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NUMERICAL MODELING OF

GEOSYNTHETIC REINFORCED

FLEXIBLE PAVEMENTS

FHWA/MT-01 -003/991 60-2

Final Report,
prepared for

THE STATE OF MONTANA
DEPARTMENT OF TRANSPORTATION

in cooperation with

THE U.S. DEPARTMENT OF TRANSPORTATION
FEDERAL HIGHWAY ADMINISTRATION

and the
Idaho, Kansas, Minnesota, New York, Texas, Wisconsin
and Wyoming Departments of Transportation and the

Western Transportation Institute at Montana State University

November 2001

prepared by

Dr. Steven W. Perkins

Montana State University

RESEARCH PROGRAM

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

NUMERICAL MODELING OF GEOSYNTHETIC
REINFORCED FLEXIBLE PAVEMENTS

FH WA/MT-01 -003/991 60-2

Final Report

Prepared for the

STATE OF MONTANA

DEPARTMENT OF TRANSPORTATION

RESEARCH, DEVELOPMENT AND TECHNOLOGY TRANSFER PROGRAM

in cooperation with the

U.S. DEPARTMENT OF TRANSPORTATION

FEDERAL HIGHWAY ADMINISTRATION

and the

Idaho, Kansas, Minnesota, New York, Texas, Wisconsin and Wyoming

Departments of Transportation

and the

Western Transportation Institute at Montana State University

October 1, 2001

Prepared by

Dr. Steven W. Perkins

Associate Professor

Department of Civil Engineering

Western Transportation Institute

Montana State University - Bozeman

Bozeman, Montana 59717

Office Telephone: 406-994-61 1 9

Fax:406-994-6105
E-Mail: stevep@ce.montana.edu

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

TECHNICAL REPORT STANDARD PAGE

1 . Report No.

FHWA/MT-01-003/99160-2

2. Government Accession No.

3. Recipient's Catalog No

4. Title and Subtitle

Numerical Modeling of Geosynthetic Reinforced Flexible
Pavements

5. Report Date

October 1, 2001

6. Performing Organization Code MSU G&C #428573

7. Author

Steven W. Perkins, Ph.D., P.E.

S . Performing rganization Report No.

9. Performing Organization Name and Address

Department of Civil Engineering
205 Cobleigh Hall
Montana State University
Bozeman, Montana 59717

10. Work Unit No

1 1 . Contract or Grant No.

99160

12. Sponsoring Agency Name and Address

Montana Department of Transportation

Research Section

2701 Prospect Avenue

P.O. Box 201001

Helena, Montana 59620-1001

1 3. Type of Report and Period Covered

Final: October 1, 1998 - October 1, 2001

1 4. Sponsoring Agency Code

5401

1 5. Supplementary Notes

Preparation in cooperation with the U.S. Department of Transportation, Federal Highway Administration

16. Abstract

Experimental studies conducted over the course of the past 20 years have demonstrated both general and specific benefits of using geosynthetics
as reinforcement materials in flexible pavements. Existing design solutions are largely empirically based and appear to be unable to account for
many of the variables that influence the benefit derived from the reinforcement. Advanced numerical modeling techniques present an opportunity
for providing insight into the mechanics of these systems and can assist with the formulation of simplified numerical methods that incorporate
essential features needed to predict the behavior of these systems.

Previous experimental work involving the construction of geosynthetic reinforced test sections has shown several difficulties and
uncertainties associated with the definition of reinforcement benefit for a single cycle of load application. Even though many reinforcement
mechanisms are apparent and often times striking during the application of the first load cycle, the distinction between reinforced test sections is
not nearly so clear as that which is seen when examining long term performance, where long term performance is defined in terms of permanent
surface deformation after many load cycles have been applied.

This indicates the need for an advanced numerical model that is capable of describing the repeated load behavior of reinforced pavements. In
particular, models for the various pavement layers are needed to allow for a description of the accumulation of permanent strain under repeated
loads. To meet these needs, a finite element model of unreinforced and geosynthetic reinforced pavements was created. The material model for
the asphalt concrete layer consisted of an elastic -perfectly plastic model where material property direction dependency could be added. This
model allowed for the asphalt concrete layer to deform with the underlying base aggregate and subgrade layers as repeated pavement loads were
applied.

A bounding surface plasticity model was used for the base aggregate and subgrade layers. The model is well suited for the prediction of
accumulated permanent strains under repeated loading and is most suitable for fine-grained materials. A material model containing components
of elasticity, plasticity, creep and direction dependency was formulated for the geosynthetic and calibrated against a series of in-air tension tests.
A Coulomb friction model was used to describe shear interaction between the base aggregate and the geosynthetic. The model is essentially an
elastic-perfectly plastic model, allowing for specification of the shear interface stiffness and ultimate strength. This model was calibrated from a
series of pull out tests.

Finite element models were created to match the conditions in pavement test sections reported by Perkins (1999a). Membrane elements
were used for the geosynthetic and a contact interface was used between the geosynthetic and the base course aggregate. Models of unreinforced
and reinforced pavement sections were created and compared to test section results.

The results showed the model's ability to describe an accumulation of permanent strain and deformation in the system. The models were also
capable of qualitatively showing mechanisms of reinforcement observed from pavement test sections. Exact predictions of pavement system
response were difficult to achieve because of several deficiencies in the material models used and because of the run times needed for the
models. The overriding model deficiency appears to be related to the model for the base course aggregate, which did not appear to be sufficiently
sensitive to effects of restraint of the lateral motion of the material. The observation of certain reinforcement effects on response measures from
the pavement system, such as vertical strain in the top of the subgrade and mean stress in the base course layer, indicate the model's suitability
for use within the context of a mechanistic-empirical modeling approach. This approach requires that the model be used for one load cycle

application with certain stress and strain response measures being used outside the model within empirical damage models to predict long-term
pavement performance. This approach is taken in a companion report for this project (Perkins, 200 1) whose focus is on the development of a
design model for this application.

17. Keywords

Pavements, Highways, Geogrid, Geotextile, Geosynthetic,
Reinforcement, Base Course, Finite Element Model, Numerical
Modeling

1 8. Distribution Statem ent

Unrestricted. This document is available through the
National Technical Information Service, Springfield, VA
21161.

1 9. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

22. Price

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

PREFACE

DISCLAIMER

The opinions, findings and conclusions expressed in this publication are those of the authors and
not necessarily those of the Montana Department of Transportation or the Federal Highway
Administration

ALTERNATE FORMAT STATEMENT

MDT attempts to provide reasonable accommodations for any known disability that may
interfere with a person participating in any service, program or activity of the department.
Alternative accessible formats of this document will be provided upon request. For further
information, call (406) 444-7693 or TTY (406) 444-7696.

NOTICE

The authors, the State of Montana, and the Federal Highway Administration do not endorse
products or manufacturers. Trade and manufacturers names appear herein solely because they are
considered essential to the object of the report.

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

ACKNOWLEDGMENTS

The author gratefully recognizes the generous financial and technical support of the Montana,
Idaho, Kansas, Minnesota, New York, Texas, Wisconsin and Wyoming Departments of
Transportation and the Western Transportation Institute at Montana State University. The
technical contribution of Mr. Yan Wang and Dr. Mike Edens is gratefully recognized. The
Amoco Fabrics and Fibers Company and Tensar Earth Technologies, Incorporated graciously
donated geosynthetic materials for preceding projects leading up to this work. Dr. Muralee
Muraleetharan of the University of Oklahoma and Dr. Kim Mish of Lawrence Livermore
National Laboratory generously provided source code and support for development of a user
defined material model.

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

TABLE OF CONTENTS

LIST OF TABLES viii

LIST OF FIGURES ix

CONVERSION FACTORS xii

EXECUTIVE SUMMARY xiii

1.0 INTRODUCTION 1

2.0 LITERATURE REVIEW 2

2.1 Numerical Modeling of Flexible Pavements 2

2.2 Numerical Modeling of Geosynthetic Reinforced Pavements 5

2.3 Tension Testing and Material Modeling of Geosynthetics 10

2.4 Soil-Geosynthetic Interface Interaction Testing and Modeling 12
3.0 PRIOR TEST SECTION WORK 13

3.1 Test Sections Constructed 14

3.1.1 Test Box and Loading Apparatus 1 5

3.1.2 Pavement Layer Materials 1 6

3.1.3 Instrumentation 19

3.1.4 As-Constructed Pavement Layer Properties 20

3.2 Summary of Results 22
4.0 PAVEMENT LAYER MATERIAL MODELS AND CALIBRATION TESTS 26

4.1 Asphalt Concrete 26

4.2 Base Aggregate and Subgrade 29

4.3 Geosynthetics 33

4.3.1 Uniaxial Tension Tests 34

4.3.1.1 Fast Monotonic Tension 36

4.3.1.2 Creep Tension 36

4.3.1.3 Slow Monotonic Tension 37

4.3.1.4 Cyclic Tension: Series I 37

4.3.1.5 Cyclic Tension: Series II 37

4.3.2 Constitutive Model Formulation 37

4.3.2.1 Elasticity 39

4.3.2.2 Plasticity 40

4.3.2.3 Creep 42

4.3.3 Results 43

4.3.3.1 Fast Monotonic Tension 43

4.3.3.2 Creep Tension 45

4.3.3.3 Slow Monotonic Tension 45

4.3.3.4 Cyclic Tension: Series I 48

4.3.3.5 Cyclic Tension: Series II 48

4.4 Soil-Geosynthetic Interaction 50

4.4.1 Pull Out Tests 51

4.4.2 Determination of Interaction Parameters via Simplified

Numerical Solution 53

4.4.3 Geosynthetic/ Aggregate Interaction Model (GAIM) 55

4.4.4 Calibration of GAIM Via Finite Element Model Simulation of

Pull Out Tests 57

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

5 .0 PAVEMENT TEST FACILITY FINITE ELEMENT MODEL 68

5.1 Unreinforced FE Model 68

5.2 Perfect Reinforced FE Model 70

5.3 Reinforced FE Model 70
6.0 FINITE ELEMENT MODELING RESULTS 71

6.1 Unreinforced Pavements 71

6.2 Reinforced Pavements 78
7.0 CONCLUSIONS 87
8.0 REFERENCES 89
APPENDIX A: NOTATION 94

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

vii

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements

Final Report S. W. Perkins

LIST OF TABLES

Table 2.2. 1 Summary of finite element studies of

geosynthetic reinforced pavements. 6

Table 3.1.1 Comparison test section variables. 14

Table 3.1.2 Geosynthetic material index properties. 18

Table 3.1.3 As-constructed asphalt concrete properties. 21

Table 3.1.4 As-constructed base course properties. 21

Table 3.1.5 As-constructed subgrade properties. 22

Table 3.1.6 Test section loading conditions. 22

Table 4.1.1 Indirect tension resilient modulus test results. 28

Table 4.2.1 Listing of bounding surface model material constants. 32

Table 4.2.2 Material model parameters for base aggregate and subgrade soils. 33

Table 4.3.1 Orthotropic elastic material properties. 40

Table 4.3.2 Anisotropic yield stress ratios. 42

Table 4.3.3 Creep material properties. 42

Table 4.3.4 Anisotropic creep ratios. 43

Table 4.4.1 GAIM material parameters. 60

Table 6.1.1 Material parameter values used for the AC of unreinforced test sections. 7 1

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

viii

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

LIST OF FIGURES

Figure 2.1.1 Cyclic behavior of unbound aggregate a) conventional plasticity models,

b) idealized actual behavior and kinematic hardening models. 4

Figure 2.3.1 Illustration of a) elastic -plastic, b) thermo-visco, c) anisotropic and

d) ratcheting stress- strain behavior. 10

Figure 3.1.1 Schematic diagram of the pavement test facility. 1 5

Figure 3. 1 .2 Input load pulse and corresponding load cell measurement. 16
Figure 3.1.3 Grain size distribution of hot- mix aggregate, base course aggregate

and silty sand subgrade. 17

Figure 3.1.4 CBR versus compaction moisture content for the clay subgrade. 19

Figure 3.2.1 Permanent surface deformation versus load cycle (CS2, 5, 6, 7, 8, 11). 23

Figure 3.2.2 Permanent surface deformation versus load cycle (CS9, 10). 24

Figure 3.2.3 Permanent surface deformation versus load cycle (SSS1, 2, 3, 4). 24

Figure 3.2.4 TBR for sections CS5, 6, 7 and 1 1 relative to section CS2. 25

Figure 3.2.5 TBR for section CS10 relative to section CS9. 25

Figure 4.2.1 Schematic illustration of the bounding surface plasticity model. 30

Figure 4.3.1 Schematic of uniaxial tension specimen configuration. 35

Figure 4.3.2 Boundary conditions for membrane element used in FE analysis. 38

Figure 4.3.3 Tabular data for isotropic hardening rule for the geo synthetics. 41
Figure 4.3.4 Experiment and prediction for fast monotonic uniaxial tension for the

geogrid in the a) machine, b) cross-machine and c) 45° directions. 44
Figure 4.3.5 Experiment and prediction for fast monotonic uniaxial tension for the

geotextile in the a) machine and b) cross-machine directions. 45
Figure 4.3.6 Experiment and prediction for creep uniaxial tension for the geogrid in

the a) machine, b) cross-machine directions and c) 45° directions. 46
Figure 4.3.7 Experiment and prediction for creep uniaxial tension for the geotextile

in the a) machine and b) cross-machine directions. 47
Figure 4.3.8 Experiment and prediction for slow monotonic uniaxial tension for the

geogrid in the a) machine and b) cross-machine directions. 47
Figure 4.3.9 Experiment and prediction for slow monotonic uniaxial tension for the

geotextile in the a) machine and b) cross-machine directions. 48
Figure 4.3.10 Experiment and prediction for series I cyclic uniaxial tension for the

geogrid in the a) machine and b) cross-machine directions. 49
Figure 4.3.1 1 Experiment and prediction for series I cyclic uniaxial tension for the

geotextile in the a) machine and b) cross-machine directions. 49
Figure 4.3.12 Experiment and prediction for series II cyclic uniaxial tension for the

geogrid in the a) machine and b) cross-machine directions. 50
Figure 4.3.13 Experiment and prediction for series II cyclic uniaxial tension for the

geotextile in the a) machine and b) cross-machine directions. 50

Figure 4.4. 1 Schematic drawing of the pull out apparatus. 5 1
Figure 4.4.2 Sleeves used to form the gap interface at the front of the pull out apparatus. 52

Figure 4.4.3 Plan view of in- soil specimen arrangement. 53
Figure 4.4.4 Shear stress vs shear displacement relationship for the simplified numerical

solution of interaction in the pull out test, a) 5 kPa, b) 15 kPa, c) 35 kPa. 55
Figure 4.4.5 Experimental and predicted pull out load-displacement curves for

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

a) geogrid A and b) geotextile A. 56

Figure 4.4.6 Geo synthetic/aggregate interaction model. 57

Figure 4.4.7 Finite element model of pull out box. 59

Figure 4.4.8 FEM and pull out test results for Geogrid A, MD, a = 35 kPa. 61

Figure 4.4.9 FEM and pull out test results for Geogrid A, XMD, a = 35 kPa. 61

Figure 4.4. 10 FEM and pull out test results for Geogrid A, MD, a = 15 kPa. 62

Figure 4.4. 1 1 FEM and pull out test results for Geogrid A, XMD, a = 15 kPa. 62

Figure 4.4. 12 FEM and pull out test results for Geogrid A, MD, a = 5 kPa. 63

Figure 4.4. 13 FEM and pull out test results for Geogrid A, XMD, a = 5 kPa. 63

Figure 4.4. 14 FEM and pull out test results for Geotextile, MD, a = 35 kPa. 64

Figure 4.4. 15 FEM and pull out test results for Geotextile, XMD, a = 35 kPa. 64

Figure 4.4.16 FEM and pull out test results for Geotextile, MD, a = 15 kPa. 65

Figure 4.4.17 FEM and pull out test results for Geotextile, XMD, a = 15 kPa. 65

Figure 4.4.18 FEM and pull out test results for Geotextile, MD, a = 5 kPa. 66

Figure 4.4. 19 FEM and pull out test results for Geotextile, XMD, a = 5 kPa. 66
Figure 4.4.20 FEM and pull out test displacement results at various load levels

for Geotextile, MD, a = 35 kPa. 67

Figure 5.1.1 Finite element model of unreinforced pavement test sections. 69
Figure 6.1.1 Permanent surface deformation from FEM and experiments for

unreinforced SSS test sections. 72
Figure 6.1.2 Permanent surface deformation from FEM and experiments for

unreinforced CS test sections. 72
Figure 6.1.3 Dynamic vertical stress versus depth along the load plate centerline

for test section SSS 1. 73
Figure 6.1.4 Dynamic vertical stress versus depth along the load plate centerline

for test section CS2. 73
Figure 6.1.5 Permanent vertical strain versus radius in the bottom of the base

(z = 160 mm) for test section SSS1. 74
Figure 6.1.6 Permanent vertical strain versus radius in the top of the subgrade

(z = 350 mm) for test section SSS1. 74
Figure 6. 1 .7 Permanent vertical strain versus depth along the load plate

centerline for test section CS2. 75
Figure 6.1.8 Permanent horizontal strain in the bottom of the base

(z = 215 mm) versus radius for test section SSS1. 75
Figure 6.1.9 Permanent horizontal strain in the top of the subgrade

(z = 310 mm) versus radius for test section SSS1. 76
Figure 6.1.10 Permanent horizontal strain in the bottom of the base

(z = 325 mm) versus radius for test section CS2. 76
Figure 6.1.11 Permanent horizontal strain in the top of the subgrade

(z = 415 mm) versus radius for test section CS2. 77
Figure 6.1.12 Dynamic vertical stress versus depth along the load plate

centerline for test section CS2 using a revised model. 78

Figure 6.2.1 Lateral permanent strain in the bottom of the base versus

lateral distance after 10 cycles of load. 80

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

Figure 6.2.2 Lateral permanent strain along the load plate centerline

versus depth after 10 cycles of load. 80

Figure 6.2.3 Mean stress at peak load along the bottom of the base. 81

Figure 6.2.4 Mean stress at peak load along a line 70 mm above the bottom of the base. 81
Figure 6.2.5 Vertical stress at peak load in the top of the subgrade. 82

Figure 6.2.6 Lateral permanent strain in the top of the subgrade

versus lateral distance after 10 cycles of load. 82

Figure 6.2.7 Vertical permanent strain along the load plate centerline

versus depth after 10 cycles of load. 83

Figure 6.2.8 Permanent surface deformation versus applied load

cycles for reinforced sections. 83

Figure 6.2.9 Relative displacement between the base aggregate

and the geosynthetic interface. 84

Figure 6.2.10 Interface shear stress between the base aggregate and the geosynthetic. 84

Figure 6.2.1 1 Permanent vertical strain in the top of the subgrade for various

values of geosynthetic modulus and interface elastic slip (E s i ip ). 86

Figure 6.2.12 Average mean stress in the base aggregate for various values

of geosynthetic modulus and interface elastic slip (E s i ip ). 86

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

CONVERSION FACTORS

The following conversion factors are required for interpretation of results contained in this
report.

1 m = 3.28 ft

1 mm = 0.0394 in

1 kN = 225 lb

1 kN/m = 68.6 lb/ft

lkPa = 0.145 psi

lMN/m 3 = 7.94xlO" 6 lb/ft 3

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

xii

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

EXECUTIVE SUMMARY

Experimental studies conducted over the course of the past 20 years have demonstrated both
general and specific benefits of using geosynthetics as reinforcement materials in flexible
pavements. Existing design solutions are largely empirically based and appear to be unable to
account for many of the variables that influence the benefit derived from the reinforcement.
Advanced numerical modeling techniques present an opportunity for providing insight into the
mechanics of these systems and can assist with the formulation of simplified numerical methods
that incorporate essential features needed to predict the behavior of these systems.

Previous experimental work involving the construction of geosynthetic reinforced test
sections has shown several difficulties and uncertainties associated with the definition of
reinforcement benefit for a single cycle of load application. Even though many reinforcement
mechanisms are apparent and often times striking during the application of the first load cycle,
the distinction between reinforced test sections is not nearly so clear as that which is seen when
examining long term performance, where long term performance is defined in terms of
permanent surface deformation after many load cycles have been applied.

This indicates the need for an advanced numerical model that is capable of describing the
repeated load behavior of reinforced pavements. In particular, models for the various pavement
layers are needed to allow for a description of the accumulation of permanent strain under
repeated loads. To meet these needs, a finite element model of unreinforced and geosynthetic
reinforced pavements was created. The material model for the asphalt concrete layer consisted of
an elastic -perfectly plastic model where material property direction dependency could be added.
This model allowed for the asphalt concrete layer to deform with the underlying base aggregate
and subgrade layers as repeated pavement loads were applied.

A bounding surface plasticity model was used for the base aggregate and subgrade layers.
The model is well suited for the prediction of accumulated permanent strains under repeated
loading and is most suitable for fine-grained materials. A material model containing components
of elasticity, plasticity, creep and direction dependency was formulated for the geosynthetic and
calibrated against a series of in-air tension tests. A Coulomb friction model was used to describe
shear interaction between the base aggregate and the geosynthetic. The model is essentially an
elastic -perfectly plastic model, allowing for specification of the shear interface stiffness and
ultimate strength. This model was calibrated from a series of pull out tests.

Finite element models were created to match the conditions in pavement test sections
reported by Perkins (1999a). Membrane elements were used for the geosynthetic and a contact
interface was used between the geosynthetic and the base course aggregate. Models of
unreinforced and reinforced pavement sections were created and compared to test section results.

The results showed the model's ability to describe an accumulation of permanent strain and
deformation in the system. The models were also capable of qualitatively showing mechanisms
of reinforcement observed from pavement test sections. Exact predictions of pavement system
response were difficult to achieve because of several deficiencies in the material models used
and because of the run times needed for the models. The overriding model deficiency appears to
be related to the model for the base course aggregate, which did not appear to be sufficiently
sensitive to effects of restraint of the lateral motion of the material. The observation of certain
reinforcement effects on response measures from the pavement system, such as vertical strain in
the top of the subgrade and mean stress in the base course layer, indicate the model's suitability
for use within the context of a mechanistic -empirical modeling approach. This approach requires

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

xiii

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

that the model be used for one load cycle application with certain stress and strain response
measures being used outside the model within empirical damage models to predict long-term
pavement performance. This approach is taken in a companion report for this project (Perkins,
2001) whose focus is on the development of a design model for this application.

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

xiv

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

1.0 INTRODUCTION

Experimental studies conducted over the course of the past 20 years have demonstrated both
general and specific benefits of using geosynthetics as reinforcement materials in flexible
pavements (Berg et al. 2000; Perkins and Ismeik, 1997). Existing design solutions are largely
empirically based and appear to be unable to account for many of the variables that influence the
benefit derived from the reinforcement. Advanced numerical modeling techniques present an
opportunity for providing insight into the mechanics of these systems and can assist with the
formulation of simplified numerical methods that incorporate essential features needed to predict
the behavior of these systems.

Previous experimental work reported by Perkins (1999a) involving the construction of
geosynthetic reinforced test sections has shown several difficulties and uncertainties associated
with the definition of reinforcement benefit for a single cycle of load application. These results
are summarized in Section 3 of this report. Even though many reinforcement mechanisms are
apparent and often times striking during the application of the first load cycle, the distinction
between reinforced test sections is not nearly so clear as that which is seen when examining long
term performance, where long term performance is defined in terms of permanent surface
deformation after many load cycles have been applied. For example, an examination of the
dynamic surface deformation or the permanent surface deformation during the first load cycle
often times does not show a clear distinction between reinforced test sections whose long term
performance is dramatically different. In addition, reinforcement benefit, defined in terms of the
increase in the number of load cycles that can be applied to a reinforced section as compared to
that of an identical unreinforced test section, may increase as permanent surface deformation
increases. For this reason, numerical models demonstrating purely elastic response and/or those
models incapable of showing an accumulation of permanent surface deformation and strain
within the pavement layers will require the use of certain simplifying assumptions regarding the
use of empirical damage models relating short term or elastic response to long term behavior.

To allow for the modeling of growth of permanent surface deformation with applied load
cycle, material models for the base aggregate, subgrade soils and most likely the geosynthetic
need to be capable of exhibiting an accumulation of permanent strain with increased load cycle.
The material model for the asphalt concrete needs to contain components allowing it to
permanently deform and conform to the deformed upper surface of the base aggregate. In the

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

absence of this feature, the asphalt layer would attempt to rebound upwards upon the removal of
load and would thereby create artificial tensile stresses acting upwards on the top of the base
aggregate.

The finite element type chosen for the geosynthetic is a critical feature. Geosynthetics in
this application do not offer reinforcement because of a resistance to bending, as would a sheet
of material such as steel. Geosynthetics have essentially zero bending resistance. As such, a
membrane element is the most appropriate element for the geosynthetic as these elements are
formulated to have no in-plane bending resistance. To accurately model the effect of lateral
restraint of base aggregate, a contact or interface model governing shear behavior between the
geosynthetic and the surrounding soil is required.

The purpose of the research described in this report was to formulate a numerical model
(finite element model) that contained these advanced features. Through this work, several critical
modeling features have been noted and have been incorporated into a companion report whose
focus is the development of a design model for reinforced pavements (Perkins, 2001).

2.0 LITERATURE REVIEW

The purpose of this literature review is to present material pertaining to finite element modeling
of flexible pavements, finite element modeling of geosynthetic reinforced flexible pavements,
geosynthetic tension testing and material modeling methods, and soil-geosynthetic interface
interaction testing and modeling methods. This material is presented such that the modeling
needs, as described in Section 1, and direction of this research can be placed within the context
of existing work.

2.1 Numerical Modeling of Flexible Pavements

Numerical modeling of flexible pavements through the use of the finite element method has
developed as the general finite element method has evolved. Early programs commonly used in
practice typically consist of two-dimensional, axisymmetric models with linear or nonlinear
elastic material properties for the various pavement layers (asphalt concrete, base, subbase and
subgrade). Programs such as ILLI-PAVE, MICH-PAVE and ELSYM5 have been developed
within this framework. Models using nonlinear elastic material models generally express the
elastic modulus, or resilient modulus, as a function of stress state, whereas linear elastic models

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

treat the elastic modulus of the materials as a constant for all stress states. These programs
typically apply load to the pavement surface uniformly over a circular area. Two-dimensional
axisymmetric programs can only model a single wheel load application. Three-dimensional
programs are capable of accounting for multiple wheel loads as well as moving wheel loads.
Two-dimensional programs, such as KENLAYER, can also account for multiple wheel loads and
moving wheel loads, but do so by superposition techniques, which are possible only for elastic
material models. Chen et al. (1995) has provided a summary of programs commonly used for
pavement modeling.

Programs developed using elastic material models are incapable of showing permanent
deformation of the asphalt concrete surface as no permanent strains can develop in any of the
material layers upon removal of the traffic load. These programs are typically used to evaluate
the tensile strain at the bottom of the asphalt concrete layer and the vertical compressive strain in
the top of the subgrade when traffic load is applied. Empirical expressions are then used to relate
asphalt concrete tensile strain to fatigue and subgrade vertical compressive strain to permanent
surface deformation.

Finite element programs capable of predicting permanent surface deformation due to the
development of permanent vertical compressive strain in the base and subgrade layers generally
must contain plasticity based constitutive models for these materials. Conventional plasticity
models with isotropic hardening rules are well suited for the prediction of permanent strain under
a single cycle of load application. Under uniform stress and strain conditions, such as that found
in a triaxial test, these models typically show a response illustrated in Figure 2.1.1a where an
elastic -plastic response is seen during the application of load and a purely elastic response is seen
during unloading. Repeated application of a stress to the same level as that experienced during
the initial load cycle results in purely elastic behavior with no accumulation of permanent strain.
Actual material behavior under this type of repeated stress would be as shown in Figure 2.1.1b.
Plasticity based material models with kinematic hardening rules can be formulated to match this
type of material behavior.

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Final Report

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements

S.W. Perkins

CO
CO
CD

-•— '

o
o

-•— '
CC

Q
15

CO
CO
CD

-•— '

o
cs
>

Axial Strain

Figure 2.1.1 Cyclic behavior of unbound aggregate a) conventional plasticity models,
b) idealized actual behavior and kinematic hardening models.

Finite element programs with plasticity models for the base and subgrade exhibiting the
type of behavior illustrated in Figure 2.1.1a are capable of predicting permanent surface
deformation after the application of the first traffic load (Bonaquist and Witczak, 1996; Kirkner
et al., 1994, 1996) but, then tend to not predict well the accumulation of permanent deformation
with increased load cycles. These types of models can show an accumulation of permanent
surface deformation, as illustrated by Zaghloul and White (1993) and White et al. (1998), if the
asphalt concrete layer is allowed to experience a decrease in thickness by virtue of being loaded,
as is possible if a viscoelastic or an elastic -plastic model is used for this layer. Thinning of the
asphalt concrete layer under a given load cycle allows the stress transmitted to the base and
subgrade materials to be greater during the next load cycle, which then allows for additional
plastic strains to develop.

The use of plasticity models with kinematic hardening rules allows for the growth of
permanent surface deformation to be better predicted. Plasticity models of this type have been
available since the 1970' s (Dafalias, 1975) but have only recently been applied to pavement
modeling. McVay and Taesiri (1985) described a bounding surface plasticity model that was
developed and compared to results from repeated load triaxial tests. Ramsamooj and Piper
(1992) described a model that was based on the model originally proposed by Prevost (1978).
This model incorporated a kinematic hardening rule along with routines for pore water pressure
generation and dissipation. The model was compared to cyclic triaxial tests on sands and clays.
The model was used to show the importance of pore water dissipation through drainage on the

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

development of rut depth in a flexible pavement. Desai et al. (1993) and Wathugala and Desai
(1993) have described a hierarchical plasticity model that accounts for cyclic loading. Research
efforts at the US Army Corp of Engineers Waterways Experiment Station (Rollings et al., 1998)
are focused on the implementation of models such as these in finite element codes for the
prediction of permanent deformation of flexible pavements.

2.2 Numerical Modeling of Geosynthetic Reinforced Pavements

A number of studies have been conducted to examine the utility of finite element programs to
predict the response of roadways reinforced with geo synthetics. Several of these studies have
been performed in conjunction with experimental studies such that comparisons between model
predictions and experimental results could be made. For the studies discussed below, Table 2.2.1
has been created to summarize the major features associated with each study's model.

Barksdale et al. (1989) adapted an existing finite element model to predict the response
seen in the experimental portion of their study. The prediction of tensile strain in the base
material was essential in determining the level of tensile strain developed in the geosynthetic,
which in turn determined, in part, the benefit provided by the reinforcement. The cross-
anisotropic linear elastic model used for the base was the only model capable of simultaneously
predicting the lateral tensile strains in the bottom of the base and the small vertical strains in the
bottom and upper part of that layer, as observed in the laboratory experiments.

The finite element model was calibrated and verified by using data from an unreinforced
pavement section from a previous study and from the test data generated from one of the
experimental test series of their study. The unreinforced pavement section used for calibration
was strong in comparison to the sections described for their study. The finite element model was
capable of predicting measured variables to within +/- 20 % for the strong unreinforced section.
For the weaker sections used in the study described as part of their work, the finite element
predictions were not as good. The strain in the geosynthetic was over predicted by about 33 %
when the geosynthetic was located in the bottom of the base. It was under predicted by about 14
% when located in the middle of the layer. The vertical stress and vertical strain on the top of the
subgrade was under predicted by about 50 %. The lateral strains were also under predicted by
about 50 %. The model was not capable of predicting permanent strain or deformation in that all
layers were linear elastic.

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Final Report

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements

S.W. Perkins

Table 2.2.1 Summary of Finite Element Studies of Geosynthetic Reinforced Pavements.

Author

Barksdale et al. (1989)

Burd&Houlsby (1986)

Burd & Brocklehurst
(1990)

Burd & Brocklehurst
(1992)

Dondi (1994)

Miura et al. (1990)

Wathugala et al.
(1996)

Analysis Type

Axi-symmetric

Plane strain

Three-dimensional

Axi-symmetric

Constitutive
Model

Isotropic, non-linear
elastic

None

Isotropic, linear
elastic

Isotropic
elastoplastic, D-P

AC Thickness
(mm)

Variable

None

120

Base

Constitutive

Model

Anisotropic, linear
elastic

Isotropic, elastoplastic,
Matusoka

Isotropic,
elastoplastic, D-P

Isotropic, linear
elastic

Isotropic,
elastoplastic, D-P

Base

Thickness

(mm)

Variable

300

150

140

Geosynthetic

Constitutive

Model

Isotropic, linear elastic

Isotropic, linear
elastic

Isotropic,

elastoplastic, von

Mises

Geosynthetic
Element Type

Membrane

Truss

Solid continuum

Geosynthetic

Thickness

(mm)

None

Interface

Elements &

Model

Linear elastic-
perfectly plastic

None

Elastoplastic, Mohr-
Coulomb

Linear elastic joint
element

None

Subbase

Constitutive

Model

None

Isotropic, linear
elastic

Isotropic,

elastoplastic,HiSS

5„

Subbase

Thickness

(mm)

None

200

165

Subgrade

Constitutive

Model

Isotropic, non-linear
elastic

Isotropic, elastoplastic,
von Mises

Isotropic,
elastoplastic, Cam-
Clay

Isotropic, linear
elastic

Isotropic,

elastoplastic, HiSS

Load
Application

Monotonic

Monotonic, footing width
= 75 mm

Monotonic, footing width
= 500 mm

Monotonic, footing
width = 500 mm

Monotonic, two
rectangular areas,
240 mm x 180 mm

Monotonic, 200 mm
diameter plate

Single cycle, peak

pressure = 725 kPa

on a 1 80 mm

diameter plate

Remarks on

Observed
Improvement

Base layer could be
reduced in thickness
by 4-18%. Greater
improvement seen for
sections with weak
subgrade

Improvement seen after a

penetration of 4 mm.

Model overprecited

improvement beyond a 4

mm displacement

Improvement seen after a
penetration of 12 mm.
Improvement increased

with increasing
geosynthetic stiffness.

Improvement seen

after a penetration of

25 mm.

15-20 % reduction in
vertical displacement,
fatigue life of section
increased by a factor
of 2-2.5

5 % reduction in

vertical displacement.

Improvement level

did not match
experimental results.

20 % Reduction in

Permanent

Displacement

D-P: Drucker-Prager

Department of Civil Engineering,

Montana State University - Bozeman,
6

Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

A parametric study was conducted with the finite element model to calculate the lateral
tensile strain at the bottom of the AC layer and the vertical strain at the top of the subgrade for a
single load application. This was used for evaluations of fatigue resistance and to indicate the
degree of rutting that would occur, which in turn was used to evaluate improvement in pavement
performance for unreinforced and reinforced sections. Reinforcement improvement was
quantified as the reduction in aggregate base thickness for a reinforced roadway giving the same
tensile strain (fatigue) and vertical strain (reflecting permanent deformation) as that for the
unreinforced section. Improvement was seen to increase with increasing geosynthetic stiffness,
and to decrease with increasing subgrade stiffness and asphalt thickness. Optimal improvement
was seen when the geosynthetic was placed between the bottom of the base and 1/3 up into the
base layer.

Barksdale et al. (1989) used the 1972 AASHTO design method to determine design
thickness for the sections with subgrade CBR strengths ranging from 3 to 10 and for two
different traffic loading conditions. Using the more stiff geosynthetic, reductions in base course
thickness ranged between 4 to 16 % when improvement was based on equal lateral strain in the
bottom of the AC layer, and 6 to 18 % when improvement was based on equal vertical strain at
the top of the subgrade. In general, more improvement was observed for sections with a weak
subgrade and a thinner AC layer. The magnitude of the benefits defined in this study are less
than those for a preponderance of experimental studies as summarized by Berg et al. (2000).
Barksdale et al. (1989) felt that the mechanisms modeled were more suited for geotextiles and
that additional research was needed to define the mechanisms of improvement associated with
geogrids and to develop suitable models.

Burd and Houlsby (1986) developed a large strain finite element model for the purpose of
examining experimental results of reinforced unpaved roads, but could be extended to include
material elements representing an asphalt layer. A large strain formulation was included to
account for the extensive rutting that can take place in unpaved roads. Interface elements were
not included in the formulation, which implies perfect fixidity between the soil layers and the
geosynthetic. The model was used to predict the response of a footing resting on a base layer
with a geosynthetic layer placed between the base and the underlying subgrade. The model
predictions were compared to experimental results and were shown to match reasonably well.
The experimental results showed a slight improvement in the load-displacement curve for the

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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reinforced footing for footing penetrations less than 4 mm, while the model did not show
improvements of this kind until the footing penetration exceeded 4 mm. Beyond a penetration of
4 mm, the improvement exhibited by the reinforced footing became significant for both the
model and the experimental results, with the model over predicting the experimental results at
larger displacements and with this over prediction becoming more significant as the footing
displacement increased.

Burd and Brocklehurst (1990) applied this same model to a larger footing. Similar to the
results of Burd and Houlsby (1986) the model did not show improvements in the load-
displacement curve until a settlement of 12 mm was reached. The model was used in a
parametric study to demonstrate the importance of the geosynthetic stiffness on improvement
levels.

Burd and Brocklehurst (1992) extended this model to include interface elements. The
model was used to predict the response of a footing placed on a base material over top a
subgrade with reinforcement between the base and subgrade. The finite element analyses
predicted negligible improvement in the load versus displacement response until a displacement
of over 25 mm was reached. In general, the model with interface elements tended to show less
improvement than the earlier version without these elements. In light of the results of Burd and
Houlsby (1986), where model results were compared to experimental results, it appears that
interface elements were needed only when large footing displacements were present.

Dondi (1994) used the commercial program ABAQUS to model a geosynthetic reinforced
pavement. Load was applied to the pavement surface by two rectangular areas measuring 240
mm by 180 mm and representing a single pair of dual wheels. The wheels were separated by a
distance of 120 mm. Each rectangular area experienced a peak loading pressure of 1500 kPa.
Due to the loading geometry, a three-dimensional finite element analysis was performed. A
cohesion of 60 kPa was assigned to the base course soil to avoid numerical instabilities. Different
friction coefficients were used between the geosynthetic and the base and subgrade soils.
Sections were analyzed with and without the geosynthetic layer and for two geosynthetics of
differing elastic modulus.

The evaluation of stress and strain measures for elements in the base and in the subgrade
indicated that the base layer experienced moderate increases in load carrying capacity for the
reinforced cases while the strain in the subgrade was seen to decrease substantially for the

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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reinforced cases. The model indicated that the geosynthetic layer reduced the shear stresses and
strains experienced by the subgrade. Vertical displacement of the loaded area was reduced by 15
to 20 % by the inclusion of the geosynthetic. The displacement of the unreinforced section was
not indicated. An empirical power expression involving tensile strain in the AC layer was used
to evaluate the fatigue life of the sections, showing that the life of the reinforced sections could
be increased by a factor of 2 to 2.5 as compared to the unreinforced section.

Miura et al. (1990) performed a finite element analysis of a reinforced paved road in
support of a laboratory and field experimental program. The section layer thicknesses were
chosen to match the laboratory test sections. The results from the analysis of reinforced and
unreinforced sections showed general agreement with results from the laboratory test sections
where surface displacement and strain in the geosynthetic were plotted against distance from the
centerline of the load. The improvement in the surface displacement for the reinforced section as
compared to the unreinforced section was greatly underestimated by the finite element model as
compared to the experimental results. The finite element model showed a reduction in
displacement of 5 % while the experiment showed a 35 % reduction. The monotonic loading
results from the finite element analysis were compared to the experimental results at 10,000
cycles of applied load. In this light, the finite element model was not intended to be an exact
representation of the experiments but were intended more to shed light on the mechanisms
involved in reinforcement.

Wathugala et al. (1996) used the commercial program ABAQUS to formulate a finite
element model of a geogrid reinforced pavement. The base aggregate and subgrade soils were
modeled using the hierarchical constitutive model developed by Desai et al. (1986) and
Wathugala and Desai (1993). This model can account for non- linear behavior during non- virgin
loading, which is particularly appropriate for cyclic loading applications. This feature was not
used, however, with non-virgin loading modeled by a linear elastic response. No special
interface models were used between the geogrid and the surrounding soil. The geosynthetic was
given a thickness of 2.5 mm. The pavement section was analyzed with and without the geogrid
layer. The addition of the geogrid was shown to reduce the permanent rut depth by
approximately 20 % for a single cycle of load. This level of improvement was most likely due to
the flexural rigidity of the geosynthetic, which is an artificial feature arising from the material
and element model used for the geosynthetic.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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2.3 Tension Testing and Material Modeling of Geosynthetics

Geosynthetic materials are known to exhibit thermo-visco-elastic-plastic, direction-dependent,
and in some cases, normal stress dependent behavior. Elastic -plastic stress- strain behavior is
illustrated schematically in Figure 2.3.1a where a non- linear response is seen during loading. A
stiffer response is observed during unloading and is often approximated by a linear response
indicative of the elastic behavior of the material. Otherwise, kinematic hardening concepts can
be used to account for hysteretic behavior observed during unloading-reloading cycles. Thermo
and visco behavior are illustrated in Figure 2.3.1b where decreasing temperature (T) or
increasing strain rate result in a stiffer stress-strain response. Direction-dependent or anisotropic
behavior implies a difference in stress-strain response depending on the direction that load is
applied (Figure 2.3.1c). Ratcheting is often observed when constant load amplitude cyclic
tension tests are performed (Figure 2.3. Id), where ratcheting refers to the accumulation of
permanent strain with applied load cycle. Ratcheting is typically described by the incorporation
of kinematic hardening concepts that allow the elastic region to grow, contract and shift with
loading and unloading. Ratcheting may also be viewed as a viscous process where creep strains
develop during each load cycle. Creep and stress relaxation are also material responses that are
commonly associated with geosynthetic materials.

fast or
lowT

slow or

Figure 2.3.1 Illustration of a) elastic -plastic, b) thermo-visco, c) anisotropic and
d) ratcheting stress- strain behavior.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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A number of studies are available that show the characteristics described above. Monotonic
and cyclic tensile tests performed on geogrids (Bathurst and Cai, 1994; Ling et al., 1998; Moraci
and Montanelli, 1997) have shown that tensile stress-strain behavior is non-linear and that
significant plastic strains develop. Constant load amplitude cyclic tests have shown that
permanent strains accumulate with applied load cycle. Bathurst and Cai (1994) have shown that
tensile stress-strain behavior is strain rate dependent. Ashmawy and Bourdeau (1996) have
shown that for a nonwoven geotextile, stress-strain behavior is highly non-linear and that
significant ratcheting occurs with cyclic loads. In contrast, a woven geotextile was shown to
exhibit essentially linear elastic behavior during loading and unloading once the initial crimp is
removed from the material. Additionally, ratcheting was seen to be relatively minor. A number
of studies have shown that geosynthetics exhibit time-dependent creep behavior. Leshchinsky et
al. (1997) have shown both creep and stress-relaxation behavior for geogrids, with stress-
relaxation being observed to be as great as 50 % of the initial load for a polyethylene geogrid.

The above characteristics are complicated by the fact that most geosynthetics exhibit
significant direction-dependent properties. Ingold (1983) has shown that strength anisotropy
exists for a geonet product while many manufacturers commonly report different values for
strength and tensile modulus in the machine and cross-machine directions of a given product.
McGown et al. (1982) has shown that normal stress confinement of certain geosynthetics has an
influence on load-strain behavior. In general, effects of confinement are significant for
nonwoven geotextiles, much less significant for woven geotextiles and non-existent for geogrids.

The finite element method has been used for modeling the response of roadways and
reinforced walls where in the course of this modeling, constitutive models for the geosynthetic
have been implemented. As discussed in Section 2.2, for reinforced roadways, Barksdale et al.
(1989), Miura et al. (1990), Burd and Brocklehurst (1992) and Dondi (1994) have used isotropic,
linear elastic models for the geosynthetic, while Wathugala et al. (1996) used an isotropic,
elastic -perfectly plastic model where plasticity corresponded to a von Mises strength criterion.
For the dynamic analysis of reinforced walls, Yogendrakumar and Bathurst (1992) used a non-
linear hyperbolic model that was capable of describing hysteretic behavior seen during
unloading-reloading cycles. For the static analysis of reinforced walls, Karpurappu and Bathurst
(1995) used a non-linear equation developed from isochronous load- strain- time test data.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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From the laboratory tensile testing data summarized above, it is clear that the stress- strain
behavior of geosynthetic materials is complex and that a general purpose material model must
contain a number of components to describe this behavior. In Section 4.3, a material model for
geosynthetic materials is presented that accounts for elastic, plastic, viscous and anisotropic
behavior.

2.4 Soil-Geosynthetic Interface Interaction Testing and Modeling

Soil-geosynthetic interface interaction properties are commonly evaluated by performing direct
shear tests (ASTM D 5321) and/or pull-out tests (McGown, 1978; Gourc et al, 1980; Ingold,
1983; Jewell et al. 1984; Bonczkiewicz et al., 1988). Direct shear tests are generally thought to
be appropriate for situations where a block of soil moves relative to an essentially stationary
geosynthetic and where the normal stresses on the geosynthetic are relatively low. Common
applications for direct shear tests include covered side slopes for liners and soil block sliding
along a geosynthetic layer for a reinforced slope. These situations correspond to conditions
where extensibility of the geosynthetic does not play a significant role. Pull-out tests are
appropriate for situations where interface shear resistance is governed by the extensibility of the
geosynthetic and where the geosynthetic moves relative to the surrounding soil on both of it's
sides. Common applications for pull-out tests include situations where the geosynthetic is
anchored into a soil mass as loads are applied to the unanchored end, as in a reinforced wall or
slope.

Shear strength parameters are the most common properties determined from these tests
since the designs for which these properties are used are focused primarily on the limit state of
the structure. An interface friction coefficient or angle is generally calculated from direct shear
tests by dividing the ultimate shearing resistance by the normal pressure applied for the test. For
pull-out tests, the ultimate shearing resistance is determined by dividing the ultimate pull-out
load by two times the surface area of the embedded geosynthetic. The ultimate shearing
resistance is then divided by the normal pressure to compute an interface friction coefficient.
This approach assumes that the entire length of the geosynthetic is mobilized when ultimate pull
out load is reached.

For design solutions providing a description of displacements for service loads less than
limit state loads, information describing the shear load - displacement behavior of the interface

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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is necessary. This information is generally expressed in terms of an interface shear modulus
defined as the ratio of mobilized shear resistance to shear displacement. Interface shear modulus
can be defined directly from the initial part of the shear stress versus shear displacement curve
from direct shear tests. For pull-out tests, the definition of interface shear modulus is more
complex. The extensibility of the geosynthetic implies that the distribution of mobilized shear
resistance varies along the geosynthetic and with displacement of the geosynthetic 's loaded edge.
These conditions imply that the pull-out test must be analyzed as a boundary- value problem with
appropriate assumptions made regarding the constitutive relationship of the geosynthetic itself
and for the interface interaction. Adjustment of parameters contained within the material model
for the interface and subsequent comparison of the analysis to the pull-out results allows for the
determination of the interface shear modulus.

For pavement system base reinforcement applications, it is not entirely clear which test is
more appropriate for defining interface shear properties. On the one hand, the lateral movement
of base aggregate atop the geosynthetic appears to be a condition of direct sliding as
approximated by direct shear tests. On the other hand, strains in the geosynthetic can become
appreciable after many cycles of load, meaning that extensibility of the geosynthetic becomes
important and results from pull-out tests may be more appropriate. It is clear that an adequate
description of the small displacement shear stress - displacement relationship is necessary to
describe interaction, particularly for the early part of pavement loading.

Material models have been presented to describe ultimate shear resistance as a function of
normal pressure and geosynthetic grid structure (Koerner et al., 1989; Jewell, 1990; Giroud et al.,
1993). Interface shear stress - displacement relationships have been proposed for the purpose of
evaluating pull-out test results (Juran and Chen, 1988; Yuan and Chua, 1991; Bergado and Chai,
1994; Abramento and Whittle, 1995; Ochiai et al, 1996; Sobhi and Wu, 1996; Alobaidi et al,
1997; Madhav et al, 1998; Gurung and Iwao, 1999; Perkins and Cuelho, 1999). Several finite
element models have been developed to describe pull-out loading of geosynthetics (Wu and
Helwany, 1987; Wilson-Fahmy and Koerner, 1993; Yogarajah and Yeo, 1994).

3.0 PRIOR TEST SECTION WORK

Previous work supported by the Montana Department of Transportation focused on the
construction and evaluation of geosynthetic reinforced pavement test sections constructed in a

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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facility located at MSU. Test sections were constructed for the purpose of providing data to
evaluate the mechanisms by which geosynthetics serve to reinforce flexible pavements and to
provide data to which the numerical model, developed as part of this work, could be compared.
Perkins (1999a,b) provides detailed information describing the pavement test facility, the
construction process, instrumentation used and results obtained. Other papers related to this test
section work are given in Perkins et al. (1998a,b, 1999). The purpose of Section 3 is to briefly
describe the pavement test facility, the materials used, and to summarize the results from this
previous study that are used for comparison to the numerical model.

3.1 Test Sections Constructed

The test sections used for comparison of the numerical model are given in Table 3.1.1. Of these
test sections, 5 are control sections with no reinforcement and 7 are test sections with either a
geogrid or geotextile reinforcement. The geosynthetic products used are described in Section
3.1.2. Two types of subgrade were used for the test sections listed in Table 3.1.1. A clay
subgrade represents a weak subgrade with a CBR of approximately 1.5. The silty sand subgrade
is a more competent material with a CBR of approximately 15. Additional details for these and
the other pavement layer materials are given below.

Table 3.1.1 Comparison test section variables.

Section 3

Nominal

Base

Thickness

(mm)

Subgrade
Type

Geosynthetic

Position

CS2

300

Clay

Unreinforced

CS5

300

Clay

Geogrid B

Base/subgrade interface

CS6

300

Clay

Geotextile

Base/subgrade interface

CS7

300

Clay

Geogrid A

100 mm above base/subgrade interface

CS8

300

Clay

Unreinforced

CS9

375

Clay

Unreinforced

CS10

375

Clay

Geogrid A

Base/subgrade interface

CS11

300

Clay

Geogrid A

Base/subgrade interface

SSS1

200

Silty- sand

Unreinforced

SSS2

200

Silty- sand

Geogrid A

40 mm above base/subgrade interface

SSS3

200

Silty- sand

Geotextile

40 mm above base/subgrade interface

SSS4

200

Silty- sand

Unreinforced

a Nominal ,

AC thickness =

75 mm for al

sections.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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3. 1. 1 Test Box and Loading Apparatus

A test box was constructed having inside dimensions of 2 m in width and length and 1.5 m in
height and was constructed of reinforced concrete. Figure 3.1.1 shows a schematic of the
pavement test facility. A load frame was constructed to rest and ride on I-beams set into the
concrete walls. A load actuator, consisting of a pneumatic cylinder with a 305 mm diameter bore
and a stroke of 75 mm, is used to apply a cyclic load to the pavement. A 50 mm diameter steel
rod 300 mm in length extends from the piston of the actuator. The rod is rounded at its tip and
fits into a cup welded on top of the load plate that rests on the pavement surface.

Load actuator

Rollers

Surface
LVDT

— ^| 6 305 mm LQL

::::::::::::::::: Geosynthetfc : :

Subgrade- :::::::::::::::::::::::

77777777777777777777777777777777777777777777777777777777777777777777777777777777/

1.50 m

2 m '

Figure 3.1.1 Schematic diagram of the pavement test facility.

The load plate consists of a 305 mm diameter steel plate with a thickness of 25 mm. A 4
mm thick, waffled butyl-rubber pad was placed beneath the load plate in order to provide a
uniform pressure and avoid stress concentrations along the plate's perimeter.

A binary solenoid regulator attached to a computer controlled the load-time history applied
to the plate. The software controlling the load pulse was set up to provide the load, or plate
pressure pulse shown in Figure 3.1.2. This pulse has a linear load increase from zero to 40 kN
over a 0.3 second rise time, followed by a 0.2 second period where the load is held constant,
followed by a load decrease to zero over a 0.3 second period and finally followed by a 0.7

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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second period of zero load before the load cycle is repeated, resulting in a load pulse frequency
of 0.67 Hz.

T5
CO
O

45
40
35
30

° 25

>
CC
Q_

20
15
10

Input pulse

-*— Load cell

0.2 0.4 0.6

Time (sec)

0.8

Figure 3.1.2 Input load pulse and corresponding load cell measurement.

The prescribed maximum applied load of 40 kN resulted in a pavement pressure of 550
kPa. This load represents one-half of an 80 kN axle load from an equivalent single axle load
(ESAL) and hence represents one ESAL. The load frequency was selected to allow the data
acquisition system time to store data before the next load pulse was applied. The average peak
plate pressure and standard deviation over the course of pavement loading is given in Section
3.1.4 for each test section reported. The pavement load typically did not return to zero following
the application of each load cycle. The average minimum load over the course of pavement
loading is also given in Section 3.1.4 for each test section. Also shown in Figure 3.1.2 is the
corresponding output from the load cell for a typical load application. The hump seen on the
descending branch of the curve is due to back venting of air pressure into the solenoid and was
characteristic of all load pulses.

3. 1.2 Pavement Layer Materials

Hot-mix asphalt concrete was used for the test sections listed in Table 3.1.1. The aggregate

gradation meets the Montana Department of Transportation specifications for a Grade A mix

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design. Asphalt cement used was PG-58/28 and asphalt content was approximately six percent.
A grain size distribution for the hot-mix aggregate is shown in Figure 3.1.3. As-constructed
properties of the AC for each test section are given in Section 3.1.4. Results from indirect tension
resilient modulus tests are presented in Section 4.1.

100 -I

90 -

80 -

70 -

60 -

ffl

50 -

-t— •

!_
CI)

40 -

30 -

20 -

10 -

asphalt aggregate
e
Subgrade

in 1 1 1 i — i — n ii 1 1 i i — i — mi 1 1 1 i — i — inn 1 1 i — i — iiii 1 1 1 i — i — i

100 10 1 0.1 0.01 0.001

Particle size (mm)

Figure 3.1.3 Grain size distribution of hot- mix aggregate, base course aggregate and silty sand
subgrade.

The geosynthetics used for the test sections shown in Table 3.1.1 and their index properties
as reported by the manufacturers are listed in Table 3.1.2. A series of tension tests were
performed on these two materials and are reported in Section 4.3.1. Pull out tests were also
performed on these two materials with the surrounding soil being the base aggregate used in
these test sections with results presented in Section 4.4.1.

A crushed- stone base course was used for all experimental test sections. The base course
grain size distribution is shown in Figure 3.1.3, where it is seen that 100 % passes the 19 mm
sieve. The material is classified as an A-l-a or a GW. Specific gravity of the material is 2.63.
Modified Proctor tests resulted in a maximum dry unit weight of 21.5 kN/m 3 at an optimum
moisture content of 7.2 %. This material was typically compacted at a moisture content of 6.3 %
and to a dry unit weight of 21 kN/m 3 . As-constructed properties of the base course for each test
section are given in Section 3.1.4. A series of triaxial tests were performed on this material and

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are presented in Section 4.2 when the constitutive model for this material is described and
calibrated. The triaxial tests yielded a drained friction angle of approximately 48 degrees.

Table 3.1.2 Geosynthetic material index properties.

Geogrid A:

Geogrid B:

Geotextile:

Tensar BX- 1100

Tensar BX- 1200

Amoco 2006

Material

Polypropylene

Structure

Punched

Woven

Drawn, Biaxial

Mass/Unit Area (g/m 2 )

215 1

309 1

250 J

Aperature Size (mm)
Machine Direction

25'

None

Cross-Machine Direction

33'

Wide- Width Tensile Strength

at 2 % Strain (kN/m)
Machine Direction

5.06 2

7.32 2

4.25 4

Cross-Machine Direction

8.50 2

11.9 2

13.6 4

Wide- Width Tensile Strength

at 5 % Strain (kN/m)
Machine Direction

9.71 2

13.4 2

11.9 4

Cross-Machine Direction

16.5 1

22.9 2

26.4 4

Ultimate Wide- Width

Tensile Strength (kN/m)
Machine Direction

13. 8 2

21. I 2

40.2 4

Cross-Machine Direction

21.2 2

31.3 2

42.9 4

1 IFAI, 1994; 2 Tensar, 2001; 3 i

VMOCO, 1996; 4 A

MOCO, 2001

To provide information on the influence of subgrade strength on reinforcement benefits,
two subgrade materials were used in this study. A highly plastic clay subgrade was used to
represent a soft subgrade while a silty-sand was used to represent a hard subgrade. The soft
subgrade consisted of a CH or A7-(6) clay, having a liquid limit of 100 % and a plastic limit of
40 %. One hundred percent of the clay material passes the #200 sieve. Specific gravity of the
clay is 2.70. Modified Proctor compaction tests resulted in a maximum dry density of 16.0
kN/m 3 occurring at an optimum moisture content of 20.0 %. The clay was compacted at a
moisture content of approximately 45 % in order to obtain a California bearing ratio (CBR) of
approximately 1.5.

The target moisture content of 45 % was established by conducting laboratory, unsoaked
CBR tests. Figure 3.1.4 shows the variation of CBR with compaction moisture content. On this

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figure, it is noted that only a relatively small change in CBR results between a moisture content
range of 43 to 46 %.

"i 1 1 1 1 1 1 1 1 1 r

36 37 38 39 40 41 42 43 44 45 46 47 48
Water content, %

Figure 3.1.4 CBR versus compaction moisture content for the clay subgrade.

The hard subgrade (approximate CBR=15 at a moisture content of 14%) consisted of fines
trapped from the baghouse of a local batch hot-mix plant. The material is classified as a SM or
A-4, with 40 % non-plastic fines and a liquid limit of 18 %. Specific gravity of the silty-sand is
2.68. Modified Proctor tests resulted in a maximum dry density of 18.2 kN/m 3 occurring at a
moisture content of 1 1.5 %. This material was typically compacted at a moisture content of 14.8
% and a dry unit weight of 17.5 kN/m 3 .

As constructed properties of the compacted clay and silty sand subgrade in the test sections
are given in Section 3.1.4. Shelby tubes were pushed into the subgrade during excavation of the
sections for each test section. Undisturbed samples were used to conduct triaxial tests, with
results presented in Section 4.2 where the constitutive model for the subgrade materials is
presented and calibrated.

3.1.3 Instrumentation

An extensive array of instrumentation was used in the test sections to quantify the mechanical
response of the pavement materials to pavement loading. This data has allowed for the
description of reinforcement mechanisms and has provided data to which the numerical model

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has been compared, as described in Section 5. The test sections contained instruments to measure
applied pavement load, surface deflection, and stress and strain in the various pavement layers.
Instrumentation has been categorized into sensors measuring applied pavement load, asphalt
surface deflection, tensile strain in the asphalt concrete, stress and strain in the base course and
subgrade, and strain on the geosynthetic. Data acquisition software was configured to record
information on the full time-history of response for prescribed load cycles and maximum and
minimum sensor response for other load cycles. A full description of the type of sensors used,
installation techniques and the data acquisition used is given in Perkins (1999a).

3.1.4 As-Constructed Pa vement Layer Properties

Perkins (1999a) has described the construction techniques used for the test sections and the
quality control measures taken to collect data during and after construction. Quality control
measures were taken to provide information on the consistency of the pavement layer materials
between test sections. These measures included measurement of in situ water content and dry
density in the subgrade and base course layers during construction and during excavation, DCP
tests on the compacted subgrade during construction and during excavation, measurement of in-
place density of the compacted AC, and measurement of in-place density of the AC from 100
mm and 150 mm diameter AC drill cores. Additional tests were performed on both bulk AC
samples and the 100 mm diameter cores. These tests included determination of asphalt cement
content, air voids, rice specific gravity, Marshall stability, penetration and kinematic viscosity. A
statistical analysis of these measures was provided and discussed by Perkins (1999a) and
illustrated which sections could be directly compared. The purpose of this section is to
summarize those properties which impact input parameters to the numerical model presented in
Section 5.

As-constructed asphalt concrete properties for the test sections are given in Table 3.1.3.
Test section temperature is determined from average room temperature over the course of the
test. Thickness, density and air voids were determined from direct measurements on 100 mm and
150 mm diameter cores taken from the test sections. Asphalt content was determined from bulk
samples. Marshall stability and flow were determined from 100 mm cores taken from the test
sections.

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Table 3.1.3 As-constru

;ted asphalt concrete properties.

Section

Test Section
Temperature

(°Q

Thickness
(mm)

Density
(kN/m 3 )

Air

Voids

(%)

Asphalt

Cement

(%)

Marshalls

Stability
(lb)

Flow

CS2

23.1

3.3

6.8

2013

CS5

22.6

5.6

6.1

2292

CS6

23.3

3.1

6.6

2471

CS7

22.9

4.3

6.6

1979

CS8

23.1

3.3

6.1

2527

CS9

22.7

5.2

6.3

2167

CS10

22.9

4.3

6.5

2190

CS11

23.4

1.9

6.0

2480

SSS1

23.0

4.1

5.4

2956

SSS2

22.6

6.3

5.7

2043

SSS3

22.4

6.7

6.2

1372

SSS4

22.8

4.4

6.1

2125

As-constructed measurements of the base aggregate and subgrade are listed in Table 3.1.4 and
Table 3.1.5, respectively. Table 3.1.6 provides information on loading conditions for each test
section.

Table 3.1.4

As-constructed

:>ase course properties.

Section

Thickness (mm)

Dry Density (kN/m j )

CS2

300

20.6

CS5

300

20.6

CS6

300

21.0

CS7

300

20.6

CS8

300

20.7

CS9

375

20.9

CS10

375

20.5

CS11

300

20.5

SSS1

210

20.6

SSS2

205

20.7

SSS3

200

20.8

SSS4

200

21.1

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Table 3.1.5

As-constructed subgrade properties.

Section

Thickness (mm)

Moisture Content (%)

Dry Density (kN/m 3 )

CS2

1045

44.8

11.4

CS5

1045

44.9

11.4

CS6

1045

44.4

11.1

CS7

1045

44.2

11.4

CS8

1045 j

44.8 J

11.5

CS9

970

44.9

11.4

CS10

970

44.9

11.3

CS11

1045

45.1

11.4

SSS1

1128

14.7

17.0

SSS2

1131 j

14.9

17.0

SSS3

1147 J

14.8 J

17.1

SSS4

1145

14.8

17.1

Table 3.1.6

Test section loading

conditions.

Section

Average Peak Load

(kN)

Peak Load Standard
Deviation (kN)

Average Minimum Load (kN)

CS2

40.1

0.27

1.0

CS5

40.1

0.34

1.2

CS6

39.9

0.37

1.3

CS7

40.0

0.22

1.3

CS8

40.1

0.21

1.2

CS9

39.9

0.26

1.6

CS10

40.1

0.32

1.2

CS11

40.0

0.44

1.0

SSS1

40.1

0.89

2.2

SSS2

40.3

0.34

1.2

SSS3

40.2

0.73

1.3

SSS4

40.5

0.47

1.0

3.2 Summary of Results

Presented in Figures 3.2.1, 3.2.2 and 3.2.3 are results of permanent surface deformation versus
load cycle applied to each of the test sections. Sections CS2 and CS8 are duplicate unreinforced
test sections with identical pavement layers. Test sections CS5, 6, 7 and 11 can be compared to
CS2 and 8 to evaluate TBR. Similarly, test section CS10 can be compared to CS9 for evaluation
of TBR. Test sections SSS1 and 4 are duplicate unreinforced test sections. These test sections
showed a better performance, as defined in terms of permanent surface deformation, in
comparison to the two reinforced test sections (SSS2 and SSS3). As described in Perkins

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(1999a), the principal reason for this observation was the higher air voids of the asphalt concrete
in test sections SSS2 and 3 as compared to SSS1 and 4 and the resulting reduced stiffness of this
layer. Had the asphalt concrete been more comparable between these sections, it is believed that
little differences in pavement performance would have been seen between reinforced and
unreinforced sections, meaning that reinforcement had little impact for sections with this
structural section and subgrade strength.

200000

400000
Cycle number

600000

800000

Figure 3.2.1 Permanent surface deformation versus load cycle (CS2, 5, 6, 7, 8, 11)

Figures 3.2.4 and 3.2.5 provide values of TBR computed at permanent surface deformation
values ranging from 1 mm to 25 mm. In Figure 3.2.4, sections CS5, 6, 7 and 11 were compared
to section CS2 to calculate TBR values. In Figure 3.2.5, section CS10 was compared to CS9.
Perkins (1999a) provides further results from the instrumentation contained in these sections.
These results are shown as needed in Section 5.3 when predictions from the model are compared
to test section results.

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100000 200000 300000

Cycle number

400000

Figure 3.2.2 Permanent surface deformation versus load cycle (CS9, 10)

SSS4

250000 500000 750000
Cycle number

1000000

Figure 3.2.3 Permanent surface deformation versus load cycle (SSS1, 2, 3, 4).

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100 3

DC
DQ

O 10

° 1 J

0.1

-i 1 1 1 1 1 1 1 r

5 10 15 20

Rut depth (mm)

Figure 3.2.4 TBR for sections CS5, 6, 7 and 1 1 relative to section CS2.

100 3

DC
DQ

.2 10
DC

0)
DQ
O

1 =

0.1

10 15

Rut depth, mm

Figure 3.2.5 TBR for section CS 10 relative to section CS9.

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4.0 PAVEMENT LAYER MATERIAL MODELS AND CALIBRATION TESTS

The numerical finite element model developed as part of this work was designed to match stress,
strain and displacement measurements in test sections described in Section 3. These
measurements describe the dynamic stress response, the accumulation of permanent strain in the
pavement layers and the development of permanent surface deformation for unreinforced and
reinforced flexible pavements. To accomplish these objectives, material models that allow for the
development and accumulation of permanent strain with applied load cycle were required. In
addition, a contact or interface model was required to describe shear behavior between the
geosynthetic and the surrounding soil. This section describes the various material models used to
satisfy these objectives.

4.1 Asphalt Concrete

Measurements from test sections described in Section 3 indicated that less than 15 % of the
permanent surface deformation at the end of a test was due to permanent vertical compression of
the asphalt concrete (AC) below the load plate. Given that asphalt concrete is a viscous material
and that it exhibits permanent strain, ideally a visco-plastic material model would be used. A
number of factors precluded the use of a model of this type. These factors include the relatively
small contribution to permanent deformation due to the AC layer, the lack of relevance of
properties pertaining to the development of permanent deformation in this material on benefits
derived from the reinforcement, the difficulty in determining visco-plastic material parameters
through established laboratory tests, the complexity of material models used for the other
pavement layers and the desire to increase computational efficiency.

Initially, a simple linearly elastic model was selected. After initial use of this material
model in the finite element model, it was observed that the rebound of this elastic layer after the
applied load was returned to zero created vertical tensile stresses on the top of the base layer. For
this reason, the model was extended to include a plasticity component. The plasticity was
introduced by specification of an ultimate yield stress corresponding to a perfect plasticity
hardening law.

Incorporation of this material model into the finite element model described in Section 5
showed that vertical stresses in the subgrade close to the centerline of the load plate tended to be
under predicted, while vertical stresses at a radius greater than approximately 300 mm tended to

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be over predicted. In addition, the predicted deflected shape of the asphalt surface tended to be
more flat than that seen from test section results. These findings suggested that the use of
isotropic elastic and plastic properties for the asphalt concrete tended to cause this layer to act
too much like an elastic slab distributing the stress too broadly. For these reasons, direction
dependency, or anisotropy, was added for the elastic and plastic properties. The addition of
anisotropy essentially allowed for the reduction of the flexural stiffness of the asphalt layer while
maintaining the vertical stiffness in compression.

Direction dependence of elastic properties was prescribed though the use of a linear,
orthotropic elastic constitutive matrix. Orthotroic linear elasticity is described by three moduli
(Eij), three independent Poisson's ratios (Vy), and three shear moduli (G,j), resulting in the elastic
constitutive matrix (Note: Appendix A contains a listing of all symbols and notation used in the
report)

e x

£ y

£ z

> =

• xy

• xz

J yz

l/£,

-v„'£,

-v*/£«

v„/£,

UE y

-%/£«

V IE

xz x

-V IE

yz y

\IE z

1/G„

1/G A ,

\IG

yz)

(4.1.1)

where the subscripts x and y denote the in-plane horizontal directions, and z denotes the vertical
direction. Plasticity was described in terms of an ultimate yield stress, o\c and six plastic
potential ratios, i?,j, given in Equation 4.1.2, whose values are typically less than one and
describe the reduction in yield stress in each respective direction.

®x

R y

°y

°z

> = i

i — >

R *y

O AC

P^xy

V3 I" z

V3X-J

(4.1.2)

Values of elastic modulus (E z ), Poisson's ratio, and yield stress (c° AC ) in the principal

direction of loading (i.e. the z-direction) were determined by conducting indirect tension resilient

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modulus tests per ASTM D4123, where these tests were performed at the Asphalt Institute,
Lexington, Kentucky. Tests were performed on 150 mm diameter samples cored from the test
sections described in Section 3. Four 150 mm diameter cores were typically taken from each test
section upon the completion of the tests and were taken from areas outside the footprint of the
load plate. An additional six 100 mm cores were also obtained. Percent air voids was determined
for each core with the average air voids computed. Resilient modulus was typically determined
for two 150 mm cores from most sections. These cores were chosen to bracket as closely as
possible the average air voids for the test section. Resilient modulus tests were performed at the
average room temperature existing during the time the corresponding test section was loaded and
were performed at three frequencies of loading (0.33, 0.5 and 1 Hz) and at two test positions
corresponding to a 90 degree rotation. At the end of resilient modulus testing, the samples were
loaded to failure to determine the ultimate strength of each core. Values of resilient modulus and
Poisson's ratio are reported in Table 4.1.1 and are average values from each test rotation and
testing frequency.

Table 4.1

.1 Indirect tension resilient modulus test results.

Test
Section

IDT Test

Temperature

(degree C)

Specimen

Air Voids

(%)

Average

Resilient

Modulus (MPa)

Average

Poisson's

Ratio

Ultimate

Tensile

Strength (kPa)

SSS3-1

2.93

4583

0.34

1273

SSS3-3

4.17

3748

0.34

1009

SSS3-4

5.32

3032

0.30

938

SSS4-1

4.25

4519

0.26

1218

SSS4-2

3.38

4596

0.31

1288

SSS4-4

5.57

2826

0.15

888

CS2-3

1.97

3668

0.42

828

CS2-4

1.97

4094

0.36

901

CS5-3

5.12

1150

0.22

585

CS6-2

2.47

1934

0.35

604

CS7-1

4.46

1741

0.31

538

CS7-3

2.66

2049

0.41

581

CS8-2

2.87

1723

0.38

567

CS9-2

4.09

1356

0.41

447

CS9-4

6.10

1188

0.33

468

CS10-2

2.96

2944

0.42

977

CS11-1

1.23

1796

0.95

449

CS11-4

1.88

1538

0.32

441

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The IDT test results show a strong dependency on test specimen temperature and a lesser
dependence on specimen air voids. The dependence on air voids appears to become stronger as
the test temperature decreases. Given the consistency between certain sets of results from the test
sections, it would appear that the difference in actual temperature in the asphalt concrete during
the period over which pavement loading occurred between test sections is less than that implied
by the values of room temperature reported in Table 3.1.3. For instance, test sections CS2 and
CS8 were identical unreinforced test sections that displayed nearly identical pavement loading
performance. The difference in room temperature for the two tests was reported as 7 degrees C.
The IDT tests performed at this temperature difference resulted in a significant difference in
modulus of the AC, which did not appear to be evident from the test section results. It is believed
that the difference in actual AC temperature in the test sections was moderated by the presence
of the large body of soil upon which AC rested and is less than that implied by room temperature
measurements. Values for the AC properties listed in Equations 4.1.1 and 4.1.2 are provided in
Section 6 for the models analyzed.

4.2 Base Aggregate and Subgrade

The constitutive model used for both the base aggregate and the subgrade soil is based on the
bounding surface concept originally developed by Dafalias (1975) and extended for the
description of isotropic cohesive soils by Dafalias and Hermann (1982) and later updated by
Dafalias and Hermann (1986). The model is described in terms of two surfaces represented in the
stress space shown in Figure 4.2.1. The parameters / and / represent the first stress invariant and
the square root of the second deviatoric stress invariant, respectively, and, in general terms, are
reflective of mean normal stress and shear stress, respectively. These surfaces are also a function
of the lode angle, a, defined in terms of the third deviatoric stress invariant. The lode angle
reflects stress paths ranging from triaxial compression to triaxial extension.

The larger surface shown in Figure 4.2.1 represents the bounding surface, which in a
conventional plasticity model is equivalent to a yield surface. The second surface shown in
Figure 4.2.1 denotes an elastic zone. Stress states within the elastic zone produce purely elastic
behavior. Stress states lying between the elastic zone and the bounding surface are capable of
producing both elastic and inelastic behavior. As the stress state approaches the bounding
surface, the rate of plastic strain increases. In a conventional plasticity model, the surface for the

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elastic zone is coincidental with the bounding surface, meaning that stress states lying below the
current yield surface always produce purely elastic behavior. This feature of conventional
plasticity models limits their use for predicting the accumulation of permanent strain in
pavement layers under the application of repeated traffic loads, as explained in Section 2.1.

Ellipsel

Ellipse2

Figure 4.2.1 Schematic illustration of the bounding surface plasticity model.

A radial mapping rule is used to locate a point on the bounding surface corresponding to
some state of stress inside or on the bounding surface. This mapping rule is illustrated by the
dashed line in Figure 4.2.1 having an origin on the / axis at the value CI , where C is a material
parameter and I is defined below. This mapping rule is necessary to prescribe yielding
characteristics determined from the image point on the bounding surface to the current state of
stress.

The bounding surface concept is general and permits the inclusion of any type of

formulation for a yield surface, which is taken to represent the formulation for the bounding

surface. The bounding surface used in the current model consists of three segments, as illustrated

in Figure 4.2.1. The adoption of a combined surface allows greater flexibility in assigning

behavior within the tension region of the stress space, the importance of which is discussed

below. The bounding surface model used for the base aggregate and the subgrade soil uses a

yield surface formulation extended from critical state soil mechanics models (Schofield and

Wroth, 1968). A critical state line, defining the failure state of the material, is given by a line

with a slope of N, where N is a. function of the lode angle and is related to the slope of the critical

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state line, M, in p-q stress space, and where M is given in terms of the material's drained friction

angle as

M= 6sln ^ (4.2.1)

3 - sin (j>

This formulation specifies the current size of the bounding surface in terms of the parameter I ,
the value of which reflects the amount of preloading or preconsolidation of the material. The
value of IJR represents the value of / at the intersection of the bounding surface and the critical
state line. The parameter R defines the ratio of the major to minor axes of ellipse 1 and is a
material constant.

The quantity TI„ defines the intersection of ellipse 2 with the / axis in the tension region
and dictates the tensile strength of the material, with the tensile strength changing depending on
the value of I as dictated by overconsolidation. The parameter T is a material constant and can
be set to a low value to model materials with little tensile strength.

The remaining point defining the shape of the bounding surface is the intersection of the
surface with the / axis. This intersection point is governed by the material constant A. Small
values of A pertain to materials with little cohesion. The parameters R, A and T are known as
shape factors. The parameter s p defines the size of the elastic zone. A value of 1 means that the
elastic zone shrinks to a point located at the projection center, CI . As s p increases to infinity, the
elastic zone becomes larger and approaches the bounding surface.

The model contains another five material parameters in addition to those listed above. The
first two (m and h) are associated with the hardening rule. The next two (k and k) are associated
with the critical state soil mechanics definition of compression behavior in a void ratio vs.
natural logarithm plot and are related to the compression index, C c , and the swelling index, C s , as
defined from consolidation tests, by Equations 4.2.2 and 4.2.3.

* = — (4 2 2)

2.303 K }

K = — (4 2 3)

2.303 K '

The last parameter is Poison's ratio, v. The shear modulus, G, and elastic modulus, E, are then

determined from Equations 4.2.4 and 4.2.5.

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G =

3(l-2v)(l + g/ „)
6k(1+v)

«/-/,} + /J

(4.2.4)

E =

3(l-2v)(l+ g/ „)
3k

((/-/,WJ

(4.2.5)

where / is the current mean hydrostatic stress, h is taken as a constant equal to atmospheric
pressure and e in is the initial void ratio of the material. According to these equations, shear
modulus and elastic modulus will increase as the mean normal stress increases. The model
contains the ability to define separate material constants for M, R, A and h for stress paths in
compression and extension. In the absence of data to support a proper selection of these terms,
values of these parameters were taken to be equal in extension and compression. A summary of
material parameters for the model is given in Table 4.2.1, where all parameters except h are
dimensionless. Steps required for the calibration of these constants is described by Kaliakin et al.
(1987).

Table 4.2.1

Listing of bounding surface model material constants

Parameter

Name

Range of Values

Virgin compression slope

0.1-0.2

Swell/recompression slope

0.02-0.08

Slope of critical state line in p-q stress space

0.8-1.4

Poisson's ratio

0.15-0.3

Atmospheric pressure

101 kPa

Shape parameter

2-3

Shape parameter

0.02-0.2

Shape parameter

0.05-0.15

Projection center parameter

0.0-0.5

Elastic zone parameter

1-2

Hardening parameter

0.02

Hardening parameter

5-50

The model is not ideally suited for the description of granular soils since it has been
formulated in terms of critical state soil mechanics concepts. In particular, the parameters X and
K often times do not adequately define the compression behavior of granular soils. While the
shape parameters describing the cohesion and tensile strength (A and T) can be set low to mimic
the lack thereof in granular soils, some finite level of cohesion and tensile strength is always

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predicted. In addition numerical instabilities can sometimes result when A and T are given low
values. This model, however, was viewed as adequate for the description of the base aggregate
for the purposes of this research. Any limitations associated with the use of this model will be
explored later in this report.

A series of isotropically consolidated undrained conventional triaxial compression tests
were performed on the base aggregate and subgrade materials to calibrate the material properties
contained in Table 4.2.1. Additional isotropically consolidated drained conventional triaxial
compression tests were performed on the base aggregate material. Data was collected during
consolidation for all tests to aid in calibration of the model parameters describing compression
behavior. Tests were performed at overconsolidation ratios of 1, 2 and 6 as needed for calibration
of material parameters (Kaliakin et al., 1987). These tests resulted in the parameters listed in
Table 4.2.2 for the base aggregate and two subgrade soils.

Table 4.2.2

Material model

parameters for base agg

regate and subgrac

Parameter

Values

Clay Subgrade

Silty Sand Subgrade

Base Aggregate

0.236

0.022

0.02

0.15

0.005

0.0018

0.65

1.6

Me/Mc

1.0

0.1

0.2

0.15

h (kPa)

101.4

1.75

1.4

1.5

0.03

0.02

0.015

0.03

0.01

0.0

1.1

1.2

0.02

15.0

20.0

h (kPa)

315

750

3900

soils.

4.3 Geosynthetics

In Section 2.3, it was shown that the stress-strain behavior of geosynthetic materials exhibits
components of elasticity, plasticity and creep and is direction, time and temperature dependent.
Of interest to this project is whether and by how much these material features influence the
performance of geosynthetics used for reinforcement in flexible pavement systems. This question

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has several levels of consideration. The overall objective of the numerical model is to provide a
reasonable match of the results from test sections described in Section 3.0. In light of only this
consideration, the model for the geosynthetic would need only to account for conditions present
in this test facility. Since temperature was relatively constant for all test sections constructed,
incorporation of the dependence of geosynthetic properties on temperature was not necessary and
the model for the geosynthetic was calibrated from tension tests conducted around the same
temperature as that in the test sections. Measurement of strain on the geosynthetics from the test
sections indicated permanent strain as high as 2.5 %. Measurement of dynamic strain indicated
an induced load in the material as high as 2.6 kN/m. These results indicate that plastic strains
occur in the materials and that these strains accumulate with applied load cycle. The latter
observation suggests that ratchetting occurs, as defined in Section 2.3.

These observations suggest that all factors of elasticity, plasticity, creep and direction
dependence are potentially important. The philosophy taken in this work was to assume that each
of these properties was important and that a material model for the geosynthetic should be
formulated to account for each.

Presented in the sections that follow is a constitutive model for geosynthetic materials that
accounts for elastic, plastic, viscous and anisotropic behavior. The incorporation of isotropic -
hardening plasticity allows for non- linear stress- strain behavior to be modeled. Anisotropy is
provided to account for direction dependency of stiffness (elasticity), yield (plasticity) and creep.
The inclusion of creep is provided as an attempt to model ratcheting behavior seen during cyclic
loading. This model is calibrated from and compared to several types of uniaxial tension
experiments described in the following section. Implicit in this approach is the assumption that
the geosynthetic can be treated as a continuum. No attempt has been made to account for the
discontinuous nature of these materials with respect to the theories and models used to describe
stress-strain behavior. As such, this work should be viewed as an examination of the suitability
of continuum-based constitutive models to describe observed geosynthetic stress-strain behavior.

4.3. 1 Uniaxial Tension Tests

To calibrate the components of the geosynthetic constitutive model, several types of uniaxial
tension tests were performed on the geogrid and geotextile described in Section 3.
Manufacturer's properties for each of these materials were listed in Table 3.1.2. For all tension

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tests performed, samples were prepared to a minimum length to width ratio of 2.5, with specimen
length being approximately 750 mm. This configuration is different from commonly used wide-
width specimens and was chosen such that a condition of uniaxial tension, rather than plane
strain tension, occurred in the interior portion of the sample. Conditions of uniaxial tension were
necessary to calibrate material properties contained in the constitutive models used. Specimens
were gripped by gluing the ends of the material between two sheet metal plates. Holes were then
drilled in the plates and mounted to a load cross-arm. An electric gear motor was used to provide
load for constant rate of deformation tests. For load control tests, a pneumatic actuator was used.
Axial and lateral strain was measured on points interior to the sample in order to avoid
lateral restraint effects from the gripped ends. Axial displacement of two points each
approximately 250 mm from the ends of the specimen was measured using displacement pots
fixed to the load frame and attached to the specimen through slightly tensioned, thin wire cables.
The gage length between the two axial displacement points was approximately 250 mm. Lateral
strain was calculated in a similar way by measuring lateral displacement for two points directly
across from each other in the middle of the specimen. Figure 4.3.1 provides a drawing of the
specimen configuration and measurement locations.

A A A A A

Figure 4.3.1 Schematic of uniaxial tension specimen configuration.

The applied line load on the sample was determined by dividing the uniaxial load by the

current width (W c ) of the specimen as determined by

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W e =^(l-8 ; ) (4.3.1)

where W,- is the initial width of the specimen and 8/ is the lateral strain. The initial width of the
geotextile specimens was directly measured, while that for the geogrid was calculated from
Equation 4.3.2, which has been derived from ASTM D5262 (1995)

W : =W_

-^l) (43 - 2)

where W m is the measured width between outside ribs of the sample and iV is the number of ribs
contained across the sample. Geogrid samples typically contained 8 to 11 ribs depending on
whether it was oriented in its machine or cross-machine direction, respectively.

4.3.1.1 Fast Monotonic Tension

The pneumatic actuator was used to apply relatively rapid loads to the geosynthetic specimens
oriented in their machine and cross-machine directions. Rate of strain application was on the
average of 10 % strain per second. Relatively rapid loads were used to collect load-strain data
where creep strains were minor. This data was used to calibrate elastic and plastic material
properties, which is more easily done in the absence of creep. Geogrid specimens were taken to
rupture. Limitations in the load-transfer mechanism prevented taking the geotextile specimens to
rupture. Loads of approximately 75 % of the manufacturer's rated ultimate strength of the
geotextile were applied.

4.3.1.2 Creep Tension

Constant load creep tension tests were performed to calibrate creep properties of the
geosynthetics. To expedite testing time, tests were performed by applying tensile loads to
specimens in stages. Five stages of load were applied to a single specimen with values ranging
from 0.16 to 0.8 kN. Relatively light loads were used to calibrate creep parameters for an
application where loads in this range were anticipated. Each of the five loads was allowed to
remain on the sample for approximately 30 hours prior to the addition of the next load.

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4.3.1.3 Slow Monotonic Tension

The electric gear motor was used to apply load at a relatively slow rate of strain. Strain rates of
approximately 0.25 % strain per minute were used. Data from these tests was not used for direct
calibration of material parameters but was used to assess the ability of the model to account for
differences between fast and slow monotonic loading, where modeled differences were due to
the development of creep strains during slow loading. Model predictions were made using the
actual displacement versus time record from the test being predicted. As with the fast monotonic
tension tests, geogrid specimens were taken to rupture while nearly 100 % of the manufacturer's
rated ultimate strength was applied to the geotextile.

4.3.1.4 Cyclic Tension: Series I

Cyclic uniaxial tension tests were performed where 12 load cycles were applied at increasing
stress amplitudes. The duration of each test ranged from 22 to 27 seconds. Loads for the last load
cycle ranged from 60 to 85 % of the manufacturer' s rated ultimate strength for the geogrid and
45 % of that for the geotextile. Prediction runs for these tests used the actual load time history
from the test being predicted. These tests were performed to allow for the examination of the
suitability of the constitutive model for describing one class of cyclic loads.

4.3.1.5 Cyclic Tension: Series II

A second series of cyclic uniaxial tension tests was performed where cycles of load were applied
at 12 increasing levels of load amplitude and where multiple cycles were applied at each load
amplitude. The number of load cycles applied at each load amplitude ranged from 100 to 700
with the larger number of load cycles applied for the higher levels of load amplitude. Load
cycles were applied at a period of approximately 1.8 seconds. For modeling purposes, the actual
shape of the load pulse was approximated by a flat-topped triangular shaped pulse and applied at
the average pulse frequency observed in the test being predicted. These tests were performed to
determine if the addition of creep in the model could predict observed ratcheting behavior.

4.3.2 Constitutive Model Formulation

Components of the constitutive model were formulated within the context of the commercially

available finite element (FE) package ABAQUS (Hibbitt et al., 1998) used for the entire

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numerical modeling effort. While it is not necessary to formulate and carry out computations
within the framework of a finite element model, this was done in order to utilize the constitutive
drivers contained within the FE program. Predictions made within the FE program required that
an element type and corresponding boundary conditions be selected. A membrane element type
(9-node quadratic) with the boundary conditions shown in Figure 4.3.2 was selected. A 4-node
quadratic element was also used and shown to produce predictions no different than that with the
9-node element. The 4-noded elements were used later in the analysis of the pavement test
facility as these elements were more computationally efficient.

f*i.>

Am •

► xm

1 =Ar _^^ _jjj

Figure 4.3.2 Boundary conditions for membrane element used in FE analysis.

The membrane element type is formulated to possess in-plane tensile and shear stiffness
and strength while containing no resistance to bending or compression. Selection of a membrane
element type requires that a thickness of the membrane be selected. A thickness of 1 mm was
used for both the geogrid and the geotextile. Experimental values of line load (determined as
discussed in Section 4.3.1) were divided by a thickness of 1 mm to obtain experimental values of
uniaxial stress for purposes of calibration. Specification of a membrane element also requires
input of the membrane section's Poisson's ratio. This Poisson's ratio is used to determine
changes in the membrane thickness as load is applied and does not influence in-plane Poisson
effects, which are dictated by specified material properties. A default section Poisson's ratio of
0.5 was used, which implies overall incompressible behavior, meaning that the membrane
thickness decreased in all cases where uniaxial loads were applied. As described in Section 4.3.3,

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results were obtained from FE analyses in such a way as to be comparable to the manner in
which results were derived from experiments.

4.3.2.1 Elasticity

Direction dependence of elastic properties were prescribed though the use of a linear, orthotropic
elastic constitutive matrix. Orthotroic linear elasticity is described by three moduli (Eij), three
independent Poisson's ratios (v,y), and three shear moduli (G/j), resulting in the elastic
constitutive matrix

^ xm

e„

i xm-m

* xm-n
I m-n ,

-v

m-xm

IE m

-v

n-xm

IE n

xm-m

IE xm

-v

n-m

IE n

■v

xm-n

IE xm

-v

m-n

'E,n

xm-m

1/G xm _„

\IG

(4.3.3)

where the subscripts xm and m denote the in-plane cross-machine and machine directions, and n
denotes the direction normal to the plane of the geosynthetic. Elastic constants were calibrated
from the fast monotonic tension and unloading-reloading portions of the cyclic tension tests. The
in-plane elastic parameters (£„ , E m , V xm . m ) were determined directly from tests performed in the
machine and cross-machine directions of the material. Poisson's ratio in the m-xm direction is
related to these other constants by the equation

v =v

m—xm xm—m

(4.3.4)

The in-plane shear modulus (G xm . m ) was calibrated from uniaxial tension tests performed on
samples of the dimensions shown in Figure 4.3.1 and where the samples were cut in a direction
45° to the machine and cross-machine directions. Measurement of uniaxial tensile stress (cT),
uniaxial tensile strain (e~), and lateral strain (e ) allows for the in-plane shear modulus to be
calculated from Equation 4.3.5, which results from a simple stress and strain transformation of
the element.

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2(8-8)

(4.3.5)

The measured stresses and strains in Equation 4.3.5 are from the initial portion of the test.

Properties involving the out-of-plane normal direction, n, were selected only to provide for
stability of the constitutive matrix and are immaterial with respect to subsequent predictions due
to the element type used in the FE analysis for this material. Table 4.3.1 provides a summary of
elastic values calibrated for the geogrid and geotextile materials. As can be seen from Table
4.3.1, the geogrid product has significantly greater shear stiffness as compared to the geotextile.
The geotextile has an in-plane shear stiffness of essentially zero, however a value of zero is not
numerically permissible.

Table 4.3.1 Orthotropic elastic material properties.

Parameter

Geogrid

Geotextile

E xm (kPa)

645,000

960,000

E m (kPa)

600,000

239,000

E n (kPa)

1,000,000

^Jxm -m — *-Tvffl -n — ^Jm-n \&£ &)

30,000

1.0

*xm-m

0.03225

0.5

*m-xm

0.03

0.1245

* xm-n — * n-xm — *m-n — * n-m

4.3.2.2 Plasticity

Plasticity was modeled by the use of the Hill yield criterion with isotropic hardening (Lubliner,
1990). The Hill yield criterion allows for the specification of anisotropic yield. An associated
flow rule was used. The isotropic hardening rule is specified by providing tabular data of
uniaxial yield stress versus plastic strain. Data corresponding to either the machine or cross-
machine directions of the geosynthetic can be used. This data was obtained from the fast uniaxial
tension tests described in Section 4.3.1, where plastic strain was determined by subtracting
elastic strain from the total strain. Figure 4.3.3 illustrates the data used to specify the isotropic
hardening rule for the geogrid and geotextile, where this data corresponds to the cross-machine
direction of the materials.

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40000

Plastic Strain (%)

Figure 4.3.3 Tabular data for isotropic hardening rule for the geo synthetics.

Anisotropic yield was specified by the use of Hill's stress function (Hibbitt et al., 1998),
which serves to modify the amount of yield that takes place in different directions of the
material. These constants are expressed in terms of six yield stress ratios defined as

R xm

R m

v m

xm-m

a/3t"

V xm—m

xm-n

a/3t"

v xm—n

„ R m-n ,

^ m -n\

(4.3.5)

where a ° is a reference yield stress taken to be the tabular data provided for specification of the
isotropic hardening rule (which describes yield in the cross-machine direction), and Oy is the
measured yield stress in each respective direction. Table 4.3.2 provides a summary of the yield
stress ratios for the geogrid and geotextile products, where it is seen that values of 1 for R sm
result from this being the reference direction of the material. Values listed in Table 4.3.2 were

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determined by comparing ultimate yield stress values in non-cross machine material directions to
that in the cross machine direction from fast monotonic tension tests. As with the elastic
properties, yield stress ratios provided for directions involving the out-of-plane direction are
immaterial. The values selected for the yield stress ratio R xm . m reflect the relatively weak in-plane
shear strength possessed by both materials, and where this strength for the geotextile is
essentially zero. This stress ratio was determined by comparison of results from uniaxial tension
tests performed on specimens oriented in a direction 45° to the machine and cross-machine
directions to FE results on similarly oriented materials.

Table 4.3.2 Anisotropic yield stress ratios.

Yield Stress Ratio

Geogrid

Geotextile

Kxm

1.0

0.584 j

0.74

0.7

1.0

Kxm-m — Arm -n — ^m-n

0.091

lxlO" 7

4.3.2.3 Creep

Creep behavior of the geosynthetics was modeled by a strain hardening form of a creep power

law (Hibbitt et al., 1998), where the creep strain rate is given by

(Ao"[(m + l)e c 'fp

(4.3.6)

where A, n and m are material constants, a is the uniaxial tension stress and 8" is the creep strain
in the material. Calibrated values for A, n and m from creep tension tests are listed in Table 4.3.3.

Table 4.3.3 Creep material properties.

Yield Stress Ratio

Geogrid

Geotextile

l.OxlO" 8

l.OxlO 8

-0.8

1.22

1.13

Anisotropic creep was specified in a manner similar to that for anisotropic yield. Six creep
stress ratios were specified to modify or scale the amount of creep taking place in each material
direction. These creep stress ratios are listed in Table 4.3.4.

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Table 4.3.4 Anisotropic creep ratios.

Creep Ratio

Geogrid

Geotextile

J^xm

1.0

0.5

0.55

0.5

Kxm-m — Arm -n — ^m-n

0.3

4.3.3 Results

As noted in the section above, the fast monotonic and creep tests were used for calibration of the
model. Predictions of all the tests described in Section 4.3.1 were made using the model
described above. In general, displacement or load was applied to the three upper nodes shown in
Figure 4.3.2. Figure 4.3.2 shows the material orientation when predictions were made of
response in the machine direction of the geosynthetic. The material axes were rotated 90° when
predictions were made for the cross-machine direction and rotated 45° when in-plane shear
behavior was examined. Predicted axial load was determined by summing the reaction forces for
the three bottom nodes and dividing by the current width of the sample. Axial and lateral strain
were determined by averaging the three top or three side nodal displacements, respectively, and
dividing by the original height and width of the element.

4.3.3.1 Fast Monotonic Tension

Predictions were made of the fast monotonic tension tests with the model containing elastic and
plastic material components and not the creep component described by Equation 4.3.6. Since the
material was not time-dependent for these analyses, displacement was applied according to an
automatic increment scheme. Figures 4.3.4 and 4.3.5 show a comparison of experiments and
predictions for the geogrid and geotextile materials oriented in various directions, where both
axial (positive strains) and lateral strains (negative strains) are plotted against the applied axial
load. For the geogrid, predictions of elastic-plastic response and ultimate strength are well
predicted in the machine and cross-machine directions of the material. For response in the 45°
direction, the majority of the elastic -plastic response is well predicted. Predictions were forced to
produce a greater ultimate strength than that exhibited in the experiments. Ultimate strength in
the experiments was accompanied by significant twisting of geogrid ribs that allowed for
substantial axial displacements to occur. Given that geogrid materials confined by soil would be

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largely prevented from twisting, it was believed that higher ultimate strengths should be modeled
in the predictions.

14 T

Strain (%)

Figure 4.3.4 Experiment and prediction for fast monotonic uniaxial tension for the geogrid in
the a) machine, b) cross-machine and c) 45° directions.

For the predictions of the geotextile material shown in Figure 4.3.5, the majority of the
load-strain curve for the cross-machine direction is well predicted. Given that experiments were
not carried out to failure, ultimate strength in the predictions was selected from manufacturer's
data. Load- strain behavior in the machine direction of the geotextile is not predicted particularly
well. The experimental curve shows a behavior of increasing secant modulus with increasing
strain and is due to removal of the crimp imposed in the material during the manufacturing
process. The scaling of the hardening rule as established from results in the cross-machine
direction by a anisotropic stress ratio to model behavior in the machine direction prevents exact
prediction of this type of behavior. Lateral strain is generally under predicted for the geotextile

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material and is due in part to the relatively low Poisson's ratio used. Poisson's ratios as great as 2
were permissible given elasticity stability constraints but produced numerical instabilities in the
FE program.

Strain (%)

Figure 4.3.5 Experiment and prediction for fast monotonic uniaxial tension for the geotextile
in the a) machine and b) cross-machine directions.

4.3.3.2 Creep Tension

Predictions of the creep tension tests are shown in Figures 4.3.6 and 4.3.7 where axial creep
strain is plotted against time of the applied load. Creep strain from the experiments was
determined by subtracting the instantaneous strain for each load application. Creep strain in the
cross-machine direction of the geogrid is very well predicted by the model and is over predicted
in the machine direction. Over prediction of creep strain in the machine direction was allowed to
better model creep behavior in the slow- monotonic and cyclic tests where results indicated that
creep was under predicted, as will be shown later in this section. For the geotextile, predictions
are seen to be good for both the machine and cross-machine directions.

4.3.3.3 Slow Monotonic Tension

Figures 4.3.8 and 4.3.9 provide predictions of monotonic tension tests performed at a slow strain

rate and where creep was included in the model. Predictions for the geogrid materials are

generally very good. For the machine direction, it appears that greater creep strains are needed to

model behavior, while from Figure 4.3.6a it is seen that a lower creep rate is required. For the

geotextile material in the cross-machine direction, behavior is matched well for low load levels,

however strain is under predicted for higher loads. This is in contrast to Figure 4.3.7b where

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creep strain is seen to be well-predicted. Response in the machine direction of the geotextile is
reasonably well predicted with the exception of lateral strain predictions. The reason for the poor
prediction of lateral strain is most likely due to the same reasons for poor predictions of lateral
strain seen in the fast-monotonic test series, as described in Section 4.3.3.1.

2e+5 4e+5 6e+5
Time (sec)

8e+5

Time (sec)

2.0 t

Time (sec)

Figure 4.3.6 Experiment and prediction for creep uniaxial tension for the geogrid in the
a) machine, b) cross-machine directions and c) 45° directions.

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Time (sec)

.0 -i 1 1 1 1 1 1

1e+5 2e+5 3e+5 4e+5 5e+5 6e+5

Time (sec)

Figure 4.3.7 Experiment and prediction for creep uniaxial tension for the geotextile in the
a) machine and b) cross-machine directions.

14 T

2 4 6
Strain (%)

10 12

Strain (%)

Figure 4.3.8 Experiment and prediction for slow monotonic uniaxial tension for the geogrid in
the a) machine and b) cross-machine directions.

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-4 -2

Strain (%)

■ i i

8 10 12 14

Strain (%)

Figure 4.3.9 Experiment and prediction for slow monotonic uniaxial tension for the geotextile
in the a) machine and b) cross-machine directions.

4.3.3.4 Cyclic Tension: Series I

Figures 4.3.10 and 4.3.11 illustrate predictions of the series I cyclic tension tests. For the geogrid
materials, the loading portion of the curve (the backbone curve) is reasonably well matched. The
unloading-reloading behavior does not show, however, the hysteresis seen in the experimental
results since the model predicts linear-elastic behavior during unloading and reloading to the
previously established yield surface. The relatively stiff, nearly elastic behavior of the geotextile
is seen in Figure 4.3.11.

4.3.3.5 Cyclic Tension: Series II

Predictions of multiple cycle tension tests are illustrated in Figures 4.3.12 and 4.3.13. Two sets
of curves are provided for each material direction. The upper curves correspond to axial strain at
the peak load for the loading cycle plotted, while the bottom two curves correspond to the axial
strain at the end of the load cycle when the applied load is zero. Results have been plotted for the
first and last load cycle for each load increment. For the geogrid materials, predictions of both
maximum and minimum strain are seen to be reasonably good, particularly for the lower load
levels. For the geotextile, predictions of maximum strain are reasonably good while minimum

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strain is under predicted. An accelerated creep strain rate at higher load levels is most likely
needed with the geotextile material in order to provide a better match to the minimum strain
response.

1 2

Strain (%)

1.0 1.5 2.0
Strain (%)

i
3.0

Figure 4.3.10 Experiment and prediction for series I cyclic uniaxial tension for the geogrid in
the a) machine and b) cross-machine directions.

Strain (%)

Figure 4.3.11 Experiment and prediction for series I cyclic uniaxial tension for the geotextile in
the a) machine and b) cross-machine directions.

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2000 3000

Cycle Number

1
4000

I i i i i i i

1000 2000 3000 4000 5000 6000

Cycle Number

Figure 4.3.12 Experiment and prediction for series II cyclic uniaxial tension for the geogrid in
the a) machine and b) cross-machine directions.

4000

Cycle Num be r

1000 2000 3000 4000 5000 6000
Cycle Number

Figure 4.3.13 Experiment and prediction for series II cyclic uniaxial tension for the geotextile in
the a) machine and b) cross-machine directions.

4.4 Soil-Geosynthetic Interaction

Pull out tests were conducted to provide a means of calibrating a model used for interaction
between the geosynthetics and the base course aggregate. Since the pull out test involves not
uniform displacement and strain conditions, the test must be analyzed as a boundary value

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problem in order to extract material properties. Two methods were used to analyze the pull out
experiment. The first is described in Section 4.4.2 and consists of a simplified numerical solution
of the problem. The second involves using the finite element program and material models used
in this research for the base aggregate and geosynthetic materials. The first solution provides
initial values of parameters that are later updated in the second method.

4.4.1 Pull Out Tests

The pull out apparatus used to generate data to which an interaction model could be compared
was built following guidelines established by ASTM (1995). The box is similar in design to that
reported by Farrag (1991) and is shown schematically in Figure 4.4.1. The inside dimensions of
the box are 1100 mm high, 900 mm wide and 1250 mm long. The box was fabricated from 6.35
mm thick steel plate reinforced by flat steel stiffeners running vertically along the outside of the
box's walls.

Reaction Frame

Air Bag (Normal Confinement)

Sheet Metal Grip

Loading Device

1100 mm

Load Transfer Sleeves

Figure 4.4.1 Schematic drawing of the pull out apparatus.

The gap interface at the front of the box was designed to minimize the development of

lateral earth pressure induced by soil movement toward the front wall as geosynthetic pull out

progressed. This was accomplished by the use of two sleeves as shown in Figure 4.4.2 that

extended into the box. The upper surface of the top sleeve along with the top half of the front

wall, the top half of the rear wall and the entire height of the side walls were lined with

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lubrication layers to reduce friction as soil moved towards the front wall. A smooth, semi-rigid
geomembrane was first attached to these wall surfaces with rivets. Low-adhesion silicone grease
was applied on the exposed surface of the geomembrane. A latex rubber membrane was then
placed over top of the greased surface prior to soil placement. Vertical normal stress was applied
to the top of the soil mass with a flexible bladder fitting the plan area of the pull out box, as seen
in Figure 4.4.1. The bladder was controlled by regulated air pressure and could be inflated to a
maximum pressure of 200 kPa.

Pullout box
front wall

Direction of
pull

mm - 100 mm
adjustable

Figure 4.4.2 Sleeves used to form the gap interface at the front of the pull out apparatus.

Figure 4.4.3 shows the arrangement of the geosynthetic sample as placed in the pull out
box. The geosynthetic was gripped by gluing it between two sheet metal plates that extended out
though the gap in the box. Five extensometers (Celesco Transducer Products, Model PT-101,
Canoga Park, CA) were used to monitor displacement along the length of the geosynthetic
during pull out. The cables for the extensometers were enclosed in a rigid housing. The ends of
the cables were attached to the geogrid at the rib junctions using metal clips. For the geotextile, a
low-profile nut and bolt assembly wedged through the weave of the material was used to hold the
end of the cable. The length of the geosynthetic samples ranged from 300 mm to 715 mm with
the shorter samples being used for the higher confinement pressures.

Pull out force was provided by a screw jack driven by an electric motor that was set at a
displacement rate of 1 mm per minute and was measured by a load cell. Prior to conducting a
pull out test, the force versus displacement relationship needed to overcome friction between the

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sheet metal and the confining soil was determined by conducting pull out tests on the sheet metal
alone. These tests were conducted at each of the confining stress levels used in the pull out tests
with the results being used to adjust pull out load measurements from tests on the geo synthetics.
Pull out tests were conducted at confining stress levels of 5, 15 and 35 kPa.

Geosynthetic
Sample

Load Transfer
Sleeve

Extensometers

150 mm
■* *

900 mm

I e? 150 mm

y2 600 mm

"Tt-

300 - 715 mm

1250 mm

Sheet Metal Grip

Figure 4.4.3 Plan view of in- soil specimen arrangement.

Geogrid A and Geotextile A were used to conduct the pull out tests. Properties of these
materials were summarized in Table 3.1.2. Pull out tests were performed on each material
oriented in the machine and cross-machine directions. The base aggregate described in Section
3.1.2 was used as the confining soil. The aggregate was placed in the pull out box at a water
content of 5 % and compacted in 30 mm lifts to a dry density of approximately 20.5 kN/m 3 ,
which represented 95 % of the modified Proctor density. A hand-held vibrating plate was used to
compact the material. Results from the pull out tests are presented in Section 4.4.2 when
compared to the simplified numerical solution.

4.4.2 Determination of interaction Parameters Via Simplified Numerical Solution

The boundary conditions described for the pull out test preclude the use of a simple calculation

for the determination of interaction parameters. A numerical solution was developed to describe

the pull out process. Perkins and Cuelho (1999) have described the development of this solution

in detail. This solution is a simplified version of that developed through the finite element

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method as described in Section 4.4.3 in that a number so simplifying assumptions were made.
These assumptions include:

• The surrounding soil was regarded as a stationary rigid body such that absolute
movement of the geosynthetic was equivalent to relative movement between the
geosynthetic and the soil. This assumption also meant that a material model was not
required for the soil.

• A simple, non-linear load-strain relationship was assumed for the geosynthetic. This
relationship does not contain the features described in Section 4.3, which are included in
the finite element model of the pull out test.

In essence, the simplified model does not contain the complex material descriptions for the soil
and geosynthetic described in Sections 4.2 and 4.3.

The expression used to describe the relationship between shear stress (x) and shear
displacement (u) between the geosynthetic and the surrounding soil is given in function form as:

x=/(«,G i)¥p , ¥r ,o„) (4.4.1)

where G, is the initial interface shear modulus defined as the initial slope of the shear stress vs.
shear displacement curve, \\f p and \\f r are the peak and residual friction angles for the interface,
and G„ is the normal stress on the interface. This relationship allows for a non-linear curve of
shear stress versus shear displacement to be specified.

This solution was applied for the conditions present in the pull out tests described in
Section 4.4.1. The parameters G„ \\f p and \\f r were varied until a reasonable match was achieved
between the experiments and the predictions. Figure 4.4.4 shows the shear stress versus shear
displacement curves resulting for geogrid A and geotextile A when pulled in their machine and
cross-machine directions under normal stress confinements of 5, 15 and 35 kPa. A comparison of
the pull out force measured at the front of the geosynthetic versus the pull out displacement at
this same point between predictions and experiment is shown in Figure 4.4.5. The values of G„
\\f p and \\f r from this approach were used as starting values for input parameters into the finite
element model of the pull out test.

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GG-XMD

Q.
OT

(0
CD

GG-MD

GT-MD

GT-XMD

10 20 30 40 50
Shear Displacement (mm)

— i

10 20 30 40 50
Shear Displacement (mm)

GG-XMD

10 20 30 40 50
Shear Displacement (mm)

Figure 4.4.4 Shear stress vs. shear displacement relationship for the simplified numerical
solution of interaction in the pull out test, a) 5 kPa, b) 15 kPa, c) 35 kPa.

4.4.3 Geosynthetio 'Aggregate Interaction Model (GAIM)

The finite element model contained an interaction material model for the interface between the
base aggregate layer and the geosynthetic. The model consisted of Coulomb friction model with
direction and normal stress dependent friction coefficients (Hibbitt et al. 1998). In its simplest
form, the model contains two material properties, a friction coefficient, (I, and a parameter E s n p .
The model is illustrated with the aid of Figure 4.4.6. Shearing resistance, x, is a function of the
amount of shear displacement, A, the latter being the relative displacement between the
aggregate layer and the geosynthetic. The initial part of the x vs. A curve is elastic, with the slope
of the curve dictated by specification of E s n p . Ultimate shearing resistance is reached according
to the relationship between x and a, which is specified by the friction coefficient, (I. From Figure

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4.4.6, it is seen that the shear stiffness of the interface, given by the elastic part of the x vs. A
curve, is not constant but increases as normal stress on the interface increases.

5kPa

Experimental

A ♦ • Predicted
-i 1 1 1 1 1 1 1 1 1

10 20 30 40 50

Displacement (mm)

8 10

^35kPa

/ _^-15kPa

A /

5kPa

^ — •

""" *

- HI/

Experimental
Predicted
i i

A ♦ •

1 1

Displacement (mm)

Figure 4.4.5 Experimental and predicted pull out load-displacement curves for
a) geogrid A and b) geotextile A.

The friction coefficient can take on different values for the two principal in-plane directions
of the contact interface. The friction coefficient can also be specified as a function of normal
stress on the interface by listing values of friction coefficient and normal stress

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X 2

X 3

.Oi

a 2

a 3

-slip

x 2

X 3

Oi CJ

Figure 4.4.6 Geosynthetic/aggregate interaction model.

4.4.4 Calibration of GAIM Via Finite Element Model Simulation of Pull Out Tests
Calibration of the material parameters contained in the Geosynthetic/Aggregate Interaction
Model (GAIM) was accomplished by creating a finite element model of the pull out box
described in Section 4.4.1. The GAIM described in Section 4.4.3 was used for the contact
interfaces between the geosynthetic and the aggregate. Initial values for the material parameters
contained in the GAIM were assigned from information obtained from the simplified numerical
solution described in Section 4.4.2. Material parameters were then adjusted until predictions
from the finite element model matched those from the pull out tests.

The finite element model developed for the pull out box is shown in Figure 4.4.7.
Symmetry of the box was recognized such that one-half of the box could be modeled. Three
views of the pull out box model are shown. The box top view shows the plan view of the box

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looking down on the box, where the centerline of the box is shown. The centerline represents the
plane of symmetry for the two box halves. The side of the box corresponding to the centerline
had boundary conditions where displacement in the y direction was constrained. Displacement in
the x and z directions was unconstrained. The remaining three sides of the box contained
boundary conditions corresponding to constraining displacement perpendicular to the face of the
box while allowing displacement in the plane of the face. This condition models the lubricated
sides used in the pull out box. A uniform mesh size of 12 elements in the x direction and 5
elements in the y direction was created.

The box side view shows the height of the two halves of aggregate above and below the
plane containing the geosynthetic. Three elements were contained in the height above the
geosynthetic and 4 in the height below. The height of these elements became finer as the plane
containing the geosynthetic was approached, as noted in Figure 4.4.7.

The geosynthetic was modeled using 4 noded membrane elements and used the material
properties described in Section 4.3. The membrane was placed in a position corresponding to that
used in the pull out test being modeled. The width (y dimension) of the geosynthetic was
typically 0.3 m. The front edge of the geosynthetic was 0.4 m from the front face of the box. A
uniform mesh was used for the geosynthetic. Six elements were contained across the width (y
dimension) of the geosynthetic, with 6 to 14 elements contained along the length (jc direction)
and depending on the length of the geosynthetic. The edge of the geosynthetic along the
centerline of the box was constrained from displacement in the y direction. No other boundary
conditions were applied to the geosynthetic sheet. Contact interfaces were established above and
below the geosynthetic to describe interaction between the geosynthetic and the aggregate.

The base aggregate material model corresponded to that described in Section 4.2. The
aggregate was given a density and therefore exerted a self-weight normal pressure on the
geosynthetic. Additional normal pressure was applied along the top surface of the upper
aggregate layer to produce the desired normal stress (a) on the surface of the geosynthetic.
Displacement was applied to the leading edge of the geosynthetic at a rate of 1 mm per minute.
Displacement in the y direction of the leading edge was constrained as this displacement rate was
applied in the x direction.

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Box Top View:

0.45 m

}IL

Mesh:
12(x)-5(y)

-1 .25 m.

C/L

-> x

Box Side View:

0.38 m

0.62 m

A
Mesh: 3 Elements

Mesh: 4 Elements

Geosynthetic Plan View:

0.15 m

0.1 m

0.2 m

-> x

Figure 4.4.7 Finite element model of pull out box.

Reaction forces for the nodes along the leading edge where displacement was applied were
summed for a range of displacement values. This allowed the pull out force to be plotted against
the pull out displacement. The displacement along the length of the geosynthetic could also be

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plotted at different pull out load levels. Adjustment of the GAIM parameters to provide a match
between finite element model predictions and pull out test results yielded the parameters
summarized in Table 4.4.1 for Geogrid A and the Geotextile. Direction dependency of the
friction coefficient (|l) was used for both geo synthetics. Normal stress (a) dependency on the
friction coefficient was used for the geotextile.

Table 4.4.1 GAIM material parameters.

a (kPa)

Eslip (m)

Geogrid A

1.376

1.570

0.001

1.376

1.570

1.376

1.570

Geotextile

0.840

0.750

0.001

1.050

1.020

1.270

1.150

Results from the finite element model are compared to the pull out test results in Figures
4.4.8 - 4.4.19 where it is seen that generally good agreement is seen between predictions and test
results. Figure 4.4.20 shows an example from one pull out test of the development of
displacement at different load levels during the test. In this figure, displacement at various
positions along the length of the geosynthetic is plotted for six different load levels.

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Geogrid A

MD
a = 35 kPa

■ FEM Prediction
• Experiment

20 30 40

Displacement (mm)

Figure 4.4.8 FEM and pull out test results for Geogrid A, MD, a = 35 kPa.

Geogrid A

XMD
a = 35 kPa

•Experiment

■ FEM Prediction

20 30

Displacement (mm)

Figure 4.4.9 FEM and pull out test results for Geogrid A, XMD, a = 35 kPa.

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E 10

re
o

Geogrid A

MD
o= 15kPa

■ FEM Prediction
•Experiment

20 30 40

Displacement (mm)

Figure 4.4.10 FEM and pull out test results for Geogrid A, MD, a = 15 kPa.

Geogrid A

XMD
o= 15kPa

ju -

25 -

-. 20 -
E
z
&15-

*
*

* ^^^

/ >^

.'/

* * .

J 10-

if
i/
it
if

- - - -FEM Prediction

»/

5 -

20 30

Displacement (mm]

Figure 4.4.11 FEM and pull out test results for Geogrid A, XMD, a = 15 kPa.

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Geogrid A

MD
g = 5 kPa

12 i

FEM Prediction
Experiment

20 30

Displacement (mm)

Figure 4.4.12 FEM and pull out test results for Geogrid A, MD, a = 5 kPa.

Geogrid A

XMD
g = 5 kPa

FEM Prediction
Experiment

20 30

Displacement (mm]

Figure 4.4.13 FEM and pull out test results for Geogrid A, XMD, a = 5 kPa.

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Geotextile

MD
a = 35 kPa

FEM Prediction
Experiment

20 30

Displacement (mm)

Figure 4.4.14 FEM and pull out test results for Geotextile, MD, a = 35 kPa.

Geotextile

XMD
a = 35 kPa

FEM Prediction
Experiment

20 30

Displacement (mm]

Figure 4.4.15 FEM and pull out test results for Geotextile, XMD, a = 35 kPa.

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Geotextile
MD
25 c= 15 kPa

f 15

§ 10

5 -

FEM Prediction
Experiment

10 20 30 40 50 60 70 80

Displacement (mm)

Figure 4.4.16 FEM and pull out test results for Geotextile, MD, a = 15 kPa.

Geotextile

XMD
o= 15kPa

FEM Prediction
Experiment

10 20 30

Displacement (mm;

Figure 4.4.17 FEM and pull out test results for Geotextile, XMD, a = 15 kPa.

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7 i

Geotextile

MD
g = 5 kPa

S 3
o

1 fi

- -FEM Prediction

— Experiment

20 30

Displacement (mm)

Figure 4.4.18 FEM and pull out test results for Geotextile, MD, a = 5 kPa.

Geotextile

XMD
g = 5 kPa

■ FEM Prediction
• Experiment

20 30

Displacement (mm)

Figure 4.4.19 FEM and pull out test results for Geotextile, XMD, a = 5 kPa.

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Pull Out Load:
2.9 kN/m

100 150 200

Distance (mm)

300

Pull Out Load:
4.6 kN/m

100 150 200

Distance (mm)

300

Pull Out Load:
7.2 kN/m

100 150 200

Distance (mm)

300

Pull Out Load:
12.4 kN/m

100 150 200

Distance (mm)

300

E 6.0

n. 4.0 -

Pull Out Load:
17.7 kN/m

100 150 200

Distance (mm)

300

Pull Out Load:
25.4 kN/m

100 150 200

Distance (mm)

300

Figure 4.4.20 FEM and pull out test displacement results at various load levels for Geotextile,
MD, a = 35 kPa.

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5.0 PAVEMENT TEST FACILITY FINITE ELEMENT MODEL

A finite element model was created to simulate the pavement layer thicknesses, boundary
conditions and loading present in the pavement test sections described in Section 3. All modeling
was done using the commercial program ABAQUS (Hibbitt et al. 1998). Three types of models
were created. The first is a model of pavement test sections without reinforcement and is
described in Section 5.1. The second is a model where reinforcement is described in such a way
that it represents the maximum amount of reinforcement benefit that could be expected with a
"perfect" reinforcement product. This model was created to provide a means of comparison to
the 3 rd type of model where the geosynthetic reinforcement layer was explicitly included. Since
effects of the reinforcement are ultimately expressed in terms of prevention of lateral movement
of the base aggregate at the level of the geosynthetic, perfect reinforcement is simulated by
modifying the unreinforced model by preventing all in-plane or lateral motions of the base
aggregate element nodes at the level of the geosynthetic. This in effect simulates reinforcement
with an infinitely stiff geosynthetic and an infinitely stiff contact shear interface between the
geosynthetic and the aggregate. This model is described in Section 5.2. The third type of model
created is one where a separate material layer corresponding to the geosynthetic is added to the
unreinforced model and is described in Section 5.3.

5.1 Unreinforced FE Model

The finite element model of unreinforced pavements is a 3-dimensional model created to match
the conditions for the pavement test sections described in Section 3. A two-dimensional axi-
symmetric model was not used because of the potential influence of the box's square corners and
for the later inclusion of a layer of geosynthetic reinforcement that has direction dependent
material properties. Symmetry of the box was recognized such that a model of one-quarter of the
box was created. Figure 5.1.1 illustrates the geometry and boundary conditions used for the
development of the model. Actual layer thicknesses for the AC and base aggregate correspond to
the test section being modeled and were given in Tables 3.1.3-3.1.5.

The width in the x and y directions of the l A box modeled was 1 m. The pavement load was
applied as a uniform pressure equal to the values given in Table 3.1.6 for each test section over
one-quarter of a circular plate having a radius of 152 mm. The time history of the pavement load
was applied to approximate the curve given in Figure 3.1.2. For several models, the load plate

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and rubber pad were modeled using additional material elements. For these cases, the load plate
and rubber pad were modeled by plates having a radius of 152 mm. The load plate had a
thickness of 25 mm and was given isotropic elastic properties with a Young's modulus of 2xl0 8
kPa and a Poisson's ratio of 0.33. The rubber pad had a thickness of 4 mm and was also given
elastic properties with a Young's modulus of 400 kPa and a Poisson's ratio of 0.

' Sym ^try^

Figure 5.1.1 Finite element model of unreinforced pavement test sections.

The vertical edge directly beneath the load plate centerline was a symmetry line and was
therefore constrained from motion in the x and y dimensions and free from constraints in the z
direction. The four faces of the box were constrained in a direction perpendicular to the box face
and in the second horizontal direction parallel to the box wall, and otherwise free of constraint in
the z direction. The nodes along the perimeter of the asphalt concrete layer directly adjacent to
the box walls were free of all constraints such that the nodes were free to move in from the box
wall as pavement load was applied. This boundary condition removed an artificial attachment of
the asphalt concrete to the walls of the box and thereby prevented tensile loads from developing
in the asphalt concrete. The symmetry planes of the model were unconstrained in the z direction

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and in the horizontal direction parallel to the plane. Motion in the horizontal direction
perpendicular to the plane was constrained.

Eight-noded hexagonal solid elements were used for all material layers. Approximately 42
elements were used for each of the load plate and rubber pad while 230, 570 and 1710 elements
were used for the asphalt concrete, base aggregate and subgrade layers, respectively. The nodes
between the material layers were equivalenced and therefore connected.

5.2 Perfect Reinforced FE Model

A FE model was created where the reinforcement was modeled in such a way as to provide for
the maximum effect on pavement performance. Within the context of the material and finite
element models developed for this project, the principal effect of reinforcement on the
performance of the pavement is the prevention of lateral strain or displacement of the base
aggregate at the interface with the geosynthetic. Maximum effect of a reinforcement layer could
thereby be simulated by preventing all lateral motion of the base course aggregate at the level
where it would be in contact with the geosynthetic. This was accomplished by modifying the
unreinforced model described in Section 5.1 by prescribing boundary conditions to the nodes at
the bottom of the base aggregate, where these boundary conditions prevented all x and y motion
of the nodes. For these models, the simulated reinforcement effectively has an infinite tensile
stiffness and an infinitely stiff contact interface with the base aggregate.

5.3 Geosynthetic Reinforced FE Model

A third type of finite element model was created where a sheet of geosynthetic reinforcement
was included as part of the pavement cross-section. The geosynthetic was modeled by 4 noded
membrane elements that have the property of containing tensile load carrying capacity, but have
no resistance in bending or compression. Membrane elements are two-dimensional elements that
are commonly used for describing flexible sheets having tensile load carrying capacity. The
material model described in Section 4.3 was used for the geosynthetic. In all cases, the
geosynthetic was placed between the base aggregate and the subgrade. The contact interface
model described in Section 4.4 was used between the base course aggregate and the
geosynthetic.

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6.0 FINITE ELEMENT MODELING RESULTS

6.1 Unreinforced Pavements

FE models were created to match conditions in unreinforced test sections described in Section 3.
Layer thicknesses, density and void ratio for the materials used in test sections SSS1, SSS4, CS2,
CS8 and CS9 were modeled. Table 6.1.1 provides a summary of the properties used for the AC
layer for each test section. Material model parameters for the bounding surface plasticity model
used for the clay and silty sand subgrade and the base aggregate were listed in Table 4.2.2.

Table 6.1.1

Material parameter values used for the AC of unreinforced test sections.

Parameter

SSS1

SSS4

CS2

CS8

CS9

E x (MPa)

3150

3400

3920

2980

1710

E y (MPa)

3150

3400

3920

2980

1710

E z (MPa)

3150

3400

3920

2980

1710

G xy (MPa)

1167

1259

1219

1103

633

G^(MPa)

1167

1259

1219

1103

633

G v ,(MPa)

1167

1259

1219

1103

633

v, v

0.35

v>.

0.35

gV (kPa)

780

880

940

740

540

1.0

Rxy

0.7

Rx Z

0.7

Ryz

0.7

Figures 6.1.1 and 6.1.2 show a comparison of the permanent surface deformation from the
finite element models compared to data from the test sections for 1000 load applications. Figures
6.1.3 and 6.1.4 show a comparison of dynamic vertical stress along the load plate centerline for
test sections SSS1 and CS2, respectively. Figure 6.1.5 shows the permanent vertical strain
towards the bottom of the base at a depth of 160 mm below the pavement surface for test section
SSS1 plotted against radius from the load plate centerline for load cycles 1, 10, 100 and 1000. A
similar plot is shown in Figure 6.1.6 for permanent vertical strain in the top of the subgrade at a
depth of 350 mm below the pavement surface for test section SSS1. Figure 6.1.7 shows the
permanent vertical strain versus depth along the load plate centerline for test section CS2.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

Figures 6.1.8-6.1.11 show the permanent horizontal strain in the bottom of the base and in the
top of the subgrade for test sections SSS1 and CS2.

SSS4-TS

200

400 600

Load Cycles

800

1000

Figure 6.1.1 Permanent surface deformation from FEM and experiments for unreinforced SSS
test sections.

CS8-FEM

200

400 600

Load Cycles

800

1000

Figure 6.1.2 Permanent surface deformation from FEM and experiments for unreinforced CS
test sections.

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Dynamic Vertical Stress (kPa)

100 200 300 400

500

Subgrade

1.6 J

Figure 6.1.3 Dynamic vertical stress versus depth along the load plate centerline for test
section SSS1.

Dynamic Vertical Stress (kPa)

100 200 300 400

500

o -

^ AC

0.2 -

FEM^ — "

Base

0.4 -

Subgrade

.»-«.

Depth (it

o o
bo b)

1 -

1.2 -

1.4 -

1 R -

Figure 6.1.4 Dynamic vertical stress versus depth along the load plate centerline for test
section CS2.

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Cycle 100
-Cycle 10

Cycle 1

Cycle 1000

0.6

0.8

Radius (m)

Figure 6.1.5 Permanent vertical strain versus radius in the bottom of the base (z = 160 mm) for
test section SSS1.

0.6 n

Cycle 1000

0.8

-0.1 J

Radius (m)

Figure 6.1.6 Permanent vertical strain versus radius in the top of the subgrade (z = 350 mm)
for test section SSS1.

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Permanent Vertical Strain (%)

0.5 1 1.5

Cycle 100 Subgrade

Cycle 1000

1.6 J

Figure 6.1.7 Permanent vertical strain versus depth along the load plate centerline for test
section CS2.

0.1

^ 0.05

I o

i_
*->

« -0-05

*->

C
O
N "0.1

S -0.15

| -0.2

0- -0.25

-0.3

Cyiffl/o2

i i

0.4 0.6 (

c/c/JlO

Radius (m)

c/cfe 100

Cycle 1000

0.8

Figure 6.1.8 Permanent horizontal strain in the bottom of the base (z = 215 mm) versus radius
for test section SSS1.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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0.1 -i

-0.25 J

Figure 6.1.9 Permanent horizontal strain in the top of the subgrade (z = 310 mm) versus radius
for test section SSS1.

i_
*->

£
O
N

X
£
£

Figure 6.1.10 Permanent horizontal strain in the bottom of the base (z = 325 mm) versus radius
for test section CS2.

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Final Report

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements

S.W. Perkins

i_
*■>

c
o

E
i-

Figure 6.1.11 Permanent horizontal strain in the top of the subgrade (z = 415 mm) versus radius
for test section CS2.

The above results show the general ability of the FE model to predict the accumulation of
permanent strain in the pavement layer materials and the accumulation of permanent surface
deformation under repeated load. The dynamic vertical stresses predicted by the model were
generally less than those seen from the measurements made in the test sections. The under
prediction of dynamic vertical stress by finite element response models appears to be a common
weakness inherent to many programs using continuum-based material models (BRRC, 2000).
Permanent vertical strains were generally under predicted in the base aggregate layer and over
predicted in the subgrade layer. Permanent horizontal strain in the bottom of the base aggregate
and top of the subgrade generally compared well to results from test sections. The observation of
extension (negative horizontal strain) beneath the projection of the load plate and compression
beyond a radius of 200 to 300 mm was also observed from test section measurements.

The poor agreement of dynamic vertical stress was improved by including additional
elements for the stiff steel load plate and compressible rubber pad beneath the load plate and
additional anisotropy for the AC layer. In particular, the in-plane elastic moduli and the out-of-
plane shear moduli for the asphalt concrete were reduced from the values given in Table 6.1.1
while the other values were kept the same. This caused the asphalt concrete layer to behave less

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like a continuous slab and allowed the load to be more localized. Figure 6.1.12 shows results
from test section CS2 illustrating how the vertical stress distribution could be improved. The
models described above were not rerun for multiple load cycles with these new parameters.

Dynamic Vertical Stress (kPa)

100 200 300 400 500

1.6 J

Figure 6.1.12 Dynamic vertical stress versus depth along the load plate centerline for test
section CS2 using a revised model.

6.2 Reinforced Pavements

The finite element models described in Sections 5.2 and 5.3 were created to examine the effect of
a simulated perfect reinforcement condition and the effect of a geosynthetic material having the
properties described in Section 4.3 and 4.4. Three models corresponding to an unreinforced
model, a model with perfect reinforcement (as described in Section 5.2) and a model containing
geosynthetic reinforcement were created with layer properties similar to those from test section
CS2. The models were run for 10 applications of load. The modulus of the geosynthetic was
approximately 15 times greater than that which was reported for Geogrid A in Section 4.3.

Figure 6.2. 1 illustrates the permanent horizontal strain along a line emerging from the load
plate centerline and passing through the bottom of the base aggregate and along the symmetry
plane after 10 cycles application of load. As expected, the fixed base case shows no lateral strain
as the nodes along the bottom of the base are fixed from motion in the x and y directions. The

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reinforcement considerably limits the amount of lateral strain at this depth in the base and
corresponds qualitatively to behavior seen in experimental test sections described in Section 3.

Figure 6.2.2 shows the permanent horizontal strain along a vertical line extending through
the load plate center and plotted against depth throughout the pavement section after 10 cycles of
load. The results show that the effect of restricting lateral motion of the base aggregate at the
geosynthetic interface is seen by a reduction of lateral strain further up in the base aggregate and
well into the subgrade soil with this effect being most pronounced for the fixed base case.

Figure 6.2.3 shows the mean stress, defined as the average of the three principal stresses,
along the same horizontal line in the bottom of the base as used in Figure 6.2.1 and at the point
where the peak pavement load was applied for the first load cycle. The results show that a
restriction of lateral motion of the base aggregate results in an increase in mean stress, with this
effect being most significant for the fixed base case. For these analyses, the increase in modulus
of the base for the reinforced case is approximately 1.5 to 3 times that of the unreinforced case at
this location. This effect begins to diminish for points higher in the base, as illustrated in Figure
6.2.4 for a position 70 mm above the bottom of the base. In the companion report for this project,
the increase in mean stress for a predefined volume of aggregate was as much as 2.5 for
comparison unreinforced and fixed base cases.

Figure 6.2.5 shows the vertical stress along the top of the subgrade at peak load for the first
load cycle. The effect of confinement and subsequent increase in modulus of the base is to
reduce the maximum vertical stress occurring under the load plate. Figure 6.2.6 shows data
similar to Figure 6.2. 1 showing that the lateral strain in the top of the subgrade is reduced with
reinforcement. The effect of these mechanisms is to reduce the permanent vertical strain beneath
the load plate centerline and to reduce the amount of permanent surface deformation of the
pavement, as illustrated in Figures 6.2.7 and 6.2.8.

Figures 6.2.9 and 6.2.10 show the relative displacement between the interface contact
surfaces and the interface shear stress versus lateral distance along the x-axis extending through
the contact surface between the base aggregate and the geosynthetic. Results are shown for load
cycles 1 and 10 for the point at which the load is a maximum and a minimum for the cycle.
Figure 6.2.9 indicates that for this analysis, the value of E s np= 0.1 mm is not exceeded but is
being approached for 10 cycles of load application. This figure also shows the ability of relative
displacement to accumulate with applied load cycle even though E s n p is not exceeded.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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0.1 n

-0.1

1—

*->

(/)

-0.2

1—

*->

-0.3

-0.4

-0.5 J

0.4

0.6

0.8

Lateral Distance (m)

• Unreinforced

• Fixed Base

• Reinforced

Figure 6.2.1 Lateral permanent strain in the bottom of the base versus lateral distance after 10
cycles of load.

Lateral Strain (%)

-0.50

-0.30

-0.10

Subgrade

0.10

Fixed Base

1.4 L

Figure 6.2.2 Lateral permanent strain along the load plate centerline versus depth after 10
cycles of load.

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Unreinforced
Fixed Base
Reinforced

0.4 0.6

Lateral Distance (m)

Figure 6.2.3 Mean stress at peak load along the bottom of the base.

Unreinforced
Fixed Base
Reinforced

0.2 0.4 0.6

Lateral Distance (m)

0.8

Figure 6.2.4 Mean stress at peak load along a line 70 mm above the bottom of the base.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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(0
Q.

</>

*-»
</)

75
o

r
a>

50
45
40
35
30
25
20
15
10
5

Unreinforced
Fixed Base
Reinforced

0.2

0.8

0.4 0.6

Lateral Distance (m)

Figure 6.2.5 Vertical stress at peak load in the top of the subgrade.

4-*
(/)

re
a>

4-*

0.6 0.8

Lateral Distance (m)

• Unreinforced

• Fixed Base

• Reinforced

Figure 6.2.6 Lateral permanent strain in the top of the subgrade versus lateral distance after 10
cycles of load.

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-0.20

Vertical Strain (%)
0.30 0.80

1.4 J
Figure 6.2.7 Vertical permanent strain along the load plate centerline versus depth after 10

cycles of load.

^.0 •

2 •

Ctf

H —

I—

—j

1.5 «

■*—>

-1— '

1 •

CD
Q_

CD
Q

0.5«

Unreinforced

4 6

Load Cycles

Figure 6.2.8 Permanent surface deformation versus applied load cycles for reinforced sections.

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0.10 n

■ — Cycle 1 Load
a — Cycle 1 Unload
A— Cycle 10 Load
A— Cycle 10 Unload

0.2 0.4 0.6 0.8

Lateral Distance (m)

Figure 6.2.9 Relative displacement between the base aggregate and the geosynthetic interface.

ou n

•

25 ■

20 ■

C/J

•

CO
CD

15 •

•

10 •

■

■ — Cycle 1 Load
e — Cycle 1 Unload
A— Cycle 10 Load
A— Cycle 10 Unload

0.2 0.4 0.6 0.8

Lateral Distance (m)

Figure 6.2.10 Interface shear stress between the base aggregate and the geosynthetic.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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Several problems were noted with the contact model used to describe interaction between
the geosynthetic and the surrounding base aggregate. It would generally be expected that as the
value of E s u p reduced, the benefit provided by the geosynthetic would increase. This expected
relationship was not always observed. Figures 6.2.11 and 6.2.12 show values of permanent
vertical strain in the top of the subgrade after one cycle of load and average mean stress in the
base aggregate when peak pavement load is applied for the first load cycle. These measures are
an indicator of pavement performance. These results are plotted against a modulus multiplier,
which is a number by which the calibrated elastic modulus described in Section 4.3 for the
geosynthetic materials used was multiplied. Results are plotted for several different values of
E s u p . The results in Figure 6.2.11 show that the vertical strain on the top of the subgrade
increases as E s u P decreases, meaning that pavement performance, defined in terms of this
response measure, decreases as E s ii P decreases. The results in Figure 6.2.12 indicate more
expected results, showing that mean stress generally increases as E s i ip decreases, although the
results are not completely consistent. Pavement surface deformation was generally seen to
increase as E s i ip decreased, indicating that the negative effect of decreasing E s u P on subgrade
strain tended to control behavior of the pavement system.

The amount of vertical strain in the top of the subgrade is influenced by the level of shear
in the top of the subgrade. As E s u P decreases, more tensile load is transferred to the geosynthetic
through interface friction, thereby creating more tensile strain in the geosynthetic and hence
more shear strain that acts upon the top of the subgrade. Only when the geosynthetic becomes
very stiff in tension does this effect begin to diminish. On the other hand, decreasing E s u p
provides more lateral constraint on the base aggregate and generally provides for increased
confinement. These results indicate the complexity of interaction between the various
components of the reinforced pavement system. The lack of overall expected benefit as a
function of E s u P is believed to be due primarily to a material model for the base course aggregate
which is not sufficiently sensitive to the effect of mean stress on layer stiffness.

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Final Report

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements

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^ 1.2 n
I 1.1 H

re
u

1 -

> 0.9 H

*■•
C

ro 0.8 -

i_
a>
a.

0.7

Eslip = 5 mm
Eslip = 1 mm
Eslip = 0.1 mm

10 20 30

Modulus Multiplier

- 1
50

Figure 6.2.11 Permanent vertical strain in the top of the subgrade for various values of
geosynthetic modulus and interface elastic slip (E s i ip ).

(0
Q.

V)
<D

*->
(f)

C
(0
d)

bU -

♦ Eslip = 5 mm
■ Eslip = 1 mm
A Eslip = 0.1 mm

45 -

40 -

^-± *

*■

35 -

♦^ #

Modulus Multiplier

Figure 6.2.12 Average mean stress in the base aggregate for various values of geosynthetic
modulus and interface elastic slip (E s i ip ).

The models described above all employed only one contact interface, namely that between
the top of the geosynthetic and the bottom of the base aggregate. The nodes of the geosynthetic
were equivalenced and therefore connected to the underlying subgrade material. The above
results indicate the importance of shear transmitted to the subgrade and suggest that the contact
interface between the geosynthetic and the underlying subgrade may need to be specified.

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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Several models were created where a second contact interface was added for the interface
between the geosynthetic and the subgrade. Interface shear strength and stiffness were given
relatively low values to model the contact that would be expected between a weak subgrade and
the geosynthetic. From these models, anticipated results were not generally observed.

Similar difficulties were encountered for models where the geosynthetic was elevated into
the base course aggregate layer. The effect of the geosynthetic on vertical strain throughout the
section and on mean stress in the base was not always predictable for these cases. These cases
point to improvements required for the base aggregate material model to account for the effects
of the reinforcement and for further examination of the contact interface model on system
performance.

The reinforcement functions and benefits illustrated in Figures 6.2.1 - 6.2.10 were seen
when the geosynthetic modulus for Geogrid A was increased by a factor of 15. When the
properties listed in Section 4.3 for Geogrid A were used directly in the model, negligible
reinforcement benefit was observed. In the companion report (Perkins 2001) for this project, a
factor of 4.4 was applied to the geosynthetic tensile modulus (as determined from ASTM D 4595
at 2 % strain) to match reinforcement benefit seen from comparison test sections. The general
observation of the need to increase the measured geosynthetic modulus in order to derive
expected benefits points to deficiencies in the numerical model used and may suggest that
traditional measurement techniques for geosynthetic tensile properties may be inappropriate for
this application. For instance, the rate of loading in a roadway application may be as great as 40
times that employed in the ASTM D 4595 test method, which may account for an effectively
higher modulus in the application. Normal stress confinement by overlying roadway materials
and vehicle loading may also cause an effectively higher modulus in some geosynthetic
materials.

7.0 CONCLUSIONS

The material modeling and finite element modeling work described in this report allows the

following conclusions to be made:

1. An elastic -perfectly plastic material model for the asphalt concrete layer was necessary to
allow this pavement layer to permanently deform with the underlying base layer and to
prevent artificial tensile loads from being applied to the base layer when pavement load

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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was returned to zero. The use of isotropic material properties for the AC layer resulted in
an underprediction of vertical dynamic stress under the load plate centerline. The
introduction of direction dependency of elastic and plastic properties (material anisotropy)
allowed the pavement load to be more localized and produced improved predictions of
vertical stress beneath the load plate centerline and an improved deflected shape of the AC
surface.

2. A bounding surface plasticity model was used for the base aggregate and subgrade layers.
The model showed elastic -plastic behavior with isotropic hardening. The bounding surface
concept allowed for permanent strains to be predicted under repeated pavement loading.
Comparison of permanent strain in the aggregate and subgrade layers from test section
results to FEM predictions showed the general ability of the model to describe the
accumulation of permanent strain under repeated load. The model was well suited for the
subgrade material while improvements are needed for modeling the base aggregate layer.
In particular, the small level of tensile strength predicted for the aggregate layer and the
apparent lack of sensitivity of material stiffness on mean stress confinement created
limitations in its use to describe the effects of geosynthetic reinforcement.

3. A material model for the geosynthetic was formulated to include components of elasticity,
plasticity, creep and direction dependency. The model provides reasonable predictions of
various types of in-air tensile tests.

4. A relatively simple Coulomb friction model was used to describe interaction between the
geosynthetic and the base aggregate layer. The model provides reasonable predictions of
pull out response as compared to test conducted using the base aggregate and geosynthetics
used in the test sections available to the project.

5. Finite element models of reinforced pavements were capable of qualitatively showing
mechanisms of reinforcement previously observed from instrumented test sections. In
particular, the reinforced models showed a reduction of lateral strain at the bottom of the
base, an increase in mean stress confinement for a zone of aggregate adjacent to the
geosynthetic, an improved vertical stress distribution on the subgrade, and a reduction of
shear in the top of the subgrade. In order to see appreciable effects from the reinforcement,
the elastic modulus of the material needed to be increased by approximately an order of
magnitude. This may be due to the manner in which elastic modulus is determined from

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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
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common tension tests but is more likely due to deficiencies in the material model used for
the base course aggregate.

6. While providing reasonable predictions of pull out behavior, the interface contact model
produced several unexpected results in the finite element models of reinforced pavements
that require further examination. In particular, increasing the shear modulus of the interface
appeared to increase the amount of shear transmitted to the subgrade and hence increased
the vertical strain in the top of the subgrade. For the models examined, the strain in the
subgrade tended to control the overall deformation behavior of the pavement. This result
may also be due to a material model for the base aggregate that is not sufficiently sensitive
to effects of confinement caused by the geosynthetic.

7. The complexity of the models and the necessity to run the models for many load cycles
caused excessively long run times and limited the amount of cases that could be examined.
Future work in this area will require more computationally efficient models and projection
methods that can be used to project stress and strain measures forward over steps of load
cycles prior to running a new load step.

8.0 REFERENCES

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Monotonic and Cyclic Simple Tension", Geosynthetics International, Vol. 3, No. 4, pp. 493-

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ASTM (1995), Proposed Standard Test Method for Measuring Geosynthetic Pull-out Resistance

in Soil, Draft No. 6, ASTM, Philadelphia, PA, USA.
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Flexible Pavement Systems", National Cooperative Highway Research Program Report No.

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Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

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Structures, Report RO-97-SC.2137, Belgian Road Research Center.
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Using Finite Element Analysis", Proceedings of the Conference on Computer Methods and

Advances in Geomechanics, Balkema, pp. 1783-1788.
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Reinforced Unpaved Roads", Proceedings of the 4 International Conference on Geotextiles,

Geomembranes and Related Products, The Hauge, Netherlands, pp. 217-221.
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Dimensional Membrane Elements", Computers and Geotechnics, Vol. 2, pp. 3-22.
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for Analysis of Flexible Pavement Structure", In Transportation Research Record 1482,

TRB, National Research Council, Washington, DC, USA, pp. 123-133.
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Isotropic Cohesive Soils", Journal of Engineering Mechanics, ASCE, Vol. 112, No. 12, pp.

1263-1291.
Dafalias, Y.F. and Hermann, L.R. (1982), "Bounding Surface Formulation of Soil Plasticity",

Soil Mechanics-Transient and Cyclic Loads, Chapter 10, John Wiley and Sons, Ltd., pp. 253-

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Dafalias, Y.F. (1975), On Cyclic and Anisotropic Plasticity, Ph.D. Thesis, University of

California at Berkeley.
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Pavements", Proceedings of the Conference Airport Pavement Innovations - Theory to

Practice, Vicksburg, MS, pp. 46-60.
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Constitutive Modeling of Geologic Materials", International Journal of Numerical Methods

in Geomechanics, Vol. 10, pp. 25-257.
Dondi, G. (1994), "Three-Dimensional Finite Element Analysis of a Reinforced Paved Road",

Proceedings of the Fifth International Conference on Geotextiles, Geomembranes and

Related Products, Singapore, pp. 95-100.
Farrag, K. (1991), Pull-Out Testing Facility For Geosynthetics, Louisiana Transportation

Research Center, Project No. 87-1GT, 225 p.
Giroud, J.P., Darrasse, J. and Bachus, R.C. (1993), "Hyperbolic Expression for Soil-

Geosynthetic or Geosynthetic -Geosynthetic Interface Shear Strength", Geotextiles and

Geomembranes, Vol. 12, No. 3, pp. 275-286.
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Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

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Final Report S. W. Perkins

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Final Report S. W. Perkins

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Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

APPENDIX A: Notation

Provided below is a list of symbols and their definitions. Given the fact that certain symbols have
been used more than once for different material definitions, the notation list below is broken
down by various categories. Duplicate definitions for the same symbol was necessary to avoid
confusion with symbols used in original references.

General

BCR Base course reduction ratio (%)

CBR California bearing ratio (%)

TBR Traffic benefit ratio (unitless)

Asphalt Concrete Material Model

E x Elastic modulus in the x direction (MPa)

E y Elastic modulus in the y direction (MPa)

E z Elastic modulus in the z direction (MPa)

G X} Shear modulus in the x - y plane (MPa)

G xz Shear modulus in the x - z plane (MPa)

G yz Shear modulus in the y - z plane (MPa)

R x Yield stress ratio for the x direction (unit less)

R y Yield stress ratio for the y direction (unit less)

R z Yield stress ratio for the z direction (unit less)

Rxy Yield stress ratio for the x - y plane (unit less)

R xz Yield stress ratio for the x - z plane (unit less)

R yz Yield stress ratio for the y - z plane (unit less)

Vjcy, Vy, Poisson's ratio in the x - y plane (unitless)

V u , Vj, Poisson's ratio in the x - z plane (unitless)

V yz , V zy Poisson's ratio in the y - z plane (unitless)

g°ac Ultimate yield stress (kPa)

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

Base Aggregate and Subgrade Bounding Surface Material Model

A Shape parameter (unitless)

C Projection center parameter (unitless)

C c Compression index (unitless)

C s Swelling/recompression index (unitless)

E Elastic modulus (kPa)

e in Initial void ratio (unitless)

G Shear modulus (kPa)

h Hardening parameter (unitless)

/ First stress invariant (kPa)

h Atmospheric pressure (kPa)

I Size of ellipse 1 of the bounding surface (kPa)

/ Square root of the second deviatoric stress invariant (kPa)

m Hardening parameter (unitless)

M Slope of critical state line in p-q stress space (unitless)

N Slope of critical state line in /-/ stress space (unitless)

R Shape parameter (unitless)

s p Elastic zone parameter (unitless)

T Shape parameter (unitless)

a Lode angle (degrees)

K Swell/recompression slope (unitless)

X Virgin compression slope (unitless)

V Poisson's ratio (unitless)

(J> Drained soil friction angle in triaxial compression (degrees)

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

Geosynthetic Material Model

A Creep material parameter (unit less)

E m Elastic modulus in the machine direction (kPa)

E n Elastic modulus in the direction through the thickness of the material (kPa)

E xm Elastic modulus in the cross-machine direction (kPa)

G m .„ Shear modulus in the machine - normal to the geosynthetic plane (kPa)

G xm . m Shear modulus in the cross-machine - machine plane (kPa)

G xm - n Shear modulus in the cross-machine - normal to the plane direction (kPa)

m Creep material parameter (unit less)

n Creep material parameter (unit less)

N Number of geogrid ribs contained across the width of a sample

R m Yield and creep stress ratio for the machine direction (unit less)

R n Yield and creep stress ratio for the normal to the plane direction (unit less)

R xm Yield and creep stress ratio for the cross-machine direction (unit less)

R m -n Yield stress ratio for the machine - normal to the geosynthetic plane (unit less)

R xm - m Yield and creep stress ratio for the cross-machine - machine plane (unit less)

R xm - n Yield and creep stress ratio for the cross-machine - normal to the geosynthetic

plane (unit less)

W c Current width of a geosynthetic sample loaded in uniaxial tension (m)

Wi Initial width of a geosynthetic sample (m)

W m Physically measured width of a geogrid sample from rib to rib (m)

y m . n Shear strain in the machine - normal to the geosynthetic plane

Jxm-m Shear strain in the cross-machine - machine plane

y xm . n Shear strain in the cross-machine - normal to the plane direction

8 ; Lateral strain across the width of a geosynthetic sample

e m Normal strain in the machine direction

8„ Normal strain in the direction through the thickness of the material

e xm Normal strain in the cross-machine direction

e~ Uniaxial strain on a sample oriented 45° with respect to its principal directions

8 Lateral strain on a sample oriented 45° with respect to its principal directions

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

Numerical Modeling of Geosynthetic Reinforced Flexible Pavements
Final Report S. W. Perkins

Z cr Creep strain

e cr Creep strain rate

Vm-n, V n -m Poisson's ratio in the machine - normal to the geosynthetic plane

V m - xm , V xm . m Poisson' s ratio in the machine - cross-machine plane

V n . xm , V xm .„ Poisson' s ratio in the cross-machine - normal to the geosynthetic plane
G m Normal stress in the machine direction (kPa)

G„ Normal stress in the direction through the thickness of the material (kPa)

c xm Normal stress in the cross-machine direction (kPa)

o~ Uniaxial stress on a sample oriented 45° with respect to its principal directions

(kPa)

a ° Reference yield stress describing yield in the cross-machine direction (kPa)

x m .„ Shear stress in the machine - normal to the geosynthetic planet (kPa)

% xm . m Shear stress in the cross-machine - machine plane (kPa)

x xm .„ Shear stress in the cross-machine - normal to the plane direction (kPa)

Geosynthetic/Aggregate Interaction Simplified Numerical Model

G, Initial interface shear modulus (kN/m 3 )

u Interface shear displacement (m)

G„ Interface normal stress (kPa)

x Interface shear stress (kPa)

\\f p Peak interface friction angle (degrees)

V|/ r Residual interface friction angle (degrees)

Geosynthetic/Aggregate Interaction Model (GAIM)

E s i if Elastic slip (m)

A Interface shear displacement (m)

a Interface normal stress (kPa)

x Interface shear stress (kPa)

(I Interface friction coefficient (unitless)

Department of Civil Engineering, Montana State University - Bozeman, Bozeman, Montana 59717

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