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THESIS

 

2 A
200' LIBRARY
Mlchigan State
University

 

 

 

This is to certify that the
thesis entitled

EFFECT OF TEMPERATURE AND LOADING TIME ON THE STIFFNESS
PROPERTIES OF HMAC IN FLEXIBLE PAVEMENTS

presented by

Dong-Yeob Park

has been accepted towards fulﬁllment
of the requirements for

Ph.D. degree in Civil Engineering

 

 

 

 

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EFFECT OF TEMPERATURE AND LOADING TIME ON THE STIFFNESS
PROPERTIES OF HMAC IN FLEXIBLE PAVEMENTS

By

Dong-Yeob Park

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

2000

ABSTRACT

EFFECT OF TEMPERATURE AND LOADING TIME ON THE STIFFNESS
PROPERTIES OF HMAC IN FLEXIBLE PAVEMENTS

By

Dong-Yeob Park

Surface deﬂections and backcalculated layer moduli of ﬂexible pavements are
signiﬁcantly affected by the temperature of the asphalt concrete (AC) layer. The
correction of these deﬂections and moduli to a reference temperature requires the
determination of an effective temperature of the AC layer. In light of this, a new
temperature prediction model for determining the AC temperature on the basis of a
database approach is presented, and temperature correction factors for AC modulus are
developed. Temperature data points (317) and deﬂection proﬁles (656) were collected
from six in-service test sites in Michigan. Temperature data points (197) from three of the
test sites were used to develop the temperature prediction model, and data from the
remaining sites were used for validation. The developed temperature prediction model
has a R2 greater than 90% and an F-statistic signiﬁcantly greater than 1.0. For further
validation of the temperature prediction model, temperature data points (18444) ﬁ'om
seven Seasonal Monitoring Program (SMP) sites (Colorado, Connecticut, Georgia,
Nebraska, Minnesota, South Dakota, Texas) were obtained ﬁ'om the LTPP Database
(DATAPAVE 2.0). The validation results suggest that the model could be adopted to all
seasons and other climatic and geographic regions. The major improvements over

existing models are: (a) the model does not require temperatures for the previous 5 days,

(b) it takes into account temperature gradients due to diurnal heating and cooling cycles,
and (c) it needs fewer parameters than other published models. The effect of temperature
prediction error on the performance prediction was also investigated. Temperature
proﬁles obtained ﬁ'om the temperature prediction and correction study were used in the
following structural analysis.

The temperature-dependent behavior of ﬂexible pavement is due to viscoelastic
properties of the AC layer. Hence, in the second part of this study, 2-D and 3-D ﬁnite
element analyses (FEA) of ﬂexible pavements were performed to investigate the
inﬂuence of realistic temperature distributions and dynamic loads on pavement responses
(mainly, stress, strain, and dissipated energy). Parametric studies (AC thickness, base
stifﬁiess, loading condition, and temperature distribution across the AC layer) were ﬁrst
conducted with a 2-D axisymmetric ﬁnite element (FE) model. Effects of three
temperature distributions (night, morning, and day) and three loading types (load case I -
uniform vertical load over the entire load area, load case 11 - uniform vertical load only
under tire treads, and load case 111 - measured vertical and lateral stresses under tire
treads) on the structural response were further investigated with a 3-D FE model. The
evaluations ﬁ‘om 2-D and 3-D analyses were consistent. These results could explain the
occurrence of top-down cracking in AC pavements under certain conditions, and
contribute to the development of an improved performance model and/or asphalt

pavement design program based on advanced material characterization and dynamic

loads.

Copyright by
Dong-Yeob Park
2000

To my family, for their support and love

ACKNOWLEDGMENTS

The author would like to express his deepest gratitude to his major advisor Dr.
Neeraj Buch for his patience, guidance, and encouragement throughout this research.
Gratitude is also expressed to the doctoral committee members Dr. Karim Chatti, Dr.
Ronald Harichandran, and Dr. Comic Page for their valuable advice. The author
expresses special thanks to Tom Hynes, Kurt Bancroft, Frederick Carian, and the trafﬁc
control crew of the Michigan Department of Transportation (MDOT) for their assistance
during ﬁeld data collection. The authors are grateful to Dr. Dar-Hao Chen and Dr. Mike
Murphy of Texas DOT and Mr. John Rush of LTPP Customer Support Service Center for
providing temperature data.
Finally, the author grateﬁilly acknowledges the ﬁnancial support provided by the

IVIDOT, Pavement Research Center of Excellence (PRCE) and Composite Materials and

Structures Center (CMSC) at Michigan State University.

vi

TABLE OF CONTENTS

LIST OF TABLES ................................................................................................ ix
LIST OF FIGURES ............................................................................................... x
CHAPTER
1 INTRODUCTION ..................................................................................... 1
1.1 BACKGROUND ............................................................................ 1
1.1.1 Temperature Prediction Model and Correction Factor
for AC Modulus .................................................................. 1
1.1.2 Theoretical Analysis of Flexible Pavements ........................ 2
1.2 OBJECTIVE. . .............................................................................. 3
1.3 SCOPE ........................................................................................... 4
1.4 ORGANIZATION OF THESIS. .................................................... 4
2 LITERATURE REVIEW I - TEMPERATURE PREDICTION MODEL
AND CORRECTION FACTOR FOR AC MODULUS ............................. 6
2.1 GENERAL ..................................................................................... 6
2.2 REVIEW OF TEMPERATURE PREDICTION AND
CORRECTION PROCEDURE FOR AC MODULI ....................... 7
2.2.1 AASHTO Guide Temperature Prediction and Correction
Procedure. .......................................................................... 7
2.2.2 BELLS Temperature Prediction Model. .............................. 8
2.2.3 BELLSZ Temperature Prediction Model. ............................ 13
2.2.4 LTPP AC Pavement Temperature Models ........................... 14
2.2.5 Asphalt Institute (AI) Model. .............................................. 15
2.2.6 Temperature Prediction Model Based on Heat Transfer
Theories and Correction Procedure. .................................... 16
2.2.7 Temperature Correction Procedure Based on the Theory
of Linear Viscoelasticity. .................................................... 17
2.2.8 Other Temperature Correction Methods for AC moduli. ..... 18
2.2.9 Climatic Model. .................................................................. 19
2.2.10 Summary of the Existing Temperature Prediction Models... 19
2.3 APPLICATIONS TO THE FIELD/PAVEMENT DESIGN. ........... 20
LITERATURE REVIEW 11 — OVERVIEW OF THEORETICAL
ANALYSIS OF FLEXIBLE PAVEMENTS .............................................. 22
3.1 INTRODUCTION OF VISCOELASTICITY TO THE FLEXIBLE
PAVEMENT ANALYSIS .............................................................. 22
3.2 VISCOELASTICITY. .................................................................... 24
3.3 AXISYMMETRIC FINITE ELEMENT ANALYSIS. .................... 27
3.4 3-D HEXAHEDERAL AND TRIANGULAR ISOPARAMETRIC
ELEMENTS. ................................................................................. 29

3.5 SUBGRADE CHARACTERIZATION. ......................................... 30

3.5.1 Winkler Foundation. ........................................................... 30
3.5.2 Elastic Solid Foundation. .................................................... 31
3.6 APPLICATIONS OF STRUCTURAL ANALYSIS TO THE
FIELD. ........................................................................................... 31
3.7 SURFACE-INITIATED CRACKING. ........................................... 34
DATA COLLECTION — TEMPERARTURE PREDICTION MODEL
AND CORRECTION FACTOR ............................................................... 38
4.1 TEST SITES AND DATA COLLECTION. ................................... 38
4.2 TEMPERATURE PREDICTION MODEL AND CORRECTION
FACTOR DEVELOPMENT. ......................................................... 41
4.3 INFLUENCE OF TEMPERATURE PREDICTION ERROR ON
PERFORMANCE PREDICTION. ................................................. 41
FINITE ELEMENT MODEL .................................................................... 47
5.1 GENERAL ..................................................................................... 47
5.2 MODELING. ................................................................................. 49
5.2.1 Viscoelastic Material Model for the AC Layer. ................... 49
5.2.1.1 Time Integration Procedure. ................................... 52
5.2.1.2 Implementation of Temperature Effects. .................. 55
5.2.2 Boundary Conditions (Temperature). .................................. 57
5.2.3 Loading. ............................................................................. 59
5.2.4 Two-Dimensional FE Model ............................................... 61
5.2.5 Three-Dimensional FE Model. ............................................ 67

RESULTS AND DISCUSSION I — FIELD DATA ANALYSIS AND
DEVELOPMENT OF TEMPERATURE PREDICTION MODEL AND

CORRECTION FACTOR .......................................................................... 75
6.1 TEMPERATURE MEASUREMENTS. ......................................... 75
6.2 TEMPERATURE PREDICTION MODEL. ................................... 78
6.3 VALIDATION OF THE TEMPERATURE MODEL. .................... 80
6.4 TEMPERATURE CORRECTION FACTOR FOR AC

MODULUS .................................................................................... 91
6.5 INFLUENCE OF TEMPERATURE PREDICTION ON

PERFORMANCE PREDICTION. ................................................. 98
6.6 SUMMARY. .................................................................................. 99
RESULTS AND DISCUSSION II - FINITE ELEMENT ANALYSIS ...... 101
7.1 TWO-DIMENSIONAL FE MODEL .............................................. 101
7.2 THREE-DIMENSIONAL FE MODEL. ......................................... 110
7.3 SUMMARY. .................................................................................. 122
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ............... 124
8.1 SUMMARY AND CONCLUSIONS .............................................. 124
8.2 RECOMMEDATIONS FOR FUTURE RESEARCH. ........... ' ......... 1 27

viii

APPENDIX

A ADDITIONAL FIGURES AND TABLES. ............................................... 130
B EXAMPLE OF A FINITE ELEMENT ANALYSIS INPUT FILE ............. 149
REFERENCES ...................................................................................................... 156

LIST OF TABLES

Table .................................................................................................................. Page
4-1 Test Site Location and Inventory Data ....................................................... 39

4-2 Location of SP8] and SMP Sections, Sensor Location, and Pavement

Cross-Section Data. ................................................................................... 39
4-3 Measured Temperature. ............................................................................. 42
4-4 Database from Stubstad et aL (1994) .......................................................... 45
5-1 Experimental Design matrix for the Parametric Study. ............................... 62
6-1 Statistical Results of Nonlinear Regression Analysis. ................................. 79
6-2 Statistical Results of Regression Analysis for Constant “a”. ....................... 94
6-3 Regression Constant “a” at Each Test Site. ................................................ 97
A-l Parametric Study: Stresses and Strains at the Peak Load. ........................... 131

LIST OF FIGURES

Figure .................................................................................................................. Page
2-1 Predicted pavement temperature ................................................................ 8
2-2 Asphalt modulus temperature adjustment factor. ........................................ 9

2-3 Temperature variations for a day at three different depths of an AC layer... 11

2-4 Relationship between the AC modulus and temperature at three different

depths. ....................................................................................................... 11
3-1 A typical axisymmetric ﬁnite element: (a) isometric view (b) the

four-node isoparametric element. ............................................................... 28
3-2 Comparison of Winkler Foundation with elastic solid foundation. .............. 32
3-3 Deﬂection-time history plot of computed dynamic deﬂection bowl under

the FWD loading plate using viscoelastic parameters for asphalt layers ...... 34
3-4 Overview of surface- initiated longitudinal wheel path cracks. .................... 35
3-5 Core showing surface-initiated longitudinal wheel path cracks. .................. 35

3-6 Vertical contact stresses measured for bias ply, radial, and wide-base

radial tires at appropriate rated load and inﬂation pressure. ........................ 36
4-1 Schematic of manual temperature measurements. ...................................... 40
5-1 Nonuniform vertical stress distribution under a wide-base tire. .................. 48
5-2 An example of stress-relaxation data obtained at different temperature. ..... 50

5-3 Master relaxation modulus and shift factor of the asphalt mixtures used

in the analysis. ........................................................................................... 51
5-4 Curve ﬁt of master relaxation modulus to the Prony series. ........................ 53
5-5 Prony series strains. ................................................................................... 54
5-6 Temperature distributions along the AC layer used in the analysis. ............ 58
5-7 FWD load pulse used in the analysis. ......................................................... 60

xi

5-8

5-9

5-10

6-3

6-4

6-5

6-6

6-8

6-9

6-10

6—11

6-12

Two-dimensional model for a three-layer system. ...................................... 63

Illustration of applied loads in the 2-D axisymmetric analysis. ................... 65
Applied nonuniform pressure versus time and location ............................... 66
Three-dimensional model for a three-layer system . ................................... 69
Illustration of applied loads in the 3-D analysis. ......................................... 71
Measured tire pressure used in the analysis. ............................................... 72

Variation of air, surface, and subsurface temperature over a 24 hour
period at site 45 (Traverse City, MI). ......................................................... 76

Measured temperature at the surface (a) and the mid-depth (b) vs. AC
modulus at site 25 (New Buffalo, MI). ....................................................... 77

Temperature change as a function of pavement depth and time of
measurement at site 25 (New Buffalo, MI) ................................................. 78

Measured temperature vs. predicted temperature at test sites (88, 45, 25)... 80

Comparison of measured and predicted temperature changes as a
ﬁinction of pavement depth and time of measurement at SPSl site

K24 (St. Johns, MI). .................................................................................. 81
Validation: measured temperature vs. predicted temperature ...................... 82

Validation: measured temperature vs. predicted temperature
(site 8-1053-1, Colorado). .......................................................................... 83

Validation: measured temperature vs. predicted temperature
(site 9-1803-1, Connecticut) ....................................................................... 84

Validation: measured temperature vs. predicted temperature
(site 13-1005-1. Georgia). .......................................................................... 85

Validation: measured temperature vs. predicted temperature
(site 31-0114-1, Nebraska). ........................................................................ 86

Validation: measured temperature vs. predicted temperature
(site 27-6251-1, Minnesota). ...................................................................... 87

Validation: measured temperature vs. predicted temperature
(site 46-9187-2, South Dakota). ............... 88

xii

6—13

6-14

6-15

6-16

6-17

7-1

7-2

7-3

7-4

7-6

7-7

7-9

‘7-10
7-11
7-12

7-13

Validation: measured temperature vs. predicted temperature
(site 48-1068-1, Texas). ............................................................................. 89

Validation: measured temperature vs. predicted temperature
(U 828181 , Texas). ..................................................................................... 90

Comparison between the MSU model and other models using data from
Stubstad et a1. (1994). ................................................................................ 91

Measured mid-depth temperature vs. backcalculated modulus of each
layer and peak deﬂection at site 88 (Deerton, MI). ..................................... 92

Performance (rutting) prediction error caused by middepth temperature
prediction deviation. .................................................................................. 100

Illustration of critical stress/strain locations in a typical pavement
Structure. ................................................................................................... 102

Typical stress pulse for morning temperature distribution and uniform

load.. ....................................................................................................... 103
Typical example of stresses for uniform temperature and uniform

FWD load. ................................................................................................. 104
Effect of base modulus. .............................................................................. 105
Effects of temperature distributions on horizontal stresses

under FWD load. ....................................................................................... 107
Effects of temperature distributions on horizontal strains

under FWD load. ....................................................................................... 108
Effect of AC thickness on stress response. ................................................. 109
Positions at which pavement responses are evaluated. ................................ 1]]
Transverse hysteresis loop for day temperature distribution. ...................... 112
Calculated dissipated energy for the load case I. ........................................ 115
Calculated dissipated energy for the load case II. ....................................... 116
Calculated dissipated energy for the load case 111. ...................................... 117

Typical contour lines of the horizontal stress under the load for load

xiii

7-14

7-15

7-16

A-l

case I and day temperature distribution. ..................................................... 118

Typical contour lines of the horizontal stress under the middle row
for the load case I and day temperature distribution .................................... 1 19

Typical contour lines of the horizontal stress under the load for load
case II and day temperature distribution. .................................................... 120

Typical contour ;lines of the horizontal stress under the middle row
for the load case 11 and day temperature distribution. ................................. 121

Calculated dissipated energy for the load case I and night temperature
distribution. ............................................................................................... 135

Calculated dissipated energy for the load case I and morning temperature
distribution. ............................................................................................... 136

Calculated dissipated energy for the load case I and day temperature
distribution. ............................................................................................... 138

Calculated dissipated energy for the load case 11 and night temperature
distribution. ............................................................................................... 139

Calculated dissipated energy for the load case 11 and morning temperature
distribution. ............................................................................................... 141

Calculated dissipated energy for the load case 11 and day temperature
distribution. ............................................................................................... 142

Calculated dissipated energy for the load case III and night temperature
distribution. ............................................................................................... 144

Calculated dissipated energy for the load case 111 and morning temperature
distribution. ............................................................................................... 145

Calculated dissipated energy for the load case 111 and day temperature
distribution. ............................................................................................... 147

xiv

CHAPTER 1
INTRODUCTION

1.1 BACKGROUND
1.1.1 Temperature Prediction Model and Correction Factor for AC Modulus

The use of falling weight deﬂectometer (FWD) data has become one of the
primary means of characterizing in situ structural properties of ﬂexible pavements.
Structural capacity (deﬂection and modulus) of the asphalt concrete (AC) layer is
strongly inﬂuenced by ambient and pavement temperatures. In order to accurately
determine or backcalculate the AC modulus, a two-step correction procedure needs to be
applied. Typically the ﬁrst step consists of predicting the effective temperature of the AC
layer, and the second step consists of adjusting the FWD deﬂection or the computed
modulus to a reference temperature using a correction factor.

The 1986 AASHTO Guide for Design of Pavement Structures (AASHTO Guide,
1986) presents a temperature correction protocol for FWD deﬂections. This procedure
requires the use of average air temperature for the previous S—days to predict pavement
temperature in selected depths. However, pavement engineers lmve challenged its
accuracy and practicality; this procedure does not account for the temperature gradient
effects due to diurnal heating and cooling cycles, which have a signiﬁcant effect on the
effective pavement temperature and its relationship with the AC modulus and the surface
deﬂection (Inge and Kim, 1994; Stubstad et al., 1994).

Recognizing the urgent need to develop a more accurate and practical

temperature-modulus correction procedure for pavement rehabilitation designs in the

state of Michigan, the Michigan Department of Transportation MDOT) funded a study
in 1996 to develop a new temperature prediction model. The resulting model has proven
to be both robust and widely applicable. The temperature and deﬂection data used in this

study were obtained ﬁ'om ﬁeld measurements.

1.1.2 Theoretical Analysis of Flexible Pavements

Recently mechanistic analysis has become more and more essential to the design
and performance evaluation of ﬂexible pavements. With the emergence of inexpensive
microcomputers and the ﬁnite element analysis (FEA), many issues associated with
analyzing pavements have been overcome with numerical modeling. This capability has
facilitated the transition from empirical methods to mechanistic methods in pavement
design.

Flexible pavements are classiﬁed into those pavements that have an asphalt
concrete (AC) surface. Flexible pavements consist of a surface course built over a
base/subbase course, and compacted roadbed soil. Although one of the important
characteristics of asphalt concrete (AC) is the time and temperature dependent behavior
due to its viscoelastic properties, many ﬁnite element programs and methods of pavement
analysis relied on elastic analysis due to their relative simplicity.

Surface deﬂection data collected from the FWD, which is a dynamic load, have
mostly been used and analyzed to evaluate an existing pavement. Therefore, simulation
of asphalt concrete pavement subjected to FWD load is the ﬁrst step to understand the
deﬂection and stress-strain behavior of ﬂexible pavements. In the FEA, the structural and

other auxiliary conditions such as material models, load model, boundary conditions,

element type, and geometry have to be correctly modeled to obtain reasonable response
results.

Nonetheless, many realistic problems such as temperature variation across the AC
layer and dynamic loading were simpliﬁed or not included in many viscoelastic analyses
even though complex interactions can affect structural analysis and pavement
performance. For instance, pavement temperature varies across the depth of AC layer
with time. During heating, the lower part of the AC layer is still cool, whereas, when the
surface of the asphalt is cooling, the heat transmission still causes an increase in
temperature at the lower part of the asphalt layer several hours aﬁer the top layer has
started to cool down. Although these temperature variations can signiﬁcantly affect the
analysis, it has not been carefully studied together with the FWD load pulse.

In this part of the study, structural analyses were performed based on
viscoelasticity of the AC layer subjected to an FWD pulse and various temperature
distributions across the AC layer. The inclusion of dynamic load effect for routine
pavement design involving viscoelastic materials is yet to be realized in the future.
Hence, this study can contribute to the development of an improved asphalt pavement

design program based on advanced material characterization and the FWD pulse.

1.2 OBJECTIVE
The objectives of this research are to:

1. Develop a practical and sound model to predict the effective temperature in the AC
layer and temperature correction factor for backcalculated AC moduli.

2. Investigate temperature effects on structural analysis and performance evaluation.

3. Simulate selected AC pavements using viscoelastic/elastic composite layer theories.

4. Better understand stress-strain behaviors of ﬂexible pavements subjected to FWD
load pulse and various temperature distributions across the AC layer.

1.3 SCOPE

The temperature prediction and correction factor part of this study addresses the
development of a practical yet robust temperature prediction model and correction factor
for AC modulus, which can be adopted in the ﬁeld. For this reason, the developed model
was validated with various methods and ﬁeld data. Characterization of the base and
subgrade were not within the scope of the research.

The theoretical analysis part addresses only structural analysis of ﬂexible
pavements to understand the effect of pavement temperature and FWD pulse on structural
responses. Additionally, the inﬂuence of nonuniform load pressures, including the truck
tire tread effect, is investigated in association with surface-initiated longitudinal wheel
path cracks, in an effort to relate analysis results to ﬁeld observations/applications.
Development of pavement design algorithm/program, performance model, and speciﬁc
mixture characteristics (mixture composition, gradation, etc.) that may affect the

analysis/problem are beyond the scope of this study.

1.4 ORGANIZATION OF THESIS
This thesis is divided into eight chapters, including the introduction.
CHAPTER 2: Comprehensive literature review on temperature prediction models and
temperature correction factors for AC modulus.
CHAPTER 3: Comprehensive literature review on structural analysis of ﬂexible

pavements, and overview of axisymmetric ﬁnite element model, the

f}

CHAPTER 4:

CHAPTER 5:

CHAPTER 6:

CHAPTER 7:

CHAPTER 8:

viscoelastic characterization of the AC layer, the subgrade
characterization, and applications of structural analysis to the ﬁeld.

Field data collection procedure for the development of temperature
prediction model and correction factor and preliminary analysis of data.
Introduction of boundary conditions, material properties and FEA of
two- and three-dimensional models, and parametric study approach.
Development and validation of temperature prediction model and
correction factor and investigation of inﬂuence of effective temperature
prediction errors on structural analysis and performance model.
Structural analysis results of 2-D and 3-D models and discussion.
Summary of conclusions on the research results, comments on

applications to the ﬁeld, and recommendations for future research

CHAPTER 2

LITERATURE REVIEW I — TEMPERATURE PREDICTION MODEL AND
CORRECTION FACTOR FOR AC MODULUS

2.1 GENERAL

Falling Weight Deﬂectometer (FWD) data is widely used to evaluate in-situ
material properties of pavement layers. Backcalculated AC moduli are inﬂuenced by
pavement temperature because asphalt concrete is viscoelastic. Therefore, an effort is
usually made to obtain representative AC layer moduli of ﬂexible pavements, in which
backcalculated AC moduli are adjusted to a reference temperature. This temperature
correction can be achieved in two steps: (1) determination of the temperature, the so
called effective temperature, which represents the temperature of the AC layer at the time
of FWD testing and (2) adjustment of the backcalculated AC moduli to a reference
temperature using correction factors.

Determination of the effective temperature is currently accomplished in many
ways. Southgate (1968) suggested that the surface temperature plus the 5-day mean air
temperature should be used as partial inputs to determine the temperature at various
depths within the AC layer. Other methods include the use of a single measured
temperature either at the top or at a certain depth within the AC layer. For routine
deﬂection testing and analysis for state highway agencies (SHA’s), it is desirable, from a

practical view point, to devise a nondestructive prediction method for determining the

effective AC layer temperature. Hence, numerous temperature prediction models and

correction procedures have been developed.

2.2 REVIEW OF TEMPERATURE PREDICTION AND CORRECTION
PROCEDURE FOR AC MODULI

2.2.1 AASHTO Guide Temperature Prediction and Correction Procedure

The 1993 AASHTO Guide does not include any speciﬁc temperature prediction
procedure for the effective AC layer temperature. The 1986 AASHTO Guide presents a
temperature prediction procedure that allows for the estimation of the average pavement
temperature during a deﬂection test.

This procedure requires as input (1) the pavement surface temperature dining the
deﬂection test, and (2) the mean air temperature for the previous ﬁve days to predict
pavement temperatures with depth Using this information, the pavement temperature at
various depths within the total asphalt layer thickness is obtained from Figure 2-1. The
average of these values is taken as the pavement temperature during the deﬂection test.
The AC modulus adjustment factor (Fe) is a function of this average pavement
temperature, as illustrated in Figure 2-2. The corrected asphalt modulus value, at 70 °F

(21 °C), is then determined by:

E700]: : Fe X Btp (2-1)
where:
E70°F = corrected asphalt layer modulus used to determine the effective structural
layer coefﬁcient for the existing in situ asphalt material
Fe asphalt modulus adjustment factor

Esp = Uncorrected asphalt layer modulus determined from the interpretation of
deﬂection basin measurements. (This predicted modulus corresponds to a
pavement temperature condition, tp, occurring during the deﬂection test.)

Temperature at Depth, °C

2.2.2 BELLS Temperature Prediction Model

Stubstad et al. (1994) and Balzer and Jansen (1994) developed a temperature
prediction equation to predict temperature at one-third depth of the AC layer as an
effective temperature and a temperature correction, respectively.
The BELLS temperature prediction equation is as follows (Stubstad et al., 1994):

T.,3=8.77+0.649*IR+{log(d)-1.5} {-0.503*IR+0.786*(5-day)+4.79*sin(hr-
18)}+{sin(hr-14)}{2.20+0.044*IR} (2-2)

Pavement Surface Temperature Plus 5-Day Mean Air Temperature, °F
0 20 4O 60 80 100 120 140 160 180 200 220 240 260

 

    
  
   
    
  
 

 

 

7oﬁrrrrrnlrrrll 16°
60 >- Depth in Pavement 25mm(1 in.) 140
50 __ 50mm(2in.) . ‘20 gs—
4.3.”
40 l- 100mm(4in.) 4 100 3
300mm 80 ii
I— ' 0
30 200mm (12m.) f=3
20 150m (831-) g
(6111.) u 60 g
10
4 40 E"
o
d 20
~10
I l 1 1 1 1 1 i i 1

 

 

-30 -20-10 0 10 20 30 40 so 60 i0 so 90 160
Pavement Surface Temperature Plus 5-Day Mean Air Temperature, °C

Figure 2-1. Predicted pavement temperature (AASHTO Guide, 1986)

 

100.0 I t Tm rrrrrI r

TVUVT
L111

‘1
J

 

(E70°F / Em)
'
4

H10

L0

Asphalt Concrete Modulus Adjustment Factor, F ,

 

 

 

0.1 ,. ,......L .

10 100 200

 

Pavement Temperature at Time of NDT Test, tp (T)

Figure 2-2. Asphalt modulus temperature adjustment factor (AASHTO Guide, 1986)

where:

Tm = pavement temperature at third-point in AC mat, °C,
IR = inﬁ'ared temperature reading at time of FWD test, °C,
log = base 10 logarithm,
d = depth at which mat temperature is to be determined, i.e. total AC mat
thickness divided by 3, mm,
5—day = previous mean 5-day air temperature, sum of 5 highs and 5 lows divided
by 10, °C,
sin = sin function in 24 hour system. To use the time-hr function correctly,

divide the number of hours in cycle by 24, multiply by 21:, and apply the
sin function in radians.

Due to the heating and cooling cycles of pavement, surface temperature is higher
than subsurface temperature during the daytime. On the other hand, surface temperature
becomes lower than subsurface temperature during the nighttime (Figure 2-3). The
BELLS model adequately takes into account the different temperature-depth gradients
due to this diurnal and nocturnal cycles, which have signiﬁcant effect on effective
pavement temperature. However, the BELLS temperature prediction equation predicted
temperature only at one-third of the AC layer.

Based on the temperature at the one-third depth of the AC layer, two correction
models were presented; one is a modiﬁed correction model and the other is the new
temperature correction model (Balzer and Jansen, 1994). The schematic relationship

between backcalculated AC modulus and temperature at three different depths is shown
in Figure 2-4. It can be seen that the AC modulus follows a different curve depending on
whether the asphalt is heating or cooling. During heating, the lower part of the AC layer
is still relatively stiff and has signiﬁcant inﬂuence on the composite stiffness of the AC

layer. Whereas, when the surface of the asphalt is cooling, the heat transmission still

10

AC modulus

Temperature

 

Surface Temperature

   

 

Bottom
Temperature

Morning Afternoon Evening Night Morning

 

[ Mid-depth Temperature

 

 

 

Time

Figure 2-3. Temperature variations for a day at three different depths of an AC layer.

 

 

 

 

‘n Heat'n m
Heating '3 I g g
'8 '3 Heating
E E
L)
< 2
\ Cooling
Cooling Cooling
Temperature Temperature Temperature
(3) Top of AC layer (b) Bottom of AC layer (c) Appr. one third of
AC layer

Figure 2—4. Relationship between the AC modulus and temperature at three different

depths (Modiﬁed ﬁ‘om Balzer and Jansen (1994)).

lb]

causes an increase in temperature and softness of the lower part of the asphalt layer
several hours after the top part has started to cool. This results in stagnation of the AC
moduli before it starts to increase due to the lower temperature. Judging from Figure 2-4,
(a) and (b), it would be very difﬁcult to establish a temperature correction model based
on the temperature at either the top or the bottom of a thick asphalt layer, as there is no
deﬁnite AC modulus at any temperature chosen as standard temperature. Figure 2-4 (c)
shows that temperatures at the depth of one third of the AC layer thickness exhibits a
deﬁnite correlation between the temperature and the AC modulus. According to Kim et
a1. (1996), the middepth temperature also provides a reasonable relationship between
temperature and AC modulus. At this depth (one third depth or middepth), it would be

possible to calculate a deﬁnite reference AC modulus at a standard temperature.

 

The adjusted model is
5., = E“ (2-3)
1 — 2.210g(TAC / Tref)
where:
Tmf = reference temperature, °C,
EM — reference AC modulus, MPa,
TAC = AC temperature measured during the FWD test, at one third of the total
AC thickness, °C,
EAC = AC modulus found from FWD testing and backcalculation, MPa
The new temperature correction model is
Eref = 1 00.018 (Trcf- TAC) EAC (2-4)
where:
EM, Tm; TAc, EAC = same as above.

Similarly, a plot of backcalculated asphalt modulus vs. temperature was also

developed by Briggs and Lukanen (2000), along with a regression equation which can be

12

used to determine the asphalt modulus at any temperature. The form of the regression

equation is:
E. = E..fe°‘“"~" (2-5)

where:
E. = predicted asphalt modulus at temperature, t
Enf = asphalt modulus at reference temperature, tmf
e = exponetial function
or = regression coefficient
t = temperature of asphalt, degrees C
trey = reference temperature of asphalt (25 degree C)

2.2.3 BELLSZ Temperature Prediction Model

Lukanen et a1. (2000) developed new coefﬁcients for BELLS temperature
prediction model using temperature data from 40 sites monitored in the Seasonal
Monitoring Program for the Long Term Pavement Performance (LTPP) program. In the
BELLSZ model, the previous 5-day average air temperature, which was difficult to
obtain, was replace by the previous day’s air temperature, and the sine functions of the
BELLS model were replace by two sine functions based on an18-hr cycle. The form of
the resulting equation is:

Td=2.78+0.9l2*IR+{log(d)-l.25} {-0.428*IR+0.553*(1-day)+2.63*sin(hr13-

15.5)}+0.027*IR*sin(hrlg-l3.5) (2-6)
where:
Td = pavement temperature at depth (1, °C
IR = infrared surface temperature, °C
log = base 10 logarithm
d = depth at which mat temperature is to be predicted, mm
l-day = average air temperature the day before testing
sin = sine function on an 18-hr clock system, with 2n radians equal to one 18-hr

cycle

13

hm = time of day on a 24-hr clock system, but calculated using an l8-hr asphalt
concrete (AC)

This new BELLS temperature prediction model became more practical by
employing more easily obtainable variables such as the previous l-day average air
temperature instead of 5-day average air temperature. The model covered a wide
geographical area since it used the largest and most diverse set of data from the FHWA
LTPP program’s SMP

The BELLSZ model was also used for temperature adjustment of backcalculated
asphalt moduli. The semi-logarithmic format of the equation relating the asphalt modulus
to the mid-depth asphalt temperature allows for a simple means of adjusting the
backcalculated asphalt modulus for the effects of temperature. The approach is to

calculate a modulus temperature adjustment factor using the following equation:

A TAF = los’ope"T" 1“" (2-7)
Where:
A TAF = asphalt temperature adjustment factor
Slope = slope of the log modulus versus temperature equation
(-0.0195 for the wheelpath and —0.021 for mid-lane are recommended)
Tr = reference mid-depth hot-mix asphalt (HMA) temperature
Tm = mid-depth HMA temperature at time of measurement

2.2.4 LTPP AC Pavement Temperature Models

Mohseni and Symons (1998; 1998a) presented improved LTPP low and high
temperature models using the LTPP seasonal data.
The improved LTPP low temperature model is

T," = -1.56+0.72 r... — 0.004 Lat2 + 6.26 log.o(H+25) — z(4.4+0.52 02...)” (2-8)

14

Tm = low AC pavement temperature below surface, °C,

Tm = mean low air temperature, °C,

Lat = latitude of the section, degrees,

H = depth to surface, mm

om = standard deviation of the mean low air temperature, 0C,

2 = from the standard normal distribution table, z=2.055 for 98% reliability

The improved LTPP high temperature model is

T,av = 54.32+0.78 T... — 0.0025 Latz —1 5.14 log.o(H+25) - z(9+0.6l 02.9”2 (29)

where:

Tm = high AC pavement temperature below surface, 0C,

T... — high 7-day mean air temperature, °C,

Lat = latitude of the section, degrees,

H = depth to surface, mm

(rag, = standard deviation of the high 7-day mean air temperature, 0C,

2 = ﬁ'om the standard normal distribution table, z=2.055 for 98% reliability

2.2.5 Asphalt Institute (AI) Model

In order to account for the effect of temperature on moduli of asphalt mixtures,
the relationship between mean pavement temperature, Mp, and mean monthly air
temperature, M., based on the depth below the pavement surface was developed and

implemented in the DAMA computer program (Hwang and Witczak, 1979).

 

 

l 34
M = M 1+ - + 6 2-10
p a( z + 4) z + 4 ( )
where:
z = depth below surface, inches.

The temperature at the upper third point of each layer is used as the weighted average

pavement temperature.

15

2.2.6 Temperature Prediction Model Based on Heat Transfer Theories and
Correction Procedure

Shao et a1. (1997) developed an asphalt pavement subsurface temperature
prediction procedure which was based on heat transfer theories and used the surface
temperature history from the previous morning. It consisted of three major steps:

1. Predicting yesterday’s maximum and today’s early morning minimum pavement
surface temperatures on the basis of the maximum and minimum air temperature
and weather conditions.

2. Generating the surface temperature history from yesterday morning to today’s
ﬁrst temperature measurement.

3. Estimating AC layer mid-depth temperature as a function of the surface
temperature history using heat conduction theory; the solution to a one-

dirnensional heat conduction problem is as follows:
r (At + Bit)eth(Xi)

X- _ .2
x n +ZBi(t-ti)(Xi2etﬁ:(X,-)-—7;-e X’ )
T , =T If 2: U
(x 0 0e (Nani; (Ai+Bit)€IjC(/Yi+l) i

 

 

 

 

- X - _ .2
+ 23:1! -ti+1XXi2+le'fC(Xi+1)- :1 e X'“ )
(2-11)
where
T (x, t) = predicted temperature at a certain depth of AC, x, and a
current time, t,
To = constant (the initial temperature distribution)
t = current time
x = the mid-depth length of AC
on = the thermal diffusivity (m2/sec or mZ/hr)
7} = measured surface temperature at the corresponding time t,-

16

Ti+1 “Ti

ti+1 "‘1'
A,“ =Ti-'Bi(ti—tl) (no sumon i;i=l,2,...,n)

13,-:

(no sum on i; i = 1,2,...,n)

 

X
X- =
’ 2,/a(t—t,-)
x
Xi+1 =

 

24620 - 1,41)

2 X _ 2
erf(X) = 37 e 5 d: error function, and

2 _ 2
etfc(X) = T; J: 8 6 d5 complementary error function.

Based on mid-depth temperatures, correction factors were obtained as follows (Kim et

al., 1996):
71., = 10'“"”°’ (2-12)
where:
M; = correction factor
m = regression constant; statewide m-value is 0.0262.
T = effective temperature at the time of the FWD testing
To = reference temperature such as 25 °C

2.2.7 Temperature Correction Procedure Based on the Theory of Linear
Viscoelasticity

Park and Kim (1997) analytically developed a correction model based on the
theory of linear viscoelasticity and the time-temperature superposition principle. The

model is based on a simple temperature -modulus correction factor:
M = E m/ Er (2-13)
where:

Ar = temperature-modulus correction factor,

17

E n = corrected modulus at temperature To,
Er = backcalculated modulus of the AC ﬁ'om deﬂections at temperature T

The theoretical correction factor is deﬁned in Equation (2-13), and the relaxation
modulus E(§) and the time-temperature shift factor ar(T) for the asphalt mixture are

deﬁned follows:
A. = E(§o) / 15(5) (2-14)

where the reduced times £0 and E, are deﬁned, respectively, by

5,0 = t./ ar(To) and I; = t1/ a1(T) (2-15)
in which t1 is the loading duration. In this correction procedure, viscoelastic properties of
the AC layer were applied to the correction factor modeL However, it is questionable

whether it works at low temperatures because AC behaves elastically at low temperatures

rather than viscoelastically.

2.2.8 Other Temperature Correction Methods for AC Moduli
Johnson and Baus (1992) suggested that the following formula which is based on

the work done by Lytton et a1. (1990) who took an approximation ﬁ'om the Asphalt

Institute (1982):
_ —0.0002175(70“886—T"886)
II E — 10 (2-16)
where:
M; correction factor

.4
II II

temperature at the time of the FWD testing, °F
Ullidtz (1987) developed a model based on backcalculated moduli ﬁ'om AASHO

Road Test deﬂection data. The correction model is as follows:

18

1
,1, =
3.177-1.67310gT

 

(2-17)

for T>1 °C.

2.2.9 Climatic Model

The climatic-material-structures model (CMS model) was developed by Dempsey
et al., 1985. In the case of ﬂexible pavements, it is generally assumed that the stiffness of
the AC layer depends on pavement temperature, while unbound materials such as
granular bases and subgrades depend on moisture content. This climatic model which
was used to analyze ﬂexible pavements simulates climatic conditions that control
temperature and moisture conditions in the pavement layers and in the subgrade. It used
sunshine percentage, windspeed, air temperature and solar radiation to compute the heat
ﬂux boundary condition on the roadway surface and the resulting temperature proﬁle
throughout the asphalt pavement. A subroutine computes accompanying changes in
asphalt stiffness, resilient modulus and Poisson’s ratio of the base, subbase and subgrade
with time. Input of material properties, pavement geometry, pavement inﬁltration

parameters, and many other data are required.

2.2.10 Summary of the Existing Temperature Prediction Models
All the models summarized in this chapter have advantages and disadvantages.
Predictions of some models such as the BELLS model and the heat transfer theory based
model are accurate. However, they require various input variables, or the prediction
procedures are rather complex, which make them impractical in the ﬁeld. For example,

heat transfer theory based prediction is very reliable; prediction deviations are within i2

19

°C. On the other hand, it needs many input variables and a computer program due to the
iterative procedure. To obtain a previous mean 5-day air temperature in every routine
test is not practical. Relatively simple (statistical) models reduce the accuracy or do not
account for temperature gradient effects due to diurnal heating and cooling cycles (e.g.,
AI model) although the AI model is capable of accounting for monthly temperature

variations.

2.3 APPLICATIONS TO THE FIELD/PAVEMENT DESIGN

In a mechanistic-empirical (M-E) ﬂexible pavement design procedure, the two
most critical components are structural models and transfer functions (Thompson, 1995).
Development of structural models and/or a mechanistic analysis procedure is a major task
in M-E pavement design. The measured deﬂections ﬁ'om a FWD are commonly used
along with loading and pavement layer thickness information as input to backcalculation
programs to compute a modulus for each layer of the pavement structure. The pavement
layer moduli are then used in a forward calculation routine to estimate the stress/strain
distributions in each of the pavement layers under expected traffic volumes and loadings.
Transfer functions (distress models) relate these pavement responses to pavement
performance as measured by the type and severity of distress (rutting, fatigue cracking,
and so on). For ﬂexible pavements, it is common practice to relate the horizontal strain at
the bottom of the AC layer to the number of applications of that strain level to cause
failure in cracking (Bonnaure et al., 1980; Buch et al., 1999; Thompson, 1987). Similarly,
the vertical strain at the top of the subgrade is related to permanent deformation (rutting)

(Buch et al., 1999; Kenis, 1977; Owusu-Antwi, 1998). These transfer functions are quite

20

sensitive to small changes in strain levels (Briggs and Lukanen, 2000). Therefore, it is
crucial to accurately determine the effective (pavement) temperature and adjust the
moduli to the reference/ standard temperature because the stress/strain distribution in the
pavement structure is dependent on the stiffness or moduli of each of the pavement

layers, and the stiffness of the AC layer varies with pavement temperature.

21

CHAPTER 3

LITERATURE REVIEW II — OVERVIEW OF THEORETICAL ANALYSIS OF
FLEXIBLE PAVEMENTS

3.1 INTRODUCTION OF VISCOELASTICITY TO THE FLEXIBLE
PAVEMENT ANALYSIS

The structural analysis of layered systems such as layered soil deposits and
pavements have long been pursued by many geotechnical and pavement engineers. Based
on Love’s stress ﬁmction (Love, 1927) and a Bessel function expansion of the load
applied on a ﬁnite boundary surface, Burmister (1943, 1945) developed solutions for a
two-and three-layer systems. Using Burmister’s analytical solutions, afterwards, tabular
and graphical summaries of stresses and displacements in two- and three-layer systems
for various combinations of geometrical and material parameters were presented (Acum
and Fox, 1951; Jones, 1962; Peattie, 1962; Huang, 1969).

Due to their relatively simple analysis, many ﬁnite element programs and
methods of pavement analysis for mechanistic pavement design rely on elastic analysis,
for example, of layered systems in the Shell Pavement Design Manual. Design to limit
permanent deformation is often related to vertical elastic strain on the top of the
AC/subgrade layer(s). Similarly, fatigue cracking for design criterion is related to tensile
strain at the bottom of the AC layer. The use of elastic analysis methods alone has
limitations and can result in incorrect characterization of pavement performance unless

material temperatures as well as loading time are speciﬁed (Lu and Wright, 2000).

22

Huhtala et al. (1990) reported strains measured under moving wheel loads in a test track.
Yun and Chatti (1996) compared viscoelastic and dynamic responses obtained by a
moving load solution very well with ﬁeld data. It was observed that, in the longitudinal
direction, compressive strains occur which are followed by a tensile peak and then
compressive strains again, whereas, in the transverse direction, the strain is all tensile.
Viscoelastic responses led to asymmetric strain pulses, but elastic analysis predicted
symmetric strain pulses (Huhtala et al., 1992; Nilsson et al., 1996). Good agreement was
found between the measured deﬂections and computed deﬂections by using viscoelastic
analysis (U ddin, 1998). It has been reported that the resilient modulus of asphalt mixtures
is very sensitive to pavement temperature. It was also estimated that the difference in
asphalt mixture stiffness on the top and bottom of the AC layer can be as high as a factor
of 4 if temperatures at the top and bottom are 44 and 30 °C, respectively (Van de L00,
1978). Brown and Snaith (1978) found that strain of asphalt mixture is much more
sensitive to the variation of experiment temperature than to the change of loading
magnitude.

Hence, many researchers have considered the viscoelastic analysis in ﬂexible
pavements. Perloff and Moavenzadeh, (1967) studied the surface deﬂection of
homogeneous viscoelastic halfspace under moving loads. Stresses and displacements in
viscoelastic multi-layer systems were also investigated (Chou and Larew, 1969; Elliot
and Moavenzadeh, 1971; Huang, 1973). In order to predict the structural responses of
layered viscoelastic systems and the integrity of ﬂexible pavement, a computer program
named VESYS (Kenis, 1977 and 1978), was developed by the Federal Highway

Administration. Huang (1993) also presented a computerized procedure for the analysis

23

of linear elastic, nonlinear elastic, and/or viscoelastic layer systems. Kim et a1. (1996)
developed a procedure for the analysis of stresses and displacements in a viscoelastic
layered system subjected to transient loading and transient and spatially homogeneous
temperature conditions. They used the viscoelastic solutions through a linear convolution
integral of the unit response functions and the time rate of loading and the existing elastic
solutions to determine the unit response functions for the corresponding viscoelastic
problem.

Subsequent sections of this chapter brieﬂy review the viscoelastic characterization
of the AC layer and the axisymmetric ﬁnite element model and the subgrade
characterization typically used for the analysis of ﬂexible pavement. The ﬁnal section

addresses how this analytical evaluation can be used in the ﬁeld.

3.2 VISCOELASTICITY

The mechanical behavior of elastic materials is independent of loading time and
temperature and obeys Hooke ’3 law: the stress is proportional to the strain. On the other
hand the behavior of viscoelastic materials is that the response to an applied stress or
strain depends upon the rate or time period of loading. For linear viscoelastic materials,

the uniaxial stress-strain relationship can be expressed by

de(r)dr

 

0(1): [E(t — 1) (3-1)

where o(t) is the stress at time t, E(t) is the uniaxial relaxation modulus, and s(t) is the
strain at time t.

The relaxation modulus can be obtained from a one-step strain test.

24

If 8(t) is a step function:
3(t) = 0 if t S 0, (3—2)
8(t) = so if t > 0, (3-3)
where 8(t) is strain at time t, and so is constant.

Then, the relaxation modulus is obtained as follows:

 

E(t) = a“) (3‘4)

30

Similarly, under a constant stress 00, the creep compliance, D(t), is deﬁned as

00) = 3‘9 (3-5)

0'0

In general, the mechanical property of viscoelastic materials depends not only on
loading time but also on temperature. Especially some of the mechanical properties of
amorphous polymers have a strong dependence upon temperature. There exists a special
class of viscoelastic materials whose temperature dependence of mechanical properties is
amenable to analytical description. This class is referred to as being thermorheologically
simple materials. The simplifying feature of the thermorheologically simple materials is
that, when the unit response ﬁmction (e.g., creep compliance or relaxation modulus)
curve measured at different constant temperatures are all plotted against time on a
logarithmic scale, the curves can be superposed so as to form a single curve, deﬁned as a
master curve, corresponding to an arbitrary ﬁxed temperature (i.e. reference temperature)
by means of horizontal translation only. This makes it possible to cover times outside the
range that is not easily accessible by practical experiments. The horizontal distance
(representing the shift factor, 37) between the master curve and any one of the isothermal

curves is independent of time.

25

This feature has a very signiﬁcant consequence in that the dependence of the
material property upon both time and temperature can be represented by dependence
upon a single variable called reduced time, and the feature is often referred to as time-
temperature superposition or reduced-time methods. For example, the uniaxial relaxation
modulus in mathematical notation is expressed as:

E(t, T) = 1311400 (3-6)
where EM(t,) is the master relaxation modulus corresponding to a reference temperature

(TR), and for constant temperature,
I = —— (3-7)

where tr and aT are the reduced time and the time-temperature shift factor, respectively.
The aT is a temperature dependent material function that reﬂects the inﬂuence of
temperature on internal viscosity of the material. One may obtain the relaxation modulus
E(t,T) if the master relaxation modulus EM(t,) and the shift factor a—r(T) are given; or,
conversely, one may obtain EM(t,) when E(t,T) and aT(T) are given.

The method of relating the horizontal shifts along log time scale to temperature
changes was developed by Williams, Landel, and Ferry so called the WLF equation as
shown in Equation 3-8. For temperature above the glass transition temperature of the
material, the shift factor aT for thermorheologically simple materials is usually expressed

in the following form:
__ 01(T—T) (3-8)

where aT is the shift factor, c. and 02 are constants, and Tr is the reference temperature.

26

The constants, c1 and c2, can be obtained by plotting (T-T,) against (T-T,)/log aT. The
slope and the intercept of the plotted line are -c1 and -c2, respectively.
By Equations 3-6 and 3-7, the uniaxial stress-strain relationship of Equation 3-1 at

a given time, t, and a given temperature T can be rewritten as

t— r )dde(z')z_
0(t)= gm) 0T d1 (3-9)

 

where C(t) is the stress at time t and temperature T, E(t,) is the uniaxial relaxation
modulus which becomes a temperature independent relaxation modulus of a temperature
dependent “reduced time”, and 8(t) is the strain at time t.

(Young et al.,-1991; Sperling, 1993; Kim et al., 1996)

3.3 AXISYMMETRIC FINITE ELEMENT ANALYSIS

Solids and shells whose geometry is axisymmetric (bodies of revolution), which
have material properties and loads that also are axially symmetric, can be modeled using
cylindrical coordinates r, 0, and 2. With these assumptions, a three dimensional problem
can be reduced to a two dimensional one. However, ﬂexible pavements must be assumed
to be continuous (no cracks) and inﬁnitely wide. Therefore the edge (shoulder) effect

cannot be considered.

In an axisymmetric state of stress, only four stress components, 6,, 69, oz, and In,
are non-zero (Figure 3-1). The major and minor principal stresses along the axis of
symmetry (r=0) coincide with either 0', (:69) or oz depending on the vertical position 2.
Cook (1989) formulates the plane linear isoparametric element which can be revised to

obtain the axisymmetric element. The global coordinates (radial and vertical), and the

27

 

 

 

 

 

 

 

 

 

 

 

l I
Ml"——"—‘—’|K
ﬁt
.1 1
;1 >é
1 .11 1
r . 1 e

 

(b) the four-node isoparametric element

Figure 3-1. A typical axisymmetric ﬁnite element

28

corresponding displacements at an internal point can be related to the corresponding

nodal quantities through shape functions:

u NiO NjO NkO NmO
{}= 4 (3-10)

<u
w 0 NiONjONkONm :1

 

 

where the individual shape functions are

 

+ +
NI=(l—§I4(1—7])’ whereI=i,j,k,m

(Cook et al., 1989)

3.4 3-D HEXAHEDERAL AND TRIANGULAR ISOPARAMETRIC ELEMENTS

3-D hexahederal (brick) and triangular elements are rarely used in pavement
analysis because 2-D axisymmetric elements usually produce satisfactorily accurate
results with greater economy. However, 3-D elements may still be appropriate to
investigate the complicated interactions among environmental factors, nonuniform loads
and pavement layers which are not necessarily treated appropriately by 2-D elements. As
long as the mesh is properly designed, reliable and valuable results are produced ﬁ'om the
3-D model.

In three dimensions, the isoparametric procedure is similar to the 2-D

development. For brick element geometry and the ﬁeld quantity 0 of a solid

29

isoparametric element, x, y, and z are coordinates interpolated ﬁ'om nodal values of x,-, y,,
and zr:
x=ZN;x;, y=ZN;y,-, z=ZN121, ¢=2Ni¢p, (3-11)
where i ranges over the number of nodes in the element. Shape functions N,- are ﬁinctions
of isoparametric coordinates 4", r), and ;2 Similarly, for triangular element:
x=ZNgx,-, y=ZN,-yr, ¢=2N;¢,, (3-12)

(Cook et al., 1989)

3.5 SUBGRADE CHARACTERIZATION
In order to mechanically or mathematically model the particular characteristics of
soil behavior, the linear elastic idealization is usually adopted. The Winkler and elastic

solid foundations are the most commonly used linear foundation models. In the analysis

of this study, the subgrade is characterized as an elastic solid foundation.

3.5.1 Winkler Foundation

The Winkler foundation model represents soil as closely spaced linear elastic
springs (Winkler, 1867) (Figure 3-2 (a)). The Winkler foundation was ﬁrst applied in the
pavement ﬁeld in order to determine the critical stresses in an inﬁnitely large concrete
slab under a single load for three cases of loadings: corner, edge, and interior
(Westergaard, 1926). Since the Winkler foundation models the subgrade with only one
support, k, and is capable of modelling edge or comer loading more realistically than the

elastic solid model, it was commonly used in concrete pavement analysis.

30

3.5.2 Elastic Solid Foundation

The elastic solid foundation is another well-known model for pavement support.
This is ideal for cases where there is no joints or cracks, and for a single uniform slab
thickness and subgrade support. Hence, it is widely used for ﬂexible pavement modeling.
The soil is modeled as a linear elastic, isotropic, homogeneous solid. The elastic solid
model may be more realistic than the Winkler foundation in some aspects since real
subgrade supports experiences shear stresses to some extent. The signiﬁcant difference

between an elastic solid foundation and the Winkler foundation is shown in Figure 3-2.

3.6 APPLICATIONS OF STRUCTURAL ANALYSIS TO THE FIELD

As pointed out in the previous chapter, the estimated stress/strain distributions for
each layer of the pavement structure are related to pavement performance by transfer
functions (distress model), and this is a major task in mechanistic-empirical (M-E)
pavement design.

The structural analyses based on viscoelasticity can be also related to the design
and/or performance evaluation. Some applications of the viscoelastic analysis to the ﬁeld
were reviewed in the earlier section. Besides them, recently, Rowe et al. (1995 and 1997)

and Collop et al. (1995) presented viscoelastic approaches to fatigue cracking/rutting.

31

 

 

(a) Winkler Foundation

 

 

 

 

(b) Elastic Solid Foundation

Figure 3-2. Comparison of Winkler Foundation with Elastic Solid Foundation (After
Tabatabaie-Raissi, 1977)

32

Rowe et a1. predicted fatigue performance by considering the energy dissipated
(work done) in asphaltic materials under loading with the damage being proportional to
the cumulative dissipated energy. The calculation of the fatigue life was proposed in the

previous work as follows (Rowe, 1993):

N1 = 205x Vb‘w x wo'Z-m x L11...L64 (3-13)
where:
N1 = number of load cycles to “crack initiation”
Vb = volume of binder (%)
w, = dissipated energy per cycle at start of test (J/mz)

LI’m = work ratio, calculated from the initial complex modulus and phase angle
and input into the statistical analysis as a calculated parameter.

This dissipated energy fatigue criterion replaced the “(tensile) strain criterion” currently
used for fatigue damage.

Lee et a]. (1998) investigated a mechanistic approach to uniaxial viscoelastic
constitutive modeling of asphalt concrete that accounts for damage evolution under cyclic
loading conditions. The constitutive equation developed ﬁ'om the controlled-strain fatigue
tests satisfactorily predicted the stress-strain behavior of asphalt concrete all the way up
to failure under the controlled-stress mode as well as under the monotonic loading with
varying strain rates. Uddin (1998) also found good agreement between the measured ﬁeld

deﬂections and computed deﬂections by viscoelastic analysis (Figure 3-3).

33

Deﬂection. inch

Deﬂection Time history under FWD Loading Plate, Node 796 um

 

  
    
     
   
    

  

 

 

 

° 00‘ 25 4
1
0 ‘ O
'0'001 '. _ Elastic properties of asphalt -25.4
0.002 . layers -503
-0.003 'I- . . . Viscoelastic properties of -76.2
'l asphalt layers
-0.004 -- , 402
v I ‘ Peak FWD measured
.0005 ‘U' deﬂection = -0.0053 inch '127
-0.006 : : "HAL : e .- r : : a _1 52
0 0.01 0.02 0.03 0.04 0.05

Time, seconds

Figure 3-3. Deﬂection-time history plot of computed dynamic deﬂection bowl under the
FWD loading plate using viscoelastic parameters for asphalt layers (Uddin,
1998)

3.7 SURFACE-INITIATED CRACKING

It has been reported that surface-initiated cracking had become more prevalent in
recent years (CROW, 1990; Myers et al., 1998; Uhlmeyer et al., 2000). Figures 3-4 and
3-5 show surface- initiated longitudinal wheel path cracks and core with surface-initiated
longitudinal wheel path cracks, respectively. However, the mechanism for this type of
failure has not been clearly identiﬁed.

Gerritsen et al. (1987) concluded that both thermal and load related effects caused
the surface initiated cracks that did not extend into the lower bituminous base layers.
Dauzats et a1. (1987) noted that surface cracking were initially caused by thermal stresses
and then further propagated by traffic loads, and a rapid hardening of the mix binder

likely contributed to this type of cracks. Nunn (1998) suggested from ﬁeld observations

34

  

Figure 3-4. Overview of surface- initiated longitudinal wheel path cracks
(Myers et al., 1998)

 

Figure 3—5. Core showing surface-initiated longitudinal wheel path cracks (Myers et a1,
1998)

35

that a very different failure mechanism occurs at the “breakpoint” thickness, and the
cracks propagated from the top of the pavement surface downward for full depth cracks
in thinner pavements. Recently, Uhlmeyer et a1. (2000) reported ﬁeld observations in
Washington State. They claimed that these top-down cracks were typically longitudinal,

appeared in or near the wheelpaths, and decreased in width with depth.

 

 

-2000 1
—- Bias Ply
' " " Radial
-1800 A -— WideBase Radial
«4600

 

 

-1400 fl

 

’C

I [10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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0 50 100 150 200 250 300 350

Transverse Location, X (m)

Figure 3-6. Vertical contact stresses measured for bias ply, radial, and wide-base radial
tires at appropriate rated load and inﬂation pressure (Myers et al., 1999)

It should be noted that the predominant tire used by trucks on the roads today is

the radial over the bias ply tire. This is due to their increased load carrying capacity,

36

better traction, handling, and fuel efficiency (Myers et al., 1998). The main structural
differences between the radial and bias ply tires are that radial tires have a more ﬂexible
wall and rigid tread structure. Figure 3-6 shows the vertical stress distribution across the
tire for three tire treads: bias ply, radial, and wide base radial (Myers et al., 1999).
Hirneno et al. (1997) and de Beer et al (1997) presented measurements of tire forces that
were non-uniform contact pressures. Numerous researchers investigated the effects of
nonuniform vertical stress distributions on pavement performance (Marshek et at., 1986;
Marshek et at., 1986a; Chen et al.,1986). However, their work was limited to the eﬁ‘ects
of stresses and strains at the bottom of the AC layer. Myers et al. (1998) investigated the
effects of tire treads and combination of vertical and lateral tire-pavement interface
stresses and concluded that tensile stresses under the treads of tire were the primarily

cause of the cracks.

37

CHAPTER 4

DATA COLLECTION — TEMPERARTURE PREDICTION MODEL AND
CORRECTION FACTOR

4.1 TEST SITES AND DATA COLLECTION

The objective of this study was to (a) develop a model to predict effective
temperature of the AC layer at any depth based on ﬁeld data, and (b) adjust the
backcalculated modulus to a reference temperature using the predicted effective AC layer
temperature and modulus correction factors.

Six in-service Michigan ﬂexible pavement sites were selected in 1997, 1998, and
1999. The inventory data for the Michigan sections is summarized in Table 4-1 and the
data from the special pavement studies (SPS-l) is summarized in Table 4-2. All the
Michigan test sites were instrumented with temperature sensors connected to a OM-220
Portable Data Logger (Omega Engineering, INC).

In order to proﬁle the temperature gradient, test holes (approximately 2.5 cm in
diameter) were drilled and partially ﬁlled with a viscous ﬂuid (mineral oil). A digital
thermometer probe was inserted, and temperature readings were recorded at 30 minute
intervals. Figure 4-1 shows a schematic of the manual temperature measurement layout.

MDOT’s KUAB FWD was used to determine the deﬂection basins. In order to
investigate temperature effects on deﬂections and backcalculated AC moduli and develop
correction factors, deﬂection and temperature data were collected simultaneously from

the FWD and sensors/thermocouples. These tests were conducted throughout the day.

38

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39

 

 

Approximately 2.5cm diameter

10E:> 19"— e e ®

Traffic Direction Hole 1 Hole 2 Hole 3 FWD

 

 

(a) Top View Digital Thermometer

 

 

 

 

..: H\ Li
’1 2
‘4 \\ /

Base Oil in Holes

 

 

 

 

D‘—

Subgrade

 

(b) Cross-Section View

Figure 4-1. Schematic of manual temperature measurements.

40

4.2 TEMPERATURE PREDICTION MODEL AND CORRECTION FACTOR
DEVELOPMENT

Data ﬁ'om three of the six test sites (sites 25, 45, 88) were used to develop the
temperature prediction model. The model was validated and calibrated using data ﬁom
site 69 in Brevort, MI, two Special Pavement Studies (SPSl) sites near St. Johns, MI, and
eight Seasonal Monitoring Program (SMP) sites in Colorado, Connecticut, Georgia,
Nebraska, Minnesota, South Dakota, and Texas. The SMP data was obtained ﬁ'om
DATAPAVE 2.0 (1997). Inventory and temperature sensor location information for these
sites is summarized in Table 4-2. Data presented by Stubstad et al. (1994) was also used
for comparing the newly developed model with other popular models.

Temperature correction factors for backcalculated AC layer moduli of all four test
sites (sites 25, 45, 88, 69) and two SPSl sites were computed based on measured and
predicted middepth temperature of the AC layer, using 317 temperature data points and
656 deﬂection basins (Table 4-3). Table 4-4 suMarizes the data ﬁ'om Stubstad et al.

(1994), which was used for the validation.

4.3 INFLUENCE OF TEMPERATURE PREDICTION ERROR ON
PERFORMANCE PREDICTION

For structural (performance prediction) models and/or mechanistic analysis
procedure in M-E pavement design, the measured deﬂections from a FWD are commonly

used along with loading and pavement layer thickness information as input to

4]

Table 4—3. (a) Measured Temperature at Site 25 (New Buffalo, MI)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I MeasmedTemperature
Time | Air Surface 3" (7.62 cm) I 5.1" (12.95 cm) I
| °c °F °c °1= °c I °F °c °F
8:30 AM 18.0 63.9 22.0 71.1 24.5 * 75.6 26.0 782
9:00 AM 20.0 67.5 23.0 72.9 24.1 74.8 25.0 76.5
9:30 AM 21.0 69.3 23.0 72.9 24.0 74.7 24.8 76.1
10:00 AM 21.0 69.3 24.0 74.7 24.3 75.2 24.8 76.0
10:30 AM 21.0 69.3 24.0 74.7 24.5 75.6 24.8 76.1
11:00 AM 22.0 71.1 24.0 74.7 24.7 75.9 24.9 76.3
11:30 AM 23.0 72.9 24.0 74.7 25.0 76.5 25.1 76.6
1:00 PM 26.0 78.3 29.0 83.7 27.0 80.1 26.3 78.8
1:30 PM 25.0 76.5 29.0 83.7 27.5 81.0 26.9 79.8
2:00 PM 25.0 76.5 29.0 83.7 27.9 81.7 27.2 80.3
2:30 PM 25.0 76.5 30.0 85.5 28.6 82.9 27.7 81.3
3:00 PM 27.0 80.1 34.0 92.7 30.3 86.0 28.2 82.2
3:30 PM 28.0 81.9 38.0 99.9 32.0 89.1 29.5 84.5
4:00 PM 27.0 80.1 36.0 96.3 33.3 91.4 30.1 85.6
4:30 PM 27.0 80.1 38.0 99.9 34.0 92.7 31.1 87.4
5:00 PM 26.0 78.3 36.0 96.3 34.8 94.1 31.7 88.5
5:30 PM 26.0 78.3 35.0 94.5 34.7 93.9 32.1 892
6:00 PM 25.0 76.5 35.0 94.5 352 94.8 32.5 90.0
6:30 PM 25.0 76.5 34.0 92.7 35.2 94.8 33.0 90.8
7:00 PM 24.0 74.7 33.0 90.9 34.9 94.3 32.9 90.6
7:30 PM 24.0 74.7 32.0 89.1 34.1 92.8 32.6 90.1
8:00 PM 23.0 72.9 30.0 85.5 33.3 91.4 322 89.4
8:30 PM 23.0 72.9 27.0 80.1 31.9 88.9 31.7 88.4
9:00 PM 22.0 71.1 27.0 80.1 31.1 87.4 31.0 87.3

 

 

 

 

42

Table 4-3. (b) Measured Temperature at Site 45 (Traverse City, MI)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measured Temperature
Time Air Surface 2.25” (5.72 cm) 5.0" (12.7 cm)
°c I °F °c I °F °c I °F °c °1=

10:00 AM 22.0 71.1 29.0 83.7 25.6 77.5
10:30 AM 23.0 72.9 31.0 87.3 27.9 81.7
11:00 AM 24.0 74.7 32.0 89.1 29.4 84.4 26.0 78.3
11:30 AM 26.0 78.3 35.0 94.5 30.8 86.8 26.9 79.9
12:30 PM 27.0 80.1 37.0 98.1 33.6 91.9 28.5 82.8
1:00 PM 28.0 81.9 39.0 101.7 35.4 95.2 29.4 84.4
1:30 PM 27.0 80.1 40.0 103.5 36.6 97.3 30.7 86.7
2:00 PM 28.0 81.9 41.0 105.3 37.7 99.2 31.1 87.4
2:30 PM 29.0 83.7 41.0 105.3 38.9 101.4 32.0 89.1
3:00 PM 29.0 83.7 42.0 107.1 39.4 102.4 32.9 90.7
3:30 PM 28.0 81.9 43.0 108.9 39.8 103.0 33.5 91.8
4:00 PM 29.0 83.7 42.0 107.1 39.8 103.0 34.0 92.7
4:30 PM 28.0 81.9 42.0 107.1 39.4 102.3 34.0 92.7
5:00 PM 30.0 85.5 40.0 103 .5 40.1 103.6 34.6 93.7

 

Table 4-3. (c) Measured Temperature at Site 69 (Brevort, MI)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measured Temperature
Time Air Surface 5.0” (12.7 cm)
°c I °F °c I °F °c I °1=

5:57 AM 7.2 44.4 14.1 56.8 17.8 63.5
6:27 AM 8.1 46.0 14.4 57.4 17.5 63.0
6:57 AM 9.2 48.0 14.4 57.4 17.3 62.6
7:57 AM 17.9 63.7 15.8 59.9 17.1 62.2
8:27 AM 18.7 65.1 18.8 65.3 17.4 62.8
8:57 AM 21.4 70.0 21.8 70.7 18.2 64.2
9:27 AM 23.3 73.4 24.3 75.2 19.1 65.8
9:57 AM 22.9 72.7 26.5 79.2 20.2 67.8
10:27 AM 19.6 66.7 28.6 82.9 21.2 69.6
10:57 AM 22.5 72.0 30.8 86.9 22.6 72.1
11:27 AM 22.3 71.6 31.5 88.2 23.7 74.1
12:27 PM 21.6 70.3 35.4 95.2 25.8 77.9
12:57 PM 26.2 78.6 36.3 96.8 27.1 80.2
1:27 PM 27.9 81.7 37.4 98.8 27.9 81.7
1:57 PM 25.2 76.8 38.2 100.2 28.9 83.5
2:27 PM 30.1 85.6 38.8 101.3 29.8 85.1
2:57 PM 27.8 81.5 38.3 100.4 30.3 86.0
3:27 PM 33.4 91.6 37.3 98.6 30.8 86.9
3:57 PM 29.6 84.7 37.6 99.1 31.2 87.6
4:27 PM 25.0 76.5 36.8 97.7 31.4 88.0
4:57 PM 29.8 85.1 35.2 94.8 31.6 88.3

 

 

43

 

Table 4-3. ((1) Measured Temperature at Site 88 (Deerton, MI)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measured TWC J
Time Air Surface 2.5" (6.35 cm) 3.5" (8.89 cm) 6.5" (16.5 cm)J
°c I °F °c I °1=
7:58 AM 15.0 58.5 19.0 65.7
8:30 AM ‘ 20.0 67.5 22.0 71.1
9:01 AM 26.0 78.3 22.0 71.!
9:30 AM 25.0 76.5 26.0 78.3
10:00 AM 25.0 76.5 29.0 83.7 25.3 77.0 24.9 76.2 24.9 76.3
10:30 AM 24.0 74.7 31.0 87.3 26.4 79.0 26.2 78.6 24.8 76.1
11:01 AM 24.0 74.7 32.0 89.1 28.6 82.9 27.2 80.4 252 76.8
11:30 AM 26.0 78.3 35.0 94.5 29.9 85.3 28.6 82.9 26.0 78.3
12:59 PM 26.0 78.3 40.0 103.5 35.0 94.5 32.7 90.2 28.5 82.8
1:34 PM 26.0 78.3 41.0 105.3 36.6 97.3 34.3 93.2 29.2 84.0
2:00 PM 27.0 80.1 43.0 108.9 38.3 100.4 35.3 95.0 30.2 85.8
2:30 PM 26.0 78.3 43.0 108.9 38.5 100.8 36.2 96.6 30.8 86.9
3:01 PM 27.0 80.1 44.0 110.7 40.4 104.2 37.7 99.2 32.1 89.2
3:33 PM 25.0 76.5 43.0 108.9 42.4 107.8 38.4 100.6 32.4 89.8
4:00 PM 27.0 80.1 43.0 108.9 41.4 106.0 39.1 101.8 33.1 91.0
4:30 PM 26.0 78.3 43.0 108.9 41.5 106.2 39.4 102.3 33.6 91.9
5:00 PM 25.0 76.5 43.0 108.9 41.8 106.7 39.5 102.5 34.2 93.0
5:31 PM 25.0 76.5 42.0 107.1 41.5 106.2 39.9 103.2 35.1 94.6
5:59 PM 24.0 74.7 41.0 105.3 41.6 106.3 39.8 103.1 35.1 94.6
Table 4-3. (e) Measured Temperature at SPS Site K24
MeasuredTemperature
Time Air Surface 2.0" (5.1 cm) 3.5" (8.89 cm) 525" (13.4 cm)
°c I °1= °c I °1= °c I °F °C I °F °c I °1=
7:30 AM 21.0 69.3 23.0 72.9 22.8 72.5 24.0 74.6 24.5 75.6
8:00 AM 22.0 71.1 24.0 74.7 23.6 73.9 23.6 73.9 23.7 74.1
8:30 AM 24.0 74.7 27.0 80.1 24.8 76.1 24.4 75.4 24.0 74.7
9:00 AM 26.0 78.3 29.0 83.7 26.8 79.7 25.6 77.5 24.8 76.1
9:30 AM 27.0 80.1 32.0 89.1 28.5 82.8 27.1 80.2 25.5 77.4
10:00 AM 28.0 81.9 34.0 92.7 30.6 86.5 28.4 82.5 26.5 79.2
10:30 AM 28.0 81.9 35.0 94.5 32.4 89.8 30.0 85.4 27.7 81.3
11:40 AM 27.0 80.1 38.0 99.9 36.7 97.5 33.6 91.9 30.4 86.2
12:00 PM 28.0 81.9 40.0 103.5 37.8 99.5 34.4 93.4 31.2 87.6
12:30 PM 28.0 81.9 42.0 107.1 39.3 102.2 35.3 95.0 31.7 88.5
1:00 PM 29.0 83.7 42.0 107.1 40.5 104.4 36.8 97.6 33.4 91.6
1:30 PM 29.0 83.7 43.0 108.9 41.3 105.8 38.0 99.8 34.3 93.2
2:00 PM 30.0 85.5 44.0 110.7 42.5 108.0 39.2 102.0 35.3 95.0
2:30 PM 31.0 87.3 46.0 114.3 43.1 109.0 39.9 103.2 35.9 96.1
3:00 PM 30.0 85.5 45.0 112.5 43.4 109.6 40.3 103.9 36.5 97.2
3:30 PM 29.0 83.7 44.0 110.7 44.6 111.7 41.5 106.1 37.6 99.1
4:00 PM 29.0 83.7 44.0 110.7 44.4 111.4 41.6 106.3 38.0 99.9

 

44

 

Table 4-3. (f) Measured Temperature at SPS Site K59

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measured Temperature
Time Air Surface 1.75" (4.4 cm) 3.25" (8.3 cm) 5.0” (12.7 cm)
°c I °F °C I ‘T °c I °F °c L °F °c I °F

7:30 AM 24.0 74.7 26.0 78.3 24.6 75.7 24.5 75.5 25.0 76.5
8:00 AM 26.0 78.3 27.0 80.1 25.8 77.9 25.2 76.8 25.1 76.6
8:30 AM 28.0 81.9 29.0 83.7 27.6 81.1 26.0 78.2 25.3 77.0
9:00 AM 30.0 85.5 33.0 90.9 30.0 85.5 27.6 81.1 26.3 78.8
9:30 AM 30.0 85.5 32.0 89.1 30.5 86.4 28.5 82.8 26.8 79.7
10:00 AM 32.0 89.1 35.0 94.5 32.4 89.8 29.4 84.4 27.5 81.0
10:30 AM 31.0 87.3 38.0 99.9 34.3 93.2 31.3 87.7 28.8 83.3
12:00 PM 33.0 90.9 43.0 108.9 39.8 103.1 35.6 95.5 32.2 89.4
12:30 PM 34.0 92.7 44.0 110.7 41.5 106.2 36.9 97.8 33.3 91.4
1:00 PM 33.0 90.9 46.0 114.3 41.7 106.5 38.0 99.9 34.2 93.0
1:30 PM 34.0 92.7 45.0 112.5 41.6 106.3 38.7 101.1 35.1 94.6
2:00 PM 34.0 92.7 45.0 112.5 41.8 106.7 39.4 102.4 35.7 95.7
2:30 PM 34.0 92.7 45.0 112.5 42.0 107.1 39.8 103.0 36.4 97.0
3:00 PM 33.0 90.9 43.0 108.9 41.6 106.3 39.6 102.7 36.8 97.7
3:30 PM 34.0 92.7 43.0 108.9 41.2 105.6 39.9 103.3 37.3 98.6
4:00 PM 32.0 89.1 41.0 105.3 40.7 104.7 40.1 103 .6 37.5 99.0

 

Table 4-4. Database from Stubstad et al. (1994)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AC Measured TLemeprature
Location Time Thickness Surface One-Third Depth
(inches/cm) °C I °F °C I °F
INcbraska 10:00 AM 6.7 43.5 8.6 46.9
Arapaho 11:34 AM 18.1 64.0 13.9 56.5
12:52 PM 21.9 70.9 18.5 64.8
2:34 PM 26.1 78.4 22.6 72.1
9:37 AM 8.9 47.5 13.4 55.6
11:00 AM 7.2/18.3 11.2 51.6 14.4 57.4
12:54 PM 18.6 64.9 18.0 63.9
3:07 PM 23.1 73.0 21.4 70.0
9:10 AM 33.1 91.0 29.8 85.1
10:20 AM 37.8 99.5 33.0 90.9
12:10 PM 45.0 112.5 38.0 99.9

 

 

45

 

backcalculation programs to compute the modulus for each layer of the pavement
structure. The backcalculated modulus of the AC layer is then adjusted to a reference or
standard temperature. The adjusted backcalculated layer moduli are then used in a
forward calculation routine to estimate the stress/strain distributions in each of the
pavement layers and structural performance under expected trafﬁc volumes and loadings.
Hence, the sensitivity of temperature prediction on performance prediction was also
investigated because the temperature prediction is eventually used for temperature
correction and structural performance prediction. Cross-section and deﬂection data
collected from sites 25, 45, 88, 69 and two SPSl sites were used as input to the
MICHBACK program (Harichandran et al., 1994) to compute layer moduli. The
computed AC layer modulus at each site was temperature—corrected based on both
measured and predicted layer middepth temperatures. This layer modulus information
along with pavement cross-section information was then used to compute layer responses
such as, surface deﬂection, horizontal strain at the bottom of the AC layer, vertical strain
at the base, and vertical strain at the subgrade. The CHEVRONX program (Warren et al.,
1963) was used for determining the structural responses. The layer moduli, pavement
responses, and mixture properties were incorporated into a rut prediction model (Kim et
al., 2000) to compute predicted rut depth.

The results ﬁ'om the data analysis will be presented and discussed in Chapter 6.

46

CHAPTER 5

FINITE ELEMENT MODEL

5.1 GENERAL
Due to the relatively simple geometry, ﬂexible pavement analyses are usually
approached by two-dimensional (2-D) axisymmetric models. Many ﬁnite element
programs based on 2-D models have been satisfactorily used in analyzing ﬂexible
pavement problems. However, many realistic problems are beyond the capability of 2-D
models. A three-dimensional model is needed to solve these problems.
The 2-D ﬁnite element analysis/program for ﬂexible pavements has the following
advantages:
0 The model is efficient and relatively simple.
0 Such models/programs are widely used and available.
0 Most 2-D pavement programs have been satisfactorily validated with theoretical
solutions and ﬁeld data.
However, realistic pavement behavior can be far more complex than the 2-D model
idealization. Among the problems that 2-D models/programs cannot accurately and
conveniently model are the following:
o Nonuniform contact pressure distribution under the load (Figure 5-1)
0 Realistic lateral forces pressure (e. g., lateral stresses in various directions)

between tire and pavement surface.

47

These limitations can be solved by multiple 2-D analyses and superposition; however

this usually leads to complicated and time-consuming procedures.

  
  
   
   

2000
ma)“ : 1563 mi Tire Load = 45kN

1500 Tire Pressure = 850-900 kPa
1 cm

1000

Bridgestone 425/65 R225
(Wide-Base)

500 I

Vertical Contact Stress, ozz (kPa)

 

Longitudinal (x)
Vehicle Path

Figure 5-1. Nonuniform vertical stress distribution under a wide-base tire (De
Beer et al., 1996)

In this study, 2-D analysis was performed to carry out parametric studies in which
temperature distribution across the asphalt concrete (AC) layer, AC thickness, base
stiffness, and load pressure distribution were varied. The 3-D analyses were used to
investigate the effect of temperature distribution and nonuniform load pressure such as
tire tread effects as shown in Figure 5-1.

The following sections describe the viscoelastic model for the AC layer, boundary
conditions and inputs, application of the load pulse, and two-dimensional and three-

dimensional models used in this study.

48

5.2 MODELING

It is assumed that only the mechanistic properties of the viscoelastic (AC) layer
depend on temperature and those of the elastic (base and subgrade) layers are not affected
by temperature. Base/subbase and subgrade were modeled as a linearly elastic, isotropic,
and homogeneous solid. Because of the relatively small effect of Poisson ratio 0 on
pavement behavior, constant Poisson’s ratios were used in the analysis. The 2-D and 3-D
ﬁnite element models (FEM) were developed using the I-DEAS graphical software (I-
DEAS Master Series, 1997) and the analysis was performed using the ABAQUS FEM

software package (ABAQUS User Manual, 1993).

5.2.1 Viscoelastic Material Model for the AC Layer

In this model, the relaxation at a given temperature, T, is described by a
temperature-independent relaxation function in terms of a temperature dependent
“reduced time (t,)” which is deﬁned as tr = t/ar, as follows:

t— r )dds(r)r

a(t)= ;IE( (IT) dr

 

(3-9)

where o(t) is the stress at time t and temperature T, E(t,) is the uniaxial relaxation
modulus which becomes a temperature-independent relaxation modulus by using the
“reduced time”, and 8(1) is the strain at time t.

The shift factor a1 is deﬁned in the following form (Williams, Landel, and Ferry (WLF)

equation):

c1 (T—Tr) (3'3)
02 +(T—Tr)

 

h(T) = —log a7 =

49

where h(T) is a time shiﬁ function, a1 is the shift factor, C; and C2 are constants, and Tr is

a reference temperature.

The time shift ﬁmction h(T) can be used to extrapolate the relaxation data to very short or

long times. This study requires the relaxation data of very short times due to the relatively

short duration of the FWD load (approximately 30 milliseconds). The extrapolation

procedure is described below.

Relaxation experiments are carried out with the stress measured in a given time

range under the applied constant strain, for example between t=l sec and t=1000 sec, at

diﬂ‘erent temperatures, e. g., at T = To and T = T0 i AT. Let T0 be selected as the reference

temperature: h(To) = 0 and results can be plotted on a logarithmic time scale as shown in

Figure 5-2. The master curve is made by shifting the data along the horizontal axis.

00)
01'
E(t)

A

 

 

 

 

 

l
e M >I<————>
Extrapolated I Measured : Extrapolated >
10° 103 Logt (sec)

Figure 5-2. An example of stress-relaxation data obtained at different temperature

(modiﬁed from ABAQUS User Manual, 1993)

50

The measured curves make it possible to determine the shift functions h(To—AT) and
h(To+AT). The shiﬁ ﬁmctions can in turn be used to extrapolate the relaxation curves
beyond the measured domain as shown in Figure 5-2. By carrying out stress relaxation
tests over a wide enough range of temperatures, a complete relaxation curve spanning
many decades in time can be obtained. Figure 5-3 shows master relaxation modulus and

shift factor of the asphalt mixtures used in the analysis.

1.E+05

47.8(T— 25)
392 + T _ 25

logaT =

1 .E+04

1 .E+03

1.E+02

Relaxation Modulus. E(t). (MPa)

 

1.901
1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+0 1.E+0 1.E+0 1.E+0 1.E+0
0 1 2 3 4
Reduced Time, tIaT, (see)

Note: 1 MPa = 145 psi

Figure 5-3. Master relaxation modulus and shift factor of the asphalt mixtures used in the
analysis (Kim et al., 1996)

51

5.2.1.1 Time Integration Procedure

The basic hereditary integral formulation for linear isotropic viscoelasticity is:

co) = $200, - t})édt' + 1pm,. — t;)¢dt' (5-1)

where e and 45 are the mechanical deviatoric and volumetric strains; I is a unit matrix, K
is the bulk modulus and G is the shear modulus, which are functions of the reduced time
t,; and ° denotes diﬂ'erentiation with respect to t'.

With a series of exponentials known as the Prony series, the relaxation ﬁmctions K(t) and

G(t) can be deﬁned individually:

"K
K(t,) = 1r,o + 2 K,- exp(—t,. /z,,-K)

i=1 (5-2a)

"G
G(tr) = Goo + Z G,- exp( —t,. /t,,-G)

i=1
where K... and G.o represent the long-term bulk and shear moduli. In general, the
relaxation times triK and trio need not be equal to each other; however, it is assumed that tri
= triK = tﬁG. In many practical cases, it can be assumed that nK = 0.

For this analysis, the master relaxation modulus in Figure 5-3 was curve ﬁtted to
the Prony series using the statistical program, SYSTAT. The regression ﬁts well up to
approximately 10 seconds, which is a period long enough to cover the analysis since the
period of FWD pulse is 30 milliseconds (Figure 5-4). The regression equation for the
master curve is as follows:

E(t) =14285-2209*(1-exp(-t,/1. l447957))-4427.6*(l -exp(—t,/0.0006052))-4454.6‘(1-
exp(-t,/0.0173438))-2637.2*(l-exp(-t,/0.000013l)) (5-2b)

with R2 = 99%.

52

1.E+05

1.E+04

1.E+03

Relaxation Modulus, E(t), (MPa)

. 7 f : -Master Relaxatlon Modulus
9f“ II-efl ,‘ . +P Series

 

1.E+02
1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.501 1.900 1.901
Reduced Time, tlat, (see)

Note: 1 MPa = 145 psi

Figure 5-4. Curve ﬁt of master relaxation modulus to the Prony series

The deviatoric integral equation is
t "G
s = [02(000 + Z Giexp(( t', — tr)/tr,-)e'dt' (5-3)
i=1

This equation can be written in the form
n
S = 200 e— Zaiei (5-4)
i=1

where G0 = G.o + 2%, G,- is the instantaneous shear modulus, (x,- = Gg/G0 is the relative

modulus for term i and

t, , de .
e.- = I, (1—exp<( tr -t..)/tn-))-c—i,;;—altr <55)

53

incr

0h

is the viscous strain in each term of the series of spring-dashpot elements (Figure 5-5).
For ﬁnite element analysis, this equation must be integrated over a ﬁnite

increment of time. To carry out this integration, it is assumed that during the increment,

the total strain varies linearly with time, and hence de/dtr' = Ae/At,. To use this relation,

Equation 5-5 is broken up into two parts:

 

 

 

 

 

 

[7+1 _ I: 1 r n / . de 61 r
e. —I, (—exp«tr—tr) tn»——,— tr
dtr
n+1 d (5-6)
t r +1 8 I
+ I; (1‘ exp(( tr 'tr’} l/tri )l—Tdtr
, d1,
8
81
82 83 84
run—d ——— ——— -—— ——-— b—
Figure 5-5. Prony series strains (ABAQUS User Manual, 1993)
Observingthat
I l I
l—e(t’_t;’+ Mr" =1—e‘A’r/’ri +e‘A’r/’"' (1-e(t’_t;')/t'i) (5-7)

use of this expression and the approximation for de/dtr' during the increment yields

54

I

_ . t" de
e57+1=(1—e At, /t,, )IOr ——,dt,

 

dt,
_ . n I _ n . d ,
+3 Afr/l“ jar (1_e(tr tr)/trl) e,dt,. (5-8)
dt,
Ae tn+l t! _tn+l)/t . p
r (1_e(r r I'l )dtr

A t , 1,7
The ﬁrst and last integrals in this equation are readily evaluated, whereas the second

integral represents the viscous strain in the ith term at the beginning of the increment.

Hence the clunge in the 1"" viscous strain is

_ . _ . ._ . A
Aei =(1—e At’ H" )e" + (e Mr H” -l)e;' + (At, —t,.i(1-—e At,/t,, )) *Ate
r

(5-9)
In an increment, the above equation is used to compute the new value of the viscous

strains. Equation (5-4) is then used subsequently to obtain the new value of the stresses.

(ABAQUS User Manual, 1993)

5. 2. 1.2 Implementation of Temperature Eﬂects
In order to relate the reduced time increment, At,, to the actual time increment, At,

it is assumed that the (time) shift factor (ar) applies to the rate of change of reduced time

when the temperature changes in the analysis:

dt, = —l—dt (510)
0r

where tr is the reduced time corresponding to the change in temperature and t is the actual

time at the reference temperature.

55

For a ﬁnite time increment, the increment of reduced time, At,, is calculated based on the
assumption that the shift function, h(T), is generally a smoothly varying function of
temperature which is well approximated by a linear ﬁinction of temperature over an

increment, and temperature T is a linear function of time t. Then, one ﬁnds the

 

relationship to be:
or
GT —1 ___ eA+Bt
with
l
A = E IMTha‘) — tTh(T + AT)] (512)
l
B = — [h(T + AT) — h(T)]
At
This yields to the following relationship
T AT
Atr = 177+ eA+Btdt =%(8A+BIT+AT _eA+BtT) (5-13)
As a result, the increment of reduced time is calculated as
r=1/aT+AT_l/aT (5-14)
h(T + AT) - h(T)
It is readily veriﬁed that
At, ——) ~1—At if At—>O (5-15)

ar

(ABAQUS User Manual, 1993)

56

5.2.2 Boundary Conditions (Temperature)

Temperature was speciﬁed at each node or node sets. It was assumed that
temperature varied only with depth and does not change with time because of the short
time period. Thus, each layer of nodes across the AC layer was assigned a temperature
value. The temperature distribution was obtained from ﬁeld data and the developed
temperature prediction model in this study (as explained in Chapters 4 and 6).

In order to investigate temperature gradient effects on the structural analysis, four
typical temperature distribution cases were considered (Figure 5-6):

0 morning (uniform temperature distribution),

- daytime (temperature at the surface is higher than at the bottom),

0 average temperature of daytime (uniform temperature distribution of average

temperature of daytime), and

o nighttime (temperature at the surface is lower than at the bottom)
Very low temperature distribution cases were not selected in this study because
viscoelastic materials behave like elastic materials at very low temperatures. Figure 5-6
shows temperature distributions of three AC thickness cases: 3” (7.6 cm), 7” (17.8 cm),
and 15” (38 cm) thick AC layers. These three different AC thickness cases were used in

the analyses for the parametric study.

57

Depth (Inches)

Depth (Inches)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.0 : 1:: I Hj~ 0.0
i ': ‘ ' 1 : I” 1.0
9.5 , --------- r, ----------- r:--- ANight(1:30 All) are
I 1 allowing (7:00 AM) I ~ 2-0
1.0 ' ----------- ﬂ: ------- ”Nu """"" L--_---_-__-._*. ------ 1
- ' oDay (1:30 PM) . ; t 3,0
1.5 Imus. --------- u ————— xAgraee Dgy Ian --------- - M
2.0 I. ------- u ----------- iota: --------- “ 5-0
; I . T 1 : I0.0
2.5 a ----- u ---------- —-. ------------ ,- --------- 01-31 -------- I
. , . . . — 7.0
3.0 ------ amt: ----- ------------ E ------- a—ﬁ-x ---------
. , . ; ~ 8.0
3.5 - . _ , . 1,
10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0
Temperature (°C)
(a) 3” (7.6 cm) thick AC layer
0.0 = ; r; r ; o 0.0
A: C: ' ' x E e
1 0 I ANiohtl1=30 Ml) . ............ ; ____________
' A :1 x : o
; - nﬂorning (7:00 AM) ;
20 I 1‘ D__ D 13°F" *. Io -.,.L-5.o
E A r: . "I ' I x e'
3 0 “I .................. A ”1:11-- *Aj'mﬁ' 0'! ...... 1-1-1 ............
, ‘ C} : : Z ’ :
4.0) ------------ ‘WD ----------- ex ----- E ----------- L 10-9
ﬁn ‘ ‘ ”0 x '
‘-° 4 6. , , .3 x .
M ........... 9!---4 ....... ....... 9.--.f-----.x ..... i ............ Hm
I I? A ' f o L x ;
7.0 ----------- dim-a ----- e ----- ----- at ----- g ------------
8.0 l 3 i l f g - 20.0
10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0

Tem peratu re (°C)

(b) 7” (17.8 cm) thick AC layer

Figure 5-6. Temperature distributions along the AC layer used in the analysis

58

009th (cm)

Depth (cm)

 

 

 

 

 

 

 

 

 

M F' 3 Am 111(1'30 AM) r” ' r 41 M
A: D 9 ' x : : o
A D _ x : : e
20 IA ________ 1;: _____ DMornmg(7:00 AM)_:.It-;-_-_-__-___LQ ......... . 5°
' ‘ g x I q a
; “ a oDay(1:30 PM) :5; 0";
4.0 r ----------- : ---------- D-u »x--:---c ------ r ----------- ~ 10.0
: CI xAverage Day 31! : O :
A : CI A‘ . . g :0 .' 1
I 4 ........... -- ........... ........... ~1 . A
2 0.0 I g ‘ ‘ . ,3 . 50 E
0 . u A : : a: : ' o
c D A : : ox v
= 8.0 .. ------ ----------- o ------- A—-,~ ----------- ;*”-.--x ------------------------- 20-0 g
‘ D A; ; O X
i 3 i ‘3 § : 250 5
8 1M ------------ w ----------- a -------- , ----------- r. ------ ,. 1 ---------------------- I -
: D . ,9. g =
12.0 9 ............ 1 ........... 3 .......... L: ...... 1,..: ........ x4 ....................... 300
a i: : ' t 5
. ............ ; ........... E1---_----__IA---Q ............. x“: ....................... 35.0
14.0 : D M . x : :
. C1 :A 0 : X : :
10.0 . é i i i ; ~40.0
10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0

Tem peratu re (°C)

(c) 15” (38 cm) thick AC layer

Figure 5-6. Temperature distributions along the AC layer used in the analysis (continued)

5.2.3 Loading

When the applied excitation is time-varying, the dynamic effects (damping) need
to be considered. However, the damping effects can be neglected if the time rate of input
is reasonably small. In this study, the damping effects were not considered since the
FWD load time history based on a typical FWD pulse of 33msec duration was used to
simulate the FWD impact load. The peak FWD load was taken as 40 kN (9000 lbs)

distributed over an area of 707.9 sq. cm (109.7 sq. inches). Figure 5-7 shows the FWD

load pulse used in this study.

59

 

1.2 g , 3 f
1.0

0.6 7*

Magnitude of the Load (PIPmax)

 

 

 

 

1 i

0.0 A.
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Time (Milliseconds)

Figure 5-7. FWD load pulse used in the analysis (Uddin, 1998)

In order to implement the FWD pulse in the analysis, it was divided into many
steps. The load magnitude was varied linearly over the step ﬁ'om the value at the end of
the previous step (or zero, at the start of the analysis) to the value at the end of the current
step. This pulse was also adopted for the tire pressure, and the peak load pressure was
taken as 115 psi (793 kPa) for the uniform load pressure distribution. Detailed load

pressure values are presented in the later sections.

60

5.2.4 Two-Dimensional Finite Element (FE) Model
In the ﬁrst step of this research, 2-D models were developed and a parametric

study was conducted to gain experience in correctly building a model. The following
factors relative to the viscoelastic response were investigated:
(a) The effect of temperature distribution across the AC layer
(b) The effect of AC thickness
(c) The effect of rigidity of the base
((1) The effect of loading condition.
This experience facilitated the use of a more complicated 3-D model. For the 2-D model,
the axisymmetric model was used, which assumes that the pavement geometry and
loading are both axisymmetric. Figure 5-8 and Table 5-1 present the geometry and mesh
of the 2-D model and the experimental design for the parametric study, respectively. The
range and type of variation of the above four parameters were determined on a realistic
assessment of practical pavement applications:
0 Temperature distribution

— Details were presented in the earlier section.
0 AC layer thickness

— Thin (7.6 cm (3”)) and thick (38 cm (15”)) AC layer were analyzed besides the

typical 17.8 cm (7”) layer.

- Rigidity of the base

— Modulus varied between 138 MPa (20 ksi) and 6895 MPa (1000 ksi), to separate

granular, asphalt and cement-stabilized bases.

61

an m: u £2 _
.s 32 u so _ .soz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N: 5 o: as as 5 as we a: me a. 03m .3: 5.5....

E as a: m: E e: E E m: N: :_ .359 ﬁlm. «5.52

o: a: 8. S. 8. 2: a: n2 2: z: 9: 52...:

a «a a 8 a a a a a 8 a 8mm =2:

3 a 8 a 3 S S a 8 R an .350 an: :32

R 2 R E a 2 K E a. 8 S 532.5

8 8 3 8 no 3 8 a an R on 8mm :2: 532....

a x mm an a on a. we S 8 a. .356 ﬁlm a: 8222

3. 2. a. s. 2. a an a on 2 x .5825

mm a a on a «N a 8 n a a 3mm .3:

mm a on e 2 t 2 2 2 2 2 .359 EM 5.

: 2 a m a e m e m N _ Bea:

. i I. .. _. r :1. ,. i 4. 1. . . . . .. . . neg—Ecuﬁaﬂonﬁoh
., . .. . i. ., ,_ ., L : : . , . Seasons—:3
as b R f a: _ as m as 1 m3 _ n2 _ maeJ a8, 1— 3 ﬂ 3 322.3223...

ﬂan aﬁ— _ w.» _ A53 unoaonh U<

 

 

33% 05253.“ 05 go.“ 592: :wﬁou Ecogcomxm .Tm £an

62

Load

 

 

 

 

 

7».......... VIIIIIIIII
additional 3” & --
15” 7.6cm& . .
( 38cm) AC, Viscoelastic, E(t)
for the parametric
study
Base, Elastic,
,, E=50 ksi (345 MPa) and additional 20, 100,
20 50.8
( cm) and 1000 ksi (138, 689.5, 6895 MPa) for the
parametric study
I
r Y
466” (1183.6 cm)
Subgrade, Elastic, E=10 ksi (69 MPa)
X

 

 

(a) Geometry

Figure 5-8. Two-dimensional model for a three-layer system

63

 

 

(b) Mesh

Figure 5-8. Two-dimensional model for a three-layer system (continued)

64

9000 lbs (40 kPa) at the
peak of the pulse

.IIIIIIIII

‘5.91”(15 cm)

 

 

 

 

 

(a) Uniform load

 

 

 

 

 

 

 

 

61.5 psi (424 kPa) ~ 82 psi (565 kPa) 82 psi (565 kPa) ~ 61.5 psi (424 kPa)
at the peak of the pulse at the peak of the pulse
I ll
j9l”(15cm)> 3.91”(15 cm)r
(b) Nonuniform load (c) Nonuniform load
(higher pressure at the load center), (higher pressure at the load edge)

Figure 5-9. Illustration of applied loads in the 2-D axisymmetric analysis

Loading condition

Nonuniform load cases (higher pressures at the load center and edge) were
analyzed as well as a standard uniform FWD pressure of 565 kPa (82 psi) at the
peak of the pulse (Figure 5-9). Figure 5-10 shows nonuniform pressures versus

time and location. The FWD pulse was adopted for the pulse of nonuniform.

65

Load (kPa)

600

 

500

400 AAA, ' *

300 - ’ ’ T

200 If"

100 ‘ ”

 

0.0000
0.0031

0.0050

   
    

 

    
 

   
  
     
      
  
     

[III '6’“ \
Ill" .§“\\\\\\\

"O k ‘7 7 .
\R\\\\

_..”'v
Ill. ““\““

   

    

 

00119
0.0144
0.0188 IT“:

 

 

g g ‘g .. m w Edge
0
- o 2’,- § §5 0‘ ‘0 Element No.
Time (8°C) ‘5 3 Load Center
(a) Higher pressure at the load center
600
500 “A
“ 400 ’ ,
,. , "9‘“
V300~**”"”' \‘A‘ K
E ’ ‘fe‘v' “‘0
.1 200 \\\\,.t\‘ A
100
o ,_ . . ..-
O O . 4,... ‘ ‘v’ﬁ ‘
8. g g § § 93 3 co 1”" 5, 3\Load
0 d d g g 5 5 2 a 37’ g ”Element Edge
Time (sec) ° °' 3 g 3. No
0 -

   
   
  

  

 
   
  

 

 

  

Load Center

(b) Higher pressure at the load edge

Figure 5-10. Applied nonuniform pressure versus time and location

66

5.2.5 Three-Dimensional FE Model

The development of a 3-D model allows the investigation of the more realistic
problem that is not easily simulated with a 2—D model. Hence, the effects of temperature
distribution and different load type were mainly investigated with the 3D model. A
typical 7” (17.8 cm) thick pavement was selected (Figure 5—11).

Three temperature distribution cases (Morning, Daytime, and Nighttime) were simulated.

Details of temperature distribution were presented in the earlier section. In order to

investigate the effects of load, three types of load cases were applied (Figure 5-12):

(a) Load case I: uniform vertical load over the entire load area (no lateral stress),

(b) Load case H: uniform vertical load only under tire treads (no lateral stress), and

(0) Load case 111: measured vertical and lateral stresses under tire treads (as measured by
Myers et al., 1998).

The uniform vertical load for the load cases I and II is 793 MPa (115 psi) at the peak of

the pulse. Measured tire-pavement interface stresses presented by Myers et al., 1998 were

modiﬁed for the load case III. The load case 111 includes vertical and lateral tire-

pavement interface stresses. Details of measured stresses are illustrated in Figure 5-13.

The FWD pulse was adopted for the pulse of all loads applied in this analysis.

Because there is no distributed tangential load option in ABAQUS, an alternative
method was used to apply lateral stresses to the model. A layer of membrane elements
was ﬁrst added to the loaded area. The membrane elements were parallel to the global XZ
plane as shown in Figure 5-11. Body forces equivalent to the lateral stresses were then

applied to these elements in the desired lateral direction. The membrane elements had

67

negligible stiffness (for instance, 6.9E—6 kPa(1.0E-6 psi)) in order to avoid restricting the
responses of the pavement. However, applied loads are transmitted to the pavement since
loads computed at nodes (i.e., nodal loads) are not associated with the element stifﬁiess

(Cook et al., 1989). Discussion of results will be presented in Chapter 7.

68

144” (366 cm) 4

Trafﬁc Direction
L ad ,
144” (366 cm) 0 "£5,
' 8.2”
Im (20.8 cm)
7.8” (19.8 cm) ’0
Tl

 

 

      

 

7* (1'73 cm) AC, Viscoelastic, E(t)
20,, (50.8 cm) Base, Elastic, E=50 ksr (345 MPa)

l

 

 

Subgrade, Elastic, E=10 ksi (69 MPa)
466” (1 184cm)

 

 

 

 

(a) Geometry

Figure 5-11. Three-dimensional model for a three-layer system

69

 

      

It?“

 

H

I

                

l0
\
"I

i‘
I.-

               

n
i

 

Ira“.
O.

    

 

(b) Mesh

layer system (continued)

Figure 5-11. Three-dimensional model for a three

70

  
 

(a) Load case I

[Tread 1
I Tread 2
ITread 3
ITread 4

(b) Load case 11

 

I% Uniform vertical load

Measured (non-uniform)
vertical load

/ Measured lateral stress

(c) Load case III

Figure 5-12. Illustration of applied loads in the 3-D analysis

71

7.8” (19.8 cm)

 

IA
I

 

‘1
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.82”(2.lcm) 0.4l”(1cm) 0.82”(2.lcm) 0.41”(lcm) 0.41”(l§n) 0.82”
I I l (2.1cm)
l.23"(3.lc (‘ V I I‘ ’I‘ ’I
l I
34
69 69 J 138 207 t
200 200 110 110 133
103
290 290 414 696
503
386 496 331 331 276
483 483 689 1034 ——
745 745 1034 1089 —-I
552 724 496 496 414
703
965 827 E 1103 1103
3 689 689 800 503
662 807 :N
1048 869 g 1103 1103
917 917 896 517
772 772 1089 1089
614 752
1055 1055 800 434
676 676 1034 1034
945 945 703 290
469 469
483 483 696 696
E g 614 614 503 276
200 200 E 207 207 f .00 .00
:- 3: 207 207 303 207

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I Tread 1 I ITread 2 I I Tread 3 I I Tread 4I

Note: Numbers in bricks are vertical stresses in kPa. (1 kPa = 0.145 psi)

(a) Vertical contact stress

Figure 5-13. Measured tire pressure used in the analysis
(modiﬁed from Myers et al., 1998)

72

I Tread 5 I

8.2” (20.8 cm)

 

IL

[I

303 407

 

338 448
386 414
283 310
110 159
90
48 48
Tread 1 read 2 Tread 3 Tread Tread 5
Note: Numbers in bricks are lateral stresses in kPa. (l kPa = 0.145 psi) Z I X

(b) Lateral tire-pavement interface stresses

Figure 5-13. Measured tire pressure used in the analysis
(modiﬁed from Myers et al., 1998) (continued)

73

 

a.

 

\

 

%\

212

5.317;?

 

0

7‘80
/7

 

meme 2

 

180

180

 

«.1
180 ///a;3
_

(—

 

180
1‘6:

Tread 5

 

 

 

\ .V MT. 8T. 8.1 m_ A

0

0

20

r

 

:3
¢\

220 297

73282823282

170

\—>

 

165

K/Y

1§j

"

Tread 4

 

 

307

301

\V

2—
9
2

0
9

A

31

)40

 

 

/\

 

 

 

813%

0
0
8

 

 

 

<——/7

 

7

Tread 3

 

 

 

 

 

 

 

Tread 2

 

 

 

 

 

 

 

 

\
n m m m 1 m m .1.
\4m_\m\vM.\wm\<m\v.m—\wm max/m:
/m_/ m K m K Mum—K m. K m_r\m_s\m\m_r\w
m_ w m m m m w
m m m m m m m
.x K K A A \\ \

Tread l

 

Note: Numbers in bricks are angles of lateral stresses in degree.

(c) Orientation of lateral tire-pavement interface stresses

(modiﬁed from Myers et al., 1998) (continued)

Figure 5-13. Measured tire pressure used in the analysis

74

CHAPTER 6

RESULTS AND DISCUSSION I — FIELD DATA ANALYSIS AND
DEVELOPMENT OF TEMPERATURE PREDICTION MODEL AND
CORRECTION FACTOR

6.1 TEMPERATURE MEASUREMENTS

Data from Site 45 in Traverse City, Michigan (Figure 6-1) shows that the
temperature at the bottom of the AC layer as well as the air and surface temperatures vary
during the day. Figures 6—2 (a) and (b) illustrate backcalculated AC modulus-temperature
data based on temperatures measured at the surface and mid-depth for Site 25 in New
Buffalo. Based on these three illustrations it is evident that the AC moduli follows a
different curve depending on whether the AC is heating or cooling. During the heating
cycle, the AC sublayers are still relatively stiff; whereas, when the AC surface is cooling,
the AC sublayers are relatively soft. These cycles have signiﬁcant impact on the
backcalculated modulus of the AC layer, making it difficult to establish a temperature
correction model based on the temperature recorded at the top or the bottom of the AC
layer.

Figure 6—3 presents a change in temperature gradients recorded in July at site 25.
The air and AC surface temperatures ﬂuctuated whereas temperatures at lower depths
were relatively stable, resulting in cone-shaped temperature gradient plots. Temperature
gradient slopes changed as the pavement alternated between heating and cooling cycles.
For example, even though the surface temperatures measured at 3:00 PM and 7:00 PM

were close, the subsurface temperatures differed by more than 5 °C (9 °F).

75

 

50.0 : T
45.0 «
40.0 —
35.0 J
30.0 ~
25.0 I

20.0 ~ -

Temperature (°C)

15.0 1

 

I I I I
I """"""""""""" T """""""""

10.0

 

 

 

 

 

 

: . . . Ix Air I Surface 0 Subsurface (14 cm)I
5-01 " ‘ ""3“" T """""""" I """"""" i" """""" I """""
0.0 i i i i I ; , . e
0200 2224 4:48 7212 9236 12200 14224 16248 19212 21236 0200
Time

Figure 6-1. Variation of air, surface, and subsurface temperature over a 24 hour period at
site 45 (Traverse City, MI)

76

AC Modulus (MP8)

AC Modulus (MPa

3100

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

Measured Hysteresis 0 Heating eCooling_I
2900 1 ------------------ g ---------- 1. ------- ; ------------------ ------------------
I Heaﬁng I
2700 « ---------------------------------------
..‘e..
2500 - ------------ 5 -------------- t ----- n ----------------- 3.: ------------------
I “°C.. 1 .
2300 - ----------------------------------------- ------------- ‘ 31.;‘3' ------------------
2100 .. 5 ————— 3'3, ———————— ' ~~~~~~
: 3 go
1900 -1- .5-
1 .34
""""" bu...--.eeooh".. 1
1700~ g -------- -~°j~~~
: e '
1500 «» ~ ~: ~ 5 —————————
Theoretical Hysteresns ---------- I 3
1300 . . ,
15 20 25 30 35 40
Surface Temperature(°C)
(a)
3100 7
2900 I --------- e ------- “mm-~7--------4{Measifred Hysteresis :‘ 0 Heating. Coolian
...... 9399-...
270° I 6‘. "*--::-:.. 1
a 7% '
2500 “I """" °t°‘ """ Q """ 8:3... --.,U“‘Heatmg """""""""""""""""""""""
2300 ~ I I ...................... . . ..........
2100 ,H_---h,5. I -_h ___________________ I __________
I I '0 .::e... . I
1900 1» ~ . g 391224;“; -----
[Theoretical Hysteresis. ---------------- I ; ; 9 ~89
1700 eeeeee 1 ~~~~~~~~ : ' A I ~: ------- I -----------------------------
e
1500 ----------- f ------------------------
1300 , ' . . , . i . .
24 25 26 27 28 29 30 31 32 33 34
Measured Mid-depth Temperature (°C)

 

(b)

Figure 6-2. Measured temperature at the surface (a) and the mid-depth (b) vs. AC

modulus at site 25 (New Buffalo, MI)

77

 

 

 

Air I
Surface

 

 

54
101
15~

201

Depth (cm)

 

251

 

 

30

 

15 20 25 30 35 40

Tom peratu re (°C)

 

I+9z00 +1030 +13200 +15:00 +17:00 +19:00 +21:00I

 

Figure 6—3. Temperature change as a ﬁmction of pavement depth and time of
measurement at site 25 (New Buffalo, MI)

Therefore, both pavement depth and time of FWD testing should be included as input

variables in a temperature prediction model.

6.2 TEMPERATURE PREDICTION MODEL

In light of the above discussion, a temperature prediction model was developed
using 197 data points from the three sites summarized in Tables 4-1 and 4-3. Using non-
linear regression techniques and numerical optimization, pavement depth, time of FWD
testing, and pavement surface temperature were selected as input variables for the model,

and the following model was developed:

7; = TM+ (0.34512 - 0.043222 + 00019623) x sin(-6.3252t+5.0967) (6—1)

78

where: T: = AC pavement temperature at depth 2, °C

Tm; = AC pavement temperature at the surface, °C
z = Depth at which temperature is to be determined, cm.
sin = Sine ﬁmction, radians

t = Time when the AC surface temperature was measured, days
(0<t <1, e.g., 1:30 pm. = 13.5/24 = 0.5625 days)
The developed temperature prediction model has a R2 of 99.34%.. Table 6-1 summarizes
the statistical parameters associated with this model. Figure 6-4 compares the measured
temperature to the predicted temperature for sites 25, 45, and 88. Results ﬁom the model
overlap the 1:1 or 45° line indicating a very good ﬁt. Site 25 (New Buffalo) showed a
slight noise compared to other sites. This was because of ﬁeld conditions on the test day:

heavy traffic and slight rain before measurements.

Table 6-1. Statistical Results of Nonlinear Regression Analysis

SUM-OR DEGREE OF MEAN-

 

 

 

 

 

 

 

 

SOURCE FREEDOM F P-value
REGRESSION .206E+06 5 .412E+05 5882.346 6.334E-208
RESIDUAL .134E+04 192 6.997
TOTAL .2035+06 I97
CORRECTED .412E+04 I96
RAW R-SQUARED (l-RESIDUAL/TOTAL) = 0.9934
CONFIDENCE INTERVAL (95%)
PARAMETER ESTIMATE A.S.E. LOWER
lOUND UPPER BOUND
a] -6.325 0.175 ~6.670 -5.980
8.2 5.097 0.104 4.892 5.302
a3 -0.877‘ 0.436 -l.737 -0.016
a4 -0.279" 0.125 —0.525 ~0.033
35 0.032‘ 0.009 0.015 0.049

 

 

 

 

 

 

 

*6, a4, and a5 are based on inches. In Equation 6-1, they were converted for centimeters
(cm).

79

 

45

1 1 1 I ,‘ |. 1 .s
I ' I O
1 1 1 I .0 , I . \a
1 1 1 1 .' I o
.
‘0 ,. .............................................................. , .---,*_ _.. -- - ________
. 1 1 1 b' a .-
l . ‘
1 1 i .‘I I c l
n
1 1 1 1 ‘_ ' 1

   

35 --------

 

 

x911. 3910.4 cm)

----------

mm 9919.9 cm)

Predicted Temperature (C)
(a)
O

 

 

 

 

 

 

25 , L _ _' _ ‘i __ ’f'j’j' ____________ §_ xsm ”(10.5 cm),
j , ' ' " . . 5 0311545151 cm)
20 » ' l 1 ....... i » lslte 45112.7 cm)»
. . _ " . I I E Oslts 2517.0 cm)
1 ' ' - ' : : :
15 1 **** ; "; +311. 25 (13.0cm)’
. " ' ' ' ' E -s|te 25(25 cm)
10 z . 1, . 1 1 1 I 1
10 15 20 25 30 35 40 45 50

Measured Tem perature (°C)

Figure 6-4. Measured temperature vs. predicted temperature at test sites (88, 45, 25)

6.3 VALIDATION OF THE TEMPERATURE MODEL

To test the accuracy and robustness of the model presented in Equation 6-1
independent temperature and cross-section information from site 69, two SPSl sites,
eight SMP sites, and one additional site from Arapaho, Nebraska (data obtained ﬁ'om
Stubstad et al. (1994)) were acquired. Data from the SMP sites covers a one year period
(one week of each month). The data for the site in Texas covers a period from September
through January. Figure 6-5, for example, shows a comparison of measured and predicted
temperature changes as a ﬁmction of pavement depth and time of measurement at SPSl
site K24 (St. Johns, MI). Figures 6-6 through 6-14 illustrate the comparison between the

measured and predicted at some of the above validation sites.

80

 

 

 

 

 

 

 

 

 

° 1
2 IU— +Measured at 9:00 »
—a— Predicted at 9:00
4 ---1 -- +Measured at 13:00. _____________
-9— Predicted at 13:00
A +Measured at 17:00
E 6 1~— +Predicted at 17:00 +
g . -1 ................
1o . ..... .
12 »— » ——————— [LIN—mm
14 1 ﬂ

 

 

 

Temperature (°C)

Figure 6—5. Comparison of measured and predicted temperature changes as a function
of pavement depth and time of measurement at SPSl site K24 (St. Johns,
MI)

81

 

60

Y

.............

  

50 1

Predicted Temperature (°C)

 

I I I
. . l
3': . .'...7. .
- - 1
:': :_:"‘f~: .
3‘: . . . . .. " lf': . I
*1. "I 1 | l l
’ 1 “1 1 I
‘ I '
- g 2.0 (5.1 cm) -------- . , .
I. i 1 1 I a o
,.l U, 1 1 r
/ l / 1 .
I/ / o
/
n
I
I

____________

...............

------ 7e

 

 

 

 

10 20
Measured Temperature (°C)

60

40

30

Figure 6-6 (a). Validation: measured temperature vs. predicted temperature at site 69

 

 

 

 

 

 

 

(Brevort, MI)

60 r : : . ,-’
.0 E
V 'Q ----------------- -I
i E """" 1.111.; 6r """
§ ' ...... €9.99"! ......
,. E
2 ', I35.1cm 08.9 cm 0 13.3 cm

‘40 v i I 1 I F I 'r 1' 1

-40 -30 ~20 -10 0 10 20 30 40 50 60

Measured Temperature (°C)

Figure 66 (b). Validation: measured temperature vs. predicted temperature at SPSl site

K24 (St. Johns, MI)

82

Predicted Temperature (°C)

60

50

40

30

20 -.

10

 

 

 

 

 

 

 

A. l
‘ I I T I I .-,
1 1 1-':/‘ -<'
6e5- (16e5 cm, : . ":I v’I”
4. 1 . . -1- -,.-,'-— _ +’ -----
' ' : -.‘/' 4'
9.0-(20.3 cm) ; ,,;.’« ;
4>-- ., - 1 1 -------------------------------- .--- ’-/ --------------
772/(g7 7 . 7 i . In
r v I 1 f’I l
- ////,///// 39.11.191.931») .............. 6...---1 ..................
///////% 5 E f 15" ,4“? '
7 7 - -'L ----- - 11111 I ~~~~~ ,vf." ’e"," -’- - -:- - A — I ————————————————
= . 1 L,.;-:,’,y' : Llne lot
1 1 1 .'1./ . , 1 1 1 I
4* r : 1 f ,.-:'-’,wt:’"' 1' ‘ 5‘1”...”
' : l'-."’,',’.’." : :
I :,.A. -_'/J’..’::: I _______________________________
I 1 1 , 'v,’ Al,’
i C '-"v."""."
1 '. P’ It)" I
: . 9:7— .6' * :* *****
I 1.-'/ . f 1
1 .I'/. .’_ 1 1 1
1 'o"’ 'I,’ 1 l t
A 77777777777777 ’317’11 .f,- 1 — 1» ——--6 ~~~~~~~ .‘..,.__-___,_ -----------------
I .-' ' .’.' ' +-8 C’
',.;,-:.e1, +401; [0 4.4 cm 08.3 cm e12.7 cml
..',ﬁ’xr-" : ........ 1 .1 : ........ : ........ z .......
.u”’ ’l-4 1 1 I I
o',‘ 'l,‘ I 1 1
.4" : :
0

10 20 30 40 50 60

Measured Temperature (°C)

Figure 6-6 (c). Validation: measured temperature vs. predicted temperature at SPSl site

Predicted Temperature (°C)

60~

50

40

30

20

10*

K59 (St. Johns, MI)

Slte 8-1053-1 (Delta, Colorado)

 

  

'1 1 1 1 E E I..".:.";':
‘ .. ' : : : : .-:’ ‘ ..’."
~.-T.mp. .t __I 25.: _ 4e6 (11.68 cm) _'L _________ : _________ I ......... r--...;:‘d.‘ (fl: -
2.4“ ' - . 3 E I E .-:"" ,; .
-1- (0.1 cm) _--_ _ 4.5 (11.43 cm) '1‘ _________ : _______ ."1" (5; ’11:" ”J. ........
I : 1 : .. .y. ‘1" :
.1 ............... -235" 159.09 cm) ......... ~j ,6 tv ........ ,1 ________
l : 0“ ~ .1", i I
l I ' .1 I i
4 """"""""""""""""" 1 """""" , ' "':""""_:"""i.ii{e":3€ """
1 ........ . ......... 1 .................. -- ......... 1 ......... 1----.5311111 .....
I .‘. E I I I

 

1
1 1 1
. 1 f / f 1 1
1 1 1
+4 0 3'"! I ' 1
..A ________ a. - '-’ ..'
‘It “ .’. " 1 1 1 1
1 .',’ at ' 1 1 1
o ..'I ,’,r 1 1 1 1
v ' l,‘ 1 1 1 1 1
+‘2 c ,“r’ /:o' 1 1 1 1 1 1
e / .
—1 AAAAAA c-o —1 -------------- O I O t I —————
.I I a
'0' I. .‘ l l I I I
' I . 0‘ 1 1 1 1 1
b I. "'-' 1 1 1 1 1 1 1
‘.')- "- 1 1 1 1 1 1 1 1
a. ‘3. ‘ _ - .I, _ ,1 _, 1, - L1, 1_. . ,1_,_-,,,1 _________
.'.’ fan-t 1 1 1 I 1 1 1
.‘l . l 1 1 1 1 1 1 1 1
o',’ ‘/,' 1 1 1 1 1 1 1 1
" / .' I 1 1 1 1 1 l 1
1' a. 1 1 1 1 1 1 1 1 1
r f f r f 1

 

 

-10 0 10 20 30

Measured Temperature (°C)

Figure 6-7 (a). Validation: measured temperature vs. predicted temperature at the

middepth of 6.1 cm (site 8-1053-1, Colorado)

83

60

60 Site 8-1053-1 (Delta, Colorado)

 

 

  
 

50+ Temp.at .I_‘§;§i§:§2 3333;32:- E‘J" (11.88 cm) -:r~—-——-~-E---------i----~-—-:r __-._ , ________
4.29" ' Z ,_ ‘ 5 5 E - i}

40 1 110.9 cm) 4 ~:‘-5 ("-‘3 W" —1 --------- : --------- +135; ‘ ------ 1 ---------
3 3 WW : i ' In? - .4 ° 5
1.. 30 -1 ......... ; ...... /- ;23.5' (511.19 cm). ..... 7‘ ...................
3 z W ; a s " ' .-:"' e
g 20 1 --------- 3 ------ -------- -:— ‘° --------- : -------------------
3. 2 . : : . : . .
g 101 ---------- : --------- 1 --------- g --------- .—---'--: --------- 1 --------- ---------
I- 5 ' .° : :
3 o «1 ------------------ 1 ----------------- ‘9“: --------- : --------- 1 ------------------ J ---------
2 5 ' E 5 3
g -10 ~ ------------------- g ---------------- J ————————— ,1 ------------------ 1 --------- L -------- J ---------
h . I

~20 .1 ------ , ------- 1 . ------------

-30 ~ ' ***** . E

-40 L :

 

 

 

—40 -30 -20 -10 0 1O 20 30 40 50 60
Measured Tern peratu re (°C)

Figure 6—7 (b). Validation: measured temperature vs. predicted temperature at the bottom
(site 8-1053-1, Colorado)

 

60 Site 9-1803-1 (New London, Connecticut) ' ' ' ‘ ".

\e

  

D I
I I a
O
1 1 'v o "
e f '
5o _._ - - - . - - - .2 . V - L :1- - .. . . _ E -g--:1. ..‘;1'2:’::’,;:._ - _
Temp. at s 2 I t _1-"./ ,-,’.-‘
E.::'..:'::-f 22".?! 1 r 1 . 1 e 1
40 ~> 3.5. --- -::-:" --------------------- :— --------- :---------.:--- :j;/ ;t:--’ -—: ---------
A -,.:_ I“.H"- ‘ ‘ v p
o (8.89 cm) . ' r 1 ‘,.,’.'1' .
o i ..‘ . ’a'_'
.— 30 1------4 ...... 1 ......... :+ 4...: ........ 1 ........
1 1 .n 1
5 : . . : a : :
1 1 I O a' I 1 i
20«-----~-~,L -------- ----------- ..r- ------ ”poof-"-
1 1 " I 1
i : '('o. I : : Equ.l|ty
5 10 ~ ---------- --------- --------- ---------
1 1 1 ' 1 1 1 1
I l ’. I I I I
h ' f-soc . .' 1 : l
o ............................... _- J ................................................
: +-‘°c: I. ,r i
1 1 ‘1’ I. 1
1 1 _' .1 0' 1
-10 ..... 1 6‘ -1- - ;l ’ﬁlg J .....
.2 +152 C ;._:.".:'{‘-b i .
‘l : .-;:’.z ' :
'20 A __________ 1'“w ./' '7. R’
1 ' ‘/1.‘ 1
.e'.’ ‘f’ .’ i
-30 41"., 1 '
' ’0’ ’o": i
, I'O'. 1 1 1 1 1 1 . 1
g .0 l l | i l I I 1 1
40 ’ + 1 7 ‘ﬁ ‘ r 1 ?

 

 

 

40 -30 -20 -10 0 10 20 30 40 50 60

Measured Temperature (°C)

Figure 6-8 (a). Validation: measured temperature vs. predicted temperature at the
middepth of 8.89 cm (site 9-1803-1, Connecticut)

84

 

 

 

 

 

60 «——8lte 9-1803-1 (New London, Connecticut) : I .'
.. ———————— --------------------------
o3, _________________ : _____ , ° ‘1‘ _______
8 r 1 . .1 h : r w
i ' s° _______ 3 ____________________________
E w} _________ 3 _________________________
l- . a 1
E ; a s

-30 4 .............. L

-40 . T . i . I . i ,

-40 -30 -20 -10 0 10 20 30 40 50 60

Menu red Temperature (°C)

Figure 6-8 (b). Validation: measured temperature vs. predicted temperature at the bottom
(site 9-1803-1, Connecticut)

 

60

Site 1 3-1005-1 (Houston, Georgla) E

-,----
t
‘. \ t‘..

 

,,,,,,,,,,,,,,,,,,

Predlcted Temperature (°C)

,,,,,,,,,,,,,,,,,

 

 

 

40 -30 -2o -10 0 1o 20 30 4o 50 60
Measured Temperature (°C)

Figure 6-9 (a). Validation: measured temperature vs. predicted temperature at the
middepth of 8.4 cm (site 13-1005-1, Georgia)

85

60

 

    
 

Site 13-1005-1 (Houston, Georgia) :

5° .. ’ see-2.33:.;-:.:-:.:‘zs-s.:-=..:- ; Z l """"
Temp. at I E: :3"1:'1‘333__.__,: 7.7" (19.56 cm) 1 '3'

40 + 5.63" ‘--- Ej. '§.5j';.?' .- . -r ------------------ r -------- 1 ---------

(14.3 cm) "

30 I S,9.1'(23.11CM) i ___________ " a3. .............. z _________

20 «I --------- 3 ------

i I ' l

i i | I I

10% I I O l I l
.....................................................................................

I I I I I I I l

I I I 1 I I

i | | I I l

I
—-'-—---

Predicted Temperature (°C)

- I
I l i I I I I i
I l I | I I I v
-30., --_ . __ .1. ..... . ..-...I,_..._ ___L,_ . ,,,..|__. I ____________________________
I l - I I l I .
I I i I I l I l

 

 

 

-4O -30 -20 -10 0 10 20 30 4O 50 60
Measured Tem peratu re (°C)

Figure 6-9 (b). Validation: measured temperature vs. predicted temperature at the bottom
(site 13-1005-1, Georgia)

60 ..

 

 

    
 
 
 

Site 31-0114-1 (Thayer. Nebraska) .- I

5 3:2?"31, :. 3:721:12. . I ' 157:4};

50 4' Temp. at I :g- ' 33:33:? 3 . . """"" 7" ’.{,?'(" "
3.9” ' ' 1 : "

«- . “ -12'(3o.« cm) ......... - ”a -4 .........

-24'(eo{.ee cm) ‘ ------- . 53'3"} ........ J ........

‘t
9.

- -----L.---

10 --------- ----- --------- ---------

Predicted Temperature (°C)

 

 

o ‘i ----------------- r ---------- ' f. "-r ----------------- 1 --------- Ir -----------------
+4°C ..f) ’ I r i 1
10 I (Q, I I I I I
- . ..‘..- . V _ 7” -,_ ; ~-.‘ .................................
.' I -

+-2.“C .--,-’ It“ : :
i o',’ I I r
' a ' / I

-20 -< ~~~~~~~ .,, o. I I. - I» ___________

.‘I

" .l' .’. i
.' I 0’: I
'/. I.“ I r
‘30 _+. 7 o ‘lt2" I ‘ i
0" . " . l
I‘-/. .’-. I
I" -’.- ‘ ' 1

 

-4o -30 -20 -1o 0 10 20 30 4o 50 60
Measured Temperature (°C)

Figure 6-10. Validation: measured temperature vs. predicted temperature at the middepth
of 9.9 cm (site 31-0114-1, Nebraska)

86

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

60 Site 21-6251-1 (Beltrami, Minnesota) 1 ; ”:7?
50 I . .5743“. I.
z"."‘."r
A 40 13'", ----------
8
30
2
3 1 * : 1 .' . ' 1
g. 20 i """"" Soil """"" 1L” " “i """"""" L‘rh‘e‘é'of """
‘ ' ' -‘ ' Equ'lity
E 10+ a --------
I! 5 i
'D 0 -------------- _» 0' rrrrrrrrrrrrrr r 777777777777777777777777777
g : '5?
a -10 _ .......... f4..c._1. , I ..J ...............
E . 1 .
-20 _..+'2.p. .' - ..................................................
-30 gr}: ; T 777777777777777777 . ————————— r ——————————————————————————
4o - ’-'" : i I L . , .
-40 -30 -20 -10 0 10 20 30 4O 50 60

Measured Temperature (°C)

Figure 6-11 (a). Validation: measured temperature vs. predicted temperature at the
middepth of 8.3 cm (site 27-6251-1, Minnesota)

 

 

 

 

 

 

60 . ‘ Site 27-6251-1 (Beltram|.llnneeota) :7
5° I'm...“ I museum '
5.51" - I ,
4o ----- ~ ‘ eeeeeeeee :~ --------- ---------------------
8 ‘1‘; cm) . 10.2" (25.9 cm) 3 "'3 , ,
0" 307 _________________ ' . . VVVVV I _________ - 777777777 IA 3". A: ______ .> __________________
2 ‘ i I 1: :r 1' °.' t :
g 20 .. ,__.' ............ If ...............................
g. 10 77777777777777777 . 3 ..J‘. . f: .......... I ............................
I- : '
E 0 """""""""" Y """"" E"#. """"""""" I """""""""" ‘l """""
-10 ---------------------- . ~' A: ——————————————————————————————————————————————————————————
I f I
2 ‘ ..a' (if ' i
-20 7777777777777777 : e» -----------------------------------------------
:81' :
-30 ,. ~~~~~~~~ -----------------------------------------------
-40 f 3
~40 -30 —20 -10 O 10 20 30 40 50 60

Measured Tern perature (°C)

Figure 6-11 (b). Validation: measured temperature vs. predicted temperature at the
bottom (site 27-6251-1, Minnesota)

87

    
       
       
     

 

 

 

 

 

 

60 . Slte 46-9187-2 (Meade, South Dakota)
50 Temp. at "I 5'9. (15cm) ~~~~~~~ AAAAAAAAA L~ “‘{;2f:;;_.’-'-_'_'_’
2.56" V, .. % » ‘ $1.221: -'
40 (3.5 cm) . .. ,5 (15-24 cm) i '3; ‘,_ .-‘ --J VVVVVVVVV
o ¢:T”/-’{f’//// 1 1 i ' ,4?" '
o. ......... 1 ......... Wis-u.» . ........
5 3 M a s . .. 3 i
a 20' """"""""" ‘ ' ,.-"’ "I """"" 7"“[ruie’6r """
3' ‘ : 5 : Equality
E ‘ """""
o
'— l
'5 k
3 r
.2
'5 ......................................
2
n.
40 -30 -20 -10 0 10 20 30 40 50 60

Measured Temperature (°C)

Figure 6-12 (a). Validation: measured temperature vs. predicted temperature at the
middepth of 6.5 cm (site 46-9187-2, South Dakota)

 

   
  

  

 

  
 

 

 

 

60 i_,___sno 46-9187-2 (Meade. South Dakota)
5° ' ’ min.“ I 5-9'(‘5 cm)
4.17" .. i i r

A 40 a (10“ cm) ~ » 6 (15.24 cm) . I... m ‘
0 W WWW E E E I E
‘5 30 --------- g --------- /]La"t7l-62cm), : ---------
a 20 ; W 5 g ' ‘
E 3 m 1 A
0 r w .y;
E 10 --------- : 777777777777 ' 7777777777777 , ............................
O ' 4.
I- .-r
u o , ................................. - ....................................................
8
2
'u 40 A .........................................................................................
s : /

-2o 77777 ‘ 1,/

30 / 3

/ 1 1
.40 ; ; : 1 : : : ' :
-40 -3o -20 -10 o 10 20 30 4o 50 60

Measured Temperature (°C)

Figure 6-12 (b). Validation: measured temperature vs. predicted temperature at the
bottom (site 46-9187-2, South Dakota)

88

 

 

 

 

 

 

6° ‘ ->- .
50 710.9- (21.7 cm) ____________________________ _ :25?
4o ' ‘ , a 3" (i514 cm) ~ .........
'3 ' 126:2” ‘ 3 5 . .--" 1
1. 3° . Z??? rues cm) ........
g W/A ; s ' . 1
E 20 . Q ________ ' 4r “1 QQQQQQQQQQQQQQQQ LIICIO-of....
n. ". 1 Eon-my
g 10 QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ -_-'Q _________
3
3 °'
2
'5 .10
l
.20
-30
4o ' - - , . .
4o -30 -20 .10 o 1o 20 so 40 so so

Measured Temeprature (°C)

Figure 6-13 (a). Validation: measured temperature vs. predicted temperature at the
middepth of 12.8 cm (site 48-1068-1, Texas)

Slte 48-1068-1 (Lamar, Texas)

 

 

 

 

60a . .
50 10.9r(27.7cni) _______ _________ L _________
4o , 6'(1‘6.24cm)l VVVVVVV ,3“ .. __________________
5 IWIWCW ' Y I -' '3 1
a. 307 1 ;/ s'(zo.3 cm)l _,__-:;_.L-2 _________
a , m rrrrrrrrrrrrrrr ...‘.r~~; ----------------------------
3' 2 ‘ e,‘ '
E 10 ----------------- u" 7777777777777777777777777777777777777
.2
v o 7777777777777777777777777777777777777 if 77777777777777777777777 . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
3 3
% -10 rrrrrrrrrr ,
2
n.
20
-30 L ,,,,,
-40 , ; 4 a a? A ‘ r
-40 -30 -20 -1o 0 1o 20 30 4o 50 60

Measured Temeprature (°C)

Figure 6-13 (b). Validation: measured temperature vs. predicted temperature at the
bottom (site 48-1068-1, Texas)

89

 

 

 

 

 

‘50 ~F———. sue uszo1s1 (Texas) , . ,3: .
v .'./ a,
. l l ‘,v" ‘ ’
50.--“--“4---” W. ...... 1. ................... V- .;(",’ '
Temp. at 7.25'(18.42 cm) ' ' ,:".'
3.5. ‘ ' ' . ’:"' '
40 ~ - (8 es m)- --------- r -------- 1 ------------ .rﬁr‘ "1 --------
A I I I
0 » 15" (38.1 cm) - .
qua ' ' ' ‘
2 30 --------------------------- 1 --------- r --------- : ----- r -------- -: --------
g 2., -l ____________________________________ _________ : _________ l.- 199-93---
3 '. : : Eﬂlﬂmy
-i l l |
E 10 4 ------------------------------------ +-- - -. --------- 4 ------------------ 4 ---------
'— ."("‘-’ i I '
‘0 ------------------- -z': . —r ————————— i ——————————————————————————————————————
3 o if 1":-.’- : : :
.2 .z:"' . - ~
1: -10 4 ’ ' ,: L
E : I 1 I
E : : r .
-20 .i ........ . : : ..I ................
..I-v,‘ I : : :
-30 « - ,.~_-';¢’,r.- ---------------- t : --------- 1 -------------------------
.'/ ;t c - I r r
4 4’: ‘l . . r
y 'l. I . I
40 ’x‘— % l 'r , . + Y S r
40 -30 -20 -10 0 10 20 30 40 50 60

Measured Temperature (’C)

Figure 6-14 (3). Validation: measured temperature vs. predicted temperature at the

      
 
 

 

 

middepth of 8.89 cm (US281S1, Texas)

 

 

 

‘0 sm uszusr (Texas) . ;
”7.2%" ii‘st" em y“ """""" g """""""" : """"
6.5" - = . - A - - i
‘0 ‘”" " """“I """"" I' """"" : """" I ' " I"'""”"‘I """"
.3 ""53"" 15-(3o.1 cm) ; .. . - ;
.. so .. --------- --------- *- --------
5 . ‘ ‘
2 . ........................................................ Mamet---
i o Eiquellty
§ 1o4 ------------------- T ---------
I- : . - :
g o ..p ..... t I- ._ _ .._ _ L. I ......
3 .10 +7» , g - , 3 ----------
a : : 1
-2o l—A- : . ----- J.
.30 .L ..... .
-4o 1. 3 : : 4L : 4 f L
.40 .30 -2o .10 o 1o 20 so 40 so so

Measured Temperature (°C)

Figure 6-14 (b). Validation: measured temperature vs. predicted temperature at the

bottom (US281S1, Texas)

90

The results from the developed model again overlap the 1:1 or 45° line, indicating that
this model has wide-ranging applicability. The newly developed model was comparable

to other popular models even though fewer input variables and a simpler procedure was

 

 

 

 

 

 

 

adopted (Figure 6-15).
45 . . . . 1 ,
eBELLS MODEL ; : g g 2 g
5 4° ‘ " AAASHTO MODEL """"" ' """"" g """"" g """" : “““““
E 35. , DAI MODEL ..... I ......... ‘3. ..........
“ oMSU MODEL ; i d E E
I 304 T I , *’ :_ t ---------- §~~:~- u : rrrrrrrrrr
5 . r z ; ,= a s
E 25 4i ' ' Ir; “““ I """ It I I
g 20 J---—~—-~,~~ ------- ,- --------- LL eeeee -_ 1- ------- --------- ;L -------- 1 ---------
o 1 l3'3: : D : : :
l,— L , , A . . L
3 15 ---------- 3 ----------------- :----'A--s- --------- --------- a ---------- r ---------
o 3 : E ? E E E E
3 1° “““ *. """"" r """"" """""
o. a 5 D; D D . r .
5 . ;
o . » - .
O 5 1O 15 20 25 30 35 40 45

Measured Temperature at hl3 (°C)

Figure 6-15. Comparison between the MSU model and other models using data from
Stubstad et al. (1994). (h/3 represents one third depth of the AC layer)

6.4 TEMPERATURE CORRECTION FACTOR FOR AC MODULUS
The deﬂection proﬁles and pavement cross-sections were used as inputs to
backcalculate the layer moduli for the four test sites (sites 25, 45, 69, 88) and two SPSl

sites (K24, K59). Maximum deﬂections were measured at a load level of 40 RN (9000

91

lbs). Figure 6—16 shows the dependence of AC modulus and peak deﬂection and the

independence of base and roadbed modulus on mid-depth AC temperature.

 

 

 

 

 

1oooo .. I , . .4 V. , . 1000
I
o.
E
s a
'3 d
'5 C
o 3
2 0
'u a“
a .
- D
3
o 8
-— I
8 o
x n.
O
I
m r l 7 it
fffff 5 Quantum ACé-B???:9~§ﬂhs@¢9
,0 ; L .: ; 100

 

20 25 30 35 40 45
Measured Mid-depth Temperature (°C)

Figure 6-16. Measured mid-depth temperature vs. backcalculated modulus of each layer
and peak deﬂection at site 88 (Deerton, MI)

The backcalculated AC modulus is corrected to a reference pavement temperature

Tr (25°C) in accordance with Equation 6-2:

131‘, = ET X CF (6-2)
where:
ET, = Corrected AC modulus to the reference temperature Tr
such as 25 °C
E1 = Backcalculated AC modulus at measured middepth temperature T
CF = Correction factor
92

The general relationship between mid-depth temperature and backcalculated AC modulus

can be described as follows (e.g., backcalculated AC modulus-middepth temperature data

in Figure 16):
log ET = b + a *T (6-3)
where:
ET = Backcalculated AC modulus at mid-depth temperature T
T = Middepth temperature
a, b = Regression constants

By substituting Equation 6-3 into Equation 6-2, the correction factor is derived in terms
of “a” (regression constant in Equation 6-3):

CF = 10““) (64)
Therefore, the correction factor is determined by the slope, “a”, of the linear ﬁt of
temperature versus the log of the backcalculated modulus relationship (Kim et al., 1996).
Table 6-2 presents the statistical parameters associated with regressions for constant “a”.
Table 6-3 summarizes the computed values for constant “a” at each site based on
measured and predicted mid-depth temperature. Values of “a” obtained using measured
and predicted mid-depth temperature at each site are consistent, which also indicates that

the temperature prediction and correction procedures were valid.

93

Table 6-2. Statistical Results of Regression Analysis for Constant “a”

Site 25 - Rsressiou based on Predicted Mid-depth Temperautre

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SOURCE :UMIU A328 DF :53; F p-value
REGRESSION 268.47 2 134.24 L 71174.58 1.19952E-42J
RESIDUAL 0.04 22 0.0019 '
TOTAL 268.5 1 24 [—
CORRECTED 0. 13 23 L

 

 

RAW R-SQUARED (l-RESlDUAL/TOTAL) =

0.99985

Site 25 - Regression based on Measured Mid-depth Temperature

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SOURCE 33319325 DF ;ACELIJ‘gE F p-value
REGRESSION 268.50 2 134.25 223005.6 4.20349E-48
RESIDUAL 0.01 22 0.0006 t i I i I

TOTAL 268.51 24 - 1 " _...
CORRECTED 0.13 23 1 .
RAW R-SQUARED (l-RESIDUAL/I‘OTAL) = 0.99995

 

 

 

 

 

 

 

Site 45 - R mien based on Predicted Mid-depth Temperautre
l SOURCE Egg/1:328 DF KESA‘JRE F p-value 1
lREGRESSION 188.77 2 94.38 38242.825 1.49004E-23J
RESIDUAL 0.03 12 0.0025 T b I a
TOTAL 188.80 14 F
CORRECTED 0.03 13 ;I

 

 

 

 

 

 

 

 

RAW R-SQUARED (l-RESIDUAL/TOTAL) =

0.99984

94

Table 6-2. Statistical Results of Regression Analysis for Constant “a” (Continued)

Site 45 - R ession based on Measured Mid—depth Temperautre
SUM-OF- MEAN-
I SOURCE DF SQUARE F p-value

SQUARES

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[REGRESSION 188.77 2 94.38 39774.067 1.17737E-23
RESIDUAL 0.03 12 0.0024 ;1 f w i
TOTAL 188.80 14 f
CORRECTED 0.03 13 L
RAw R-SQUARED (l-RESIDUAL/TOTAL) = 0.99985

 

Site 69 - Rgreesion based on Predicted Mid-depth Temperautre

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SOURCE :aEAOLIE-S DF ISAESERE F p-value I
REGRESSION 249.06 2 124.53 59272.102 4.28486E-3j
RESIDUAL 0.04 18 0.0021 5 i A) w i
TOTAL 249.10 20 r A 3
CORRECTED 0.37 19 i .
RAw R—SQUARED (I-RESIDUAL/TOTAL) = 0.99985
Site 88 - ion based on Predicted Mid-depth Temperature
I SOURCE 238;; DF ngIiAqRE F p-value
GRESSION 192.09 2 96.05 60339.324 1.83839E-33
RESIDUAL 0.03 17 0.0016 ‘7 . I M f f,
TOTAL 192.12 19 M I.
CORRECTEDI 0.49 18 I
RAw R-SQUARED (l-RESIDUAUTOTAL) = 0.99986

95

Table 6-2. Statistical Results Of Regression Analysis for Constant “a” (Continued)

Site 88 - Rsreesion based on Measured Mid-depth Temperautre

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RAW R-SQUARED (l-RESIDUAL/TOTAL) =

K24 - Regression based on Predicted Mid-depth Temperautre

SUM-OF- MEAN-
SOURCE S CU gs DF S CU AJ__R__E_ F p-value
REGRESSION 145.73 2 72.87 464114.78 2.82379E-32
RESIDUAL 0.00 13 0.0002 . :
TOTAL 145.73 15 *
CORRECTED 0.22 14 '
‘ H
0.99999

 

 

 

 

 

 

 

 

 

 

 

 

 

RAW R-SQUARED (l-RESIDUAL/TOTAL) =

Mien based on Measured Mid-depth Temperautre

 

 

0.99967

SUM-OF- MEAN-
SOURCE S CU A=R__E S DF S ; U ARE F p—value
REGRESSION 185.01 2 92.50 22405.577 8594061343]
ﬁr 7*"; 7T 7?; 7 , T
RESIDUAL 0.06 15 0.0041 ,5
TOTAL 185.07 17 E; 5
CORRECTED 0.50 16

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RAW R—SQUARED (l-RESIDUAL/TOTAL) =

96

SOURCE EEEES DF :[ESEéE F p—value 1
REGRESSION 185.03 2 92.51 33737.168 3.99412E-28
RESIDUAL 0.04 15 0.0027 . . '
TOTAL 185.07 17 r
CORRECTED 0.50 16 L
0.99978

Table 6-2. Statistical Results of Regression Analysis for Constant “a” (Continued)

 

 

K59 - Regression based on Predicted Mid-depth Temperantre
1 —

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SUM-OF- MEAN-
SOURCE S U ARES DF S QU Ag F p-value
REGRESSION 138.82 2 69.41 15840.069 3.28128E-24
RESIDUAL 0.06 14 0.0044 ,-.
TOTAL 138.88 16
CORRECTED 0.36 15 .
a. «4.
RAW R-SQUARED(1-RESIDUAUTOTAL) = 0.99956
Wion based on Measured Mid-depth Temperautre
SUM-OF- MEAN-
SOURCE S ; U ARES DF S ; U ARE F p-value
REGRESSION 138.86 2 69.43 39657.278 5.33199E-27
RESIDUAL 0.02 14 0.0018 F
TOTAL 138.88 16
CORRECTED 0.36 15
RAW R-SQUARED ( l-RESIDUAUTOTAL) = 0.99982
Table 6-3. Regression Constant “a” at Each Test Site
“3” value
Site
No. Based on Measured Middepth Based on Predicted Middepth
Temperature Temperature
25 -0.0230 -0.0140
45 -0.0037 -0.0039
69 ‘ T ”T“ ‘ -0.0176
88 -0.0231 -0.0231
K24 -0.0256 -0.0265
K59 -0.0250 -0.0257

 

 

 

97

 

6.5 INFLUENCE OF TEMPERATURE PREDICTION ON PERFORMANCE
PREDICTION

In order to investigate effects of middepth temperature prediction error (AT) on
performance prediction, a three step procedure was applied. First, AC moduli based on
measured and predicted middepth temperatures were backcalculated using MICHBACK
and adjusted to a reference pavement temperature (25 °C). Second, these backcalculated
moduli were used to compute structural responses using CHEVRONX. Finally, based on

these analysis results, rut depths were calculated using the following performance (rut)

model (Kim et al., 2000).

RD = (— 0.016HAC + 0.033 ln(SD)+ 0.01 170M — 0.01]n(KV))-

[— 2.703 + 0.657(gvm )“97 + 0.2710,,” )0“ + O.2581n(N) — 0.0341n[%)]

SC
(6-5)
where:
RD = average rut depth along a speciﬁed wheel path segment
(inch)
SD = pavement surface deﬂection (in.),
KV = kinematic viscosity (centistroke),
Tm“. = annual ambient temperature (°F),
HAC = thickness of asphalt concrete (in.),

N = cumulative trafﬁc volume (ESAL),

amuse = vertical compressive strain at the top of base layer (103),
8,35 = vertical compressive strain at the top of subgrade (10'3),
EAC = resilient modulus of asphalt concrete (psi), and

Egg = resilient modulus of subgrade (psi).

Middepth temperature difference of ’14 °C resulted in a 3% performance (rutting)
error in rut prediction (Figure 6-17). The error for temperature corrected AC modulus

ranges from 10% to 20% based on a i2 °C and :4 °C middepth temperature difference,

respectively. These results imply that middepth temperature predictions within :4 °C

98

deviation does not signiﬁcantly affect the performance prediction. This was further
investigated statistically, using t-tests performed for the two groups of computed ruts
based on measured and predicted middepth temperatures. A temperature deviation of :4
°C and an a risk value of 0.02 (risk of error) were used. It was concluded that means of
two groups are not different with P-value for two-tail tests equal to 0.973. If the p-value
is much larger than the Speciﬁed a risk, it indicates that the means of two groups are not
signiﬁcantly different. This t-test was carried out for corrected AC moduli and structural
responses as well, and it was found that a middepth temperature prediction deviation
within :4 °C does not signiﬁcantly inﬂuence them. t-test for temperature prediction
deviation within i2 °C produced similar results. Therefore, :4 0C of middepth
temperature prediction deviation can be used as a criterion for the accuracy of the model.
Accordingly, bandwidths of i2, i4, and :6 °C are shown in the validation plots of
middepth temperature prediction (Figures 6-6 through 6-14) to illustrate the accuracy of

the model. It can be seen that most data resides within :4 °C bandwidth.

6.6 SUMMARY

A practical and accurate subsurface temperature prediction model for asphalt concrete
pavements was developed. Temperature correction factors for AC modulus were
computed based on measured and predicted mid-depth temperatures, which resulted in
consistent values. The form Of the new temperature prediction model for AC pavements
accounts for the temperature gradients which vary with time of day when FWD proﬁles
are measured. Predicted temperatures at various depths were in good agreement with

measured temperatures, demonstrating an acceptable degree of accuracy for the model,

99

thus promising potential use by state highway agencies. The model’s robustness and
accuracy were validated using data from SPSl and SMP sites, and an investigation of the
inﬂuence of temperature prediction errors on performance prediction was conducted. The
validation results conﬁrmed that the model could be adopted in other climatic and

geographic regions.

 

Performance Prediction Error (%)

 

 

 

 

-8 -8 4 .2 0 2 4 e 8
Difference between Measured and Predicted Middepth Temperature (°C)

Figure 6-17. Performance (rutting) prediction error caused by middepth temperature
prediction deviation

100

CHAPTER 7

RESULTS AND DISCUSSION II — FINITE ELEMENT ANALYSIS

7.1 TWO-DIMENSIONAL FIN ITE ELEMNET (FE) MODEL

The tensile strain at the bottom Of AC layer has traditionally been used as a design
criterion to assess the number of load applications likely to initiate fatigue cracking.
Recently, Briggs and Lukanen (2000) presented the relationship between the number of
applications of a given strain level to achieve failure. They concluded that the estimated
number of applications to failure was very sensitive to changes in strain. Likewise, the
vertical compressive stress/strain on top of AC, base, and subgrade layers has been“ used
to prevent rutting (Figure 7-1).

In an axisymmetric state of stress, only four nonzero stress components, 0}, 09,
oz, and In, exist. Along the axis of symmetry (r = 0), 1,, = 0 and O, = 0'9. The major and
minor principal stresses along the axis of symmetry (r = 0) coincide with either the
vertical stress Oz or the horizontal stress (I, (= 0'9) depending on the vertical position z
(Figure 3-1). Figures 7-2 and 7-3 Show stress plots with time and depth for a case of 18
cm (7”) thick AC, base modulus of 345 MPa (50 psi), morning temperature, and uniform
FWD load. These plots Show typical stress behaviors during FWD loading. Maximum
stresses are found at the surface Of the AC layer during loading. Bending stresses result in
compression in the horizontal direction at the surface of the layer in the vicinity of the

load, and tension at the bottom.

101

Load

 

lHH

 

AC Compressive stress at the top of AC 43:52:32?
I l of bases
Tensile stress at the bottom of AC
Base
Compressive
1H Huntiimmm.
of subbase

 

>

Subbase
Compressive Stress at the top of the roadbed

Figure 7-1. Illustration Of critical stress/strain locations in a typical pavement structure.

 

 

 

  
 

 

 

\\

102

use. Sacha: was cotsnﬁmﬁ oSHSOqES wﬁEoE Sm 839 $23 329$. .mé. 2:me

 

Eouom 65 an $25 .mo_te>1lr Eouom e5 um 39% .3558: .7
oomtam o5 «a macaw .mo_to> Lwi Oomtzw e5 “a macaw .8553... no:

 

 

 

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Nomad Nommd mowed momme Nome; mo-mo.m oo+mo.o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

camp- 0 H A n ” com-
............. ,- gammy—9:00 om?
cum- . , H m
................ _ L ............... - ooT
n.6,. o .............................. 1- _ n 1 ,- ............. - 8- m...
M _ ,_ n n , . ﬂ
9 o m - o s
x m
d - S
on I.
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omm 4.. ma 948$: 24m . 1.311%, :2th ....... 7 09.
mo 9::on Oman 98 U< x02“ ., m
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103

Stress (kPa)

 

 

 

 

 

 

 

 

 

  
 

 

 

 

 

-1380 -920 -460 0 460 920 1380
0 . . r 0
1 r -------------------- ' ---------------------------------------------
2 1 ------------------------------------------------------------------------- 5
‘5 3 1 Compression ------------------------------- TenS10n -------------- ‘5
= 1 e
4 .............................. 10
i 1 z : . 1 1 . a
8 5 ‘ """ Analysiscondition:l8cm(7”) ” """""" E """""" 3
6 ,,,,, thick AC,basemodulusof345 ' - ”j ___________ i ___________ r 15
MPa (345 psi) 1 3 1
7 _ t I I I l
a 4 ; i i L i 20
-200 -150 -100 -50 O 50 100 150 200

Stress (psi)
[,0 At 3.1 millisec +At 6.9 millisec +At 13.1 millisec (Peak Load)l

 

 

(a) horizontal stress

Stress (kPa)

 

 

 

 

 

 

 

 

 

-828 -690 -552 -414 -276 -138 0

0 t L i 1 ‘ O

1 «1 ----------------------------

2 1 ------------------------------ 1 5
E 3 1 -------------------------------------------- E
3 9.
ii 4 1 ----------------- ~10 ii
8 5 ~ ---------------- 3

6 0 ----------------------------------------------------------------------- ”15

7 1 i : : 1

8 3 1 1 1 1 20

-120 -100 -80 -60 -40 -20 0

Stress (psi)

 

F044 At 3.1 millisec +A16.9 millisec +A113.1 millisec (Peak Load)l
(b) vertical stress

Figure 7-3. Typical example of stresses for uniform temperature and uniform FWD load

104

I Strain

8

Horizon

Compressive Stress at the Surface
(kPa)

1

 

 

§§§§§§§§§§

 

 

 

I Horizontal Stress -----
[:1 Vertical Stress

 

 

 

 

 

 

 

 

oooooooooooooooooooooooooooooooooo

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L...
o .1
689 3447 6895 24132
Base Modulus (MPa)
(a) Compressive stress at the surface
Base Modulus (ksi)
0 725 1450 2176 2901 3626
2.50E-04 J A 1 f
2005-04 1‘
1.50504 44 -------------------------------------------------------------------------------
1005-04A\ Tension ----------------------------
5005-03 ------- \ ---------------------------------------- M.
%_L_________.__
0.00900 3 g - " A
-5.00E-05 ~~ , 1 Compression
4,005.04 __ ...............................................................................
4.50504 A ~~~~~~~~~~~~ : ~~~~~~~~ - -------
-O—At the surface oftheAC layer
2005-04 . ......
—A—AtthebottomoftheAC layer
4.50504 7 r i If
0 5000 10000 15000 20000 25000
Base Modulus (MPa)

(b) Horizontal strains

Figure 7-4. Effect of base modulus (Analysis condition: 17.8 cm thick AC, day
temperature, and uniform load)

105

However, the high stiffness of the base has the effect of reducing the compressive
stresses in the horizontal direction at the top of the AC layer, which eventually causes
Slight tensile strains at the surface. On the other hand, tensile strains in the horizontal
direction at the bottom of the AC layer decreases as base stiffness increases (Figure 7-4).

Distress models are sometimes called transfer functions that relate structural
responses to various types of distress such as rutting and fatigue cracking. For fatigue
distress models, traditional transfer functions relate tensile strains at the bottom Of the AC
layer to the allowable number of load repetitions (Huang, 1993). Several agencies
presented models that relate the allowable number of load repetitions, Nf, to tensile strain,
8., at the bottom of the AC layer based on laboratory fatigue tests, for example,

the Asphalt Institute equation for 20% of area cracked is (A1, 1982)

Nf: 0.0796 (So-3.291 (ED-0.854 (7-1)
where E; is the HMA modulus.
According to the concept of design criteria using tensile strain, results from a parametric
study indicate that cracking could be initiated at the surface of the pavement where a high
stiffness of sublayer is underlain (e.g., asphalt overlaying a concrete pavement).

Typical examples of effects of temperature distributions on horizontal stresses and
strains evaluated at the surface and bottom of the AC layer under the load center are
plotted in Figures 7-5 and 7-6. The effects of temperature of the AC layer on stress and
strain distributions are Signiﬁcant. AS temperature increases, stress decreases and strain
increases due to the stress relaxation in the AC layer. However, it is noted that strain
variations of non-uniform and uniform temperature distributions were close especially at

low temperatures, e.g., strains at night and in the morning.

106

 

 

 

 

 

 

 

 

 

 

 

 

 

50 j j T 1 1' 345
,_ 0 ._ 1 o ._
= r
5 5
O ‘ . I
O _50 L__,_,__-, .. : -345 O
5 4’1
’5 -100 ,_ »~ -690 g
N
g a
I :‘z’
-150 ~ -1035
-200 i 1 . 4 5M -1380
0.0E+00 505-03 105-02 1.5E-02 2.05-02 2.5E-02 3.05-02
Tim e (sec)
(a) at the surface of the AC layer under the load center
A 100 1 - 689 A
'g E
" 8° 1 551 £5.
a a
g 60 A ~ 413 g
‘0 re
3 40 . , 276 3
S S
.g 20 1 ,_ 138 ,2
c c
I o 0 I
'20 T 1 f I T h ‘138
0.0E+OO 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02
Time (sec)
+ Night (14~23 deg C) -I— Morning (uniform 20 deg C)
—A— Day (43~32 deg C) eAveragi Day (uniform 38 deg C)

 

 

 

(b) at the bottom of the AC layer under the load center

Figure 7-5. Effects of temperature distributions on horizontal stresses under FWD load

107

 

2.0E-05

 

0.0E+00

-2.0E-05

-4.0E-05 -*

-6.0E-05

-8.0E-05

Horizontal Strain

-1.0E-04

 

-1 .ZE-O4 *

 

 

 

-1.4E-04 r . . .
0.0E+00 5.0E-03 1.0E-02 1.5E-02 2.0E-02 2.5E-02 3.0E-02

Time (see)

(a) at the surface of the AC layer under the load center

 

1.8E-04 M 7 r ; M
1.6E-04 1 1 1 1 1 1
145-04
1.2E-04
1.05-04
8.0E-05
6.0E-05
4015-05 «
2.05-05 ~«

0.0E+OO « ‘ - w .
0.oe+00 505-03 105-02 1.55-02 205-02 255-02 305-02

Time (sec)

Horizontal Strain

    

A

 

 

   

V

 

+ Night (14~23 deg C) +Morning (uniform 20 deg C)
—A— Day (43~32 deLC) o AverageDay (uniform 38 de C)

 

 

(b) at the bottom of the AC layer under the load center

Figure 7-6. Effects Of temperature distributions on horizontal strains under FWD load

108

The inﬂuence of the AC thickness on the structural response was also
investigated. As the AC thickness increased, horizontal stresses evaluated at the peak
loading time decreased until they reached an asymptotic value. However, vertical stresses
did not Signiﬁcantly change (Figure 7-7). Effects of different tire pressure distributions
excluding tire tread loads were also investigated. It was found that these had a negligible
effect on trends of the stress and strain distributions in the pavement. Effects of tire tread
pressure will be discussed in the subsequent sections.

Detailed tables of stress and strain evaluated at the peak loading time are

 

presented in Appendix A.
AC Thickness (in)
0 2 4 6 8 10 12 14 16
20m :1? '1 L ' 1 1 1 1 1
‘ Anal is condition: base modulus .
1500 » ys 238

   
  
 

""" of 345 MPa (50 ksi), morning

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 1i, temperahueandmiformload . 138

- 500 - _ ..... a ........ ; : ........ ' ......... ; ......... ,._.
r 1 a a = ~38 8.
i O 1 1 I v
o M u
g -5001 a; . .1. ~ ------- . ---------- ~ -63 3
:7. ' : 2 .‘5
m 4000 1 Compression ' ; -163 rn

-1500 1 ' ,,,,,,,,

-2000 4 1 , ‘ . . | ---~« ”263

-2500 1 1 1 1 1 1 M1 1* -363

0 5 10 15 20 25 30 35 40 45

AC Thickness (cm)

-0- Horizontal stress at the surface +Vertical stress at the surface
-o— Horizontal stress at the bottom +Vertical stress at the bottom

 

 

 

 

Figure 7-7. Effect of AC thickness on stress response

109

7.2 THREE-DIMENSIONAL FE MODEL

Although the inﬂuence of tire contact pressure on the initiation and propagation of
pavement distress is signiﬁcant, very sparse amount of actual data have been reported in
the literature (Myers et al., 1998). Hence, effects of more realistic distribution of tire
contact pressures are investigated, using tensile strains and dissipated energy.

In order to investigate the effects of temperature distribution and load type, three
temperature distribution cases (Morning, Daytime, and Nighttime) were Simulated, and
three types of load cases were applied:

(a) Load case I: uniform vertical load over the entire load area (no lateral stress),

(b) Load case 11: uniform vertical load only under tire treads (no lateral stress), and

(c) Load case HI: measured vertical and lateral stresses under tire treads (as measured by
Myers et al., 1998).

Dissipated energy is deﬁned as the area within a stress-strain hysteresis loop
under cyclic loading and represents the energy lost at a ﬁnite location in the AC layer.
Figure 7-9 Shows typical stress-strain hysteresis loops during a single loading cycle at
two different positions (#1 and #2 in Figure 7-8). The dissipated energy per unit volume
in the tensile direction was calculated ﬁ'om horizontal tensile stress and strain hysteresis

loops (Chatti and Kim, 1996):

 

 

_ end—1 _ 7-2
VV’ = (Gem! arm') X (gent! + 8m: + Z (CHI 01) X (8114 + 8i) ( )
2 2 ,=,.,,, 2 2
where:
Gin), sin, = initial stress and strain amplitudes
om, send = ending stress and strain amplitudes
oi, 81 = stress and strain amplitude at a particular time during the cycle

110

Tread 1 T Tread 3 Tread 4

Tread 5

    
 

Traffic
Middle Row Direction

Top View of Loading Area

Figure 7-8. Positions at which pavement responses are evaluated

111

-1379

-1034

-690

Stress (kPa)

-345 0

345 690 1034 1379

 

2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00

Strain

-5.00E-05 ~

-1.00E-O4
-1.50E-04
-2.00E-04

AP7<»

4

 

 

1

.1

 

.....................

 

4 ........................................... -3
.
1.

   
 

1 ------------- Compression: —

 

 

Tension: +

 

 

 

 

 

-200

-150

-100

-50 O

50 100 150 200

Stress (psi)

 

 

~o— Load case I - Top

0 Load case i - Bottom -8— Load case it - Top
~I— Load case ii - Bottom + Load case Ill - Top —A— Load case Ill - Bottom

 

 

-1379
2.00E-04

1.50E-04 1
1.00E-04

5.00E-0
0.00E+0
-5.00E-0

Strain

-1.00E-O
-1.50E-0
-2.00E-0

-1034

-690

(a) at #1

Stress (kPa)

-345

0

345 690 1034 1379

 

5 s

 

0
5
4 1
4

 

 

  

 

—--n--~-—-----~.------

 

 

Compression: —
Tension: +

 

 

 

 

 

4
-200

-150

-100

-50

0

50 100 150 200

Stress (psi)

 

 

+ Lead case I - Top

6— Load case I - Bottom B- Load case II - Top
+ Load case ll - Bottom + Load case Ill - Top 4+ Load case Ill - Bottom

 

 

(b) at #2

Figure 7-9. Transverse hysteresis loop for day temperature distribution

112

Many researchers have applied energy concepts to pavement analysis including
the prediction of pavement fatigue life and permanent deformation (Chatti and Kim,
1996; SHRP A-404, 1994; Van Din and Visser, 1997; Rowe et al., 1995 and 1997).
Fatigue of viscoelastic materials subjected to repeated dynamic loading can be associated
with the energy loss such that the fatigue life can be related to the total energy dissipated
during the period before failure.

SHRP (1994) established a relationship between dissipated energy and fatigue

life, Nf:
Nf = 2. 3 65 expoowvra (%)-1.332 (7_3)
where:
VFB = percentage of voids ﬁlled with asphalt
w = initial dissipated energy per cycle, psi

with R2 = 0.76

The fatigue life, Nf, decreases with increasing initial dissipated energy per cycle.

For load cases I and II, tensile stresses and strains in the horizontal direction at the
bottom of the AC layer and compressive stresses and strains at the surface were always
present. It should be noted that tensile strain at the surface of the AC layer occurred for
the load case III (measured vertical and lateral stresses under tire treads) and day
temperature distribution (Figure 7-9). Similarly, Myers et al. (1998) reported transverse
tensile stresses induced by radial truck tires at the tire-pavement interface ﬁ'om elastic
analysis. The authors concluded that these tensile stresses appeared to provide the most
viable mechanism of surface-initiated longitudinal wheel path cracks, perhaps combined

with thermal stresses during cooling.

113

Results from 3-D analyses for load cases I and II were consistent with those from
2-D analyses. The peak value of dissipated energy generally occurred at the bottom of the
AC layer under the uniform load. At high temperature, the dissipated energy increased
especially at the bottom of the AC layer (Figure 7-10). According to the concept of the
viscoelastic theory/dissipated energy, as the amount of dissipated energy increases, the
fatigue life is expected to decrease. Similar analyses were reported by Rowe and Brown
(1997); as temperature was increased, fatigue life decreased. Cracking could be initiated
at the bottom under uniform loading since the peak value of dissipated energy occurs at
the bottom. For high temperature and tire tread loads (load case II and HI), dissipated
energy at the surface became higher than at the bottom at certain positions (Figures 7-11
and 7-12). These results from load cases H and III indicate that these surface conditions
can become critical so that cracking could be initiated at the surface as explained in the
earlier section. Detailed plots of dissipated energy are presented in Appendix A.

Figures 7-13 and 7-14 show horizontal stress distributions under the load area for
the load case I. The stress distribution follows typical bending stress distribution with
compression at the surface and tension at the bottom. However, for load case II (uniform
vertical load only under tire treads), peak compressions at the surface were localized
under tire treads, and tension at the bottom was similar to that for load case I. The
inﬂuence of the tire tread load on stress distribution is signiﬁcant at the surface of the AC
layer and reduced with depth. (Figures 7-15 and 7-16). No unusual peak stresses were

observed due to the numerical formulation.

114

Dissipated Energy Density (psi)

Dissipated Energy Density (psi)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.50E-03 2.40E-02
3.005-03 ——————————————— _ 2.00502
[mm the surface IN the bottom
2.505-03 « -------------- - ————————————————————————————————————————————
- 1.60E-02
2.005-03 - -----------------------------------------------------------
~ 1.20E-02
1.50E-03 ~ ------------------------------------------------------------
~ 8.00E-03
1.005-03 ~ ------------------------------------------------------
5.00E-04 e ________________ 4.00E-03
I“
0.00E+00 “" ‘ *4 - 0.00E+00
Night Morning Day
(a) at #1
3.505-03 2.40E-02
30%” -, " — 2.005-02
2.505-03 J -------------------------------------------------------------
[EM the surface IN the bottorﬂ ’ 1-605'02
2.005-03 . -------------------------------------------------------------
— 1.20E-02
1.505-03 ---------------------------------------------------------
_ 8.00E-03
1.00E-03 -_
5.005-04 —~ -- " 4005-03
0.00E+00 _ _ ‘ > 0.00E+00
Night Morning Day
(b) at #2

Figure 7-10. Calculated dissipated energy for the load case I

115

Dissipated Energy Density (kPa)

Dissipated Energy Density (kPa)

Dissipated Energy Density (psi)

Dissipated Energy Density (psi)

3.50E-03

3.00E-03 ,

2.50E-O3 4

2.005-03 ~

1.50E-03 -‘

1.00E-03 -

5.00E-04 ~

0.00E+OO

 

 

 

.....................................................

...................................

.....................................................

 

 

 

 

 

 

 

 

 

2.4OE-02

2.005-02

'- 1 .60E-02

~ 1.20E-02

. 8.00E-03

~ 4.00E-03

 

— 0.00E+00

 

3.50E-03

3.00E-03 .-...

2.50E-03 4

2.00E-03 ~

1.50E-03 4

1.00E-03 --

5.00E-04 ~

.............

 

 

.77V7-_-_-_-------__--.4

.............................................................

......................................

 

 

 

 

 

 

2.4OE-02
— 2.00E-02
«L 1.60E-02
— 1.20E-02
~ 8.00E-03

1 4005-03

 

— 0.00E+00

Figure 7-11. Calculated dissipated energy for the load case 11

Dissipated Energy Density (kPa)

Dissipated Energy Density (kPa)

Dissipated Energy Density (psi)

Dissipated Energy Density (psi)

3.50E-03

3.00E-03 '

2.50E—03

2.00E-03

1.50E-03

1.00E-03

5.00E-04

0.00E+00 -

 

 

 

.....................................................

.....................................................

" z 1
1 1‘ -
‘e

Morning

 

 

 

 

(a) at #1

 

Day

- 2.4OE-02

4 2005.02

r 1.60E-02

- 1.20E-02

l 8.00E-03

-_ ~ 4.00E-03

 

 

- 0.00E+00

 

3.505-03

3.00E-03

2.50E-03 .

2.005-03

1.50E-03

1.00E-03 .

5.00E-04

l ........... . .........
[DAt the surface IN the bottom I

 

 

1

 

......................................................

 

0.00E+OO '

 

 

 

2.4OE-02

- 2.00E-02

r- 1.60E-02

. 1.20E-02

~ 8.00E-03

__~ 4.005-03

 

 

 

   

Morning

(b) at #2

Day

~ 0.00E+00

Figure 7-12. Calculated dissipated energy for the load case 111

117

Dissipated Energy Density (kPa)

Dissipated Energy Density (kPa)

Sll

Value(psi)
-151
-129
' -107
-85.7
-64
-42.3
-20.6
L12
22.3
44.6
66.3
88.0
1 10
191

Loading Area

      

Figure 7-13. Typical contour lines of the horizontal stress under the load for the load case
I and day temperature distribution

118

\O/
811 Trafﬁc DiI’CCtiOIl Middle Row under the Load
Value(psi)
-l 51 _— ..

-129 11
-107 ‘
-85.7

-64.1

-42.4

-20.7

0.959

2.26

449 AC Layer
66.0

87.7

109

191

 

 

      

 

Figure 7-14. Typical contour lines of the horizontal stress under the middle row for the
load case I and day temperature distribution

119

Value(psi)
.-129
y .-111
‘ .-99.9
--75.5
--57.8
--40.0
--22.9 .e
--4.55 Trafﬁc Direction
-l9.2
'90.9

'48.7 I
'66.4
. ‘84.2

“102

Loading Area

 
 
  

   

AC Layer

Figure 7-15. Typical contour lines of the horizontal stress under the load for the load case
11 and day temperature distribution

120

0/
Traffic Direction Middle Row under the Load l
S] 1 .
Value(psr) _

1129 1
'-111
; --99.2
- 275.5
'-57.8
, -40.1
4 --22.4
. --4.67

' 19'0 AC Layer

 

 

      

 

Figure 7-16. Typical contour lines of the horizontal stress under the middle row for the
load case 11 and day temperature distribution

121

7 .3 SUMMARY

Parametric study (AC thickness, base stiffness, loading condition, and
temperature distribution across the AC layer) was conducted with a 2-D axisymmetric FE
modeL Bending stresses resulted in compression in the transverse direction at the surface
of the AC layer in the vicinity of the load, and tension at the bottom. However, the base
stiffness had the effect of decreasing the compressive stresses in the transverse direction
at the surface and eventually induced slight tensile strains. As the AC thickness
increased, horizontal stresses evaluated at the peak loading time decreased until they
reached an asymptotic value. However, vertical stresses did not signiﬁcantly change. The
effects of temperature of the AC layer on stress and strain distributions are signiﬁcant. As
temperature increased, stresses decreased and strains increased due to the stress
relaxation in the AC layer.

Effects of three temperature distributions (night, morning, and day) and three
loading types (load case I - uniform vertical load over the entire load area, load case 11 -
uniform vertical load only under tire treads, and load case 111 - measured vertical and
lateral stresses under tire treads) on the structural response were investigated with the 3-D
FE model. For most cases of loading, compressive stresses and strains in the transverse
direction at the surface of the AC layer and tensile stresses and strains at the bottom were
always present. However, for the load case III, tensile strain at the surface of the AC
layer occurred at high temperature. For the load case 1, higher dissipated energy was
magnitudes resulted at the bottom of the AC layer than at the surface. As temperature

increased, the dissipated energy increased especially at the bottom of the AC layer.

122

However, for load cases 11 and III, as temperature increased, dissipated energy also
increased at both the surface and the bottom of the AC layer, and dissipated energy
became greater at the surface than at the bottom in some locations. This could explain the

occurrence of top-down cracking in AC pavements under certain conditions.

123

CHAPTER 8

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

8.1 SUMMARY AND CONCLUSIONS

The objectives of this research were to:

0 Develop a practical, but fundamentally sound model to predict the effective
temperature in the AC layer and temperature correction factor for backcalculated AC
moduli.

0 Investigate temperature effects on structural analysis and performance evaluation.

0 Simulate selected AC pavements based on the viscoelasticity/elasticity composite
layer system.

0 Better understand stress-strain behaviors of ﬂexible pavements subjected to FWD
load pulse and various temperature distributions across the AC layer.

A comprehensive literature review regarding ﬂexible pavement temperature prediction

model and correction factors, structural analysis of ﬂexible pavements, applications of

temperature correction procedure and structural analysis to the ﬁeld was presented.

Axisymmetric ﬁnite element model and viscoelasticity were also overviewed.

Based on the ﬁeld data collected from in—service ﬂexible pavements in Michigan,
a practical and accurate subsurface temperature prediction model for asphalt concrete
pavements was developed. Temperature correction factors for AC modulus were
computed based on measured and predicted mid-depth temperatures, which resulted in
consistent values. Temperature data points (317) and deﬂection proﬁles (656) were

collected from the six in-service test sites in Michigan. Temperature data points (197)

124

from three of the test sites were used to develop the temperature prediction modeL and
data ﬁ'om the remaining sites were used for validation. The developed temperature
prediction model has a R2 greater than 90% and an F-statistic signiﬁcantly greater than
1.0. The form of the new temperature prediction model for AC pavements accounts for
the temperature gradients which vary with time of day when FWD proﬁles are measured.
Predicted temperatures at various depths were in good agreement with measured
temperatures, demonstrating an acceptable degree of accuracy for the model, thus
promising potential use by state highway agencies. For further validation of the
temperature prediction model, temperature data points (18444) ﬁ'om seven Seasonal
Monitoring Program (SMP) sites (Colorado, Connecticut, Georgia, Nebraska, Minnesota,
South Dakota, Texas) were obtained from the DATAPAVE 2.0 (LTPP Database). The
validation results suggested that the model could be adopted to all seasons and other
climatic and geographic regions. The major improvements over existing models are (a)
the model does not require temperatures for the previous 5 days (b) it takes into account
temperature gradients due to diurnal heating and cooling cycles (c) it needs fewer
parameters than other published models and (d) the effect of temperature prediction error
on the performance prediction were also investigated.

Middepth temperature difference of :4 °C resulted in a 3% performance (rutting)
error in rut prediction. The error for temperature corrected AC modulus ranges from 10%
to 20% based on a :2 °C and i4 °C middepth temperature difference, respectively. These
results imply that middepth temperature predictions within i4 oC deviation does not
signiﬁcantly affect the performance prediction. This was further investigated statistically,

using t-tests performed for the two groups of computed ruts based on measured and

125

predicted middepth temperatures, and it was concluded that the means of two groups are
equal. Accordingly, bandwidths of i2, i4, and 21:6 °C were applied to the validation plots
of middepth temperature prediction, and most data resided within :4 °C bandwidth.

The developed temperature prediction and correction procedure have been

incorporated into revised versions of the programs MICHPAVE and MICHPACK.

Parametric study (AC thickness, base stiffness, loading condition, and
temperature distribution across the AC layer) was conducted with a 2-D axisymmetric FE
mode]. Bending stresses resulted in compression in the transverse direction at the surface
of the AC layer in the vicinity of the load, and tension at the bottom. However, the base
stiffness had the effect of decreasing the compressive stresses in the transverse direction
at the surface and eventually induced slight tensile strains. As the AC thickness
increased, horizontal stresses evaluated at the peak loading time decreased until they
reached an asymptotic value. However, vertical stresses did not signiﬁcantly change. The
effects of temperature of the AC layer on stress and strain distributions are signiﬁcant. As
temperature increased, stresses decreased and strains increased due to the stress
relaxation in the AC layer.

Effects of three temperature distributions (night, morning, and day) and three
loading types (load case I - uniform vertical load over the entire load area, load case 11 -
uniform vertical load only under tire treads, and load case 111 - measured vertical and
lateral stresses under tire treads) on the structural response were investigated with the 3-D
FE model. The evaluations ﬁ'om 2-D and 3-D analyses were consistent. For most cases of

loading, compressive stresses and strains in the transverse direction at the surface of the

126

AC layer and tensile stresses and strains at the bottom were always present. However, for
the load case III, tensile strain at the surface of the AC layer occurred at high
temperature. For the load case 1, higher dissipated energy was magnitudes resulted at the
bottom of the AC layer than at the surface. As temperature increased, the dissipated
energy increased especially at the bottom of the AC layer. However, for load cases 11 and
III, as temperature increased, dissipated energy also increased at both the surface and the
bottom of the AC layer, and dissipated energy became greater at the surface than at the
bottom in some locations. This results could explain the occurrence of top-down cracking
in AC pavements under certain conditions, and contribute to the development of an
improved performance model and/or asphalt pavement design program based on

advanced material characterization and dynamic FWD.

8.2 RECOMMEDATIONS FOR FUTURE RESEARCH

o In order to achieve higher suitability for various site and environmental conditions,
the calibration work for the pavement temperature prediction model and correction
factors for AC modulus should be continued with updated version of the LTPP
database.

0 The inﬂuence of temperature prediction error on the shape of the deﬂection basin
should be studied.

0 The effect of repeated cyclic loads on the structural amlysis needs to be investigated
based on viscoelasticity, realistic temperature distribution, and dynamic loads.

0 An attempt needs to be made to implement the ﬂexible pavement analysis for

pavement designs and performance models.

127

o Discontinued ﬂexible pavement (e.g., pavement with cracks) should be simulated to

study the inﬂuence of viscoelasticity, temperature distribution, and dynamic loads.

128

APPENDICES

129

APPENDIX A

ADDITIONAL FIGURES AND TABLES IN CHAPTER 7

Additional tables and ﬁgures for the chapter 7 are here presented. Table A-1
shows stresses and strains evaluated at the peak loading from the parametric study.
Figures A-l through A-9 present calculated dissipated energy at various locations for

different types of loading and temperature distributions form the 3-D analysis.

130

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63:82. :23». 0.8 vague: mm 5.89.950 .BoZ

 

 

 

 

  

 

 

 

 

      
           

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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134

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Dissipated Energy Density (psi)

Dissipated Energy Density (psi)

Figure A-l.

3.50E-03

3.00E-03

2.50E-03

2.005-03

1.50E-03 -

1.00E-03

5.005-04

0.00E+00 <

3.50E-03

3.00E-03

2.50E-03

2.005-03

1.50E-03

1.005-03

5.005-04

0.00E+00

Transverse Distance, X (cm)

 

 

 

 

 

 

 

 

 

0 5 10 15 20
Uniform Load (No Tire Tread) — ....... .» 1005-02
— 1.50E-02
: : : *"*:* :’* ‘ :" ;“‘ “‘+1.00E-02
4i ........ J
. 5.00E-03
~ 0.00E+OO
0 1 2 3 4 5 6 7 8 9
Transverse Distance, X (in)
|+Surface -e—Middle -e-Bottom I
(a) at the ﬁ'ont row
Transverse Distance, X (cm)
0 5 10 15 20
+L """" é" ‘ ' " uniform Load (NO THC Tread) """ E """" t 2.00E’02
i #2 ----- é ------
r 1.50E-02
..L, I ...............
4* 1.00E-02

 

 

*- 5.00E-03

 

0 1 2 3 4 5 6

Transverse Distance, X (in)

7 8

 

[+Surrace -B—Middle +eottom]

 

(b) at the middle row

43W 0.00E+00
9

Dissipated Energy Density (kPa)

Dissipated Energy Density (kPa)

Calculated dissipated energy for the load case I and night temperature

distribution

135

Dissipated Enemy MW (9")

Transverse Distance, x (cm)

 

 

 

 

 

 

 

 

 

 

o 5 1o 15 20
3'50E-03 ﬂ . -....-... ---- L..- ...- ..---1 ‘f T L
3.00E-03 Uniform Load (NoTireTrwd) - __ 100502 is
2.505-03 A ' a E ' ' 53‘
4.50502 8
2.00E-03 E
1.505-03 ----- woos-02 8
III
1.005-03 4 - g
5.005-03 _
5.005-04 « a
0.005+oo ~ o.ooe+oo
o 1 2 3 4 5 a 7 8 9
Transverse Distance, X (In)
[+Surface -a—Middle +Bottom]
(c)attherearrow
Figure A-l. Calculated dissipated energy for the load case I and night temperature
distribution (continued)
Transverse Distance, X (cm)
0 5 1o 15 20
3.505-03 ar——~.——L 2. f g
8 3.00E-03 3 Uniform Load (No Tire Tread) I r 2.00E-02 %
€ 2,505-03 ; e g - g . g g 3 5 f
C . . . . . .
8 : ‘ : : ' . : : 1.50E-02 3
2005-03 : I E : I I F ,,
g s ; 3 2 s 2 : a
N 150503 V 5 I : 5 5 5 : 1 1‘005'02 lg
3 1.005-03 g 1 « 3
g : : : : j g : »5.ooe-03 g
5.005-04 ' ' ' 'D
a . s a : a . ; s 3
0,005+oo W— (magma

 

 

3
Transverse Distance, X (in)

4 5 6

 

b—Surface -B—Middle +Bottom]

 

(a) at the front row

Figure A-2. Calculated dissipated energy for the load case I and morning temperature

distribution

136

Transverse Distance, X (cm)
0 5 10 15 20

 

 

 

 

 

 

 

 

 

 

3.505-03 ‘r—'—-T— 7
A . “““““ ““‘“ ““‘ : ‘3
"' : . o ; l
g 3.00E-03 1 Uniform Load(NoT1reTread) ~~ 3 777777 100502 :1
E , . . . . .
g 2.50E-03 g
E : E E E E 1.505-02
,. 2.005-03 ; - : , ; ;~
g a x s s s 2 w E
E, 1-505-03 ‘ " f ****** ““““ ~1.00E-02 5
g 1.00E-03 A B
.. » 5.005-03 g
3 5.005-04 3
a a
0.00E+00 * ‘ a '- 0.00E+00
0 1 2 3 4 5 6 7 8 9
Transverse Distance, X (In)
[+Surface -a—Middle +Bottom]
(b) at the middle row
Transverse Distance. X (cm)
0 5 10 15 20
3.50E-03 '—-—~—. '- - _- -- - ..r ‘ ‘
i 3,005-03 (t . . Uniform Load (No Tire Tread) ...... _ 200E-02 é:
2.50E-03
a ' l ‘ l I ' 2
a : : : : : : : ~1.50E-02 8
>. 2-005'03 3 ’3 ” 5‘ 3’ ’ 5 * E" "
g : : : : : : 1 . E
m 1.50E-03 — i I - 5 ~ 5 g ‘ w1.ooe-02 5
g. 1.00E-03 * 3
.. 5.00E-03 ,3-
5 5.00504 3
D . , , , , , . , o
0.00E+00 (W 0.00E+oo

 

 

0 1 2 3 4 5 6 7 8 9
Transverse Distance, X (In)

[+Surface -8- Middle +Bottom]

 

 

(c) at the rear row

Figure A-2. Calculated dissipated energy for the load case I and morning temperature
distribution (continued)

137

Transverse Distance, X (cm)

 

 

 

 

 

 

 

 

 

 

0 5 1o 15 20
3.50E-03 : 1 a . e A. v 5
Uniform Load (NOETIIO Tread) A :
g 3.005-031» ------------ :- ﬂ r I T F 2.00502
‘3 2.50E-03
8 ~ 1.505-02
E 2.008-03 A
5 1-505'03 % 1.00E-02
g 1.005-03
.. » 5.00E-03
g 5.00E-04
0.00E+00 T 7 i v i ’- 0.00E+00
o 1 2 3 4 5 6 7 8 9
Transverse Distance, X (in)
[+Sur1ace -a—Middle +eottom]
(a) at the ﬁont row
Transverse Distance, X (cm)
0 5 1O 15 20
3.50E-03 a . L. , n, f r
Uiniform (.oad (Mire tread) j 1
g 3.00E'03 4‘ """"""""""""""" . """"""""""""""""""""" _ 2.00E'02
5 2505-03 l --------------- 1 ------ J; ------- L ------------- J ---------------
1.50E-02
é, 2.001503 1
E' .
5 1.50m « ------- : ------ i ------- f ------ 1 ------- f ------- i ----- ------- a woe-oz
g 1.00E—03
.. ~ 5.008-03
g 5005-04 4 4-
0.00E+00 : : , 0.00E+00
0 1 2 3 4 5 6 7 8 9

 

 

 

 

Transverse Distance, X (in)

 

her-Surface -B-Middle +Bottom)
(b) at the middle row

 

Dissipated Energy Density (kPa)

Dissipated Energy Density (kPa)

Figure A-3. Calculated dissipated energy for the load case I and day temperature

distribution

138

Transverse Distance, X (cm)

 

 

Dissipated Energy Density (psi)

 

 

 

 

 

 

0 5 10 15 20
3.50E-O3 : _ v v _ v ‘v ‘
- Limform Load (Np Tire Tread) i a,
3-005'03 """" 3 """" 3 """ ‘1 ' 3 """"""" '- 2,005-02 2
2505-03 E 1f ~ g
‘ ' ' ' ' - 1505-02 3
2005-03 ,.
5'
1505-03 l 1.00E-02 5
1005-03 3
5005-03 §
5.00E-04 z
a
0.00E+00 '- T T , T — 0.00E+00
0 1 2 3 4 5 6 7 8 9
Transverse Distance, X (in)
[+Surface -a—Middle +Bottom]
(c) at the rear row
Figure A-3. Calculated dissipated energy for the load case I and day temperature

distribution (continued)

Transverse Distance, X (cm)

 

Figure A-4. Calculated dissipated energy for the load case H and night temperature

 

distribution

3 4 5
Transverse Distance, X (In)

6

 

 

 

Liv-Surface -B-Middle +30%,“ 1‘;

 

(a) at the front row

139

0 5 10 15 20

. 3505'” ‘Tread a freed 2 g Tread 3 g Twain Tre'ad 5 ‘
n. ' ' ' : ' '
V 3.005-03 « ------------------ , ,
g . ; , ' ' . . . 2.005 02
5 2505-03 : y r 1 ~-: 1
o : ‘ : : : : ' ‘ r1.50E-02
>~ 2.00E-03 1 : ,, : ~ : : :7 - ,1
P : : : : :
2
m 1.50E-03 1.005-02
1’ : ; : : : :
g 1.00E-031rrw :7 :--~-—-: -----------
e i i ' ' ' ' L 5005-03
3 5005-04
a

0.00E+00 0.00E+00

Dissipated Energy Density (kPa)

Dissipated Energy Density (psi)

Dissipated Energy Density (psi)

3.505-03

3.00E-03

2.50E-03 A

2.005-03

1.50E-03

1.00E-03 1*

5.005-04 1

0.00E+00

3.50E-030

3.00E-03

2.50E-03

2.00E-03

1.50E-03

1.00E-03

5.00E-04

0.00E+OO

i——_...

Transverse Distance, X (cm)

 

10

 

15

 

 

 

 

 

O 1 3

 

4

 

5

 

Tread] Tread2 Tread3 Tread4 Treads ------
l
1’ -. __________________
0 1 2 3 4 5 6 7 8
Transverse Distance, X (in)
[+Surface -9-Middle +Bottom]
(b) at the middle row
5Transverse Distance, X (cm)
10 15 20
Tread] TreadZ Tread3 Tread4 TreadS
Cl

 

6

Transverse Distance, X (in)

 

[+Surface -B—Midd|e +Bottom]

 

(c) at the rear row

 

- 2.00E-02

~ 1.50E-02

~ 1.00E-02

5.00E-03

U ‘ 2.00E-02

~ 1.50E-02

~ 1.00E-02

~ 5.00E-03

5 0.00E+OO

Dissipated Energy Density (kPa)

- 0.00E+OO

Dissipated Energy Density (kPa)

Figure A-4. Calculated dissipated energy for the load case H and night temperature

distribution (continued)

140

Dissipated Energy Density (psi)

Dissipated Energy Density (psi)

3.505-03

3.00E-03

2.50E-03

2.00E-03

1.50E-O3 ‘*

1.00E-03 *

5.00E-04

0.00E+OO

3.50E-03

3.00E-03

2.50E-03 '

2.00E-03

1.50E-03

1.00E-03 ~

5.00 E-04

0.00E+OO '*

Transverse Distance, X (cm)

 

 

 

 

 

 

 

 

5
Transverse Distance, X (in)

6

 

[+Surface -a—Middle +Bottom I

(b) at the middle row

 

 

 

9

Dissipated Energy Density (kPa)

Dissipated Energy DesIty (kPa)

0 5 1O 15 20
Tread] Tread2 Tread3 Tread4 Treads , 200502
E 1 150502
E »- 1.005-02
' «» 5.005-03
'- 0.00E+00
0 1 2 3 4 5 6 7 8 9
Transverse Distance, X (in)
[+Surface ~B—Middle -e-Bottoml
(a) at the front row
Transverse Distance, X (cm)
0 5 10 15 20
Tread] Tread2 Tread3 Tread4 TreadS ------- b 2 00E _02
' » 1505-02
1 ''''' - 1.005412
~ 5.00E-O3
- 0.00E+00

Figure A~5. Calculated dissipated energy for the load case H and morning temperature
distribution

14]

Transverse Distance, X (cm)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 5 10 15 20
3.50E-03 t
c 1 —: . . — ' 1 E
.8: 3-005-03 Tread] Tread2 Tread3 Tread4 Tread5 " " ‘ 1005-02 5
g. 2 2 . 1 ; ' 5‘
: .505-03 )1 ~ g
8 ' , 1505-02 3
,, 2.00E-03
? a : 2 s s 2 s ; E
'5 1.50E-03 1» - ' ' 2 """"" 2 1.00E-02 If.
3 1005-03 3 g g 3. 3
g ' 1 ' ' ' ' ‘ i ~ 5005-03 5
3 5.00E-04 — 3
O o
0.00E+OO - - 0.00E+00
0 1 2 3 4 5 6 7 8 9
Transverse Distance, X (In)
[+Surface -a-Middle -e—BottomJ
(c) at the rear row
Figure A-S. Calculated dissipated energy for the load case 11 and morning temperature
distribution (continued)
Transverse Distance, X (cm)
0 5 10 15 20
3505-03 1 1——~
é 3-00E'03 Tread] Tread2 Tread3 Tread4 TreadS 2_ooe-02 g
3 : - - ‘ ' , ;
g 2.50E-03 “4 g
8 1505-02 3
>. 2.00E-03 1
g E
5 1505-03 « »- 1005-02 5
3 1.005-03 3
.5 — 5.005-03 ﬁt
3 5005-04 3
o 6
0.00E+00 — — 0.005+00
0 1 2 3 4 5 6 7 8 9
Transverse Distance, X (In)
[+Surface -a—Middle +Bottom]
(a) at the front row
Figure A-6. Calculated dissipated energy for the load case 11 and day temperature

distribution

142

Dissipated Energy Density (psi)

Dissipated Energy Density (psi)

3.50E-03

3.00 E-03

2.50E-03

2.00E-03 1»

1.50E-03

1.00E-03 -

5.005-04

0.00E+00 *

3.50E-03

3.00E-03

2.505-03

2.005-03

1.50E-03 «

1.00E-03 a»

5.00E-04 a-

0.00E+00

Figure A-6

 

Transverse Distance, X (cm)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O 5 10 15 20
‘ Tread] Tread2 Tread3 Tread4 TreadS ----- , 2.005-02 g
a, _ * ‘ ; ,,,,,,,,,,,,,,,,,,, 5
e
C
- 1.50E-02 8
. ’ >.
: E
: a
1 1.00E-02 m
i 5.005-03 g
. .3
: n
1 1 1 r % 0.00E+00
O 1 2 3 4 5 6 7 8 9
Transverse Distance, X (in)
[+Surface -a—Middle +Bottom]
(b) at the middle row
Transverse Distance, X (cm)
0 5 10 15 20
Tread] ”l‘readZ. Tread3 ITreadl4 Tree . ----- -_ 2.00502 5
' ; ' ' ++++++++ ' 5
...... .1; ______ ................................... ”5°54” 0
______________ E
. ; -1.005-02 m
‘ = ‘ 55.005-03 :-
§ 0
c_:_:_:_:,_:_:_:_:_:_:_:_:_:_:_:_: a ; 000E+00
0 1 2 3 4 5 6 7 8 9

Transverse Distance, X (in)

 

Pal—Surface —a—Middle +Bottom]

 

(c) at the rear row

. Calculated dissipated energy for the load case 11 and day temperature
distribution (continued)

143

Transverse Distance, X (cm)
10 15 20

0 5
3.50E-03 l: . . . —-

 

 

 

 

 

 

 

 

 

 

i 3,005-03 Tread] Tread2 Tread3 Tread4 TreadS -- p 2 00502 g
g awe-ca » ~ 5 _ é --------- s- 5 —-:—~ 5 ------- 5
3 1505-02 3
>. 2005-03 . ,~
2' . . . - - . - 1 r
0‘3 1.505-03 w E *3 § *3 , 1005-02 5
3: 1005-03 § § 3:
a ' ' , 5.00E-03 —
.. 5.005-04 - 3
O D
0.00E+00 a . 0.00E+OO
9
Transverse Distance, X (in)
[+Surface -a-Middle +Bottom I
(a) at the front row
Transverse Distance, X (cm)
0 5 10 15 20
3.50E-03 A
A — —g . g —_ ‘ ' ‘a
- a.
g 3.00E-03 Tread] Tread2 Tread3 Tread4 TreadS 2.00502 5
3' : : : : : : : :
g 2-506-03 «l- s- + ’:" » a ------ -- 5
8 ' ' ' ' ' ' ' ' 1505-02 3-
,, 2005-03 41 « ,,
§' : : : : : : : : E“
15 1505-03 § , g - g ~ g ~ ;~ g ~ 5 ,- 51005-02 :5
'0 : 3 : : 5 : :
E g ' ' ' ' ' ' — 5005-03 3,
8 5005-041 w ; 3
a ' 1 a
0.00E+00 a 1* 0.00E+00

 

 

 

 

0 1 2 3 4 5 6 ' 7 8 9
Transverse Distance, X (in)

 

F—t— Surface -9- Middle -s- Bottomj

 

(b) at the middle row

Figure A-7. Calculated dissipated energy for the load case HI and night temperature
distribution

144

Transverse Distance, X (cm)
10 15 20

0 5
3.50E-03 ‘ .-' ’ . ... ~7——

 

 

: . « i?
I)
9, 3005-03 Tread] Tread2 Tread3 Treed4 Treads 2.00502 3
E 1 I I I i ' ' E
g 2.50E-03 - : : - : J.~ ‘ t n
: : : : : 5
3 1505-02 3
,. 2.005-03 «r >.
g i I i I I I . i g
:5 ‘-5°E'°3' : a i z : § z """ ’1-005-02 .5
'3; 1005-03 1 ~~~~~ § i J} E - E g
2- ' ' ‘ ' ‘ ' ' r 5005-03 ,9.
3 5005-04 » 8
o E
0.005+00 - ~ 0,005+00

 

 

 

0 1 2 3 4 5 6 7 8 9
Transverse Distance, X (in)

 

[+Surface -a-Middle +Bottom]

 

(c) at the rear row
Figure A-7. Calculated dissipated energy for the load case III and night temperature
distribution (continued)

Transverse Distance, X (cm)

 

 

 

 

 

 

 

 

 

0 5 1O 15 20
3.50E-03 ‘ ‘ ‘ ‘
A # :—: . : —I . . A
i 3.00E-03 Tread] Tread2 Tread3 Tread4 TreadS _ 2.00503 it:
g 2.505-03« f
3 : : : 3 : ; 1.50E—O3 8
>. 2005-03 . qr. ,_ s «-
5 2 2 s s a 2 ; : 9
... ‘-5°E-°3 “ : " é” ‘ ‘: “ ”064335
.3; 1.005-03- § ~ §~ § 3
g. ' ' ' ‘ ' r 5.005-04 g-
.! 5.00E-04 a 3
o a
0.00E+OO — — - f 0.00E+00
O 1 2 3 4 5 6 7 8 9
Transverse Distance, X (in)
[-t— Surface -E—Middle +Bottom]
(a) at the front row
Figure A-8. Calculated dissipated energy for the load case 111 and morning temperature

distribution

145

Transverse Distance, X (cm)
5 10 15 20

0
3.50E-03 t

 

 

 

 

 

 

 

 

 

 

 

 

A _ ' 3
- a.
‘3: 3005.03 . Tread] Tread2 Tread3 Tread4 TreadS ; 200502 a:
g 2.50E-03 g
5 E E 1. 1505-02 °
é, 2.005-03 :~ : : ~~~~~~~~~~ a
g s s 5 l E
m 1.50E-03 L , L '1—1-00E-02 '5
3 1.00E-03* ' 3
.§- ~ 5.005-03 §.
3 5.005-04- , 3
0 a
0.00E+00 L _ 0.00E+OO
0
Transverse Distance, X (in)
[+Surface -B-Middle +Bottom]
(b) at the middle row
Transverse Distance, X (cm)
0 5 10 15 20
3505-03 ' ‘ . '
i L—.— ._'—-. -.——. _.— '-.'—: E
v 3.005-03 Tread] Tread2 Tread3 Tread4 TreadS -_ 100502 5
‘3 2.505-03 - g
3 g » 1505-02 3
5 2005-034 ~ — E
e : : : : z : : : .
If] 150303" ‘ T " ’ ’ :"' f" ' 3 ‘ *1-005-02 .5
E 100503 3
3 . 2 55005-03 .3-
1 5.005-04-» j» ----- j —- 3
D ' a
0.005+00 .- _ _ _ _ - _ =3 ~0.005+00
0 1 2 3 4 5 e 7 8 9

Transverse Distance, X (in)

 

|+Surface -a—Middle +Bottom]

 

(c) at the rear row

Figure A—8. Calculated dissipated energy for the load case IH and morning temperature
distribution (continued)

146

Transeverse Distance, X (cm)
10 15 20

o 5
3.502-03 l a—

3.oos-03 Tread] Tread2 Tread3 Tread4 Treads

 

 

2.50E-03

L

2.005s : a a: E ++++++++ a

mos-ow é     —————— 2 ————————

5.00E-04

Dissipated Energy Denslty (psl)

 

 

 

0.00E+00 a
0 1 2 3 4 5 6 7 8 9

Transverse Distance, X (in)
Lt-Surface -B-Midd|e +Bottom]

 

 

(a) at the front row

Transverse Distance, X (cm)

0 5 10 15 20
3.50E-03 I

300503 Tread] Tread2 Tread3 Tread4 TreadS ......

 

 

' —

 

,.

2.505-03 .

2.00E-03

l

1.50E-03 ~

1.00E-03

l

,—

5.00E-04 4»

Dissipated Energy Denslty (psl)

l

 

 

 

0.00E+00

 

o 1 2 3 4 5 6 7 a 9
Transverse Distance, X (In)

 

[-A- Surface -3- Middle —o— Bottom I

 

(b) at the middle row

Figure A-9. Calculated dissipated energy for the load case 111 and day
distribution

147

y-

_ 2.00E-02

1.505-03 i i § § . gr; ..... 1 ....... F

L 5.00E-03

._

1 .5OE-02

1.00E-02

Dlsslpated Energy Denslty (kPa)

0.00E+OO

2.00E-02

1.50E-02

1.00E-02

5.00E-03

Dlsslpated Energy Denslty (kPa)

0.00E+00

temperature

Transverse Distance, X (cm)
5 10 15 20

0
3.50E-03 1 —. | — y--—--

 

 

Dissipated Energy Denslty (ps1)

' ' ‘ 3
...... n.
3.005-03 Treadl Tread2 Tread3 . Tread4 TreadS , 2.00502 5
2.50503 «wt; , ,. ‘ g g uuuuu ‘ cccccccc g
E E E E E I i 4.505412 3
2005-03 A A; g 3 £5.12”... 3 ‘ ,,,,,, i . >
I i I I i I I i g
1'505'03 5'7 5 **** """ 5 ‘r”'"'i““-“l~Loos-02 .5
1.005-03«-~m g ssssss g ...... g VVVVVV g "T“ g ..... g
' ' ' ' ' ' ‘ 5.005-03 g
5.005-04 3
a

o.ooe+oo ~ » o.ooe+oo

 

 

 

o 1 2 3 4 5 6 7 8 9
Transverse Distance, X (in)

 

[+Surface -a—Middle +Bonom]

 

(c) at the rear row

Figure A-9. Calculated dissipated energy for the load case III and day temperature
distribution (continued)

148

APPENDIX B

EXAMPLE OF A FINITE ELEMENT ANALYSIS INPUT FILE

An example ABAQUS input ﬁle of a 3-D analysis is here presented. Mesh and
elements were generated and imported ﬁ'om I-DAES. Because the generated input model
deﬁnes nodes with each nodal coordinates instead of keyword commands, node and
element deﬁnition in the input ﬁle therefore are huge. Due to the space limitations, some
parts skip.

Note. Lines starting with multiple * are comment lines.

*****************t*************************************

*HEADING

SDRC I-DEAS ABAQUS FILE TRANSLATOR

*" This input ﬁle is for the 3-D model: uniform load and day temperature distribution.
**"‘ Detailed geometry is presented in Chapter 5.

*******************#**#***********************t********

*** Node Deﬁnition

*****#**********#*t*t*******¥*#*****************#******

*NODE, SYSTEM=R
1, 5.0000000E+00, 2.46SOOOOE+02,-5.0000000E+OO
2, 5.0000000E+OO, 2.4650000E+02,-4.5900000E+00
3, 5.0000000E+00, 2.4650000E+02,-4.1800000E+00
4, S.OOOOOOOE+OO, 2.4650000E+02,-3.77000OOE+OO
5, 5.0000000E+OO, 2.4650000E+02,-3.3600000E+OO
6, 5.0000000E+OO, 2.4650000E+02,-2.9500000E+00
7, 5.0000000E+OO, 2.4650000E+02,-2.5400000E+00
8, 5.0000000E+00, 2.4650000E+02,—2.1300000E+00
9, 5.0000000E+OO, 2.4650000E+02,-1.7200000E+00
10, 5.0000000E+OO, 2.4650000E+02,-1.3 100000E+00

***t#**********$*#************

**** skip due to the space limitations
*******#**********************

149

34970, 6.3050468E+00, 2.3820433E+02, 5.7219956E-01
3497l,-4.4805292E+01, 2.3950000E+02, 2.8351542E+01

#*******************************************************

*** Element Deﬁnition
**¥********************t********#*****************t*t***
*ELEMENT,TYPE=C3D4 ,ELSET=BASE

21 l49,26693,26698,267] 3,26162

21 150,25496,28901,25472,28633

21151,31028,31033,9265,33168

****#*#****#******************

”** skip due to the space limitations
*t*******************#¥*******
73789,28262,28267,30763,28322
*ELEMENT,TYPE=C3D4 ,ELSET=SOILT
20021 ,24241 ,243 17,24244,24240
20022,24241 ,24244,243 1 7,24243
20023,24237,24278,24238,24243
20024,24237,24238,24278,24255
20025,24226,24224,24273,24277

*****ttit*********************

**** skip due to the space limitations
**************###*************
21 147,24240,24244,243 1 1,243 1 7

21 148,24244,243 17,24243,24238
*ELEMENT,TYPE=C3D8 ,ELSET=AC
1 ,1 ,2,23,22,442,443,464,463
2,2,3,24,23,443,444,465,464
3,3,4,25,24,444,445,466,465
4,4,5,26,25,445,446,467,466
5,5,6,27,26,446,447,468,467
6,6,7,28,27,447,448,469,468

***********************#******

**** skip due to the space limitations
**#***********#******##****tit
l9997,23556,23557,8838,8837,23997,23998,9279,9278
19998,23557,23558,8839,8838,23998,23999,9280,9279
19999,23558,23559,8840,8839,23999,24000,928l,9280
20000,23559,23560,8841,8840,24000,24001,9282,9281
*ELEMENT,TYPE=C3D8 ,ELSET=SOIL

20001 ,24002,24003,24006,24005,2401 1,24012,24015,24014
20002,24003,24004,24007,24006,24012,24013,24016,2401 5
20003,24005,24006,24009,24008,24014,24015,2401 8,2401 7

t***************#*************

**** skip due to the space limitations
*****¥************************

2001 9,24041 ,24042,24045,24044,24050,2405 1,24054,24053

150

20020,24042,24043,24046,24045,2405 1 ,24052,24055,24054

************************#**t*#*¥*********************

""El sets and Node sets for analysis
*#¥$*****************************¥******************$
*NSET, NSET=NDTOPMID, GENERATE

1 0,409,21

*ELSET, ELSET=ELTOP, GENERATE
2,362,20

1 0,3 70,20

12,372,20

13,373,20

19,379,20

*ELSET, ELSET=ELMIDDLE, GENERATE
1602,1962,20

l610,1970,20

1612,1972,20

1613,1973,20

1619,1979,20

*ELSET, ELSET=ELBTTM, GENERATE
3602,3962,20

3612,3972,20

3614,3974,20

3615,3975,20

3619,3979,20

*ELSET, ELSET=ELANLY
ELTOP,ELMIDDLE,ELB”ITM,

*ELSET, ELSET=ELTOTALTOP, GENERATE
1,380,]

*ELSET, ELSET=ELTREAD1, GENERATE
1,60,1

*ELSET, ELSET=ELTREAD2, GENERATE
8 1,120,]

*ELSET, ELSET=ELTREAD3, GENERATE
141 ,220,1

*ELSET, ELSET=ELTREAD4, GENERATE
241 ,280,1

*ELSET, ELSET=ELTREAD5, GENERATE
301,380,]

*ELSET, ELSET=ELTREAD

ELTREADI ,ELTREAD2,ELTREAD3,ELTREAD4,ELTREAD5
*******

*ELSET, ELSET=ACALL, GENERATE
1,3980,1

*ELSET, ELSET=ACCROSS, GENERATE
12,372,20

412,772,20

151

812,1 172,20

1212,1572,2o

1612,1972,20

2012,2372,2o

2412,2772,20

2812,3172,20

3212,3572,2o

3612,3972,20

********

*ELSET, ELSET=V1, GENERATE
1,3,1

21,23,1

*ELSET, ELSET=V2, GENERATE
4,6,1

24,26,1

***#***********************#*t

**** skip due to the space limitations
********#*****#***************

*ELSET, ELSET=S9 1 , GENERATE

100356,100358,1

100376,100378,1

"ELSET, ELSET=S92, GENERATE

100359,100360,1

100379,100380,1
*#*#**¥*******#***#**#****#****************************
* “*Membrane Element for lateral stress applications

*"The membrane elements have negligible stiffness
*******************#*******#***#**t***#******#**#******
*ELEMENT,TYPE=M3D4,ELSET=MEMB1

100001,22,23,2,1

*ELGEN, ELSET=ALLMEMB

100001,20,1,1,19,21,20
**********tttttttttt*ttt*#*********#************#******
***Material Deﬁnition
*#***#********#****************************************
******#********************

*MEMBRANE SECTION, ELSET=ALLMEMB, MATERIAL=MEMBRANE1
l

*MATERIAL,NAME=MEMBRANE1

*ELASTIC,TYPE=ISOTROPIC

0.000001

*ttttttttitttttttttttttttt

*SOLID SECTION,ELSET=BASE,MATERIAL=BASE
*MATERIAL,NAME=BASE

*ELASTIC,TYPE=ISOTROPIC

5.0E+4, 0.35

152

ans psi

*SOLID SECTION,ELSET=SOILT,MATERIAL=SOIL
*MATERIAL,NAME=SOIL
*ELASTIC,TYPE=ISOTROPIC

1.0E+04, 4.500E-01

*SOLID SECTION,ELSET=SOIL,MATERIAL=SOIL
*SOLID SECTION,ELSET=AC,MATERIAL=AC
*MATERIAL,NAME=AC
*ELASTIC,TYPE=ISOTROPIC

2071851, 0.3

****psi

*VISCOELASTIC,TIME=PRONY
0.1546401,,1.1447957

0.3099463,,0.0006052

0.3118353,,0.0173438

0.1846149,,0.0000131

***¥#***##**

*TRS

25.,47.8,392

"degree C
*##**#****##*********#*##*#*********¥******
*** Boundary Condition

*** Temperature is speciﬁed to each node or node sets.
*** It is assumed that temperature varies only with depth (Y-direction).
tttt**#*******#**t*tit*****#***##*#******#t
*INITIAL CONDITION, TYPE=TEMPERATURE
***** unit degree C

1,43.00

2,43.00

3,4300

4,4300

5,4300

6,4300

#****##*****¥#**#*************

**** skip due to the space limitations
******************************

34941,34.22
34946,34.22
34951,34.22
34956,34.22
34961,34.22
34966,34.22
34971,34.22
***¢***********
*BOUNDARY
4872,]

153

5292,]
5313,]
5733,]

#***#****#*****************#**

**** skip due to the space limitations
******************************
9766,3
9745,3
9724,3
9703,3

****************#****t************************

*** Apply Load

**********************************************

*RESTART,WRITE,FREQ=10000

*****#*********

t t * t t 4' STEP 1

* STEP,INC=10,AMPLITUDE=RAMP

*VISCO,CETOL=O. 1

0.001 ,0.001250

*DLOAD

ELTOTALTOP,P1 ,2.76
*ELPRINT,ELSET=ELANLY,TOTALS=YES,POSITTON=CENTROIDAL,FREQ=1 00
00

S11,S22,833,E11,E22,E33,S13,E13,S23

*NODE PRINT,NSET=NDTOPMID,TOTALS=YES,FREQ=10000
U,RF

*NODE FILE, NSET=NDTOPMID, FREQ=1

U,RF

*END STEP

* t i 1! i * t * t i 1!

II t * t * STEP 2

* STEP,INC=10,AMPLITUDE=RAMP

*VISCO,CETOL=0. 1

0.001,0.001875

*DLOAD

ELTOTALTOP,P1 ,19.32

* ELPRINT,ELSET=ELANLY,TOTALS=YES,POSITION=CENTROIDAL,FREQ=1 00
00

S11,S22,833,E11,E22,E33,Sl3,E13,S23

*NODE PRINT,NSET=NDTOPMID,TOTALS=YES,FREQ=10000
U,RF

*NODE FILE, NSET=NDTOPMID, FREQ=1

U,RF

*END STEP

***********
********#**********#****#*****

154

**** skip due to the space limitations
ti****************************

*************

"*"STEP 18

*STEP,INC=10,AMPLITUDE=RAMP

*VISCO,CETOL=O.1

0.001,0.002969

*DLOAD

ELTOTALTOP,P1,0
*ELPRINT,ELSET=ELANLY,T0TALS=YES,P0SIIION=CENTROIDAL,FREQ=100
00

Sl1,822,S33,E1I,E22,E33,SI3,E13,SZ3

*NODE PRINT,NSET=NDTOPMID,TOTALS=YES,FREQ=10000
U,RF

*NODE FILE, NSET=NDTOPMID, FREQ=1

U,RF

*END STEP

***********

"*"STEP 19

*STEP,INC=10

*VISCO,CETOL=0.1

0.1,0.0228437
*ELPRINT,ELSET=ELANLY,TOTALS=YES,POSITION=CENTROIDAL,FREQ=100
00

S]l,S22,S33,E1I,E22,E33,S13,E13,S23

*NODE PRINT,NSET=NDTOPMID,TOTALS=YES,FREQ=10000
U,RF

*NODE FILE, NSET=NDTOPMID, FREQ=1

U,RF

*END STEP

t******#***
*#*#******tt**ttttiit!*tﬁttit#*#******$#*******#*******

155

REFERENCES

156

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