minimum Average
|Gb*| (psi)
2755
53
424
δ b (degree)
65
30
55
Va (%)
12.5
0.7
7.0
Vbeff (%)
24.5
6.1
10.4
ρ200 (%)
11.8
1.8
5.0
ρ4 (%)
74.0
30.0
50.2
ρ38 (%)
43.0
5.0
28.0
ρ34 (%)
26.1
0.0
3.2
188 TABLE 8.4 Range of E* Model Variables at T = 130°F and f = 10 Hz Parameters
8.6.2
Maximum Minimum Average
|Gb*| (psi)
960.2
0.3
2.9
δ b (degree)
81
55
75
Va (%)
12.5
0.7
7.0
Vbeff (%)
25.1
6.1
10.5
ρ200 (%)
11.8
1.8
5.0
ρ4 (%)
74.0
30.0
50.4
ρ38 (%)
43.0
5.0
28.0
ρ34 (%)
26.1
0.0
3.2
Model Sensitivity to Gradation As earlier explained, aggregate gradation plays an important role in the E*
stiffness of a HMA mixture. Preliminary analyses showed that ρ200 , ρ4 , ρ38 and ρ34 are highly correlated with E* and hence, these variables have been used in the new E* model. The model sensitivity analyses to each of them are presented in the following paragraphs.
8.6.2.1 Model Sensitivity to ρ200 The average subdivision values of passing #200 sieve (% ρ200 ) for the predictive model with respective average observed E* value and other input variable values are shown in Table 8.5. Both the observed E* data and predicted E* values were plotted against ρ200 for its full range at T = 14°F, 70°F and 130°F and f = 10 Hz. The best- fit trend lines are shown in Figure 8.14.
189 TABLE 8.5 Input Data with Observed and Predicted E* Data for ρ200 T
o
f
Va
Vbeff
%
Gradation
Gb* Data
Observed Predicted
ρ34 ρ38
ρ4
ρ200
|Gb|*
δb
Ε∗
E*
%
%
%
psi
deg
105 psi
105 psi
F Hz
%
%
14 10
6.9
10.7 3.5 28.5 50.9
2.6
1494
18.2
39.45
37.26
14 10
6.9
10.7 3.5 28.5 50.9
3.2
1494
18.2
38.01
36.67
14 10
6.9
10.7 3.5 28.5 50.9
5.1
1494
18.2
45.73
35.64
14 10
6.9
10.7 3.5 28.5 50.9
6.9
1494
18.2
47.44
35.73
14 10
6.9
10.7 3.5 28.5 50.9 11.7 1494
18.2
44.10
41.40
70 10
7.0
10.4 3.2 28.0 50.2
2.6
424
54.6
16.00
13.67
70 10
7.0
10.4 3.2 28.0 50.2
3.3
424
54.6
12.06
13.43
70 10
7.0
10.4 3.2 28.0 50.2
5.0
424
54.6
14.20
13.08
70 10
7.0
10.4 3.2 28.0 50.2
6.6
424
54.6
15.39
13.07
70 10
7.0
10.4 3.2 28.0 50.2 11.7
424
54.6
16.28
15.20
130 10
7.0
10.5 3.2 28.0 50.4
2.6
3
74.8
1.29
1.29
130 10
7.0
10.5 3.2 28.0 50.4
3.2
3
74.8
1.13
1.27
130 10
7.0
10.5 3.2 28.0 50.4
5.0
3
74.8
1.15
1.24
130 10
7.0
10.5 3.2 28.0 50.4
6.6
3
74.8
1.30
1.24
130 10
7.0
10.5 3.2 28.0 50.4 11.7
3
74.8
1.21
1.44
190
100 T = 14o F, fc = 10 Hz
T = 70o F, fc = 10 Hz
5
E* (x10 psi)
10
T = 130o F, fc = 10 Hz 1
Observed |E*| Predicted |E*| 0.1 0
2
4
6
8
ρ200 (%) FIGURE 8.14 E* Model Sensitivity to ρ200
10
12
191 Mathematically, the model assumes a second order polynomial curve for the ρ200 variable with an initial lowering of E* value with increasing ρ200 up to ρ200 ≈ 3%. After that, the E* value increases with increasing values of ρ200 up to the practical maximum value (ρ200 ≈ 12%) and beyond that point. The predicted and observed E* data, being practically very close, showed the same trend for the full range of ρ200 . The implication of this trend is that in dense graded HMA mixtures, if the amount of passing #200 sieve material (generally around 5%) is slightly increased; the minimum E* (i.e. the “δ” in the sigmoidal function) will also increase (for ρ200 > 3%). It was observed that at very high temperatures, the new model slightly over-predicts the E* stiffness although the predictive trend is quite accurate for all temperature ranges.
8.6.2.2 Model Sensitivity to ρ4 The average subdivision values of the cumulative percentage retained on the #4 sieve (ρ4 ) for the predictive model, with respect to the average observed E* values and other input variable values, are shown in Table 8.6. Both the observed E* data and predicted E* values were plotted against ρ4 for its full range at T = 14°F, 70°F and 130°F and f = 10 Hz. The best-fit trend lines are shown in Figure 8.15. Similar to the ρ200 variable, the E* model assumes a second order polynomial curve for the ρ4 variable with an initial increase of E* value with increasing ρ4 up to a value of ρ4 ≈ 25%. After that, the E* value increases with increasing values of ρ4 up to the practical maximum value (ρ4 ≈ 74%) and beyond that point. Mathematically, this change due to ρ4 occurs in the
192 minimum E* (i.e. the “δ ” in the sigmoidal function). Note that for the E* master database, the minimum and maximum values of ρ4 are 30% and 74%, respectively, with an average ρ4 ≈ 50%. Hence for the practical range of ρ4 (30% to 74%), the new E* model will predict an increase of E* stiffness with an increase of ρ4 and vice versa. It was noticed that for the full range of ρ4 at very high temperatures; while the observed data slightly increased with increasing ρ4 values, the E* prediction remained more or less unchanged. The average error from the prediction, however, remained small. On the other hand, at other temperature ranges, the prediction is highly accurate and reflected the observed trend.
TABLE 8.6 Input Data with Observed and Predicted E* Data for ρ4 T o
f
F Hz
Va
Vbeff
Gradation ρ34
ρ38
ρ4
Gb* Data ρ200 |Gb|*
Observed Predicted
δb
Ε∗
E*
%
%
%
%
%
%
psi
deg
105 psi
105 psi
14 14 14 14 70
10 10 10 10 10
6.9 6.9 6.9 6.9 7.0
10.7 10.7 10.7 10.7 10.4
3.5 3.5 3.5 3.5 3.2
28.5 28.5 28.5 28.5 28.0
33.0 46.6 53.7 63.6 34.9
4.9 4.9 4.9 4.9 5.0
1494 1494 1494 1494 424
18.2 18.2 18.2 18.2 54.6
44.57 47.84 41.72 41.47 13.15
35.28 36.00 35.30 33.20 13.00
70 70 70 130 130 130 130
10 10 10 10 10 10 10
7.0 7.0 7.0 7.0 7.0 7.0 7.0
10.4 10.4 10.4 10.5 10.5 10.5 10.5
3.2 3.2 3.2 3.2 3.2 3.2 3.2
28.0 28.0 28.0 28.0 28.0 28.0 28.0
45.3 54.3 68.1 34.9 45.3 54.3 67.6
5.0 5.0 5.0 5.0 5.0 5.0 5.0
424 424 424 3 3 3 3
54.6 54.6 54.6 74.8 74.8 74.8 74.8
14.78 13.37 13.85 0.91 1.21 1.16 1.22
13.19 12.88 11.64 1.23 1.25 1.22 1.10
193
100 T = 14o F, fc = 10 Hz
T = 70o F, fc = 10 Hz
5
E* (x10 psi)
10
T = 130o F, fc = 10 Hz 1
Observed |E*| Predicted |E*| 0.1 30
40
50 ρ4 (%)
FIGURE 8.15 E* Model Sensitivity to ρ4
60
70
194 8.6.2.3 Model Sensitivity to ρ38 The average subdivision values for the percentage retained on the 3/8 inch sieve (ρ38 ) for the predictive model, with respect to the average observed E* value and other input variable values, are shown in Table 8.7. As before, both the observed E* data and predicted E* values were plotted against ρ38 for its full range at T = 14°F, 70°F and 130°F and f = 10 Hz. Figure 8.16 shows the best-fit trend lines for these plots. In the analysis of the model variables, ρ38 was found important for both the minimum and maximum values of E* (i.e. “δ” and “α” in the sigmoidal function). Mathematically, the model assumes a second order polynomial curve for the ρ38 in the δ and α sub- models. From Figure 8.7, it is evident that the new model predicts E* very accurately and rationally over the full range of ρ38 . The predictive accuracy of this parameter is highest in the low and very high temperature region.
8.6.2.4 Model Sensitivity to ρ34 The average subdivision values for the percentage retained on the ¾ inch sieve (ρ34 ) for the predictive model, with respect to the average observed E* value and other input variable values, are shown in Table 8.8. Both the observed E* data and predicted E* values were plotted against ρ34 for its full range at T = 14°F, 70°F and 130°F and f = 10 Hz. The best- fit trend lines are shown in Figure 8.17. In the analysis of model variables, it was found that the ρ34 value contributes significantly to the change of the maximum value of E* (i.e. “α” in the sigmoidal function). The E* model assumes a
195 TABLE 8.7 Input Data with Observed and Predicted E* Data for ρ38 T
o
f
Va
Vbeff
%
Gradation ρ34
ρ38
ρ4
%
%
%
7.7
Gb* Data ρ200 |Gb|*
δb
Ε∗
E*
psi
deg
105 psi
105 psi
F Hz
%
14 10
6.9
10.7 3.5
50.9 4.9
1494
18.2
43.18
25.33
14 10
6.9
10.7 3.5 14.7 50.9 4.9
1494
18.2
44.36
29.70
14 10
6.9
10.7 3.5 24.4 50.9 4.9
1494
18.2
47.51
34.40
14 10
6.9
10.7 3.5 36.6 50.9 4.9
1494
18.2
40.94
36.72
70 10
7.0
10.4 3.2
50.2 5.0
424
54.6
9.69
9.05
70 10
7.0
10.4 3.2 15.4 50.2 5.0
424
54.6
14.68
11.40
70 10
7.0
10.4 3.2 23.6 50.2 5.0
424
54.6
15.88
12.64
70 10
7.0
10.4 3.2 36.9 50.2 5.0
424
54.6
13.77
13.31
130 10
7.0
10.5 3.2
50.4 5.0
3
74.8
0.82
1.00
130 10
7.0
10.5 3.2 15.4 50.4 5.0
3
74.8
1.17
1.16
130 10
7.0
10.5 3.2 23.5 50.4 5.0
3
74.8
1.28
1.22
130 10
7.0
10.5 3.2 36.9 50.4 5.0
3
74.8
1.24
1.21
4.3
4.3
%
Observed Predicted
196
100 T = 14 oF, fc = 10 Hz
10
5
E* (x10 psi)
T = 70 oF, fc = 10 Hz
T = 130o F, fc = 10 Hz 1
Observed |E*| Predicted |E*| 0.1 0
10
20 ρ38 (%)
FIGURE 8.16 E* Model Sensitivity to ρ38
30
40
197 general increase of E* with an increase of the amount of aggregate retained on ¾ inch sieve in the HMA mix and vice versa. The observed E* data is not quite sensitive to ρ34 at very low temperatures, while the predicted E* data was found more sensitive. However, the observed E* data is sensitive to ρ34 at all other temperature. It is clear from Figure 8.17 that the E* prediction is quite accurate and has the same trend as the observed data for all practical ranges of temperatures. In other words, the predictive accuracy of the model in terms of sensitivity to ρ34 is high.
8.6.3
Model Sensitivity to Mix Volumetric Properties Volumetric properties play a very important role in the stiffness characteristics of
the HMA mix over a wide range of temperature and loading rate. In practice, mix designs are highly sensitive to the volumetric properties of the mix. The most important mix volumetric properties found in this study are air voids (Va) and effective volume of binder or bitumen (Vbeff). Other properties such as volume concentration (C v ), voids in mineral aggregates (VMA) and voids filled with asphalt (VFA) are some combination of these two parameters (Va and Vbeff). It can be observed from the model structure that a form of a non-linear model was used for Va and Vbeff to reflect the influence of the mix density on the E* stiffness response. The model sensitivity analysis with respect to mix volumetric properties follows.
198 TABLE 8.8 Input Data with Observed and Predicted E* Data for ρ34 T
o
f
Va
Vbeff
Gradation ρ34
ρ38
ρ4
%
%
%
Gb* Data ρ200 |Gb|* %
psi
Observed Predicted
δb
Ε∗
E*
deg
105 psi
105 psi
F Hz
%
%
14 10
6.9
10.7
0.0 28.5 50.9 4.9 1494
18.2
45.21
38.05
14 10
6.9
10.7
1.4 28.5 50.9 4.9 1494
18.2
41.27
37.08
14 10
6.9
10.7
6.0 28.5 50.9 4.9 1494
18.2
52.63
34.04
14 10
6.9
10.7 10.0 28.5 50.9 4.9 1494
18.2
35.67
31.64
14 10
6.9
10.7 26.1 28.5 50.9 4.9 1494
18.2
41.06
23.51
70 10
7.0
10.4
0.0 28.0 50.2 5.0
424
54.6
14.53
13.73
70 10
7.0
10.4
2.3 28.0 50.2 5.0
424
54.6
11.98
13.25
70 10
7.0
10.4
6.0 28.0 50.2 5.0
424
54.6
18.39
12.51
70 10
7.0
10.4 10.0 28.0 50.2 5.0
424
54.6
12.65
11.77
70 10
7.0
10.4 23.0 28.0 50.2 5.0
424
54.6
10.42
9.63
130 10
7.0
10.5
0.0 28.0 50.4 5.0
3
74.8
1.15
1.27
130 10
7.0
10.5
2.3 28.0 50.4 5.0
3
74.8
1.28
1.24
130 10
7.0
10.5
6.0 28.0 50.4 5.0
3
74.8
1.79
1.21
130 10
7.0
10.5 10.0 28.0 50.4 5.0
3
74.8
0.84
1.17
130 10
7.0
10.5 23.0 28.0 50.4 5.0
3
74.8
0.95
1.06
199
100 T = 14o F, fc = 10 Hz
T = 70o F, fc = 10 Hz
1
T = 130o F, fc = 10 Hz
5
E* (x10 psi)
10
Observed |E*| Predicted |E*| 0.1 0
10
20 ρ34 (%)
FIGURE 8.17 E* Model Sensitivity to ρ34
30
200 8.6.4
Model Sensitivity to Mix Volumetric Properties Volumetric properties play a very important role in the stiffness characteristics of
the HMA mix over a wide range of temperature and loading rate. In practice, mix designs are highly sensitive to the volumetric properties of the mix. The most important mix volumetric properties found in this study are air voids (Va) and effective volume of binder or bitumen (Vbeff). Other properties such as volume concentration (C v ), voids in mineral aggregates (VMA) and voids filled with asphalt (VFA) are some combination of these two parameters (Va and Vbeff). It can be observed from the model structure that a form of a non-linear model was used for Va and Vbeff to reflect the influence of the mix density on the E* stiffness response. The model sensitivity analysis with respect to mix volumetric properties follows.
8.6.4.1 Model Sensitivity to Mix Air Voids (Va) The average incremental values of air voids (%Va) over its full observed range along with other input variable values needed as input for the new E* predictive model with respect to the average observed and calculated predicted E* values, are summarized in Table 8.9. Both the observed E* data and predicted E* values were plotted against Va for its full range at T = 14°F, 70°F and 130°F and f = 10 Hz. The best-fit trend lines are shown in Figure 8.18. Noteworthy, based on the analysis of model variables, Va has been used in both the δ and α parts of the model to make sure that the influence of Va on E* of the mix is properly reflected for the whole range of temperature and loading rate that a mix may experience in its service life. It can be observed from Tables 8.2 through 8.4
201 that for the master E* database used in this research, the value of %Va ranged from 0.7 to 12.5 with an average value of about 7. It is clear from Figure 8.18 that the new model showed exactly the same sensitivity as the observed data. The trend of the predicted data followed the trend of the observed data very closely. In general, like the observed data, the E* stiffness decreased as the Va values were increased. The model could predict the E* stiffness of HMA mix very accurately over the full range of Va parameter.
TABLE 8.9 Input Data with Observed and Predicted E* Data for Va T
o
f
Va
Vbeff
%
Gradation ρ34
ρ38
ρ4
%
%
%
Gb* Data ρ200 |Gb|* %
Observed Predicted
δb
Ε∗
E*
psi
deg
105 psi
105 psi
F Hz
%
14 10
2.7
10.7 3.5 28.5 50.9 4.9
1494
18.2
42.19
44.71
14 10
5.3
10.7 3.5 28.5 50.9 4.9
1494
18.2
48.49
39.16
14 10
6.9
10.7 3.5 28.5 50.9 4.9
1494
18.2
46.06
35.77
14 10
8.8
10.7 3.5 28.5 50.9 4.9
1494
18.2
37.77
31.96
14 10 11.1 10.7 3.5 28.5 50.9 4.9
1494
18.2
32.40
27.77
70 10
2.1
10.4 3.2 28.0 50.2 5.0
424
54.6
12.06
16.77
70 10
5.1
10.4 3.2 28.0 50.2 5.0
424
54.6
14.89
14.57
70 10
6.9
10.4 3.2 28.0 50.2 5.0
424
54.6
14.07
13.14
70 10
8.8
10.4 3.2 28.0 50.2 5.0
424
54.6
14.18
11.67
70 10 11.1 10.4 3.2 28.0 50.2 5.0
424
54.6
11.28
10.07
130 10
2.1
10.5 3.2 28.0 50.4 5.0
3
74.8
0.93
1.59
130 10
5.1
10.5 3.2 28.0 50.4 5.0
3
74.8
1.27
1.39
130 10
6.9
10.5 3.2 28.0 50.4 5.0
3
74.8
1.26
1.24
130 10
8.8
10.5 3.2 28.0 50.4 5.0
3
74.8
0.92
1.09
130 10 11.0 10.5 3.2 28.0 50.4 5.0
3
74.8
0.91
0.91
202
100 T = 14o F, fc = 10 Hz
10
5
E* (x10 psi)
T = 70o F, fc = 10 Hz
1
T = 130o F, fc = 10 Hz
Observed |E*| Predicted |E*| 0.1 0
3
6 Va (%)
FIGURE 8.18 E* Model Sensitivity to Va
9
12
203 8.6.4.2 Model Sensitivity to Mix Effective Binder Content As noted, the new E* model uses a combination of Va and Vbeff (Vbeff /[Va + Vbeff]) to reflect the influence of the mix density on the E* stiffness response. Table 8.10 summarizes the average incremental values of effective binder content (%Vbeff, expressed by percentage of volume by volume) over its full observed range along with other input variable values needed as input for the new E* predictive model. The table also summarizes the respective average observed and calculated predicted E* values. Both the observed E* data and predicted E* values were plotted against Vbeff for its full range at T = 14°F, 70°F and 130°F and f = 10 Hz. The best- fit trend lines are shown in Figure 8.19. The value of %Vbeff ranged from about 6 to 25 with an average value of about 10.5. As can be observed from Figure 8.19, the predicted E* value generally decreased as the Vbeff values increased. This predicted trend was exactly similar to the trend exhibited by the observed E* data. Compared to the predicted E* values, the observed data, however, showed less sensitivity to Vbeff at very high temp erature. Generally, for all temperature ranges, the new E* model could predict the E* stiffness of HMA mix very accurately over the full range of the Vbeff parameter.
8.6.5
Model Sensitivity to Binder Stiffness The viscoelastic behavior of HMA mix is greatly attributed to the rheological
properties of the asphalt binder mixed with the aggregates. Binder rheology can be accurately mirrored by its complex modulus, expressed by the shear modulus (|Gb*|) and associated phase angle (δ b). As a result, the new model uses both |G b*| and δ b to
204 TABLE 8.10 Input Data with Observed and Predicted E* Data for Vbeff T
o
f
Va
Vbeff
Gradation ρ34
ρ38
ρ4
%
%
Gb* Data ρ200 |Gb|* %
Observed Predicted
δb
Ε∗
E*
psi
deg
105 psi
105 psi
F Hz
%
%
%
14 10
6.9
8.7
3.5 28.5 50.9 4.9
1494
18.2
45.38
36.46
14 10
6.9
10.8 3.5 28.5 50.9 4.9
1494
18.2
44.91
35.65
14 10
6.9
13.1 3.5 28.5 50.9 4.9
1494
18.2
38.37
34.96
14 10
6.9
17.4 3.5 28.5 50.9 4.9
1494
18.2
29.93
34.06
14 10
6.9
24.4 3.5 28.5 50.9 4.9
1494
18.2
36.65
33.14
70 10
7.0
8.1
3.2 28.0 50.2 5.0
424
54.6
16.16
13.60
70 10
7.0
10.2 3.2 28.0 50.2 5.0
424
54.6
13.55
13.12
70 10
7.0
11.7 3.2 28.0 50.2 5.0
424
54.6
12.28
12.83
70 10
7.0
15.3 3.2 28.0 50.2 5.0
424
54.6
11.39
12.32
70 10
7.0
24.1 3.2 28.0 50.2 5.0
424
54.6
13.14
11.62
130 10
7.0
8.1
3.2 28.0 50.4 5.0
3
74.8
1.37
1.34
130 10
7.0
10.2 3.2 28.0 50.4 5.0
3
74.8
1.19
1.25
130 10
7.0
11.7 3.2 28.0 50.4 5.0
3
74.8
1.00
1.20
130 10
7.0
15.3 3.2 28.0 50.4 5.0
3
74.8
0.86
1.11
130 10
7.0
24.4 3.2 28.0 50.4 5.0
3
74.8
1.26
0.99
205
100 T = 14o F, fc = 10 Hz
T = 70o F, fc = 10 Hz
5
E* (x10 psi)
10
T = 130o F, fc = 10 Hz 1
Observed |E*| Predicted |E*| 0.1 5
10
15 Vbeff (%)
FIGURE 8.19 E* Model Sensitivity to Vbeff
20
25
206 capture the E* characteristics partially controlled by the rheological properties of the binder. It was further observed in the analysis of the model variables that the minimum mix stiffness (expressed by the “δ” function of the sigmoidal model structure) is significantly sensitive to |G b*|. Hence, |G b*| is important in determining the minimum mix stiffness as well as the rate of change of stiffness with loading rate and temperature. The model sensitivity analysis with respect to binder stiffness follows.
8.6.5.1 Model Sensitivity to |Gb*| Table 8.11 summarizes the average incremental values of the shear modulus (|Gb*|) of the binder used in the HMA mixture over the full observed range along with other fixed average input variable values needed as input for the new E* predictive model. The table also summarizes the respective average observed and calculated predicted E* values. Both the observed E* data and predicted E* values were plotted against |Gb*| for its full range at T = 14°F, 70°F and 130°F and f = 10 Hz. The best- fit trend lines are shown in Figure 8.20. It can be noticed that the range of |G b*| sharply changes with temperature, which is obvious. For example, at 14°F, the value of |Gb*| ranged from 1214 to 3,858 psi with an average of 1,494 psi. On the other hand, at 130°F, the value of |G b*| ranged from 0.31 to 96 psi with an average of 2.93 psi. The trend of the observed data, presented in Figure 8.20, shows that at very low and intermediate temperatures, the model slightly under-predicts the E* values, while the predicted trend is highly accurate. The prediction is simply perfect at very high temperature range. It is clear from the plot that the prediction accomplished by the new
207 E* model is very accurate and the predicted trend closely imitates the trend set by the observed E* values for all practical limits of temperatures.
TABLE 8.11 Input Data with Observed and Predicted E* Data for |G b*| T
o
f
Va
Vbeff
%
Gradation
Gb* Data
Observed Predicted
ρ34
ρ38
ρ4
ρ200
|Gb|*
δb
Ε∗
E*
%
%
%
%
psi
deg
105 psi
105 psi
F Hz
%
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1348.8 18.2
47.72
34.77
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1393.5 18.2
45.26
35.06
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1415.0 18.2
41.41
35.19
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1442.3 18.2
41.19
35.36
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1508.8 18.2
42.42
35.76
70 10
7.0
10.4 3.2 28.0 50.2 5.0
109.9 54.6
8.78
7.57
70 10
7.0
10.4 3.2 28.0 50.2 5.0
243.9 54.6
12.38
10.53
70 10
7.0
10.4 3.2 28.0 50.2 5.0
356.2 54.6
14.25
12.23
70 10
7.0
10.4 3.2 28.0 50.2 5.0
419.3 54.6
15.03
13.02
70 10
7.0
10.4 3.2 28.0 50.2 5.0
778.4 54.6
18.60
16.35
130 10
7.0
10.5 3.2 28.0 50.4 5.0
0.9
74.8
0.76
0.78
130 10
7.0
10.5 3.2 28.0 50.4 5.0
1.5
74.8
1.22
0.95
130 10
7.0
10.5 3.2 28.0 50.4 5.0
1.8
74.8
1.04
1.02
130 10
7.0
10.5 3.2 28.0 50.4 5.0
2.1
74.8
1.17
1.07
130 10
7.0
10.5 3.2 28.0 50.4 5.0
7.1
74.8
1.96
1.79
208
100 T = 14o F, fc = 10 Hz
10
5
E* (x10 psi)
T = 70o F, fc = 10 Hz
T = 130 oF, fc = 10 Hz
1
Observed |E*| Predicted |E*| 0.1 0.1
1
10
100 |Gb|* (psi)
FIGURE 8.20 E* Model Sensitivity to |Gb*|
1000
10000
209 8.6.5.2 Model Sens itivity to δ b The average incremental values of the binder phase angle (δ b) used in the HMA mixtures over the full observed range of the master database, along with other fixed average input variable values needed as input for the new E* predictive model, have been summarized in Table 8.12. The table also summarizes the average observed and predicted E* values. Both the observed E* data and predicted E* values were plotted against δ b for its full range at T = 14°F, 70°F and 130°F and f = 10 Hz. The best- fit trend lines are shown in Figure 8.21. The trend of the observed data shows that at intermediate to very high temperature regions, the mix E* value decreases with an increase in the phase angle (δ b) value associated with the complex modulus (Gb*) of the binder, while the observed E* values are almost insensitive to the δ b at the very low temperature region. As observed from Figure 8.21, the prediction is slightly more sensitive to δ b when compared with the observed data. The figure, however, clearly shows that the prediction is quite accurate and the predicted trend is close to the observed trend for low to very high temperature ranges.
8.6.6
Model Sensitivity to Temperature To further evaluate the sensitivity of each predictor variable to temperature, the
difference between log of the predicted moduli (Log E* pred) and log of the observed moduli (Log E* obs) was denoted by ∆LogE*. Similar to the previous analysis, each predictor variable was varied over their full range within the E* master database, keeping
210 TABLE 8.12 Input Data with Observed and Predicted E* Data for δ b T
o
f
Va
Vbeff
%
Gradation
Gb* Data
Observed Predicted
ρ34
ρ38
ρ4
ρ200
|Gb|*
δb
Ε∗
E*
%
%
%
%
psi
deg
105 psi
105 psi
F Hz
%
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1493.7 17.1
42.63
36.71
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1493.7 17.2
43.78
36.62
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1493.7 17.2
42.89
36.58
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1493.7 17.3
38.45
36.53
14 10
6.9
10.7 3.5 28.5 50.9 4.9 1493.7 18.3
39.89
35.59
70 10
7.0
10.4 3.2 28.0 50.2 5.0
424.0 47.7
15.94
14.27
70 10
7.0
10.4 3.2 28.0 50.2 5.0
424.0 53.5
11.61
13.24
70 10
7.0
10.4 3.2 28.0 50.2 5.0
424.0 55.2
17.71
12.98
70 10
7.0
10.4 3.2 28.0 50.2 5.0
424.0 55.7
13.58
12.91
70 10
7.0
10.4 3.2 28.0 50.2 5.0
424.0 59.2
9.99
12.40
130 10
7.0
10.5 3.2 28.0 50.4 5.0
2.9
68.8
1.58
1.31
130 10
7.0
10.5 3.2 28.0 50.4 5.0
2.9
73.5
1.16
1.25
130 10
7.0
10.5 3.2 28.0 50.4 5.0
2.9
75.5
0.95
1.23
130 10
7.0
10.5 3.2 28.0 50.4 5.0
2.9
76.7
1.23
1.21
130 10
7.0
10.5 3.2 28.0 50.4 5.0
2.9
78.1
0.82
1.20
211
100 T = 14 oF, fc = 10 Hz
T = 70o F, fc = 10 Hz
5
E* (x10 psi)
10
T = 130o F, fc = 10 Hz 1
Observed |E*| Predicted |E*| 0.1 0
20
40 δb (degree)
FIGURE 8.21 E* Model Sensitivity to δ b
60
80
212 other variables constant at respective average values. For f = 10 Hz and T = 14°F, 70°F and 130°F, the ∆LogE* variable was plotted against each predictor variable. Ideally, for a constant variance case, the value of ∆LogE* should be about zero. While none of the candidate E* models had an ∆LogE* ≈ 0 for all temperature conditions, the model being evaluated in this chapter showed the least sensitivity of ∆LogE* (the best performance) towards temperature. Figures 8.22 through 8.29 show plots of ∆LogE* versus ρ200 , ρ4 , ρ38 , ρ34 , Va, Vbeff, |Gb*| and δ b. It was found that there were non-constant variances involved with the predictor variables used in the predictive model. None of them, however, showed any specific bias. In general, the model performed well in terms of sensitivity to temperature.
Log E*pred - Log E*obs
0.6
0.3
0 T = 14F T = 70F T = 130F
-0.3
-0.6 0
2
4
6
8
10
12
ρ200 (%) FIGURE 8.22 ∆LogE* Versus ρ200 at fc = 10 Hz and T = 14, 70 and 130°F
213
Log E*pred - Log E*obs
0.6 T = 14F T = 70F T = 130F
0.3
0
-0.3
-0.6 30
40
50 ρ4 (%)
60
70
FIGURE 8.23 ∆LogE* Versus ρ4 at fc = 10 Hz and T = 14, 70 and 130°F
Log E*pred - Log E*obs
0.6 T = 14F T = 70F T = 130F
0.3
0
-0.3
-0.6 0
10
20
30
ρ38 (%) FIGURE 8.24 ∆LogE* Versus ρ38 at fc = 10 Hz and T = 14, 70 and 130°F
40
214
Log E*pred - Log E*obs
0.6
0.3
0 T = 14F T = 70F
-0.3
T = 130F -0.6 0
10
ρ34 (%)
20
30
FIGURE 8.25 ∆LogE* Versus ρ34 at fc = 10 Hz and T = 14, 70 and 130°F
Log E*pred - Log E*obs
0.6
0.3
0 T = 14F T = 70F T = 130F
-0.3
-0.6 0
3
6
9
Va (%) FIGURE 8.26 ∆LogE* Versus Va at fc = 10 Hz and T = 14, 70 and 130°F
12
215
Log E*pred - Log E*obs
0.6
0.3
0 T = 14F T = 70F T = 130F
-0.3
-0.6 5
10
15
20
25
Vbeff (%) FIGURE 8.27 ∆LogE* Versus Vbeff at fc = 10 Hz and T = 14, 70 and 130°F
Log E*pred - Log E*obs
0.6 T = 14F T = 70F T = 130F
0.3
0
-0.3
-0.6 1
10
100
1000
10000
|Gb*| (psi) FIGURE 8.28 ∆LogE* Versus |Gb*| at fc = 10 Hz and T = 14, 70 and 130°F
216
Log E*pred - Log E*obs
0.6 T = 14F T = 70F T = 130F
0.3
0
-0.3
-0.6 0
20
40
60
80
δb (degree) FIGURE 8.29 ∆LogE* Versus δ b at fc = 10 Hz and T = 14, 70 and 130°F
8.7
Response of Predicted E* to Mix Volumetrics It can be observed from the model structure that air voids (Va) and effective
binder volume (Vbeff) were combined to Vbeff/(Va + Vbeff) to reflect the influence of the mix density on the dynamic stiffness response. Although the range of Vbeff is generally restricted to a tight tolerance around the optimum for a majority of mix responses in the master database; the wide range of Vbeff values in the database allowed for a mathematical formulation that provided reasonable predictions for a wider range of Vbeff values. The E* moduli values of a HMA mix subjected to a load frequency of fc = 1 Hz for five different temperatures (T = 14, 70, 70, 100 and 130°F) were predicted using the new model. The values of Va were varied from 1 to 12 (constant Vbeff) and the values of Vbeff were varied from 4 to 16 (constant Va). The average values of other input variables
217 (ρ200 , ρ4 , ρ38 , ρ34 , |Gb*| and δ b), as obtained from the master database, were used as input values. The input values are shown in Table 8.13. TABLE 8.13 Input Data for Model Response to Mix Volumetrics E* Test Condition
Gradation
Va
Vbeff
Gb* Data
Temp.
Freq.
ρ34 ρ38
ρ4
ρ200
(o F)
f (Hz)
%
%
%
14
1
3.5 28.5 50.9 4.9 2 to 12 4 to 16 3123 30.9
40
1
5.0 24.3 42.1 5.2 2 to 12 4 to 16 1790 47.0
70
1
5.0 24.3 42.1 5.2 2 to 12 4 to 16 103
100
1
5.0 24.3 42.1 5.2 2 to 12 4 to 16 4.95 80.3
130
1
3.2 28.0 50.4 5.0 2 to 12 4 to 16 0.50 84.7
%
(%)
(%)
|Gb*|
δb
psi
deg
68.4
The rational behavior of the new model relative to both Va and Vbeff is shown in Figures 8.30 through 8.34 through the use of contour plots of the predicted E* at different temperatures. From these contour plots, it is clear that regardless of temperature, the E* values increase with decreasing air voids (constant bitumen volume) and decrease with increasing effective binder volume (constant air voids). This behavior is exactly how an actual HMA mix behaves in the real world. The model was found to be rationally sensitive to changes in mix volumetrics characterized by the Va and Vbeff parameters. In other words, the predictive model is capable of accurately predicting the changes in E* stiffness due to any change in mix volumetrics. This is undoubtedly a highly desirable property of a stiffness predictive model of HMA mixtures. This property reinforces that the model can be accurately and confidently used in the mechanistic-empirical pavement design.
218
12
E* = 2,500 ksi
VMA = 16
VMA = 18
VFA = 50
10 E* = 3,500 ksi VMA = 14 VFA = 60
8 Va (%)
E* = 4,000 ksi VFA = 70
VMA = 12
6
E* = 4,500 ksi
4
VFA = 80
E* = 4,500 ksi
2 4
6
8
10
12
14
16
Vbeff (%) FIGURE 8.30 Response of Predicted E* to Mix Volumetrics at T = 14°F and f = 1 Hz
219
12 VMA = 18
VFA = 50
VMA = 16
E* = 1,800 ksi 10
E* = 2,100 ksi
VMA = 14
VFA = 60
Va (%)
8 E* = 2,400 ksi VMA = 12
6
VFA = 70
E* = 2,700 ksi
4
VFA = 80
E* = 3,000 ksi
2 4
6
8
10
12
14
16
Vbeff (%) FIGURE 8.31 Response of Predicted E* to Mix Volumetrics at T = 40°F and f = 1 Hz
220
12 VFA = 50
VMA = 18
E* = 500 ksi VMA = 16
10
VMA = 14
8 Va (%)
VFA = 60
E* = 600 ksi
VMA = 12
VFA = 70
E* = 700 ksi
6 VMA = 10
E* = 800 ksi 4
VFA = 80
2 4
6
8
10
12
14
16
Vbeff (%) FIGURE 8.32 Response of Predicted E* to Mix Volumetrics at T = 70°F and f = 1 Hz
221
12 E* = 100 ksi
VMA = 16 VMA = 18
VFA = 50
10 E* = 125 ksi VMA = 14 VFA = 60
E* = 150 ksi
Va (%)
8
VMA = 12
6 E* = 175 ksi
4
VFA = 70
VFA = 80
E* = 200 ksi
2 4
6
8
10
12
14
16
Vbeff (%) FIGURE 8.33 Response of Predicted E* to Mix Volumetrics at T = 100°F and f = 1 Hz
222
12 VMA = 18 VFA = 50
E* = 50 ksi VMA = 16
10
VMA = 14 VFA = 60
E* = 60 ksi
Va (%)
8 VFA = 70
E* = 70 ksi VMA = 12
6
E* = 80 ksi 4
VFA = 80
E* = 90 ksi
2 4
6
8
10
12
14
16
Vbeff (%) FIGURE 8.34 Response of Predicted E* to Mix Volumetrics at T = 130°F and f = 1 Hz
223 8.8
Comparison with Previous Models In the final step for selecting the recommended E* model, it is important to check
if the selected model really makes an improvement in the E* moduli prediction of HMA mixtures. One obvious method to assess this result is to check the model with previously existing models. This check is most logically based on statistical goodness of fit parameters as well as visually assessing the plots of predicted versus observed values of E* moduli. Both methods were carried out for this part of analysis.
8.8.1
Comparison of Statistical Goodness of Fit Table 8.14 shows the comparison of statistics among the five most prominent
models: Shell Oil (2nd version), Shell Oil (1977 version), current Witczak model (1999), Hirsch model (2003), and the new revised Witczak model. To be consistent with the comparison, the full range of the master E* database was applied to all models. Goodness of fit parameters (Se/Sy and R2 ) were calculated in both arithmetic (normal) and logarithmic scale. It should be noted that due to the inability of some models to predict over a practical input range; they give irrational values of the goodness of fit parameters or no value at all. This was the case for the Shell Oil (2nd version) model, where no rational Se/Sy or R2 could be calculated. For the 1977 version of the Shell Oil model, Se/Sy (= 0.93) and R2 (= 0.14) could be calculated only in the arithmetic scale. Goodness of fit is “poor” for this statistics.
224 The current version (1999) of the Witczak E* predictive model has “excellent” goodness of fit statistics for its original database of 2750 data points. The Se/Sy = 0.25 and R2 = 0.94 results in logarithmic scale (Se/Sy = 0.34 and R2 = 0.89 in arithmetic scale). However, when this model is applied to the master database having 7400 data points; the goodness of fit statistics decreases with Se/Sy = 0.35 and R2 = 0.88 in logarithmic scale (Se/Sy = 0.60 and R2 = 0.65 in arithmetic scale). This still places the model in the “good” category but clearly illustrates the statistical fact that goodness of fit parameters are highly dependent upon the number of observations or the range of database variables encountered. It is extremely critical to observe that the Hirsch E* Predictive model had “excellent” goodness of fit statistics for the original database having 206 data points. For these conditions, an R2 = 0.98 in logarithmic scale was found. However, when applied to the expanded master database having 7400 data points; the goodness of fit statistics deteriorated drastically to a R2 = 0.61 in logarithmic scale. When applied in the arithmetic scale, the model showed even poorer goodness of fit statistics with R2 = 0.23. It is therefore a major conclusion that the use of the Hirsch predictive model results in extremely poor predictive accuracy when fully evaluated under a comprehensive set of HMA mixes and series of E* data points. Finally, the new E* predictive model showed “excellent” goodness of fit statistics with Se/Sy = 0.32 and R2 = 0.90 in logarithmic scale (Se/Sy = 0.45 and R2 = 0.80 in arithmetic scale). It is clear from the tabular data that in both arithmetic and logarithmic
225 scale, the new E* model has the best goodness of fit statistics among all of the major models evaluated.
TABLE 8.14 Goodness of Fit Statistics of E* Models Parameters Total Number of Mixes Number of Modified Mixes Data Points Va (%) Vbeff (%) Temperature (°F) Loading Frequency, fc (Hz) E* (ksi)
Master Database Range 346 17 7400 0.1 to 18.1 6 to 25 0, 14, 15.8, 40, 100 and 130 0.1, 0.5, 1, 4, 5, 10, 16 and 25 9 to 9350
Goodness of Fit Parameters Scale Se/Sy
Shell Shell Current Witczak Hirsch New Witczak (2nd Ver.) (1977) (1999) (2003) (This Study) Arithmetic 1.3 0.93 0.60 0.88 0.45
8.8.2
R2 Scale Se/Sy
-
0.14
1.53
1.95
R2
-
-
0.65 0.23 Logarithmic 0.35 0.62 0.88
0.61
0.80 0.32 0.90
Comparison of Plots for Predicted versus Observed Data Figures 8.35 through 8.40 present plots of the predicted versus observed E*
values for the entire master E* database (7400 data points from 346 mixtures) applied to these models. The Shell Oil (2nd version) model showed the poorest prediction (Figure 8.35); while the 1977 Shell Oil ve rsion was more precise but had a heavily biased predictive capability (Figure 8.36). Figure 8.37 clearly shows that the current version of Witczak E* Predictive model has excellent prediction capability over its original database
226 having 2750 data points. But, when applied to the master database having 7400 data points, the predicting capability was reduced as shown in Figure 8.38. However, except for a small nearly constant bias, the prediction made by this model can still be considered quite reasonable. Figure 8.39 shows that the Hirsch E* predictive model has a poor to fair predictive capability for the master database. In addition, the model has a very visible bias at both very low and very high temperature regions. Figure 8.40 shows that among the models evaluated, the new Witczak E* model provided the best plot of predicted versus observed E* in terms of precision, bias and accuracy.
1000
Predicted E*, 10 5 psi
100
Arithmetic scale: R 2 = -0.68, Se/Sy = 1.30 Log scale : R 2 = -1.33, Se/Sy = 1.53
10 1 0.1 0.01
0.001 0.001
0.01
0.1
1
10
100
1000
5
Observed E*, 10 psi FIGURE 8.35 Predicted Versus Observed E* for Shell Oil (2nd Version) Model (Expanded Data)
227 1000
Predicted E*, 10 psi
100
Arithmetic scale: R 2 = 0.14, Se/Sy = 0.93 Log scale : R2 = -2.78, Se/Sy = 1.95
5
10 1 0.1 0.01
0.001 0.001
0.01
0.1
1
10
100
1000
5
Observed E*, 10 psi
FIGURE 8.36 Predicted Versus Observed E* for Shell Oil (1977) Model (Expanded Data) 1000
5
Predicted E*, 10 psi
100
Arithmetic scale: R2 = 0.89, Se/Sy = 0.34 Log scale : R2 = 0.94, Se/Sy = 0.25
10 1 0.1 0.01
0.001 0.001
0.01
0.1
1
10
100
1000
5
Obdesrved E*, 10 psi
FIGURE 8.37 Predicted Versus Observed E* for Current Witczak (1999) Model (Original 2750 Data)
228 1000 Arithmetic scale: R2 = 0.65, Se/Sy = 0.60 Log scale : R 2 = 0.88, Se/Sy = 0.35
Predicted E*, 105 psi
100 10 1 0.1 0.01
0.001 0.001
0.01
0.1
1
10
100
1000
Observed E*, 10 5 psi
FIGURE 8.38 Predicted Versus Observed E* for Current Witczak (1999) Model (Expanded Data)
1000
Predicted E*, 10 5 psi
100
Arithmetic scale: R 2 = 0.23, Se/Sy = 0.88 Log scale : R 2 = 0.61, Se/Sy = 0.62
10 1 0.1 0.01
0.001 0.001
0.01
0.1
1
10
100
1000
5
Observed E*, 10 psi FIGURE 8.39 Predicted Versus Observed E* for Hirsch (2003) Model (Expanded Data)
229
1000
5
Predicted |E*|, 10 psi .
Arithmetic Scale: R2 = 0.80, Se/Sy = 0.45 100
Log Scale
: R2 = 0.90, Se/Sy = 0.32
10 1 0.1 0.01
0.001 0.001
0.01
0.1
1
10
100
1000
Observed |E*|, 105 psi FIGURE 8.40 Predicted Versus Observed E* for New Witczak (2005) Model (Expanded Data)
8.8.3
Comparison with Hirsch Model It has been discussed in the previous section that the Hirsch model (one of the
most recent model forms) possessed excellent goodness of fit characteristics (R2 = 0.98 in log scale) for its original calibration database. However, the accuracy of the Hirsch model dramatically decreased when it utilized the comprehensive E* database of 346 mixtures and 7400 data points. The results of using the master E* database were found to be R2 = 0.23 in arithmetic scale and 0.61 in log scale. When the same database was applied to the new Witczak E* model developed in this study, it provided an R2 = 0.80 in arithmetic scale and 0.90 in log scale.
230 One of the most important criteria for developing a predictive model is to have a high quality database, which is as large and as versatile possible. Table 8.15 shows a comparison of the range of variables used in the development of the Hirsch model and the new Witczak E* model. It is very clear that in terms of number of data points and observed data range of the variables used in the model, the expanded E* database used in the development of the new E* model is much larger and more versatile than that of the Hirsch model. The new model is based on 7,400 data points obtained from 346 different mixes. As a result, the use of a high quality and very comprehensive database has made the new Witczak E* model more robust and accurate compared to other E* predictive models, including the Hirsch model, evaluated in this study. On the other hand, the Hirsch model, which was developed based on a rather narrow range of predictor variables of a small database (only 206 data points from 18 different mixes), could not fully capture the E* stiffness characteristics of HMA outside its original database range.
TABLE 8.15 Range of Variables Used in the Development of Hirsch and New Model Parameters Data Points AC Mixtures Aggregate Gradations Binders Va % VMA, % VFA, % Temperature, °F Loading Frequency, Hz |E*|, ksi
Hirsch Model (2003) 206 18 5 8 5.6 to 11.2 13.7 to 21.6 38.7 to 68.0 40, 70 and 100 0.1 and 5 27 to 3031
New Witczak Model (from this study) 7400 346 136 124 0.1 to 18.1 10.3 to 34.6 32.8 to 99.4 0, 14, 15.8, 40, 70, 100 and 130 0.1, 0.5, 1, 4, 5, 10, 16 and 25 9 to 9350
231 8.9
Final New Witczak E* Model A number of candidate E* models were developed and evaluated in this Ph.D.
research. Each candidate was tested for rationality, accuracy, precision, bias, trend, sensitivity and overall performance. While all candidate models were evaluated in the same fashion, the evaluation of the most promising model has been presented in this chapter. The final model found to be the best in terms of accuracy, precision and rationality was as follows:
(
log 10 E* = −0.349 + 0.754 | Gb * | −0 .0052
)
6 .65 − 0 .032 ρ 200 + 0.0027 ρ 200 2 + 0.011 ρ 4 − 0.0001 ρ 4 2 Vbeff + 0.006 ρ 38 − 0.00014 ρ 38 2 − 0.08 V a − 1.06 Va + V beff
Vbeff + 0 .0124 ρ 38 − 0 .0001 ρ 38 2 − 0.0098 ρ 34 2.558 + 0 .032 Va + 0 .713 V a + Vbeff + (−0.7814−0 .5785log|G b *| + 0. 8834 log δ b ) 1+ e
(8.1) where, E*
= dynamic modulus, psi
ρ200
= percentage of aggregates (by weight) passing through no. 200 sieve, %
ρ4
= aggregates (by weight) retained on no. 4 sieve, %
ρ38
= aggregates (by weight) retained on the 3/8 inch sieve, %
ρ34
= aggregates (by weight) retained on the 3/4 inch sieve, %
Va
= air voids (by volume of the mix), %
Vbeff
= effective binder content (by volume of the mix), %
|Gb*|
= dynamic shear modulus of binder, psi
δb
= phase angle of binder associated with |G b*|, degree
9
CONCLUSIONS AND RECOMMENDATIONS
9.1
Introduction The main goal of the research study presented in this Ph.D. dissertation was to
develop an enhanced version of the Witczak dynamic modulus (E*) predictive model for HMA mixtures capable of estimating changes in modulus as a function of changes in mixture volumetrics, aggregate gradation, binder properties, temperature and loading frequency (or time) for the combined database of E* lab results. Of equal importance was the goal of incorporating the complex shear modulus (G b*) parameter as the primary source of quantifying the asphalt binder stiffness, in lieu of using asphalt binder viscosity. In order to achieve this goal, a master database of mixture dynamic modulus (E*) test data, binder complex (shear) modulus (G b*) test data, binder viscosity and Ai-VTS i data, and all relevant material and mixture data was compiled and updated first. Next, a set of binder stiffness predictive models capable of accurately predicting viscosity, shear modulus and phase angle of an asphalt binder from given “A” and “VTS” values, as obtained from the conventional ASTM Ai-VTSi relationship, was developed. Finally, using the new set of binder stiffness models, a new revised enhanced version of the Witczak E* predictive model for HMA mixtures was eventually developed. The findings of the research were thoroughly analyzed. The conclusions and recommendations follow.
9.2
Conclusions The following conclusions have been drawn from this Ph.D. research: 1. In this research, two huge complex shear modulus and viscosity (Gb*-η) databases, one collected from research studies by Dr. Witczak at the University of
233 Maryland (UMD) and the other from research studies by Dr. Witczak at the Arizona State University (ASU), were combined into one master η-Gb* database. The database contains Gb* data obtained from Dynamic Shear Rheometer (DSR) test and ASTM Ai-VTS i data obtained from a range of conventional and Superpave binder tests conducted on 41 different types of asphalt binders having a wide range of modifications and aging (9 modified binders and 5 aging conditions). The types of modifier used in the binders include: lake asphalt, ge ltype modifier, and polymer- modifier (plastomeric, elastomeric and SBS type). The master database contains binder data for five different aging conditions: tank (original), RTFO, PAV at 100°C, PAV at 110°C, and recovered. The test temperatures ranged from 15°C to 177°C (59°F to 350°F) and the test loading rates ranged from 1 to 100 radian per second (0.2 to 16 Hz). In total, there are 8,940 data points in this master Gb*-η database. 2. The master Gb*-η database was used to develop a comprehensive set of new asphalt binder stiffness (η-|Gb*|-δ b) models. These models include a modified ASTM Ai-VTS i model, a new |Gb*| model and associated new phase angle (δ b) model. 3. A fully revised version of the widely known ASTM Ai-VTS i viscosity model has been developed. As opposed to the original viscosity based model, which ignores the effect of loading rate, the new model is capable of taking care of both the temperature and loading rate. The final modified ASTM Ai-VTSi model equations developed are as follows:
234 log log η f , T = A'+VTS ' log TR
(9.1)
A' = 0.9699 f − 0.0527 × A
(9.2)
VTS ' = 0.9668 f − 0. 0575 × VTS
(9.3)
where, ηf, T
= viscosity of asphalt binder as a function of both loading frequency (f) and temperature (T), cP
A
= regression intercept from the conventional ASTM Ai-VTS i equation
VTS
= slope from the conventional ASTM Ai-VTS i equation
A'
= adjusted “A” (adjusted for loading frequency)
VTS' = adjusted “VTS” (adjusted for loading frequency) TR
= temperature in Rank ine scale, °R
4. A new rational model for predicting shear modulus (|Gb*|) of asphalt binders from typical viscosity data has been developed. The model equation is as follows: 7 .1542 − 0. 4929 f s + 0 .0211 f s 2
| Gb * | = 0.0051 f η f , T (sin δ b )
(9.4)
where, |Gb*|
= dynamic shear modulus, Pa
fs
= loading frequency in dynamic shear loading mode as used in the DSR test to measure |Gb*| and δ b, Hz
ηf, T
= viscosity of asphalt binder as a function of both loading
235 frequency (f) and temperature (T), cP δb
= phase angle (predicted or measured), deg
The model is fully optimized. The new η-|Gb*| model is based on 8940 data points from 41 binders (including 9 modified binders). The model has excellent goodness of fit statistics. In arithmetic scale, the R2 = 0.83 and Se/Sy = 0.41; while in logarithmic scale, the R2 = 0.99 and Se/Sy = 0.12. The model was critically tested for accuracy and rationality, and has been found to have a very high level of accuracy and rationality over the full range of the master η-Gb* database evaluated in this research. 5. During the same study, a new rational model for predicting phase angle (δ b) associated with the complex shear modulus (Gb*) testing was also developed. The new phase angle (δ b) model equations are as follows:
δ b = 90 + (b1 + b2VTS ' ) × log ( f s × η f , T ) + (b3 + b4VTS ' ) × log ( f s ×η f , T ) 2 (9.5) log log η f , T = A'+VTS ' log TR
(9.6)
A' = c0 f s 1 × A
(9.7)
VTS ' = d0 f s 1 × VTS
(9.8)
c
d
where, δb
= phase angle, deg
A'
= adjusted A (adjusted for loading frequency)
VTS' = adjusted VTS (adjusted for loading frequency)
236 fs
= loading frequency in dynamic shear loading mode as used in the DSR test to measure |Gb*| and δ b, Hz
ηf, T
= viscosity of asphalt binder as a function of both loading frequency (f) and temperature (T), cP
TR
= temperature in Rankine scale, °R
b1 , b2 , b3 , b4 , c0 , c1 , d0 , and d1 = fitting parameters = -7.3146, -2.6162, 0.1124, 0.2029, 0.9699, -0.0527, 0.9668, and -0.0575, respectively. Like the η-|Gb*| models developed in this research, this new phase angle (δ b) model also possesses very high goodness of fit. In arithmetic scale, the R2 = 0.81 and Se/Sy = 0.44; while in logarithmic scale, the R2 = 0.82 and Se/Sy = 0.42. The model was also critically tested for accuracy and rationality, and has been found to exhibit high accuracy and rationality over the full range of the master η-Gb* database evaluated in this research. One very important difference of this model from any previous model is that it can predict δ b without the input value of |Gb*|. 6. Similar to the binder databases, two complex dynamic modulus (E*) databases, one obtained from prior UMD research studies and the other from ASU research studies (both overviewed by Dr. M. W. Witczak), were combined into one master E* database. As part of this research, the ASU database was created from raw laboratory data, checked thoroughly for the quality of data and revised. The revised master database, which was eventually used in the research, contains 7400 E* data points obtained from Complex Dynamic Modulus tests conducted at a very wide range of temperature and loading conditions. A total of 346 different
237 types of HMA mixtures, having a wide range of aging, volumetric properties, aggregate gradation and binder characteristics comprises the database. Modified binders were used in 17 of these mixtures. The database contains data related to E* test that include: laboratory E* values, temperature (T), loading frequency (f), geometry of test specimen, type of mixture (aged, un-aged, lab blended, plant mixed, field cores etc.), aggregate gradation (ρ200 , ρ4 , ρ38 and ρ34 ), mixture volumetrics (Va and Vbeff), and binder characteristics (PG grading, Ai and VTS i). The related binder shear modulus (Gb*) data was obtained by the use of the comprehensive set of asphalt binder stiffness (η-|Gb*|-δ b) models developed in the earlier stage of this research. The E* test temperature ranged from 0 to 130°F and loading rate ranged from 0.1 to 25 Hz. The database contained HMA mixtures of a wide variety of aging conditions: fresh mix (un-aged), short-term laboratory aged mix, plant mix and field aged core. The database also has a wide variety in terms of mixing process; there are laboratory blend mixes, fresh and stored plant mixes, and field cores. The aggregate gradation includes dense, open and gap gradations. In general, the master E* database is made of a very wide variety of HMA mixtures. 7. Based on the master E* database, a new comprehensive dynamic modulus (E*) stiffness predictive model has been developed in this research. The model is a completely new revised version of the existing Witczak E* Predictive Model used in the new NCHRP 1-37A M-E Pavement Design Guide (M- E PDG). The new E* model equation, still based on the sigmoidal function, is as follows:
238
(
log 10 E* = −0.349 + 0.754 | Gb * | −0 .0052
)
6 .65 − 0 .032 ρ 200 + 0.0027 ρ 200 2 + 0.011 ρ 4 − 0.0001 ρ 4 2 Vbeff + 0.006 ρ 38 − 0.00014 ρ 38 2 − 0.08 V a − 1.06 Va + V beff
Vbeff + 0 .0124 ρ 38 − 0 .0001 ρ 38 2 − 0.0098 ρ 34 2.558 + 0 .032 Va + 0 .713 V a + Vbeff + (−0.7814−0 .5785log|G b *| + 0. 8834 log δ b ) 1+ e
(9.9) where, E*
= dynamic modulus, psi
ρ200
= percentage of aggregates (by weight of the total aggregates) passing through no. 200 sieve, %
ρ4
= aggregates (by weight) retained on no. 4 sieve, %
ρ38
= aggregates (by weight) retained on the 3/8 inch sieve, %
ρ34
= aggregates (by weight) retained on the 3/4 inch sieve, %
Va
= air voids (by volume of the mix), %
Vbeff
= effective binder content (by volume of the mix), %
|Gb*|
= dynamic shear modulus of binder, psi
δb
= phase angle of binder associated with |G b*|, degree
8. As can be seen from the model equation, the new E* model uses a range of aggregate gradation parameters (ρ200 , ρ4 , ρ38 and ρ34 ), mix volumetric parameters (Va and Vbeff), and loading time-temperature dependent binder rheological parameters (|G b*| and δ b) as direct input. The use of |Gb*| and δ b as direct input is a major enhancement over the current Witczak E* Predictive Model. It was earlier
239 pointed out that the data required to establish the ASTM Ai-VTS i relationship may no longer be routinely collected because of the adoption of the Superpave Performance Grading (PG) system and its associated testing. A major limitation associated with the use of the ASTM Ai-VTS i relationship is that it does not incorporate changes of binder viscosity under dynamic loading with changing loading time (or frequency). Hence, the use of |G b*| and δ b as direct input in the new E* model makes the model much more rational and consistent with future technology from the SHRP (Superpave) studies compared to the existing models. 9. The newly developed E* model has a mathematical structure of a sigmoidal function similar to the current Witczak E* Predictive model. This model form is also used in the new NCHRP 1-37A M-E Pavement Design Guide (M-E PDG). Therefore, it can be easily implemented within the current structure of the new ME PDG with a minimum additional effort. 10. The new E* model possesses excellent statistics: high accuracy and almost no bias. It was critically evaluated over a wide range of practical limits of aggregate gradation, volumetric and rheological predictor variables. The prediction showed very good to excellent agreement with the observed E* values across all data ranges. 11. The prediction of E* stiffness showed a completely rational response to air voids (Va) and effective binder volume (V beff). The rational behavior of the new model, relative to both Va and Vbeff, was observed from the contour plots of the predicted E* at different temperatures. From the contour plots, it was clear that regardless of temperature, the E* stiffness increased with decreasing air voids (constant
240 bitumen volume) and decreased with increasing effective binder volume (constant air voids). This behavior is identical with the behavior of actual HMA mixtures. The model was found rationally sensitive to changes in mix volumetrics characterized by Va and Vbeff parameters. In other words, the predictive model is capable of accurately predicting the changes in E* stiffness due to any change in mix volumetrics. 12. It was found that the new E* model showed the best the best goodness of fit statistics, least bias and highest accuracy over previous E* models for the full range of the E* master database evaluated in this research. In fact, the newly developed model has been developed on the largest and most varied E* database ever assembled in the literature. Almost all of the previous predictive models were based on a much smaller database, comprised of a much smaller number of mixtures. Hence, it captures the dynamic modulus characteristics of various HMA mixtures in a much more accurate manner.
9.3
Recommendations for Future Research Based upon this study, the following major future research recommendations are
presented: 1. The new models developed for the prediction of dynamic modulus for asphalt binders and dynamic modulus for HMA mixtures have similar structures as the models used in the new NCHRP 1-37A M-E Pavement Design Guide (M-E PDG) procedures. This will allow the easy implementation of the new models, developed in this research, into the new design guide. Therefore, it is strongly
241 recommended that future research efforts should focus upon implementing the new models into future versions of this design guide. 2. The new models have been developed on binder stiffness and mixture modulus databases available at the time this research was carried away. Nonetheless, a concerted national effort should be undertaken to continuously expand the current database. It is recommended that these databases are continuously expanded and periodically revised (re-calibrated) models be developed. 3. The models developed in this research are capable of accurately predicting the stiffness characteristics of most binders and HMA mixtures. However, it is highly recommended that further test results be obtained on special binders and mixtures such as binder or mixture modified with polymer modifier, rubber and lime, and for special aggregate gradation such as stone mastic asphalt (SMA), gap graded mix and open graded mixtures. These “specialty” mixtures should be given uppermost priority in expanding the databases (noted in Recommendation No. 2). 4. While the new E* model showed excellent predictive strength over a wide range of temperature and loading rates, it was also noted that the weakest predictive accuracy is present in the very cold temperature region. Future research should focus on eliminating this problem from new models. 5. The new revised version of the Witczak E* predictive model has exclusively been based upon the prediction of dynamic modulus, experimentally obtained from unconfined uniaxial compressive E* testing. One potential future area of research should be directed to developing a similar E* predictive model incorporating the results of confined uniaxial E* compressive testing. The “capturing” of the non-
242 linear (stress dependent) E* response will be most important for the high temperature response of HMA mixtures and almost mandatory to properly evaluate the most accurate models for open/gap graded aggregate AC mixtures.
REFERENCES 1.
Roberts, F. L., P. S. Kandhal, E. R, Brown, D. Y. Lee, and T. W. Kennedy. Hot Mix Asphalt Materials, Mixture Design, and Construction, NAPA Education Foundation, Lanham, Maryland, 1996.
2.
Van der Poel, C. A General System Describing Visco-Elastic Properties of Bitumens and Its Relation to Routine Test Data. Journal of Applied Chemistry, Vol. 4, Part 5, pp. 221-236, and Shell Bitumen Reprint No 9, 1954.
3.
2002 Design Guide: Design of New and Rehabilitated Pavement Structures. NCHRP 1-37A Project, National Cooperative Highway Research Program. TRB, Washington, D.C., March 2004.
4.
Witczak M. W. et al. Simple Performance Test: Test Results and Recommendations. Interim Task C Report, Superpave Support and Performa nce Models Management (NCHRP 9-19) Project. Arizona State University, Tempe, AZ, 2001.
5.
Witczak, M.W., K. Kaloush, T. Pellinen, and M. El-Basyouny. Simple Performance Test for Superpave Mix Design. NCHRP Report 465, National Cooperative Highway Research Program, Washington, D.C., 2002.
6. Andrei, D., M. W. Witczak, and M. W. Mirza. Development of a Revised Predictive Model for the Dynamic (Complex) Modulus of Asphalt Mixtures. Inter Team Technical Report prepared for the NCHRP 1-37A Project. Department of Civil Engineering, University of Maryland, College Park, MD, March 1999. 7.
Fonseca, O. A. Development of a Time-Dependent Model for the Dynamic Modulus of Asphalt Mixes. PhD. Dissertation, University of Maryland, College Park, 1995.
244 8.
Witczak M. W., J. Bari, and A. Sotil. Dynamic Modulus (E*) Test and Master Curve Database. Inter Team Technical Report prepared for the Superpave Support and Performance Models Management (NCHRP 9-19) Project. Department of Civil and Environmental Engineering, Arizona State University, Tempe, AZ, February 2004.
9.
Viscosity-Temperature Chart for Asphalts. American Society for Testing and Materials (ASTM). 1998 Annual Book of ASTM Standards. Vol. 4.03, pp. 230-234, 1998.
10. Performance Graded Asphalt Binder Specification and Testing. Superpave Series No. 1 (SP-1), Asphalt Institute, Lexington, Kentucky, 1996. 11. Bonaquist, R., T. K. Pellinen, and M. W. Witczak. Development of Relationship Between Binder Viscosity and Stiffness. University of Maryland, College Park, MD, 1998. 12. Witczak, M. W., J. Bari, K. Kaloush, and T. K. Pellinen. ADOT Asphalt Cement Binder Characterization Database. Project 2, ASU-ADOT Research Program, Development of Performance Related Specifications for Asphalt Pavements in the State of Arizona. Arizona State University, Tempe, AZ, December, 2000. 13. Witczak, M. W., and J. Bari. ADOT Asphalt Cement Binder Characterization Database - Volume II. Supplementary to ADOT AC Binder Characterization Database Project, ASU-ADOT Research Program, Development of Performance Related Specifications for Asphalt Pavements in the State of Arizona. Department of Civil and Environmental Engineering, Arizona State University, Tempe, AZ, March, 2001.
245 14. Bari, J. Investigation of the Rheological Properties of Typical Asphalt Binders Used in Arizona. M.S. Thesis. Arizona State University, Tempe, AZ, May 2001. 15. Bonnaure, F., G. Guest, A. Gravois, and P. Uge. A New Method of Predicting the Stiffness of Asphalt Paving Mixtures. Journal of the Association of Asphalt Paving Technologists. Vol. 46, 1977. 16. Pellinen, T. K. Investigation of the Use of Dynamic Modulus As An Indicator of HotMix Asphalt Performance. Ph.D. Dissertation. Arizona State University, Tempe, Arizona, May 2001. 17. Dongré, R. L. Myers, J. D’Angelo, C. Paugh, and J. Gudimettla. Field Evaluation of Witczak and Hirsch Models for Predicting Dynamic Modulus of Hot-Mix Asphalt. Journal of the Association of Asphalt Paving Technologists (AAPT), Vol. 74, 2004. 18. Mirza, M. W., and M. W. Witczak. Development of a Global Aging System for Short and Long Term Aging of Asphalt Cements. Proc., Association of Asphalt Paving Technologists, AAPT, Vol. 64, pp. 393-431, 1995. 19. Kandhal, P.S. Low-Temperature Properties of Paving Asphalt Cements, Transportation Research Board, State-of-the-Art Report 7, 1988. 20. Finn, F. N. Factors Involved in the Design of Asphaltic Pavement Surfaces. HRB, National Cooperative Highway Research Program (NCHRP) Report 39, 1967. 21. Vallerga, B.A., C.L. Monismith, and K. Granthem. A Stud y of Some Factors Influencing the Weathering of Paving Asphalts. Proc., Association of Asphalt Paving Technologists, Vol. 26, 1957. 22. Traxler, R.N. Durability of Asphalt Cements. Proc., the Association of Asphalt Paving Technologists, AAPT, Vol. 32, 1963.
246 23. Kandhal, P.S. and M. E. Wenger. Asphalt Properties in Relation to Pavement Performance. In Transportation Research Records (TRR) 544, TRB, National Research Council, Washington, D.C., 1975. 24. Christensen, D. W. Analysis of Creep Data From Indirect Tension Test on Asphalt Concrete. Journal of the Association of Asphalt Paving Technologists (AAPT), Vol. 67, pp. 458-492, 1998. 25. Goodrich, J. L. Asphaltic Binder Rheology, Asphalt Concrete Rheology and Asphalt Concrete Mix Properties. Journal of the Association of Asphalt Paving Technologists, AAPT, Vol. 60, pp. 81-121, 1991. 26. Witczak, M. W., D. Z. Sybilski, R. Bonaquist, and T. K. Pellinen. Development of Relationship Between Binder Viscosity and Stiffness. Interim Report, Superpave Support and Performance Models Management (NCHRP 9-19) Project. Arizona State University, Tempe, AZ, 2000. 27. Mix Design Methods for Asphalt Concrete and Other Hot-Mix Types. Manual Series No. 2 (MS-2), 6th Edition, Asphalt Institute, Lexington, Kentucky, 1993. 28. AASHTO Guide for the Design of Pavement Structures. American Association of State Highway and Transportation Officials (AASHTO), Washington, D.C., 1993. 29. AASHTO Standard Method of Test for Determining Dynamic Modulus of Hot-Mix Asphalt Concrete Mixtures. AASHTO Designation: TP 62-03. American Association of State Highway and Transportation Officials (AASHTO), Washington, D.C., 2003.
247 30. Christensen Jr., D. W., T. K. Pellinen, and R. F. Bonaquist. Hirsch Model for Estimating the Modulus of Asphalt Concrete. Journal of the Association of Asphalt Paving Technologists (AAPT), Vol. 72, 2003. 31. Huang, Y. H. Pavement Analysis and Design. Prentice Hall, Englewood Cliffs, New Jersey, 1993. 32. Shell Pavement Design Manual – Asphalt Pavements and Overlays for Road Traffic. Shell International Petroleum, London, 1978. 33. Shook, J. K., and B. F. Kallas. Factors Influencing the Dynamic Modulus of Asphalt Concrete. Proc., Association of Asphalt Paving Technologists, Los Angeles, CA, 1969. 34. Witczak, M. W. Design of Full- Depth Asphalt Airfield Pavements. Proc., Third International Conference on Structural Design of Asphalt Pavements, Volume I, London, 1972. 35. Thickness Design – Asphalt Pavements for Air Carrier Airports. Manual Series No. 1 (MS-11), 3rd Edition. Asphalt Institute, College Park, MD, September 1981. 36. Miller, J. S., J. Uzan, and M. W. Witczak. Modification of the Asphalt Institute’s Bituminous Mix Modulus Predictive Equation. In Transportation Research Records (TRR) 911, TRB, National Research Council, Washington, D.C., 1983. 37. Akhter, G. F., and M. W. Witczak. Sensitivity of Flexible Pavement Performance to Bituminous Mix Properties. In Transportation Research Records (TRR) 1034, TRB, National Research Council, Washington, D.C., 1985.
248 38. Witczak, M. W., R. B. Leahy, J. Caves, and J. Uzan. The Universal Airport Pavement Design System, Report II: Asphaltic Mixture Material Characterization. University of Maryland, College Park, MD, May 1989. 39. Fonseca, O. A., and M. W. Witczak. A Predictive Methodology for the Dynamic Modulus of In-Place Aged Asphalt Mixtures. Journal of the Association of Asphalt Paving Technologists, Vol. 65, 1996. 40. A Guide to Calibration and Adaptation of HDM-4. The World Road Association (PIARC). The International Study of Highway Development and Management (ISOHDM), August 2001. 41. Lasdon, L.S., A.D. Waren, A. Jain, and M. Ratner. Design and Testing of a Generalized Reduced Gradient Code for Nonlinear Programming. ACM Transactions on Mathematical Software. Association for Computing Machinery, New York, New York, Volume 4, March 1978. 42. Gershkoff, D. Polymer Modified Bitumens. Performance in Empirical and Rheological Tests. Proc., First European Workshop on the Rheology of Bituminous Binders, Eurobitume, Brussels, 5-7 April 1995. 43. Molenaar J. Rheology of Bitumen. Eurobitume Workshop 99. Performance Related Properties for Bituminous Binders. Eurobitume, Luxembourg, 1999. 44. Heukelom, W. An Improved Method of Characterizing Asphaltic Bitumens with the Aid of Their Mechanical Properties. Journal of the Association of Asphalt Paving Technologists, AAPT, Vol. 42, 1973. 45. Cox W.P., E.H. Merz. Correlation of Dynamic and Steady State Flow Viscosities. Journal of Polymer Science, Vol. 28, 1958.
249 46. Standard Test Method for Dynamic Modulus of Asphalt Concrete Mixtures. National Cooperative Highway Research Program 1-37A. Provisional Test Method DM-1. Arizona State University, Tempe, AZ, June 2002. 47. Montgomery, D. C. Design and Analysis of Experiments. John Wiley & Sons, Inc., Fifth Edition, 2001.
BIOGRAPHICAL SKETCH
Javed Bari was born in Dhaka, Bangladesh, on April 27, 1969. He received his elementary education at Gopalgonj Model School, Rangamati Government School and Ideal Primary School. Then he received secondary educatio n at Ideal High School and Barisal Zilla School. His secondary education was completed at Dhaka College. In 1988, Javed entered Bangladesh University of Engineering and Technology (BUET), Dhaka, majoring in Civil Engineering. Upon graduation in 1993, he joined the Local Government Engineering Department (LGED) of the Government of Bangladesh. As an officer of Bangladesh Civil Service (BCS), Javed joined the Public Works Department (PWD) in March 1998. Later, in January 1999, he joined the Roads and Highways Department (RHD) of the Government of Bangladesh. While in service, he completed the required course works for the Master of Science in Engineering (MSE) in Environmental Engineering at BUET, Dhaka. In January 2000, Javed entered the Graduate College at Arizona State University (ASU), Arizona, USA, to pursue a Master of Science (MS) in Civil Engineering with a concentration in the field of Transportation Engineering. After obtaining the MS degree in May 2001, he joined the Civil and Environmental Engineering Department at ASU as a Senior Research Specialist. He started his Ph.D. studies in 2002 with a concentration in the field of Pavement Materials. During his graduate studies at ASU, he was heavily involved in advance pavement materials testing and research under the guidance of Dr. Matthew W. Witczak, a nationally renowned professor. Javed made a good number of technical publications; some were presented and published in several internationally renowned conferences and journals such as TRB, AAPT, ASC etc. In early 2005, Javed obtained the Professional Engineering (P.E.) certification in Arizona. He joined Arizona Department of Transportation (ADOT) in May 2005. Currently he is working as a Pavement Design Engineer in the Pavement Design Section within the Materials Group of ADOT. Javed enjoyed scholarships at all levels of his elementary, secondary, college and graduate studies. He received the “Chancellor’s Award” given by the President of Bangladesh for outstanding achievement in the secondary level-education. Javed is married with Rozina Ahad for more than ten years. The couple has a seven year-old daughter, Faria Tabassum. Javed is heavily involved in various cultural and literary activities in the society. Currently he is an active member and General Secretary of a charitable non-profit organization “Bangladesh Theater of Arizona” (BTA), Director and teacher of a language school “Shikor Bangla School” and member of the editorial board of a literary magazine “Shiri”.
