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dissertation entitled

DYNAMIC ANALYSIS OF MULTI—LAYERED PAVEMENT
SYSTEMS UNDER MOVING ARBITRARY LOADS

presented by
Kyong Ku Yun

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DYNAMIC ANALYSIS OF MULTI-LAYERED PAVEMENT
SYSTEMS UNDER MOVING ARBITRARY LOADS

By

Kyong Ku Yun

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1 995

ABSTRACT

DYNAMIC ANALYSIS OF MULTI-LAYERED PAVEMENT
SYSTEMS UNDER MOVING ARBITRARY LOADS

By

Kyong Ku Yun

A new solution for the dynamic analysis of multi-layered asphalt concrete pave-
ment systems under moving arbitrary loads has been developed. The method builds on an
existing model for stationary loads, SAPSI, which uses the complex response method of
transient analysis with a continuum solution in the horizontal direction and a ﬁnite-ele-
ment solution in the vertical direction. The structural model is a n-layered damped—elastic
medium. The subgrade can be modeled as either a rigid base or a semi-inﬁnite halfspace.
The loads are surface circular loads which may be constant in magnitude or may vibrate as
an arbitrary time function, and the analysis is under axisymmetric conditions. The moving
loads are modeled as a series of pulses with a duration equal to the time required for the

wheel to pass by a ﬁxed point in the pavement.

The proposed method of analysis in the new version of the program, SAPSI-M, is
an improvement over the existing methods because: (1) it treats moving arbitrary loads on
a n-layered damped-elastic system; (2) it incorporates such important factors as wave

propagation, inertia and damping effects of the medium as well as frequency-dependent

asphalt concrete properties; (3) it can handle any load conﬁgurations, thus making it possi-

ble to model multiple wheel conﬁgurations of truck axles and airplane landing gears.

The moving load solution of the developed program SAPSI-M has been validated
with the PACCAR full-scale ﬁeld truck tests, which was conducted on an instrumented
asphalt concrete section on a test track at the PACCAR Technical Center in Mount Vernon,
Washington. The transient analysis of the original SAPSI program has been validated
using the PACCAR’s results for Falling Weight Deﬂectometer (FWD) tests. Further, an
approach based on energy dissipation and visco-elastic properties of an asphalt concrete
pavement for predicting the fatigue life of asphalt concrete pavements has been presented
using the moving load solution in SAPSI-M computer program. The effect of truck speed
on the response of asphalt concrete pavements is signiﬁcant, and the frequency—dependent
properties of the.asphalt are a major contributor to the speed effect. A simpliﬁed solution
where the moving loads are assumed to be non-vibrating seems to be sufﬁcient for accu-
rately predicting the pavement response. Static analysis using statically back-calculated

layer moduli seems to be sufﬁcient in analyzing FWD ﬁeld tests.

Dedicated to

the memory of my father

iv

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Dr. Karim Chatti for providing the

research topic to me, for his guidance, critique and discussion during this research project.

I also wish to thank Dr. Ronald Harichandran for being a Ph.D. committee member
and my academic advisor in the MS. program. Appreciation is extended to Dr. Gilbert

Baladi and Dr. Harvey Davis for being on my Ph.D. guidance conunittee.

Financial support for this research was provided by the Great Lakes Center for
Truck and Transit Research and the Research Excellence Fund of the State of Michigan.

Their support is gratefully acknowledged.

An expression of immeasurable gratitude is given to my mother, parents—in-law,
brother and sisters, brothers and sisters-in—law for their understanding, encouragement and

support through the years of my studies.

Finally, but not least, I wish to express my sincere appreciation to my wife, Jung,
for her continuous understanding, patience and love, and to my son, Byong-Wook, for

having been a source of joy.

TABLE OF CONTENTS

 

LIST OF TABLES ................................................................................................. xi
LIST OF FIGURES ............................................................................................. xii
CHAPTER 1. INTRODUCTION .......................................................................... 1
1.1 Problem Statement .................................................................... 1

1.2 Dynamics of Pavements ........................................................... 2

1.3 Research Objectives and Organization ..................................... 3

1.3.1 Objectives ............................................................................. 3

1.3.2 The Proposed Model ............................................................. 3

1.3.3 Organization ......................................................................... 4

CHAPTER 2. LITERATURE REVIEW--.-.-- - - -- - -- “8
2.1 Introduction .............................................................................. 8

2.2 Closed-Form Solutions ............................................................. 9

2.2.1 Static Analysis .................................................................... 10

2.2.2 Dynamic Analysis ............................................................... 10

2.2.3 Moving Load Solutions ...................................................... 11

2.3 Numerical Solutions ............................................................... 13

2.3.1 Layer Theory ...................................................................... 13

2.3.1.1 Static Analysis Models ......................................... 14

2.3.1.2 Dynamic Analysis Models ................................... 16

2.3.2 Finite Element Method ....................................................... 19

2.4 Summary ................................................................................. 21

vi

CHAPTER 3.

CHAPTER 4.

RESPONSE OF VISCO-ELASTIC MULTI-LAYERED
SYSTEM TO STATIONARY DYNAMIC LOADS ................... 23
3.1 Introduction ............................................................................ 23
3.2 Equation of Motion ................................................................. 23
3.3 Multi-Layered Model ............................................................. 24
3.4 Simulation of Half Space ........................................................ 26
3.5 Complex Modulus Representation ......................................... 29
3.6 Method of Analysis ................................................................ 33
3.6.1 Complex Response Method ................................................ 33
3.6.2 Dynamic Analysis ............................................................... 33
3.7 Steady-State Solution ............................................................. 34
3.7.1 Introduction ........................................................................ 34
3.7.2 Stiffness Matrix Approach .................................................. 35
3.7.3 Layer Stiffness Matrix ........................................................ 38
3.7.4 Eigenvalue Problem ............................................................ 39
3.7.5 Green’s Functions ............................................................... 41
3.7.6 Validation of Steady State Solution .................................... 46
3.8 Transient Solution ................................................................... 47
3.8.1 Introduction ........................................................................ 47
3.8.2 Interpolation Scheme .......................................................... 48
3.8.3 Algorithm and Validation of Transient Analysis ................ 50
3.9 SAPS] Computer Program ..................................................... 51
DYNAMIC RESPONSE OF VISCO-ELASTIC MULTI-
LAYERED SYSTEM TO MOVING ARBITRARY LOADS...54
4.1 Introduction ............................................................................ 54
4.2 Assumptions ........................................................................... 55
4.2.1 Material Properties ............................................................. 55
4.2.2 Geometry ............................................................................ 56
4.2.3 Loads .................................................................................. 57
4.3 Moving Load Representation ................................................. 57

vii

4.3.1 Response to Moving Loads ................................................ 58

4.3.2 Response to Moving Constant Loads ................................. 62

4.3.3 Response to Moving Arbitrary Loads ................................. 64

4.4 SAPSI-M Computer Program ................................................. 65
4.4.1 Program Layout .................................................................. 65

4.4.2 Algorithm ............................................................................ 70
4.4.2.1 Steady-State Solution ........................................... 70

4.4.2.2 Moving Load Solution ......................................... 72

4.4.3 Uses and Limitations .......................................................... 75
4.4.3.1 Capabilities ........................................................... 75

4.4.3.2 Limitations ........................................................... 75
CHAPTER 5. FIELD VERIFICATION.... .................................................... .....77
5.1 Introduction ............................................................................ 77
5.2 Previous Full-Scale Field Test Studies ................................... 77
5.2.1 Test Facilities ...................................................................... 77

5.2.2 Previous Flexible Pavement Tests ....................................... 78

5.3 Description of PACCAR Field Test ........................................ 84
5.3.1 Test Section ......................................................................... 84

5.3.2 Instrumentation ................................................................... 84

5.3.3 Test Procedure .................................................................... 86

5.3.4 Load Measurements ............................................................ 89

5.3.5 Strain Measurements .......................................................... 89

5.3.6 Correcting for the Effect of Lateral Offset ......................... 93

5.4 Material Characterization of Cores ........................................ 94
5.4.1 Pavement Layer Characterization ....................................... 94

5.4.2 Back-calculation of Layer Moduli ...................................... 94

5.4.3 Frequency-dependent AC Properties .................................. 97

5.5 Veriﬁcation of Moving Constant Load Solution ..................... 99
5.5.1 Modeling of the Moving Wheel Load ................................ 99

5.5.2 Modeling of Pavement Cores ............................................. 99

5.5.3 Results of SAPSI-M Analysis .......................................... 101

viii

5.5.3.1 Typical Response Curves of Moving Constant

Loads ................................................................. 101

5.5.3.2 The Effect of Frequency-Dependent AC Properties

......................................................................... 104

5.5.3.3 Effect of Truck Speed ......................................... 106

5.6 Veriﬁcation of Moving Arbitrary Load Solution .................. 109

5.6.1 Moving Arbitrary Load Time History .............................. 109

5.6.2 Typical Response Curves of Moving Arbitrary Loads ..... 111

5.7 Veriﬁcation of SAPSI Transient Analysis Using FWD Tests .....

........................................................................................... l 14

5.7.1 Characteristics of FWD Field Data .................................. 114

5.7.1.1 Linearity of FWD Field Data ............................. 114

5.7.1.2 Isotropy of FWD Field Data ............................... 116

5.7.2 Static and Dynamic Predictions of Strains ....................... 116

5.7.2.1 October 1991 FWD Tests ................................... 119

5.7.2.2 June 1992 FWD Tests ........................................ 119

5.7.2.3 February 1993 FWD Tests ................................. 125

5.8 Summary ............................................................................... 126
CHAPTER 6. APPLICATION OF SAPSI-M PROGRAM TO PREDICTING

FATIGUE DAMAGE ..... ...... . ........................ «128

6.1 Introduction .......................................................................... 128

6.2 Review of Fatigue Laws in Flexible Pavements ................... 129

6.2.1 Prediction of Fatigue Life ................................................. 129

6.2.2 Accelerated Performance Tests ......................................... 130

6.2.3 Factors Affecting Fatigue Life .......................................... 131

6.2.4 Existing Fatigue Models ................................................... 133

6.3 Review of Energy-Based Fatigue Damage Models .............. 136

6.3.1 Road Damage Prediction .................................................. 136

6.3.2 Energy Concepts for Fatigue Damage .............................. 139

6.3.3 Previous Studies ................................................................ 141

6.3.4 Dissipated Energy versus Fatigue Life ............................. 144

6.4 Prediction of Pavement Fatigue Damage Using SAPSI-M

 

Program ............................................................................... 147

6.4.1 Calculation of Dissipated Energy in SAPSI-M ................ 148

6.4.1.1 Derivation of Energy Dissipation in 1-D ............ 148

6.4.1.2 Derivation of Energy Dissipation in 3-D ............ 150

6.4.2 Strain, Dissipation Energy and Fatigue Life ..................... 152

6.4.3 The Effect of Speed on Energy Dissipation ...................... 156

6.4.4 Load Equivalency Factors from Energy Dissipation ........ 157

6.5 Summary ............................................................................... 165

CHAPTER 7. CONCLUSION“-.. -- -------- 167
7.1 Summary and Conclusion ..................................................... 167

7.2 Recommendations for Future Research ................................ 171

LIST OF REFERENCES ................................................................................... 172
APPENDIX A: PARAMETRIC STUDY .......................................................... 182
APPENDIX B: SAPSI-M USER GUIDE ......................................................... 199

TABLE 2.1

TABLE 3.1

TABLE 5.1

TABLE 5.2

TABLE 5.3

TABLE 5.4

TABLE 5.5

TABLE 5.6

TABLE 5.7

TABLE 5.8

TABLE 6.1

TABLE 6.2

TABLE 6.3

TABLE 6.4‘

LIST OF TABLES

Computer Program used in Asphalt Pavement Analysis ........................... 22
Comparison of Transient Analysis ............................................................. 51
Summary of Various Instrumented Flexible Pavement Tests .................... 79

Range of Experimental Conditions from Various Instrumented Flexible

Pavement Tests ........................................................................................... 83
Eﬁ’ective Pavement Layer Thickness of Axial Cores ................................ 95
Layer Properties for October 1991 FWD Testing ...................................... 96
Layer Properties for February 1993 FWD Testing .................................... 96

Comparison of Measured and Calculated Strains - October 1991 FWD
Testing ...................................................................................................... 123

Comparison of Measured and Calculated Strains - June 1992 FWD Testing
................................................................................................................. 123

Comparison of Measured and Calculated Strains - February 1993 FWD
Testing ...................................................................................................... 124

Estimated Fatigue Life of FHWA-ALF Section using Elastic Analysis ..
............................................................................................................. 154

Fatigue Life based on Dissipated Energy from Moving Load Solution ..155
Observation of Speed Effect by Energy Dissipation ................................ 156

Load Equivalency Factor by Dissipated Energy ...................................... 164

xi

FIGURE 1.1
FIGURE 3.1
FIGURE 3.2
FIGURE 3.3
FIGURE 3.4
FIGURE 3.5
FIGURE 3.6
FIGURE 3.7
FIGURE 3.8
FIGURE 4.1
FIGURE 4.2

FIGURE 4.3

FIGURE 4.4
FIGURE 4.5
FIGURE 4.6
FIGURE 4.7

FIGURE 5.1

LIST OF FIGURES

Schematic View of the SAPSI-M Model ..................................................... 5
Orientation of Stress Components in Cylindrical Coordinate ................... 25
Layer System Subjected to Uniform Circular Load .................................. 25
Simulation of Halfspace ............................................................................. 27
Representation of Stress and Strain for Linear Elastic Material ............... 30
Representation of Stress and Strain for Linear Visco-Elastic Material ..... 30
Boundary Condition at Layered Interfaces due to Rayleigh Wave ............ 37
Assembling of Submatn'ces of Each Layer ................................................ 37
Validation of Interpolation Scheme by Maple ........................................... 52
Interpretation of Stationary Loading Condition ......................................... 59
Interpretation of Moving Load Condition .................................................. 60
Response Pulses at a Speciﬁc Location of AC Bottom as the Load

Approach .................................................................................................... 61
Discretized Moving Loads ......................................................................... 63
The Flow Chart of SAPSI-M Program ...................................................... 66
Algorithm for Steady-State Solution ......................................................... 71
Algorithm for Moving Load Solution ........................................................ 73
Plane View of PACCAR Technical Center ................................................ 85

xii

FIGURE 5.2 Cross Section of the PACCAR Test Section ............................................. 85

FIGURE 5.3 Instrumentation Layout of the PACCAR Pavement Test Track ................. 87
FIGURE 5.4 The Peterbﬂt 359 Test Truck ...................................................................... 88
FIGURE 5.5 Effect of Truck Speed on Axle Load ......................................................... 90
FIGURE 5.6 Effect of Pressure on Axle Load ................................................................ 91

FIGURE 5.7 Typical Time Histories of Measured Strain at the Bottom of AC Layer ...92
FIGURE 5.8 Asphalt Concrete Properties Versus Frequency ......................................... 98
FIGURE 5.9 SAPSI-M Model Used in the Analysis .................................................... 100

FIGURE 5.10 Typical Response of Longitudinal Strain for Moving Constant Loads ...102

FIGURE 5.11 Typical Response of Transverse Strain for Moving Constant Loads ....... 103

FIGURE 5.12 Comparison of Calculated and Measured Strains at Different Speeds
Core-l ...................................................................................................... 105

FIGURE 5.13 Comparison of Calculated and Measured Strains at Different Speeds
Core-3 ...................................................................................................... 107

FIGURE 5.14 Comparison of Calculated and Measured Strains at Diﬂerent Speeds
Core-4 ...................................................................................................... 108
FIGURE 5.15 Measured Wheel Loads ............................................................................ 110

FIGURE 5.16 Comparison of Measured and Predicted Response Curves of Moving
Arbitrary Dynamic Solution .................................................................... 112

FIGURE 5.17 Comparison of Measured and Predicted Curves of Moving Arbitrary
Quasi-Dynamic Solution .......................................................................... 113
FIGURE 5.18 Linearity of Strain Measurement in FWD Tests ...................................... 115
FIGURE 5.19 Ratio of Transverse to Longitudinal Strains in FWD Tests ..................... 117
FIGURE 5.20 Actual and Simulated FWD Load Pulses ................................................. 118
FIGURE 5.21 Comparison of Measured to Calculated Strains - Oct. 1991 FWD Tests ......
................................................................................................................ 120

xiii

FIGURE 5.22 Comparison of Measured to Calculated Strains - June 1992 FWD Tests ......

................................................................................................................ 121
FIGURE 5.23 Comparison of Measured to Calculated Strains - Feb. 1993 FWD Tests ......

................................................................................................................ 122
FIGURE 6.1 Comparison of Fatigue Models at Various Test Methods ........................ 137
FIGURE 6.2 Typical Stress-Strain Hysteresis Loop .................................................... 140
FIGURE 6.3 Typical Curve of Dissipated Energy versus Number of Cycles ............... 142
FIGURE 6.4 In Situ FHWA-ALF Fatigue Test for Surface Cracking ......................... 154
FIGURE 6.5 Types of Axle Conﬁguration for Load Equivalency Factor ..................... 158
FIGURE 6.6 Stress-Strain Hysteresis Loop -Ax1e Type-l ............................................ 159
FIGURE 6.7 Stress-Strain Hysteresis Loop -Ax1e Type-3 ............................................ 160
FIGURE 6.8 Stress-Strain Hysteresis Loop -Axle Type-4 ............................................ 162
FIGURE 6.9 Stress-Strain Hysteresis Loop -Axle Type-7 ............................................ 163

xiv

CHAPTER 1
INTRODUCTION

1.1 PROBLEM STATEMENT

Over the past thirty years, developments in both analysis techniques and material
testing procedures have made it possible to better simulate the response of asphalt con-
crete pavements to trafﬁc loads and environmental conditions. This has allowed the use of
mechanistically based design methodologies in the form of design charts, computer solu-
tions and catalogues of design. Numerous pavement response models have been generated
since the early sixties, ranging from an inﬁnite beam on a kaler foundation to a multi-
layered system over an inﬁnite half space. Current analytically-based procedures, how-
ever, assume that the load is applied statically. In recent years, there have been some anal-
yses which study the response of pavements to dynamic loads. Dynamic models vary in
complexity according to the structure analyzed (ﬁnite or inﬁnite beam/plate; elastic or
damped-elastic Winkler foundation, or damped-elastic layers) and the loading (stationary,

moving, constant, harmonic, random).

A number of methods have been used to account for the moving effect of the load.
They range from using the Dirac function or the Kronecker delta operator to pre-multiply-
ing the load by time dependent deflection shape functions (ﬁnite element analysis) or
applying a pulse load with a duration equal to the time taken to travel one tire contact

length.

There is only a small number of studies which have considered the validation of
these models by comparison with ﬁeld measurements. The agreement between experiment

and theory has been qualitative and for low speed [40, 105] because of the difﬁculties in

measuring the response of roads accurately and the ﬂuctuation of wheel loads, if they are

not measured [40].

1.2 DYNAMICS OF PAVEMENTS

Most current models of pavement analysis assume static loads. However, peak
dynamic vehicle loads may be considerably higher than static loads, and thus dynamic
loading may have a considerable impact on pavement performance. The following is a list
of reasons for considering dynamic methods of pavement analysis [18]:

(1) Trafﬁc loads are inherently dynamic since they move over the pavement and

vibrate.

(2) The pavement may respond “dynamically” if either inertial or viscous forces are
significant.

(3) Propagation of waves induced by dynamic wheel loads from the pavement
surface through the different pavement layers may signiﬁcantly affect the
response.

(4) Dynamic interactions between truck axles and the pavement surface could be
signiﬁcant.

(5) Materials may respond differently to static and dynamic stresses if the effect of
rate or frequency of loading is important.

(6) Field observations may be strongly inﬂuenced by dynamic factors, as in the
cases of strain gauge measurements, accelerometer readings, non-destructive
testing, and weigh-in-motion measurements.

All of the above considerations lead to the conclusion that it may be necessary to
use dynamic methods in analyzing pavement structures. Fluctuations in wheel loads due to
axle vibrations may cause a signiﬁcant increase in the damage incurred by pavements sys-
tem. In order to quantify this effect, it is necessary to accurately predict the primary

responses of pavements (stresses, strains and deﬂections) due to moving arbitrary loads.

1.3 RESEARCH OBJECTIVES AND ORGANIZATION

1.3.1 Objectives

The objectives of the research are as follows:
(1) To develop a model for the response of multi-layered system subjected to
moving arbitrary dynamic loads applied on the surface.

(2) To develop an efﬁcient computer program to evaluate the response of a general
multi-layered system over a half space excited by dynamic surface loads.

(3) To verify the model and the corresponding computer code against full-scale
field truck tests.

(4) To verify the stationary transient analysis against FWD (Falling Weight
Deﬂectometer) tests.

(5) To investigate possible applications of fatigue damage models for asphalt
concrete pavements.

1.3.2 The Proposed Model

The proposed model and the related computer program SAPSI-M allows for a real-
istic dynamic analysis of general multi-layered system. The following is a list of its capa-
bilities:

(l) Considers dynamic loads.

(2) Sirnulates multiple surface moving arbitrary loads.

(3) Uses of linear visco-elastic theory.

(4) Analyzes a multi-layered system.

(5) Obtains a continuum solution in the horizontal direction.

(6) Obtains a ﬁnite element solution in the vertical direction.

The proposed method of analysis in the new version of the program, SAPSI-M, is
an improvement over the existing methods because: (1) it treats moving ﬂuctuating loads

on a n-layered damped-elastic system; (2) it incorporates such important factors as wave

propagation, inertia and damping effects of the medium as well as frequency-dependent
asphalt concrete properties; (3) it can handle any load conﬁguration, thus making it possi-

ble to model multiple wheel conﬁgurations of truck axles and airplane landing gears.

When modeling a moving wheel load as a stationary haversine pulse with a dura-
tion equal to the time it takes the wheel to roll over a ﬁxed point in the pavement, the pres-
sure exerted by the tire is applied only when the tire is in direct contact with the point. This
means that the effect from the wheel load as it approaches (and leaves) the response point
is not accounted for. In reality, the moving wheel load sends stress waves as it approaches
the ﬁxed point, and theses waves (in the form of body and surface waves) propagate so that
the effect of the load is felt by the ﬁxed point in the pavement before and after the load is
directly on top of it. These waves could be a combination of direct waves from the moving
load and reﬂected waves from a layer boundary below the surface such that the net pave-
ment response may be ampliﬁed or de-ampliﬁed, depending on radiation damping, relative
to the stationary response. Thus, in general, a moving load cannot be modeled by a station-
ary transient haversine pulse. The SAPSI-M computer program computes the response of
the multi-layered asphalt concrete pavement to moving transient loads. Figure 1.1 shows

the schematic view of the proposed model.

1.3.3 Organization

An introduction with the objective of this research is presented in Chapter 1. After
reviewing the related literature and discussing the existing methods of analyses, the
remaining chapters of this dissertation describe in detail the theoretical formulations,

implementation, ﬁeld veriﬁcations and applications.

In chapter 2, the review of relevant literature on closed-form and numerical solu-

tions for the analysis of asphalt concrete pavements is presented. The currently available

3]

Fixed Layers

Variable Depth Layers

 

Pi(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

///F1/xed]Ba§el/////

FIGURE 1.1 Schematic View of the SAPSI-M Model

closed-form solutions are mostly static and based on an elastic multi-layer system. Most
of the numerical solutions involve the static analysis of a beam or plate supported by a
Mnkler foundation or a layered structure supported by a half-space based on the ﬁnite
method or layer theory, respectively. However, no analysis was found in the literature for
the dynamic response of a realistic layered pavement system to moving steady or random

forces.

In chapter 3, the theoretical back ground for calculating the response of a visco-
elastic multi-layered system subjected to stationary loads is presented. It is developed for
the axisymmetric coordinate system, and is based on the linear visco-elastic layer theory
in the horizontal direction and the ﬁnite element method in the vertical direction. The
model consists of subdividing the natural layers within the pavement proﬁle into sublayers

supported by a rigid base or a half-space.

In chapter 4, a new method for the analysis of multi-layered pavement systems
subjected to moving loads is presented. Assumptions for the new proposed model are
listed, and the moving load representation is described. Then, the response to moving con-
stant and arbitrary loads is presented in detail. Finally, the implementation of the proposed
method is described by presenting the layout and algorithm of the SAPSI-M computer

program.

In chapter 5, the validation of SAPSI-M computer program against full-scale ﬁeld
data is presented. First, the PACCAR ﬁeld test program is described including instrumen-
tation, measurements and material characterization. Then, the validation of SAPSI-M is
presented for both the moving constant and arbitrary load solutions. The veriﬁcation of
SAPSI transient analysis is also presented with a Falling Weight Deﬂectometer (FWD)

tests.

In chapter 6, the application of SAPSI-M program into the prediction of pavement
fatigue damage using the dissipated energy concept is described. The comparison of
fatigue lives from the conventional fatigue damage law using peak strain only and from
the dissipated energy law using the entire time history of the response due to moving loads
is made. This chapter also includes a section on the development of load equivalency fac-

tors using the dissipated energy approach.

In chapter 7, the results of the research are summarized and a number of conclu-

sion are put forward. Finally, some recommendations for future work are suggested.

CHAPTER 2
LITERATURE REVIEW

2.1 INTRODUCTION

An asphalt pavement may consist of a thin covering surface course over a base
course, subbase course, and compacted roadbed soil. Thus, the term pavement herein
implies all the layers in the pavement structures. The load carrying capacity of a ﬂexible
pavement is brought about by the load distribution characteristics of the layered system.
The highest quality layer is placed at or near the surface. Hence, the strength of the pave-
ment is the result of building up thick layers and, thereby, distributing the load over the

relative weak road bed soil [124].

The design and analysis of the pavement system have been approached from two
main diﬂ‘erent points of view: Empirical and mechanistic. Initially, design engineers have
approached the problem from the empirical method. As theory, researchers started analyz-

ing the problem using mechanistic concepts.

Empirical procedures in pavement design greatly rely the past experience and lab-
oratory or ﬁeld tests. Empirical relationships have also been used to assess the structural
capacity of pavements and to predict pavement strength from simple ﬁeld tests. Empirical
and semi-empirical procedures can be advantageous because they are usually simple and
are derived from actual ﬁeld data. On the other hand, the disadvantage of the empirical
approach is that it can be applied only to a given set of environmental, material, and load-
ing conditions. If these conditions are changed, the design is no longer valued and a new
method should be developed through a new trial to account for a new set of conditions.
AASHTO, NSA (National Stone Association), and Califonia Design methods may be the

most popular empirical design methods for ﬂexible pavements.

Mechanistic ﬂexible pavement design procedures are typically based on the
assumption that a pavement can be modeled as a multi-layered elastic or visco-elastic
structure on an elastic or visco-elastic foundation. Assuming that pavements can be mod-
eled in this manner, it is possible to calculate stresses, strains, or deﬂections due to trafﬁc
loading and environmental conditions at any point in the structure. An important advan-
tage of this design method is the ability to analyze a pavement for several different distress
modes such as cracking and permanent deformation. This allows the engineer to adjust the
pavement design and produce a cost-effective pavement section that does not fail prema-
turely. The main disadvantage of this design method is that it requires more comprehen-
sive and sophisticated data than empirical design methods. Extensive laboratory and ﬁeld
testing may be required to determine the design parameters such as the resilient modulus,

creep compliance, and others.

During the past few decades, a signiﬁcant improvement in the area of pavement
design and analysis has been established using theoretical mechanics. The development of
theoretical methods. especially the layer theory and its numerical development, are

reviewed and discussed in the subsequent sections.

2.2 CLOSED-FORM SOLUTIONS

Civil engineers have traditionary considered the structures of road surface to
behave as continua, and have used the methods of continuum mechanics. The ﬁrst such
analysis was performed in 1926 by Westergaard who modeled a rigid pavement as a elastic
plate supported by a Mnkler foundation [120]. In the 1940’s the theory of elastic layered
systems was published by Burmister [12], which established the analysis of system with a
2 or 3 layers of inﬁnite extent and accounted for the eﬂects of the interface conditions

between layers.

10

2.2.1 Static Analysis

The currently available closed-form solutions mostly use static analysis and are

based on an elastic multi-layer system.

Burmister [12] developed the ﬁrst general equations and solution to obtain deﬂec-
tions and stresses in two and three-layer systems based on the theory of elasticity. The
solution assumed a uniform vertical static load distributed over a circular area and each
layer is continuous in the lateral direction. Bunnister obtained his solution by assuming a
stress function involving Bessel functions and the exponential. Comprehensive tables and
charts of inﬂuence values for two and three-layer systems subjected to uniform circular
loading were produced [13]. To use these solutions easily, he developed charts for practi-

cal pavement design applications.

The application of the theory of elasticity in the design and analysis of ﬂexible
pavements raises some questions such as whether a ﬂexible pavement is an elastic body
and Hooke's law is in fact satisﬁed. Considerable research and theoretical studies have
been done on this point by several authors who considered the eﬂ’ects of visco-elasticity or

non-linearity.

2.2.2 Dynamic Analysis

The inertia effect in pavement analysis has been modeled using single and multi-
degree of freedom systems. When a dynamic load is applied at the surface of a homoge-
neous half-space, three main types of waves are formed; primary (compression), second-
ary (shear), and surface (Rayleigh) waves. Waves result in three-dimensional states of

stress, strain, and deﬂection at various points in the half space.

ll

Kausel and Peek [56] presented an explicit and closed-fonn solution for the
Green’s functions (displacements due to unit loads) corresponding to dynamic loads act-
ing on layered system. These functions embody all the essential mechanical properties of
the medium and can be used to derive solutions to problems of elasto-dynamics, such as
scattering of waves by rigid inclusions, soil-structure interaction, seismic sources, etc. The

solution is based on a discretization of the medium in the vertical direction.

One method of dynamic analysis is based on Fourier transformation. To simplify
the analysis, a transient mode of loading can be represented by a series of harmonic loads
with different frequencies and magnitudes using Fourier transformation. Once the
response to steady-state loading is obtained as a function of frequency and magnitude, the
response to a transient load in the time domain can be obtained through the inverse Fourier
transform. The governing equation for steady-state elastodynamics is the Helmholtz equa-

tion [45].

2.2.3 Moving Load Solutions

The moving problem was investigated by Cole and Huth [25] for the special case
of a vertical line load moving at constant velocity over the surface of a uniform elastic half
space. They found that the major effect of load movement is an increase in the maximum
stresses that occur in the pavement. Unfortunately, the above result is not directly applica-

ble to pavement structures, which behave like layered systems.

Pister and Westmann [81] modeled an asphalt pavement as a visco-elastic beam,
inﬁnite in extent, supported on a Winkler foundation subjected to a concentrated moving
load at constant velocity. The solution, which was written as the product of a number of
constants and a dimensionless coefﬁcient dependent upon the space and the time vari-

ables, was obtained with the initial and boundary conditions through the use of iterated

12

Laplace and Fourier transform. They observed the deﬂection and curvature occurring
directly under the moving load for different velocities and relaxation time. They con—
cluded that the relaxation time of the material is quite important. The lower relaxation
times tend to extend the viscous behavior of the beam over a larger range of velocities as
compared to higher relaxation times. For lower velocities, the visco-elastic effects are
more important so there is a considerable difference obtained by elastic and visco-elastic

analysis.

The computer program MOVE was developed by Chen [21], for dynamic analysis
of a linear elastic layered system resting on a rigid base subjected to a vertical line load
that moves with a constant velocity over the surface of the system. In MOVE, the wave
equation of displacement potentials is converted to an equivalent Laplace’s equation by
the Galilean transformation. The general solution of the equivalent Laplace’s equation is
then solved by specifying appropriate boundary conditions and transforming the load into
a Fourier integral. It should be noted that both Cole and Huth [25] and MOVE [21] solu-
tions assume elastic properties of materials. Therefore, the response increases with an
increase in velocity. Under actual ﬁeld conditions, internal damping in the materials as
well as radiation damping in the structure will cause the response to decrease as the veloc-

ity of the load is increased [ 18].

Fryba [34] solved the problem of an inﬁnite beam on an elastic foundation sub-
jected to a constant force moving at constant speed. He calculated the covariance of the
response of a beam on a \Vrnkler foundation that had statically varying stiffness to a stati-
cally stationary force moving along it at constant velocity. However, Fryba did not extend

the model to two or three dimensional problems.

Cebon [14] presented a numerical convolution method of the applied loads with a

ﬁeld of road surface impulse response functions, for calculating the time domain response

13

of an inﬁnite beam on a damped Mnkler foundation subjected to moving random loads.
The impulse response can be calculated for an idealized road response model, or deter-
mined by ﬁeld measurements subjected to an impulse. He also validated his model by
comparing theoretical results with ﬁelds measurements.The model, however, is restricted

to two-dimensional models.

Chatti [17] extended an existing solution for a moving harmonic load on a ﬁnite
beam resting on an elastic Winkler foundation, published by Boquenet [11]. The improved
solution allows for a load of arbitrary time history through the use of complex number rep-
resentation and the Fast Fourier Transform algorithm, and accounts for energy dissipation

through the foundation.

2.3 NUMERICAL SOLUTIONS

The currently available methods of pavement analysis are mostly based on static
analysis and may be subdivided into two main groups: Layer theory based on continuum
mechanics and the ﬁnite-element method. A few analytical methods for solving dynamic

problems in pavements are now available.

2.3.1 Layer Theory

Flexible pavements are layered system with better materials on top and cannot be
represented by a homogeneous mass, thus the use of Burmister’s layered theory is more

appropriate [49]. The basic assumptions for layer theory to be satisﬁed are:

(1) Each layer is homogeneous, isotropic and elastic or visco-elastic.
(2) Each layer is infinite in the horizontal direction.

(3) Each layer has a ﬁnite thickness, but the lowest layer is a half-space having
infinite thickness.

l4

(4) Continuity conditions are satisﬁed at the layer interface, as indicated by the
same vertical stress, shear stress, vertical displacement, and radial
displacement. For a frictionless interface, the continuity of shear stress and
radial displacement is replaced by zero shear stress at each side of the interface.

The use of a layer theory in modeling pavements has been quite extensive and
many presently available design methods for ﬂexible pavements are based on this theory.
These models are based upon the assumption of linear elastic or visco-elastic material
properties. A brief literature review on the numerical solution with layer theory will be

made in the following sections.

2.3.1.1 Static Analysis Models

Elastic layer theory has been used successfully for the analysis of ﬂexible pave-
ments since the 1940’s, when it was introduced by Burmister [12]. Initially, the use of the
method was restricted to systems with two or three layers that extend to inﬁnity in the hor-
izontal directions [13]. More recently, a number of computer models have been developed
to handle more layers [6, 30, 57]. Surprisingly, there is only a small number of studies
which have considered the validation of these models by comparison with ﬁeld measure-

ments.

The elastic multi-layer system solution and some of its modiﬁed versions such as
CHEV [117] and ELSYMS [6] have been extensively used in the area of pavement evalua-
tion. The program CHEV can be applied only to linear elastic materials but was modiﬁed
by the Asphalt Institute in the DAMA program to account for nonlinear elastic granular
materials [52]. ELSYMS was originally developed at the University of California, Berke-
ley, for analyzing ﬁve-layer systems under multiple wheel loads [6]. The response of the
multi-layer pavement system to uniform circular loading was analyzed by Michelow [69].

The model characterized the pavement as a semi-inﬁnite solid with n-layers with their

15

elastic characteristics being different from one layer to another. Another well known pro-
gram is BISAR [30], developed by the Shell company, which considers both vertical loads
and horizontal loads. Based on the nonlinear theory, Finn et a1. [33] developed a computer
program named PDMAP (Probabilistic Distress Model for Asphalt Pavements) for pre-

dicting the fatigue cracking and ruttin g in asphalt concrete pavements.

The VESYS 11 computer program was developed at the Massachusetts Institute of
Technology by Soussou [103] in 1973. The program models a three layer system; the
upper two layers have ﬁnite thicknesses over one inﬁnite layer. All Layers have inﬁnite
dimension in the horizontal direction and have visco-elastic properties. The VESYS
model consists of a primary response model, a damage model, and a serviceability model.
A closed-form probability solution of the response of the layered system to an axisymmet-
ric stationary load applied at the surface is obtained. The probabilistic estimates of stress,
strain and deﬂection, the loading characteristics, and the temperature history are used as
inputs to the damage model to obtain the extent of cracking. The distress indicators
obtained from the damage model are combined using the AASHTO equation to provide a

subjective measure of the present serviceability index (PSI).

The computer program, KENLAYER, was developed by Huang [49], based on
Burmister’s layer theory. It solves the response for a elastic multi-layered system under
stationary single or multiple circular loads. The AC layer can be treated as linear elastic or
visco-elastic. The moduli vary in the vertical direction but remain constant along the hori-
zontal plane for each sublayer. The base and the subgrade can be divided into several sub-
layers and the stress-dependent moduli can be estimated for each sublayer through an

iterative procedure.

Both the linear and non-linear visco-elastic material properties have been modeled

in an effort to recognize the time-dependent response of the asphaltic materials. The use of

l6

non-linear visco-elastic theory has been very limited because of the complexity of material
characterization. The validity of using a linear visco—elastic analysis to evaluate the pave-
ment response was thoroughly studied by Nair et al. [77]. Their investigation consisted of
performing laboratory tests on paving materials to determine the degree of linearity in the
response function. The study concluded that linear visco-elastic theory can be acceptably
used in pavement evaluation analysis; in order to reduce the inﬂuence of stress level, stress
state, anisotropy and other effects to a level that will make linear isotropic visco-elasticity
an acceptable characterization of asphalt concrete, a limit must be placed on the strain

level that can occur in the asphalt concrete.

2.3.1.2 Dynamic Analysis Models

Loading from a truck consists of a set of moving, time-varying surface stresses.
These stresses represent the static load carried by each wheel as well as the dynamic ﬂuc-
tuations generated by the roughness of the pavement sm'face proﬁle. A number of truck
simulation models for predicting dynamic wheel loads have been developed by several
research organizations, including MIT [44], UMTRI [31] and the University of Cambridge
[14]. These models are planar, with pitch (rotation along the longitudinal direction of the
truck) being the only form of rotation allowed. Key factors affecting the accuracy of these
models include the proper modeling of non-linear properties of the suspension springs,
kinematics of tandem axles and the sequential input of the road proﬁle into the different
axles [34]. Truck simulation models have by-in-large been validated by comparison with

full-scale tests [30].

Many dynamic models have been developed for the analysis of pavements and can

be largely classiﬁed into two main categories as follows [14]:

17

(l) A beam or plate supported by massless springs (Winkler foundation) [11, 14,
17, 34, 42, 51, 59, 60, 125] or supported by a half-space [3, 17]. The foundation
may be modiﬁed to include inertial effects [48, 96].

(2) A layered structure of elastic or visco-elastic solids [21, 34, 54, 97, 126].

Harr [42] presented a simple model for the dynamic response of a pavement sub—

jected to a rectangular uniform pulse of duration t1 seconds required for the passage of the

vehicle. His model consists of a mass, spring and damping to consider the inertia, elastic
deformation and time-dependent deformation of the pavement, respectively. By suitable
selection of model parameters, he was able to achieve close results with the AASHTO

road test [1].

Moavenzadeh and Elliott [72] presented a solution technique for the investigation
of the stresses and displacements induced in a three layer visco-elastic system subjected to
uniformly distributed normal loads. A steady localized load is made to move at a constant
velocity in a straight line on the initially ﬂat surface of a three layer system. They investi-
gated the response of the system to a load that is repeatedly applied to the same region on
the surface at a speciﬁed frequency. Actually, they extended Burmister’s elastic solution to
visco-elastic material and a moving load by using the Heaviside step function and

Duhamel superposition/integral, respectively.

Chen [21] developed the SAPSI computer program in 1987. SAPSI is a PC-based
FORTRAN computer program which calculates the dynamic response of a n-layered
visco-elastic system to multiple surface loads. Material properties for each layer may be
varied with frequency. Chen veriﬁed the program against analytical solutions [21], and

Tabatabaie validated it by comparison with ﬁeld measurements [105].

Sebaaly [91] modiﬁed a computer program DYNAMIC which was developed by

Kausel [56] to compute stresses and strains in addition to displacements due to both har-

l8

monic and impulse loadings using the Fourier transform. Peak theoretical vertical pave-
ment deﬂections were validated by comparison with ﬁeld data from the AASHTO road
test. He could not, however, validate the speed effect on strain under a moving load
because he simulated the response of the road to a falling weight rather than to a moving

load.

Zaﬁr et al. [127] developed a computer program DYNPAVE to evaluate dynamic
pavement strains under moving trafﬁc load based on a continuum-based ﬁnite-layer
approach. The model is based on a modiﬁed plane-strain model, in which three-dimen-
sional effects are taken into account by using two special viscous boundaries (front and
back) connected to the two-dimensional plane-strain model. The dynamic material proper-
ties of the asphalt concrete layer can be varied as a function of the loading frequency. They
concluded that the pavement strains are strongly functions of the pavement thickness and
the vehicle speed, and they reduce substantially with the increase in the speed of the vehi-
cle. Also, a substantial compressive strain component is present at the bottom of AC layer

that may be important for any realistic fatigue life estimation of the pavement.

A new numerical method was introduced by Hanazato et al. [38] to analyze mov-
ing trafﬁc-induced ground vibrations three-dimensionally on the basis of a dynamic elastic
ﬁnite element method. Moving load is expressed to be composed of a constant static com-
ponent and dynamic component, which is expanded into the Fourier series as the sum of
harmonic motions. This is a new prismatic model combining the ﬁnite element method
and visco-elastic layer theory. The ﬁnite element formulation is used to model an irregular
region which includes the pavement structure and the soils underneath it. The visco-elastic
layer theory is used to model the far-ﬁeld region via semi-inﬁnite thin-layered elements.

The two regions are joined by energy-transmitting boundaries. This model applies only to

l9

continuous ﬂexible pavements because of its prismatic nature and its inability to handle

discontinuity.

2.3.2 Finite Element Method

The ﬁnite element method has also been used extensively and is particularly useful
for analysis of jointed rigid pavements which is not continuous in the horizontal direction
and can be prone to be thermal warping as well as being subjected to vehicle loads.
Almost all of the models assume linear material properties, although some of model
include visco-elastic effects [9, 53, 121]. The ﬁnite element method in pavement analysis
offers the capability of modeling joints in rigid pavements and incorporates the effect of
slab dimensions on the rigid pavement response. The pavement is represented by a grid of
small elements where the layers need not be inﬁnite in the lateral direction. The method
provides solutions for stresses, strains and deformations based on numerical evaluations
which makes its accuracy very much dependent on the ﬁneness of the initial grid and the

means of conducting computations.

Three kinds of ﬁnite element models may be used in the analysis of ﬂexible pave-
ments: Two—dimensional plane strain, three-dimensional and axisymmetric models. Using
the two-dimensional plane strain model of the pavement cross section, the wheel loading
would have to be considered as a line loading, inﬁnitely long in the longitudinal direction.
This would be unrealistic because of the truck’s conﬁguration and loading condition.The
most comprehensive approach would be one that uses three-dimensional ﬁnite element
analysis. In a three dimensional model, suitable boundary conditions must be considered
at some reasonable distance away from the loaded region in all three directions. The most
commonly used two-dimensional model is the axisymmetric one. This model assume that
the pavement geometry and loading are both axisymmetric. \Vrth these assumptions, a

three dimensional problem can be reduced to a two-dimensional problem. However, the

20

asphalt surface must be assumed to be inﬁnitely wide and therefore the edge effect cannot
be considered. Non-axisymmetric multiple loads can be analyzed by superposition for lin-

ear elastic or visco-elastic materials.

Duncan et al. [31] ﬁrst applied the ﬁnite element method in 1968 for the analysis
of ﬂexible pavements. The method was later incorporated in the ILLI-PAVE computer pro-
gram by Raad and Figueroa in 1980 [83], which included a non-linear ﬁnite element pro-
gram to analyze the pavement system. Raad and Figueroa used the Mohr-Coulomb failure
criterion to modify the stress calculated using a ﬁnite element method at the end of each

iteration.

Harichandran et al. [41] developed in 1989 a non-linear ﬁnite element program
called MICH-PAVE for use on a personal computer to analyze ﬂexible pavements. The
program computes the primary responses within the pavement system subjected to a single
circular wheel load. Useful design information such as fatigue life and rut depth are also
estimated through empirical equations. In the MICH-PAVE program, the pavement is rep-
resented by an axisymmetric ﬁnite element model, and the resilient modulus model
together with the Mohr-Coulomb failure criterion is used to characterize the non-linear

material response of granular and cohesive soils.

Chatti [17] developed the DYNA-SLAB computer program in 1992, which calcu—
lates the dynamic response of a concrete slab system to moving ﬂuctuating loads using the
ﬁnite-element method of analysis. The underlying soils can be modeled either as a
damped Winkler foundation or as a layered visco-elastic system. Chatti validated the

model by comparison with theoretical results as well as ﬁeld measurements.

21

2.4 SUMMARY

In this chapter, the review of relevant literature on closed-form and numerical solu-
tions for the analysis of asphalt concrete pavements is presented. The currently available
closed-form solutions are mostly static and use an elastic multi-layered system based on
Burmister’s solution [12]. Considerable research in theoretical studies have been done by
several authors using visco-elasticity or non-linearity. Most numerical solutions are for the
static analysis of a beam or plate supported by Winkler foundation or a layered structure.
A few moving load solutions were presented, ranging from a beam on a damped Winkler
foundation to a simpliﬁed plane strain layered visco-elastic system. The ﬁnite element
method has also been used extensively and is particularly useful for analysis of jointed
rigid pavements which is not continuous in the horizontal direction. The pavement system
is represented by a grid of small elements where the layers need not be inﬁnite in the lat-
eral direction. Because of the numerical nature of discretization systems, the accuracy is
greatly dependent on the ﬁneness of the initial grid and the means of conducting computa-

tions.

Hardy and Cebon [40] pointed out that no analysis was found in the literature of
the dynamic response of a realistic layered road model to moving steady or random forces
and that very few pavement response models have been validated by comparison with ﬁeld
measurements. Table 2.1 lists some of the available computer programs applicable to ﬂex-

ible pavement analysis, and brieﬂy describes their source, availability and theory.

22

 

 

 

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CHAPTER 3

RESPONSE OF VISCO-ELASTIC MULTI-LAYERED
SYSTEM TO STATIONARY DYNAMIC LOADS

3.1 INTRODUCTION

Chen [21] developed a computer program, termed SAPSI, which can be used to
calculate the response of visco-elastic multi-layered systems subjected to multiple circular
surface loads. Material properties for each layer may be varied with frequency. SAPSI was
developed for the axisymmetric case, which enables the reduction of three-dimensional
problems to two-dimensions. The analysis in SAPSI is based on the theory of linear visco-
elastic layer theory in the horizontal direction and the ﬁnite element method in the vertical
direction. The theoretical background for the solution in SAPSI is described in the follow-

ing sections.

3.2 EQUATION OF MOTION

The equilibrium conditions for an inﬁnitesimal element together with the strain-
displacement relations and the stress-strain relations yield to the equation of motion in
terms of the displacements. Using the axisymmetric coordinate system, the governing dif-

ferential equation of motion for the homogeneous case becomes as follows [61]:

A 20 3’11, 1811r ur azuz G azu, azuz _ azu,
‘* >;,a-+;a:-;2+a—.-a.+ Tia—far “P?

azu, 1311r azuz] [Bzuz 1811r azu, 1811,] [3211,]
+0 — -—————— = 32
t

(3-1)

 

(1+2G)(az—ar+;'a—z'+'a? arZ rar azar r5;

where, ur and uz are the displacements in the horizontal and vertical directions, respec-

tively, p is the mass density, and 7. and G are Lame’s constants.

23

24

The strain—displacement relations for an axisymmetric loading are derived from the

theory of elasticity as:
Bu
8, = 5}!
u
39 - f
Bu
82 = sf
_ Bu, an,
1.. 5; "a?

_ 7. 2G ur xauz ur
“r" + )a." (5”?)
r Bur an,
09 = (A+2G)—+l(‘3—r-+§z—)
uz Bu, ur
02 = (1+2G)—+x(—+_)
2 Br r

Bur Buz
1: = G(— + —)

1'2

(3-2)

(3-3)

where, e, , 89 and El are the radial, tangential and vertical strains, respectively, and 1‘rz

and In are the shear strain and shear stress as deﬁned in Figure 3.1.

3.3 MULTI-LAYERED MODEL

The model consists of subdividing the natural layers within the pavement proﬁle

into thin sublayers which are assumed to extend to inﬁnity in the horizontal direction as

depicted in Figure 3.2. The axisymmetric condition enables the reduction of the three-

dimensional problem into a two-dimensional one. The displacements within each sublayer

are assumed to vary linearly in the vertical direetion; however in the horizontal direction,

25

 

 

 

 

'2

FIGURE 3.1 Orientation of Stress Components in Cylindrical Coordinate

V

Mi Mi “7‘82“ _,
‘r’l

 

 

 

 

 

 

 
    

 

  
   

VVVVVVVVVVVVVVV 7"VVVVVVVVVVVVVVV
’9’...’9‘0’0’9’9’9’9’4’0’9’9’...” Po’o’o’o’o’o’o’o’o’o’9.9% 9%.
Fixed Base

           

       
  

 

FIGURE 3.2 Layer System Subjected to Uniform Circular Load

26

the displacements are required to satisfy the pertinent ordinary differential equations

which are obtained from Equation ( 3-1 ) by discretizing the medium into ﬁnite layers.

With the above discretization, the free motion at a given frequency consists of a
ﬁnite number of wave modes which are obtained by solving an algebraic eigenvalue prob-
lem. These wave modes are used as shape functions to expand the displacements in the
layered medium in terms of mode participation factors. Given the displacement ﬁeld, the
strains for each layer are determined by diﬁerenﬁaﬁon. The stresses in each layer are
obtained assuming linear visco-elasticity. Finally, the layer effective stiffness matrix is
obtained using the principle of virtual work, and the contributions from the individual lay-
ers are assembled to form the global stiffness matrix. The dynamic stiffness matrix is com-

plex, symmetric and frequency-dependent [21].

3.4 SIMULATION OF HALF SPACE

The approach described above was originally developed for a layered system rest-
ing on a rigid base. In many practical cases the site is a layered system which extended to
such great depth that it becomes necessary to introduce an artiﬁcial rigid boundary at some
depth. This boundary will reﬂect some energy back into the system and will cause the site
to have some erroneous natural frequencies which will affect the overall response. This
becomes especially critical for sites with low material damping. To remedy this problem,
the simulation of a semi-inﬁnite halfspace was introduced by the Variable Depth Method
which was developed by Chen [20].

The Variable Depth Method is used to simulate a site which is composed of soft
materials extending to a great depth. Figure 3.3 shows the schematic view for simulating a
halfspace [90]. According to this method, the layered system is extended by 11 additional

layers having properties equal to those of the half space. At each frequency, the total depth

Fixed Layers

Variable Depth Layers

 

 

27

 

 

 

 

 

 

RRRHRRRE

7"V'VV'V'V'VVVVV'VVVVVVVVVVV‘V"VVVVVV
O O O O O O O O O O O O O O O O O O O O...0.0.0.0.0.0...0.0.0.0.0.0.0.0.0.

Rigid Base

FIGURE 3.3 Simulation of Halfspace [90]

28

of the extended layers is set to equal 1.5 times the shear wave length, As, of the half space.

The choice of this depth is made on the observation that fundamental mode Rayleigh
waves in a halfspace decay rapidly with depth and essentially vanish at a depth corre-
sponding to one and a half wave length. The thickness of each extended layer is set to be a
function of the excitation frequency and to increase as the term of a quotient series such

that:

2 n Vs
aho+aho+ ...... +orh0=H=1.ST (34)
where, h0 is the thickness of the last top layer, Otis a constant factor, V8 is the shear wave

velocity of the half-space, and f is the frequency.

Based on the above method of subdivision, the layer thickness will increase with
depth and decrease with frequency. This means that the layer thickness becomes very large
at low frequency and remains small as compared to the wave length. At zero frequency
(static case) the method ceases to be valid. The choice of n=10 was found to be adequate

for most practical problems [90].

The site model may be further improved by replacing the rigid boundary with a
viscous boundary. For a site with very small material damping, the contribution of higher
modes to the response may produce some artiﬁcial peaks that should not appear. This
problem is handled by the use of Lysmer—Kuhlmeyer [63] viscous boundary dashpots at
the bottom of the layered system. This boundary consists of two (vertical and horizontal)
dashpots per unit area of the boundary. The dashpots will translate into the addition of an

extra stiffness matrix, [KJH which is deﬁned as:

[K]H 2 [imng 0 J (3-5)

impV

29

where, p, V8 and VI) are the mass density, s-wave and p-wave velocities of the half-

space, respectively, and 0) is the circular frequency.

3.5 COMPLEX MODULUS REPRESENTATION

Consider a problem of one-dimensional wave propagation in a linear elastic mate-

rial. The stress-strain relationship follows Hook’s law as:

0(x, t) = E - 8(X, t) (3-6)
where, a and e are the stress and strain components, respectively, and the constant value

E is the modulus of elasticity.

For a harmonic motion of frequency to the stress and strain are also harmonic
with the frequency to . Using the complex representation for harmonic oscillations the

stress and strain can be written as:

0(x, t) = (I"‘(x)eiwt

irot

(3-7)
8(X, t) = 6*(x)e

where, (u is circular frequency, 0* and 8* are the complex amplitudes of stress and

strain, respectively, and i is J: . They can be interpreted as a pair of vectors rotating at

frequency (0 about the origin of the complex plane, as shown in Figure 3.4 (a). Substitu-

tion of Equation ( 3-7 ) into Equation ( 3-6 ) yields:

0*(x) = E - 8*(x) (3-8)
The above equation implies that the stress and strain vectors are in phase, as shown in
Figure 3.4 (b), and therefore all the strain energy is stored in a recoverable form as would

be expected for elastic materials.

30

 

 

 

 

Imag 0
I I
° ,4?
e l
0)
-80
‘ > Real - >8
8O
( a ) In Complex Plane ( b) In Real Plane

FIGURE 3.4 Representation of Stress and Strain for Linear Elastic Material [106]

(I) 5 g g /
+ Real ° 4

 

 

( a) In Complex Plane (b ) In Real Plane

FIGURE 3.5 Representation of Stress and Strain for Linear Vrsco-Elastic Material
[106]

31.

However, most materials, including soils and asphalt concrete mixtures, do not
behave in this manner and some of the energy is dissipated through various mechanisms
which are collectively called internal damping. The internal damping in soil is not consid-
ered to be caused by viscosity, the stress-strain behavior of soils under vibratory loading is
similar to that of visco—elastic materials. Asphalt concrete, on the other hand, is a visco-
elastic material in large due to the viscosity of the asphalt. This pennits the use of complex
moduli in representing the stress-strain behavior of soils and asphalt concrete that are sub-

jected to loads varying harmonically in time.

The stress-strain relation under uniaxial conditions can be expressed by:

o*c“°‘ = E*e*e“°‘ ( 3-9)
where E* the is complex modulus which, in general, is frequency dependent. Due to the
viscous eﬂ'ect and consequently the complex nature of E* , the strain vector is no longer in

phase with the stress vector and lags behind it by an angle 8 , as shown in Figure 3.5 (a).
This angle is often called the loss angle and is deﬁned as follows:

= imag(E*) ___ E_2

-1
real(E*) E1 (3 0)

Therefore,

5* = 131(0)) + iE2((o) = |ra=t=|e'i6 (3-11)
The above equation implies that the real component of E* is associated with elastic phe-
nomena in which the energy is stored in a recoverable form while the imaginary compo-
nent of E* is associated with viscous phenomena in which energy is dissipated, and 5 is a

measure of this dissipation.

32

Laboratory tests seem to indicate that the stress-strain curve for most soils and for
asphalt concrete mixtures under harmonic vibration form closed loops as shown in
Figure 3.5 (b). The shape of these loops, for soils, are found to be substantially indepen-
dent of the frequency of vibration and dependent on the strain amplitude involved [106],
while they are dependent on the frequency of vibration in the case of asphalt concrete. The
elastic modulus is expressed as the secant modulus determined by the extreme points on
the hysteresis loop while damping is proportional to the area inside the hysteresis loop.

This behavior is represented by the constant hysteresis model. According to this model

E* can be written as follows:

E* = E(l +itan5) (3-12)

The loss angle 8 can be related to the fraction of critical damping [3 as follows:

tan5=2-[3 if ﬁ«l (3-13)

Substitution of Equation ( 3-13 ) into Equation ( 3-12 ) yields:

13* = 51(1 +1213) (3-14)

Equation ( 3-14 ) is valid at any frequency of vibration. The modulus and damping
ratio are independent of frequency in soils, whereas they are dependent of frequency in the
case of asphalt concrete. The use of a complex modulus implies that the actual hysteresis
loops are approximated by elliptical hysteresis loops which are equivalent with respect to
the slope of the principal axes and the areas enclosed by the loops. Thus the secant modu-
lus equals the real component of the corresponding complex modulus. The energy losses

per cycle which are proportional to the area of the hysteresis loops are also matched.

33

3.6 METHOD OF ANALYSIS

3.6.1 Complex Response Method

The complex response method uses the notation of complex algebra to express the
response to a harmonic excitation force. This method is based on the assumption that if the
forcing function is expressed in the complex form then the steady-state response to the
driving force should have the same frequency as the excitation force. From this method,
the complex amplitude is obtained as a function of frequency. The complex response

method is very convenient in solving linear continuum mechanics problems.

Fourier series are used in the complex response method because they allow for the
study of the frequency content of time dependent data in linear system. The development
of the Fast Fourier Transform (FFT) algorithm by Cooley and Tukey [27] in 1965 and the
availability of high speed computers have greatly increased the use of complex Fourier

analysis.

The advantages of the complex response method in analyzing engineering prob—
lems are: First, it can handle damping very effectively; damping can be introduced via a
complex stiffness and can be added easily where needed. Additional damping created by a
semi-inﬁnite medium can be modeled by complex dampers. Second, the transient solu-
tions can be obtained from the steady-state solutions using the inverse Fast Fourier Trans-

form algorithm.

3.6.2 Dynamic Analysis
Stationary dynamic analysis is performed in the frequency domain using the com-
plex response method; transient loading is handled by Fourier transform techniques. An

efﬁcient interpolation scheme in the frequency domain is used to reduce the computational

34

effort. Material damping is introduced by the use of complex moduli. Multiple loadings
can be applied at the surface and each loading can have a different radius and a different
time history. Static loads can be simulated by specifying a harmonic input with zero exci-

tation frequency.

The computation steps used to calculate the response of the multi-layered system
subjected to stationary surface circular loads are as follows: (1) The dynamic loads are
decomposed into harmonic components using the Fast Fourier Transform (FFT) algo—
rithm; (2) Green's functions for displacements, stresses and strains are derived in the fre-
quency domain from the solution of the eigenvalue problem. The response at each
discretized frequency can thus be obtained by multiplying the Green's function by the
complex amplitude of the loading at that frequency; and (3) the response computed in the
frequency domain is then converted back to the time domain using the inverse Fast Fourier
Transform. This technique has the advantage that the Green’s functions in the wave-num-
ber domain are algebraic rather than transcendental. Thus, the Hankel transforms required
for an evaluation of the Green’s functions in the spatial domain can be computed in
closed-form [54]. A computer program, SAPSI, was implemented following these proce-
dures by Chen [21]. Details of the above solution techniques are described in the follow-

ing sections.

3.7 STEADY-STATE SOLUTION

3.7.1 Introduction

If an impulse of short duration is created at the surface of an elastic half-space, the
body waves travel into the medium with hemispherical wave fronts, and the Rayleigh
waves will propagate radially outward along a cylindrical wave front. The ground dis-

placement due to Rayleigh wave arrival is much greater than that for body waves. Also,

35

the attenuation of the amplitude of the Rayleigh wave is slower than that for the body

waves. Hence Rayleigh wave is considered as the dominant wave near the surface.

If horizontal layering occurs in a half-space some energy originating at the surface
and traveling into the half space will return to the surface. Multiple total reﬂections within
the upper layer can generate a second type of surface wave called the Love wave. But a
Love wave will not occur if the superﬁcial layer is the higher velocity layer. Hence, the
effect of Love wave is not considered in this research because the surface pavement layer

is denser and has a higher velocity than the lower layer in the pavement system.

3.7.2 Stiffness Matrix Approach

The pavement loading problem can be characterized as a wave propagation prob-
lem, because the motion in the pavement and the layered system underneath at is domi-
nated by propagating waves. The waves are generated at the surface by the moving wheel
loads, and from there they propagate into the underlying medium. The effect of the propa-
gating waves is to disperse energy into the inﬁnite medium and at the same time to carry
disturbance a long distance away from the source. Due to the energy dissipation by the

waves, the pavement response to dynamic loads is damped.

The layered system can be treated by the ﬁnite element method in the horizontal
direction; however it is numerically more efﬁcient to use continuum theory in the horizon-
tal direction because each layer can be assumed to be homogeneous and of inﬁnite extent.
The natural layers are subdivided into thin sublayers and it is assumed that the displace-
ments within each sublayer vary linearly in the vertical direction. In the horizontal direc-
tion, the displacements are required to satisfy the pertinent ordinary differential equations

which are obtained from the governing, partial differential equations.

36

At a given frequency, the free motion in the discretized layered region consists of a
ﬁnite number of wave modes which are obtained by an algebraic eigenvalue solution.
These wave modes serve as shape functions for expanding the displacements in terms of
mode participation factors. These nodal forces which have to be applied at the boundary
are derived from the displacement expansion by observing the strain—displacement and
stress-strain relations. The nodal forces are uniquely related to the simultaneous nodal dis-
placements at the boundary through a dynamic stiffness matrix. The dynamic stiffness

matrix is complex, symmetric, and frequency-dependent.

Referring to Figure 3.6, consider the jm layer and consider the equilibrium equa-
tion at the interface. Since the system is linear, the relationship between the forces and the
displacements at the top and bottom of each layer may be written in the frequency domain

as:

[xﬂtvp = {13,-} (345)
where, K. is the effective stiffness matrix for the jth layer, {\7} and {R} are dis-
1 1

placement and load vectors in the frequency domain, respectively. The displacement and

load matrices, as shown in Figure 3.6, are expressed at the layer interfaces as:

. . . . T
{vi}T = {va ivzJ va1 ivz’H} (3 l6)
. . . . T -
{Pf = {Px’ iP’ P,” 1131”}

2

where the factor i = J: in the front of V2 and P2 has the advantage that the stiffness

matrices thus deﬁned are symmetric. In the case of a multi-layered system, the global stiff-
ness matrix [K] = {[Kj] } is constructed by overlapping the contribution of the layer

matrices at each interface of the system as depicted in Figure 3.7. The global load vector

37

 

 

 

j V2.11"

 

 

1"“ Layer v25, P 1’
j+l Vx’”. Pg" 1

 

V25“, sz+1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

K
1
K2
K3
m1
b1
_J K":
— ThisPart
“"_ l— I I Notde
L___.J

FIGURE 3.7 Assembling of Submatrices of Each Layer

38

corresponds to the prescribed external tractions at the interface. Thus, the assemblage of
stiffness and load matrices are similar to that of structural dynamic problems in the fre-

quency domain.

3.7.3 Layer Stiffness Matrix

Assuming that the displacement is the exponential function exp(-ilcx) in the x-

direction, Waas [118] expressed the element stiffness matrix as;

2 2
[K3 = [Aj]k +[Bj]k+[Gj]—o) [Mj] (347)
where, 1: is the wave number of a generated surface wave which decays or propagates

energy in the positive x-direction, and (o the is excitation angular frequency. Each subma-

trix corresponds to a pavement layer. Denoting the thickness of the jth layer from the top

by h j , the mass density by p j , the shear modulus by Gj, and the Lame’s constant by M ,

the layer submatrix is [55]:

 

 

_2(ZG]-+)tj) 0 (201.41.) 0
[All = 5 0 201- 0 G]-
6 (zojntj) 0 2(2Gj+xj) 0
_ 0 G]. 0 2c;L
_ q (3-18)
0 (Aj—Gj) 0 (—).j—Gj)
[Bil = l (lj—Gj) 0 (ll-+01) 0
2 0 (kj+Gj) 0 (41.4.0).)
_(—xj—Gj) 0 (—).j+oj) 0 _

 

 

 

 

Gj 0 (—Gj) 0
[OJ = i 0 (201.4»).1.) 0 -(2Gj+7tj)
11,- —Gj 0 G]. 0
_ 0 —(2Gj+).j) 0 (ml-mod (“9)
r2 0 1 6
[Mil = ﬂ 0 2 0 1
6 1 0 2 0
_0 1 0 g

 

 

The displacements within the layer are obtained by linear interpolation between

the interfaces:

v=Cv1+(l—C)v2 OSCSI (3-20)

As described in the previous section, the assemblage may be understood in the

sense of the ﬁnite element method, with each thin layer constituting a linear element. The
global stiffness matrix is obtained by overlapping the matrices for each layer and the same
is done for the global load and displacement vectors. For prescribed loadings, the displace-

ments are obtained by formal inversion of the stiffness matrix.

{P}=[K]{V} then {v}=[x]"{P} (3-21)

In practice, this inversion is not necessary since the spectral decomposition would
be employed. This necessitates solving the quadratic eigenvalue problem which will be

described in the following section.

3.7.4 Eigenvalue Problem
It was shown [118] that the eigenvalue problem and its solutions are the same for

the plane and syrmnetric case. Thus, the eigensolutions obtained from the plane case may

40

be directly applied in seeking the response of the layered system for the axisymmetric

C386.

To solve the site response problem for surface waves, it is necessary to form and
solve the eigenvalue problem for the pavement proﬁle model. Based on the horizontally
layered system and the linear variations of displacement within each layer, Wass [118] has
formulated the eigenvalue problem for the plane strain condition. The free vibrations that
may occur in the layered system consist of a displacement ﬁeld of generalized Rayleigh

waves of the form:

{v} = {¢}exp(iwk—ikx) (3-22)

The mode shape { 4)} and associated wave numbers, k, can be determined from an
algebraic eigenvalue problem. The natural modes of wave propagation in the stratum are
obtained from the eigenvalue problem that follows from setting the load vector equal to

zero.
(1A1k2+131k+1C1){¢} = {0} (3-23)

where, [A] = 21.41.1413] = 2113].], [C] = 2{[Gj]—m2[MJ-]},and co is the exci-
tation circular frequency, k is the eigenvalue known as the wave number, and {11)} is the

associated eigenvector with 2n components. The matrices [A], [B], and [C] are of order
2n x 2n that are assembled from submatrices for the pavement layers according to the
scheme shown in Figure 3.7. The equation constitutes a quadratic eigenvalue problem,
with eigenvalueskj and modal shapes 1p j. In this model there are 2 degree of freedom

associated with each layer interface with total of 2n degree of freedom for a n-layer sys-

tem. The solution yields 2n Rayleigh modes and Zn wave numbers.

41

While a quadratic eigenvalue problem may always be solved as a linear eigenvalue
problem, this is not necessary here because of the special form of the matrices involved.

After rearranging rows and columns by degree of freedom, Equation ( 3-23 ) can be witten

in the partioned form:
sz +C B ¢
* . . *2 { "}={°l
1:2sz k Az+Cz 1% 0

The eigenvalue problems represented by the above equation yield the eigenvalues and pro-

duce the following eigenvectors [54]:

[Y] = [BMW and [Z] = L mall] (3-25)

1<I>,1 Ell‘l’z

where, [E] is a diagonal matrix of modulus and WM] and [(1)2] are deﬁned by eigenvec-

tors {‘91:}, and {1112}s as follows:

[<Dx] = ({¢x}1.{¢x}2w~,{¢x}2N> and [(Dz] = ({¢z}l.{¢z}2,m,{¢z}m) (3-26)

The matrices [y] and [2] were shown to be orthogonal to each other such that

[54]:
[YJTIKHZJ = [E] and (YJTIEHZJ = -1513 (3-27)
_ A, o _ c, Bx,

where [A] = [8sz A2] and [C] = [0 C2] (3-28)

3.7.5 Green’s Functions

According to Kausel [S4] and Waas [118], the effective stiffness matrix and the

corresponding eigenvalue problem are the same for the plane and axisymmetric cases.

42

Therefore, the equilibrium equation in the spatial domain may be derived from the results

of Section 3.7.3 and Section 3.7.4.

For a distributed vertical pressure with intensity q acting over a circular area with
radius a, as shown in Figure 3.2, the load vector in the axisymmetric spatial domain may

be expressed by:

{P}={pr}=q{(:} OSrSa (3-29)
132

and, {p} may also be expanded into its Hankel transform as:

Pr 1 411m 0 13,
[{Pz} - EKL t”: 0 ‘Jo(kf)]{f>z}dk] (3-30)
and by its inverse transform [99]:
13’ 1 -Jl(kr) O 0
{132} — EIL) m 0 -Jo(kr)]q{1}d‘
0
=1: (—q)rJ,(kr){ 1 }dr ( 3-31 )
O
=9E‘J,(ka){l}

where k is the wave number and J m is the Bessel function of the ﬁrst kind and mth order.

For the same layered system as shown in Figure 3.2, the displacement vector {u}

may be written in terms of cylindrical coordinates as:

u
{u}={’} (3-32)

112

43

It should be noted that {u} can be expanded into a Hankel transform in the spatial domain

[55]:

ur(r’ 9, Z) 1 _Jl(kr) 0 ﬁr
{mall—15kt. 2’11. -thh

and the inverse Hankel transform:

ﬁ,(k, Z) 1 _]l (10') 0 ur
{ﬁz(k’ 2)} - 511:0 (1)][ 0 —J0(kr)]{uz }dl' (3'34)

Now consider the force equilibrium in the spatial domain. The equilibrium equa-

tion may be derived from equations ( 3-15 ), ( 3-24 ), and ( 3-28 ), which give:

([x]k2+[<'51){V1 = {F} <3-35)
where, {V} = {R7, kVZ}T {13} = {13r 1:13,} .
After some handling of the above equation, the displacements in the frequency

domain are described in terms of inverse Hankel transform as follows:

{v,}_ { 1<I>,1 }(k,[E] [E],)-1{[EM<I>,1H1‘>,} (336)
w, ’ [<I>z][E] 1<I>,1 r13,

which can be reduced to

( 3-37 )

{v,}_ 1<I>,,11D11<1>,,1T k1<b,11131"1D11d>,1T {13,}
V 1445111311131"[<I>,,1T 1<I>,11I>11<1>,1"~ 1‘2

2

where [D] = (k2[I]-[E]2)_l.

44

The displacements at the mth layer due to a vertical circular load acting at the nth
layer may then be obtained by substituting Equation ( 3-31 ) and ( 3-37 ) into equation
(3-33), which gives:

2N

ur = qanJIS ([1281 /k

s=1

“a - 0 ( 3-38 )
2N

2 «12124133. ¢2,.I /k

s=l

:
II

where:

1
1ls = E—z—k-z—Jo(kr)11(ka)dk

 

(k - 3
=71?— J0(ksr) H32)(ksa)———12— for OSrSa
1 s ks a
4
1t (2)
=2iks 11(ksa) Ho (ksr) for rZa

 

l
l s = —J 11(hr)J (ka)dk
2 r1131: )

Jl(kr r)H(2)(k a) for OSrSa

_J£
21
2% Jl(ksa a) II(12)(ks r) for rZa

and In, are the Bessel functions of the ﬁrst kind with the mth order, and Hmm are the

Hankel functions of the second kind and mth order.

45

The stresses at the center of the mth layer due to a vertical circular load applied at

the n‘h layer may be obtained by substituting Equation ( 3-38 ) into Equation ( 3-3 ), as

follows;

=(x + 2G)— + 1(3):). +95)

___ qa:2"[¢'£;”2+ <1». “M

s=l

m+l
+qa 22 th[¢l~__8___ (1)2. 8}; slls

s=1

         

 

 

I2s 1
>7]— ( 3-39 )

1. 20 1.3—‘5.- 3",
°=(+ )—+ (8rr+ 322)

 

s=l

m+l m m+1 m
¢x, s +¢x,s n 123 1 4) s ‘1) s n
+lqaL2§£ [ 2 )0“ at E + 22 N[—z"—hm:—L_}z. $118]

=1 s=l

q)m +1 +¢m
=(1 + ZG)[q32N 2 [ x s 2 x }z, ”Ii—3.] (3'40)
3

 

 

l 2G azu 1.3—u”- ur
=(+ )_+ (arr-+1.)

¢m+l $2
= 011+2G)[qa2 2 KT}; 11.] 13-41)

s=1

m+l m‘
x,s +(pres n 8128 123 1
WM 2 lash *7 .

=1

 

 

_Gau,+ Buz
1”- (32” Brz)

m+l q)m I m+l +¢ms
= anLZN [Hkg‘s—k—zjz + 2 N[_z:s_.2___’_}:.s 0-129]
= 1 . m S

s=l

( 3-42 )

46

If Green’s function in cylindrical coordinates is deﬁned as the response of the sys-
tem due to a load of unit intensity, the Green’s functions for the displacements and stresses
induced by the vertical circular load can be obtained by setting the load, q, to equal 1.0 in
the above equations. The displacements and stresses in the frequency domain may then be

computed by multiplying Green’s function by the complex Fourier amplitudes of the load.

3.7.6 Validation of Steady State Solution

The accuracy of the SAPSI program has been veriﬁed against available “exact”
solutions for the cases of a single-layered system, a homogeneous halfspace, and a two-
layered system. Tabatabaie compared SAPSI predictions with experimental results from
non-destructive tests [105]. Chatti et al. compared SAPSI results with data from FWD and

truck tests at the PACCAR Test Track [19].

A single-layered system subjected to a static uniform circular load was utilized to
compare the SAPSI solutions with those obtained by Harr [43], Ueshita and Meyerhof
[112], and Milovic [70]. The static load was simulated by specifying a zero excitation fre-
quency. The comparisons showed that the static responses of the single-layered system

computed by SAPSI are very close to those obtained by others.

The half space solution of SAPSI was veriﬁed against the solution by Sung [104].
The halfspace was simulated by frequency dependent layers and dashpots attached to the
bottom of the layered system. The vertical surface displacements at the center of the
loaded area for the two cases were computed and compared with those obtained by Sung.
The comparison showed that the static response could be approximated by calculating the

response at a very small frequency.

A two—layered system with different properties for each layer was analyzed. The

system was subjected to a surface circular load. The static results were compared, and

47

indicated that the static responses of the two-layered system computed by SAPSI are very

close to the solution by Burmister [13].

3.8 TRANSIENT SOLUTION

3.8.1 Introduction

The complex response method is formulated for steady state vibrations. However,
transient motions such as impulse loads may be analyzed using discrete Fourier transform
techniques. Using these techniques the basic input is speciﬁed at N points uniformly dis-

tributed over the period T. The given function values are:

Ps(t) = P(s-At) ; s= 1,2...N—1 (3-43)

where, At is the time interval TN. The input can be expressed using the exponential

functions:
N/2
Ps(t) = Re 2 Psexp(iwst) (3-44)
s = 0

where the frequencies are deﬁned as (08 = 21ts/(NAt) , and P5 are complex load

amplitudes deﬁned as follows:

N-l
P = £2 P(nAt)exp(—i(osnAt) for s = 0 and N/2

n = O
N_l (3-45)
2 P(nAt)exp(—iwsnAt) for s = 1, 2...N/2—1

n=0

2

Ps N

If N is chosen as a power of 2, the complex amplitude can be computed by the Fast

Fourier Transform (FFT) algorithm, developed by Cooley and Tukey [27]. Since

Equation ( 3-44 ) is a truncated Fourier series, the function Ps( t) should be periodic over

48

the period T. Actual moving pulse loads are not periodic. However, by the addition of a
“quiet zone” consisting of a limited number of trailing zeroes to the input, both of the
above requirements can be satisﬁed. The quiet zone should be chosen sufﬁciently long
such that the response occurring at the beginning of the next cycle is very small due to the
system damping. In this case, the response within each cycle is not inﬂuenced by the pre-

vious cycle.

Discrete values of the response, U(t) , in the time domain at interval At may be

computed using the inverse Fourier transform on { Us} which is the solution for a single

harmonic input.

To obtain a complete solution, the system of linear equations must be solved for all

of the following FF’I‘ frequencies.

8

ms S _
-2-1_t_T’S-l’2"”’N/2 (3-46)

This is a formidable computation task. To minimize the computational cost, a cut-
off frequency is introduced whereby the response for frequencies above a given frequency,

f are set to zero. Also, an efﬁcient interpolation scheme is used so that the complex

max ’
response amplitudes in the frequency domain are computed for a certain number of
selected frequencies and the values for the rest of the FFT frequencies are obtained by

interpolation. The interpolation method is described in the next section.

3.8.2 Interpolation Scheme

The interpolation technique was developed by Tajirian [107] to interpolate the
response for all frequencies in-between the calculated frequencies. The technique is based
on the frequency response function of a two degree of freedom system. This technique is

general in that it works for multi degree of freedom system, and it is possible to choose the

49

computed frequencies at wider intervals to further reduce the cost of analysis. The total
response of a two-degree-of freedom system subjected to a harmonic load for each degree-

of-freedom has the following general form:

clco“ + c2002 + c3

 

U((o) = (3-47)

(1)4 + c4032 + c5
where, U((u) is the response at frequency to and c1, 02, c3, c4 and c5 are constants.

Thus, if the response of the system is known at ﬁve frequencies, the ﬁve constants may be

 

 

obtained from:
- . r .
(of (012 1 —wa1 —U1 c1 ‘ (”imi
mg 03% 1 —(1)§U2 —U2 02 with
(of: (0% 1 —w§U3 -U3 4 c3 > = < w§U3 > (3-48)
wt] 0)} l —(JJ§U4 —U4 04 w2U4
Lwé (03 l —0J§U5 —U5_ LCSJ kwg‘USJ

 

 

 

 

Following the computation of the ﬁve constants from the above equation, Equation ( 3-47

) can be used to obtain the interpolated responses for all the frequencies in the range from

to1 to (05.

The interpolation techniques described above may be used for interpolating trans-
fer functions for multi-degree—of freedom when the transfer functions behave like those of
the two-degree-of-freedom system within small frequency regions. By interpolating the
transfer function in each range so as to cover the entire range, the transfer function values

are computed for all FFT frequencies shown in Equation ( 346 ).

50

3.8.3 Algorithm and Validation of 'lT'ansient Analysis

Transient dynamic analysis (sometime called time-history analysis) is a technique
used to determine the dynamic response of a structure under the action of any general
time-dependent loads. This type of analysis may be used to determine the time-varying
displacements, strains, and stresses. SAPSI uses the following procedures for transient
analysis;

(1) Decompose the dynamic load into harmonic components to obtain complex

amplitudes using the Fast Fourier Transform (FFT) algorithm.

(2) Obtain the steady-state response due to a unit harmonic load at a ﬁxed number
of frequencies.

(3) Interpolate the steady-state response for all FFT frequencies using the obtained
interpolation function as described in the previous section.

(4) Calculate the actual response by multiplying the complex load amplitudes with
the unit response.

(5) Finally, obtain the transient response in time domain using the inverse FFI‘
algorithm.

The ﬁrst procedures are well known and established; the second procedure is the

steady-state solution which was veriﬁed by Chen as described in Section 3.7.6. Thus, only

procedure (3) and (4) are required to be veriﬁed for the transient analysis in SAPSI.

To verify the coding of transient analysis, the results from SAPSI were compared
with those obtained outside the program. First, the transient solution was obtained using
SAPSI. Then, steady-state solutions at ﬁve frequencies were obtained using the SAPSI
program. With these solutions, the transient response was calculated using the mathemati-
cal program Maple (a system for mathematical computation in symbol, numerical, and

graphical) following the procedures above.

51

Figure 3.8 summarizes the algorithm of transient analysis, which was used in
Maple. Figure 3.8(a) shows the complex load amplitudes in the frequency domain. The 16
points in Figure 3.8(b) show the steady-state response as obtained from SAPSI’s steady-
state solution. Only ﬁve points among 16 points were chosen for interpolation. The solid
line in Figure 3.8(b) shows the extended steady-state solution at 512 points by using the
interpolation function. The difference between the solid line and the 16 points is very
small. Thus, the interpolation function which is based on the frequency response function
of a two-degree-of freedom system is suitable to apply in the layered pavement system.
Figure 3.8(c) shows the steady-state solution multiplied by the complex load amplitude at
each corresponding frequency. Figure 3.8(d) is the ﬁnal transient solution in the time

domain, obtained using the inverse FFT algorithm.

Table 3.1 is a comparison of the transient analysis in SAPSI and the results using
Maple. The diﬂerences between the results are within 2%. This table shows the accuracy
of SAPSI and validates the computer program coding for transient dynamic analysis in
SAPSI. The SAPSI transient analysis also was validated against the full-scale FWD ﬁeld
test, as will be shown in Chapter 5.

TABLE 3.1 Comparison of Transient Analysis

 

 

 

 

 

 

 

Uz Error (96)
Ma le—SAPSII
SAPSI 4.66E-08 I P =
SAPSI x 100 1.71
Maple 4.745-011 lMaPlc ‘ SAPS" x 100 = 1.68
Maple

 

 

 

3.9 SAPSI COMPUTER PROGRAM

The SAPSI computer program solves the dynamic problem of surface stationary

uniform distributed circular loads on a linear visco-elastic layered system overlying a

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

( A ) x 10'8 ( B)
. 5 . .
. x Selected for Inter P.
u 1 a
a 0)
3 5 4 4
”6 ‘8
Q m
2 a
£3 '32 . *
E E
1 <
A ._k L o A A
100 200 300 O 100 200 300
Frequency ( Hz) Frequency ( Hz)
x10’7 (0) x10"8 (0)
5 - . .
- 5 . 4 .
.3 .3
o 4 J o
”’6 ‘5
Q 3 . 4 (D 2
2 2
.t .2
25.2 - E.
E E
< 1 _ < o _ ,
o . L . 4
O 100 200 300 0 0.5 1 1.5
Frequency ( Hz ) Time ( Second )

FIGURE 3.8 Validation of Interpolation Scheme by Maple

(a) Fourier Transform of the Load versus Frequency

(b) Steady-state Displacement Amplitude Due to a Unit Load versus Frequency
(c) Steady-state Displacement Amplitude versus Frequency

(d) Transient Displacement Time History

53

semi-inﬁnite halfspace or a rigid bedrock. SAPSI is a FORTRAN computer program for
execution on an IBM/PC AT, developed by Chen [2]] at the University of California-Ber-
keley. The half space underlying the layered system can be simulated by vertically extend-
ing layers and a series of dashpots attached to the bottom of the extended layers. The shear
modulus, damping ratio, and Poisson's ratio for each layer may be varied with the excita-
tion frequency of the load. Multiple loads are acceptable for both the harmonic and tran-
sient cases and each loading may have a different radius and time history. A static load can
be simulated by specifying a harmonic load with zero excitation frequency. The program
uses a special scheme in the frequency domain to interpolate between results obtained for
only a few frequencies. This provides a signiﬁcant reduction in the computational effort
required. The program uses a dynamic allocation technique which automatically sets the

dimension of the program to accommodate the input data.

The program can be run interactively with the aid of menus shown on the screen of
the monitor. The input data may be created directly from the keyboard, or read from exist-
ing ﬁles. The program is designed in such a way that the results obtained in diﬁerent
phases of the execution can be saved for later use. For instance, the model for the layered
system, the eigensolution, the input loading, the output time histories, the coordinates of
the output points, and the general printout can be saved in different ﬁles for future refer-

CHCC.

CHAPTER 4

DYNAMIC RESPONSE OF VISCO-ELASTIC MULTI-
LAYERED SYSTEM TO MOVING ARBITRARY LOADS

4.1 INTRODUCTION

Various models have been developed to obtain the response of asphalt pavement,
mostly based on layer theory or ﬁnite element method. The most commonly used models
in asphalt pavement analysis are well summarized in Table 2.1 of Chapter 2. The majority
of these models are for static surface loads and use linear elastic theory. The effects of
inertia forces and of moving loads are usually ignored. Recent studies have shown the
inertia and moving load eﬁects to be signiﬁcantTheir inclusion in pavement analysis

models would, therefore, seems to be necessary in order to obtain accurate results.

The objective of a mechanistic design procedure is to use a rational approach
which would be able to consider the needed complexities in modeling. An accurate predic-
tion of the pavement response will generally require more accurate material properties and
a better representation of loading conditions. Some program has been achieved in material
characterization through the use of non-destructive testing (e. g. Falling Weight Deﬂecto-
meter Tests) and with the development of a new generation of laboratory tests through the
Strategic Highway Research Program. In recent years, there have been new attempts to
develop pavement models with more realistic loading conditions; i.e. models that able to
analyze moving loads with arbitrary time histories. The objective of this chapter is to
develop a moving load solution for the visco-elastic multi-layered system described in

Figure 1.1.

54

55

4.2 ASSUMPTIONS

Although procedures for performing dynamic analyses have become highly devel-
oped, the completely general three-dimensional dynamic analysis which would be repre-
sentative of a pavement system subjected to a moving load is very complicated and
requires too much computational effort to be an acceptable for routine pavement analysis.
The computational eﬁ'ort can be reduced through simplifying assumptions regarding mate-
rial properties, the type and distribution of the dynamic loads, and the geometry of the
problem without a signiﬁcant sacriﬁce in accuracy. The following sections brieﬂy discuss
these simpliﬁed assumptions and their incorporation into the development of a new solu-

tion and an associated computer program.

4.2.] Material Properties

Although pavement materials are essentially non-linear, deﬂections in ﬂexible
pavement are sufﬁciently small under vehicle loading for the assumption of static linearity
to be reasonable. Recently, Hardy and Cebon [39] showed that the instrumented pavement
section used in their experiments was linear for impulse loadings over a wide frequency

range.

Pavements are inherently variable in properties because of construction practices
which lead to variable layer thicknesses and material properties. In addition, compaction
being done in the longitudinal direction only, may cause some measure of anisotropy in
the pavement layer properties. However, measurements made by Hardy and Cebon [39]
indicated that the dynamic response of their test pavement was relatively isotropic over a

short distance up to 2 meters.

Hence, it is assumed that each pavement layer consists of a homogeneous, isotro-

pic, and visco-elastic material. The assumption of homogeneous and isotropic properties

56

simpliﬁes the problem and enables the use of the equation of motion described in
Equation( 3—1 ). All materials are assumed to be linear visco-elastic in response to the
loads. With this assumption, the laws of superposition are valid, which, in turn, provide
considerable computational advantages. An additional advantage of using visco-elastic
materials in the computational model is that it leads to stable algorithms and computer
codes. The assumption of viscorelastic material properties for asphalt pavement was vali-

dated by several authors [97, 100].

4.2.2 Geometry

Complex geometries will inevitably lead to complicated costly computer pro-
grams. This is especially true in the case of three-dimensional analysis with moving
dynamic loads. Signiﬁcant simpliﬁcations can be achieved by imposing restrictions on the
geometries of the pavement system and the load distribution, respectively. Many problems
which are actually three-dimensional in space can be reduced with some justiﬁcation to

either plane or axisymmetric problems.

For pavement analysis, the single most effective restriction is to allow semi-inﬁnite
horizontally layered models. This restriction effectively reduces the problem from three to
two dimensions by utilizing the axisymmetric coordinate. Each layer is inﬁnite in the hor-
izontal directions and has a ﬁnite depth, except for the bottom layer which may have an
inﬁnite depth. The principle of continuity on primary variables results in continuous con-

tact between each interface of the layered system.

The axisymmetric model assumes that both the pavement geometry and loading
are axisymmetric. In particular, the asphalt surface layer must be assumed to be inﬁnitely
wide and therefore the edge effect can not be considered. Non-axisymmetric multiple

loads can be analyzed by superposition for linear elastic or visco-elastic materials.

57

4.2.3 Loads

The road response at the surface and vehicle dynamics are essentially uncoupled
[16]. This is because (i) the displacement of the road surface is considerably smaller than
the deﬂections of truck tires and suspensions and (ii) the speed of propagation of elastic
waves in the road surface is generally signiﬁcantly faster than the speed of the vehicles. As
a result of the weak coupling between the two systems, it is reasonable to assume that the
road surface is rigid when simulating the response of a vehicle to road roughness and treat
the vehicle as a set of moving dynamic loads when simulating the response of the road

surface.

A major challenge to the development of an efﬁcient method for dynamic analysis
of pavements is the fact that actual trafﬁc loads move over the surface of the pavement.
Tire loading is usually modeled as a circular area with uniform contact pressure. It is
assumed that multiple loads that are vertical and uniformly distributed over a circular area
are moving along a single direction at a constant velocity on the surface of a layered sys-
tem. The loads may be constant in magnitude or may vibrate as an arbitrary function of

time.

4.3 MOVING LOAD REPRESENTATION

A moving load develops a response even before reaching the point of interest due
to such important factors as wave propagation, inertia and damping effects. Thus, the posi-

tion of moving load with respect to a point of interest is a function of time.

Consider a unifomrly distributed load over a circular area, moving at a constant
velocity V on the surface of a layered system. The moving load is modeled as a series of
haversine pulses with their duration equal to the time required for the wheel to pass by a

point in the pavement. In order to consider the moving load effect on the response at a

58

ﬁxed point x0 in the pavement, it is necessary to include the effect of the load as it moves

from one location in the pavement to the next and beyond the passage of the load by theses

consecutive locations in space (either approaching or leaving x0). Thus the response time
history of the pavement at the ﬁxed point xo can be obtained as the resultant of the time

histories of the responses due to the load being at these consecutive positions in space.
This insures continuity of loading in time and space as it moves from one location to the

next.

4.3.1 Response to Moving Loads

First, the response at many points along the horizontal direction due to a stationary
haversine pulse is calculated. Figure 4.1 (a) illustrates the discretized pulse, which consists
of a rest period of A - At, where At is the time step, a haversine pulse with a duration
equal to the time required for the wheel to pass by a point in the pavement, and another
rest period of duration B-5t. The total number of time steps should be a power of 2 in the
Fast Fourier Transform algorithm. Figure 4.1(b) shows the response points in relation to

the applied stationary load. The loading and all responses are in the same time domain.

From Maxwell's reciprocity theorem, these results can be converted to responses at
a ﬁxed point subjected to a single dynamic load at diﬂerent locations in space. This con-
version will make the responses corresponding to the different locations of the load in dif-
ferent time domains. In order to bring all the different responses to the same time domain
(to account for a truly moving load as opposed to a quasi-moving inﬂuence line,) each
response is shifted by a time period corresponding to the passage of the load from one
location to the next (equal to the distance between the consecutive locations divided by the
vehicle speed.) This is illustrated in Figure 4.2(a) and (b). Figure 4.3 shows the response

pulses at a speciﬁc location of AC bottom as the load approaches. Note that the response

S9

p(ti) pul) = I: <0>A9 (h), (0)8 ]

 

 

 

 

 

 

 

 

 

 

II 1..
"
”ll 1| +

 

 

 

 

 

 

 

 

IIIIIIITTI IIIIIIITITTVTITII
l 2 3 ......... Ill Ill ............................. II. ti
(a)
M)
xn
‘
‘ 1‘rr-r
X»:
‘
‘2 Symmetry
‘1
,L . , . t . J ' % ﬂ
u.(r,) U...r(tr) U...2(t.) U2(tr) Ur(tr) Uoﬂr)
(b)

FIGURE 4.] Interpretation of Stationary Loading Condition

(a) Discretized haversine pulse
(b) Stationary loading and responses

60

 

 

 

 

 

 

 

 

 

£(ti) X /V
n - an
x. At
ﬂ: p(t‘)=[<0>A,u,<h>,<0>3m]
Unﬁt) U.(t,)=[<0>..a., <H>, <0>bml
pa.)
x /V
xn—l i ‘11—‘1— _ an - 1
true.) Una.) = I < 0 he...» < H >. < 0 >b+an-l I
P(ti)
x /V
L35] At ' “1
SIM) U10.) = [ < 0 >.,a1, < H >, < 0 >5+a1 ]
MT
1 Uo(m=[<0>a,<ﬂ>,<0>b]
”0(9)
P(ti)|
x U.1(t.) = [ < 0 am], < H >, < 0 >b-a1 ]
r} (h)
(a) 1 (b)

PR(t‘)=[<0>A.m,, <IIR >, <0>3.u.]

Dk(t‘)=[<0>..u, <HR>, <0>anlRc

 

(d)

 

FIGURE 4.2 Interpretation of Moving Load Condition

(a) Quasi-moving loads and responses

(b) Time shift in response to moving load
(c) Sequence of moving load

(d) Resultants of moving load and response

61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 o 1 04 . . Xt - .4'0 n . .
a
e
4
2
0 V.
‘20 01.2 0:4 0L6 04.8 1
10 1°45 . . Xo -72.on . .
a
a
4
2
o
“—20 032 0:4 0:6 0:8 1
10 1°45 . . Xt-j'o" T .
a
a
4
2
o W—
-20 0:2 0:4 0:6 ofe 1
1 o ‘ o 5 . Xt - .0'6 n . .
a
a
4
2 /\
o
"20 0:2 0:4 0:6 0:3 i
1 o 1 0.5 . . Xt - 3'2 n . .
a
e
4
2
o
'20 :2 04.: 0:6 ofa 1
1o 1 04 r . Xt - 3'0 n v
a
a
4
2
o
“20 of2 oi4 0:6 0:8 1

Approach

FIGURE 4.3 Response Pulses at a Speciﬁc Location of AC Bottom as the Load

 

 

 

 

 

 

62

becomes larger gradually in compression as the load approaches, but it changes to tension
abruptly when the load is near to the speciﬁc location of AC. The resultant response is
obtained by adding up the response pulses corresponding to each location of the load.
However, because the load pulses have a ﬁnite width they overlap with each other, as
shown in Figure 4.2(c). This translates to an overloading condition. This overloading
depends on the duration of the haversine pulse and on the distance between consecutive
response points. The resultant response needs to be scaled back by dividing it by an over-

load factor equal to the sum of contributing loads at a ﬁxed time (see Figure 4.2(d)).

To get a smoother (and more accurate) response, the sequence of response points
may be made denser near the load. To account for the distance effect of the moving load
the farthest response point should be chosen to be at a far enough distance from the

response point in order to consider wave propagation and damping effects.

4.3.2 Response to Moving Constant Loads

Figure 4.4(a) shows the sequence of moving constant load which is discretized at
an equal time interval and numbered from left to right as a general coordinate. From the
discussion in the previous section, the time history of the response subjected to the moving

constant load becomes the following:

DR(ti) = [0,419+ U1(ti) + + Uj(ti) + + UN(ti)]§l— (41)

where, U j(ti) is the stationary response time history (after time shifting) when the station-

ary load is applied at location j, N=2*n (n is number of response points in the stationary

solution), and R0 is a load factor due to overlapping in time or space domain.

Mth this method, the model incorporates such important factors as wave propaga-

tion, inertia and damping effects of the medium.

63

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R(ti)A
1
0 1 2 3 J N ti
(a)
RG1)
R}_R
RM K\\ /RN
R
\V1
0 1 2 3 J N ti

0))

FIGURE 4.4 Discretized Moving Loads

(a) Moving Constant Load
(b) Moving Arbitrary Load

64

4.3.3 Response to Moving Arbitrary Loads
Figure 4.4(b) shows the time history of a moving arbitrary load. From the discus-

sion in the previous section, the time history of the response becomes the following:

1
DR(ti) = [RoUo(ti) + R1U1(ti) + + RjUN(ti) + + RNUN(ti)]-R— (4-2)
0
where, R j is the ratio of dynamic to static load.

With this model, the effect of magnitude variation (vibration) is considered in

addition to the factors such as wave propagation, inertia, and damping.

A simpliﬁed (quasi-dynamic) solution which accounts for vehicle speed but does
not incorporate the effect of loading frequency can be obtained if the time history of the
rCSponse due to a moving constant load is simply multiplied by the time history of arbi-
trary loading. Equation ( 4—3 ) describes the quasi-dynamic solution for the moving arbi-
trary load:

DR(ti) = [Uo(ti) + U1(ti) + U2(ti) + + Uj(ti) + + “109151;?" (4-3)

This expression is some what different from Equation ( 4-2 ) in the sense that the
effect of magnitude variation is not considered instantaneously. This quasi-dynamic solu-
tion was obtained for the purpose of comparison with the real dynamic solution. Note that
when multiple loads are applied the multiplication of the response time history for a mov-

ing unit load by the arbitrary load time history should be prior to the superposition of the

r<=Sponses due to the multiple loads.

65

4.4 SAPSI-M COMPUTER PROGRAM

A computer program, SAPSI-M, has been written in the Fortran 77 language for
the analysis of asphalt concrete pavements subjected to moving dynamic loads. The pro-
gram is an extension of the stationary program SAPSI, which was developed by Chen [21]
at the University of California-Berkeley. The computer program has the following charac—
teristics:

(l) The steady-state solutions for ﬁxed frequencies can be obtained and stored.

(2) The stationary transient solution can be obtained and stored from the steady-
state solutions and using an efﬁcient interpolation scheme.

(3) The moving dynamic solution can be obtained from the above results.

Figure 4.5 shows the ﬂow chart of the SAPSI-M program. This section will be
dedicated to the description of program layout and both steady-state and moving load

solution algorithm. Uses and limitations are also discussed.

4.4.1 Program Layout

SAPSI-M has ten main subroutines. In a typical run, not all of these subroutines
may be used, depending on the option chosen by the user. A number of these are “sister”
subroutines which can be grouped together due to their interaction. A brief description of
the function of each subroutine and its interaction with other subroutines is presented in

the following:

A. AFFI‘
Subroutine AFFI‘ is used only when the transient solution is required. It arranges
for the performance both of the forward transformation of the input load vector from the
tithe domain to the frequency domain using the Fast Fourier Transform algorithm and of

the backward transform of the solution displacement vector from the frequency domain

 

 

 

 

 

- DECOMP - FFT

 

 

 

 

 

DO I=1,NFRQ

 

 

 

 

 

- RESPL - GENEP
- SECEVA
- KAUSEL
- INTGRL
- HANKEL

 

 

 

 

 

 

 

 

 

i

 

 

 

- INTERF - RANGE
- CNSTNT
- CSOLVE
- EXPND

 

 

V

. SUPER‘I‘ J

 

i

 

 

 

. Print out responses

 

~TRANSI-IFF'I‘

 

 

i

 

 

- MOVE - CONMOV
- ARBMOV
- QUASID
-ENERGY

 

 

 

1

FIGURE 4.5 The Flow Chart of SAPSI-M Program

67

back to the time domain using the inverse FFT. This subroutine has the sister subroutines

FFI', RFFT, and RFSN.

B. DECOMP
Transient dynamic analysis is used to determine the dynamic response of a struc-
ture under the action of any general time-dependent loads. Subroutine DECOMP is used
only when the transient solution is required and the input load data are given in the time
domain. It decomposes the dynamic loads into harmonic components to obtain the com-

plex amplitude using Fast Fourier Transform algorithm.

C. FDPINT / HSPEXT
Subroutine FDPINT veriﬁes that the desired frequencies are within the input data
of the proﬁle and interpolates material properties from the input material data ﬁle using of
linear/linear, linear/log(freq.), log/linear(freq.) or log/log relationships within the subrou-
tine INTER. Subroutine HSPEXT checks the input data on the sublayers and simulates the
half-space using Newton’s iteration method. The layered system is extended by n-addi-
tional layers having properties equal to those of the half space, according to the Variable

Depth Method [20]. Refer to the section that describes the “theory” behind it.

D. INPUTS
The program SAPSI-M can be run interactively with the aid of menus to be shown
on the screen of the monitor. The input data may be created directly from the keyboard or

read from the existing ﬁles and they can also be modiﬁed and saved in either the old ﬁle or

a new ﬁle.

Subroutine INPUTS controls the main input information of loading, conﬁguration
a11d proﬁle of layers, and output inforrnations. Thus, it has many sister subroutines. The

SiStcr subroutine CONPAR controls parameters such as frequency-dependent properties,

68

half space simulation, viscous dashpots on lower boundary and eigenvalue solution. For
the loads information, subroutine INPUTS has sisters subroutines such as DEFLDG
(deﬁne load type such as harmonic or transient), HLDMOD (read harmonic load informa-
tion such as radius and location) and TLDMOD (input for transient load, and specify the
desired frequencies of steady-state solutions). For the layer information, it has sister ones
such as DEFPRO (deﬁne proﬁle such as directory and status), FILMOD (read a frequency
independent proﬁle and modify it as necessary), FDPRO (read a frequency dependent pro-
ﬁle), and HNIPUTD (print input data of materials). Sister subroutine OUTREQ deﬁnes
response components desired and locates the points where the displacements and stresses

are desired.

E. INTERF

The interpolation technique [107] interpolates the steady-state responses for speci-
ﬁed frequencies and obtains the response for all frequencies. The technique is based on the
frequency response function of a two-degree-of freedom system. Subroutine INTERF is
used only when the transient solution is required. The subroutine RANGE computes the
number of frequency ranges to be interpolated. The steady-state responses are recalled by
subroutine READTP from a temporary ﬁle, and the subroutine CNSTNT computes inter-
polation constants for each frequency range with the aid of subroutine CSOLVE. Subrou-

tine EXPND expands the ampliﬁcation vector to all frequencies.

F. MAINE
Subroutine MAINE arranges the main ﬂow of the program for computation of
desired responses in either steady-state or transient analyses. It calls subroutines such as
FDPINT, HSPEXT, RESPL and SUPERP in the steady-state case. In the transient case,
MAIN calls DECOMP, RANGE, INTERF, SUPERT and TRANSI in addition to the sub-

l‘Outines that are required to compute the steady-state response.

69

G. MOVE
Subroutine MOVE contains the moving load solution, which computes the time
history of the pavement response when subjected to moving constant or arbitrary loads.
Subroutine CONMOV and ARBMOV calculate the response due to multiple constant and
arbitrary moving loads, respectively, while QUASID calculates the quasi-dynamic

response to moving arbitrary loads. Refer to section 4.3 for the detailed solution.

H. RESPL

This subroutine the main subroutine which computes the steady-state response. It
calls a number of subroutines, including GENEP, SECEVA, KAUSEL, HANKEL,
INTGRL and BESSJ. Subroutine GENEP initiates the eigenvalue problem. Subroutine
SECEVA solves the eigenvalue problem using the generalized Rayleigh quotient iteration
method as proposed by Wass [118]. Subroutine KAUSEL calculates displacements and
stresses for the given layered system to uniformly distributed vertical circular pressure
based on modiﬁed Kausel’s solution. Subroutine HANKEL computes the Hankel function
for Kausel’s solution. Subroutine INTGRL calculates the ﬁrst and second Kausel solu-
tions. Subroutine BESS] calculates the Bessel functions by polynomial approximation for

Kausel’s solution.

I. SUPERP / SUPERT
If the layered system is subjected to multiple loadings, the responses computed in
the frequency domain are combined before they are transformed back to the time domain.
Since the responses due to each loading are computed in local cylindrical coordinates,
they have to be transformed into global cartesian coordinates prior to combination. These
Sllperpositions are done using SUPERP and SUPERT subroutines for steady-state and
tl’ansient solutions, respectively. The results of SUPERP are printed in the main output

ﬁle, but those of SUPERT are saved in a temporary ﬁles for future use.

70

J. TRANSI
Subroutine TRAN SI is used only when the transient solution is required. This sub-
routine obtains and prints out the maximum global responses of the transient solution in

the time domain from the expanded interpolation results using the inverse Fast Fourier

Transform algorithm.

4.4.2 Algorithm

4.4.2.1 Steady-State Solution
The algorithm used for the steady-state solution is shown in Figure 4.6. Note that
all analysis procedures are executed in the frequency domain. Basically, the following

steps are undertaken:

(l) 0 Input the harmonic load information such as radius and locations.

0 Input the proﬁle of the layered systems.
0 Input the requested output points and the desired frequencies of analysis.

(2) 0 Verify that the desired frequencies are within the input data of the proﬁle.

(3) 0 Check input data on the sublayers to ensure that the sublayers in the simulated
half-space are increasing.
0 Create additional layers for half-space simulation by Newton’s iteration
method.

(4) . Initiate the generalized eigenvalue problem.
- Solve the eigenvalue problem by the Wass method.
- Compute Hankel and Bessel functions for the modiﬁed Kausel solution.
0 Compute Green’s function for a given layered system to uniformly distributed
vertical circular pressures.

(5) 0Transform the responses into the global coordinate system prior to

combination.
- Superpose the responses and print out the global responses.

71

START

. Input loads, conﬁguration
and proﬁle of layers
- Input output points

 

 

 

 

DO I=l, NFRQ

 

 

 

 

 

 

. Solve eigenvalue problem
. Calculate Kausel solution
. Calculate Green’s function

 

 

 

 

 

V

. Transform into global Cartesian coordinate
. Superpose the responses

1

. Print out the global responses

 

 

 

 

 

FIGURE 4.6 Algorithm for Steady-State Solution

72

4.4.2.2 Moving Load Solution

The algorithm used for the moving load solution is shown in Figure 4.7. Note that
the basic analysis obtaining Green’s function is executed in the frequency domain. The
response is then converted back to the time domain to obtain the transient solution. Basi-

cally the following steps are undertaken:

(1) 0 Input the time load history and load information.

0 Input the proﬁle of layered systems.
0 Input the requested output points and the desired frequencies.

(2) o Decompose the general dynamic load time history into complex harmonic
components using the FF'I‘ algorithm.

(3) 0 Verify that the desired frequencies are within the input data of the proﬁle.

(4) . Check input data on the sublayers to ensure that the sublayers in the simulated
half-space are increasing
- Create additional layers for half-space simulation by Newton’s iteration
method.

(5) o Initiate the generalized eigenvalue problem.
0 Solve the eigenvalue problem by Wass’s method.
0 Compute Hankel and Bessel functions for modiﬁed Kausel solution.
0 Compute Green’s function for a given layered system to uniformly distributed
vertical circular pressure.

(6) . Calculate the interpolation function from the steady-state response at specified

frequencies.
. Expand the interpolated results into all frequencies.
. Multiply the above results by the complex amplitude of the load.

(7) oSuperpose the responses in the frequency domain.
(8) 0 Obtain the global responses in the time domain by inverse FFT.

(9) 0 Calculate the response time histories at a speciﬁed location subjected to
moving constant or arbitrary multiple loads, as developed in this study.

73

START

. Input loads, conﬁguration
and proﬁle of layers
. Input output pornts

oDecompose the loads

 

 

 

 

 

 

 

 

DO I=I.NFRQ

 

 

 

       
    
 

. Check desired

frequency
oInterpolate Mat.

 

. Check layer input

for half-space
. Simulate half-space

     
   

 

- Solve eigenvalue problem
. Calculate Kausel solution
. Calculate Green’s function

 

 

 

 

4%

. Calculate interpolation constants
. Expand for all frequencies
- Multiplify by complex amplitude of load

V

- Transform into global Cartesian coordinate
. Superpose the responses

I

. Transform the responses back to
time domain using inverse FFI‘

 

 

 

 

 

 

 

 

FIGURE 4.7 Algorithm for Moving Load Solution

74

 

DO 1:], NLOAD

 

 

 

DO J=1,NOUT I

 

 

 

oShift time of response

oSum up the shifted response

 

 

 

 

 

 

 

 

 

 

oCalculate load factor

1

Obtain moving load solution

 

 

 

 

 

 

 

 

 

oCalculate energy & Fatigue life

 

 

 

l

. Print out responses

 

 

FIGURE 4.7 Algorithm for Moving Load Solution (cont’d)

75

4.4.3 Uses and Limitations

The proposed model contains a new solution for the dynamic analysis of asphalt
concrete pavements under moving ﬂuctuating loads. The structural model is a n-layered
damped-elastic medium. The subgrade can be modeled as either a rigid base or a semi-
inﬁnite halfspace. The loads are surface pressure loads, and the analysis is under axisym-
metric conditions. The moving loads are modeled as a series of pulses with a duration

equal to the time required for the wheel to pass by a ﬁxed point in the pavement.

Some Iirnitations on the use of the model should be mentioned. This model
requires a relatively large computer memory and running time to conduct the analysis. The
preparation of the input data requires some eﬁ‘ort and knowledge of dynamic analysis and
pavement properties on the part of the user. It is always recommended to follow the guide-
lines in the manual provided in Appendix B. The capabilities and limitations of the com-
puter program SASPI-M are listed below.

4.4.3.1 Capabilities
(1) Calculates the steady-state response at any frequency.

(2) Calculates stationary transient response.
(3) Calculates the response to multiple moving arbitrary loads.

(4) The shear modulus, damping ratio, and Poisson’s ratio for each layer may be
varied with the excitation frequency of the loads.

(5) Multiple loadings are acceptable for both harmonic and transient cases and each
loading may have a different radius and time history.

(6) A static load can be simulated by specifying a harmonic load with zero
excitation frequency.

4.4.3.2 Limitations

(1) SAPSI-M deals only with a layered system subjected to uniformly distributed
circular loads.

76

(2) It does not calculate the response at zero frequency when using the half space
simulation; instead the analysis needs to be done at near zero frequency (0.001
Hz).

(3) It does not use a cutoff frequency after which the response would be set to zero.
This concept is very useful when we know that the response is essentially zero
after a certain frequency (i.e contained within a certain frequency range). Had
SAPSI have this option, the user would be able to use small frequency steps (for
accurate interpolation in the frequency domain) with a limited number of
frequencies, while keeping the time step small enough for a smooth response in
the time domain.

(4) The interpolation scheme in SAPSI-M needs a minimum of ﬁve frequencies in
a particular frequency range and a maximum of 64 frequencies.

(5) Often SAPSI-M would give an error message (“Real indeﬁnite...”) which halts
the execution. This is due to the fact that the solution is pseudo-exact and may
“blow—up” at certain frequencies. All that one needs to do is to shift the
frequency at which the response is being computed by one or a few frequency
steps. In the case of transient loading, this can occur at different frequencies.

 

 

5.1 1‘.

menu
ysis w.

miner

5.2

368

5.2.

eXar

CHAPTER 5
FIELD VERIFICATION

5.1 INTRODUCTION

Very few pavement response models have been validated by comparison with ﬁeld
measurement as discussed by Hardy and Cebon [40]. They, also, pointed out that no anal-
ysis was found in the literature of the dynamic response of a realistic layered road model

to moving steady or random forces.

In this chapter, previous studies on full-scale ﬁeld tests are brieﬂy reviewed, and
the results of SAPSI-M will be compared with ﬁeld data. The ﬁeld tests used for ﬁeld ver-
iﬁcation involve a full scale asphalt concrete pavement section on a test track in the PAC-
CAR Technical Center in Mount Vernon, Washington. The tests conducted at the
PACCAR Test Track facility took place at different times during 1991, 1992 and 1993.
They included Falling Weight Deﬂectometer (FWD) tests for both back-calculation of
layer moduli and comparison of strain data (October 1991, June 1992, and February
1993), some preliminary truck tests to check the measurement and data acquisition system

(May 1992), and the formal series of full-scale truck tests (September/October 1993).

5.2 PREVIOUS FULL-SCALE FIELD TEST STUDIES

This section describes existing ﬂexible pavement test facilities and previous analy-

ses of pavement responses due to truck loads.

5.2.1 Test Facilities

Several pavement test facilities have been built since the 1950’s for the purpose of

examining the correlation between theoretical predictions and measured results in real

77

78

pavements under actual loads. The various test facilities can be either linear or circular test
tracks or test roads with controlled or uncontrolled loading. In most of the test facilities,
the construction was controlled to allow instrumentation to be installed during the con-
struction phase. Only a relatively small number of experiments have been conducted using
instrumentation retroﬁtted into an existing pavement. In addition, only a few experiments

applied loads from real full-scale trucks [123].

The advantage of full-scale testing include the following; (1) Excellent simulation
of ﬁeld conditions, (2) Possibility of examining the effect of change in the pavement struc-
tural section on pavement performance, (3) One test track allows the study of other forms
of pavement distresses in addition to fatigue. The disadvantages include the following: (1)
The initial investment cost and annual operation and maintenance costs are very high, (2)
A parallel, supplementary laboratory testing program is still needed, since the ﬁeld track

tests do not directly measure fundamental mixture properties.

In order to obtain full-scale ﬁeld simulation, circular and longitudinal test tracks
have been designed and constructed in a number of different countries. Well-known exam-
ples include the circular tracks located at Nantes, France, and at Pullman, near the Wash-
ington State University, and the Federal Highway Administrations ALF (Accelerated
Loading Facility). Other examples of full-scale tracks include those in Australia (ARRB),
United Kingdom (TRRL), New Zealand (Canterbury), Denmark, and OECD Group

(Organization for Economic Cooperation and Development represented by 12 countries).

5.2.2 Previous Flexible Pavement Tests
Winters [123] provides a good review of the published research on full-scale ﬂexi-
ble pavement experiments. A summary of the major tests is contained in Table 5.1. The

type of test facilities include linear track, circular test track, test road, test pit, and in ser-

 

 

 

 

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82

vice pavements. Loading conditions included circular plate loading, accelerated loading
devices (ALD), linear test apparatus, multi-axle trucks, and falling weight deﬂectometer
(FWD). A variety of strain gauges have been used including, wire gauges, electric resis-
tance gauges, H-gauges, foil-type gauges, dynatest strain transducer, and core gauges.
Measured responses included longitudinal and transverse strains at the top, bottom, and at
various depth of the asphalt concrete layer. Theoretical predictions of pavement response
were obtained from closed-form solutions such as Boussinesq equation, Burmister equa-
tion and Jone’s table of stresses in 3 layer elastic system, and from numerical computer
models based on layered elastic theory such as BISTRO, CHEVRON, BISAR, ELSYMS,
and CLEVERCALC. Table 5.2 summarizes these ranges of experimental conditions
including source of loading, load magnitude, pavement structure, gauge type, and theoret-

ical computation [123].

The review showed several examples of acceptable agreement between measured
and calculated strains in the asphalt concrete layer. Some ﬁeld tests showed unacceptable
agreement between measured and calculated strains without any reasonable explanation.

The following observations were drawn from the review [123]:

(l) Reasonable agreement between measured and calculated strains translates to a

difference of about twenty percent.

(2) The longitudinal and transverse strains are not equivalent in general; this could
be due to the distribution of contact tire pressures in the case of truck tests, to
misalignment between plate and gauge in the case of FWD tests, or to the
anisotropy of asphalt pavement in.

(3) The effect of the thickness of the asphalt concrete layer is not signiﬁcant,
however the location of the strain gauge relative to the bottom of the layer is
critical.

(4) The effect of lateral offset between the position of the loading and the strain
gauge is very significant.

83

 

 

 

 

 

 

 

 

 

 

 

 

 

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84

5.3 DESCRIPTION OF PACCAR FIELD TEST

A series of full-scale ﬁeld tests of a ﬂexible pavement section were conducted at
the PACCAR Technical Center in Mount Vernon, Washington, about sixty miles north of
Seattle, from October 1991 to September 1993. It consisted of falling weight deﬂectome-

ter (FWD) and truck tests. The plane view of PACCAR test track is shown in Figure 5.1.

5.3.1 Test Section

The pavement test section is a ﬂexible pavement with a 5.4 inch surface layer of
dense graded asphalt concrete (AC, WSDOT Class B) over a 13 inch crushed stone base.
The subgrade is a sandy clay. The average cross section of the PACCAR is shown in
Figure 5.2. The water table was measured at a depth of 5.5 ft during installation of the
instrumentation. Fifteen cores were taken from the section for installation of the instru-
mentation. These cores were used to evaluate thicknesses and conduct density tests. Based
on the results the AC layer is relatively homogeneous and of generally uniform thickness.
The instrumented section is approximately 14 ft wide and 40 ft long. It is closed to trafﬁc

except during scheduled pavement testing.

5.3.2 Instrumentation

A foil-type gauge manufactured by Micro-Measurement was used to measure the
various strain responses. An Australian-made Multi-depth Deﬂectometer (MDD) with
four linear variable differential transformers (LVDTs) was used to measure pavement
layer deﬂections. For temperature data, a multi-sensor temperature probe manufactured by

Measurement Research Corporation was used.

A total of 102 strain gauges were installed in the pavement section: Nenty (20)

gauges were installed in ﬁve axial cores to measure longitudinal and transverse strains at

85

-’
'1":;:"s";';:':: Zf 3:;an ‘

C \— High Speed

H Track
Buddings

FIGURE 5.1 Plane View of PACCAR Technical Center

 

 

 

 

 

     

 

 

Asphalt Concrete /
5.4 inch
Cmshed Stone

Base
13.0 inch

 

Sandy Sub
41.0-51.8 inch

.7 Z

 

 

Water Table

FIGURE 5.2 Cross Section of the PACCAR Test Section

 

 

 

86

the top and bottom surfaces of the AC layer; forty (40) shear strain gauges were mounted
on ten cores and a long transverse slot extending from the center line to the shoulder of the
pavement; and forty two (42) surface gauges (longitudinal and transverse) were placed

along the wheel path at about one foot intervals.

The physical layout of these gauges is shown in Figure 5.3. The layout was
designed to ensure the collection of critical pavement responses: The axial cores were cho-
sen laterally to allow collection of strain measurements from both wheel paths and the
approximate center line of the wheel base. The longitudinally oriented surface strain
gauges were speciﬁcally designed to evaluate the dynamic response of a truck as it travels
down the pavement section. The details of the instrumentation procedure can be found in

Reference [123]. The results reported in this study concern only the axial core gauges.

Two types of epoxy were used: One type (Micro Measurement M-Bond AE-lO)
was used to glue the gauges to the pavement, and the other type (Sikadur 32 Hi Mod 2)

was used to bond cores to the pavement or to ﬁll the voids in the cut.

5.3.3 Test Procedure

The truck used was a Peterbilt 359 truck with a load frame and instrumented axles
to calculate tire forces. The photograph and plan views of a Peterbilt 359 are shown in
Figure 5.4. The front and rear axles were equipped with leaf-spring suspensions. Testing
was conducted in three blocks: Mid-moming and afternoon of September 28, and mid-
morning of September 29 1993. Each test block consisted of three sets of tests correspond-
ing to three different tire pressures: 90, 58 and 31 psi. The maximum safe speed for testing
on this track section was 45 mph; so for each tire pressure, three truck speeds were used:
1 -7. 20 and 40 mph. The tests were conducted in triplicates and according to a random

order. The total number of truck runs was seventy.

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(a) Photograph (b) Plan View

 

89

Because some preliminary results from earlier tests showed the effect of lateral
offset (between the center of the tire and the gauge) to be very signiﬁcant, special care was
taken in marking the pavement and reading the tire imprint. Lime dust was used to show

the tire imprint. If the offset was greater than 4 inch the test was repeated.

5.3.4 Load Measurements

Measured loads were used to investigate the variability of tire loads with runs of
equal truck speed and tire pressure and to study the effects of speed and pressure on tire
loads. The results indicated that the variability of the load at constant truck speed and tire
pressure was within 5 percent. Figure 5.5 and Figure 5.6 show the eﬁects of truck speed
and tire pressure, respectively, on the load. Both ﬁgures show insigniﬁcant variations with
no particular trend. Accordingly, it was concluded that there was no need for correcting
the measured response for the level of load applied. Note that this was expected because

the pavement test section was newly built and its surface was fairly smooth.

5.3.5 Strain Measurements
Strain measurements were used to investigate the effects of truck speed and tire
pressure on pavement response. Figure 5.7 shows typical time histories of measured

strains in the AC layer.

The discussion above indicated that the variability of the load was insigniﬁcant in
these tests. Other signiﬁcant variables which should be accounted for in the analysis are
the pavement temperature and the offset distance between the truck wheel and the strain
gauge. It was noticed from the ﬁeld measurements that the pavement's surface temperature
did not vary much within a subset of tests with constant tire pressure; this should mean
that the temperature at the bottom of the asphalt concrete layer should be very close to a

constant within each subset of tests. Accordingly, no temperature correction was used in

 

IAve, IStdev,

 

 

 

 

 

 

 

Steer Axle
woo
soon
«too
3
G
E soon
2000
mo
0
1.7 an 40
Speed (mph)
Front Axle
9000
soon
7000
woo
3 5000
v
3 4000
3000
2000
woo
o
1.1 20 40
Speed (mph)
Reer Axle
m
7m
0000
A 5000
a
to
woo
E soon
2000
1000
o
1.1 20 40
Speed (mph)

 

 

FIGURE 5.5 Effect of Truck Speed on Axle Load

91

 

I Ave. I Stdev.

 

 

 

 

 

 

 

 

 

 

Steer Axle
5000
4000
g 3000
e

3 2000
1000
0

31 50 00

Tire pressure (psi)
Front Axle

0000
0000
7000
E 0000
V 5M
'3 4000
3 3000
2000
1000
o

31 50 so

Tire pressure (psl)
Resr Axle
0000
7000
0000
3 5000
-: 4000
I

,3 3000
2000
1000
o

31 so 00

Tire pressure (psi)

 

FIGURE 5.6 Effect of Pressure on Axle Load

 

Micro Strain

Micro Strain

Micro Strain

92

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

31 PSI - 1.4 MPH 31 PSI - 1.4 MPH
50 r A 4
C
“E
a
O U 2
.9
2
-50I 1 -50» i
15 20 25 30 15 20 25 30
Time(sec) Time(sec)
58 PSI - 20 MPH 58 PSI - 20 MPH
50 > i 50'
.E
E
«‘5
0 MN We.“ 2 0
.9
2
-50r —50' i
o i 5 {a 4 o i é :5 4
Time ( sec) Time ( sec)
90 PSI — 40 MPH 90 PSI - 40 MPH
50 . 50- <
C:
6 ,
O o 0
.§
2
-50~ -50.
0 0.5 1 1.5 2 0 0.5 1 1.5 2
Time(sec) Time(sec)

(a) (b)

FIGURE 5.7 Typical Time Histories of Measured Strain at the Bottom of AC Layer

( a) Longitudinal strain ( b ) Transverse strain

93

analyzing the eﬁect of truck speed for a given subset of tests with constant tire pressure.
On the other hand, the effect of the offset between the applied loads and the recording

strain gauge should be accounted for. This was done using the computer program SAPSI.

5.3.6 Correcting for the Effect of Lateral Offset

The computer program SAPSI was used to calculate correction factors to account
for the difference in the strain response due to the lateral offset between the wheel load
and the pavement strain gauge. This was done by calculating the response of the pavement
at diﬂ‘erent offsets along a transverse axis underneath each axle. The SAPSI results were
then normalized with respect to offset by calculating an “offset factor” as described in

Equation ( 5-1 ):

e'x==0
Offset factor = ( 5-1 )

offset

 

where e 0 is response right underneath the load and some, is response at an offset dis-

’ x:

tance from the load.

Offset factor curves for both surface and bottom strains were generated for the

three cores at the three different pressures and at a constant temperature of 68°F. The
effect of temperature on the offset factor curves was checked by obtaining offset factors
using the procedure above for different temperatures and then comparing curves. The
analysis showed that the effect of temperature on the normalized offset factor curves is

negligible.

94

5.4 MATERIAL CHARACTERIZATION OF CORES

5.4.1 Pavement Layer Characterization

The WSDOT Dynatest 8000 FWD was used to conduct deﬂection testing over the
entire test section. Testing was done in 61 locations totaling 130 drops with more tests on
the ﬁve instrumented cores. The computer program EVERCALC was used for backcalcu-
lation of layer moduli. The applied FWD loads varied from 4,874 to 17,880 pounds and

sensor spacings were set at 0, 8, 12, 24, 36, and 48 in. The measured average mid-depth

temperature of the AC layer during testing was 68°F. Layer thicknesses were obtained by
coring. In addition, the subgrade soil was saturated at a relatively shallow depth. The shal—
low water table translates into the saturated subgrade soil behaving as a “stiﬁ” layer.
EVERCALC was used to back-calculate an appropriate value for the stiff layer modulus
as well as to calculate the depth to the stiff layer. The approach consisted of varying the
value of the stiff layer modulus from 10 ksi to 1,000 ksi and choosing the value which
gave the lowest root mean square (RMS) value and the most reasonable AC modulus value
(based on laboratory tests). A stiff layer modulus of 40 ksi was obtained from the conver-
gence procedure. Because of the inability to physically measure the epoxy thicknesses,
“effective” thicknesses were determined by varying the thickness of the epoxy on top and
below each core until the theoretical strains (calculated by CHEVPC) matched the mea-
sured strains. Table 5.3 shows the effective layer thicknesses. The details of all of the

above calculations are presented in Reference [123].

5.4.2 Back-calculation of Layer Moduli
Recently many methods for the back-calculation of layer moduli using nondestruc-
tive deﬂection testing have been developed. Some of the methods use the elastic layer the-

ory while others use the ﬁnite element method. Conceptually, all methods are based on

95

TABLE 5.3 Effective Pavement Layer Thickness of Axial Cores (inch)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Axial Core

Pavement Layer Core 1 Core 3 Core 4 Core 5

Epoxy 0.40 0.25 0.00 0.60
AC 4.90 4.90 4.90 4.90
Epoxy 0.40 1.250 0.50 0.60
Base 12.70 12.00 13.00 12.30
Subgradc (Oct’91) 42.70 46.00 46.10 43.80
Subgradc (Feb’91) 57.10 60.40 60.50 58.40
Stiff Layer semi-inﬁnite semi-inﬁnite semi-inﬁnite semi-inﬁnite

 

iterative routines whereby layer moduli are assumed and the pavement surface deﬂection
is computed. If the computed values match the ﬁeld measured ones, then the calculation is
terminated. It should be noted that the calculated layer moduli are not unique; they depend
on the assumed values of the seed moduli. Moreover, various combinations of layer modu-
lus values may exist such that the calculated deﬂections match the measured ones. Never-

theless, the methods are still in the developmental stage and they can be used to estimate

the material properties of each layer of an existing ﬂexible pavement

As mentioned above, the EVERCALC program was used to backcalculate the
layer moduli. Table 5.4 and 5.5 show the backcalculated modulus values and Poisson’s

Ratio values for October 1991 and February 1993 tests, respectively. The average pave-

ment temperature was 68 0F in October and 44.5°F in February.

 

96

TABLE 5.4 Layer Properties for October 1991 FWD Testing

 

 

 

 

 

 

 

Pavement Layer Modulus (psi ) Poisson Ratio
Epoxy 500,000 0.35
AC 562,000 0.35
Base 14,800 0.40
Subgradc 10,200 0.45
Stiff Layer 40,000 0.35

 

 

 

TABLE 5.5 Layer Properties for February 1993 FWD Testing

 

 

 

 

 

 

 

 

Axial Core 1 Axial Core 3, 4 & 5

Pavement Layer Modulus Poisson Modulus Poisson
( psi ) Ratio (psi ) Ratio
Epoxy 500,000 0.35 500,000 0.35
AC 1,575,700 0.35 1,510,000 0.35
Base 20,300 0.40 27,500 0.40
Subgradc 10,700 0.45 13,400 0.45
Stiff Layer 50,000 0.35 50,000 0.35

 

 

 

 

 

 

 

97

5.4.3 Frequency-dependent AC Properties

Two sets of proﬁles were used in the analysis: One set with the properties of the
asphalt concrete layer held constant with frequency and a second set where the AC proper-
ties were varied with frequency. The reason for using frequency-dependent AC properties
stems from the fact that asphalt is a visco-elastic material and its properties (modulus,
damping ratio and Poisson’s ratio) have been shown to be strongly frequency-dependent
[100]. In the frequency-independent proﬁles, the value for the AC modulus is the one
back-calculated from FWD deﬂection data. In the frequency-dependent proﬁles, the
curves for the AC properties were developed using a similar procedure to the one reported
by Siddhartan et al. [97]. In this procedure, the curves reported by Sousa [100] are used to
describe the variation with frequency of the dynamic Young and shear moduli (IE*| and
|G*|) as well as the damping ratio of the AC layer (see Figure 5.8(a) and (c)). The intercept
for IE*I is obtained by iteration until the peak transient strain due to a unit FWD pulse is

equal to the static strain which corresponds to the back calculatedAC modulus:

N/2

= max 82 IE* I

(5-2)

 

 

EFlwo

where, P8 is Fourier transform of the load and sis frequency number.

Convergence was reached for IEO*| = 273,500 psi. The initial Poisson’s ratio was

obtained from the measured initial values of |E*| and IG*I, as reported by Sousa, using
elasticity. The computed value was 0.537 which is close but exceeds the maximum allow-
able value in elasticity of 0.5. Therefore a value of 0.499 was assumed. The initial shear

modulus, IGO*I, was then computed from elasticity. The variation with frequency of Pois-

son’s ratio was made to ﬁt the observed trends in IE*I and IG*I. Figure 5.8 (b) and ((1)

shows the ﬁnal curves used for the moduli and Poisson’s ratio respectively. To check the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{510, (A)
3
(D
a /
l4110‘5 _,...-——-t'.:i1.".5-’~
us- ........... ’ ’ ’
2 ’ ’ a
a ’ ’ a
g 5 —E:11 C

10 --E:25 C 1
E ---G:11 C
2 - (3:25 C
>~ 4
o 10 ‘ 4

10" 10° 10‘ 10
Frequency (Hz)

10, (C)

o -beta:11 C
2110 --beta:250 1
E? 1
0
Tu
E 1
2
C
_ I

10" . - -
O 10 20 30
Frequency (Hz)

98

N

(B)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r$10 -
e
o
05
10106.
a; /
2 ,,,,,
a 1...»!
E1051 """" 1
E —E:PACCAFI
g ~-G:PACCAR
>~ 4
o 10 4 -
10° 102
Frequency (Hz)
(D)
04 FPACCAR J .

.2
‘5
I
§ 0 2
.‘1‘.’
O
m

0 .

0 100 200 300 400

Frequency (Hz)

FIGURE 5.8 Asphalt Concrete Properties Versus Frequency

(a) Laboratory-measured Dynamic Moduli [100]
(b) Iterated Dynamic Moduli Used In The Analysis
(c) Laboratory Measured Material Damping [100]
(d) Poisson Ratio Used In The Analysis

99

validity of the procedure the proﬁle was subjected to the FWD pulse, and the tensile strain
at the bottom of the AC layer was compared with the measured value. The agreement was

excellent: 133 micro-strains versus 130 measured.

5.5 VERIFICATION OF MOVING CONSTANT LOAD
SOLUTION

5.5.1 Modeling of the Moving Wheel Load

Consider a uniformly distributed load over a circular area, moving at a constant
velocity on the surface of a layered pavement system. In order to consider the moving load
effect on the response at a ﬁxed point in the pavement, it is necessary to include the eﬂect
of the load as it moves from one location to the next in the pavement and before and after
the load passage by theses consecutive locations in space. The moving load is modeled as
a series of haversine pulses with their duration equal to the time required for the wheel to
pass by a point in the pavement. Thus the response time history of the pavement at the
ﬁxed point can be obtained as the resultant of the time histories of the responses due to the
load being at these consecutive positions in space. This insures continuity of loading in

time and space as it moves from one location to the next.

5.5.2 Modeling of Pavement Cores

The same proﬁles for October 1991 were used in the analysis of September 1993
truck tests because the weather conditions were nearly identical. Theoretical strains were
calculated for Axial Cores l, 3 and 4 using the back-calculated frequency-independent
proﬁles as well as proﬁles with frequency-dependent AC layer properties, as described in
the previous section. The numerical discretization of the different pavement layers into
sublayers was done according to the accuracy guidelines provided in previous research

[19]. Figure 5.9 shows the modeling of the different cores. The temperature at the bottom

100

 

Fixed Layers

10 Variable Depth Layers

‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

— : Location of calculated strain

 

 

 

 

 

 

 

 

Base

Subgrade

Stiff Layer

Layer Thickness (inch)

Layer Core 1 Core 3 Core 4
Epoxy 0.40 0.25 0.00
AC 4.90 4.90 4.90
Epoxy 1.25 1.25 0.50
Base 12.70 12.00 13.00
Subgrade 42.70 46.00 46.10

 

 

 

 

 

FIGURE 5.9 SAPSI-M Model Used in the Analysis

 

101

of asphalt concrete layer was assumed to ﬂuctuate less than the surface temperature

because of the thickness of asphalt concrete. The temperature was assumed to be uniform

through the asphalt concrete and to be 68°F. Theoretical strains and deﬂections were cal-
culated at the bottom of the asphalt concrete, not at the bottom of epoxy, because the strain
gauges were glued to the asphalt concrete with epoxy which was then used to ﬁll the voids

in the core.

5.5.3 Results of SAPSI-M Analysis

SAPSI-M was used to predict the horizontal strains in the asphalt concrete layer at
the three different cores, and the theoretical predictions were compared with experimental
results. Most of the analysis shown in this study considers only a single wheel for the pur-
pose of simplicity. A few runs were done with a half-truck in order to compare strain

pulses from single and tandem axles.

5.5.3.1 Typical Response Curves of Moving Constant Loads

Typical response curves for longitudinal and transverse strains are shown in
Figure 5.10 and Figure 5.11 respectively. Note that the steer and drive axles have different
offsets from the measuring point (three inch) difference between the center of steer's tire
and that of the outside dual tire. For this reason the peak strain magnitudes from the steer
axle and the drive (dual tires) axle are very similar. If the strains (transverse direction, in
particular) were measured right underneath the tire the strains due to the single tire of the
steer axle would have been signiﬁcantly higher than those due to the dual tires of the drive
axle [36].

Figure 5.10 shows excellent agreement in magnitude and shape between the ﬁeld
measurements and the strains calculated using SAPSI-M. Of particular interest is the

response curve for longitudinal strain at the bottom of the asphalt concrete layer. The

Micro Strain

102

LONGITUDINAL STRAIN, V220 MPH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

80 I I If I
— FieId Measured
—- - - SAPSI—M
60 - 4
I 1
l
r
I .' ‘
! .045
l
!
I .
i!
'I
_40 l l l l
0.5 1 1.5 2 2.5

Time ( sec )

FIGURE 5.10 Typical Response of Longitudinal Strain for Moving Constant Loads

Micro Strain

103

TRANSVERSE STRAIN, V-20 MPH

 

80 I f I l
— FieId Measured
-- - - SAPSI—M
60 r .

 

 

 

 

 

 

 

 

 

\. I t k.
\ I \-_. -1
\- “ - — I). “\.y.“._~.~.’.’
-20 r 4
_40 J l 1 l
0 0.5 1 1.5 2
Time (sec)

2.5

FIGURE 5.11 Typical Response of Transverse Strain for Moving Constant Loads

104

curve shows a strain reversal from compression to tension as the load passes over the mea-

suring point, and it is asymmetric.

The response curves for the transverse strain, on the other hand, do not show com-
pressive strains as the load approaches and leaves the measuring point. Figure 5.11 shows
good agreement between measured and calculated values, although not as good as for lon-
gitudinal strains. This could be due to the fact that the model in SAPSI-M is axisymmetric,
whereas the pavement is constructed in lanes less than 12 ft wide. The horizontal bending
strains in the “short” (transverse) direction should be less than those in the “long” (longi-
tudinal) direction due to smaller constraint. Another factor could be the uncertainty in the
measured dynamic loads at the peak strain points as the truck moves, i.e. matching the

load time history to the position of the load relative to the pavement.

5.5.3.2 The Effect of Frequency-Dependent AC Properties

SAPSI-M results using both frequency-dependent and frequency-independent AC
layer properties were compared with ﬁeld data. Figure 5.12 compares measured longitudi-
nal strains and predicted values using both frequency-dependent and frequency-indepen-
dent AC layer properties in Core 1, with the truck moving at increasing speeds. The ﬁeld
data and the calculated values show a consistent decrease as the truck speed increases.
Increasing the speed from creep to 40 mph causes the strain to decrease by about only
10% for the frequency-independent proﬁle, while it is 28% for the frequency-dependent
proﬁle, and 32% for the actual tests. Clearly, the proﬁle with frequency-dependent AC
properties gives much closer results to the ﬁeld data than does the proﬁle with frequency-
independent properties. This is an indirect ﬁeld veriﬁcation of the laboratory test results
obtained by Sousa [100] and others indicating that the asphalt concrete properties are

strongly dependent on the frequency of loading.

105

 

 

 

 

 

 

 

 

 

 

LONGI. STRAIN TRANS. STRAIN
200 . . 200 . . -
3 IE
“I. 150- 1 "5150’
E E
9100’ 5100’
£75 :0
._° (Ii
2° 50 5 50»
3 1:
0 ‘ ‘ ‘ O ‘ * *
0 20 40 60 O 20 4O 60
Speed (mph) Speed ( mph)
LONGI. STRAIN TRANS. STRAIN
.E 1 \
E . ...............
5,,- o.al x
30...
M
§0-4’ -SAPSI-M F—Dep
z --SAPSI-M F-Indep
02' b Field Data
0 A . A O a A A
0 20 40 60 0 20 40 60
Speed (mph) Speed ( mph)

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE 5.12 Comparison of Calculated and Measured Strains at Different Speeds

-Core 1

 

 

106

5.5.3.3 Effect of Truck Speed

The eﬂ'ect of truck speed on longitudinal and transverse strain at the bottom of the
AC layer is shown in Figure 5.13 and Figure 5.14 for Cores 3 and 4 respectively. Both
measured and calculated strains (using the frequency-dependent proﬁle) show a consistent
decrease as the truck speed increases. For Core 3, increasing the speed from creep to 40
mph causes the predicted longitudinal strain to decrease by about 27% as compared to
30% in the actual tests. Core 4 results show a predicted decrease in strain of 24% versus an
average of 32% in the ﬁeld. The agreement in the speed effect is excellent. However, the
absolute values of the measured and predicted response are signiﬁcantly different for Core
4. Considering that both Cores l and 4 have the same properties and very similar thick-
nesses of the AC and epoxy layers, and the fact that Figure 5.12 and Figure 5.14 show the
calculated strains for Cores 1 and 4 to be very close to each other and that measured
strains from FWD tests also showed the responses of Cores 1 and 4 to be very similar in
magnitude [18], calculated strains are thought to be more reasonable than the ﬁeld data in
the case of Core 4. This discrepancy could be due to calibration after two years have

passed since the installation of strain gauges.

In the case of transverse strains, increasing the speed from creep to 40 mph causes
a decrease of 20% for both calculated and measured values in Core 3 (Figure 5.13),
whereas Core 4 shows a decrease of 22% in calculated strains versus 40% in the ﬁeld. This
means that measurements of transverse strains show larger variability than those of longi-

tudinal Strains.

_. re
a 8

Longi. Strain ( E-6)
8

LONGI. STRAIN

 

ﬁi

' 1
M

 

 

 

 

 

 

 

50» 8
0 A A L
0 20 40 60
Speed (mph)
LONGI. STRAIN
e: 1 °
5 I
a 0.3 1
8
E 0.6
(U
E 0.4 . «
z 0.2 1
O _L A A
0 20 40 60
Speed ( mph)

107

 

 

 

 

 

 

 

 

 

 

 

TRANS. STRAIN
200 . . -
:3 150 .
7%
vi 5 8\
5 50»
*—
0 A A L
0 20 4O 60
Speed (mph)
TRANS. STRAIN
1
.g o
a; 0.8 ’ O
5 0.6 + 8
a
E 0.4 .
2 —SAPSI-M F—Dep
0.210 Field Data
0 A A A
O 20 4O 60
Speed I mph)

FIGURE 5.13 Comparison of Calculated and Measured Strains at Different Speeds
-Core 3

 

 

Longi. Strain ( E-6)

 

 

 

 

 

 

 

 

FIGURE 5.14 Comparison of Calculated and Measured Strains at Different Speeds

LONGI. STRAIN
200 - .
150 b \
100 lo
0 o
50 °
0 A A A
O 20 40 60
Speed ( mph )
LONGI. STRAIN
1
0.8 I o
0.6» g
0.4
0.2
O A A A
O 20 40 60
Speed ( mph )

108

 

 

 

 

 

 

 

 

 

 

TRANS. STRAIN
200 . - -
71‘
m 150 \__\
.E
g 100 10
w
(I)
S 50’ 8
'—
0 A A A
0 20 40 60
Speed ( mph)
TRANS. STRAIN
1 P
E o o o
3,,- 0.6 - 0
go...
I“
E 0.4»
o —SAPSI-M F—Dep
Z .
0.2 ’ o F1e|d Data
0 A A 4A;
0 20 40 60
Speed ( mph)

Core 4

 

 

109

5.6 VERIFICATION OF MOVING ARBITRARY LOAD
SOLUTION

5.6.1 Moving Arbitrary Load Time History

Ramp test were conducted to investigate the theory of “spatial repeatability”. This
concept states that for any given truck speed, the wheel load time histories generated by a
particular heavy vehicle are repeated closely on successive passes over a given stretch of
road. Since all heavy commercial vehicles have approximately the same natural frequen-
cies and are driven at approximately the same speed on highways, then for a given pave-
ment the dynamic wheel load peaks would always occur within a relatively narrow band of
road sections [36]. Accordingly, the issue of spatial repeatability is central to truck-pave-
ment interaction because some portions of the road may incur much larger damage than
other portions. The excitation source was a test ramp designed to excite the rigid body
sprung mass modes of the test vehicles and not the unsprung mass modes (i.e. axle hop).
This was done to help simplify and analyze the pavement strain response data. The ramp
design consisted of a 4 ft by 2 inch ramp-up, a 7 ft ﬂat section, and a 4 ft by 2 inch ramp.

down.

The load measurement from the ramp tests was taken to simulate the moving arbi-
trary load. Figure 5.15(a) shows the measured time histories of the vibrating component of
the loads for left front steer, left front drive and left real drive axles. They show a large
variation in magnitude due to the ramp which was installed just ahead of the instrumented
track. Because these loads are moving, it is necessary to shift the time history such that
they match the locations of the wheels relative to the pavement. Figure 5.15(b) shows the
load history in space, which was obtained from Figure 5.15(a) by adding the static loads
and converting the load history in time to the history in space. Note that these load histo-

ries in space are used in the SAPSI-M analysis.

LF STEER

110

 

1000’

-1000 '

Load ( ID )

—2000'

 

 

 

 

 

 

5
Time ( sec)
LF DRIVE

10

 

6000
4000 .
2000

1

Load I 10 )
O

-2000 ’
-4000 '

 

 

 

r

 

 

O

5
Time ( sec)

10

 

6000 .
4000 ’

2000 '

Load ( ID )
O

A:
53
c.’

-4000 '

 

 

 

 

LR DRIVE

‘

 

 

-6000
O

5
Time ( sec)

(a)

10

FIGURE 5.15 Measured Wheel Loads

LF STEER

 

8000
7000'
Asooo -
35000 »
8
-I 4000 »
3000 -

 

r

 

 

 

2000

5

10

Location ( 11)

LF DRIVE

15

2C

 

1 4000

12000 '

Load(|b)

6000 *

 

1 0000 -

°§°

 

 

 

4000

5

10

Location ( 11)
LR DRIVE

15

2C

 

10000

9000 ’

Load ( lb)

6000 >

8000 .

7000 ’

 

5000
0

(a) in Time (b) in Space Domain

7

 

 

 

5

10

Location ( II)

(b)

15

2(

111

5.6.2 Typical Response Curves of Moving Arbitrary Loads

The measured time history of surface longitudinal strain due to one truck passage
was compared to the predicted response using the solution for moving arbitrary loads.
Typical response curves for longitudinal strain using moving arbitrary dynamic loads is
shown in Figure 5.16. The ﬁgure shows excellent agreement in magnitude and shape
between the ﬁeld test and calculated from SAPSI-M computer program. This response

curve, also, shows a strain reversal when the load passed over the measuring point.

Figure 5.17 shows a typical response curves for longitudinal strain using moving
arbitrary quasi-dynamic loads (non-vibrating moving loads). Although the agreement in
magnitude and shape between the ﬁeld response curve and the calculated curve is less than

Figure 5.16, it is still very good.

Agreement between SAPSI-M’s predictions and ﬁeld measurements of longitudi-
nal strain was excellent in the moving arbitrary dynamic analysis and very good in the
quasi-dynamic analysis in both the shape and magnitude of the response. These results
show that although the amplitude of load vibrations is high, neglecting the effect of fre-
quency, while maintaining correct allowance for truck speed and the magnitude of the
dynamic wheel loads, does not change the response in any signiﬁcant way. Therefore, a
simpliﬁed solution where the moving loads are assumed to be non-vibrating seems to be
sufﬁcient for accurately predicting the pavement response. A similar conclusion was

reached by Hardy and Cebon using a simpliﬁed beam on Mnkler foundation model [40].

Micro Strain

112

TIME HISTORY

 

40 T I I l

 

 

 

 

 

 

 

 

 

 

 

 

 

1.4

-20 I-
-40 P
—60 r ‘
-- ' - SAPSI-M
— Measured
_80 1 1 I L I I
O 0.2 0.4 0.6 0.8 1 1.2

Time ( sec)

FIGURE 5.16 Comparison of Measured and Predicted Response Curves of

Moving Arbitrary Dynamic Solution

Micro Strain

113

TIME HISTORY

 

40

 

 

 

 

 

 

 

 

 

 

 

 

 

 

J
-- . - SAPSI-M
— Measured
0 0.2 0.4 0.6 0.8 1.2 1.4
Time ( sec)

FIGURE 5.17 Comparison of Measured and Predicted Curves of Moving

Arbitrary Quasi-Dynamic Solution

114

5.7 VERIFICATION OF SAPSI TRANSIENT ANALYSIS USING
FWD TESTS

The response of AC pavement to Falling Weight Deﬂectometer (FWD) tests was
examined by Hoffman and Thompson [47] as part of an extensive investigation on a num-
ber of nondestructive testing devices. They concluded that the FWD deﬂection pulse width
is of the order of 30 millisecond. Thus, the analysis of FWD pulse tests consists a good

transient analysis of pavement.

The asphalt concrete test section was tested using WSDOT's Kuab 8000 Falling
Weight Deﬂectometer (FWD). The FWD tests were conducted in October 1991, June
1992 and February 1993. This section describe the validation of SAPSI transient analysis

against the full-scale FWD test and static predictions.

5.7.1 Characteristics of FWD Field Data

5.7.1.1 Linearity of FWD Field Data

To investigate the linearity of FWD ﬁeld data, the measured longitudinal strain val-
ues from all three FWD test series were plotted against the applied FWD load for all cores.
Figure 5.18 indicates that the FWD measured strain data does not exhibit any signiﬁcant
non-linearity for the range of applied loads for all three seasons and for all four cores. The
ﬁgure also shows that the stiffness (which is obtained from the slope of the curve) is high-
est in February 1993 and has about the same value in October 1991 and June 1992. This is

expected and is due to the pavement temperature: The average mid-depth temperature at
the test site was measured to be about 44.5 °F in February 1993 versus 68°F in October

1991 and 71 °F in June 1992.

X 104 CORE No.1

 

   

 

 

 

 

 

 

 

2 4 a
:3
v I
'8 1.5 .’°
:3 ./
/
E 1’ z
5 .’
2 ! -Oct 91
O 51 --Feb 93 l
0 100 200 300 400

Longitudinal Strain (E-6)
X 10‘ CORE No.4

 

  

 

 
 

 

 

 

 

 

 

2
g /
V1 5» ’
‘o - /
8 /
-J I
‘3 /
OJ 1 r /
S /
0) , —OCt 91
5 , - Feb 93
0.5_ . - "JU1192
0 100 200 300 400

Longitudinal Strain (E—6)

115

 

 

 

 

 

 

 

 

 

 

 

  
  

 

 

 

 

 

 

x 10 CORE No.3
2 - .

"g
g 1.5 ’
O
.1
8
a 1 '
B
2 ‘5‘”3’3

0.5 1 ‘ ‘ "°

0 100 200 300 400
Longitudinal Strain (E-6)
X 10‘ CORE No.5
2 .

g I
:1 5 l I
g . [I
O
.1
g 1 i II
0)) I
< I "ECI 91

0.5 L _ 9° 93

0 100 200 300 400

Longitudinal Strain (E—6)

FIGURE 5.18 Linearity of Strain Measurement in FWD Tests

116

5.7.1.2 Isotropy of FWD Field Data

To investigate the isotropy in strain measurement, the ratio of transverse to longitu-
dinal strains was plotted as a function of the applied load for all cores and seasons
(Figure 5.19). The ﬂatness of the curves indicates that the transverse strain data also shows
a linear behavior. Ideally, the ratio should be equal to one if the material is perfectly isotro-
pic and the boundary conditions are perfectly symmetric. Figure 5.19 shows that the trans-
verse to longitudinal strain ratio varies from about 0.65 to 1.6 considering all tests. Within
the same core, i.e. for the same strain gauges, the ratio can vary by as much as 50% in
either direction from one test to another, indicating that the error is probably due to mea-
surement and not to any fundamental physical behavior. This could also be an indication
to the fact that measurement should be done as soon as possible after installing strain

gauge in the ﬁeld.

5.7.2 Static and Dynamic Predictions of Strains

The SAPSI transient solution was predicted and compared with FWD ﬁeld test,
together with static prediction using CHEVRON computer program. The computer pro-
gram CHEVRON was used to calculate the horizontal strains at the bottom of the AC
layer using the backcalculated moduli. The program is very well known; it uses static
analysis and the linear-elastic layered theory. In the dynamic analysis, the SAPSI com-

puter program was used.

The FWD load is modeled as a haversine pulse with a duration of 30 milli seconds.
Figure 5.20 shows the closeness of the match between the recorded pulse from FWD test-
ing provided by WSDOT and the simulated pulse. Using effective layer thicknesses for
Axial Cores 1, 3, 4 and 5, and back-calculated layer properties, theoretical transient strains

were calculated by SAPSI. Figure 5.9 shows the SAPSI model used in the analysis. The

117

 

 

CORE No.1 CORE No.3

' -'~
I
‘-

-L
0’1
\
\
I
..L
0|

Normalized Strains
Normalized Strains

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 - l 0.5 » .
—Oct 91 -Feb 93
"-Feb 93 --Jun 92
O A A 0 A A
5000 10000 15000 5000 10000 15000
Averaged Load ( IbI) Averaged Load ( lbt)
CORE No.4 CORE No.5
3 1.5 - g 1.5 »
0 a -------------
a a
.8 1 L - " " m .8 1 ' 1
E ’’’’’ T L4 — j
a ............. —— E ‘
§051 -Oct 91 305 4
Z - Feb 93 2 -Oct 91
--Jun 92 - Feb 93
0 ‘ ‘ 0 4 ‘
5000 10000 15000 5000 10000 15000
Averaged Load ( Ibf) Averaged Load ( Ibl)

FIGURE 5.19 Ratio of Transverse to Longitudinal Strains in FWD Tests

1 5000

Load ( lbf)

118

FWD PULSE IN TIME DOMAIN

l I I

 

1 0000

5000

 

 

r I I I

 

— Actual FWD Load
-— - - Simulated Harversine Load

 

 

 

A

 

 

0.005 0.01 0.015 0.02 0.025 0.03 0.035
Time ( Second)

FIGURE 5.20 Actual and Simulated FWD Load Pulses

0.04

119

results for all three test series are shown in Table 5.6 through Table 5.8. Figure 5.21
through Figure 5.23 show comparisons of measured strains with calculated static strains

from CHEVRON and dynamic strains from SAPSI.

5.7.2.1 October 1991 FWD Tests

SAPSI predictions were obtained using proﬁles with both frequency—independent
and frequency-dependent AC layer properties. The results for all cores are shown in
Table 5.6 which also includes CHEVRON (static) predictions for comparison purposes.
Figure 5.21 shows a graphical comparison between predicted and measured strains. Very
good agreement exists between measured and predicted (both static and dynamic) strains:
Ninety percent of measured strains are within 21:10 10 percent of their calculated values.
Static and dynamic predictions using both frequency-dependent and frequency-indepen-
dent AC layer properties are within 10%. SAPSI's predictions using frequency-dependent
AC properties gave the best ﬁt to ﬁeld data. CHEVRON's static predictions gave a some—
what better ﬁt to ﬁeld measurements than SAPSI's calculations using frequency-indepen-
dent AC properties. Nonetheless, very good agreement with ﬁeld measurement was

obtained in all cases.

5.7.2.2 June 1992 FWD Tests

The same proﬁles as in October 1991 FWD tests were used because conditions
were nearly identical for both tests. The dynamic load consisted of the same haversine
pulse with a duration of 30 milliseconds. The results are shown in Table 5.7. In the case of
longitudinal strains, agreement with measured values is very good for both static and
dynamic analyses. All measured longitudinal strains are within :I:10 percent of their cal-
culated values. Figure 5.22 shows a graphical comparison between predicted and mea-

sured longitudinal strains. The measured transverse strains are about seventy percent of

CORE No.1

 

2

5

Calculated Strains (E-6)
8 '8
9 o

 

 

 

O
O

100 200 300 400
Measured Strains (E-6)

CORE No.4

 

§

§

Calculated Strains (E-6)
a g

 

T

 

 

O
O

100 200 300 400
Measured Strains (E—6)

(a) h
a 8

Calculated Strains (E-6)
N
8

CORE No.3

 

 

a

 

 

 

 

 

 

 

 

 

 

1 00 .
O A A A
0 100 200 300 400
Measured Strains (E-6)
CORE No.5
400 .
’“f
m
3:300 '
.E
E
a 200+
32.
-“-’ _ o Chevron
g 1°° x SAPSI F-Ind
o + SAPSI F—Dep
0 A r A
0 100 200 300 400
Measured Strains (E-6)

FIGURE 5.21 Comparison of Measured to Calculated Strains - Oct. 1991 FWD Tests

 

 

 

 

400

CORE No.3
400 .

‘1‘
ur
3300»
.E
g
(0200 1
3..
2
610° 1
m
o

O A A A

0 100 200 300

Measured Strains (E-6)

121

400 f -
9
m
3300'
.E
2
m 200
g .
53 100. K3 Chevron
% F< SAPSI F-Ind
o + SAPSI F-Dep
o A
0 100 200 300
Measured Strains (E-6)

CORENQ4

 

 

 

 

 

 

 

 

400

FIGURE 5.22 Comparison of Measured to Calculated Strains - June 1992 FWD Tests

 

 

 

 

 

CORENo1
400 . -
It
75300
5 200 9 .
E 9
3 1001
(0
o
o A_ A A
0 100 200 300 400
Measured Strains (E-6)
CORENo4
400 -
300 -
$2

‘3

 

Calculated Strains (E-6)
to
8

 

 

O
O

100 200 300 400
Measured Strains (E-6)

CORE No.3

 

(D h
a 8

Calculated Strains (E-6)
N
8

 

 

 

 

 

 

 

 

 

 

 

100- 9 1
o A A A
0 100 200 300 400
Measured Strains (E-6)
CORE No.5
400 .
9
%300» +
.E
9
1’0" 200*
E
to
3100+
a R p Chevron
o X SAPSI F—lnd
O A r A
0 100 200 300 400

Measured Strains (E-6)

FIGURE 5.23 Comparison of Measured to Calculated Strains - Feb. 1993 FWD Tests

123

TABLE 5.6 Comparison of Measured and Calculated Strains - October 1991 FWD
Testing

Axial Averaged Microstrain Ratio (mess/cue)

Load
(lb!) SAPSI SAPSI
Measured CHEVRON CHEVRON

F-I-dep-I- F-Dm- ”Id-ec- F-Dur-

Core

 

TABLE 5.7 Comparison of Measured and Calculated Strains - June 1992 FWD Testing

Axial Averaged Micrmtrain Ratio (mess/calc)

Load
SAPSI SAPSI
Core (lb!) Measured CHEVRON CHEVRON

r-I-depc- F-Dcpo- Mud-ec- “he.

 

124

TABLE 5.8 Comparison of Measured and Calculated Strains - February 1993 FWD
Testing

Axial Averaged Microstrain Ratio (meas/calc)

Load

SAPSI
Core (“)0 CHEVRON CHEVRON
Mud-p-

 

L1: Longitudinal Strain
T2: Transverse Strain

125

the longitudinal strains, and since SAPSI's solution is axisymmetric the predicted values
for transverse strains are about 30 percent higher than the measured values. This ratio
between transverse and longitudinal strains is different from the ratio reported in October

1991 which was closer to one.

5.7.2.3 February 1993 FWD Tests

The same effective layer thicknesses as in the October 1991 proﬁles were used in
the February 1993 calculations with the exception of the depths to stiff layer. The new
back calculated depths are shown in Table 5.3. Table 5.5 shows the back calculated layer
moduli for Axial Cores 1, 3, 4, and 5 as well as the assumed values of Poisson's ratio. The
same haversine pulse with a 30 millisecond duration was applied. The results are shown in
Table 5.8. Figure 5.23 shows a graphical comparison between predicted and measured
longitudinal strains. Agreement with measured values is not as good as in the October
1991 tests for both static and dynamic analysis. About 70 percent of the measured strains
are with in 120 percent of their calculated values. This could be due to measurement
errors. As time passes after the initial installation of strain gauges and the reﬁtting of the
cores into the pavement section, exposure to moisture and temperature ﬂuctuations in the
pavement causes the sensitivity and reliability of the strain gauges to decrease. Also, the
fact that the pavement is signiﬁcantly cooler in February could have an effect on the mea-
sured response because strain gauges, and any instrumentation in general, are very sensi-
tive to hot and cold weather especially for dynamic measurements. Another indication of
questionable measurement is the difference between longitudinal and transverse strains

which was greater in February 1993 than in October 1991 tests.

Considering all the possible sources of error in measurement such as temperature
and moisture conditions, linearity of measurement and performance of the bonded pave-

ment cores, as well as the errors that accompany any numerical modeling, these results

126

constitute a very good ﬁeld veriﬁcation for both CHEVRON'S closed-form solution and
SAPSI's transient solution. The results also imply that static analysis of pavements using
statically back-calculated layer moduli seems to be sufﬁcient for accurately predicting the

pavement's ﬁeld response under stationary dynamic FWD pulse loads.

5.8 SUMMARY

A new method and associated computer program, SAPSI-M, for the analysis of
asphalt concrete pavements under moving transient loads have been developed. The pro-
gram was veriﬁed with ﬁeld data from full-scale pavement tests on an instrumented
asphalt concrete section on a test track in the PACCAR Technical Center in Mount Ver-

non, Washington. The analysis has led to the following conclusions:

(1) Agreement between SAPSI-M's predictions and ﬁeld measurements was
excellent for longitudinal strains and good for transverse strains, in both shape
and magnitude.

(2) The effect of truck speed on the response of asphalt concrete pavements is
signiﬁcant. The test results showed that increasing truck speed form creep
speed to 40 mph reduced the peak longitudinal strain by about 35 percent.
Transverse strain showed more variability with the effect of speed ranging ﬁ'om
20 to 40 percent.

(3) The analytically-predicted horizontal strains in the asphalt concrete layer using
frequency-dependent properties for the asphalt concrete layer were closer to the
ﬁeld measured values than those using frequency-independent AC properties.
This is an indirect field veriﬁcation of the strong dependency of asphalt
concrete properties on the frequency of loading and an indication that the speed
effect on horizontal strain is in part due to the frequency-dependent visco-
elastic properties of asphalt concrete.

(4) A simpliﬁed solution where the moving loads are assumed to be non-vibrating
seems to be sufﬁcient in accurately predicting the response.

127

A series of Falling Weight Deﬁectometer (FWD) tests was also conducted. The
comparison of the measured strain from FWD tests with static and dynamic theoretical
predictions were made using CHEVRON and SAPSI computer programs, respectively.

The following conclusions can be made:

(1) Ninety percent of the measured strains in October 1991 FWD tests and all
measured strains in June 1992 FWD tests were within :ile percent of their
calculated values, using both static and dynamic analyses. Seventy percent of
the measured strains in February 1993 were within :l:20 percent. These results
constitute a very good ﬁeld veriﬁcation of the stationary transient solution in
the SAPSI computer program.

(2) Static analysis using statically back-calculated layer moduli seems to be
sufﬁcient in analyzing FWD ﬁeld tests, despite the fact that static back-
calculation using FWD (dynamic) deflections will lead to "stiffened" elastic
properties.

CHAPTER 6

APPLICATION OF SAPSI-M PROGRAM TO
PREDICTIN G FATIGUE DAMAGE

6.1 INTRODUCTION

Distress in pavements is a form of failure with respect to certain criteria. It is a
phenomenon that causes considerable maintenance or loss of serviceability in a structure
system. Most commonly, the forms of distress in ﬂexible pavements are fatigue cracking,
permanent deformation or rutting and thermal cracking. This study is mainly concerned
with fatigue distress. Prediction of fatigue distress for an asphalt concrete pavement is

possible through the use of dissipated energy concepts.

An energy-based fatigue method incorporates visco-elastic properties of materials
and establishes relationships between the repetitive stresses and strains and the pavement’s
performance or fatigue life via the dissipated energy. Dissipated energy is deﬁned as the
area within a stress-strain loop and represents the energy lost (being dissipated) over one
cycle or summed over a number of cycles to obtain the cumulative dissipated energy in the

pavement as a result of a trafﬁc passage.

With the SAPSI-M computer program, it is possible to solve for the primary
response (stresses, strains, displacements) of asphalt pavement layered systems with
visco-elastic material properties. Using the program, the stress and strain time histories
and the corresponding energy dissipated at any point within the asphalt concrete layer due
to the passage of a moving load may be determined. This dissipated energy will be linked

to the fatigue distress by a relationship between the fatigue life and the energy dissipated.

128

129

The objective of this study is to present an approach based on energy dissipation
and visco-elastic properties of an asphalt pavement for predicting the fatigue life of

asphalt concrete pavements.

6.2 REVIEW OF FATIGUE LAWS IN FLEXIBLE PAVEMENTS

Modern design of pavements should incorporate a mechanistic analysis to predict
distresses in pavements so that damage and life of pavements would be assessed in a ratio-
nal way. It is important to know the fatigue characteristics of asphaltic materials over a
wide range of mixture and environmental conditions in order to be able to incorporate
sound fatigue criteria in the design method. Many research studies have been done on the
fatigue characteristics of asphalt concrete. This section will describe some of these previ-

ous studies.

6.2.1 Prediction of Fatigue Life

Fatigue cracking is considered to be a tensile phenomenon. It is the repetitive
application of tensile strain or stress, at levels considerably below that required to induce
immediate fracture, which is responsible for the initiation and propagation of fatigue
cracks. Fatigue cracking is usually related to the magnitude of tensile strain at the bottom
of the asphalt pavement layer. Sometime, however, fatigue cracking may be initiated at a
diﬁ'erent location, even at the pavement surface. Estimating the potential for such cracking

has been a challenge facing the pavement researchers [76].

Fatigue in pavements has been studied through ﬁeld observations and accelerated
performance tests in the laboratory. Published results from ﬁeld surveys along with theo-
retical and empirical models are used to predict fatigue in pavements by use of material

properties and other engineering parameters.

130

Material properties inﬂuence the occurrence of signiﬁcant distresses. It is very
important to characterize asphalt pavement materials in terms of fundamental properties,
since design systems are becoming more and more based on elastic and visco-elastic theo-
ries. Mathematical models of pavements predict the response of pavement systems based
on linear elasticity or visco—elasticity under repeated loading. The prediction of fatigue life
of pavement systems can be obtained from the relationship(s) between material properties

and distress at different stress or strain levels.

Tensile strain is the major pavement response related to fatigue damage or crack-
ing. Most of the research conducted in this area has so far singled out peak tensile strain as
the cause or determinant of fatigue damage. There is a tendency, however, to use the dissi-
pated energy concept that relates the number of cycles to failure to the amount of energy
dissipation during repetitive loading. In a later section, comparison between the fatigue
life predicted using conventional fatigue models based on peak tensile strain and that pre-

dicted from energy dissipation will be made.

6.2.2 Accelerated Performance Tests

The study of in-service pavements to formulate fatigue model, is often difﬁcult and
time consuming. Many procedures and test methods have been published for deﬁning the
fatigue characteristics of asphalt mixtures from an available information obtained from
both laboratory and ﬁeld tests. These procedures involve a variety of test techniques,
equipment and test conditions. Different laboratory tests, however, can provide widely dif-
ferent results, which can inﬂuence the predicted structural performance of the pavement.
The differences in fatigue test results should be evaluated and explained in order to com-
pare different test results and relate them to actual conditions, before applying them to

design.

131

Tests which have been used for studying fatigue in asphalt concrete mixtures
include the simple ﬂexure test (center and third point bending), two point trapezoidal test,
rotating cantilever test, direct axial load test, tri-axial test, wheel-tracking test, fracture
mechanics test and repeated load indirect tensile test. Criteria used to evaluate each
method for its potential use as a laboratory standard include: (i) The ability to simulate
ﬁeld conditions, (ii) the applicability of test results with performance of in-service pave-
ments. Factors affecting fatigue response include specimen fabrication, mode of loading,

mixture variables, and loading and environmental conditions.

For most of the laboratory test types, two basic modes of loading can be used, con-
trolled-strain which is applicable to, or representative of, thin ﬂexible pavements, and con-
trolled-stress for materials in thicker pavements. As the number of load applications
increases, the stress remains constant and the strain increases as a specimen is damaged in
a controlled-stress test, whereas strain stays constant while the stress decreases in a con-
trolled-strain test. In controlled-stress tests, stiffer mixtures exhibit longer fatigue lives
while in controlled-strain tests, the more ﬂexible mixtures have longer fatigue lives. For
the same mixture type, controlled-stress loading results in a shorter life than controlled-

strain loading [76].

6.2.3 Factors Affecting Fatigue Life

Many factors have been found to affect the fatigue behavior of asphalt mixtures.
These factors may be categorized into loading, mixture and specimen, and environmental

variables [29].

The factors related to load include magnitude, mode, frequency, duration and rest
period. It should be noted that rest period, load duration and frequency are dependent on
each other such that the load duration added to the rest period determine the frequency of

132

loading. Hence the separate effects of these variables on fatigue life might be difﬁcult to

ascertain in a conclusive manner.

Epps et al. [32] and Kennedy et al. [58] have discussed the effect of mode of load-
ing and have concluded that longer fatigue life is Obtained with materials tested under con-
trolled-strain mode. Monismith has explained this result in terms of energy dissipation,

where it was found that it is larger for controlled-stress tests, resulting in reduced fatigue

life.

Though van Dijk [116] claimed that there was no apparent effect produced by fre-
quency of loading on fatigue life, Deacon and Monismith [29] reported that increasing the
frequency of load application in the range of 30 to 100 cycles per minute signiﬁcantly
decreased fatigue life by about 20 percent. In another study, Pell and Taylor [80] found a
signiﬁcant effect of frequency, with higher frequencies producing a longer fatigue life.
Hence it can be argued that the effect of frequency of loading on fatigue life is not well

established and further investigations are needed to verify previous claims.

Fatigue life is signiﬁcantly affected by duration of load. Laboratory test results
have indicated that reduced duration of applied stresses lead to an increased fatigue life
[58]. Craus et al. [28] reported that a loading time in the range of 0.04 to 0.1 second is
appropriate for fatigue testing, in a study aimed at investigating the relationship between

loading time and thickness of asphalt layer for various vehicle speeds.

Mixture variables include asphalt content, asphalt type, aggregate type and grada-
tion and air void content. Increasing the asphalt content increases fatigue life; however,
there is an optimum asphalt content for a mix which maximizes fatigue life without com-

promising stability and consistency.

133

Porter and Kennedy [82] indicated that though the effects of aggregate type and
gradation are not fully understood, tests have shown that mixtures containing aggregates
with increased roughness and angularity and ﬁner gradation have longer fatigue lives.
Aggregate gradation and type also have a complex effect on air void content, which is a
function of mixture composition and compaction. There is evidence that increased air void

content results in a decreased fatigue life.

Craus et al. [28] reported the fabrication of specimens to be an important factor
when studying fatigue characteristics of asphalt mixtures from laboratory specimens. The
primary objective of specimen compaction is to produce realistic test specimens which
reasonably duplicate corresponding asphalt paving in all major respects including corrmo-

sition, density and engineering properties.

Environmental factors inﬂuence the fatigue resistance of asphaltic pavements.
There are immediate effects and long—term changes produced by temperature and moisture
variations, in fatigue characteristics of asphalt mixtures. Temperature is the more impor-
tant variable as most studies indicating that fatigue life increase with decreasing tempera-

ture.

6.2.4 Existing Fatigue Models

The fatigue properties of materials are expressed as a relation between an arbi-
trarily chosen criterion of fatigue life and the corresponding stress or strain. Usually, the
fatigue characteristics of a bituminous material are expressed as a relation between the ini-
tial strain at the beginning of the experiment and the number of load repetitions to failure.
The number of load applications to failure, N, is deﬁned in a number of ways, depending
upon the nature of test such as controlled-stress or controlled-strain. Fatigue life may be

deﬁned as the number of load applications up to the point when the stiffness value of a

134

material is reduced to half the initial value in the case of controlled-strain, while it may be
deﬁned as the number of applications at which a full crack develops in the case of con-
trolled-stress beam tests or permanent deformations reaches 0.25 inches at indirect tensile

tests.

Laboratory results are usually presented as plots of strain versus number of load
repetitions to fatigue failure. Strain is the repeated constant strain in controlled-strain tests
while it is usually taken to be the initial strain in controlled-stress tests. Pell [79] theorized
that much of the variation caused by the state of stress, temperature and mixture properties
can be explained and accounted for by considering strains rather than stresses from tests
when establishing fatigue life relationships. Strain-fatigue life relationships have been

shown to be fairly independent of the variables mentioned.

Early fatigue studies have found that fatigue life is often better correlated with ten-
sile strains than with tensile stress. Statistical analysis is used to ﬁt lines through scatter
test data points, and the slope and relative level of the plots characterize the fatigue behav-
ior of materials. A linear relationship is assumed to exist between the logarithm of applied

tensile strains and the logarithm of fatigue life which is expressed as follows [58]:

Nf = re)" (6-1)

where, Nf is the fatigue life, 8, is the applied tensile strain, and k and n are material

constants determined from laboratory testing. The coefﬁcients k and it completely

describe the relationship between fatigue life and strain and can be used to describe the
fatigue properties of asphalt mixtures. The value of n normally varies between 4 and 6.
Values of 3 and 7 may occasionally be found under extreme measuring conditions of tem-

perature and frequency. In order for Equation ( 6-1 ) to be used in the analysis and design

135

of pavement structures, et is assumed to be the maximum principal tensile strain, a quan-

tity identical to the maximum applied strain in uni-axial laboratory tests and determined

from the complex, multi-dimensional stress state imposed by trafﬁc on pavements.

In an attempt to account for differences observed in the fatigue life-strain relation-
ship due to added factors such as loading frequency and temperature variation, a mixture

stiffness term may be added to the previous equation as follows [73]:

Nf = an’wmixf (6-2)

where, Smi is the stiffness modulus of the asphalt mixture and c is a third calibration

X

material constant obtained from laboratory.

The mode of loading, air-void content have also been found to be important to the
fatigue behavior of asphalt mixture. Monismith et al. [76] proposed a model which

includes these factors as well as the initial strain and the mix stiffness, as follows:

cVo
Nf = aebMFe 23's“ (53)

0
where, MF is the mode factor assuming values of l and -1 for controlled-strain and con-
trolled-stress, respectively, V0 is the initial air-void content in percent, so is the initial

strain, S 0 is the initial mix stiffness in psi, and a, b, c, d, e are regression constants.

Baladi [7] proposed a fatigue model considering that temperature, air void, viscos-

ity and aggregate angularity are also important to the fatigue behavior, as follows:

ln(Nf) = a—b(TT)—-cln(CL)—d(AV)—e(KV)+f(ANG) (6'4)

136

where, TT is the test temperature, CL is the cyclic load, AV is the percent air void, KV
is the kinematic viscosity at 275°F, ANG is aggregate angularity, and a, b, c, d, e are

regression constants.

In practice, fatigue models incorporate several of the important variables men-
tioned earlier as affecting fatigue characteristics and these equations derived from experi-
mental results and regression analyses are expressed in terms of the relevant independent
variables. Figure 6.1 shows the comparison of fatigue models from different test methods
[37]. It is necessary to compare values estimated or predicted by the regression equations
with measured values from an independent set of data which was not used to derive the

equations in order to provide some validation of the equations.

A major difﬁculty with fatigue testing is developing a relationship between the
results of the laboratory tests and ﬁeld performance. To account for differences between
laboratory and ﬁeld response, shift factors are necessary to translate laboratory fatigue
characteristics to those considered to be representative of in-situ performance. Unfortu-
nately, there is no unique relationship. The shift factor appears to be dependent on the
asphalt characteristics, test types, test conditions, and the ﬁeld conditions to which the lab-

oratory test results are being compared.

6.3 REVIEW OF ENERGY-BASED FATIGUE DAMAGE
MODELS

6.3.1 Road Damage Prediction
Road damage refers to degradation of the structural integrity or surface proﬁle of a
road by trafﬁc passage. Current mechanistic pavement design practice in many countries is

to optimize resistance to fatigue and rutting. Analytical models are used to determine the

Fatigue Life

137

   
    
  
  
 

Monismith et at

n93.5l
x4455 x no"
Reitnby .3 T =68‘F
Sterling

 

n93.87 “

K3’3.65 x 10

T: 77‘
Pelt et at
my 5.3

\\ xg-u 1 IO’’
\\ T = SZ'F
\ Monismith et at
\r n, - 5.09 is
\ Ki-iJB x IO
\ T = 40° F

Perl et at

"1' 3.9
K.‘-3.0 i 10‘2
T8 SO‘F

El Kennedy et at
0 Pelt et at

 

 

O
Omniemitn et at
Aﬂoitnb a Slut m e' m
V "'9 1' 3.38 "
(2:103:10
T = 75' F
Kennedy et el)\
~3.ee ,
szJG 3 l0
1’ 8 73° F
l I I r r
I I I I ﬁr l r I l I T I 1
'0 to: to3
Streee billet-leemei

FIGURE 6.1 Comparison of Fatigue Models at Various Test Methods [37]

138

primary responses of a layered pavement structure due to a static, standard wheel load.
The fourth power law is frequently used to convert the estimated trafﬁc during the service
life into an equivalent number of standard wheel loads. The fourth power law stemmed
from the AASHTO road test [1, 45, 46] which showed that the decrease in pavement ser-
viceability caused by a heavy vehicle axle could be related to the fourth power of its static
load. Experimental fatigue and pavement deformation characteristics of the road material
are then combined with the calculated primary response to evaluate suitable pavement

layer thicknesses and material property speciﬁcations.

For in-service pavements, strains induced in the structure vary widely as a result of
variations in the types of axles, their loaded weight, tire pressure, lateral placement, etc.
Miner's hypothesis [71] seems to be a good means for accumulating the damaging effects

of mixed loading. This linear summation of cycle ratios is given by:

n.
—+——+...+—‘+...+——'151 (6-5)

where i is the ith level of applied strain at a critical point within the pavement structure,
ni is the actual number of load application at strain i, and Nif is the number of expected

fatigue failure under the application of strain. Failure is expected to occur when the linear

summation of cycle ratios reaches one.

For the analysis of fatigue damage, the most commonly used primary responses
are the horizontal tensile stress or strain at the bottom of the asphalt layer, since analytical
models generally predict the maximum tensile strain occurring at this location under the

wheel load.

An alternative to using these horizontal tensile stress or strain is that of considering

the constancy of dissipated energy. The number of cycles to failure is related to the amount

139

of energy dissipated during repetitive loading. The fact that the total energy dissipated by a
material which fails under repetitive loading could turn out to be constant irrespective of
test conditions is an important consideration for identiﬁcation of materials and prediction

of performance, which could be incorporated in design.

Several investigators have concluded that a unique relationship between the num-
ber of load applications to failure and the corresponding total dissipated energy per unit

volume exists for a particular asphalt concrete mixture.

This study involves using the entire response time history of stress and strain at a
point in the pavement to predict fatigue damage, rather than using just the peak tensile
strain. This is relevant because ﬁeld measurements have shown that a moving load will
generate a ﬂexural strain inﬂuence line with one tension peak and two compression zone

(beyond and after the peak) at the bottom of the asphalt concrete layer.

6.3.2 Energy Concepts for Fatigue Damage

The use of energy methods to describe and study fatigue response of asphaltic
materials has gained ground in recent years. One such method has been the concept of
energy dissipation. Fatigue of visco-elastic materials like asphalt concrete subjected to
repeated dynamic loading has been associated with the energy loss such that the fatigue

life can be related to the total energy dissipated during the period before failure.

Dissipated energy is deﬁned as the area within a stress—strain hysteresis loop and
represents the energy lost at a ﬁxed point in the pavement as a result of trafﬁc passage.
Figure 6.2(a) and (b) show a typical stress-strain hysteresis loops obtained in controlled-
strain and controlled-stress tests, respectively [102]. The energy can be considered as
being dissipated over one cycle or summed over a number of cycles to Obtain the cumula-

tive dissipated energy. Fatigue tests of asphalt concrete mixtures under laboratory condi-

Stress ( psi)

Stess ( psi)

140

 

250*

200*

150-

100-

50-

I
..e
O
O

r

 

 

 

50

1 00 150
Micro Strain

(a)

200

250 300

 

150*

100L

50*

—100l-

 

l J

 

 

 

200 300
lMeanmmn

(b)

400

500

FIGURE 6.2 Typical Stress-Strain Hysteresis Loop [102]

(a) Strain Control Test
(b) Stress Control Test

141

tions have demonstrated that there is a unique relationship between the fatigue life of the

mixture and the cumulative energy dissipated by the mixture during the fatigue test.

Figure 6.3(a) shows the variation of dissipated energy per cycle with the number of
load repetitions [102]. The dissipated energy per cycle decreases with increasing load rep-
etitions in the controlled-strain fatigue tests; whereas, for the controlled-stress tests, the
dissipated energy per cycle increases as the number of load repetitions increases.
Figure 6.3(b) shows the increase of the total energy dissipated during fatigue tests as a
function of the number of load repetitions [114]. It shows how the cumulative dissipated
energy changes at the controlled-strain and stress test modes. It is seen that the total

energy dissipated in practically the same in both cases.

6.3.3 Previous Studies

Early researchers noticed that a unique relationship might possibly exist between
the number of cycles to failure and the cumulative energy dissipated to failure. If so, labo-
ratory testing could be abbreviated, and compound loading could be handled more
directly. This means that loading mode, temperature, and frequency of loading dot not
have a signiﬁcant inﬂuence on the total energy dissipated prior to failure. Accordingly, it is
argued that this approach permits prediction of the fatigue life of a mixture over a wide
range of conditions based on a few simple fatigue tests. Because of these advantages, con-
siderable effort has been made to investigate possible relationships between cycles to fail-
ure and cumulative dissipated energy. In this method the number of cycles to failure is

related to the amount of energy dissipated during repetitive loading.

Van Dijk[115] is one of the ﬁrst researchers who employed energy methods to
fatigue tests. From tests conducted with different modes of loading, he concluded that the

total energy dissipation due to repeated loading at failure for a certain type of asphaltic

 

I

0.035

.0

8
I

L

O
O
N
01
r
r

Dissipated Energy ( psi)

.0
O
N
r
I
r

 

°'°‘5 x Controlled—Stress Test ‘

O Controlled-Strain Test

 

 

 

 

 

 

A A A_ AAAAA L A A A A A A A A l A A A A A A A A
0.01 5
Number of Cycles

(a)

 

.0
(D
r

  
 
 
 
 
 
 
   
 
 
  
 
 

.° 9
V G
r r

.0
a)
I

.0
b
1

Total Energy dissipated ( J/cm"3 )
.0 .0
Q 0'!

 

.0
N
r

x Controlled-Stress Test
0 Controlled-Strain Test 4

.0
u-A

 

 

 

 

 

.-
p

3 4 5 6
Number of Cycles x 104

(b)

FIGURE 6.3 Typical Curve of Dissipated Energy versus Number of Cycles
( a ) Dissipated energy per cycles [102] ( b) Total dissipated energy [114]

143

material remains constant irrespective of the test mode used. Nevertheless, he indicated
that the controlled-stress mode of loading dissipates energy much faster than the con-
trolled-strain mode. Later, Van Dijk [116] also reported that the cumulative dissipated

energy versus the number of cycles relationship is not independent of the mix formulation
but is independent of test methods (two and three point bending), temperature (50 °F to

104 0F), mode of loading (controlled-stress, controlled-strain), and frequency (10 to 50
Hz). Chomton and Valayer [22] also concluded that cumulative energy is the sole indepen-
dent factor that predicts fatigue life, and that energy seems to be independent of the mix-

“HOS.

Monismith et al. [76] reported that the uniqueness of this relationship for different
types and conditions of testing could not be sustained. In fact, detailed investigation
revealed that these relationships are different for different mixes and are aﬁected by both
test temperature and mode of testing. Despite this disappointment, dissipated energy
remains a useful concepts in fatigue investigation, because the energy dissipated during
the initial loading cycle which captures not only the effect of the imposed strain level but
also of the dynamic mix properties, is a good predictor of cycles to failure and is thus a
key component of surrogate models. Furthermore, dissipated energy is highly correlated
with stiffness decrements during fatigue testing and helps to explain the effects of mode of

loading on mix behavior.

Tangella et al. [108] summarized well the advantages and disadvantages of the dis-

sipated energy methods. The following are considered to be advantages of this approach:

(1) According to Van Dijk [116], the major advantage of this method is that
loading, temperature, frequency of loading, and occurrence of rest period do
not have a signiﬁcant inﬂuence on the total dissipated energy. The number of
cycles to failure is mainly related to the amount of energy dissipated during the
test. If validated, this could lead to dramatic reduction in laboratory testing, and

144

avoidance of the mode of loading issue in laboratory work would be a great
advantage.

(2) This method is based on a physical phenomenon which explains the fatigue
behavior of visco-elastic materials through the accumulation of the distortional
energy resulting from load repetitions.

(3) For both stress and strain controlled ﬂexural tests, there exists a unique relation
between the total dissipated energy per unit volume and the number of load
applications to fatigue failure.

(4) Prediction of fatigue life is possible as a ﬁrst approximation if initial stiffness
and phase angle are known.

(5) Structural design of an asphalt concrete layer to consider fatigue effect is
possible as a ﬁrst approximation.

The disadvantages of this method include:

(1) Accurate prediction of fatigue behavior is not possible without conducting
detailed fatigue tests.

(2) The procedure proposed in this method can not be considered as a design
technique: rather, it serves to indicate the general magnitude of the fatigue life
of a given asphalt mixture.

6.3.4 Dissipated Energy versus Fatigue Life

Several investigators have proposed dissipated energy as a controlling parameter

for the fatigue of asphalt mixes.

Van Dijk [115] suggested that the initial dissipated energy per unit volume per

cycle for a sinusoidal loading can be obtained as follows:

We = 1:6on sin to ( 6-6 )

where, (30 , so are initial stress and strain amplitudes respectively, and do is a phase angle

between stress and strain wave signals.

145

Since the phase angle keeps changing during fatigue test, the fatigue life may be
divided into intervals in which the phase angle is relatively constant and total dissipated

energy is found by summing the dissipated energy of each interval i, as follows:

[I
wfatigue = 2 Wi (6-7)

i= 1
The result of these energy calculations for different asphalt mixes may be
expressed as a function of the number of strain repetitions to fatigue. An independent

energy relationship for each mix has been found between the number of load applications

to fatigue, Nf, and the total dissipated energy per unit volume, Wfatigue, expressed by:

Wfatigue = Aer (6‘8)
where, A and z are material parameters. For an asphalt concrete Van Dijk found that

zwas equal to 0.63 and A was equal to 6.76x104 J/m3 [115].

Sousa et al. [102] proposed a fatigue model based on the dissipated energy from
the ﬂexural beam test that consisted of a l/2 factorial experimental design for mixtures
composed of two aggregates, two asphalts, with two asphalt contents, at two compaction
levels, using two temperatures and two loading modes: Controlled-stress and controlled-
strain. They deﬁned energy dissipation as an exponential function of the number of load

repetitions:

BN
where Wf is the total dissipated energy up to failure, e is base of natural logarithms, A is

a material parameter corresponding to the energy dissipated in the ﬁrst cycle, B is a mate-

rial parameter representing how fast the energy dissipated per cycle changes, and N1: the

is number of cycles to failure. For controlled-stress tests, B is positive whereas it is nega-

146

tive for controlled-strain tests. The ratio of energy dissipated, Rf, was introduced for con-

venience, as follows:

BN,

where R, is the ratio of the energy dissipated at failure to the energy dissipated in the ﬁrst
cycle. Sousa et al. [102] used a least square technique to determine the value of R, for

controlled-stress and controlled-strain tests. They obtained R, as 2.1 for controlled-stress

tests and 0.5 for controlled-strain tests. These results indicate that, for controlled-stress
tests, failure occurs when the area of the loop doubles from the initial value, whereas for
controlled-strain tests the failure occurs when the area of the stress-strain loop reaches
50% reduction from the initial value. Combining the above results into the equations
(6-9) and (6-10) produces, the following relationship of fatigue life and the dissipated

energy in cycle one:

N, ln(2.l )/(0.0071 x Am) for controlled-stress test (6-11 )

N, ln(0.5)/(—0.00247 x A133) for controlled—strain test (6-12)

Monismith et a1. [76] evaluated the fatigue performance of a thin asphalt pavement
section (3.5inch asphalt concrete layer over 12 inch in base) at FHWA’s Accelerated Load-

ing Facility. All tests were performed under a controlled-strain mode of loading at a fre-

quency of 10 Hz under sinusoidal loading with no rest periods at the temperature of 68 °F.
Fatigue tests were summarized in the form of relationships between fatigue life and initial
strain, and initial dissipated energy per cycle. The following equations were developed

using linear regression analysis:

N, = 8.959x10—8(eo)—3'574 R2=0.987 (6-13)

147

N, = 425,31(v.,0)‘1-346 R2=0.987 (6-14)

where, N, the is fatigue life, so the is initial peak-to-peak tensile strain, and W0 is the ini-

tial dissipated energy per cycle in psi.

6.4 PREDICTION OF PAVEMENT FATIGUE DAMAGE USING
SAPSI-M PROGRAM

The prediction of ﬁeld performance using the concept of energy dissipation
requires the use of a computer program to solve the boundary value problems representa-
tive of pavement systems with visco-elastic material properties. The problem is reduced to
determining the energy dissipated at any given point in the asphalt concrete layer with the
passage of a moving wheel load. A crack will initiate ﬁ'om the point where the most

energy is being dissipated.

“With the SAPSI-M computer program, it is possible to solve the problems for lay-
ered asphalt concrete pavement systems with visco-elastic material properties. Using the
program, the stress and strain time histories and the corresponding energy dissipated at
any point within the asphalt concrete layer due to the passage of a moving load may be
determined. This dissipated energy will be linked to the fatigue distress by a relationship

between the fatigue life and the energy dissipated.

The objective of this part of the study is to investigate the possibility of using an
approach based on energy dissipation to predict fatigue damage of an asphalt concrete

pavement using the computer program SAPSI-M.

148

6.4.1 Calculation of Dissipated Energy in SAPSI-M

As discussed previously, the dissipated energy is deﬁned as the area within a
stress-strain hysteresis loop for a speciﬁc location in the pavement corresponding to an
application of moving loads. Using the SAPSI-M program, the dissipated energy at any
point in the pavement structure may be calculated by obtaining the stress-time and strain-
time histories and calculating the area within the corresponding hysteresis loop. This pro-

cedures, however, would require two SAPSI-M runs to calculate the dissipated energy per

cycle.

Given the stress time history from a single run of SAPSI-M program, the strain
time history can be predicted immediately using the appropriate stress-strain relationship.
This will simplify the procedure and will save computer run time. In the following, the
calculation of energy dissipation per cycle in the frequency domain will be shown using

the stress time history only at a point due to the passage of the load.

6.4.1.1 Derivation of Energy Dissipation in 1-D
The dissipated energy in any direction can be expressed analytically as the work

integral:

r
W = [0 o(t) - gifdt (6-15)

Using complex representation, the stress time history can be written in a Fourier

series, as follows:

149

N/2 . t
we'
ReXQs-e
s=0

<r(t)

N/2
= ROLE Qs(cos((ust) + isin(0)st))]

= 0
as = Rer+iIsz

N/2
- _ - .3.
eU)-Re};:wsﬁs e
s =

iro,t

Then the Equation (6-15 ) becomes:

W - [T ReNz/ZZQ eiw't ReNz/Ziu)£8 eim't t
0 s-0 8 s-O SE8

where:

Q = Reﬂs+ilm§2s
E = IEsleia'
E*==Eei5
z Be2iB for B « 1

and

5 = 2BJ(1-i32)
1—252
5 = 25 for [3«l

tan

with

E*

u
8

B = Damping ratio

Complex Modulus of Elasticity

Poissons ratio

Loss angle

Then

(6-16)

(6-17)

(6-18)

(6-19)

( 6-20 )

150

N/2 an N/2
Rez Os-e' ‘ = 2 [Rer- coswst—Imﬂs- sinwst] (6-21)

s=O s=0

N/2 N/2-

. as in)“ 1(08 .
Re ZorroSE—s - e = Re 20113—4 - [(Reﬂs- costust -1sz- smwst)
s = s =

+ i(Isz- coswst+ Rer- sintust)][cosﬁ-isin5]
N/2

= 2 1578 - [(Rer- sin8—Isz- cosS)coscust
s = 0 8

(6-22 )

—(Isz- sinﬁ + ReQS- c085) sincost]
Substituting the above results into Equation( 6-18 ) and taking advantage of the

orthogonality relations of the sine and cosine functions, the expression for the energy dis-

sipation in the l—dimension is expressed as following:

 

N/2
w = 2 ,gilnslz. sinﬁs
s=0 5
W2 (6'23)
= 2 ZTESI-QSQS srnﬁs
s=0 8

6.4.1.2 Derivation of Energy Dissipation in 3-D

The dissipated energy in three-dimensions can be expressed analytically in matrix

form as follows:

w = [ZlomlT- [Cl-[6(t)ldt («S-24)

where:

T T
{U(t)} = {OK oy O'z oxy crxz oyz}

151

.1 —u —u 0 0 0 1
l —u 0 0 0
[C] = l 1 0 0 0
13* (1+u) 0 0
(Hit) 0

. (1w).

 

 

The stress-strain relationship is deﬁned in the usual manner.

1 .
8,,- = §(-5,juokk+(r+u)oij) (6-25)

If only tensile stress/strain is considered in x—direction, the dissipated energy will be sim-

pliﬁed as follows:

T 1 . .
wx = JOE{OxOx—tt(oxoy+oxdz)}dt (6-26)

Substitute Equation ( 6-16 ) and ( 6-17 ) into the above equation gives the following:
0

T T T
wx = Re [0 éoxoxdt — Re [0 goxeydi — Re [0 gcxézdt (6-27 )

From the derivation of the one-dimensional problem and after taking advantage of
the orthogonality relations of the sine and cosine functions, the expression for the energy

dissipated in the x-direction is expressed as follows:

- 2 IEt—Su'lsinssuzcnxsazcﬂys + RCQZS) + ImnXS(Imst + Imﬂzs))) (6-28 )
s = 0 s
N/2

+ 2 g—s(ttcosﬁs(Rest(lmst + Iszs) — Imst(Rers + Rest)»

s=0 s

The expression in the y and 2 directions may be Obtained by substitution, as follows:

152

N/2
=32;— :(sin5s (Re 2y!) $2+Im st))

—22 OE:(ttsin58(Rers(Rers + Rest) + Imﬂysamﬂys + Iszs))) (629)

N/2
+3205: —:(tl.c035s (ReQy s(ImQys + Imﬂzs) — Imst(Rers + Reﬂzs)»

N/21t

wz = ’é—Smna (Re 2,9 + ImZQZSD
s- — 0 Es
N/2
— 2E —(ttsin5 (ReQ (Reﬂys+Rest) “manningys +Iszs))) (6-30)
:2:
+2— —().tcos5s (ReQz S(ImQys + Imﬂzs) — Imﬂzs(ReQys + Reﬂzs)»
s— - 0 Es

The energy dissipation due to shear is expressed by:

N/2
w,, = 2 Msin83(Re29ij+lmZQU-) (6-31)
s=0 Es

From the above derivations, it is possible to calculate the energy dissipation per
cycle in the frequency domain from a single SAPSI-M run with the stress time history as

output. For fatigue analysis, only the contributions from the tensile stresses/strains in the x

and y directions are of signiﬁcance.

6.4.2 Strain, Dissipation Energy and Fatigue Life

Monismith et al. [76] developed fatigue models based on both peak tensile strain
and initial dissipated energy from FHWA-ALF pavement fatigue test. The test was per-
formed on a thin asphalt pavement section (3.5inch asphalt concrete layer over a 12 inch
base) and included single and dual tire loadings consisting a full scale ﬁeld test subjected

to an uni-directional moving load in the speed of 10.5 MPH. Beam specimens (2.5 inch x

153

2.0 inch x 15.0 inch) were made from the sawed asphalt concrete slab sections which were
delivered to the University of California-Berkeley. Controlled-strain fatigue tests were

performed on these specimens at four strain levels at a frequency of 10 Hz under sinusoi-

dal loading with no rest periods, and at 68°F. The fatigue test results were summarized in

the form of equations ( 6-13 ) and ( 6-14 ).

Monismith et al. [76] also simulated the in situ mix performance using an elastic
analysis of the response of the given pavement section under single and dual tires with
12,000 lbs wheel load (24 kips axle load) and 140 psi tire pressure. The thickness were 3.5
inch and 12 inch for asphalt concrete and base, respectively. The stiffness values were
700,000 psi and 15,000 psi for asphalt concrete and base layers, respectively. The maxi-
mum tensile strain at the bottom of AC layer were determined from the elastic analysis,
and the fatigue life was determined using Equation ( 6—13 ) as shown in Table 6.1. They
compared the estimated fatigue life to the in situ FHWA-ALF fatigue test results.
Figure 6.4 shows the in situ FHWA-ALF fatigue test results [76].

As mentioned before, Monismith et al. simulated the test section using an elastic
analysis and estimated the fatigue life only from peak tensile strain. However, in this
study, the peak tensile strain and initial dissipated energy per cycle are determined for the
same test section using the SAPSI-M program in order to calculate the fatigue life. The
same layer properties are utilized: however, some properties, which are necessary for
visco-elastic analysis, are assumed based on the PACCAR ﬁeld test. The moving load is
simulated at 10.5 MPH. Results are compared to the in-situ FHWA-ALF ﬁeld fatigue tests.

These results are summarized in Table 6.2.

The initial dissipated energy per cycle is largest under the single tire conﬁguration

with subgrade stiffness of 5,000 psi, and it is smallest under the dual tire with subgrade

154

TABLE 6.1 Estimated Fatigue Life of FHWA-ALF Section using Elastic

 

 

 

 

 

 

 

 

 

 

 

 

 

Analysis [76]
Fatigue Life Fatigue Life
Tine Conﬁguration Subgradc Tensile Strain from Peak from III-situ
Stiffness (psi) at AC Bottom Strain Test
5.000 551.10-6 40,000
Srngle 10,000 526*10'6 47.000 52.000
5,000 467‘10'6 72,000
Dual 10,000 443.1045 87.000 109.000
200 -
180 -
11322.5 DUAL TIRE
.5 120 - 425/65R22.5 SINGLE TIRE
air
:
.5:
U
E 80 -
U
40 -
0 r
0

24:; Single-Axle Loads, 1000 Repetitions

FIGURE 6.4 In Situ FHWA-ALF Fatigue Test for Surface Cracking [76]

 

155

well with the ﬁeld fatigue test results as shown in Table 6.2.

stiffness of 10,000 psi. This results in the shortest and longest fatigue life, respectively.
The estimated fatigue lives from peak tensile strains compare well to the ones using linear
elastic analysis, with both showing a consistent. The fatigue lives from the initial dissi-
pated energy are underestimated at high dissipated energy level and overestimated at

lower energy level relative to those from peak strain. However, these fatigue lives compare

TABLE 6.2 Fatigue Life based on Dissipated Energy from Moving Load Solution

 

 

 

 

 

 

 

, _ Peak Fatigue Life
Initial Tensile
n... Subgradc Dissipated Strain .
Conﬁguration Stiffness Energy *106 Dissipateed Peak Sm Field
(psi) Wx (psi) 3x( 1 Energy Wx 8;
S' 1 5,000 0.0903 545 36,000 41,500 52 000
mg 6 10.000 0.0815 511 43,500 51,900 ’
5.000 0.0517 471 101,000 69,900 109 00
D "a1 10,000 0.0462 439 124,000 89,400 0’

 

 

 

 

 

 

 

The ﬁndings from this validation effort may be summarized as follows:

(1) The estimated fatigue life based on the peak tensile strain model from the
moving load solution compares well with the estimated fatigue life from linear
elastic analysis and that of the in-situ pavement. This constitutes another ﬁeld

validation of SAPSI-M’s moving load solution.

(2) The estimated fatigue life based on energy dissipation for the dual-tire
conﬁguration is approximately triple compared to that of the single-tire
conﬁguration. The actual pavement fatigue life to surface crack initiation is
approximately double that for the dual-tire configuration compared to the

sin gle-tire conﬁguration.

(3) The fatigue model based on energy dissipation of Equation ( 6-14 ) seems to
underestimate fatigue life at higher dissipated energy levels and overestimate

fatigue life at lower dissipated energy levels.

 

156

(4) The above ﬁndings constitute the possibility of using an approach based on
energy dissipation to predict fatigue damage of an asphalt concrete pavement
by computer program SAPS l-M.

6.4.3 The Effect of Speed on Energy Dissipation

The same proﬁle as in the previous section with a subgrade stiffness of 5,000 psi
was used to study the speed effect by means of initial dissipated energy per cycle and
fatigue life. Three additional speeds (4 MPH, 30 MPH and 60 MPH) were used in the
SAPSI-M simulation, in addition to the previous result of 10.5 MPH. The results are

shown in Table 6.3.

The table shows that peak tensile strains decrease by 23 percent, which leads to
increasing the fatigue life by 2.37 times, while, the initial dissipated energy decreases by
52 percent, resulting in increasing the fatigue life by 4.93 times, as speed increases from 4
to 60 MPH. This suggests that the speed effect is more signiﬁcant relative to energy dissi-
pation than to peak tensile strain, and that it aﬂects fatigue life signiﬁcantly. It should be
noted, however, that speed will generally increase the truck axle vibrations which would
lead to higher loads, and consequently lower fatigue lives. Therefore the speed effects on

load magnitude and pavement response offset each other to a certain extent.

TABLE 6.3 Observation of Speed Effect by Energy Dissipation

 

 

 

 

 

 

 

Initial . Fatigue Life
Speed Dissipated PﬂkTensﬂe .
(mph) Energy Sm“! (8x) Dissipated WSW-II
WX (PSI) Energy Wx 3x
4.0 0.1350 586*10’6 17,300 32,000
10.5 0.0903 545*10-6 36,000 41 ,500
30.0 3 0.0688 503:1: 10-6 59,600 55,100
60.0 0.0566 460* 10-6 85,300 75,800

 

 

 

 

 

 

157

6.4.4 Load Equivalency Factors from Energy Dissipation

The load equivalency factor represents the ratio of the number of repetitions of any
axle load and axle conﬁguration necessary to cause the same damage (example: reduction
in PSI, and damage in general) as one application of an 18 kips single axle load. Since PSI
(Present Service Index) is obtained from measurement of roughness and distress, e.g.,
cracking, patching and rut depth at a particular time during the service life of the pave-

ment, the load equivalency factor may be obtained using energy dissipation.

The same proﬁle as in the previous section (3.5 inch AC thickness with 5,000 psi
subgrade stiffness) was used to simulate the in-situ pavement system. A single tire conﬁg-
uration (24 kips axle load) was used with the load moving at 10.5 MPH, in order to com-
pare with the available data. An 18 kips single axle load (9 kips wheel load) with 140 psi

was also used to calculate the load equivalency factor.

Figure 6.5 shows the types of axle conﬁgurations used to investigate the eﬂ’ects of
axle number (single, tandem and tridem) and space between axles. Figure 6.6 shows the
stress-strain hysteresis loops together with the strain- and stress-time histories of axle
type-1. Due to the compressive strain/stress that the pavement experiences before and after
the moving load passes by a ﬁxed point in the pavement, the stress-strain hysteresis loops
start and ﬁnish in the negative quadrant. The four hysteresis loops are overlapped almost
exactly because the four axles are far enough apart not to interface with each other. This
axle type causes exactly the same damage as four applications of a single axle load (24

kips) based on the observations of strain and.

Figure 6.7 shows the stress-strain hysteresis loops for axle-3. The axle-2 type con-
ﬁguration is not shown because it consists of two hysteresis loops; one inside, and one out-

side (instead of two in Figure 6.7). The strain and stress time histories show the larger

 

Axle- l

Axle-2

Axle-3

Axle-4

Axle-5

Axle-6

Axle-7

158

 

10

TO

A
U
1‘

1658’

1658’

1658’

 

 

m n n
E4 1658’ ﬁg 16.58’ y

FIGURE 6.5 Types of Axle Conﬁguration for Load Equivalency Factor

 

159

 

 

 

 

 

 

 

x 10“ STRAIN-TIME HISTORY
6 T 1 g I I I
4 i-
i a
U)

-2 l l l 1— l l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O 1 2 3 4 5 6
Time ( sec)
x 104 STRESS-TIME HISTORY
8 I I I T T I
6 .—
3 4 -
i
2 i-
g H
. a H H V
_2 L l L 1 1 AL
0 1 2 3 4 5 6
Time ( sec)
x 104 STRESS—STRAIN HISTERISIS LOOP
7 I I
6 .-
5 .-
4 l-
? 3 r-
i-
5 2
1 .
0
-1 l-
_2 1 1
-2 O 2 4 6
Strain x 10-4

FIGURE 6.6 Stress-Strain Hysteresis Loop -Axle Type-l

Stress (psi)

160

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 10—4 STRAIN-TIME HISTORY
6 I I I I I I
4- i i l .
2 - .
° N U V ‘
_2 I l 1 L l
0 0.5 1 1.5 2 2.5 3 3.5
Time ( sec)
x 104 STRESS—TIME HISTORY
8 I I I I I I
6 - n n -
4 ~ -
2 ~ .
. V .
_2 l t t A 1 4L
0 0.5 1 1.5 2 2.5 3 3.5
Time ( sec)
x 104 STRESS-STRAIN HISTERISIS LOOP
7 , 1
/1
6 '- d
5 - .
4 r- .1
3 P .
2 i .
1 e .
0 / 7
-1 r / -
_2 l 1
-2 0 2 4 6
Strain x 104

FIGURE 6.7 Stress-Strain Hysteresis Loop -Axle Type-3

161

strain/stress reversal after the ﬁrst and the second load applications due to the interface
between axles. The hysteresis loop for the axle-3 type consists of some additional areas in
the negative direction compared to the axle-l type due to larger negative stress/strain after
the ﬁrst and second load applications. Due to these additional area, the dissipated energy

becomes larger and thus results in a shorter fatigue life.

Figure 6.8 shows the stress-strain hysteresis loop together with the strain and stress
time histories due to the axle-4 conﬁguration. Because of the shorter distance between the
axles, there are no stress/strain reversals between load applications; the stress/strain
remains tensile between the load applications. This result in a smaller energy dissipation
and thus longer fatigue life. This type of stress strain time history can be exhibited in
thicker, stiffer asphalt concrete pavements. It may be noted that the strain time history of
tandem axle conﬁguration is strongly dependent on the space between the axles by com-

paring the Figure 6.7 and Figure 6.8.

Figure 6.9 shows the stress-strain hysteresis loop corresponding to axle-7 conﬁgu-
ration, which consists of two outside loops and two inside loops. The axle-6 conﬁguration
consists of 3 hysteresis loops: Two inside loops, and one outside loop instead of two loops.
These axle conﬁgurations consist of one single axle with a tandem (axle-2) or a tridem

(axle-3), respectively.

A substantial presence of compressive strains at the bottom of asphalt concrete
layer is clear from the Figure 6.6 to Figure 6.9. This behavior is important, especially
when laboratory tests are needed to estimate the fatigue strength or life of AC. The most
current laboratory procedures use a pulse load that creates only tension in the specimen.
For realistic fatigue life estimation, the whole strain history (combination of tension and
compressive) may be important, instead of the current laboratory that include tensile strain

only.

Steee (psi)

162

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 STRAIN-TIME HISTORY
6x10
4 + a
2 +- .,
0 ﬁ— 4
-2 i- a
_4 l l l 1 1 l
0 0.5 1 1.5 2 2.5 3 3.5
Time ( sec)
4 STRESS-TIME HISTORY
8x10
6 " -l
4 p -l
2 e .
0 -1
_2 l l l l l J
0 0.5 1 1.5 2 2.5 3 3.5
Time ( eec)
x 104 STRESS-STRAIN HISTERISIS LOOP
7 f I I I T I I
6 - a
5 - .
/
4 . 4
3 - .
2 r .
1 " II
0
‘ -1 l- J
-2 l 1 l l l J L
-3 -2 - 0 1 2 3 4 5 6
Strain x 104

FIGURE 6.8 Stress-Strain Hysteresis Loop -Axle Type-4

Stress (psi)

163

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 10" STRAIN-TIME HISTORY

6 I I I I ' 17

4 .- I!

2 .- d

”W M Mr 1
-2 l ‘ l 1 l 1

Time ( sec)
‘ STRESS—TIME HISTORY

8x10

6 b d

4 r .1

2 '- d

0 J H U U V .1
...2 l 1 J 1 1 1

Time ( sec)
x 10‘ STRESS-STRAIN HISTERISIS LOOP

7 ' ‘

6 ' 4

5 ” 1

4 r J

3' 1

2 b ‘1

1 h '1

0
-1 . 4
-2 1 1

Strain x 104

FIGURE 6.9 Stress-Strain Hysteresis Loop -Axle Type-7

164

 

 

 

 

 

 

 

 

 

 

 

 

 

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165

Table 6.4 summarizes the results by expressing them as mean initial dissipated
energy, peak tensile strain, fatigue life and load equivalency factor for each axle type. The
initial dissipated energy and peak tensile strain are a function of axle type and conﬁgura-
tion. The load equivalency factor is also dependent on the axle type and conﬁguration. The
LEF (Load Equivalency Factor) based on dissipated energy is more sensitive than that
from peak strain. Thus, it is believed that the LEF’s which are based on energy dissipated

are more appropriate than those for peak strain.

6.5 SUMMARY

An approach based on energy dissipation and visco-elastic properties of an asphalt
pavement for predicting the fatigue life of asphalt concrete pavements has been presented.
Using the SAPSI-M computer program, the stress and strain-time histories and the corre-
sponding energy dissipated at any point within the asphalt concrete layer due to the pas-
sage of a moving load was determined. This dissipated energy was linked to the fatigue
distress by a relationship between the fatigue life and the energy dissipated. The analysis
has led to the following conclusions:

(1) The estimated fatigue life based on the peak tensile strain model from the

moving load solution compares well with the estimated fatigue life from linear

elastic analysis and that of the in-situ pavement. This constitutes another field
validation of SAPSI-M’s moving load solution.

(2) The estimated fatigue life based on energy dissipation for the dual-tire
conﬁguration is approximately triple compared to that of the single-tire
conﬁguration. The actual pavement fatigue life to surface crack initiation is
approximately double that for the dual-tire conﬁguration compared to the
single-tire conﬁguration.

(3) The fatigue model based on energy dissipation of Equation ( 6-14 ) seems to
underestimate fatigue life at higher dissipated energy levels and overestimate
fatigue life at lower dissipated energy levels.

166

(4) The above ﬁndings constitute the possibility of using an approach based on
energy dissipation to predict fatigue damage of an asphalt concrete pavement
using the computer program SAPSI-M.

(5) The speed effect on dissipated energy is higher than that on peak tensile strain,
and may be even more signiﬁcant in pavement systems based on laboratory
fatigue models. However the effect of in-situ speed on the load vibration will be
an offset factor.

(6) The initial dissipated energy and peak tensile strain are functions of axle type
and conﬁguration. The load equivalency factor is also dependent on the axle
type and configuration. It appears that the load equivalency factor using the
energy dissipation method are more sensitive than those using the peak tensile
strain approach.

CHAPTER 7
CONCLUSION

7 .1 SUMMARY AND CONCLUSION

A new method of analysis to study the dynamic response of horizontally n-layered
damped-elastic systems to surface moving arbitrary loads under axisymmetric conditions
has been developed. An associated new computer program called SAPSI-M has been writ-
ten in the FORTRAN 77 language to incorporate the proposed method. The method and
program builds on an existing model for stationary loads, SAPSI, which was developed at

the University of California-Berkeley.

The proposed method uses the complex response method of transient analysis
with a continuum solution in the horizontal direction and a ﬁnite element solution in the
vertical direction, incorporating such important factors as wave propagation, inertia, and
damping effects of the medium as well as frequency-dependent asphalt concrete proper-
ties. The method and the associated computer program SAPSI-M have the following char-

acteristics, capabilities and limitations:

1. The structural model is a n-layered damped-elastic medium. Each layer is inﬁnite in the
horizontal directions and has a ﬁnite depth, except for the bottom layer. The subgrade
can be modeled as either a rigid base or a semi-inﬁnite halfspace. With these
axisymmetric assumptions, a three dimensional problem can be reduced to a two-

dimensional one, except that the edge effect can not be considered.

2. It is assumed that multiple loads that are vertical and uniformly distributed over a
circular area are moving along a single direction at a constant velocity on the surface of
a layered system. The loads may be constant in magnitude or may vibrate as an

arbitrary function of time. The moving loads are modeled as a series of pulses with a

167

168

duration equal to the time required for the wheel to pass by a ﬁxed point in the
pavement. The proposed method can handle any load conﬁguration, thus making it
possible to model multiple wheel conﬁgurations of truck axles as well as airplane

landing gears.

All materials are assumed to be linear visco-elastic in response to the loads. It is
assumed that each pavement layer consists of a homogeneous, isotropic, and visco-
elastic material. The asphalt material properties can be frequency dependent or
independent. The shear modulus, damping ratio, and Poisson’s ratio for each layer may

be varied with the excitation frequency of the loads.

The accuracy of the proposed method and of the corresponding program has been

validated with available ﬁeld data. The moving load solution of the developed program,

SAPSI-M, has been validated with the PACCAR full-scale ﬁeld truck tests. The transient

analysis in the SAPSI program has been validated with Falling Weight Deﬂectometer

(FWD) tests. The analysis has led to the following conclusions:

1.

Agreement between SAPSI-M's predictions and ﬁeld measurements was excellent in
the moving constant dynamic analysis for longitudinal strains and good for transverse

strains, in both shape and magnitude.

Considering all the possible sources of error in measurement such as temperature and
moisture conditions, linearity of measurement and performance of the bonded
pavement cores, as well as the errors that accompany any numerical modeling, this

constitute a very good validation of the moving load solutions in SAPSI-M.

The effect of truck speed on the response of asphalt concrete pavements is signiﬁcant.
The test results showed that increasing truck speed form creep speed to 40 mph reduced
the peak longitudinal strain by about 35 percent. Transverse strains showed more

variability with the effect of speed ranging from 20 to 40 percent.

169

4. The analytically-predicted horizontal strains in the asphalt concrete layer using

frequency-dependent properties for the asphalt concrete layer were closer to the ﬁeld
measured values than those using frequency-independent AC properties. This is an
indirect ﬁeld veriﬁcation of the strong dependency of asphalt concrete properties on
the frequency of loading. The results also show that the frequency dependent properties

of the asphalt concrete are a major contributor to the speed effect.

Agreement between SAPSI-M’s predictions and ﬁeld measurements of longitudinal
strain was excellent in the moving arbitrary dynamic analysis and very good in the
quasi-dynamic analysis in both the shape and magnitude of the response. These results
show that although the amplitude of load vibrations is high, neglecting the effect of
frequency, while maintaining correct allowance for truck speed and the magnitude of
the dynamic wheel loads, does not change the response in any signiﬁcant way.
Therefore, a simpliﬁed solution where the moving loads are assumed to be non-
vibrating seems to be sufficient for accurately predicting the pavement response. A
similar conclusion was reached by Hardy and Cebon using a simpliﬁed beam on

Winkler foundation model [40].

The comparison of Falling Weight Deﬂectometer (FWD) test results with static

and dynamic theoretical predictions was made using CHEVRON and SAPSI computer

programs, respectively. The following conclusions can be made from these results:

1.

Ninety percent of the measured strains in October 1991 FWD tests and all measured
strains in June 1992 FWD tests were within ilO percent of their calculated values,
using both static and dynamic analyses. Seventy percent of the measured strains in

February 1993 were within iZO percent.

Considering all the possible sources of error in measurement, the above results

constitute a good validation of SAPSI’s transient analysis.

170

3. Static analysis using statically back-calculated layer moduli seems to be sufficient in

analyzing FWD ﬁeld tests, despite the fact that static back-calculation using FWD

(dynamic) deﬂections will lead to "stiffened" elastic properties.

An approach based on energy dissipation and visco-elastic properties of asphalt

concrete for predicting the fatigue life of asphalt concrete pavements has been presented.

Using the SAPSI-M computer program, the stress and strain time histories and the corre-

sponding energy dissipated at any point within the asphalt concrete layer due to the pas-

sage of a moving load were determined. This dissipated energy was linked to the fatigue

distress by a relationship between the fatigue life and the energy dissipated. The analysis

has led to the following conclusions:

1.

The estimated fatigue life based on the peak tensile strain model from the moving load
solution compares well with the estimated fatigue life from linear elastic analysis and
that of the in-situ pavement. This constitutes another ﬁeld validation of SAPSI-M’s

moving load solution.

The estimated fatigue life based on energy dissipation for the dual-tire conﬁguration is
approximawa triple compared to that of the single-tire conﬁguration. The actual
pavement fatigue life to surface crack initiation is approximately double that for the

dual-tire conﬁguration compared to the single-tire conﬁguration.

The fatigue model based on energy dissipation of Equation( 6-14 ) seems to
underestimate fatigue life at higher dissipated energy levels and overestimate fatigue

life at lower dissipated energy levels.

The above ﬁndings constitute the possibility of using an approach based on energy
dissipation to predict fatigue damage of an asphalt concrete pavement using the

computer program SAPSI-M.

171

5. The speed effect on dissipated energy is higher than that on peak tensile strain, and may
be even more signiﬁcant in pavement systems based on laboratory fatigue models.
However increased speed and pavement roughness will cause higher dynamic loads,

thus potentially offsetting the speed effect.

6. The initial dissipated energy and peak tensile strain are functions of axle type and axle
conﬁguration. The load equivalency factor is also dependent on the axle type and axle
conﬁguration. It appears that load equivalency factors using the energy dissipation

method are more sensitive than those using the peak tensile strain approach.

7 .2 RECOMMENDATIONS FOR FUTURE RESEARCH

Based on the results and ﬁndings of this study, the following general recommenda-
tions are suggested for future research on the analysis of asphalt concrete pavement

response:

1. Further research into the effect of moving loads, pavement roughness, axle

conﬁgurations and suspension types on pavement damage should be undertaken.

2. Further research in using the energy approach to assess load equivalency and pavement

damage is called for.

3. There is a need for assessing the SAPSI-M model in terms of its degree of efﬁciency

and accuracy in the future.

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APPENDICES

APPENDIX A

PARAMETRIC STUDY

A.1 INTRODUCTION

A.l.l DESCRIPTION OF SAPSI-M COMPUTER PROGRAM

SAPSI-M is a FORTRAN computer program to calculate the response of a visco-
elastic layered system subjected to moving arbitrary loads at surface. The half space
underlying the layered system may be simulated by vertically extending layers and a series
of dashpots attached to the bottom of the extended layers. The shear modulus, damping
ratio, and Poisson's ratio for each layer may be varied with the excitation frequency of the
load. Multiple loads are acceptable for the harmonic, transient and moving load cases and
each loading may have a different radius and time histories. A static load can be simulated
by specifying a harmonic load with zero excitation frequency. The program uses a special
scheme in the frequency domain to interpolate between results obtained for only a few fre-
quencies. This provides a signiﬁcant reduction in the computational effort required. The
program uses a dynamic allocation technique which automatically sets the dimension of
the program to accommodate the input data.

The program can be run interactively with the aid of menus shown on the screen of
the monitor. The input data may be created directly from the keyboard, or read from exist-
ing ﬁles. The program is designed in such a way that the results obtained in different
phases of the execution can be saved for later use. For instance, the model for the layered
system, the eigen solution, the input loading, the output time histories, the coordinates of
the output points, and the general printout can be saved in different ﬁles for future refer-
ence.

The user's guide for informations on input and output operations in the program is
provided in Appendix B.

A.l.2 THE USE OF SAPSI-M FOR PAVEMENT ANALYSIS

The SAPSI-M computer program is used to predict the primary response (stress,
strain, displacement) of a pavement system subjected to truck moving loads. Current

182

183

design methods treat the pavement as a quasi-static system choosing material characteris-
tics which are related to the speed of the vehicle and the expected temperature conditions.
However, the actual dynamic effects due to moving loads are speciﬁcally considered.

The passage of a truck over a ﬁxed point in the pavement can be modeled in a
number of ways. The truck load can be treated as:

l-a vertically vibrating force which moves in the horizontal direction;
2-a vertically vibrating force which is stationary in space;
3-a constant force which moves in the horizontal direction;

4-a static force.

At present the SAPSI-M computer program can treat the load all the cases. Some-
time, the movement of the load in the horizontal direction may be approximated by using
the concept of an inﬂuence line. This concept is useful in determining the general shape of
the response of a ﬁxed point in the pavement as a truck passes by it. An inﬂuence line
gives the variation in the static response at one (ﬁxed) point due to a unit load traversing
the pavement. It has been shown that this simpliﬁed method can produce essentially the
same results as does the much more complicated dynamic analysis.

A.2 NUMERICAL CRITERIA ON ACCURACY

A.2.l COMPARISON WITH AN " EXACT" SOLUTION

The validation of the SAPSI computer program was done by comparing its predic-
tions with those from an exact solution by Sung. A homogenous half space which has a

shear modulus, G = 1.49x106 psf, a shear wave velocity, Vs = 619.5 fps, a Poisson's ratio,

it = 0.25 and a mass density, p = 3.8824 lb.sec2/ft4, was considered. The half space was
subjected to a surface disk load with a uniform pressure, q = 1000 psf.

Two cases corresponding to different loading radii were studied. The half space
was simulated by frequency-dependent layers and dashpots attached to the bottom of the
layered system. For the case in which the loading radius, r0, was 0.5 ft, the frequency was
varied from 0.01 to 200 Hz and for the case in which r0 was 1.0 ft, the frequency was var-

ied from 0.01 Hz to 100 Hz in the exact solution. Since the program SAPSI-M uses the
same routines to perform both static and dynamic analyses, only dynamic loads were con-

184

sidered in the veriﬁcation example. The vertical surface displacements at the center of the
loaded area were computed and compared with those obtained by Sung.

A.2.2 ANALYSIS

Using the above layer properties, a one-layer proﬁle was designed with the only
layer being the response layer, since the response was sought at the surface. For each fre-
quency, and using 10 layers for half space simulation the response thickness was reduced
in each run until it converged to the exact solution. The variation in response thickness for
each frequency is in Figure A-l. The ﬁgure shows that the curve for the static case is not
smooth. The reason for this is that the sublayer thicknesses of the simulated half space
(generated by the geometric series described above) do not match the attenuation curve for
the Rayleigh wave motion. Therefore the number of layers in the half space was varied as
10, 15, 20 and 25. As seen from Figure A-l, the 20-layer half space gives the best results.
Thus we can conclude that for static analysis, more than 10 layers are needed to accurately
simulate a halfspace.

For higher frequencies (> 50 Hz, in this case), increasing the number of layers to
simulate the half space from 10 to 15, 20 and 25 did not make any difference, as shown in
Figure A-2. The analysis shows that for the percent error to be less than about 10%, the
response thickness should be less than about 0.2 ft for the static case and about

0.001% f max for the dynamic case. Table A-1 contains the wavelength, for each fre-

quency, h = Vs/f.

A.3 DETERMINATION OF LAYERED PROFILES FOR USE IN
THE ANALYSIS

The objective of this part of the study was to determine layer thickness require-
ments for the accuracy of SAPSI-M and to design the optimum proﬁles for Cores 1,3,4
and 5. Table 5.3 in Chapter 5 gives the effective layer thicknesses used for Axial Cores
1,3,4 and 5; and Table 5.4 gives the material properties for each layer in the ﬂexible pave-
ment. The load applied on the ﬂexible pavement was taken as 10,000 lbs and the area of

Table A-1 Wave Length at Different Frequencies

 

I Frequency (Hz) 1 0.01 I 50 | 100 [ 200]
I Wave Length (ft) ] 61,950] 12.4 | 6.2 I 3.1 |

 

 

185

 

 

[+rreq=o.ornz +freq=50Hz +freq=100Hz +rreq-2oonzj

 

 

Percent Error (96)

 

10 -hyer hit-space

 

 

 

 

 

 

 

 

 

 

0 4 * t : ‘ ‘ t : 4 .
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Thickness of response layer (1!)
L—o—nom Hz +st50Hz +1.: room]
16
14 i. use radius -1 n
10-lsyer Isl-specs
12 .4.

Percent Error (7.)

 

 

 

 

 

 

0.25 0.3 0.35 0.4 0.45 0.5
Thickness of Response layer (ft)

 

Figure A-l Effect of the Thickness of the Response Layer and the Number of Layers

for Half-space Simulation

 

186

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+10 layers +15 layers +20 layers +25 layers
(a) FREQ - 50H:
so
_ r. - 12.3 a
E °° “
rd
‘3 20
a 10 .t
o . .
o 5 1 1.5 2.5
Thickness of the Response layer(ft)
(h) FREQ - 100Hz
4o
1. - 6.2 it
E 30 ..
iii
a 10 «-
0 + : t t
o 0'5 Thickness ‘of the Resporill’e layer(ﬂ) 2 2'5
(e) mo - zoom
40
E 30 .. r. - 3.1 ii
iii
3 20 .1
a 10 4»
o .h—_ i i 4‘
o 0.5 1 1.5 2.5
Thickness of the Response layer (ft)

 

 

 

Figure A-2 Effect of the Thickness of the Response Layer and the Number of Layers
Simulating for Half-space

187

the circular load was assumed to be 109in2. This translates into a radius of the loaded area
being equal to 0.49 ft and an applied pressure equal to 4217 psf.

A.3.l MATERIAL PROPERTIES

The ﬁrst step is to ﬁnd the cutoff frequency beyond which the response does not
need to be calculated. The determination of f cutoff is based on the input load and the
response. The cutoff frequency is the maximum frequency (fmax) at which the load or the
response dies out. Therefore no analysis is needed beyond the cutoff frequency. The corre-
sponding time step was calculated as dt = 1/ (2 - f max) and the frequency step as
df = 1/ (N - dt) . The dynamic analysis within SAPSI-M is performed in the frequency
domain and transient loadings are handled by the Fourier transform technique, as men-
tioned before.

For each layer, the shear modulus, G, and the shear wave velocity, Vs, were calcu-

lated from the given values of Young's Modulus, Poisson's ratio and mass density using

elasticity equations:
E
0-2—(1+u) (A-l)

G
V = — A-2
s ,l p ( )

The maximum wavelength corresponding to the cutoff frequency is: h = V s/ f

A.3.2 PROCEDURE

The procedure outlined below was used to design the best proﬁle for Axial Core
No. 1. The same number of layers for asphalt concrete, base and subgrade, were used for
the other axial cores (3, 4 and 5).

A.3.2.1 Maximum allowable thickness of the response layer:

For a particular proﬁle, the thicknesses of base, subgrade and epoxy were kept
constant while varying the thickness of the response layer in the asphalt concrete. The
response layer was varied from 0.003ft to 0.2ft. The response was calculated for each of

these at different frequencies. A plot of 82/8” versus W is shown in Figure A-3 (where

188

 

+at20Hz +at30Hz +at64Hz —)(-at80Hz

 

 

 

 

 

 

I
‘9.
N
w

0.4 «-

0.3 4»

0.2 -~

0.1 ~+

0 4. .L i i : t i t ¢
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

W

Figure A-3 Normalized Strain Versus Thickness of Response Layer in AC

189

H is the total thickness of the AC layer, h is the response layer thickness, and z is the depth

at which the response is sought). It can be seen that the ratio of 82/8” approaches unity

when the response layer thickness is less than about 0.02ft. To get better than 10% accu-
racy, the thickness should be less than H/6. A thickness equal to 0.01ft was chosen for the
AC layer, and the subsequent analyses were performed using this value.

A.3.2.2 Maximum allowable sublayer thickness for pavement layers:

For each material type (AC, Base, Subgradc), the minimum number of sublayers
required for an accurate response was determined. One constraint had to be imposed: The
total number of layers was limited to 16 in SAPSI-M.

(a) Asphalt Concrete

The proﬁle was started with 3 layers, having response layers of 0.005ft as the top
and bottom layers. In the consecutive runs the thickness of the layers other than the

response layers were equally subdivided until a convergent response (ex) was reached.

The response was sought at the bottom of AC and therefore the depth at which the
response was sought did not change for the different proﬁles. It was found that the

response converged when the number of sublayers were 121/db = 1000 . (see Figure A-
4.)

From Figure A-4, it can be seen that an AC proﬁle with four layers (h/dh = 200)
is sufﬁcient. One can reasonably choose the 4 layer proﬁle, since the percent error for this
proﬁle is only about 2 percent.

(b) Base

The proﬁle was started with 3 layers, with equal thicknesses and the response (ex)
was sought at mid-depth. The response approximately converged at 14 layers. A plot of
l/dh versus percent error is shown in Figure A-4. It can be seen that a 3-layer base
h/dh = 20 gives an error of 9%, and a 5-layer base (h/dh = 30) gives an error of 3%.

Therefore a 4-layer base of equal thicknesses was chosen (h/dh = 25 . A graph of h/dh
versus percent error was plotted including this 4-layer proﬁle. The percent error was
within 5%. Hence the 4-layer proﬁle was selected.

(c) Subgradc

1‘ error

16 em»

‘1. error

190
Minnow

 

14
‘21-
1°!»

 

 

t 4.00

12.00 ..
10.00 0
0.00 r
sea .
4.00 ..
7.90 0

0.00

BASE layer. mu;

 

 

 

 

suaeme layer. may;

 

 

 

30 4p
:5 is
2° ,
15 J.
10.1)
5 J.

 

 

Figure A-4 Effect of Sublayer Thickness

 

 

 

 

 

+E:

191

The proﬁle was started with 3 layers of equal thickness and the response (8) was
sought at the center of the total subgrade thickness. (similar procedure to that performed

for the base). It was found that the response (8,) approximately converged at 15 layers.

Therefore the percent error was calculated with respect to the lS-layer proﬁle. A plot for
l/dh versus percent error (ex) is shown in Figure A-4. It can be seen from the ﬁgure that

the percent error is 11% for the 3-layer proﬁle h/dh = 5 and is 3% for the 7-layer proﬁle

(h/dh = 7). The ﬁgure shows that a 5-layer proﬁle is quite appropriate since it approxi-
mately produces a 5% error in the tensile strain response.

A.3.2.3 Accuracy of selected proﬁles (AC, Base, Subgradc) at different
frequencies:

The accuracy of the proﬁles for AC, base and subgrade as decided upon was

checked at different frequencies. Comparing the converging response (ex, 8,, uz) to that

obtained using the selected proﬁles at different frequencies, a plot of percent error versus
frequency was plotted in Figure A-5. The graphs show that the percent error remained less
than 10% at different frequencies. Therefore the selected sublayer thicknesses were justi-
ﬁed.

A.3.3 PROFILE EFFECTS ON PAVEMENT DYNAMIC RESPONSE

A.3.3.1 The Effect of Stiff Layer

There was evidence from the back-calculation analysis, using the computer pro-
gram EVERCALC and FWD deﬂection data, that the subgrade soil was saturated at a rel-
atively shallow depth. The shallow water table translates into the saturated subgrade soil
behaving as a “stiff” layer. EVERCALC was used to back-calculate an appropriate value
for the stiff layer modulus by an iteration process.

Figure 5-9 shows the model for the soil proﬁles of the three cores. The layer thick-
nesses were obtained by coring and layer properties were obtained from back-calculation
analysis as described in Chapter 5.

192

 

 

(+15): +15: +Uz]

 

AC layer

 

%Error
0

 

 

 

 

 

 

  

o 10 20 so so so so 10

Freq-curd!»
BASEIayer

12

1“ M.,—7+ : A - -

on v ' TV

3‘ ,.
2+
0 . a

 

 

 

 

0 10 20 N 40 50 00 70

Frequency (Hz)

 

 

SUBGRADE layer

 

 

96 Error

 

 

 

70

 

 

 

Figure A-5 Accuracy of the Decided Proﬁles at Different Frequencies

193

3.3.1.1 Effect of Depth to Stiff Layer

To investigate the effect of the depth to stiff layer on the vertical deﬂection and lon-
gitudinal strain, the thickness of subgrade layer was varied from 2 ft to 20 ft. The proﬁle
used was that of Core 1. Transient analysis in SAPSI was performed at the three speeds,
1.7 mph, 20 mph and 40 mph, and using frequency-independent as well as frequency-
dependent proﬁles. Figure A-6 shows the results of the analysis. The ﬁgure indicates that
the tensile strain at the bottom of the asphalt concrete layer is independent of the depth of
the stiﬁ layer for all three speeds and for both frequency-dependent and frequency-inde-
pendent proﬁles. The vertical deﬂection is affected by the depth of the stiff layer only at
very low speed, with the deﬂection decreasing with decreasing depth to stiff layer. This is
expected since at very low speed, the behavior approaches static conditions where the
shallower the stiff layer the stiffer the foundation soils get and the lower the vertical
deﬂection becomes. Note that for the PACCAR site the depth to stiff layer is about 5 feet.
It is interesting to note that at this depth, the displacement curves for creep speed and 20
mph intersect. Since a truck speed of 20 mph is basically equivalent to an FWD pulse, this
could explain the ﬁnding that the results of static and dynamic analyses were practically
identical in the FWD analysis. If the stiff layer were deeper, the back-calculated moduli
could have been stiffer, which could result in lower dynamic strains.

3.3.1.2 Effect of Stiff Layer Modulus

To investigate the effect of the stiff layer modulus on the pavement response, the
above analysis was repeated with the modulus of the stiff layer being that of rock. The
rock layer was assumed to have a Young’s modulus of 1,000 ksi (versus 40 ksi for PAC-
CAR site) and a Poisson’s ratio of 0.25. Figure A-7 shows the results. Again, only the ver-
tical deﬂection at very low speed is affected by the modulus (stiffness) of stiff layer, with
the rock layer causing smaller deﬂections than in the PACCAR case for both frequency-
dependent and frequency-independent proﬁles.

3.3.2 The Effect of Base Layer

The eﬁect of the base layer on the vertical deﬂection and the tensile strain at the
bottom of the asphalt concrete layer was investigated. The exercise consisted of calculat-
ing the response with and without a base layer. In addition, the response was sought at the
bottom of the AC layer and at the bottom of the epoxy. This lead to four cases, (A) through
(D) as shown in Figure A-8. The proﬁle of Core 1 (except for base and subbase thick-

194

FRE. lN-DEPENDENT FRE. lN-DEPENDENT

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15 200
’5; 6
= I 150 .............................
S10 . 9‘;
St a
g (.3. 100* \
p \ “J ;
'13) 5 F g 50) §
g m- PACCAR site 3 ,\
§ \
O ‘\ ‘ ' * 0 ‘\ ‘ ‘ ‘
O 5 10 15 20 0 5 10 15 20
Depth to Stiff Layer (it) Depth to Stiff Layer (it)
FRE. DEPENDENT FRE. DEPENDENT
15 . ~ . . 200 - - e -
g a .....
._ 150 r ,. ................
E10 1 L2; l _ .4 I . .. _ _ _ - _
{33' g 100 r
E 5 S é; 1% —V=1.7 mph
E 3 so . \ ---v.-.2o mph
S s --V=4o mph
0 4 a .a . o A L . A
O 5 10 15 20 0 5 10 15 20
Depth to Stiff Layer (ft) Depth to Stiff Layer (it)

Figure A-6 Effect of Depth to Stiff Layer on Longitudinal Strain and Vertical
Displacement

Vertical Disp.( mils)

Vertical Disp.( mils)

FRE. lN—DEPENDENT

195

 

 

 

 

 

 

 

 

 

 

 

 

15 -
10-
5' S
Q— PACCAR site
§
0 . . I .
0 5 10 15 20
Depth to Stiff Layer (ft)
FRE. DEPENDENT
15 . . . .
5 . l
0 A . . a
0 5 10 15 20

Depth to Stiff Layer (ft)

200

Longi. Strain (E-6)
8
o

200

...s
01
C

Longi. Strain (E-6)
8
O

FRE. lN—DEPENDENT

 

 

ﬁ

 

 

’/// ///

 

5 1'0 15 21)
Depth to Stiff Layer (ft)

 

 

FRE. DEPENDENT

1

 

 

 

 

 

 

 

S kV=1.7mph

50- ~-V=20mph
--V=4O

o S A - "“3

o 5 1o 15 20

Depth to Stiff Layer (ft)

Figure A-7 Effect of Stiff Layer Modulus on Longitudinal Strain and Vertical

Displacement

 

 

 

 

way
AC

Epoxy

Base

 

Subgrade

 

 

 

(CaseA)

 

 

 

 

 

 

Subgradc

 

 

 

(CaseC)

- : Location of calculated strain

196

 

 

 

Epoxy

 

Emit!

Subgrade

 

 

(CaseB)

 

 

 

 

Subgradc

 

 

(CaseD)

Figure A-8. Pavement Models to Investigate the Eﬁ'ect of Base Layer

 

 

197

nesses) was used with frequency-dependent AC properties. The epoxy was assumed to
have the same properties as the asphalt concrete. Figure A-9 shows the results. As
expected, removing the base layer increases the tensile strains (B versus A and D versus
C.) Also, the tensile strain at the bottom of the epoxy is higher than at the bottom of the
asphalt concrete (C versus A and D versus B,) since the response point is farther from the
neutral axis. However the speed effect is not affected: The normalized strain with respect
to creep speed shows the same trend with speed for all cases. It decreases by 28% as the
speed increases from creep to 60 mph.

As expected, the vertical deﬂections of Cases (A) and (C), and Cases (B) and (D)
are exactly the same because they have the same proﬁle and the location of the response is
close enough not to cause any additional deﬂection. The deﬂection in (A) and (C) are
slightly lower than in (B) and (D) because of the absence of the base. The normalized
deﬂection relative to the deﬂection at creep speed shows the same trend for all the cases at
all speeds. They decrease by as much as 50% as the speed increases from creep to 60 mph.
Therefore the speed effect on longitudinal strain and vertical deﬂection is the same with or
without the base and regardless of the tensile strain location in the asphalt concrete.

198

 

 

 

 

 

 

 

 

 

 

 

Normalized Disp.

 

 

 

 

 

 

 

 

 

VERTICAL DISP. LONGI. STRAIN
15 f . . . f
g at {of 3 i- x
510 1 9 4 21,150 ' O 5 x
a. 6 .9 3
g S
9 as a; 100
8 5 1- ...'
5 g 50 -
> .J
O ‘ 4 ‘ 0 ‘ ‘ ‘
O 20 40 60 0 20 40 60
Speed ( mph ) Speed ( mph )
VERTICAL DISP. LONGI. STRAIN
11. .1. 8
.E
0.8 1 15 s 0.8 a
5 a
o 6 1‘ E 0.6 - 0 (A)
I .
_ + B
0.4 g 0.4 » ( )
a ' (C)
0.2' 1 z 0.2 ' x (D)
O a a . O 4 . L
0 20 40 60 O 20 4O 60
Speed ( mph ) Speed ( mph)

Figure A-9 Effect of Base Layer on Longitudinal Strain and Vertical Displacement

APPENDIX B

SAPSI-M USER GUIDE

B.1 INTRODUCTION

SAPSI-M stands for a System for Analysis of Pavement - Soil Interaction for Mov-
ing Loads. The SAPSI-M code is programmed in FORTRAN 77 language. It can be used
to calculate the response of a visco-elastic layered system subjected to surface uniformly
moving loads.

This computer program is based on the theory of linear visco-elastic layer theory
in horizontally and ﬁnite element method in vertically. This program was developed on the
axisymmeu’ic coordinate, which reduces the 3-dimensional problems into 2-dimensional
one. The half space underlying the layered system can be simulated by vertically extend-
ing layers and a series of dashpots attached to the bottom of the extended layers.

B.l.l CAPABILITIES AND LIMITATIONS
3.1.1.1 Capabilities

The computer program SAPSI-M has the following capabilities and limitations:
(1) A steady-state solutions for ﬁxed frequencies can be obtained.

(2) A stationary dynamic solution of transient analysis can be obtained and stored
from the steady-state solution and interpolation scheme.

(3) The solution of multiple moving arbitrary loads can be calculated.

(4) The shear modulus, damping ratio, and Poisson’s ratio for each layer may be
varied with the excitation frequency of the loads.

(5) Multiple loadings are acceptable for both harmonic and transient cases and each
loading may have a different radius and time history.

(6) A static load can be simulated by specifying a harmonic load with zero
excitation frequency.

199

200

8.1.1.2 Limitations

(1) SAPSI-M deals only with a layered system subjected to uniformly distributed
circular loads.

(2) It does not calculate the response at zero frequency when using the half space
simulation but does the analysis at near zero frequency (0.001 Hz).

(3) It does not use a cutoff frequency after which the response would be set to zero.
This concept is very useful when we know that the response is essentially zero
after a certain frequency (i.e contained within a certain frequency range). Had
SAPSI-M have this option, the user would be able to use small frequency steps
(for accurate interpolation in the frequency domain) with a limited number of
frequencies, while keeping the time step small enough for a smooth response in
the time domain.

(4) The interpolation scheme in SAPSI-M needs a minimum of ﬁve frequencies in
a particular frequency range and a maximum of 64 frequencies.

(5) Often SAPSI-M would give an error message (“Real indeﬁnite...”) which halts
the execution. This is due to the fact that the solution is pseudo-exact and may
“blow-up” at certain frequencies. All that one needs to do is to shift the
frequency at which the response is being computed by one or a few frequency
steps. In the case of transient loading, this can occur at different frequencies.

B.l.2 ORGANIZATION OF THE MANUAL

An introduction to the SAPSI-M computer program was given in Section 1. This
section also lists the capabilities and limitations of the SAPSI-M program. Section 2 is an
application guide to SAPSI-M analysis. The guide explains the logical approach in
SAPSI-M analysis, and consideration prior to SAPSI-M analysis. Section 3 explains how
to run SAPSI-M job, and shows input format with examples. This contains numerous
comments which are helpful to the user in the day-to-day use of the program.

201
B2 APPLICATION GUIDE

B.2.l INTRODUCTION

It is important that the program users understand how to effectively utilize the
capabilities of the program in order to efﬁciently perform the SAPSI-M analysis. This
requires knowledge of the analytical procedures used in this computer program. A set of
guidelines describing procedures for the SAPSI-M analysis is considered to be very help-
ful for the users to utilize the program capabilities. It is therefore the purpose of this sec-
tion to provide such guidelines.

8.2.2 SAPSI-M ANALYSIS PROCEDURE
8.2.2.1 Steps Involved in SAPSI-M Moving Load Analysis

Step1. Select the time history of load (pulse) and compute its harmonic components
to highlight the dominant frequencies contained.

Step 2. Determine the possible important frequency range by examining the
dominant frequencies of the pavement structure.

Step 3. Based on results of Steps 1 and 2, determine the cut-off frequency of the
analysis.

Step 4. Based on laboratory tests, determine the shear modulus, Poisson’s ratio and
damping ratio at the different frequencies, for use in the frequency
dependent proﬁle.

Step 5. Select the frequencies at which the steady-state pavement responses are to
be solved. A minimum of 5 frequencies should be selected. These
frequencies are selected from Steps 1 and 2. The number of frequencies can
be increased, if necessary, to improve the accuracy of the interpolated
transfer functions.

Step 6. Perform the moving load analysis in SAPSI-M.

Note: Static analysis is performed at near zero frequency, (0.001Hz), and steady state anal-
ysis is done at selected frequencies. The pavement layer properties are independent of fre-
quency for the static analysis, but they can be either one of frequency dependent properties
or frequency independent for the steady state analysis.

202

8.2.2.2 Considerations Prior to SAPSI-M Analysis
A. Rigid Base vs. Half-space

SAPSI-M has the capability to simulate the existence of a unifonnly damped or
undamped, half-space below the top pavement layers. Therefore, it can avoid using very
deep soil models with many sublayers, which leads to additional savings when the soft soil
extends to a relatively large depth or when the bed rock boundary cannot be established.

In the case of half-space simulation, the program adds (with the given option) an
additional soil layer below the top layers with the thickness of (l.5*Vs)/f, where Vs. is the
shear wave velocity of the half space and f is the ﬁequency of analysis because of this
modeling, we can not obtain the static response at zero frequency for half-space sirnula-
tion. In addition to this, the viscous dashpots are added to the base of the new soil.

8. Cutoff Frequency (fcuttoff)

The cutoﬁ frequency is an important parameter since it not only sets an upper limit
on the number of frequencies to be analyzed but also controls the maximum allowable ele-
ment (layer) sizes, and thus, the dimension of the stiffness and mass matrix of the prob-
lem. The factors governing the choice of the cutoff frequency are:

a. The frequency content of the input load.
b. The dominant frequencies of the entire pavement system.

c. The time increment of the input time history.

C. Selection of Frequency Points

The frequencies to be selected for the SAPSI-M analysis depend on the number of
peaks in the transfer function and how close these peaks are located relative to each other.
Since an efﬁcient interpolation scheme on complex response functions has been incorpo-
rated into SAPSI-M, it is usually sufﬁcient to solve for 7 to 8 frequencies and intermediate
solutions can then be obtained by interpolation.

D. Input 01’ Frequency Points

In the SAPSI-M program, the transfer functions are computed at discrete fre-
quency points which are integer multiples of the frequency step, df. The frequency step is
calculated from

203

dt = (l/dt)NFFT (A-l)

where, the input parameters dt and NFFT are the time step and number of points to be
used in the Fourier Transform of the time history, respectively.

Once the frequency step is deﬁned, frequency points, fi, are input through the use
of integer frequency numbers, NFREQi, deﬁned as follows:

NFREQl- = fi/df i = 1,2 ...... NF (A-2)

where, NF is the total number of frequency points selected for the analysis. The maximum
frequency number to be speciﬁed in the SAPSI-M program is controlled by the cutoff fre-
quency and can be obtained as follows:

NFREQNF = fNF/df (A-3)

where, {NF is the cutoff frequency.

At present the SAPSI-M computer program can treat the load as a vertically vibrat-
ing pressure which is moving or stationary in space, or as a static pressure. However, the
movement of the load in the horizontal direction can be approximated by using the con-
cept of inﬂuence line. This concept is useful in determining the general shape of the
response of a ﬁxed point in the pavement as the truck passes by it. The inﬂuence line
determines the static effect of a point for a unit load, acting at various positions, on the
magnitude and sign of the primary response of the pavement structure (stresses, strains
and deﬂections).

8.2.3 NUMERICAL MODELING CRITERIA
8.2.3.1 Finite Element Meshing

In order to accurately transmit the waves, the ﬁnite element model should be dis-
cretized so that the thickness of each element (layer) does not exceed M8; where x is the
shortest wave length of interest in the analysis. The wavelength criteria can be relaxed to
1/5 if the mass matrix used in the analysis is constructed from the combination of consis-
tent and lump mass matrices (usually 50% each). Since the mass matrix computation in
SAPSI-M is automated to consist of 50% lump mass and 50% consistent mass, the 1/5
criteria can be used in constructing pavement models.

204

8.2.3.2 Half-space Simulation

In order to simulate the half-space condition at the bottom boundary, two tech-
niques of variable depth method and viscous boundary at the base are used. In variable
depth method, 11 extra layers with total thickness of 1.5). and with the properties of half-
space are added to those pavement proﬁle. The wavelength, 71 , is the shear wave length in
the half space and is a function of frequency. The choice of 1.5). arose ﬁom the observa-
tion that fundamental modes of Rayleigh wave in half-space decay with depth and essen-
tially vanish at a depth corresponding to 1.51.. Furthermore, the 1.51 layer thickness is
subdivided into n layers with increasing thickness with depth. The choice of n = 10 is suf-
ﬁcient for many practical problems. VVrth this technique, the layer thickness will increase
with depth and decreasing frequency. This is the desired characteristic of the model since
surface wave mode shapes decrease exponentially with depth and since their depth of pen-
etration increase with decreasing frequency. The model with extra added layers is further
improved by replacing the rigid boundary at the base of the extended layer with viscous
dashpots in horizontal and vertical directions.

B.3 INPUT GUIDE

B.3.l HOW TO RUN A SAPSI-M JOB

SAPSI-M can be run interactively or using batch ﬁles. An example of a batch ﬁle
is given below (for static and dynamic case respectively). To run SAPSI-M using a batch
ﬁle (ﬁle name: CORE1.RUN) the redirection command SAPSIM < CORE1.RUN should
be given at the DOS prompt. The input data may be created directly from the keyboard or
read from the existing ﬁles and they can be modiﬁed and saved in either the old ﬁle or a
new ﬁle. The extension *.RUN for a batch ﬁle is suggested for convenience.

8.3.1.1 Input Files

All ﬁle names for input and output are designated by the user. For convenience, the
input and output ﬁles should use an extension which designates the type of information
contained within for SAPSI-M input ﬁles.

The following are suggested extensions:

1. *.PRO - Proﬁle of layered system.

2. *.LOD - Wheel load geometry, conﬁguration and magnitude.

205

3. *.REQ - Output request points.
4. *.TIM - Time history of the load.
5. *.FRE - Frequency spectrum of the load for transient analysis.

The ﬁles are explained in detail later in this Section and an examples for each ﬁle are also
provided.

8.3.1.2 Output Files

The program SAPSI-M is designed in such a way that the results obtained in each
phase of the execution can be saved for later use. The following are suggested extensions:

1. *.BIN - Binary output ﬁle for the responses.

2. *.RES - The Output ﬁle which saves the global strains and stresses and global
responses.

3. *.FOU- This output ﬁle contains the fourier transform points of load or response
in the frequency domain.

206

B.3.2 EXAMPLES OF BATCH FILE

The program SAPSI-M can be run interactively with the aid of menus on the screen or using a batch ﬁle as
shown below. This batch ﬁle to run SAPSI-M may have extension of *RUN.

8.3.2.1 An Example of

Static Analysis Batch File

This example shows the case of frequency independent properties and static solution with steady state solu-
tion by deﬁning near zero frequency.

I .......................................
HARMONIC ....................
C .......................................
F .......................................

 

 

 

 

 

R ....... -
STATICREQ ...................

 

 

 

 

STATICRES ....................

Flag for Input parameters

Comments

Flag for controlling parameters

Flag for Frequency dependent properties

Frequency independent property (0: fre. independent, 1: frequency dependent)
Flag for Half-space simulation

- 10 layers for half-space (0: no simulation, N: number of layers for half space)

Flag for Viscous dashpot at lower boundary

Attach a viscous dashpot (0: no dashpot, I: attach viscous dashpot)
Flag for Eigenvalue solution

Solves eigen value problem without saving the solution

. This option avoids printing the eigenvalue solution

Exit to previous screen

Flag for Proﬁle

Read the input proﬁle

Filenarne of the input proﬁle, being frequency independent
Exit to previous screen

Flag for Load ﬁle

Read the load ﬁle

(0: harmonic, l: transient in time domain, -1: transient in frequency domain)
Number of ﬁequencies to be analyzed

Near zero frequency for static solution

Load input ﬁlename

Exits to the previous screen

--Flag to Deﬁne the output request point

Read the output request ﬁle

Output request ﬁlename

Strain output requested (1: the strains, 0: stresses)

Tb save the output responses

Binary output ﬁlename (This ﬁle may be used for post analysis)
Exit to the previous screen

Exit to the previous screen

Flag for Program execution

Results are saved in this output ﬁlename

8.3.2.2 An Example of

207

Steady State Analysis Batch File

This example shows the case of frequency dependent properties and steady-state solution at 4 frequencies.
The same explanations hold true as for the static analysis except for a few as shown below.

I ...... - -
TRANSIENT SOL. ..........

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

STEADFBIN ...................
X
X ......

E .......................................
STEADFRES ........... .

 

 

Flag for Input parameters
Comments

Flag for controlling parameters

Flag for Frequency dependent properties

Frequency dependent property (0: fre. independent, 1: frequency dependent)
Flag for Half-space simulation

10 layers for half-space (0: no simulation, N: number of layers for half space)
Flag for Viscous dashpot at lower boundary

Attach a viscous dashpot (0: no dashpot, l: attach viscous dashpot)

Flag for Eigenvalue solution

Solves eigen value problem without saving the solution

This option avoids printing the eigenvalue solution

Exit to previous screen

Flag for Proﬁle

Read the input proﬁle

Filename of the input proﬁle, being frequency dependent

- Exit to previous screen

Flag for Load ﬁle

Read the load ﬁle

(0: harmonic, 1: transient in time domain, -1: transient in frequency domain)
Number of frequencies to be analyzed

Frequency at which the steady state solution be obtained

-- Same as above

Same as above

Same as above

Load input ﬁlename fro Harmonic loads

Exits to the previous screen

Flag for Deﬁne the output request ﬁle

Read the output request ﬁle

Output request ﬁlename

Strain output requested (1: the strains, 0: stresses)
To save the output responses

Binary output ﬁlename (This ﬁle may be used for post analysis)
Exit to the previous screen

Exit to the previous screen

Flag for Program execution

Results are saved in this output ﬁlename

208

8.3.2.3 An Example of Moving Load Analysis Batch File

This example show the case of frequency dependent properties and transient solution at time domain. The same explana-
tions hold true as for the steady state analysis except for a few as shown below.

 

 

I ............................................. Flag for Input parameters

TRANSIENT SOL. ............... Comments

C ............................................. Flag for controlling parameters

F ............................................ Flag for Frequency dependent properties

1 ............................................. Frequency dependent property (0: fre. independent, 1: frequency dependent)

H ............................................ Flag for Half-space simulation

10 ........................................... 10 layers for half-space (0: no simulation, N: number of layers for half space)

V ............................................ Flag for Viscous dashpot at lower boundary

1 ............................................. Attach a viscous dashpot (0: no dashpot. I: attach viscous dashpot)

E ............................................ Flag for Eigen solution

0 ............................................. Solves eigen value problem without saving the solution

1 ............................................. This option avoids printing the eigenvalue solution

X ............................................ Exit to previous screen

P ............................................ Flag for Proﬁle

R ............................................ Read the input proﬁle

COREIFPRO ....................... Filename of the input proﬁle, being frequency dependent

X ............................................ Exit to previous screen

L ............................................ Flag for Load ﬁle

R ............................................ Read the load ﬁle

1 ............................................. (0: harmonic. 1: transient in time domain. -1: transient in frequency domain)

512 ........................................ .Number of discretized points in time domain

512 ........................................ .Number of discretized points in frequency domain
.............................................. A blanks is needed to skip when the input is in time domain

5 ............................................. .Number of frequencies to be analyzed.

4 .............................................. The steady state solution will be obtained at this point

55 ........... The steady state solution will be obtained at this point

112 .......................................... The steady state solution will be obtained at this point

168 .......................................... The steady state solution will be obtained at this point

256 ......................................... 512/2 (N/2 or close number is recommended for good interpolation)

( F15.5 ) ................................. Format of input loading

0.0023 .................................... Time step of input loading at ﬁle PULSEITIM

144 ........................................ Multiplier of input loading (for conversion purpose, e.g. from psi to psf)

PULSE] .TIM ........................ Filename of load-tirne history

Y ............................................ Yes, to create a Fourier Transform ﬁle

TRANSFOU ........................ Filename to save Fourier Transform results

PULSEILOD ....................... Load input ﬁlename for Transient loads

X ............................................ Exits to the previous screen

0 ............................................ Flag for Deﬁne the output request ﬁle

R - . .- - ...... Read the output request ﬁle

TRANSREQ ........................ Output request ﬁlename

1 .............................................. Strain output requested (1: the strains, 0: stresses)

8 ............................................ To save the output responses

TRANSBIN .......................... Binary output ﬁlename for using at post analysis

X ............................................ Exit to the previous screen

X ............................................ Exit to the previous screen

E ............................................ Flag for Program execution

TRANSRES ......................... Results are saved in this output ﬁlename

1 .............................................. Flag for moving load solution

B.3.3 PROFILE FILE

209

There are two types of proﬁles, depending on the material properties being either frequency- independent or
frequency-dependent. The format of each ﬁle has to comply with the following. These proﬁles may have
extension *.PRO for convenience.

8.3.3.1 Format

A. Frequency Independent Properties

Number of layers

Em

Layer identiﬁcation number

Mass density of this layer (lb.sec2/ft4)

Real part of the shear modulus of this layer (psf)
Imaginary part of shear modulus (psf)

Real part of the Poisson’s ratio of this layer
Imaginary part of the Poisson’s ratio of this layer
Thickness of this layer (ft)

Card set 1:
Columns Yan'ablL Format Notes Em
l-S NLAY 15
Card set 2:
Columns Mable. Eonnat Notes
1-5 LAY(I) 15
6-17 RO(I) E12.4
18-29 GR(I) E12.4
30-41 610) E12.4
42-53 VUR(I) E12.4
54-65 VUI(I) E12.4
66-77 mm E12.4
Provide NLAY cards for card set 2

B. Frequency Dependent Properties

Card set :1
Columns
l-S

Card set 2:
Columns
1-5
6-17
18-29

Card set 3:
Columns
1-5
6-10

Cardset4:

1-12
13-24
25-36

Yariahle.
NLAY

Yariahlc.
LAY(I)

R00)
THO)

NGU)
INTG(I)

FG(l)
GR( 1)
FG(2)

Eonnat
15

Eonnat
15

E12.4
E12.4

15
15

E12.4
1512.4
E12.4

Notes
(1)
(1)

ﬁrm
Number of layers Card set 2:

Em

layer identiﬁcation number

Mass density of this layer (lb.se02/ft4)
Thickness of this layer (ft)

Emu
Total number of shear moduli for this layer

Interpolation ﬂag of shear moduli for this layer
1- Linear/Linear interpolation

2- Linear/LOS (freq) interpolation

3- Log/Log interpolation

4- Log/Linear (freq) interpolation

Em
The ﬁrst ﬁequency of the shear modulus curve (Hz)

The ﬁrst shear modulus of the shear modulus curve (psf)
The second frequency of the shear modulus curve (Hz)

37-48 GR(2) 812.4
Provide [NG(I)/+2]/3 cards for card set 4
Note(1): Input in Ascending order
Card set 5:

Columns Yatiahlo. Emnat

1-5 NB(I) 15
6-10 INTB(I) 15
Card set 6:

Columns YariahlL Emnat
1-12 FB(1) E12.4
13-24 BR(l) E12.4
25-36 FB(2) E12.4
37-48 BR(2) 1312.4

Provide [NB(I)/+2]/3 cards for card set 6
Card set 7:
Columns YariatzIL Emnat

1-5 NU(I) 15
6-10 INTN(I) 15
Card set 8:

Columns Yaﬁahlo. Emnat
1-12 FN(1) E12.4
13-24 VU(l) E12.4
25-36 FN(2) E12.4
37-48 VU(2) E12.4

Provide [NB(I)/+2]/3 cards for card set 8

(2)
(2)

(3)
(3)

Repeat card set 2 through 8 for NLAY times

Note(2): Input in Ascending order
Note(3): Input in Ascending order

210

The second shear modulus of shear modulus curve (psf)

Em

Total number of damping ratio for this layer
Interpolation ﬂag of damping ratios for this layer
1- Linear/Linear interpolation

2- Linear/L08 (freq) interpolation

3- Log/Log interpolation

4- Log/Linear (freq.) interpolation

ﬁrm

The ﬁrst ﬁequency of the damping ratio curve (Hz)
The ﬁrst damping ratio of the damping ratio curve
The second frequency of the damping ratio curve (Hz)
The second damping ratio of the damping ratio curve

Em
Total number of Poisson’s ratio for this layer

Interpolation ﬂag of damping ratios for this layer
1- Linear/Linear interpolation

2- Linear/L08 (freq.) interpolation

3- Log/Log interpolation

4- Log/Linear (freq.) interpolation

Burn

The ﬁrst frequency of the Poisson’s ratio curve (Hz)
The ﬁrst Poisson’s ratio of the Poisson’s ratio curve
The second frequency of the Poisson’s ratio curve (Hz)
The second Poisson’s ratio of Poisson’s ratio cruve

8.3.3.2 Examples of Proﬁle

211

A. An Example of Frequency Independent Profile (undamped ease)

u—s
O‘
windows-wro—

s-nt-ds—tt—Is-It-tI-t
amnion—see

.4505E+01
.4505E+01
.4505E+Ol
.4505E+01
.4505E-l-01
.4505E+01
.4200E+01
.4200E-1-01
.4200E+01
.4200E-1-01
.4039E+01
.4039E+01
.4039E-f-01
.4039E+01
.4039E+01
.4349E+01

.2670E-1-08
.3000E+08
.3IXJOE+08
.3(X)0E+08
.3(X)0E+08
.2670E-I-08
.7600E+06
.7600E-l06
.7600E+06
.7600E+06
.5060E+06
.5060E+06
.5060E-1-06
.5060E-i-06
.5060E+06

.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00

.3500E+00
.3500E-i-00
.3500E+00

.3500E+00
.3500E+00
.3500E+00

.4000E+00
.4000E+00
.4000E+00
.4000E-i-00
.4500E+00
.4500E+00
.4500E+00
.4500E+00
.4500E-l-00

.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.OOOOE+00
.0000E+00
.00me
000013400
000013100
.OOOOE+00
000084-00
.0000E+00
.0000E+00
.OOOOE-i-OO
2.1333E+06 000084-00 .3500E+00 .0000E+00 .IOOOE+01

.0333E+(D
.0100E-1-(X)
.1942E+(X)
.l942E-1-00
.0100E+(X)
.0333E+m
.2650E+(Xl
.2650E-i-(X)
.2650E-1-(X)
.2650E+(X)
.71 16E+m
.71 16E+(X)
.71 16E+m
.71 16E+(X)
.71 16E+(X)

B. An Example of a Frequency Independent Proﬁle (damped case)

h—r
\OWQONUI-hDJNO—O‘

i—n—oi—s-bi—Is—Ir-s
O‘th-AUJN—‘C

.4505E+01
.4505E-i-01
.4505E+Ol
.4505E+01
.4505E+01
.4505E+01
.4200E+01
.4200E-i-01
.4200E-1-01
.4200E+01
.4039E+01
.4039E+01
.4039E+01
.4039E+01
.4039E+01
.4349E+01

.2670E-i-08 5.3334E-1-06
.3000E+08 6.0000E+06
.3000E+08 6.0(1)0E+06
.3000E+08 6.0000E+06
.3000E+08 6.0000E+06
.2670E+08 5.3334E-i-06
.7600E+06 7.6(X)0E+04
.7600E+06 7.6(X)0E+04
.7600E+06 7.6000E+04
.7600E+06 7.6000E+04
.5060E+06 3.0360E+04
.5060E-I-06 3.0360E+04
.5060E-1-06 3.0360E+04
.5060E+06 3.0360E+04
.5060E+06 3.0360E+04
2.1333E+06 1.2798E+05

.3500E+00
.3 500E+00
35me
35me
.3 500E+00
.3500E+00
.4000E+00
.4000E+00
.4000E+00
.4000E+00
.4500E+00
.4500E+00
.4500E+00
.4500E+00
.4500E+00

.0000E+00
.0000E+00
.OOOOE-i-OO
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00
.0000E+00

.0333E+00
.0100E+00
. 1942E+00
.1942E+00
.0100E+00
.0333E+00
.2650E+00
.2650E+00
.2650E+00
.2650E+00
.71 16E+00
.71 16E+00
.71 16E+00
.71 16E+00
.71 16E+00

.3500E+00 .0000E+00 .1000E+01

C. An Example of Frequency Dependent Proﬁle

10

l 4.50505+00 4.650050]
0.211095% 1.31295+07 3.25005+02 7.10015+07
021(0205+00 2.25005-01 3.25005+02 7.89005-15

0.1(XJOE+00 4.99905-01 3.25005+02 -7.66005-02
2 450554-00 100005-02

2 3
0.211025% 131295107 3.25005+02 7.10015+07
0.1(X)05+00 2.25005-01 3.25(IJE+02 7.89005-15

2 2
O.l(X)05+OO 4.99905-01 3.25(X)5+02 -7.66005-02
3 4.20m5-100 353335-01

2 l
0.1(X)105+00 7.6114E+05 3.25005102 7.6114E+05
0.1(005+00 5.00005-02 325(115-102 5.111005-02

2 1
0.1(X)05+00 4.00005-01 3.25(XI5+02 4.0)005-01
4 4.20(X)5+00 3.53335-01

2 1
0.21(X)105+00 7.6114E+05 3.25005+O2 7.6114E+05
0.1(X)OE+00 5.00005-02 3.25005+02 5.00005-02

2 1
0.1(XJOE1-00 4.00005-01 3.25m5+02 4.(X)005-01
5 4.20(X)E+00 353335-01

2 l
0.21(DIOE+00 7.6114E+05 3.25005+02 7.6114E+05
0.21001015+00 5.00005-02 3.25CD5+02 SLEDGE-02

0.1(D05+00 4.00005-01 3.25m5-102 403005-01
6 403905-100 8.89505-01

2 1
0.21(XJIOE+OO 5.06485+05 3.25005+02 5.06485+05
0.1(X)05+00 3.00005-02 325005102 3.00005-02

2 l
o.1ooor-:+oo 4.50005-01 3.25(XIE+02 4.50005-01
7 403905100 889505-01

2 1
0.2100105+00 5.06485+05 325005-102 506485+05
0.21(X)IOE+00 300005-02 3.25(XJ5+02 3.00005-02

0.1(X)05+00 4.5000501 3.25005+02 4.50005-01
8 4.03905+00 8.8950501

2 1
0.21m1()5+00 506485+05 325005-102 5.06485+05
0.1(X)05100 3.00005-02 325(05402 300005-02

2 l
0.1(XJ05+00 450005-01 325015402 4.50005-01
9 403905+00 8.89505-01

2 1
0.103105+00 5.06485+05 3.25005+02 506485105

2

0210010300 300005-02 325005-02 300005-02

0.10005+00 4.50005-01 3.25(XJE+02 4.50005-01

120 41.34905+(X) l.0(X)05+00

0.21m05+00 2.13335+06 3.25005+02 213335-106

1
0.21(XJIOE+OO 300005-02 3.25001-:+02 3.(l)005-02
0.10005+00 350005-01 3.25(XJE+02 350005-01

212

8.3.4 LOAD FILE

213

There are two types of load ﬁle, one is Harmonic loads for steady state solution and the other is pulse loads
for transient solution. These load ﬁles may have extension *LOD for convenience. The format of each ﬁle

has to comply with the following:
8.3.4.1 Format

A. Harmonic Loading

Card set 1:
Columns Madame. Eonnat Notes
1-5 NLOAD 15
Card set 2:
Columns Madam. Eonnat Notes
1-5 LOAD(I) 15
6-18 RL(I) 513.4
19-31 X(I) 512.4
3244 Y(I) 512.4

Provide NLOAD cards for card set 2
Card set 3:

ColumnslmnahlLEmnatNotos

1-12 SL(l) 512.4
13-24 SL(2) 512.4
3244 SL(3) 512.4

Provide (NLOAD+5)/6 cards for card set 3

A. Transient Loading

Card set 1:
Columns yariablL Eonnzu Notes
1-5 NLOAD 15
Card set 2:
Columns Yariahlo. Eonnal Notes
1-5 LOAD(I) 15
6-18 RL(I) 513.4
19-31 X(I) 512.4
3244 Y(I) 512.4
Provide NLOAD cards for card set 2

Em
Total number of loads

Ennx
Load identiﬁcation number

Radius of this load (ft)
X-coordinate of this load (ft)
Y-coordinate of this load (ft)

Em
The magnitude of the ﬁrst load (psi)

The magnitude of the second load (psf)
Y-The magnitude of the third load (psf)

Emu
Total number of loads

Em
Load identiﬁcation number

Radius of this load (ft)
X-coordinate of this load (ft)
Y-coordinate of this load (ft)

Note: The user may choose to provide the magnitudes of load in either the time-history input ﬁle (*.TIM) or

in magnitude at batch ﬁle.

5.3.4.2 Examples of Load File

A. An Example of Harmonic Loading

—
O

Note: The ﬁrst column (from 1 to S) in row 2 is the load number, the second column (from 6 to 18) gives the
radius of the load in feet, the third column (from 19 to 31) speciﬁes the x coordinate ofthe load in feet, and
the fourth column (from 32 to 44) speciﬁes the y coordinate of the load in feet. This is repeated as many

\DWQC‘M-BUJN—

.30425-I-00
315054-00
.30425+00
.31505+00
.31 1754-00
.30755+00
.31 1754-00
.307SE+00
.36925+00

000054-00
.43335+01
00005-000
.43335-I01
.(0005+00 606255-100
.43335+01 606255-100
10005-000 118755-000
.43335+01 7.18755+00
.20925+02 250054-00

.00005+00
.00005+00
.1 1255+01
.1 12SE+01

10 .36455+00 .20925+02 633755400

. 12975+05
. 1293E+05

. 1 2965-105
. l 2965+05

. 1297 5+05
. 1 2805+05

times as the number of loads speciﬁed.

The last two rows are the magnitudes of the 10 loads in psf.

B. An Example of Transient Loading

5

Note: The magnitude of the loads is given in the time history of the load either in the time domain or the ﬁe-

m-th—t

.30425+00
315054-00
.30425+00
.31505+00
369354-00

quency domain.

000054-00
.4333E+01
.(X)OOE+00
.43335+01
.20925+02

00005+OO
00005+00
.1 1255+Ol
.l 1255+01
.25005+00

. 12965-105
. 12965405

214

.12935+05 .12965+05

215

8.3.5 OUTPUT REQUEST FILE

This ﬁle deﬁne the request point by deﬁning layer number and x & y-coondinates at which the response is
sought. This output request ﬁle may have extension *REQ for convenience. The format of each ﬁle has to
comply with the following:

13.3.5.1 Format
Card set 1:
Columns YadahlL Eonnat Notes Em
1-5 NOUT 15 Total number of points where outputs are requested
Card set 2:
Columns Xan'ahls. Eonnat Notes Bun
1-5 LOAD(1) 15 Layer number where this output point is located
6-15 XO(I) 510.4 X-coordinate of this output point (ft)
16-25 Y0(I) 510.4 Y-coordinate of this output point (ft)
Provide NOUT cards for card set 2

13.3.5.2 Example of Output Request File

6
4 .0001 .2083
4 4.3333 .2083
4 20.9167 .2083
l .0001 .2083
1 4.3333 .2083
1 20.9167 .2083

Note: The ﬁrst column (1 to 5) gives the layer number at which the response is sought. The second column
(6 to 15) is the x coordinate and the third column (16 to 25) is the y coordinate of that particular response

pomt.