finite element analysis of the mechanics of viscoelastic

FINITE ELEMENT ANALYSIS OF THE MECHANICS OF VISCOELASTIC
ASPHALTIC PAVEMENTS SUBJECTED TO VARYING TIRE CONFIGURATIONS
by
Roberto Firmeza Soares
A THESIS
Presented to the Faculty of
The Graduate College at the University of Nebraska
In Partial Fulfillment of Requirements
For the Degree of Master of Science
Major: Engineering Mechanics
Under the Supervision of Professor David H. Allen
Lincoln, Nebraska
November, 2005

FINITE ELEMENT ANALYSIS OF THE MECHANICS OF VISCOELASTIC
ASPHALTIC PAVEMENTS SUBJECTED TO VARYING TIRE CONFIGURATIONS
by
Roberto Firmeza Soares
A THESIS
Presented to the Faculty of
The Graduate College at the University of Nebraska
In Partial Fulfillment of Requirements
For the Degree of Master of Science
Approved as to style and content by:
_______________________________
David H. Allen
Chair of Committee
_______________________________
Mehrdad Negahban
Member
_______________________________
Yuris A. Dzenis
Member
_______________________________
Yong R. Kim
Member
November, 2005

FINITE ELEMENT ANALYSIS OF THE MECHANICS OF VISCOELASTIC
ASPHALTIC PAVEMENTS SUBJECTED TO VARYING TIRE CONFIGURATIONS
Roberto Firmeza Soares, MS
University of Nebraska, 2005
Advisor: David H. Allen
This thesis proposes a model for predicting the mechanical behavior and
performance life of asphalt pavements subjected to various tire configurations, layer
geometries and material properties. A viscoelastic two-dimensional finite element model
is formulated in order to predict which combination of design variables increase and
improve pavement life.
The formulation utilizes recently proposed advanced algorithms to produce a
time-marching computational algorithm capable of predicting the spatial and temporal
variations in stresses, strains and displacements in viscoelastic roadways subjected to
various configurations of loads, geometry and material properties.
In this study, the effects of truck loads on the pavement performance are
simulated by a cyclic load. Three tire configurations are included in the study, including
two types of the new generation, wide-base single tire and one conventional dual tire
configuration. Also, two different types of hot mix asphalt and three variations of the
asphalt layer thickness are evaluated.

The results presented here show that the use of conventional dual tires produces
approximately 50% more service life when compared to the use of wide tires. Also, by
increasing the asphalt layer thickness, the service life is increased. For instance, a 15 cm
thick asphalt layer provides a 15% better life than a 10 cm thick layer. However, if a 20
cm thick layer is compared with a 15 cm layer, the life goes up about 18%. Comparisons
of two different asphalt mixtures, shows that one produces a lifespan six times that of the
other. These results indicate that the model proposed herein can play a fundamental role
in pavement design and hot mix selection.

i
Dedication
to God for my health;
to my wife Marilena;
to my parents Aroldo and Laís.

ii
Acknowledgments
I wish to express my sincere gratitude to Dr. David H. Allen, chair of my graduate
advisory committee for the time invested on me, for his guidance and all the
opportunities during my Master’s program. Without his expert knowledge this work
would not have been possible. Also I would like to thank all the members of my
committee, Dr. Mehrdad Negahban, Dr. Yong Rak Kim and Dr. Yuris Dzenis for all the
assistance that I have received in so many ways and for being available to share their
knowledge every time I had questions.
I also would like to thank professor Jorge B. Soares from the Federal University
of Ceará (Brasil) for introducing me to the research field of asphalt pavement.
I wish to acknowledge the financial support provided by the Army Research
Office with the contract number is W911NF-04-2-0011.
Special thanks are due to my classmates Mr. Philip Yuya, Mr. Ashwani Goel and
Mr. Kitti Ratatanadit for their friendship and shared knowledge; to Mr. Felipe Freitas,
Mr. Victor Teixeira and Mr. Sávio Câmara for all the technical support; Mr. James Nau
for setting up the supercomputer; Ms. Michelle Sitorius for taking the time to read
through my thesis; and my friends Ms. Jamilla Lutif and Mr. Thiago Aragão for their
friendship.
Finally, I wish to thank my family. My wife, Marilena, for all the support, love,
patience and encouragement throughout this effort and also for making all the pictures in
this work. There are not enough words to describe my appreciation. My mother and

iii
father whom I love so much for encouraging me to pursue my education; and to my
brother Paulo and my sister Renata, just for being part of my life.
But very special thanks are due to God for giving me my life back.

iv
List of Tables
Table 1: Viscoelastic material properties.................................................................... 54
Table 2: Prony series coefficients of relaxation modulus for AC I.......................... 70
Table 3: Prony series coefficients of relaxation modulus for AC II......................... 71
Table 4: Material properties for the road elastic layers............................................ 71

v
Figure Index
Figure 1: Pavement structure......................................................................................... 3
Figure 2: Typical asphalt pavement structure.............................................................. 3
Figure 3: Loads in Pavement ......................................................................................... 4
Figure 4: Aggregates ....................................................................................................... 6
Figure 5: Mixing ................................................................................................................ 7
Figure 6: Pouring mix into the mold .............................................................................. 7
Figure 7: Gyratory compactor......................................................................................... 8
Figure 8: Cracking.......................................................................................................... 11
Figure 9: Cracking.......................................................................................................... 11
Figure 10: Thermal cracking......................................................................................... 13
Figure 11: Rutting........................................................................................................... 14
Figure 12: Rutting........................................................................................................... 14
Figure 13: Tandem axle – wide tires (Courtesy of Michelin American)................. 16
Figure 14: Tandem axle ................................................................................................ 16
Figure 15: 18 wheeler truck .......................................................................................... 17
Figure 16: Tridem axle................................................................................................... 17
Figure 17: European truck ............................................................................................ 18
Figure 18: Truck with dual tires or wide tires configuration ..................................... 19
Figure 19: Dual tire vs. Wide base .............................................................................. 20
Figure 20: Types of tires ............................................................................................... 22
Figure 21: Measured tire pressure distribution. Courtesy Tekscan, Inc................ 23
Figure 22: Tire footprints – Hua and White, 2002..................................................... 24

vi
Figure 23: General three dimensional body............................................................... 33
Figure 24: T3 triangular element representation....................................................... 46
Figure 25: Wiechert model............................................................................................ 53
Figure 26: Rectangular bar subjected to a tip load................................................... 56
Figure 27: Loading history and response of example 1........................................... 57
Figure 28: Loading history and response of example 2........................................... 58
Figure 29: Cylinder with Internal Pressure................................................................. 59
Figure 30: Finite element mesh used in example 3.................................................. 60
Figure 31: Loading history and response for example 3.......................................... 61
Figure 32: Beam problem of example 4 ..................................................................... 63
Figure 33: Loading history and response for example 4.......................................... 64
Figure 34: Typical two-lane roadway cross section.................................................. 65
Figure 35: Right half side to be modeled.................................................................... 66
Figure 36: Boundary conditions on the pavement .................................................... 67
Figure 37: Side slopes of roadways (NDor, 2005).................................................... 67
Figure 38: Pavement structure and all layers ............................................................ 69
Figure 39: Comparison between AC I and AC II....................................................... 72
Figure 40: Weight distribution ...................................................................................... 73
Figure 41: Tire pressure distribution (Blackman, Halliday, and Merrill, 2000)...... 75
Figure 42: Pavement 3-D with forces.......................................................................... 76
Figure 43: Pavement 2-D with forces.......................................................................... 76
Figure 44: Comparison between 3D and 2D loads................................................... 77
Figure 45: First cyclic load – axle loads...................................................................... 78

vii
Figure 46: Cyclic load .................................................................................................... 79
Figure 47: Convergence of maximum displacement for 29.5 tire........................... 81
Figure 48: Mesh for the dual configuration tire – 29.5 cm ....................................... 81
Figure 49: Convergence of maximum displacement for 44.5 tire........................... 82
Figure 50: Mesh for the wide configuration tire – 44.5 cm ...................................... 82
Figure 51: Convergence of maximum displacement for 49.5 tire........................... 83
Figure 52: Mesh for the wide configuration tire – 49.5 cm ...................................... 83
Figure 53: Pavement with side slope 1:6 ................................................................... 84
Figure 54: Pavement with side slope 1:3 ................................................................... 84
Figure 55: Pavement with side slope 1:1 ................................................................... 84
Figure 56: Pavement with side slope 0.5:1 ................................................................ 84
Figure 57: Displacement on the surface for different side slopes .......................... 85
Figure 58: Failure criterion............................................................................................ 87
Figure 59: Displacement versus time.......................................................................... 88
Figure 60: Displacement versus time in log scale .................................................... 88
Figure 61: Different tire configurations for HMA thickness of 10cm....................... 89
Figure 62: Different tire configurations for HMA thickness of 15cm....................... 90
Figure 63: Different tire configurations for HMA thickness of 20cm....................... 90
Figure 64: Different HMA thicknesses for dual tires 29.5 cm.................................. 91
Figure 65: Different HMA thicknesses for dual tires 44.5 cm.................................. 93
Figure 66: Different HMA thicknesses for dual tires 49.5 cm.................................. 93

viii
Table of Contents
CHAPTER 1 .......................................................................................................................1
Introduction.................................................................................................................... 1
Asphalt Pavement Structures..................................................................................... 4
Asphalt Materials........................................................................................................ 4
Asphalt Pavement Distress ......................................................................................... 9
Truck Loading........................................................................................................... 15
Pavement Design ...................................................................................................... 24
Research Objective.................................................................................................... 29
CHAPTER 2 ..................................................................................................................... 31
Solution Method.......................................................................................................... 31
CHAPTER 3 ..................................................................................................................... 38
Finite Element Formulation....................................................................................... 38
Incrementalization of the Constitutive Equations..................................................... 38
Method of Weight Residuals ..................................................................................... 40
Displacement Formulation ....................................................................................... 44
CHAPTER 4 ..................................................................................................................... 50
Code Description and Verification ........................................................................... 50
Code Description ...................................................................................................... 50
Code Verification...................................................................................................... 52
CHAPTER 5 ..................................................................................................................... 65
Asphalt Concrete Roadway Analysis ...................................................................... 65
Roadway Geometry................................................................................................... 65

ix
Pavement Layer Characterization ............................................................................ 67
Overestimation Between 3-D and 2-D...................................................................... 74
Mesh Convergence.................................................................................................... 79
Results ......................................................................................................................... 85
CONCLUSION................................................................................................................. 94
BIBLIOGRAPHY............................................................................................................. 97

1
CHAPTER 1
The interstates and highways connecting cities and sustaining economic exchange
are integral to the infrastructure of a country. Accordingly, the characteristics—
durability, safety, and comfort—of these structures are vital not only for those who travel
on roadways but also for those involved in the actual construction. This paper evaluates
the effectiveness of the current use of asphalt surfaces and proposes to develop a model
for predicting mechanical behavior and performance life of pavements.
I begin with a general overview of pavement structures, looking at the benefits
and drawbacks of different methods of paving. Then, a comparison between wide base
tires and dual tire configurations is presented. The second section explores the solution
method. The third section develops the model for mechanical behavior and performance
prediction based on the finite element formulation. The fourth section evaluates the code
description and verification, determining the effectiveness of the entire system. The fifth
and final segment analyzes the research results from the different computational analyses
performed.
Introduction
Pavements are structures built as layered systems to sustain traffic and to diminish
environmental effects. The most common paving methods are: asphalt concrete (or
flexible pavements), Portland cement concrete (or rigid pavements) and composite
pavements. Establishing which method to use is usually determinate upon cost; however,
other variables such as durability, i.e. maintenance, and the purpose of the roadway also
play a role, necessitating a delicate balance between road life and road utility. As of

2
today, there are 2.27 million miles of paved roadways in the United States, where 94%
have surfaces of asphalt concrete. 1 Correspondingly, an evaluation of the sustainability of
these surfaces necessitates further research.
Asphalt concrete is a mixture of asphalt and mineral aggregate mixed together,
spread and compacted at high temperatures. Asphalt concrete is also known as hot mix
asphalt or HMA. The asphalt layer is built on top of several other layers, finishing the
paving process. With less rigid materials used at the bottom, the lowest layer in the
structure is the subgrade or natural terrain. The next layer is the subbase, consisting of
specified, compacted material placed on top of the subgrade to serve as a foundation for
the pavement. On top of the subbase is the base, another compacted material layer, lying
directly beneath the concrete or asphalt. Figure 1 and Figure 2 show the pavement
structure, differentiating the various layers. This configuration can vary from case to
case. In some instances, these structures require additional reinforced layers to provide
sufficient total pavement thickness and to prevent pavement distress.
1 National Asphalt Pavement Association, retrieved November 10, 2005, from:
http://www.hotmix.org/history.php

3
Figure 1: Pavement structure
Figure 2: Typical asphalt pavement structure
A basic factor in pavement design, and integral to this thesis, is the vertical
traction on the top of the subgrade (see Figure 3). Traction can be minimized by a

4
properly designed pavement structure, reducing the traction to one tenth of the tire
pressure applied at the surface of the pavement. In turn, the strain level is very low,
limiting long term deformations. ideas
Figure 3: Loads in Pavement
Asphalt Pavement Structures
Asphalt Materials
Hot mix asphalt (HMA) is composed of three components: aggregates, asphalt
binder and air voids. The interaction of these components and the viscoelastic behavior of
the binder define the thermomechanical behavior of the mixture.

5
Historically, determining which asphalt cement or binder to use has been based on
(i) phenomenological tests developed over time and (ii) experience with asphalt
pavements. Since empirical results work only when all the conditions for all the tests are
satisfied, penetration and viscosity tests are slowly being replaced by tests that more fully
characterize asphalt binders. For instance, two binders might have the same viscosity at
the test temperature, but behave differently at a very low temperature. As a result, it can
be difficult to select a binder that will consistently work under varying conditions.
In 1987, the U.S. Congress authorized the Strategic Highway Research Program
(SHRP) — a five-year, applied research initiative — to develop and evaluate techniques
and technologies to combat the deteriorating conditions of the nation's highways and to
improve their performance, durability, safety, and efficiency. The final product of the
SHRP asphalt research program is a system referred to as Superpave, which stands for
Superior Performing Asphalt Pavements. The objectives of Superpave are: to investigate
why some pavements perform well, while others do not; to develop tests and
specifications for materials that will outperform and outlast the pavements being
constructed today; and to work with highway agencies and industry to have the new
specifications put to use. This enables reduction in maintenance costs, and minimization
on the number of premature pavement failures. This program actually proposes a new
system of specifying asphalt materials, providing a new asphalt binder specification, new
tests and a new mix design and analysis system.
In this system, asphalt binders are classified based on their performance in a range
of climates and pavement temperatures, relating the properties of the asphalt mixture to

6
the conditions under which they will be used. 2
Those binders are classified by a “PG”
(Performance Grade) rating, where the first number indicates the high temperature grade
and the second number corresponds to the low temperature grade. Thus, a PG 58-22 is
intended for use where the average, seven-day, maximum pavement temperature is 58°C
and the expected minimum pavement temperature is -22°C. 3
By selecting quality
aggregates and the correct Superpave performance graded (PG) asphalt binder, it is
possible to make high quality, hot mix asphalt for a variety of conditions. Figure 4,
Figure 5, Figure 6 and Figure 7 show the Superpave procedure to produce the mixture in
the laboratory.
Figure 4: Aggregates
2 CDot, 2005.
3 Note that these numbers are pavement temperatures and not air temperatures.

7
Figure 5: Mixing
Figure 6: Pouring mix into the mold

8
Figure 7: Gyratory compactor
Although the character of the asphalt binder is influential to a hot mix asphalt, air
voids and aggregates can also influence pavement performance. Factors like gradation
and type of aggregates and/or high permeability to air or water play an important role in
pavement durability. Asphalt binders exposed to air become brittle due to oxidation and
tend to undergo cracking. High permeability to water enables stripping of the binder from
the aggregates, making it possible for the lower layers to be carried away.
In addition, another important factor to keep in mind when determining pavement
materials and structure is traffic loads. Heavy vehicles, such as trucks, significantly
influence pavement distresses and failure. Currently, the importance of roadway

9
transportation has grown as vehicles are heavier with greater load-carrying capacity and
high tire pressures. Consequently, pavement loadings have significantly increased along
with the number of heavy vehicles traveling roads across the globe, resulting in
premature failure. As a primary variable of pavement design, accurate configuration of
the truck loading and investigations on the impact of heavy traffic on pavement
performance is an important issue and, indeed, remains an issue. Although truck loads are
important, pavements are complex systems involving interaction of other variables. Their
performance is also influenced by material properties, the environment, construction
practices and geometry of tires and/or layers.
Asphalt Pavement Distress
Determining the expected service life of a pavement system is an important
parameter to be considered in the design of a pavement. Pavements constructed of hot
mix asphalt are typically designed to last at least 20 years and perform well under stress,
from the extremes of high and low temperatures to heavy traffic loads. However, it is not
unusual to see severe deterioration in those pavements well before the expected service
life is met, resulting in irregular roadways and high maintenance and rehabilitation costs.
Loads, geometry and material properties contribute to declines or increases in this
expected life, some of theses factors include, thickness of pavement layers, material
properties, tire pressures, axle loads, tire configuration and volume of traffic. The major
causes of asphalt pavement failures are cracking and rutting.
Cracking
One possibility leading to cracking occurs due to excessive traffic load, causing
pavement failure by fatigue cracking, and introducing openings on the surface allowing

10
moisture to penetrate and accelerating pavement deterioration. These cracks start as
microcracks and propagate with time due to tensile or shear stresses or both. These
cracks then form macrocracks. To accurately predict fatigue is a complex task and for the
past several decades, significant research has been done in this area to develop reliable
models.
Castell, Ingraffea and Irwin focus on identifying the differences in crack
propagation between a common laboratory fatigue test specimen - beam and layered
pavement using a 2-D finite element with fracture mechanics capabilities. 4 Lee and Kim
present a mechanistic approach to uniaxial viscoelastic constitutive modeling of asphalt
concrete mixtures, accounting for the rate-dependent damage growth. 5 Lee also
developed a fatigue performance prediction model of asphalt concrete based on the
elastic-viscoelastic correspondence principle and continuum damage mechanics.
Schapery developed a so-called work potential theory applicable to describing the
mechanical behavior of elastic media with growing damage and other structural changes. 6
He also developed a theory for predicting the time-dependent size and shape of cracks in
linear viscoelastic, isotropic media. 7 A number of studies involving theory of crack
initiation and growth in viscoelastic media can be found in the literature. 8
4 Castell, M.A., A. R. Ingraffea, L. H. Irwin. “Fatigue Crack Growth in Pavements.” Journal of
Transportation Engineering. Volume 126, Number 4 (July 2000): 283-290.
5 Lee, Hyun-Jong, Kim, Y. Richard. “Viscoelastic Constitutive Model for Asphalt Concrete Under Cyclic
Loading.” Journal of Engineering Mechanics. Vol. 124, No. 1 (January 1998): 32-40.; Lee, Hyun-Jong;
Kim, Y. Richard. “Viscoelastic Continuum Damage Model of Asphalt Concrete With Healing.” Journal of
Engineering Mechanics. Vol. 124 Issue 11 (Nov98): 1224, 9
6 Schapery, R.A. ‘‘A theory of mechanical behavior of elastic media with growing damage and other
changes in structure.’’ Journal of Mech.Phys. Solids. 38 (1990): 215–253.
7 Schapery, R.A. A theory of crack initiation and growth in viscoelastic media (Part I), International
Journal of Fracture, 11, 1975. 141–59.; Schapery, R.A. A theory of crack initiation and growth in
viscoelastic media (Part II), International Journal of Fracture, 11, 1975b. 369–88.
8 See (Schapery, 1975a; Schapery, 1975b; Schapery, 1978; Schapery, 1986; Schapery 1990; Park and
Schapery, 1997; Kim, 1995, Costanzo and Allen, 1993).

11
Figure 8: Cracking
Figure 9: Cracking

12
Cracking can also be caused by temperature, either from very large temperature
variations or very cold temperatures. Those thermal cracks occur because the shrinkage
of the surface is restrained; therefore, tensile stress is developed until it reaches the
tensile strength, forming the crack. 9
Not only is the asphalt surface subjected to this
phenomenon, but also the base, subbase and subgrade, that can potentially propagate up
to the asphalt layer by reflective cracking. In this case, tensile stress is greatest in the
longitudinal direction, explaining the fact that thermal cracks are usually transverse. Easa
provided a reliability-based model which predicts thermal cracking of pavements and
relates it to cold winters, spring thaws, and daily cyclic thermal loading. 10 and Shen and
Kirkner 11 developed a multiscale model to simulate the distributed thermal cracking
primarily for asphalt concrete pavement structures. They used this model to study the
interaction between multiple cracks and predict the crack spacing. They assumed there is
a set of fictitious cracks on the pavement. When temperature drops, some fictitious cracks
become predominant to the others and turn out to be major cracks.
9 See (Al-Qadi, I. L., et al., 2004).
10 Easa, Said M. “Reliability-Based Model for Predicting Pavement Thermal Cracking.” Journal of
Transportation Engineering. Volume 122, Number 5 (September 1996): 374-380.
11 Shen, Weixin, David J. Kirkner. “Distributed Thermal Cracking of AC Pavement with Frictional
Constraint.” Journal of Engineering Mechanics. Volume 125, Number 5 (May 1999): 554-560.

13
Figure 10: Thermal cracking
Rutting
The other cause of asphalt pavement failures is rutting. Rutting is characterized by
depressions along the wheel path caused by the permanent deformation of the asphalt
concrete mixture, base, or subgrade materials. Highway rutting is common due to high
canalizations and higher repetition traffic. Ramsamooj presented a new elastoplastic
model for predicting the stress/strain response of asphalt concrete under cyclic loading,
incorporating a complete description of the plastic deformation of asphalt concrete and
including the elastic, viscoelastic, and plastic components. 12 The plastic deformation is
calculated by a computer program and compared with experimental results. Douglas,
1997; Collop, 1995; and Chen, 2004 also studied several effects of rutting in asphalt
pavements.
12 Ramsamooj, D.V., J. Ramadan, G. S. Lin. “Model Prediction of Rutting in Asphalt Concrete.” Journal of
Transportation Engineering. Volume 124 Number 5 (September 1998): 448-456.

14
Figure 11: Rutting
Figure 12: Rutting

15
In the past 20 years traffic load as well as truck tire pressures have increased in
the US and impacted the pavement long-term life and performance; consequently, the
roadways have experienced premature failure of the asphalt layer. The effect of tire
configurations and pressure are not the only variables of interest in pavement design.
Varying material properties, such as size of the aggregate, percentage of binder in the
asphalt concrete mixture, and the thickness of asphalt layer, impact the service life of
pavement. Furthermore, by increasing the thickness of the asphalt layer, the pavement
becomes less vulnerable to high pressure tires. It should be noted that with load,
geometry and material property variation, expected pavement life is altered. Given that
the major part of paved roadways have hot mix asphalt surface, those issues are of great
interest for the Departments of Transportation around the country. Among the primary
factors that contribute to pavement deterioration are changes in truck loading, which
include axle weight and configuration, tire type, and suspension systems.
Truck Loading
Wide base tires x dual tires
Currently, a wide-base single tire has been proposed as a replacement for
conventional dual tires. They are more commonly used in Europe and Canada; however,
these configurations are slowly appearing in the US market and are intended to replace
the dual tire configuration for a larger tire. In the United States the most common type of
truck is the 18-wheeler, which is has 5 axles. The front, steer axle has one wheel at each
side and the rear has two tandem axles. Each tandem axle is composed by two pairs of
double wheels at each side. Figure 13 and Figure 14 shows a tandem axle and Figure 15
shows an 18-wheeler.

16
Figure 13: Tandem axle – wide tires (Courtesy of Michelin American)
Figure 14: Tandem axle

17
Figure 15: 18 wheeler truck
Europe uses the wide tire configuration on the majority of trucks. There, the most
common configuration also has 5 axles, again with the front, steer having two wheels,
and the rear, instead of having two tandem axles, has one single and one tridem axle with
three axles at the rear with single wheels, or 6 wheels in total. Thus, whatever the
configuration, the entire vehicle usually has 12 wheels in total. Figure 16 shows a tridem
axle and Figure 17 shows an European truck.
Figure 16: Tridem axle

18
Figure 17: European truck
Even though wide tire configuration is used more often in Europe, it is being
introduced in the US and Canada by simply replacing the dual tires by wide ones and still
carrying equal loads. This research project investigates the effect of wide base tires and
dual tires assemblies in the US and Canada. 13
Replacing both sides of dual tires with
larger tires results in a smaller total contact area with the pavement. The load weight
remains the same with the larger tires, yet, the pressure is larger in the wide tire case. 14
13 Figure 18 shows the types of truck analyzed for this research project.
14 Figure 19 shows a dual configuration compared with a wide base tire.

19
Figure 18: Truck with dual tires or wide tires configuration
While a single wide tire offers advantages for fuel efficiency, productivity and
vehicle stability, 15 its pavement wear characteristics are still being examined. Even
though the market for wide-base tires does not surpass 5% of truck applications, there is
potential for an increase in demand.
15 (Giles, 1979; John and Glauz, 1995; Sebaaly, 1998)

20
Figure 19: Dual tire vs. Wide base
Research in the past has shown that the wide-base tire can cause significantly
more damage to the pavement compared to conventional dual tires. The damage is related
to tire width and tire pressure. Sebaaly and Tabatabaee investigated the response of
various flexible pavement structures in terms of tensile strain on the bottom of the asphalt
layer, the compressive stress at the asphalt layer interface and the surface deflection. 16
For all asphalt concrete thicknesses, the tensile strain under the wide-base tires was 50%
greater than those under the dual tires. Compressive stresses at the interface were lower
for the dual tires. The wide-base tires are designed with narrow tread width, causing high
16 Sebaaly, P., N. Tabatabaee. Effect of Tire Pressure and Type on Response of Flexible Pavement.
Transportation Research Record. (1989): (1227) 115-127.

21
vertical and contact stresses between the road and tire. The US Department of
Transportation performed a comprehensive truck size and weight study showing that the
wide-base tires appear to cause 1.5 times more rutting than dual tires on flexible
pavements. This is an indication that those tires were designed with greater concern for
non-pavement costs, thus, resulting in restrictions on allowable weight on axles equipped
with wide-base single tires. Most of the studies performed comparing dual tires versus
wide tires, concluded that wide tires are more damaging to the pavement than are
conventional dual tires. 17
Recent advances in materials and tire design have led to the development of a new
generation of wide-base tires with the claim that their new technology causes no
additional infrastructure wear as compared to dual tires. This new design resulted in a
wider flatter transverse profile, providing a uniform pressure distribution (See Figure 20).
Ponniah analyzed the impact of this new technology on pavements by using the methods
of equivalent single axle loads and showed that the current axle load limit of 6000 kg in
Canada for single tires can be increased up to 8000 kg for wide-base tires with no more
damage than dual tires at 10000 kg. 18
17 (Akram, Scullion, Smith, Fernando, 1992; Huhtala, Pihlajamaki, Pienimaki, 1989; Perdomo, Nokes,
1993, Al-Qadi, Pyeong, Elseifi, 2005; Myers; Roque, Ruth, Drakos, 1999; Priest, 2005)
18 (Ponniah et al., 2003)

22
Figure 20: Types of tires
Al-Qadi developed an experimental program performed under different loading
and environmental conditions and at different speeds in order to understand the behavior
of the new tire. 19
The conclusions showed that the newly developed tires and the
conventional dual tire produce approximately the same horizontal strains under the
asphalt concrete layer. Furthermore, the fatigue damage is the same as that produced by
dual tires. However, the vertical compressive stresses induced by the wide-base are
greater on the upper asphalt concrete layers of the tested pavement. Recently, Al-Qadi
performed a three-dimensional analysis using a viscoelastic constitutive model to
simulate the pavement responses to different vehicular loadings. They used two newgeneration
wide-tires, and, in general, the wider one produced less damage than the
narrower one. For this analysis, damage was defined by four failure parameters: fatigue
cracking, primary and secondary rutting, and top-down cracking. It was noted that by
using the wide-base tires, the truck’s overall weight would be reduced by approximately
450 kg and the truck has a 4% reduction in fuel consumption. 20
19 Al-Qadi, Imad L., Pyeong J. Yoo, and Mostafa A. Elseifi. “Characterization of Pavement Damage Due to
Different Tire Configurations.” Journal of the Association of Asphalt Paving Technologists. (March 7-9,
2005).
20 (Markstaller et al., 2000)

23
The tire-pavement contact area is another concern when considering pavement
analysis. In the last decade, a series of sophisticated measuring systems have been
developed to measure these tire-pavement contact stresses. 21
It has been found that tirepavement
stress distribution is significantly affected by tire inflation pressure, tire type,
and tire load. Figure 21 depicts different pressure distributions from different tires and
Figure 22 shows different tire-pavement contact areas. Different tires have different
contact areas on the pavement, requiring sophisticated instrumentation to measure and,
consequently, to apply them. In this study, tire-pavement contact area and tire pressure
are not treated as variables, as this is a subject to be incorporated in 3D finite elements.
Papers by Blab and Harvey, 2002; Hua and White, 2002; Greer and Palazotto, 1998;
Blackman, Halliday, and Merrill, 2000; Huhtala, Pihlajamaki and Pienimaki, 1989
provide comprehensive studies of tire pressure distribution, tire contact area.
Figure 21: Measured tire pressure distribution. Courtesy Tekscan, Inc.
21 See (Siddharthan, 2002)

24
Figure 22: Tire footprints – Hua and White, 2002
Pavement Design
The primary goal of pavement engineering is to predict pavement performance. In
the past, pavement design was almost totally driven by empirical relationships based on
long-term experience and field tests, such as the American Association of State Highway
and Transportation Officials (AASHTO) Road Test. These procedures, however, are
applicable only to similar load conditions (tire types and tire pressures), pavement
materials, and environment, making this model very limited. It would require a large
number of empirically-based models to predict the roadway performance for each
different road structure subjected to variable traffic loads.
To overcome the limitations of the empirical methods, mechanistic procedures
were introduced, allowing pavement engineers to undertake design at a site taking into
consideration loading, materials and environmental conditions, other than at the road test
sites. Such procedures provide the pavement response in terms of stress, strains, and
displacements in a multilayer pavement system when subjected to a vehicle load. In
1943, solutions for a two-layered system were developed by Burmister and then extended

25
to three-layered systems and then multiple-layered systems by 1945. 22
Burmister’s
theory can be applied to a system with any arbitrary number of layers with the help of
computers. This theory is the first analytical solution to calculate layered systems
response.
Phenomenological-empirical methods, developed in the 1950’s, are the basis for
most pavement design procedures that use the AASHTO method of axle load equivalency
factor. 23 This design method is used to convert mixed traffic axle loads into standard (18-
kip) equivalent single axle loads (ESALs) and is based on the total number of passes of
an equivalent axle load factor during the design period. 24
Many load equivalency factors
have been proposed in the past, based on pavement response to load (deflection, stress,
strain) or distress manifestations (rutting, fatigue, cracking), however, the AASHTO load
equivalency factor are the most widely used conversions in the world. Some researchers
and agencies advocate the mechanistic approach, in which LEFs are defined in terms of
pavement response to load such as stress or strain 25 . Others prefer empirical models (such
as AASHTO), in which LEFs are subjectively defined in terms of user acceptance of the
pavement ride quality and fatigue or distress manifestations (See Huhtala, 1992 and
Southgate 91).
In 1995, Hajek conducted a study that proposed a general axle load equivalency
factors – LEFs, that are independent of pavement variables. Such LEFs are valid based on
22 Burmister, D. M. (1945). "The General Theory of Stresses and Displacements in Layered Soil Systems,"
Journal of Applied Physics, Vol. 16, pp. 84-94, 126- 127,296-302.
23 Helwany, Sam, John Dyer Joe Leidy. “Finite-Element Analyses of Flexible Pavements.” Journal of
Transportation Engineering. Volume 124, Issue 5 (September/October 1998): 491-499.
24 AASHTO - American Association of State Highway and Transportation Officials. AASHTO Guide for
design of Pavements Structures. Washington, D.C. 1993.
25 See (Southgate 1993; Ioannides 1991; Ioannides 1993; Papagiannakis 1990)

26
the argument that uncertainties resulting from traffic volume variables and projections are
much larger than those inherent to LEF estimates.
The methods of equivalency factor can be empirical or mechanistic and although
mechanistic based, these methods used elastic material constitutive theory and assume no
induced damage or healing during loading and unloading. 26 Many recent pavement
analysis procedures such as the SHRP Superpave use the pavement response as an input.
In order to design a pavement based on a mechanistic approach it is necessary to
use a structural analysis tool or computer program to predict the stress-strain response.
Computer programs based on the multilayered elasticity method (MLE) have been
developed and are widely used for structural analysis of asphalt pavements because of
their simplicity, but they often simplify the problem, making it unrealistic. The success of
the mechanistic approach mainly depends on how realistically it can model the pavement
materials behavior and tire-pavement interaction loading. As a result of introducing more
realistic variables, such as inelastic material behavior, as well as complex geometries and
loading conditions, the analytical solution for this problem is extremely difficult if not
impossible. Thus, numerical techniques are needed to solve it. Currently, the most
commonly used technique is the Finite Element Method (FEM), which is a method for
solving partial differential equations governing the response of objects with complicated
geometries and boundary conditions. 27
Chen’s review and evaluation of five computer
programs, including finite element and MLE programs, sought to establish the most
appropriate one for pavement structural analysis. 28 He showed that the results from
26 See (Berthelot, C.F., 1998)
27 See (Allen, David H., and Walter Haisler, 1985).
28 Chen D.H., M. Zaman, J. Laguros, and A. Soltani. “Assessment of computer programs for analysis of
flexible pavement structure.” Transportation Research Record. TRB, 1482:123-133 (1995).

27
ABAQUS program, a commercial finite element modeling program that has been widely
applied for pavement analysis were comparable to those from other programs.
Programs like ELSYM5 and BISAR also make unrealistic assumptions; they treat
the tire interaction as static, uniform and stationary circular load. Papagiannakis 29 and
Sousa 30 attempted to model loadings more realistically; however, these models still suffer
limitations. Siddharthan proposed a continuum-based finite-layer model to evaluate
pavement response, incorporating factors such as noncircular contact area, viscoelastic
material characterization and vehicle speed. 31 They document important pavement
response parameters generated from the finite-layer analytical model, 3D-moving load
analysis, under a variety of loading conditions. The loading conditions include two types
of tires (conventional and wide base) and different contact stress distributions (uniform
and nonuniform). These studies also reveal that there is a significant difference between
the responses computed with uniform (conventional assumption) and nonuniform contact
tire-pavement stress distributions. Finite element analysis of pavements is a well-studied
topic and references can be found in a simple literature review. 32
For a model to be realistic it must consider all major effects that influence the
materials response. Additional effects can also contribute to the pavement performance
and may be used as inputs in the model, such as temperature, 33 or moisture and aging of
29 Papagiannakis, A.T., N. Amoah, R. Taha. “Formulation for Viscoelastic Response of Pavements under
Moving Dynamic Loads.” Journal of Transportation Engineering. Vol. 122, No. 2 (March/April 1996):
140-145.
30 Sousa, J. B, J. Lysmer, S. S. Chen, and C. L. Monismith. “Effects of Dynamic Loads on Performance of
Asphalt Concrete Pavements.” Transportation Research Record 1207. National Research Council. (1988).
31 Siddharthan, R.V., Krishnamenon, N., and Sebaaly, P.E. “Pavement Response Evaluation using Finite-
Layer Approach.” Transportation Research Record. No. 1709 (2000): 43-49; Siddharthan, Raj V.
“Pavement Strain from Moving Dynamic 3D Load Distribution.” Journal of Transportation Engineering.
Volume 124, Number 6 (November 1998).
32 See (Mun, 2003; wang, 2001; Helwany, 1998; Erkens, Liu and Scarpas, 2002)
33 See (Park, Sun Woo and Y. Richard Kim, 1998)

28
the asphalt layer. 34 The long term response of a pavement involves reduction in stiffness
of the asphalt material throughout the life of the pavement. Collop shows that an
exponential relationship exists between the reduction in asphalt layer stiffness and
cumulative fatigue damage. 35
In a separate study, Harvey addressed one aspect of the
performance of asphalt concrete pavements—rutting of the asphalt mix under high
temperature loading conditions for both highway and airfield pavements. 36
Erik showed
that the effect of climate on pavement structures is a major contributor to pavement
deterioration in cold regions, caused by seasonal freezing and thawing. 37
Although these studies have advanced the general knowledge of pavement design,
a more accurate characterization of paving material is essential in order to develop a
comprehensive model. A time-domain model for the viscoelastic pavement response
under dynamic load should be developed, taking into account the time-dependency of the
load-frequency and material behavior, including the stress-strain-time constitutive law.
By designing a roadway without time effects, the main evaluation mechanisms of
structural performance are disregarded and the energy dissipated due to loading and
unloading during the years of service will not be counted. In addition, dynamic vehiclepavement
interaction effects are well known to be significant for the analysis of dynamic
response of pavement subjected to moving loads. 38 Computing pavement response using
34 See (Krishnan, J. Murali, and K.R. Rajagopal, 2005)
35 Collop, A. C. “Stiffness Reductions of Flexible Pavements due to Cumulative Fatigue Damage.” Journal
of Transportation Engineering. Volume 122, Number 2 (March 1996): 131-139.
36 Harvey, J.T., T.P. Hoover, N.F. Coetzee and C.L. Monismith. "Rutting Evaluation of Asphalt Pavements
Using Full-Scale Accelerated Load and Laboratory Performance Tests," 2002 FAA Airport Technology
Transfer Conference. (May 2002).
37 Simonsen, Erik. “Prediction of Pavement Response during Freezing and Thawing Using Finite Element
Approach.” Journal of Cold Regions Engineering. Volume 11, Number 4 (December 1997): 308-324.;
Simonsen, Erik. “Prediction of Temperature and Moisture Changes in Pavement Structures.” Journal of
Cold Regions Engineering. Volume 11, Number 4 (December 1997): 291-307.
38 See (Wu, Chih-Ping, 1996).

29
dynamic models provides more accurate field measurements than those computed using
static models. 39
Research Objective
The objective of the present research project is to develop a model for predicting
mechanical behavior and performance life of pavements by focusing on tire
configurations and also including loading configurations, layer geometry, and material
properties. Based on the test results, the model is utilized to predict which pavement
configuration offers the greater life. These results play a fundamental role in pavement
design as they provide important information about pavement service life. For example,
by using a certain mixture A, pavement life will be reduced by X number of years,
compared against the use of a certain mixture B. It also can be used to compare pavement
life by applying a truck load with dual tires or a truck load with wide-base tires. In this
particular case, it can be shown that by using one specific set of tires in opposition to the
other, the pavement life will be reduced. Various types of results are possible with this
model that will be useful in selecting materials and designing pavement structures with
improved performance.
This model also employs nonlinear inelastic finite element methods to layered
structures of pavements. Computational results for various tire loading conditions are also
investigated to monitor the impact of loading conditions on pavement failure and life.
Furthermore, the influence of layer thickness of pavement structure and the variation of
pavement material properties are also evaluated.
39 See (Mamlouk, Michael S., 1997)

30
In order to fully develop the potential of this model, the next section will
formulate the statement of the problem. Then, all the variables involved in the analysis
will be described and the governing field equations will be stated, based on the
continuum mechanics approach. The 2D problem here is composed of a linear
viscoelastic layer (asphalt concrete) and three elastic layers (sub-grade, sub-base and
base), constituting the initial boundary value problem (IBVP). Subsequently, the finite
element formulation will be posed and finally the results from this analysis will be
shown.

31
CHAPTER 2
Solution Method
Complex geometry and mixed boundary conditions with heterogeneous and nonelastic
materials are normally present in roadway problems, making the solution
sometimes only feasible by numerical computation. The problem to be solved in this
research may be described as a linear, two-dimensional, quasistatic initial boundary value
problem (IBVP). It can be considered quasistatic since the cyclic motion of vehicles on
the roadway is not enough to reach the fundamental frequency of the pavement, or the
lowest frequency at which the pavement can vibrate. Also, even though we are treating
with a 2-D problem, a more general case of a three dimensional domain will be
presented.
In developing the solution technique, I will first describe the complete problem
statement with all the variables involved in the analysis and the governing field
equations. This formulation follows a continuum mechanics approach, where a body is
considered to be a continuum or without any microstructure when analyzed on a small
scale. A number of variables are known a priori while others need to be determined by
field equations, which are provided by kinetics, kinematics, and material constitutive
behavior. 40 A concise formulation is shown here, similar to that found in most continuum
mechanics textbooks. 41
40 Kinetics is a field of mechanics that studies loads on a body relating force and mass. Kinematics is the
field that considers motions of bodies without regard to force and mass. The material constitutive equations
are used to relate kinetics and kinematics and provide additional equations to complete the statement of a
well posed problem.
41 (Malvern, 1969).

32
Consider a general three dimensional body Ω and its boundary∂Ω as shown in
Figure 23. Consider a small volume ∆Ω from this body, where:
∂Ω
∪ ∂Ω =∂Ω
(2.1)
1 2
∂Ω1∩ ∂Ω
2
=∅
(2.2)
Boundary ∂Ω
1
is subjected to known tractions (natural boundary conditions) and
boundary ∂Ω 2 is subjected to known displacements (essential boundary conditions).
To solve the IBVP we need to derive the governing equations from the kinetics
equations (conservation of mass, conservation of linear and angular momentum),
kinematics equations (strain-displacement) and constitutive equations. Additional
constraints can be obtained from the boundary conditions and initial conditions. The
standard summation convention and range convention are observed in the equations
shown below.

33
Figure 23: General three dimensional body
The purpose of solving an IBVP is to know how the body will deform with a load
applied on it. These mechanical responses are given in terms of stresses, strains and
displacements at each point of the body. Those variables, known as state or field
variables, may be functions of space and time and are represented by the stress tensor
σ ( x
, t)
, strain tensor ε ij
( x, t)
and displacement vector u ( x, t)
. Twenty-one unknown
ij
i

34
variables are identified, consequently making necessary the same number of field
equations to solve the problem.
The global statement of conservation of linear momentum is given by three parts:
the rate of addition of momentum to region Ω , plus the forces acting on region Ω ,
equals the rate of change of momentum in region Ω . Or it can be expressed by:
∂
−∫ ρvv ( ⋅ ndS ) + ∫ tdS+ ∫ρ fdΩ= ∫ ( ρvd
) Ω
∂t
∂Ω ∂Ω Ω Ω
(2.3)
where v is the velocity vector, ρ is the density, n is the unit outer normal, t is traction
applied on surface and f is the body force. To obtain a local statement in index notation,
first dot with base vector e i
and apply the divergence theorem, which gives the local
form of the conservation of linear momentum:
dv
σ
ji,
j
ρ f ρ
dt
i
+
i
= (2.4)
where ρ is the mass, f
i
is the body force per unit area and vi
is the velocity vector and
d
dt
is relative to material derivative. It was noted before that the quasistatic conditions
apply in this research, meaning the cyclic loads are applied slowly compared to the
fundamental frequency of the pavement. As a result, the dynamic effects of the
acceleration are negligible and can be eliminated
σ + ρ f = (2.5)
ji, j i
0
Conservation of angular momentum must be satisfied at all points in a body or the
rate of change of angular momentum crossing boundary of region Ω plus moments
applied to region Ω must equal the rate of change of angular momentum in region Ω .

35
Satisfying the conservation of angular momentum assures equilibrium of the body. The
general form of the conservation of angular momentum is given by:
e
σ + ρm
= 0
(2.6)
ijk ij k
where eijk
is the permutation symbol and ρmk
is body moment. If no body moments are
assumed, the conservation of angular momentum reduces to:
σ = σ
(2.7)
ij
ji
stating that the stress tensor is symmetric, 42 reducing the number of unknown stresses
from nine to six.
Kinematics constraints are satisfied by the strain tensor and it can be expressed in
terms of the displacement field in a body. This is called the strain-displacement tensor
and is given by:
1
ε
ij
= ( ui, j
+ uj, i
− uk , iuk , j )
(2.8)
2
When the displacement is infinitesimal, or no large deformation is considered, the
term uki
,
uk,
jcan be neglected. As a result, the strain tensor becomes the infinitesimal
linear strain tensor:
1
ε
ij
= ( ui, j
+ uj,
i )
(2.9)
2
The infinitesimal tensor is symmetric by simple observation, reducing the number
of strain unknowns from nine to six.
A material is said to possess viscoelastic behavior when the stress is a single value
function of the entire history of the strain. Thus, mathematically,
42 (Malvern,1969)

36
τ
σ () t = σ ε( = t
τ )
(2.10)
{ }
τ =−∞
where braces denote history dependence.
Asphalt concrete is a material that exhibits viscoelastic mechanical behavior; thus,
the material constitutive equations are given for a general orthotropic, linear viscoelastic
material and can be expressed in hereditary integral form or convolution integral:
t
∂εkl
( xk
, τ)
σ ( x , t) = ∫ C ( t−τ)
dτ
∂τ
ij k ijkl
0
t
∂σkl
( xk
, τ)
ε ( x , t) = ∫ D ( t−τ)
dτ
∂τ
ij k ijkl
0
(2.11)
(2.12)
where:
C ( t− τ ) is the fourth order tensor of relaxation moduli, relating stress to strain,
ijkl
D ( t− τ ) is the fourth order tensor of creep compliances, relating strain to stress.
ijkl
Constraints imposed by boundary conditions can be applied on the body as natural
(Neumann) or essential (Dirichlet). The first case refers to traction applied on the
boundary and is given by:
t ( x, t ) = known on ∂Ω
i
t ( x, t)
= σ n
i ji j
1
(2.13)
(2.14)
where t i
is the traction per unit area on the surface and n
j
is the unit outer normal. The
second one is when displacement is applied. Essential boundary conditions are given by:
u ( x, t ) = known on ∂Ω
(2.15)
where ui
are displacements known a priori on the surface.
The initial conditions are:
i
2

37
σ
ij
( xt , ) = 0⎫ ⎪
εij
( xt , ) = 0 ⎬ ∀xinΩ+∂Ω
ui
( x, t) = 0 ⎪ ⎭
(2.16)
The above problem introduces all the unknown variables and equations, providing
the solution to the IBVP. Our goal is to find σ ( x
, t)
, u ( x, t)
and ε ij
( x, t)
. The equations
provided are given by Equation (2.5), Equation (2.9), and Equation (2.11) with
constraints imposed by boundary conditions in Equation (2.13) and Equation (2.15) and
initial conditions Equation (2.16).
ij
i

38
CHAPTER 3
Finite Element Formulation
After formulating a brief statement of the IBVP, it is necessary to develop the
method for obtaining a numerical solution.
Incrementalization of the Constitutive Equations
The constitutive equation for a linear viscoelastic material is shown in Equation
(2.11). This equation cannot be implemented into the finite element formulation in the
integral form; therefore, a numerical incrementalization is necessary. Equation (2.11) is
restated here:
t
∂εkl
( xk
, τ)
σ ( x , t) = ∫ C ( t−τ)
dτ
∂τ
ij k ijkl
0
(3.1)
where
Cijkl
is the fourth order tensor of relaxation moduli relating stress to strain. The
procedure described here was developed by Zocher, Groves and Allen, 1995 and it is
shown in a concise form. In fact, in the original method developed by Zocher, Groves and
Allen, thermoviscoelasticity was considered differently from this research where
temperature terms are not included.
They subdivided the timeline into discrete intervals, such that tn+ 1
= tn
+∆ t, where
t +
corresponds to current time plus the time step. Also, the stress at current time is
n 1
assumed to be known. The goal is to express stress at t
n + 1
in a way suitable for an
incremental finite element formulation. Rewriting Equation (3.1), representing the state
of stress at time t
n + 1
, becomes:

39
σ x t C t τ dτ C t τ dτ
tn
tn+
1
∂εkl
( x, τ) ∂εkl
( x, τ)
ij
( ,
n+ 1) = ∫ ijkl
(
n+ 1− ) +
ijkl
(
n+
1−
)
∂τ
∫
∂τ
0
tn
(3.2)
Let us define
∆ Cijkl
as follows:
∆C ≡C ( t −τ
) −C ( t − τ )
(3.3)
ijkl ijkl n+
1
ijkl n
Substituting Equation (3.3) in Equation (3.2), becomes:
∆ σ = − τ τ +∆σ
tn+
1
∂εkl
( x, τ)
R
ij ∫ Cijkl ( tn+
1
) d
ij
∂τ
tn
(3.4)
where
R
ij
t n
∫
∆ σ = ∆C
0
ijkl
∂εkl
( x, τ)
dτ
∂τ
(3.5)
And
∆σ
ji
is defined to be:
∆σ ≡σ − σ
(3.6)
t+∆t t
ji ji ji
Now, considering that each element of
Cijkl
can be represented by the Wiechert model:
Mijkl
ijkl
C () t = C +∑ C e ρ m
(3.7)
ijkl ijkl∞
ijklm
m=
1
−
t
where
η
ρ = (with no sum on i, j, k and l) and
ijklm
ijklm
C ijkl m
ηijkl m
are the dashpots coefficients
and
Cijkl
are the spring constants.
ρijkl m
is referred as the relaxation time.
Also, suppose ε ( t, τ ) can be approximated over the interval t ≤τ ≤t
+ 1
by:
kl
ε (, t τ) = ε + ε ι× ( t− t ) × H( t−t
)
(3.8)
kl kln
kl n n
where ε
kl
is a constant representing the time rate of change over the interval, Ht ( − t n
) is
the Heaviside step function and the symbol “×” represents a multiplication.
Substituting Equation (3.3) in Equation (3.2), results in:
n
n

40
∆ σ = C ∆ ε +∆ σ
(3.9)
'
R
ij ijkl kl ij
where,
C
Mijkl
' 1
ijkl
= Cijkl + η
∞
ijkl
t
m
m=
1
∆t
−
( ρijkl m )
∑ (3.10)
∆ 1−
e
∆ ε = ∆
(3.11)
kl
ε
kl
t
The final steps in the incrementalization of the constitutive equation involve writing an
expression for
where,
R
∆σ ij
in a convenient form. Only the final equations are shown here:
∆t
−
( ijkl m )
Mijkl
R
ij
Sijkl 1
m
tn
− e ρ
m=
1
∆ σ =−∑ ( )
(3.12)
ijkl m n ijkl m
0
t
tn
−τ
−
( ρijkl m )
n
∂εkl
( x, τ)
S ( t ) ≡ ∫ C dτ
(3.13)
e ∂τ
Now the constitutive equation can be implemented to an incremental formulation.
The approach shown here is a brief review; a more complete and detailed formulation is
detailed by Zocher et al. 1995.
Method of Weight Residuals
A finite element formulation can be found in open literature such as Yang, 1986;
Reddy, 1993; Hughes, 1987; Zienkiewicz, 1971. The methodology shown here is a
concise formulation and it is introduced by integrating the conservation of linear
momentum over the volume against a test function v
i
. The governing differential
equation treated herein is specified by Equation (2.5), transforming it to integral form
gives

41
∫ ( σ
ji,
j
+ ρ fi ) vdV
i
= 0
(3.14)
Ωi
where
Ω
i
is the sub domain and vi
is a weight function or any arbitrary admissible
displacement vector. Equation (3.14) can be rewritten as:
∫ ( σ
jivi ), j
− σ
jivi,
j
+ ρ fvdV
i i
= 0
(3.15)
Ωi
by applying the relations in the equation above
( σ v ) = σ v + σ v
ji i , j ji, j i ji i,
j
σ v = ( σ v ) −σ
v
ji, j i ji i , j ji i,
j
or
∫( σ v ) dV − ∫σ v dV + ∫ ρ f v dV = 0
(3.16)
ji i , j ji i,
j i i
Ωi Ωi Ωi
Now, applying Gauss divergence theorem, Equation (3.16) can be rewritten as:
∫ ∫ ∫ (3.17)
σ
jivn i jdV− σ
jivi , jdV+ ρ fvdV
i i
= 0
∂Ωi Ωi Ωi
or
∫σ v dV = ∫ σ vn dV + ∫ ρ fvdV
(3.18)
ji i,
j ji i j i i
Ωi ∂Ωi Ωi
where, n
j
is the unit outer normal. Inserting Cauchy’s Formula (Equation (2.14)),
Equation (3.18) can be rewritten as:
∫σ
v dV = ∫ tvdV+
∫ ρ fvdV
(3.19)
ji i,
j i i i i
Ωi ∂Ωi Ωi
Note:

42
1 1
σ
jivi , j
= σ
ji[ ( vi, j
+ vj, i
) + ( vi, j
−vj,
i
)]
2 2
σ v = σ [ ε + ϖ ]
ji i,
j ji ij ij
where ϖ
ij
is the linear rotation tensor and σ
jiϖ ij
is the product of a symmetric and skewsymmetric
tensors, which results in zero, so:
σ v
= σ ε
(3.20)
ji i,
j ji ij
Now inputting Equation (3.20) in Equation (3.19), gives:
∫σ ε dV = ∫ t v dS + ∫ ρ f v dV
(3.21)
ji ij i i i i
Ωi ∂Ωi Ωi
If we add the traction boundary conditions applied on boundary ∂Ω
2
, Equation (3.21)
becomes:
∫σ ε dV = ∫ t v dS + ∫ ρ f v dV
(3.22)
ji ij i i i i
Ω ∂Ω2
Ω
which is the principle of virtual work.
Up to this point, the variable time had not imposed influence on the development
and the procedure presented is similar for an elastic formulation problem. By introducing
time dependency behavior, Equation (3.22) takes the form:
∫σ ε dV = ∫ t v dS + ∫ ρ f v dV
(3.23)
t +∆ t t +∆ t t +∆ t t +∆ t t +∆ t t +∆ t
ji ij i i i i
Ωi ∂Ωi Ωi
where the superscript “t
+∆ t” means “at reduced time t+ ∆ t”. It is necessary to
incrementalize Equation (3.23), since the stress-strain relations (Equation (3.9)) are
incrementalized. Let us define the following relations:

43
∆σ ≡σ −σ
t+∆t t
ji ji ji
∆ε ≡ε −ε
t+∆t t
ij ij ij
∆u ≡u −u
t+∆t t
i i i
∆v ≡v −v
t+∆t t
i i i
(3.24)
or rearranging:
σ = σ +∆σ
t+∆t t
ji ji ji
ε = ε +∆ε
t+∆t t
ij ij ij
u = u +∆u
t+∆t t
i i i
v = v +∆v
t+∆t t
i i i
(3.25)
ε (t) and v ( t)
are zero, as a consequence of u ( t)
being known. Now substituting
ij
i
Equations (3.25) in Equation (3.23), yields:
or reorganizing terms:
i
t t+∆ t t+∆t
∫( σ
ji
+∆σ ji)
∆ εijdV = ∫ ti ∆ vidS + ∫ ρ fi ∆vidV
(3.26)
Ωi ∂Ωi Ωi
∫∆σ ∆ ε dV = ∫ t ∆ v dS + ∫ ρ f ∆v dV −∫ σ ∆ε
dV (3.27)
t+∆ t t+∆t t
ji ij i i i i ji ij
Ωi ∂Ωi Ωi Ωi
Introducing Equation (3.9) in Equation (3.27), yields:
∫ C dV ∫ t vdS ∫ f vdV ∫ dV (3.28)
'
R t t t t t
(
ijkl
∆ εkl +∆σij ) ∆ εij = +∆ i
∆
i
+ ρ +∆
i
∆
i
− σ
ji
∆εij
Ωi ∂Ωi Ωi Ωi
Rearranging Equation (3.27), it can be referred to as incremental variational initial
boundary value problem:
∫C ∆ε ∆ ε dV = ∫ t ∆ vdS+ ∫ ρ f ∆vdV−∫σ ∆ε dV−∫ ∆σ ∆ε
dV (3.29)
'
t +∆ t t +∆ t t R
ijkl kl ij i i i i ji ij ij ij
Ωi ∂Ωi Ωi Ωi Ωi
Notice that the term
∆εij
is a function of vi
and
∆ε
kl
is a function of u
i
.

44
Displacement Formulation
Let us now convert the problem to a matrix notation that is easily programmed.
Notice that this research is treating a plane strain problem, which considers the third
dimension infinitely long. In this notation we could write the following matrices:
The change in displacement vector:
⎡∆u⎤
[ ∆ u]
= ⎢
∆v
⎥
⎣ ⎦
(3.30)
The change in strain tensor:
⎡∆ε
⎤
xx
[ ε ]
⎢ ⎥
∆ = ⎢ ∆ε
yy ⎥
⎢ ε ⎥
⎣∆
xy ⎦
(3.31)
The change in stress during
∆ t :
⎡∆σ
⎤
xx
[ σ]
⎢ ⎥
∆ = ⎢ ∆σ
yy ⎥
⎢ σ ⎥
⎣∆
xy ⎦
(3.32)
The stress-strain relationship can be written for a general anisotropic material as:
R
⎧∆σ ⎫
xx ⎡C' 11
C' 12
C'
13 ⎤⎧∆ε ⎫ ⎧
xx
∆σ
⎫
xx
⎪ ⎪ R
σ
⎢
yy
C' 21
C' 22
C'
⎥⎪ ⎪ ⎪ ⎪
⎨∆ ⎬= ⎢
23
⎥⎨∆ εyy ⎬+ ⎨∆σ
yy ⎬
⎪ R
σ ⎪
xy
C' 31
C' 32
C'
⎪
33
γ ⎪ ⎪
xy
σ ⎪
⎩∆ ⎭
⎢⎣
⎥⎦⎩∆ ⎭ ⎩∆
xy ⎭
(3.33)
or in symbolic form:
R
{ σ} [ C ']{ ε} { σ }
The body forces (per unit mass) at reduced time t
∆ = ∆ + ∆ (3.34)
+ ∆ t
⎡ f
⎤
t+∆t
t+∆t
x
⎡⎣f
⎤ ⎦ = ⎢ t+∆t⎥
f
y
⎢⎣
⎥⎦
(3.35)

45
The surface tractions at reduced time t
+ ∆ t
t+∆t
[ ]
t
t+∆t
⎡t
⎤
x
= t+∆t
⎢⎣
ty
⎥⎦
(3.36)
The stresses at reduced time t
t
[ σ ]
t
⎡σ
⎤
xx
⎢ t ⎥
= ⎢ σ
yy ⎥
⎢
t
σ ⎥
⎣ xy ⎦
(3.37)
The terms defined in Equation (3.12)
R
[ ∆ σ ]
R
⎡∆σ
⎤
xx
⎢ R ⎥
= ⎢ ∆σ
yy ⎥
⎢
R
σ ⎥
⎣∆
xy ⎦
(3.38)
The term
∆ and term ∆ε
kl
can be written in matrix form:
εij
1
∆ ε ( [ ][ ]
ij
= ∆ vi , j
+∆vj,
i ) →∆ εij
= B ∆ v
(3.39)
2
1
∆ ε ( [ ][ ]
kl
= ∆ uk , l
+∆ul,
k ) →∆ εkl
= B ∆ u
(3.40)
2
The principle of virtual work is in Equation (3.29) may be written as:
∫∆ε C ∆ ε dV = ∫ ∆ vt dS+ ∫∆vρ f dV −∫∆ε σ dV −∫ ∆ε σ dV (3.41)
'
t+∆ t t+∆t t R
ij ijkl kl i i i i ij ji ij ij
Ωi ∂Ωi Ωi Ωi Ωi
The element matrices are derived for a triangular T3 element. Element T3 is
shown in Figure 24.

46
y
u
3
3
v
3
2
v
2
v
1
1
u
2
u
1
x
Figure 24: T3 triangular element representation
For a 3 node element, let:
∆uxyt
( , , )
⎡
⎢
i i
⎡ ⎤ i=
1
∆ u = ⎢ ⎥ =⎢ ⎥
3
3
∑
⎤
N ∆u () t ⎥
⎣∆vxyt
( , , ) ⎦ ⎢ ⎥
⎢∑
Ni∆vi()
t ⎥
⎣ i=
1 ⎦
(3.42)
where
Ni
are the shape functions. A linear displacement function of x and y is assumed
and the shape functions can be described as:
where:
Ni = ai + bx
i
+ ciy
(3.43)

47
1
ai = ( xjyk −xkyj
)
2A
1
bi = ( yj − yk
)
2A
1
ci = ( xk −xj
)
2A
(3.44)
where i, j, k are a permutation of 1, 2, 3. So, the displacements on the element can be
written as:
e
u uiNi ujN j
ukNk
∆ =∆ +∆ +∆
e
v vN
i i
vjN j
vkNk
∆ =∆ +∆ +∆
(3.45)
can be rewritten as:
[ ][ ]
e e e
∆ u = H ∆ U
(3.46)
where:
0 0 0
e ⎡N N N ⎤
H = ⎢
0 N1 0 N2 0 N ⎥
⎣
3⎦
[ ]
1 2 3
(3.47)
is the shape function matrix for the element, and:
e
[ ]
⎡∆u1
⎤
⎢
v
⎥
⎢
∆
1
⎥
⎢∆u
⎥
2
∆ U =⎢ ⎥
∆v2
⎢ ⎥
⎢∆u
⎥
3
⎢ ⎥
⎢⎣
∆v3
⎥⎦
(3.48)
is the nodal displacement matrix. Also,

48
⎡ ∂∆u
⎤
⎢ ⎥
⎡∆ε
⎤
⎢ ∂x
⎥
xx
⎢ ⎥ e e ⎢
[ ][ ]
∂∆v
⎥
∆ ε = ⎢∆ εyy
⎥ = B ∆ U = ⎢
=
∂y
⎥
⎢ γ ⎥
⎢ ⎥
⎣∆
xy ⎦ ⎢∂∆u
∂∆v⎥
⎢ + ⎥
⎣ ∂y
∂x
⎦
∆u
⎡∂N
1
1
∂N2
∂N
⎤⎡ ⎤
3
⎢ 0 0 0 ⎥⎢
x x x v
⎥
∂ ∂ ∂ ⎢ ∆
1
⎥
⎢ ⎥
⎢ ∂ N ∂ N ∂ N ⎥⎢∆
u ⎥
0 0 0
⎢ ∂ ∂ ∂
⎥⎢ ⎥
⎢∂N ∂N ∂N ∂N
∂N
∂N
⎥ ⎢ ∆u
⎥
1 2
3 2
= ⎢
⎥⎢ ⎥
y y y ∆ v2
1 1 2 2 3 3 3
⎢ ⎥⎢ ⎥
⎣ ∂y ∂x ∂y ∂x ∂y ∂x ⎦⎣⎢
∆v3
⎥⎦
(3.49)
or
⎡b 0 b 0 b 0⎤
B c c c
1 2 3
e
=
⎢
0
1
0
2
0
⎥
⎢
3 ⎥
⎢⎣
c1 b1 c2 b2 c3 b ⎥
3⎦
(3.50)
Equation (3.41) can be rewritten in the form:
∫
T T T T
e e e e e e
⎡
⎣∆U ⎤
⎦
⎡
⎣B ⎤
⎦
⎡
⎣ C'
⎤⎡
⎦⎣B ⎤⎡
⎦⎣∆ U ⎤
⎦dV = ⎡
⎣∆U ⎤
⎦
⎡
⎣H ⎤
⎦ tdS2
Ωi
∂Ωi
(3.51)
e
T
e
T
e
T
e
T
R e
T
e
T
t
+ ⎡
⎣∆U ⎤
⎦
⎡
⎣H ⎤
⎦ ρ fdV − ⎡
⎣∆U ⎤
⎦
⎡
⎣B ⎤
⎦
⎡
⎣σ ⎤
⎦dV − ⎡
⎣∆U ⎤
⎦
⎡
⎣B ⎤
⎦
⎡
⎣σ
⎤
⎦dV
∫ ∫ ∫
Ωi Ωi Ωi
∫
and the element stiffness matrix can be defined as:
e e
T
e
⎡
⎣k ⎤
⎦ = ∫
⎡
⎣B ⎤
⎦
⎡
⎣ C'
⎤⎡
⎦⎣B ⎤
⎦dV
(3.52)
Ωi
and the element force vector, which has contributions from specified displacement
boundary conditions, concentrated nodal forces, traction boundary conditions, body
forces, stresses at start of time step and change of stresses during time step, can be
defined as:

49
∫
e
e
T
1
=
2
∂Ωi
⎡
⎣f ⎤
⎦
⎡
⎣H ⎤
⎦
⎡
⎣ t ⎤
⎦dS
∫
e
e
T
f2
= H ρ f dV
Ωi
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
∫
e e
T
R
3
= σ
Ωi
⎡
⎣f ⎤
⎦
⎡
⎣B ⎤
⎦
⎡
⎣
⎤
⎦dV
(3.53)
∫
e e
T
t
4
= σ
Ωi
⎡
⎣f ⎤
⎦
⎡
⎣B ⎤
⎦
⎡
⎣
⎤
⎦dV
e
where ⎡
⎣f
⎤
1 ⎦ , e
⎡
⎣f
⎤
2 ⎦ , e
⎡
⎣f
⎤
3 ⎦ and e
⎡
⎣f
⎤
4 ⎦are element contributions to the force vector. Now
Equation (3.51) can be rewritten as:
⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
e e e e e e
⎣k ⎦⎣∆ U ⎦ = ⎣f1 ⎦+ ⎣f2 ⎦+ ⎣f3 ⎦+
⎣f4
⎦ (3.54)
And the summation of contributions from all elements yields,
[ K]{ ∆ U} = { F}
(3.55)
where [ K ] is the global stiffness matrix, { F } is the global force vector and { U}
change in the displacement vector during the time step.
∆ is the
Note that solving the linear system in Equation (3.55) results in the displacement
contribution. Thus, one can find the stresses and strains contributions by using Equation
(3.34) and Equation (3.49) respectively.

50
CHAPTER 4
Code Description and Verification
Code Description
Our main objective is to predict the maximum displacement when a load is
applied. To conduct the analysis, two codes were used: the first code was written by the
author, to create roadway meshes – Road Mesh, and the second code is the Structural
Analysis of Damage Induced Stresses in Thermo-Inelastic Composites – SADISTIC
(Allen, 1994), to perform finite element analysis. The mesh generator provides for the
roadway geometry, loads and material properties as an input, and the resulting code gives
you, as an output, the input file for the finite element code. Both codes were written in
FORTRAN, in addition, commercial software was used as a postprocessor to visualize
the meshes, both deformed and undeformed cases. The purpose of the roadway mesh
generator code is to perform discretization of the roadway domain into triangular subdomains,
and was developed in such a way that it can be divided in three parts: i) input
roadway geometry, loads and material properties; ii) output used as input for the finite
element code; and iii) output used as input for mesh visualization. The chart below shows
how the mesh generator works in combination with the finite element code.

51
Start
Input loads
Input roadway geometry
Input material properties
• Elastic
• Viscoelastic
• Roadway width
• Tire width
• Axle dimension
Number of divisions horizontally
• Under the truck
• On the shoulder
• Number of layers
• Layer thickness
Number of divisions vertically
• In each layer
Output
Create input file for finite
element code
Create input file for mesh
visualization
SADISTIC solves two and three dimensional thermomechanical continuum
problems using the finite element method. Linear thermoelastic (both isotropic and
orthotropic), elastic-plastic, thermoviscoplastic, and linear thermoviscoelastic are
included as material models. In addition, interface elements can be used to simulate crack
growth. The code has a broad variety of options that include: plane stress, plane strain,
generalized plane strain, and penalty constraints. 43 The combination of these two codes
provides the necessary tools to perform the pavement analyses.
43 A more detailed description of the program is beyond our interest here and can found in (ZOCHER,
1995).

52
Our roadway problem involves several different materials with a total of four
layers; one layer is treated as viscoelastic and the other three as elastic. Traction (natural)
boundary conditions and displacement (essential) boundary conditions were specified
throughout the domain Ω. The temperature was considered to be constant for the entire
analysis and was not taken into consideration. A triangular element was used in the finite
element discretization.
Code Verification
In this chapter the computer program SADISTIC is verified with several two
dimensional problems. Four problems were chosen to verify the code with the purpose of
investigating the response of several viscoelastic bodies to mechanical loadings. The
verification is obtained by solving simple problems with a known analytical solution and
comparing the solutions with numerical results from the code. The roadway is modeled as
a plane strain problem, since the third dimension is considered to be infinite. Thus, the
three problems solved are also plane strain problems. This procedure verifies the code
and confirms the code output is correct, while also ensuring that more complex cases are
accurate. The examples will be presented with two graphics each, the first one showing
loading and the second one detailing the viscoelastic response (displacement). All
examples are modeled using a hypothetical viscoelastic material. The uniaxial relaxation
modulus is given in the form of the standard linear solid (a 1-element Wiechert model)
with one term in the Prony series. Below, Figure 25 shows the Wiechert model.

53
E
1
E
2
E
n−1
E
n
E ∞
η
1
η
2
η
n−1
η
n
ε
σ
Figure 25: Wiechert model
The Wiechert model can have as many spring-dashpot Maxwell elements as are
needed to approximate the distribution satisfactorily, and are given by:
n
i
Et () = E +∑ Ee ρ
(4.1)
∞
i=
1
i
t
−
however, to make the verification simpler, only one Maxwell element was used. The total
stress σ transmitted by the model is the stress in the isolated spring (of stiffness E ∞
) plus
that in each of the Maxwell spring-dashpot arms. The uniaxial relaxation modulus is
given by:
t
−
1
=
∞
+
1
(4.2)
Et () E Ee ρ
In the examples that follow, values of 0.1 MPa and 0.4 MPa were assigned
to E ∞
and E 1
respectively: ρ
1
, which is the relaxation time, is 1.0, and the Poisson’s ratio

54
is 0.3. The material description of the Wiechert model (C ij in Voigt notation) is given in
Table 1.
ij
q
C
ij q
η
ij q
11,22 ∞ 0.1346150D+06
1 0.5384620D+06 0.5384620D+06
12,23 ∞ 0.1346150D+06
1 0.2307690D+06 0.2307690D+06
44,66 ∞ 0.3846150D+05
1 0.1538460D+06 0.1538460D+06
Table 1: Viscoelastic material properties
The first two solutions are obtained for a two dimensional rectangular bar with
length 2 m as shown in Figure 26. The bar has zero displacement in the x-direction at
x = 0 . Evenly distributed load is applied to the bar at x = 2 . The input to be considered is
the load P and the output is the time-dependent displacement ux= ( 2) . Since the stresses
and strains can be shown to be spatially homogeneous, the bar is modeled exactly with
two triangular elements.
Example 1. (Creep test) The bar is subjected to a tip load
P
0
= P H()
t in a test
called a creep test, where
0
P is the load P at time equals to zero and Ht () is the Heaviside
function. The exact analytical solution to the displacement for an elastic bar under plane
strain condition is:

55
2
−(1 −ν ) PL
ux ( = Lt , ) = (4.3)
EA
whereν is Poisson’s ratio, P is the load on the end of the bar, at ( x = 2 ), E is the elastic
modulus, L is the bar length and A is the cross sectional area of the bar. The bar has a
width b. Using the viscoelastic correspondence principle it is easy to show that the
solution to the viscoelastic problem is given by:
ν
ux ( = Lt , ) = ⎜ ⎟
( )
2
−(1 − ) P ⎛ L⎞
−
E Ee t ∞
+
1 ⎝ A⎠
(4.4)
The response of the bar from 0 to 40 seconds to a load of 2000 N is shown in
Figure 27. By using a time step ∆t
of 0.1 seconds, the finite element solution produces an
error considered small or hardly noticeable in the graphic.
tip load
Example 2. (Ramp creep test) In this example the uniaxial bar is subjected to the
P=
0
P tH t
()
The analytical solution for the viscoelastic problem can be shown to be
ν
ux ( = Lt , ) = ⎜ ⎟
( )
2
−(1 − ) Pt ⎛ L ⎞
−
E Ee t
∞
+
1 ⎝ A⎠
(4.5)
Figure 28 shows the result for this problem.

Figure 26: Rectangular bar subjected to a tip load
56

57
INPUT
4000
3000
Load P (N)
2000
1000
0
0 5 10 15 20 25 30 35 40
Time (s)
OUTPUT
Displacement u at x = 2 (m)
4.00E-02
3.00E-02
2.00E-02
1.00E-02
0.00E+00
0 5 10 15 20 25 30 35 40
Time (s)
Analytical
FEM
Figure 27: Loading history and response of example 1

58
INPUT
20000
16000
Load P (N)
12000
8000
4000
0
0 2 4 6 8 10
Time (s)
OUTPUT
Displacement (x = 2) (m)
0.25
0.2
0.15
0.1
0.05
0
0 2 4 6 8 10
Time (s)
FEM
Analytical
Figure 28: Loading history and response of example 2

59
Example 3. This example is a 2-D problem and treats a thick-wall viscoelastic
cylinder problem with an internal pressure p applied to the inner diameter, as shown in
Figure 29. Consider the internal pressure to be given by
P
0
= P H()
t , similar to the creep
test. The elasticity solution can be employed along with the viscoelastic correspondence
principle and the radial displacement is given by:
0 2
Pab(1 + ν)(1−2 ν)
⎛b r⎞
ur
(,) r t = D()
t
2 2 ⎜ − ⎟
a + (1−2 ν ) b ⎝r b⎠
(4.6)
Figure 29: Cylinder with Internal Pressure

60
For this problem, a = 2 m, b = 4 m, and
0
P = 100 Pa. Figure 29 shows the radial
displacement at x = 3.00 m up to 50 seconds. The mesh used here contains 120 elements
and is shown in Figure 30. Due to symmetry conditions only one quarter of the cylinder
needed to be modeled. The analytical and finite element solutions are presented in Figure
31 for the mid-thickness displacement.
Figure 30: Finite element mesh used in example 3

61
INPUT
200
Internal Pressure (Pa)
150
100
50
0
0 10 20 30 40 50
Time (s)
OUTPUT
5.0E-04
Displacement (r = 3) (m)
4.0E-04
3.0E-04
2.0E-04
1.0E-04
0.0E+00
0 10 20 30 40 50
Time (s)
FEM
ANALYTICAL
Figure 31: Loading history and response for example 3

62
Example 4: (Beam with tip load) This last example treats a 3-D cantilever beam shown
in Figure 32 has length L = 10m and cross-sectional area of 1 m 2 . The beam is subjected
to a load at x = 10 m
P=
PH
0
() t
Where P
0
is 1 Newton. We seek the displacement v at x = 10m. This problem has
spatially varying stress and strain as opposed to the first two bar problems.
The analytical solution is derived by applying the viscoelastic correspondence
principle to the elastic solution for the tip deflection from strength of materials. This
procedure results in:
PL
= (4.7)
3I
3
0
v [ Dt ( )]
where I is the area moment of inertia of the beam and it is 1/12.
The results are calculated from o to 10 seconds. Note that the results shown in
Figure 33 are not exact, but are a good approximation, since they produce a maximum
difference of 3% between the analytical solution and the finite element code. A time step
of 0.1 seconds was used for this example.

Figure 32: Beam problem of example 4
63

64
INPUT
2.0
1.5
Load P (N)
1.0
0.5
0.0
0 2 4 6 8 10
Time (s)
OUTPUT
Displacement (x = 10) (m)
0.04
0.03
0.02
0.01
0.00
0 2 4 6 8 10
Time (s)
FEM
ANALYTICAL
Figure 33: Loading history and response for example 4

65
CHAPTER 5
Asphalt Concrete Roadway Analysis
As described before, the purpose of this analysis is to show a comparison between
different configurations in a pavement structure. Correspondingly, in this section,
pavement geometry, pavement layer characterization, overestimation between 3-D and 2-
D, and mesh convergence are evaluated.
Roadway Geometry
Consider a typical two-lane asphalt roadway described in the Roadway Design
Manual of the Nebraska Department of Roads (shown in Figure 34) that contains a
symmetry line down the middle of it. For simplicity, roadway symmetry for only the right
half of the lane will be modeled (See Figure 35).The road is 12 meters wide with two
3.50 meters traffic lanes and two shoulders of 2.50 meters. Figure 36 shows the boundary
conditions of the pavement used in the model.
Figure 34: Typical two-lane roadway cross section
A pavement design must include means of surface drainage–removal of all water
present on the pavement surface, shoulder surface or any other surface from which it may

66
flow onto the pavement. This is important to ensure a high-quality, long-lasting
pavement. Moisture accumulation is always problematic in any pavement structural
layers, and if not systematically removed, can cause problems by weakening the
pavement structure. When rutting—the permanent deformation along the wheel path—is
present, the pavement becomes more vulnerable to pooling water on the surface. When
this happens, 90% of the pavement life is depleted. Some of the ways to prevent water
infiltration and accumulation include: impermeable HMA, roadside ditch and slope. The
only one considered in this analysis is the slope of the cross section that allows rainwater
to flow to the side of the pavement where it is collected in a roadside ditch.
Figure 35: Right half side to be modeled

67
Figure 36: Boundary conditions on the pavement
Typically, the travel lane has slopes to both shoulders usually at 2% grade while
paved shoulders have slopes of 4%. The side slope grade varies according to the
classification of the roadway. It can be 1:3, 1:4 or 1:6, depending on whether the roadway
is a rural state highway, a municipal state highway or an interstate. Figure 37 shows the
different possibilities of side slopes.
Figure 37: Side slopes of roadways (NDor, 2005)
Pavement Layer Characterization
The pavement in this research is composed of four layers. The top layer is the
asphalt concrete, which varies from 10cm to 20cm for this analysis. The subsequent

68
layers with the multiple elements involved in the hot mix asphalt are shown in Figure 38,
detailing the pavement structure and all layers. This figure shows the variety of
aggregates sizes in each layer and how the asphalt matrix works as a binder with the
aggregates embedded in the asphalt matrix.

Figure 38: Pavement structure and all layers
69

70
The asphalt concrete layer is the final layer to be built on top of the other layers.
Even though the pavement geometry can vary from case to case and require additional
layers if necessary, for our analysis, the selected pavement contains four layers: a 10 cm
HMA, a 40 cm granular base, a 30 cm granular sub-base and a in-situ sub-grade, which is
the graded natural terrain and has a depth of 1.10 m in the model. The asphaltic layer is
modeled as an isotropic viscoelastic medium defined by a generalized Maxwell model
and represented by:
n
i
Et () = E +∑ Ee ρ
(4.8)
∞
i=
1
i
t
−
The remaining layers are treated as linear elastic media. Table 2 and Table 3 list the
Prony series coefficients of relaxation modulus for the two different types of hot asphalt
mixtures: AC I and AC II. All HMA are asphalt concrete type and mixed with different
binders. The Prony series constants for AC I were obtained from the study of Mun (2003)
and AC II were obtained from Wen, 2001.
AC I
Poisson ration = 0.30 E i (kPa) ρ i (s)
∞ 5.83E+04 -
1 4.91E+09 2.00E-02
2 5.74E+09 2.00E-01
3 4.96E+09 2.00E+00
4 2.96E+09 2.00E+01
5 1.26E+09 2.00E+02
6 4.47E+08 2.00E+03
7 1.58E+05 2.00E+04
Table 2: Prony series coefficients of relaxation modulus for AC I

71
AC II
Poisson ration = 0.35 E i (kPa) ρ i (s)
∞ 9.90E+04 -
1 -7.50E+05 1.00E-05
2 2.17E+06 1.00E-04
3 -5.39E+06 1.00E-03
4 8.47E+06 1.00E-02
5 4.08E+06 1.00E-01
6 1.05E+06 1.00E+00
7 1.16E+05 1.00E+01
8 5.06E+04 1.00E +02
9 3.10E+03 1.00E+03
10 1.70E+03 1.00E+04
11 -2.00E+02 1.00E+05
Table 3: Prony series coefficients of relaxation modulus for AC II
Base Sub-base Sub-grade
E (MPa) ν E (MPa) ν E (MPa) ν
4000 0.35 800 0.35 200 0.35
Table 4: Material properties for the road elastic layers
The mixture AC I was obtained at temperature 5°C and the relaxation modulus data for
mixture AC II was obtained at temperature 20°C. Figure 39 shows a comparison between
AC I and AC II.

72
AC I
1.E-07 1.E-04 1.E-01 1.E+02 1.E+05 1.E+08 1.E+11 1.E+14
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+01
1.00E+00
Relaxation Modulus (MPa)
Time (s)
AC II
1.00E+05
1.00E+04
1.00E+03
1.00E+02
1.00E+01
1.00E+00
1.E-07 1.E-04 1.E-01 1.E+02 1.E+05 1.E+08 1.E+11 1.E+14
Relaxation Modulus (MPa)
Time (s)
Figure 39: Comparison between AC I and AC II

73
The loads applied to a pavement are regulated. Most US states have limitations
for the maximum weight per axle limit and overall weight. Nebraska Department of
Roads sets limits for 34,000 lb for tandem axles and a maximum gross-vehicle weight of
80,000 lb. The 34,000 lb in a tandem axle applies a load of 151kN on the roadway
surface, where a uniform distribution to each side of the axle is considered. The front,
steer axle has 12,000 lb and applies a load of 53.4kN. Figure 40 shows the load
distribution on each axle.
Figure 40: Weight distribution
In order to accommodate for these limitations, different tire configurations are
used. Conventional dual tires are being replaced by only one wide-base single tire. By
replacing both sides of dual tires by one wide result in a smaller total contact area with
the pavement. Since they are still carrying the same load as the dual tires, the pressure is
larger in the wide tire case. In addition to this, the axle distance is not treated as a variable
and it is fixed at 1.80 meters.
The analyses performed here include three different tire types with different
overall widths: 29.5 cm (295/80R22.5) for the dual configuration and 44.5 cm
(445/50R22.5) and 49.5 cm (495/45R22.5) for the wide base configuration. As mentioned

74
before, tire inflation pressures and contact areas are not treated as variables. The standard
inflation pressure was the same for the each tire at 100 psi. One other design variable
included in the analyses is the change in the asphaltic layer thickness, using values of 10,
15 and 20 cm. In addition to tire configuration and thickness of asphalt layer, changes in
hot mix asphalt were also evaluated. The goal is to compare the pavement service life for
one variable parameter.
The analyses are done in the following way. Three parameters are treated as
variables: tire width, type of hot mix asphalt and thickness of asphaltic layer. However,
only one parameter is varied each time and all others are held fixed. For example, if
different tire widths are the selected variable, then only one type of hot mix asphalt and
one thickness of asphaltic layer are selected. Afterwards, it is possible to predict which
one gives the best service life of a pavement. Similarly, by selecting a type of tire and a
thickness of asphaltic layer would be possible to conclude which one will give the best
service life.
Overestimation Between 3-D and 2-D
The analyses are run in a plane strain configuration, assuming the configurations
shown in Figure 34 are infinitely long in the third dimension. Therefore, in these types of
analyses, the loading is applied as a strip load in the third dimension. The 2D plane strain
analyses have advantages over a 3D analysis in that the computational time is
considerably reduced. By running a 2D problem and applying a strip of load in the third
dimension, an overestimation of load is induced. Since a tire applies a non-uniform load
to a pavement, the consideration of a uniform load is considered as an approximation.

75
Figure 41 shows a real pressure distribution on a tire, showing pressure is higher in the
center and lower in the corners.
Figure 41: Tire pressure distribution (Blackman, Halliday, and Merrill, 2000)
A reasonable 3-D approximation for this pressure would be a sinusoidal curve in
both directions. By doing a 2-D analysis, only one dimension can be treated as a
sinusoidal curve, since there is no variation on the third dimension. Figure 42 shows a 3-
D pavement with a dual tire configuration load, and Figure 43 shows a 2-D with the same
load. The comparison between the two types of loads is shown in Figure 44.

76
Figure 42: Pavement 3-D with forces
Figure 43: Pavement 2-D with forces

77
Figure 44: Comparison between 3D and 2D loads
To solve this problem, a factor showing the ration between 3D and 2D has to be
found. This factor is then multiplied by the maximum displacements in the 2D to correct
and make a more realistic prediction. Two analyses are necessary to find this factor. The
first analysis is a 3-D finite element mesh with only one tire load and elastic materials as
shown in Figure 42. The other analysis shows a 2-D mesh with the same material
properties, loads and geometry. Only one time step is necessary. A maximum
displacement of 4.46E-01 m was the result for the 3-D analysis and 6.34E-01 was the
maximum displacement for the 2-D analysis. With these numbers, it is shown that the
displacements are overestimated by a factor of 1.42x and, as a result, the displacements
used in the analysis are corrected by being divided by this factor.
For this analysis, the response of a cyclic load imposed by the truck on the
pavement is simulated. The truck applies a static load of 151kN with a tandem axle. A 60
feet truck traveling at 70 mph takes less than one second to pass through a fixed point in
the pavement. However, for simplicity, one second of simulation was considered to
simulate the passage of all five axles, being 0.2 seconds the time that each axle loads the

78
pavement. Ramp functions for axle loading and unloading were implemented in the code,
with a period of 0.2 seconds for each axle passing or 0.1 seconds for loading and 0.1
seconds for unloading and a peak of 18,900kPa. The truck takes a total five cycles of 0.2
seconds or 1 second for passing all five axles followed by a 19 second interval of rest,
which is the time until the next truck passes, totaling a period of 20 seconds. Three trucks
per minute or 180 trucks per hour are considered. Figure 45 shows the five axle load and
a rest period up to 5 seconds, although the rest period still goes up to 20 seconds. For
scale reasons it would be hard to see the five cycles with a larger scale. Figure 46 shows
the first 3 cycles of trucks passing over the pavement.
0.0
0 1 2 3 4 5
-4.0
Load (KPa)
-8.0
-12.0
-16.0
-20.0
Time (s)
Figure 45: First cyclic load – axle loads

79
0.0
0 10 20 30 40 50 60
-4.0
Load (KPa)
-8.0
-12.0
-16.0
-20.0
Time (s)
Figure 46: Cyclic load
The load is applied on the pavement by increments of time and this requires an
analysis of the minimum time step that will produce accurate results. This can be
accomplished by simply running the code for a few cycles of loading and increasing the
time step for each run until the results start to change with the time step, Giving the
minimum allowable time step. After running the code with different time steps, making
them smaller and smaller, the minimum allowable time step is found to be 0.125 seconds
for the loading part and 1.0 second for the rest part.
Mesh Convergence
The process of discretizing the roadway into elements is known as mesh
generation. Fine meshes have the disadvantage of increasing the number of equations that
must be subsequently solved. However, choosing a relatively coarse mesh will result in
an inaccurate numerical solution. Therefore, in order to decide the size of the triangular
elements, an analysis of mesh convergence must be conducted.

80
The first step is to create a mesh using the fewest, reasonable number of elements
and analyze the model. Then, recreate the mesh with a denser element distribution, reanalyze
it and compare the results to those of the previous mesh. The mesh density is
increased and the model is re-analyzed until the results converge satisfactorily. One
method of establishing mesh convergence requires a curve of a result parameter (in this
case maximum displacement on the surface was used) to be plotted against some measure
of mesh density (number of elements was used). To determine when results have
converged satisfactorily and accurately, we used at least three runs to plot a curve which
can then be used to indicate when convergence is achieved or how far away the most
refined mesh is from full convergence. Notice that each tire used in the analysis
employed a different mesh and, consequently, a mesh convergence study had to be
performed for each of them.
The first step was to keep all layers with the minimum reasonable number of
elements. The subdivision was performed at each layer. A total of 5 steps were necessary
to obtain convergence with the mesh. The first one was to vertically subdivide the first
layer. When convergence was achieved, the first layer was then horizontally subdivided
completing the second step. Now, the first layer has the adequate number of elements, so
no further divisions were necessary in this layer. After this, only horizontal subdivisions
were necessary, since they share the same vertical lines. The third, fourth and fifth steps
were accomplished by keeping fixed the previous layers that already achieved
convergence and varying the horizontal divisions in each corresponding one. Figure 47
shows the mesh convergence for dual tire configuration. The final number of elements is
1423. Figure 48 shows the final mesh.

81
29.5 cm tire
1.38E-04
Maximum Displacement (m)
1.37E-04
1.37E-04
1.36E-04
1.35E-04
1.34E-04
1100 1150 1200 1250 1300 1350 1400 1450 1500 1550
Number of Elements
Figure 47: Convergence of maximum displacement for 29.5 tire
Figure 48: Mesh for the dual configuration tire – 29.5 cm
Figure 49 shows the mesh convergence for the 44.5 cm wide tire and Figure 50 shows its
respective mesh with 920 elements.

82
44.5 cm tire
1.32E-04
Maximum Displacement (m)
1.31E-04
1.31E-04
1.30E-04
1.29E-04
1.28E-04
750 800 850 900 950 1000 1050 1100
Number of Elements
Figure 49: Convergence of maximum displacement for 44.5 tire
Figure 50: Mesh for the wide configuration tire – 44.5 cm
Figure 51 shows the mesh convergence for the 49.5 cm wide tire and Figure 52 its
respective mesh with 901 elements. So, for each tire configuration used, a different mesh
must be selected.

83
49.5 cm tire
1.34E-04
Maximum Displacement (m)
1.33E-04
1.32E-04
1.31E-04
1.31E-04
1.30E-04
700 750 800 850 900 950 1000
Number of Elements
Figure 51: Convergence of maximum displacement for 49.5 tire
Figure 52: Mesh for the wide configuration tire – 49.5 cm
After the mesh convergence study was performed, another convergence had to be
done. A real roadway does not have a side slope like the one shown on Figure 52, but a
smoother one. As mentioned before, the side slope varies according to the classification
of the roadway. It can be 1:3, 1:4 or 1:6, all depending if the roadway is a rural state

84
highway, a municipal state highway or an interstate. As a result, different meshes with
the different side slopes had to be created in order to verify the accuracy of each one.
Each mesh was treated with the same materials of Table 4 and the HMA layer was also
treated as an elastic material, for simplicity purposes, with Young’s modulus of
E = 8000MPa
. Also, one tire configuration of 44.5 cm width was considered. The
purpose of this study is to show that a mesh with a side slope 1:6 does not produce a
considerable difference in the results compared to a mesh with side slope 0.5:1. Four
meshes were created, with side slopes 1:6, 1:3, 1:1 and 0.5:1 and are shown below. Note
that the pavement shown in Figure 56 is the one used in all analyses.
Figure 53: Pavement with side slope 1:6
Figure 54: Pavement with side slope 1:3
Figure 55: Pavement with side slope 1:1
Figure 56: Pavement with side slope 0.5:1

85
Figure 57 shows a graphic with the displacement on the surface for each side slope. The
analysis shows that the maximum difference between displacements in similar locations
is less than 0.1%, which gives support to use the mesh presented in Figure 50.
Different Slopes - Displacement on the surface
Displacements (m)
-5.0E-05
-7.0E-05
-9.0E-05
-1.1E-04
-1.3E-04
-1.5E-04
0 10 20 30 40 50
Node #
0.5:1 1:01 1:03 1:06
Figure 57: Displacement on the surface for different side slopes
Results
The results of this analysis and the value of this research project provide a new
potentially more reliable way to predict the life of asphalt pavements subjected to cyclic
loads, involving different variables, such as, tire widths, hot mix asphalt and thicknesses
of asphalt layer. What makes this analysis unique is the possibility to predict which
combination of design variables gives the best service life. One limitation to this analysis
is that no damage is considered. Since no damage is taken in consideration, an empirical

86
but physically meaningful criterion is adopted. In addition, the actual number obtained as
a result from the finite element code does not represent the real life of a pavement.
We start the analysis by defining the failure criterion used. The failure criterion is
not the main thrust of this research and we have chosen a simple one in order to
demonstrate the technique. A more accurate failure criterion will be a subject to be
incorporated as a future work. All pavements involved have the same base and sub-base
thicknesses, the only possibility of variation is the top layer composed by the HMA.
However, the overall thickness remains the same by changing the subgrade thickness.
The pavement is said to fail when the difference between the asphalt layer thickness and
the maximum displacement on the surface becomes less than the average size of the
aggregate in the mixture. Figure 58 shows two simple cases with different asphalt layer
thicknesses to explain the failure criterion. Consider the roadway in case b with thickness
b
h greater than the roadway in case a with thickness
a
h . Note that at time t = 0 a force
F( x, t)
is applied to the surface in both cases and the displacement are measured at
b
a
time = t 1
. In case b, the displacement ( δ ) is greater than in case a ( δ ), but
where,
Also, when
h
−δ
≤h
− δ
b b a a
( h −δ
or h −δ
) ≤ D
a a b b MAX
MAX
D is the maximum size of the aggregate in the mixture, which in this case is
considered to be 2 cm, the pavement is considered to fail. Note also that the picture is
exaggerating the displacement, however δ is not greater than the thickness ( h ) of the
roadway.

87
Figure 58: Failure criterion
This computer simulation requires an extraordinary amount of computer
processing and would make the work practically unachievable to conduct over a longer
period of time, such as one full year. Since our prediction does not involve damage, we
show that it is possible to extrapolate the results after a certain number of cycles. This can
be done by 1) running the problem up to 3000 seconds or 150 cycles of 20 seconds,
instead of the full year, and 2) adding a trend line to the data. Figure 59 and Figure 60
clarify this idea, by showing the graphic with the maximum displacement on the surface
versus time and followed by the same graphic in log scale, making it easier to see the
tendency.

88
0.00018
0.00016
Maximum Displacement (m
0.00014
0.00012
0.0001
0.00008
0.00006
0.00004
0.00002
0
0 500 1000 1500 2000 2500
Time (s)
Figure 59: Displacement versus time
Maximum Displacement (m)
0.01 0.1 1 10 100 1000 10000
0.00018
0.00016
0.00014
0.00012
0.0001
0.00008
0.00006
0.00004
0.00002
0
Time (s)
Trend line
Figure 60: Displacement versus time in log scale
As it can be seen in both graphics, the displacements are varying in a cyclic way
according to the load. The top part reflects the pavement being loaded and the lowest
part where the trend line is positioned corresponds to the pavement being unloaded. The

89
trend line is placed where a permanent deformation is observed. Notice in Figure 60 that
after around 500 seconds a linear approximation makes an appropriate choice for the
extrapolation. This choice is based on the R 2 value for each graphic. A trendline is most
reliable when its R 2 value is at or near 1. 44
The first analysis is done by keeping the HMA thickness in 10 cm and two types
of mixtures. The design variable to be compared in the next three graphics is the different
tire widths.
HMA Thickness 10 cm
250
211.4
Time (year)
200
150
100
50
34.8
26.2
140.9
28.8
158.5
0
29.5 cm 44.5 cm 49.5 cm
Tire Width
AC I (See Table 2) AC II (See Table 3)
Figure 61: Different tire configurations for HMA thickness of 10cm
44 R 2 value is a number from 0 to 1 that reveals how closely the estimated values for the trendline
correspond to your actual data. A trendline is most reliable when its R 2 value is at or near 1. It is also
known as the coefficient of determination.

90
HMA Thickness 15 cm
Time (year)
300
250
200
150
100
50
0
257.6
187.1
164.9
40.8
29.2
32.0
29.5 cm 44.5 cm 49.5 cm
Tire Width
AC I (See Table 2) AC II (See Table 3)
Figure 62: Different tire configurations for HMA thickness of 15cm
HMA Thickness 20 cm
Time (year)
350
300
250
200
150
100
50
0
300.4
190.3
203.8
46.0
31.9
34.6
29.5 cm 44.5 cm 49.5 cm
Tire Width
AC I (See Table 2) AC II (See Table 3)
Figure 63: Different tire configurations for HMA thickness of 20cm

91
As it can be seem from Figure 61, Figure 62 and Figure 63 a comparison between
tires widths is presented and the tire width 29.5 cm gives the best pavement life for all the
thicknesses and the two mixtures. The wide tires 44.5 cm always gives the least life, and
49.5 cm is always intermediate. The 29.5 cm tire width gives a pavement life
approximately 50% better than if the wide tire 44.5 cm is used and around 40% better life
if the comparison is between the dual tire 29.5 cm and wide tire 49.5 cm. If the two wide
tires are compared, the wide tire 49.5 cm gives a better life of around 10% more than the
44.5 cm wide tire. It is a good point to notice that among the wide tires, the wider the tire,
the better the pavement life. Note also, if you compare in one graphic, the AC II gives a
better life in all cases compared to AC I, as a result of being a stiffer mixture.
Tire Width 29.5 cm
Time (year)
350
300
250
200
150
100
50
0
300.4
257.6
211.4
34.8
40.8
46.0
10 cm 15 cm 20 cm
HMA Thickness
AC I (See Table 2) AC II (See Table 3)
Figure 64: Different HMA thicknesses for dual tires 29.5 cm
Now we turn our attention to comparing life for different thicknesses of asphalt
layer. Three thicknesses are presented in each graphic for two different mixtures. Three

92
graphics are presented, each one with applied from a different tire configuration. By
looking at Figure 64 with dual tire configurations, it can be seem that by increasing the
thickness of the asphalt layer, the life increases for both mixtures. If a 15 cm thick layer
is used as opposed to a 10 cm, the life goes up by 22% for AC II and 18% if AC I is used.
Figure 65 shows if a 44.5 cm wide tire is used and a 15 cm is used, instead of a 10 cm,
the life goes up about 17% for AC II and 12% for AC I. On the other hand if a 49.5 cm is
used it only increases life about 14% for AC II and 11% for AC I.
If the thickness is increases from 10 cm to 20 cm, the life increases for all cases
and mixtures. In the case of AC II it gives a better life of 42% for dual tires 29.5 cm; 35%
for 44.5 cm wide tires and 29% for 49.5 cm wide tires. Note that the dual tires always
give the better increase of life if the thickness is increased.
Overall, dual tire configurations with 29.5 cm give the best pavement life in terms
of all parameter and 44.5 cm gives the shortest pavement life. A 10 cm thickness of
asphalt layer gives the shortest life for all types of tire configurations and a 20 cm layer
gives the best life. AC II gives the best life as opposed to AC I for all cases. This is a
strong tool to be improved and used in pavement design, since it gives a reasonable
comparison of how the pavement response will over the years and how to helps to make a
decision of which geometry and materials use.

93
Tire Width 44.5 cm
200
164.9
190.3
Time (year)
150
100
50
26.2
140.9
29.2
31.9
0
10 cm 15 cm 20 cm
HMA Thickness
AC I (See Table 2) AC II (See Table 3)
Figure 65: Different HMA thicknesses for dual tires 44.5 cm
Tire Width 49.5 cm
250
200
158.5
180.7
203.8
Time (year)
150
100
50
28.8
32.0
34.6
0
10 cm 15 cm 20 cm
HMA Thickness
AC I (See Table 2) AC II (See Table 3)
Figure 66: Different HMA thicknesses for dual tires 49.5 cm

94
CONCLUSION
A finite element viscoelastic model has been developed for the purpose of
analyzing service life of asphalt pavements subjected to different tire configurations,
layer geometries and material properties. The primary motivations for this study is to
model asphalt pavements and predict which combination of design variables increases
and improves pavement life. The model gives a comparison of predicted lives for various
roadway and tire configurations.
The model compares the relative effect of wide base single tires and dual
assemblies for equal inflation pressures and equal loads. Since this new generation of
tires is already approaching the US market and is intended to replace the dual tire
configuration for a larger tire, the likelihood of damage to U.S. roadways must be
evaluated. This technique is very powerful for designing pavement, allowing engineers to
choose the best hot mix asphalt to be used. Also in pavement design, the model is able to
predict how much longer the roadway will last if a thicker layer of hot mix asphalt is
chosen.
As expected, dual tire configurations give the best life if compared to the other
two wide tire configurations for all mixtures and for all thicknesses. Additionally, for the
wide tire case, as tire size increases, the damage potential from tandem axles decreases.
By analyzing pavement response to different thicknesses, it is observed that the thicker
asphalt layer of 20 cm gives the best life for all mixtures and tire configurations
compared to the thinner layers of 10cm and 15 cm. Another advantage of this model is
the capability of minimizing the need of extensively laboratory work, since the
predictions rely upon the fundamental structural properties of each layer. However,

95
because the current generation of the model does not provide any source of energy
dissipation in the form of cracks, it has certain limitations that are left as future work.

96
FUTURE WORK
It has been shown that computational modeling for pavements has great potential and can
provide substantial benefits; however, many possibilities exist to continue to improve this
research, such as:
i. Extend the analysis capabilities to three dimensions. Even though the
computational effort will be much greater, the results will be more accurate;
ii.
Incorporate and model damage, in the form of discrete cracks, at any length scale
by incorporating a type of fracture mechanics model to the analysis;
iii.
Introduce more realistic tire inflation pressures and contact areas and their effect
on wide base single tires and dual tire configurations;
iv.
Analyze the effect on pavement life of possible under- and over- inflation in tires;
v. Perform a qualitative field validation, to compare life of wide tires against dual
tires;
vi.
Incorporate the effect of dynamic loading in the code.

97
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