Longitudinal Cracking in Pavement - R Luo

7/24/2019 Longitudinal Cracking in Pavement - R Luo

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Copyright

by

Rong Luo

2007


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The Dissertation Committee for Rong Luo Certifies that this is the approved version

of the following dissertation:

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Minimizing Longitudinal Pavement Cracking Due to

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Subgrade Shrinkage

Committee:

Jorge A. Prozzi, Supervisor

Kenneth H. Stokoe, II

C. Michael Walton

Jorge G. Zornberg

Loukas F. Kallivokas

Krishnaswamy Ravi-Chandar


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89BSubgrade Shrinkage

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by

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Rong Luo, B.E.; M.E.

91B

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

92BDoctor of Philosophy

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The University of Texas at Austin

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August 2007


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To My Parents


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v

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Acknowledgements

I would like to express my deepest appreciation to Dr. Jorge A. Prozzi, my

supervisor, for his invaluable advice, guidance and encouragement through the whole

time of my study at The University of Texas at Austin. I would have never realized my

potential or developed a strong passion to be a researcher in the area of pavement design

and modeling without his long-term support. It has been a privilege and pleasure to work

with him. I extend my gratitude to the other members of my dissertation committee, Dr.

Kenneth H. Stokoe, II, Dr. C. Michael Walton, Dr. Jorge G. Zornberg, Dr. Loukas F.

Kallivokas, and Dr. Krishnaswamy Ravi-Chandar, for their endless help and advice. My

special thanks go to Ms. Jan Slack for her kind assistance.

I am sincerely grateful to Dr. Jessica Y. Guo, who always motivates and supports

me to pursue an academic career. Many thanks go to all my friends at UT for their

encouragement, willingness to help and precious support. Last but not least, I appreciate

my parents and husband, who are always with me.


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96BSubgrade Shrinkage

Publication No._____________

Rong Luo, Ph.D.

The University of Texas at Austin, 2007

Supervisor: Jorge A. Prozzi

The State of Texas has the most extensive network of surface-treated pavements

in the nation. This network has suffered from the detrimental effects of expansive soils in

the subgrade for decades. Longitudinal cracking on the Farm-to-Market (FM) network is

one of the most prevalent pavement distresses caused by volumetric changes of expansive

subgrades. Engineering practice has shown that geogrid reinforcement and lime treatment

can effectively reduce the reflection of longitudinal cracking on the pavement over

shrinking subgrade. However, little is known about the mechanism leading to the

propagation of the shrinkage cracks to the surface of the pavement. The use of geogrid

reinforcement and lime treatment is mostly based on empirical engineering experience

and has not been addressed in depth.

This dissertation research evaluates the stress field and constitutive models of the

subgrade soil subjected to matric suction change. The non-uniform matric suction change

in the subgrade is simulated by a thermal expansion model in a finite element program,

ABAQUS, to determine the shrinkage stresses in the subgrade soil and pavement


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structure. Numerical solution by the finite element analysis shows that the most likely

location of shrinkage crack initiation in the subgrade is close to the pavement shoulder

and close to the interface of the base and subgrade. Linear elastic fracture mechanics

theory is used to analyze the crack propagation in the pavement. Compared to the fracture

toughness of the pavement materials, the stress concentration at the initial shrinkage

crack tip is large enough to drive the crack to propagate further. When the shrinkage

crack propagates through the whole pavement structure, a longitudinal crack develops at

the pavement surface close to the pavement shoulder.

Based on the analysis of shrinkage crack propagation, this dissertation

investigates the mechanism of geogrid reinforcement and lime treatment. The geogrid can

significantly reduce the stress concentration at the crack tip if the geogrid is placed at the

bottom of the base. A geogrid with a higher stiffness further reduces the stress intensity

factor at the upper tip of the shrinkage crack. The lime treatment can improve the

mechanical properties of the expansive soil in several ways. The lime-treated soil has

lower plasticity index, higher tensile strength and higher fracture toughness. The possible

location of the shrinkage crack initiation is not in the lime-stabilized soil but in the

untreated natural soil close to the bottom of the lime-treated layer, where tensile stresses

exceed the tensile strength of the untreated soil. The shrinkage crack is less likely to

develop through lime-treated soil, which has increased fracture toughness. The

combination of geogrid reinforcement and lime treatment offers the most benefit for the

control of dry-land longitudinal cracking. In a pavement with a lime-treated layer, the

best place to install the geogrid is at the interface between the lime-stabilized layer and

the untreated natural soil. If using a geogrid with high stiffness, the Mode I stress

intensity factor may be reduced to a certain level that is lower than the fracture toughness

of the pavement material.


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99B

Table of Contents

List of Tables ...........................................................................................................x

List of Figures........................................................................................................ xi

Chapter 1 Research Motivation ...............................................................................1

1.1 Background of Expansive Soils................................................................1

1.2 Engineering Problems due to Expansive Soils .........................................4

1.3 Research Objectives..................................................................................6

1.4 Dissertation Outline ..................................................................................7

Chapter 2 Stress Analysis of Pavement Subgrade...................................................92.1 Stress Analysis on Saturated Soil .............................................................9

2.2 Stress Analysis on Unsaturated Soil .......................................................11

2.3 Volumetric Change Theory of Unsaturated Soil ....................................15

2.4 Determination of Matric Suction Profile ................................................20

2.5 Summary.................................................................................................26

Chapter 3 Modeling of Pavement over Shrinking Subgrade.................................28

3.1 Model Construction ................................................................................29

3.2 Matric Suction Simulation and Model Constraints.................................32

3.3 Finite Element Mesh...............................................................................47

3.4 Simulation Results and Analysis ............................................................47

3.5 Summary.................................................................................................49

Chapter 4 Propagation of Crack in Pavement........................................................54

4.1 Fundamentals of Fracture Mechanics .....................................................54

4.2 Toughness of Pavement Materials ..........................................................58

4.3 Crack Propagation Process .....................................................................604.4 Summary.................................................................................................63

Chapter 5 Benefit of Geogrid Reinforcement........................................................66

5.1 Mechanism of geogrid Reinforcement....................................................66


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5.2 Modeling of Geogrid...............................................................................69

5.3 Summary.................................................................................................70

Chapter 6 Benefit of Lime Treatment....................................................................74

6.1 Background of Lime Treatment..............................................................75

6.2 Model Construction of Pavement with Lime-Treated Layer..................81

6.3 Crack Development in Untreated Subgrade Soil....................................89

6.4 Summary.................................................................................................97

Chapter 7 Combined Effect of Lime Treatment and Geogrid Reinforcement.......99

7.1 Determination of Geogrid Location........................................................99

7.3 Modeling of Pavement with Geogrid and Lime Treatment..................103

7.3 Summary...............................................................................................107

Chapter 8 Conclusions and Recommendations....................................................117

8.1 Conclusions...........................................................................................117

8.2 Recommendations for Further Research...............................................120

Bibliography ........................................................................................................122

Vita ..127


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List of Tables

Table 1.1 Properties of Clay Minerals.3Table 2.1 Typical Values of a and b Corresponding to Mineral Classification (Lytton,

2004) ................................................................................................................................. 20Table 3.1 Matric Suction Distribution in Wet Subgrade Soil........................................... 33Table 3.2 Matric Suction Distribution in Dry Subgrade Soil ........................................... 34Table 3.3 Logarithm of Matric Suction Change in Modeled Pavement Subgrade........... 35Table 4.1 Stress Intensity Factors of Trial Cracks............................................................ 62Table 6.1 Stress Intensity Factors of Trial Cracks in Pavement with Lime-Treated Layer........................................................................................................................................... 90Table 6.2 Stress Intensity Factors of Shrinkage Cracks ................................................... 94Table 7.1 Mode I Stress Intensity Factor of Shrinkage Cracks in Pavement with GeogridReinforcement and Lime Treatment ............................................................................... 102Table 7.2 Stress Intensity Factors of Shrinkage Cracks (Geogrid Stiffness = 400 kN/m)

......................................................................................................................................... 108Table 7.3 Stress Intensity Factors of Shrinkage Cracks (Geogrid Stiffness = 800 kN/m)

......................................................................................................................................... 109Table 7.4 Stress Intensity Factors of Shrinkage Cracks (Geogrid Stiffness = 1600 kN/m)......................................................................................................................................... 110Table 7.5 Stress Intensity Factors of Shrinkage Cracks (Geogrid Stiffness = 3200 kN/m)......................................................................................................................................... 111Table 7.6 Stress Intensity Factors of Shrinkage Cracks (Geogrid Stiffness = 6400 kN/m)......................................................................................................................................... 112Table 7.7 Stress Intensity Factors of Shrinkage Cracks (Geogrid Stiffness = 12800 kN/m)......................................................................................................................................... 113


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1B

List of Figures

Figure 2.1 Coordinates Defined for Stress Analysis of Soils ........................................... 13Figure 2.2 Chart for the Prediction of Suction Compression Index (McKeen, 1980)...... 18

Figure 2.3 Mineral Classification (Lytton, 2004) ............................................................. 19Figure 2.4 Thornthwaite Moisture Index Spatial Distribution in Texas (Wray, 1978) .... 24Figure 2.5 Variation of Soil Suction of Road Subgrade with Thornthwaite Moisture Index(Wray, 2005)..................................................................................................................... 25Figure 3.1 Pavement Structure in Finite Element Model..31Figure 3.2 Proposed Pavement Model with the First Model Constraint39Figure 3.3 Proposed Pavement Model with the Second Model Constraint...40Figure 3.4 Proposed Pavement Model with the Third Model Constraint..41Figure 3.5 Proposed Pavement Model with the Fourth Model Constraint42Figure 3.6 Definition of Thermal Expansion Coefficient in ABAQUS (ABAQUS, 2007)........................................................................................................................................... 46

Figure 3.7 Transverse Stress Distribution in Pavement without Geogrid (First ModelConstraint)..50Figure 3.8 Transverse Stress Distribution in Pavement without Geogrid (Second ModelConstraint)..51Figure 3.9 Transverse Stress Distribution in Pavement without Geogrid (Third ModelConstraint)..52Figure 3.10 Transverse Stress Distribution in Pavement without Geogrid (Fourth ModelConstraint)..53Figure 4.1 Three Fracture Modes (Lawn, 1993)55Figure 4.2 Crack Increment in Specimen of Unit Thickness............................................ 57Figure 4.3 Stress Intensity Factors of Crack in Non-Geogrid Pavement (Unit:MPam

0.5)...65

Figure 5.1 Mechanism of Geogrid Preventing Crack....................................................... 68Figure 5.2 Stress Intensity Factors of Cracks in Geogrid-Reinforced Pavement (Unit:MPam

0.5)...72Figure 5.3 Relationship between Mode I Stress Intensity Factor of Crack Tip in Base andGeogrid Stiffness .............................................................................................................. 73Figure 6.1 Reduction in Plasticity Index by Lime Treatment (Holtz, 1969) .................... 76Figure 6.2 Relationship between Plastic Index and Swelling (Seed et al., 1962) ............ 77Figure 6.3 Swell Pressure as a Function of Lime Content and Period of Curing for Irbid,Jordan, Clay (Basma and Tuncer, 1991)........................................................................... 78Figure 6.4 Shrinkage Cracks in High PI Clay Covered by Lime-Treated Layer (Courtesyof Lytton and Scullion) ..................................................................................................... 81

Figure 6.5 Model of Pavement with Lime-Treated Layer.84Figure 6.6 Transverse Stress Distribution in Pavement with Lime-Treated Layer (a)..87Figure 6.7 Transverse Stress Distribution in Pavement with Lime-Treated Layer (b)..88Figure 6.8 Single Shrinkage Crack in Subgrade Soil (Model 6.1) ................................... 91Figure 6.9 Mode II Crack in Shrinking Soil (Konrad and Ayad, 1997) ........................... 92Figure 6.10 Multiple Shrinkage Cracks in Subgrade Soil ................................................ 93


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Figure 6.11 Comparison of Mode I Stress Intensity Factor in Single Model and MultipleCrack Models.................................................................................................................... 96Figure 7.1 Pavement Model with Geogrid Reinforcement and Lime Treatment (Model7.1) .................................................................................................................................. 101

Figure 7.2 Pavement Model with Geogrid Reinforcement and Lime Treatment (Model7.2) .................................................................................................................................. 102Figure 7.3 Shrinkage Cracks in Pavement with Geogrid Reinforcement and LimeTreatment ........................................................................................................................ 104Figure 7.4 Mode I Stress Intensity Factor at Upper Crack Tip of Shrinkage Crack....... 114Figure 7.5 Mode I Stress Intensity Factor at Upper Crack Tip of Crack No. i............... 115Figure 7.6 Mode I Stress Intensity Factor at Lower Crack Tip of Shrinkage Crack...... 116


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2B

Chapter 1 Research Motivation

12B1.1BACKGROUND OF EXPANSIVE SOILS

Expansive soils are generally defined as soils that experience significant

volumetric changes when subjected to moisture variation. Expansive soils are the result

of a complex combination of conditions and processes for the parent materials, including

basic igneous rocks and sedimentary rocks containing montmorillonite (Chen, 1988).

Expansive soils found in the United States are primarily produced by the parent materials

in the second category, such as shales and claystones. Volcanic ash and glass, which arethe constituents of shales and claystones, can be weathered to montmorillonite, a so-

called swelling clay accounting for most of the expansive soil problems.

Expansive clay minerals, e.g. montmorillonite, have a large specific surface and

carry a large net negative electrical charge that attracts the exchangeable cations (positive

ions). These cations include Ca2+

, Mg2+

, H+, K

+, NH4

+, and Na

+, all of which are the most

common exchangeable cations in clay minerals. The ability of clay to absorb cations from

the solution can be quantified by the cation exchange capacity, which is defined as the

charge or electrical attraction for cation per unit mass, in milliequivalent per 100 g of

soil. Of the three most important groups of clay materials, montmorillonite, illite, and

kaolinite, montmorillonite has the largest cation exchange capacity. The Atterberg limits

of soil materials were found to be related to the type of clay mineral and the nature of the

attracted ion. The clay mineral with higher cation exchange capacity shows higher

Atterberg limit values, as shown in Table 1.1 (Chen, 1988). The clay structure is another

important property of the clay mineral. The octahedral or tetrahedral layers of

montmorillonite allow weak bonding of exchangeable cations in interlayer positions. The

crystal layer lattice may take up substantial quantities of water, with accompanying large


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changes in the clay volume (Snethen et al., 1975). As a result, expansive soils exhibit

significant volume changes with the variation of the amount of present water.


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Plasticity

index

(%)

11

44

99

Mg

2+

Liquid

limit (%

) 39 83 158

Plasticity

index

(%)

8 50

101

Ca2

+

Liquid

limit

(%)

34

90

166

Plasticity

index

(%)

7 38

104

K+

Liquid

limit

(%)

35

81

161

Plasticity

index

(%)

1 27

251

Cation

Na+

Liquid

limit

(%)

29

61

344

Cation

Exchange

Capacity

(milliequivalent

/100g)

3-15

10-40

70-80

SpecificSurface

(square

meter/g)

10-20

65-180

50-840

ClayMineral

Kaolinite

Illite

Montmorillonite

Table1.1PropertiesofClayMinerals(AfterChen,1988)


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13B1.2ENGINEERING PROBLEMS DUE TO EXPANSIVE SOILS

Expansive soils in foundations and subgrade can cause serious damage to houses,

buildings, roads and pipelines because of the soils expansion or shrinkage when

moisture levels change. Problems associated with expansive soils are common

worldwide, as have been reported in the United States, Australia, Canada, China, India,

Israel, and South Africa (Chen, 1988). Expansive soils are found in 20 percent of the

United States (Krohn and Slosson, 1980). Texas, Colorado and Wyoming have the most

severe degree of expansive soil occurrences.

In Texas, more than half of the total damage caused by expansive soils occurs on

highways and streets, which costs the Texas Department of Transportation (TxDOT)

millions of dollars to repair every year (Jayatilaka and Lytton, 1997). Longitudinal

cracking on the Farm-to-Market (FM) network is one of the most prevalent pavement

distresses due to the volumetric change of the expansive subgrade. This type of dry-land

crack initiates in the drying subgrade soil and reflects from the highly plastic subgrade

through the pavement structure (Sebesta, 2002). Pavement and geotechnical engineers

have for many years attempted to eliminate the dry-land cracking resulting from the

expansive subgrade.

A number of methods have been used to treat expansive soils, which can be

grouped into three categories: i) alteration of expansive material by mechanical, chemical

or physical means; ii) control of subgrade moisture conditions; and iii) geogrid

reinforcement. Lime stabilization is the most extensively used alteration for modifying

the expansive soils in the subgrade. The lime treatment thickness can vary from 0.25 m to

1 m. Other commercial stabilizers, for example, Roadbond EN1 and EMC Squared, have

also been used for treating the expansive soils (Rajendran and Lytton, 1997). These non-

calcium stabilizers have been shown to increase the strength and stiffness of the treated


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soil, reduce the swelling, decrease the permeability, and moderate the suction. Using

vertical barriers at the edge of the pavement is a typical method for controlling the

subgrade moisture conditions. Jayatilaka et al. (1997) found that installing impermeable

geomembranes as vertical moisture barriers in pavement sections could reduce the

moisture variation in expansive subgrade and then restrain pavement roughness. To date,

geogrid reinforcement combined with lime treatment is the most effective method to

prevent longitudinal cracking on Farm-to-Market (FM) roads caused by the shrinkage of

expansive subgrade. In particular, in the Bryan District (Texas) the geogrid is placed at

the interface of the cement-treated or lime-treated subbase and a 3 to 4 inch flexible base.

Based on the investigation of maintenance base repairs over expansive soils in the San

Antonio, Lufkin and Bryan Districts, Sebesta (2002) found that the geogrid-reinforced

pavement section in the Bryan District had the best observed performance of all the

stabilized sections with various treatments for longitudinal cracking.

However, despite the preliminary success of geogrid in limiting longitudinal

cracks, little is know about the mechanism leading to propagation of the longitudinal

cracks to the surface of the pavement. The use of lime and geogrid is mostly based on

empirical engineering experience to prevent the longitudinal cracks due to expansive

subgrade. Lime stabilization has been shown to reduce the plasticity index (PI) and swell

potential of the expansive soil. Lime-treated layers are usually found to be intact without

any shrinkage cracks, while multiple shrinkage cracks develop in the untreated natural

soil beneath the lime-treated layer. The mechanism of lime-stabilized soil preventing

shrinkage cracks has not been clarified, and the benefit of the lime treatment has not been

quantified. Significant research efforts have been spent on geogrid reinforced flexible

pavements subjected to traffic loading (Kuo et al., 2003; Kwon, et al., 2005; Tingle et al.,

2005). However, the use of geogrids to control environment-induced pavement distresses


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has never been addressed in depth to date. A theoretical approach is desirable to analyze

the mechanism of the dry-land longitudinal crack development and to minimize this type

of crack by means of lime-treatment and geogrid-reinforcement based on current

successful experience.

14B

1.3RESEARCH OBJECTIVES

This dissertation research considers the problem of dry-land longitudinal cracks

on pavements over expansive subgrade soils. The goals of this dissertation are to address

the propagation of the shrinkage cracks from the subgrade to the pavement surface and to

provide theoretical support to the current treatment methods with respect to the

propagation of shrinkage cracks. The research focus is on two of the issues identified in

the preceding section: the development of desiccation cracks and the benefit of treatment

methods.

The issue of crack development is rooted in the moisture variation in expansive

subgrade soil. The impermeable pavement surface layer has a significant impact on water

migration into the expansive subgrade beneath the pavement, which results in the non-

uniform moisture change in the subgrade. The gradients of moisture variation, together

with the soil expansive properties, determine the tensile stress distribution and the

shrinkage crack initiation. The first research objective of this dissertation is, thus, to

simulate the differential moisture change in the shrinkage subgrade soil and to model the

crack initiation and propagation.

The second research objective is to quantify the benefit of geogrid reinforcement

and lime treatment, which are two effective methods in practice to control longitudinal

cracks due to expansive subgrade. This quantification is based on the functional

mechanism of the two methods. The quantification will also provide theoretical support


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for the development of an optimal pavement design method to minimize the dry-land

longitudinal cracks.

15B

1.4DISSERTATION OUTLINE

This dissertation is organized as follows.

Chapter 2 analyzes the stresses and strains in the pavement subgrade induced by

the moisture variation. The available volumetric change theories of unsaturated soils are

introduced in order to predict the suction change associated with moisture change. This

chapter aims at finding an appropriate approach to simulate the differential suction

change in the subgrade soil under the impermeable pavement layers. The simulation of

the suction change is critical to shrinkage crack development.

Based on the stress and strain analysis in Chapter 2, Chapter 3 describes in detail

the modeling of a pavement over shrinking subgrade with expansive soils. A two-

dimensional plane strain pavement model is constructed using the finite element method.

The computer program ABAQUS is used for this purpose. The chapter further discusses

the distribution of tensile stresses in the pavement structure and subgrade soil. The

possible locations of shrinkage cracks initiation are identified under different constraints.

Chapter 4 studies the propagation of the shrinkage cracks from the shrinking

subgrade to the pavement surface. Linear elastic fracture mechanics theory is used to

simulate the desiccation crack propagation process. The shrinkage crack is modeled using

the finite element technique. The stress concentration at the crack tips is evaluated to

discuss whether the crack is stable or unstable according to the pavement material

fracture properties.

Chapter 5 describes the modeling of a geogrid-reinforced pavement. This chapter

discusses how the geogrid prevents the shrinkage crack from propagating toward the


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pavement surface. The chapter also presents the sensitivity analysis conducted to study

the effect of geogrid properties on its reinforcement benefits.

Chapter 6 reviews the properties of lime-treated soil and illustrates that the

inclusion of lime-treated layer in the subgrade changes the location and propagation of

shrinkage cracks. This chapter discusses the development of single shrinkage crack and

multiple cracks in the untreated subgrade soil beneath the lime-stabilized layer.

Chapter 7 presents the combined benefit of geogrid reinforcement and lime

treatment. The ideal installation position of the geogrid is determined based on the

analysis of the shrinkage propagation. The stress concentration at the crack tips is

compared among the studied models with different geogrid properties and the number of

shrinkage cracks in the model.

Chapter 8 summarizes the main findings and addresses the contributions of the

proposed methodologies. It concludes by outlining a number of directions in which the

proposed methodologies could be further extended.


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3B

Chapter 2 Stress Analysis of Pavement Subgrade

To study the crack initiation in expansive pavement foundations, it is necessary to

analyze the stress/strain state in the subgrade soil. The subgrade soil consists of

unsaturated soil, which is above the water table, and saturated soil that is under the water

table. This chapter provides an understanding of the state of the art in the stress analysis

and volumetric change theories of the expansive soils. Section 2.1 describes the

equilibrium conditions and constitutive relations of the saturated soil. Section 2.2

presents the stress variables and the stress-strain relations of the unsaturated soil. This

section also introduces a number of terms related to soil suction, including total

suction, matric suction and osmotic suction. Matric suction is a critical parameter

that will be used through this entire dissertation. Section 2.3 explains the available

volumetric theories of expansive soils as well as the constitutive relations that can be

used to predict the volumetric strain of the soil element. The methods used to determine

the soil suction are summarized in Section 2.4, including the experimental measurement

and theoretical prediction. Section 2.5 is a summary of this chapter.

16B

2.1STRESS ANALYSIS ON SATURATED SOIL

The equilibrium conditions for a saturated soil can be described by the effective

stress, ( )wu , in which is the total stress, and wu is the pore-water pressure. This

stress variable has the following tensor form (Fredlund and Rahardjo, 1993):

w

w

w

u

u

u

333231

232221

131211

in which

wu = pore-water pressure;


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wu11 = effective stress in the 1x direction;

wu22 = effective stress in the 2x direction;

wu

33 = effective stress in the

3x direction; and

12 , 23 , 31 , 13 , 32 , 21 = shear stress components.

The shear stress components have the following relationships under equilibrium

conditions:

2112 = (2.1)

3223 = (2.2)

1331 = (2.3)

Assuming that the saturated soil behaves as an isotropic and linearly elastic

material, the effective stress variable is used to formulate the constitutive relations with

respect to the generalized Hookes law. Equations (2.4), (2.5) and (2.6) give the elastic

constitutive relations in the 1x , 2x , and 3x directions:

( )ww u

EE

u23322

1111 +

=

(2.4)

( )ww u

EE

u23311

2222 +

=

(2.5)

( )ww u

EE

u22211

33

33 +

=

(2.6)

where

11 = normal strain in the 1x direction;



E = modulus of elasticity with respect to a change in the effective stress; and

= Poissons ratio for the soil structure.


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17B2.2STRESS ANALYSIS ON UNSATURATED SOIL

Unlike saturated soils, an unsaturated soil has more than one independent stress

variable because of the presence of soil suction. Soil suction is a measure of a soils

affinity for water (Chen, 1988). In other words, suction is a parameter indicating the

intensity with which the soil will attract water. Generally, soil with lower water content

has higher soil suction. The soil suction, commonly called total suction, can be

quantified in terms of relative humidity. Total suction consists of two components: matric

suction and osmotic suction. Matric suction is derived from the negative water pressure

associated with capillary phenomenon. Osmotic suction arises from the soluble salts in

the soil water, which produce the osmotic repulsion forces.

The total suction and its components can be measured in laboratory and in the

field (Fredlund and Rahardjo, 1993). The matric suction can also be predicted

theoretically by solving the moisture diffusion equation that governs the matric suction

distribution in the soil body (Mitchell, 1979, 1980). Based on the theoretical solution of

the diffusion equation, a number of computer programs have been developed that are

capable of estimating the matric suction profile in the pavement subgrade during different

climate seasons (Gay, 1994; Jayatilaka, 1999; Lytton et al., 2004).

Due to the existence of soil suction, more than one independent stress variable is

used to describe the equilibrium condition and to formulate the constitutive equations of

an unsaturated soil. Fredlund and Morgenstern (1976) used two stress variables in the

stress analysis of the unsaturated soil: net normal stress, ( )au , and the matric suction,

( )wa uu , in which is the total normal stress, au is the pore-air pressure, and wu

is the pore-water pressure. Consequently, the stress state of the unsaturated soil can be

expressed by two independent stress tensors:


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12

a

a

a

u

u

u

333231

232221

131211

and

wa

wa

wa

uu

uu

uu

00

00

00

.

The two stress tensors cannot be combined into one because they have different

constitutive relations which depend on the soil properties.

Assuming that the unsaturated soil is isotropic and linearly elastic, the constitutive

relations can be formulated in terms of the two stress state variables, ( )au and

( )wa uu , as shown in Equations (2.7), (2.8) and (2.9) (Fredlund and Rahardjo, 1993):

( )H

uuu

EE

u waa

a ++

= 2332211

11

(2.7)

( )H

uuu

EE

u waa

a ++

= 2331122

22

(2.8)

( )H

uuu

EE

u waa

a ++

= 2221133

33

(2.9)

where

E = modulus of elasticity for the soil structure with respect to a change in the net

normal stress, ( )au ; and

H = modulus of elasticity for the soil structure with respect to a change in matric

suction, ( )wa uu .

Every constitutive equation for the unsaturated soil can be explained as an

extension of each corresponding constitutive relation for the saturated soil because of the

additional stress variable (matric suction) in addition to the normal stress.

Equations (2.7), (2.8) and (2.9) can also be written in incremental forms:

( ) ( ) ( )waaa uudH

udE

udE

d ++=1

21

33221111

(2.10)


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udE

udE

d ++=1

21

33112222

(2.11)


udE

udE

d ++=1

21

22113333

(2.12)

Therefore, the incremental volumetric change of an unsaturated soil can be calculated by

adding the incremental strains in the three directions:

332211 ddddV

dVv ++== (2.13)

in which dV is the volume change of the unsaturated soil, V is the soil volume at

initial state, and vd is the incremental volumetric strain.

Let 1x be the transverse direction perpendicular to the vehicle travel direction on

the pavement, 2x be the longitudinal direction which is the vehicle travel direction, and

3x be the vertical direction, as shown in Figure 2.1.

x2

x3

x1

(Vehicle Travel Direction)

(Vertical Direction)

(Transverse Direction)

61B

Figure 2.1 Coordinates Defined for Stress Analysis of Soils

If the initial condition is considered after the subgrade construction when the

subgrade soil is intact without any cracks, the initial strains are zero in all three

directions. During the desiccation process of the soil in the pavement subgrade, the lateral

strains (the strain in horizontal directions, 11 and 22 ) remain zero before crack


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initiation because of lateral constraint. The field data collected by Konrad et al. (1997)

confirmed that drying soils experience a restrained desiccation so that the lateral strains

were maintained zero until a crack initiated in the soil. As a result, the incremental

horizontal strains in both transverse ( 1x ) and longitudinal ( 2x ) directions remain zero

before cracking, which means:

( ) 02332211

11 =

++

=H

uuu

EE

u waa

a

(2.14)

( ) 02331122

22 =

++

=H

uuu

EE

u waa

a

(2.15)

This fact indicates that the straining is forced to be one-dimensional. If isotropic elasticity

remains applicable, 332211 = (Morris, et al., 1992). Therefore, Equation (2.14)

can be rewritten as:

( ) 02331111

11 =

++

=H

uuu

EE

u waa

a

(2.16)

By rearranging Equation (2.16), the following equation can be reached:

( ) ( )waaa uuH

Euu

=

1

1

13311 (2.17)

in which ( )au11 is the incremental tensile stress in the transverse direction. Since the

pore-air pressure is atmospheric for most practical engineering problems, the net normal

stress in the vertical direction, au33 , equals to 33 by setting atmospheric pressure

zero. The total vertical stress, 33 , so-called the overburden pressure, is produced by the

self-weight of the soil and the pavement layers covering the soil. This stress can be

calculated by:

ssbbaa hhh ++=33 (2.18)

in which

a = unit weight of mass asphalt;

b = unit weight of mass base material;

s = unit weight of mass soil;


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ah = thickness of asphalt layer;

bh = thickness of base; and

sh = depth of point A from the top of the subgrade.

The overburden pressure results in compressive horizontal stresses, which

increase with depth. The matric suction in the soil reduces this compressive horizontal

stresses. At shallow depths, relatively small matric suction may reduce the compressive

net normal stress to zero or even make it negative. If the soil cannot sustain any tensile

stress, cracks will develop as the net normal horizontal stress, au11 , approaches zero.

However, soils are considered to have a certain amount of tensile strength, t , and this

tensile strength has been used in the crack initiation criterion that predicts the onset of

large tensile cracks by comparing the tensile strength with the net normal horizontal

stress (Lee et al., 1988; Morris et al., 1992; Ayad et al., 1997). Even though microcracks

may build up and coalesce in early stages, the criterion is well accepted that if au11

exceeds the tensile strength of the soil, t , a large scale tension crack will develop. For

example, Ayad et al. (1997) conducted an experiment to measure the tensile strength of

an intact clay deposit at the experimental site of Saint-Alban, Quebec, Canada. They

reported a tensile strength value of 9 kPa for the tested Saint-Alban clay.

18B2.3VOLUMETRIC CHANGE THEORY OF UNSATURATED SOIL

One may question the volumetric change formulation if it is based on the

assumption that the unsaturated soil is a linearly elastic material because soil behavior is

highly plastic in engineering practice. However, Equations (2.10) through (2.13) indicate

that the volumetric change of soil can be produced by either net normal stress or matric

suction or both. The relationship may not be linear between soils volume change and the

normal stress or the matric suction, but the volumetric compliances with respect to net


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normal stress and matric suction can be determined by laboratory experiments. Morris et

al. (1992) presented a constitutive equation as:

( ) ( )waaatv uudCudCV

dVd +== (2.19)

in which

v = volumetric strain of an elastic soil element;

V= overall volume of the soil element;

( )at

u

V

VC

=

1;

( )waa

uu

V

VC

=

1; and

= mean normal stress.

tC and aC are referred to as the volumetric deformation coefficients, which are

constants for linearly elastic case only. Fredlund and Rahardjo (1993) graphically

presented the constitutive surfaces for an unsaturated soil, which indicated that the

volumetric deformation coefficients vary from one stress state to another in a nonlinear

manner on the curved constitutive surface. The logarithm of the two stress variables are

found to be linearly related to the volumetric strain of an unsaturated soil. Lytton et al.

(1977; 1995; 2004) developed an empirical model to estimate the volumetric strain of an

elemental volume of soil:

=

i

f

i

f

i

f

hh

h

V

V

101010 logloglog (2.20)

where

V

V= volumetric strain;

ih = initial value of matric suction;

fh = final values of matric suction;

i = initial value of mean principle stress;


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f = finial value of mean principle stress;

i = initial value of osmotic suction;

f = finial value of osmotic suction;

h = matric suction compression index;

= mean principal stress compression index; and

= osmotic suction compression index.

The matric suction compression index ( h ) can be predicted by the empirical

procedure developed by McKeen (1980). This method estimates h using percent fine

clay, plasticity index (PI), and cation exchange capacity (CEC). The percent fine clay is

calculated by dividing the fine clay (finer than 2 microns) content by percentage passing

No. 200 sieve. The cation exchange capacity can be determined by a routine test

procedure in agricultural laboratories, or it can be estimated by empirical relationships

developed by Mojekwu (1979) as shown in Equations (2.21) and (2.22):

( ) 17.1PLCEC= gmeq 100/ (2.21)

( ) 912.0LLCEC= gmeq 100/ (2.22)

where

PL = plasticity limit, in percent; and

LL = liquid limit, in percent.

Based on the percent fine clay, PI and CEC, McKeens method calculates the activity

(Ac) and cation exchange activity (CEAc) as in Equations (2.23) and (2.24):

% clay

PIAc= (2.23)

% clayCECCEAc= gmeq 100/ (2.24)

The calculated Ac and CEAc are used to obtain a guide number of h in the Chart for

the Prediction of Suction Compression Index (Figure 2.2) developed by McKeen (1980).


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The guide numbers in Figure 2.2 are h for soils with 100 percent fine clay. To acquire

the value of h for real soil, the guide number determined by Figure 2.2 is reduced by

multiplying the percent fine clay. Finally, the obtained suction compression index may be

corrected by Equations (2.25) and (2.26) to compensate for the different initial volume of

soil mass during a wetting or drying process (Lytton, 2004):

( )hehswellh

= (2.25)

( )he

hshrinkageh

= (2.26)

62BFigure 2.2 Chart for the Prediction of Suction Compression Index (McKeen, 1980)

For lime treated soils, Lytton (2004) proposed a method for estimating the

plasticity index (PI) and the liquid limit (LL) as shown in Equations (2.27) and (2.28):

=

9

lim%9lim

ePIPI

untreatedtreatede (2.27)


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ba

PILL untreatedtreatede +=lim (2.28)

Parameters a and b in Equation (2.28) depend on soil mineral classification, as shown

in Figure 2.3. Table 2.1 shows typical values of a and b corresponding to each soil

mineral classification. Therefore, the matric suction compression index (SCI) can be

predicted by McKeens method following the above steps.

63B

Figure 2.3 Mineral Classification (Lytton, 2004)


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20

47B

Table 2.1 Typical Values of a and b Corresponding to Mineral Classification (Lytton,

2004)

Group a b

I 0.83 11

II 0.81 14

III 0.73 20

IV 0.68 25

V 0.68 25

VI 0.68 25

19B2.4DETERMINATION OF MATRIC SUCTION PROFILE

In order to study the development of desiccation cracks in the subgrade soil

during the reduction in water content and increase of matric suction, it is desirable to

estimate the shrinkage stresses generated between two steady state matric suction

profiles. If the two steady state matric suction profiles are known, Lyttons model

(Equation (2.20)) can be used to predict the volumetric strain that occurs between the two

steady states based on the matric suction changes. Consequently, the shrinkage stress

produced by the matric suction change can be estimated using the stress-strain

constitutive relationship of the subgrade soil. Based on the stress distribution, the

development of shrinkage cracks can be modeled. As a result, the determination of matric

suction is necessary for the analysis on the desiccation cracks. Previous research has

shown that matric suction can be either measured in the laboratory and the field or

predicted theoretically.


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42B2.4.1 Measurement of Matric Suction

Matric suction can be measured using filter paper in the laboratory as described in

American Society for Testing and Materials (ASTM) standard, D 5298-03 (2003), which

is a simple and economical method for the suction range from 10 to 100,000 kPa. In this

test method, the soil specimen is placed with filter papers in an airproof container for

seven days. This duration is sufficient to allow different vapor pressures inside the

container to reach equilibrium, including pore-water vapor pressure in the specimen,

pore-water vapor pressure in the filter paper, and partial water vapor pressure in the air.

Subsequently, a calibration relationship is developed between the filter paper water

content with soil suction based on the type of filter paper used and the test procedure.

Finally, the suction of the specimen can be determined using the measured mass of the

filter papers and the calibration relationship.

The axis-translation technique is another method to directly measure the matric

suction in the laboratory. This measurement was originally proposed by Hilf in 1956 for

both undisturbed and compacted soil specimens (Fredlund and Rahardjo, 1993). During

the test, a closed pressure chamber is used to contain the unsaturated soil specimen. A

pore-water pressure measuring probe connects a tube full of de-aired water and the soil

specimen. The water in the tube has a tendency to go into tension producing negative

water pressure, which is measured by a gauge. By increasing the air pressure in the

closed chamber, the water has a greater tendency to go into tension. Once equilibrium is

reached, the matric suction of the soil can be determined based on the difference between

the air pressure in the chamber and the measured negative water pressure.

The tensiometer is a device commonly used in the field to directly measure the

negative pore-water pressure in a soil. The tensiometer allows equilibrium to be achieved

between the soil and the measuring system. At equilibrium, the water in the tensiometer


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has the same negative pressure as the pore-water in the soil. Currently, there are different

types of tensiometers available for use in the field (Fredlund and Rahardjo, 1993).

43B

2.4.2 Theoretical Model of Matric Suction Prediction

Mitchell (1979, 1980) proposed a theoretical model to simulate the effects of

climate (evaporation and infiltration) on matric suction at ground surface in a sinusoidal

form with frequency n , as shown in Equation (2.29):

( ) ( )ntUUtu e 2cos,0 0+= (2.29)

in which

( )tu ,0 = matric suction at ground surface, in pF (kPa=0.098110pF);

eU = equilibrium suction, in pF;

0U = amplitude of suction change at ground surface, in pF;

n = number of suction cycles per second; and

t = time in seconds.

To study the suction not only at the ground surface but along the depth of the soil,

Mitchell developed a model to estimate the suction ( )tyu , at any time t and depth y :

( )

+= ynntynUUtyu e

2cosexp, 0 (2.30)

in which

y = soil depth;

= soil diffusion coefficient,c

p

d

w

= ,

w = water density;

d = soil dry density;

p = unsaturated permeability; and

c = inverse slope of log suction (in pF) vs. gravimetric water content.


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The equilibrium suction in Equations (2.29) and (2.30) can be estimated for

different locations based on the Thornthwaite Moisture Index (TMI) (Wray et al., 2005).

As defined in Equation (2.31), TMI is a parameter introduced by Thorthwaite (1948) to

characterize the moisture balance in a specific location taking into account rainfall,

potential evapotranspiration and the depth of available moisture stored in the rotting zone

of the vegetation. The calculation procedure of TMI includes three steps: i) determining

monthly potential evapotranspiration; ii) allocating available water to storage, deficit and

runoff on a monthly basis; and iii) totaling monthly runoff moisture depth, deficit

moisture depth and evapotranspiration to obtain annual values.

pEDEFRTMI 60100 = (2.31)

where

R = runoff moisture depth;

DEF = deficit moisture depth; and

pE = evapotranspiration.

Wray (1978) developed a TMI map of Texas based on historical means of TMI,

as shown in Figure 2.4. As the TMI value is determined, the corresponding equilibrium

suction, eU , can be estimated using Figure 2.5 (Wray, 2005) or by a regression equation,

Equation (2.32) (Lytton et al., 2004).

( )TMIUe 0051.0exp5633.3 = (2.32)


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64B

Figure 2.4 Thornthwaite Moisture Index Spatial Distribution in Texas (Wray, 1978)


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65BFigure 2.5 Variation of Soil Suction of Road Subgrade with Thornthwaite Moisture Index(Wray, 2005)

If the soil is under a flexible impermeable cover, e.g., flexible asphalt pavement,

the matric suction under the pavement center line is different from that under the

pavement edge (shoulder). Mitchell (1979) obtained the analytical solution of steady state

matric suction within the soil body under a flexible impermeable cover of length L . The

matric suction under the impermeable cover has an approximate relationship with thematric suction at the cover edge:

( ) ( )

a

La

x

UuUxu eyey

4cosh

2cosh

+ (2.33)

where

( )xuy = matric suction at the location with a distance of x from the pavement

centerline in the depth y ;

x = distance from the pavement centerline;

yu = matric suction at the pavement edge in the depth y ;

L = pavement width; and


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a = soil active zone depth, under which the soil matric suction has a constant

value of eU .

The horizontal matric suction profile can be predicted using Equation (2.33) based

on the vertical matric suction profile. Consequently, the matric suction distribution under

a flexible impermeable pavement is obtained at each steady matric suction state.

Based on Mitchells models, a number of computer programs have been

developed to predict the matric suction profiles in the pavement subgrade, such as

FLODEF (Gay, 1994), PRES (Jayatilaka, 1999), WinPRES (Lytton et al., 2004), and

SUCH (Wray et al., 2005). Lytton et al. (2004) presented matric suction data at

equilibrium, dry and wet conditions predicted by WinPRES in a number of highway

construction sites in Texas. They also showed the matric suction compression index

(SCI) for different layers of soils in the subgrade. Some of these data will be selected for

use in the proposed finite element models later in this dissertation.

20B

2.5SUMMARY

This chapter has discussed the stress and strain state in saturated and unsaturated

soils as well as the volumetric change theory of unsaturated soils. Particular attention has

been paid to Lyttons model (Equation 2.20) because it provides a reasonable and

relatively simple relation between the volumetric strain and three measurable variables.

One variable, matric suction, can also be predicted by theoretical methods. This model

will be used in the following chapters to simulate the differential matric suction change in

the subgrade soil. The available data of matric suction and matric suction compression

index in the literature make is possible to simulate matric suction change. The matric

suction data will be used as the only load on the proposed pavement model. The loading

condition differentiates the proposed model from most traditional pavement models that


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have traffic as the primary load. Chapter 3 will present the details of pavement model

construction, load simulation, and modeling results.


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4B

Chapter 3 Modeling of Pavement over Shrinking Subgrade

The stress/strain analysis in Chapter 2 offers the theoretical principles to analyze

the stress distribution in the pavement structure over a shrinking subgrade soil. The stress

distribution before crack initiation is critical in order to investigate the potential location

and propagation of the shrinkage crack. Before the analysis, an assumption is made that,

right after construction, the subgrade is intact with no macro cracks, which is the initial

condition in this analysis. In this initial condition, both pavement and subgrade are in

equilibrium condition. As the moisture content decreases in the subgrade soil, the matric

suction increases, which results in volumetric changes of the soil. If the matric suction

change is uniform and the soil is not constrained, normal strains will occur in each

direction unaccompanied by normal stresses. However, because the pavement is an

impermeable cover, the matric suction change is not uniform in the subgrade soil. In

addition, the lateral confinement does not allow the soil to have free expansion or

shrinkage. Therefore, tensile stresses will occur as the matric suction increases. As the

tensile stress reaches the tensile strength of the soil, a shrinkage crack will initiate in the

subgrade.

The aim of this chapter is to (1) establish a pavement model using finite element

techniques; (2) simulate the matric suction change in the subgrade soil beneath the

pavement layers; and (3) find possible locations of shrinkage crack initiation in the

pavement model. The chapter is divided into four sections, which are ordered as the

general finite element modeling steps. Section 3.1 focuses on the model construction in a

finite element computer program, ABAQUS. Section 3.2 explains the simulation of

matric suction change by means of temperature change in the soil body, and describes the

possible constraints at the model boundaries. Section 3.3 presents the finite element mesh


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of the model, and Section 3.4 discusses the results and findings, followed by a summary

of the chapter in Section 3.5.

21B

3.1MODEL CONSTRUCTION

A two-dimensional (2D) plane strain finite element (FE) model is developed in a

commercial computer program, ABAQUS, to simulate the stress field in the pavement

layers and the subgrade. In this dissertation research, plane strain is defined as a state of

strains in which the strain normal to the 31 xx plane, 22 , and the shear strains 12

and 32 are assumed to be zero. The coordinates are consistent with those stated in

Chapter 2: 1x is the transverse direction perpendicular to the vehicle travel direction on

the pavement, 2x is the longitudinal direction which is the vehicle travel direction, and

3x is the vertical direction (see Figure 2.1). The assumptions of plane strain are realistic

for long bodies: for example, a pavement that is infinitely long in the travel direction with

constant cross-sectional area subjected to loads that act only in the 1x or 3x directions

and do not vary in the 2x direction.

The modeled pavement structure consists of an asphalt surface layer, a granular

base and a multi-layered subgrade. Each pavement layer is assumed to be homogenous,

isotropic, linearly elastic, weightless, and bonded to the underlying layer. Because of

symmetry, a half-wide (4 m) pavement is studied to reduce computation effort. The

thickness of the asphalt layer is 0.025 m, and the thickness of the base is 0.250 m. The

subgrade of a pavement section in Fort Worth (Texas) is selected for this analysis based

on the available data in the literature (Lytton et al., 2004). With a total depth of 4.5 m,

this subgrade consists of six layers with different soils; each layer has a specific matric

suction compression index (SCI). In order to apply proper boundary conditions, one more

layer of 1.5 m without suction change is added to the bottom of the subgrade to make the

subgrade a total depth of 6 m. The width of the subgrade soil is extended to 12 m for the


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purpose of applying different model constraints that will be presented in later sections.

Figure 3.1 shows the details of the constructed pavement model in ABAQUS. In this

model, the soil under the pavement is defined as pavement subgrade, while the soil not

under the pavement (the soil under the edge CD in Figure 3.1) is defined as field soil.

Edge AF in Figure 3.1 is the centerline of the pavement. These definitions make the

following model description more clear.

The Youngs moduli of the asphalt, base and subgrade are assumed to be 2,500,

350 and 75 MPa, respectively. Since the moduli of different layers of the subgrade are

not available in the reference, a representative average modulus of 75 MPa is used for all

subgrade layers and for the field soil. The use of a constant modulus for subgrade and

field soil has a minimum effect on the accuracy level of the modeling results because the

elastic moduli of different layers in the subgrade have approximately the same order of

magnitude. The Poissons ratio is assumed to be 0.35 for every layer of the pavement

structure, the subgrade and the field soil.


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Fiure3.1PavementStructureinFiniteElementModel


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22B3.2MATRIC SUCTION SIMULATION AND MODEL CONSTRAINTS

44B3.2.1 Determination of Suction Change

For a pavement subgrade without significant content of sulfates, the osmotic

suction rarely changes in the subgrade soil. This condition is assumed for the subgrade

and field soil in the proposed model. Therefore, the volumetric change produced by the

osmotic suction variation can be ignored, which means it is not necessary to include the

osmotic suction term in Lyttons model (Equation (2.20)). In addition, considering a

newly constructed pavement structure without traffic loading, the mean principle stress

can also be neglected. Consequently, in this research, the matric suction is the only

independent variable determining the volumetric change of the subgrade soil under a

pavement, and Equation (2.20) can be simplified as:

=

i

f

hh

h

V

V10log (3.1)

Matric suction data and compression index to be applied to the pavement structure

are selected based on a previous study in Texas. The selected matric suction data were

predicted by the computer program WinPRES developed by Lytton et al. (2004). In order

to consider the most critical case (a long-term heavy rain followed by an extended dry

period), this study selects two steady state vertical matric suction profiles: one with

extremely low matric suction values, another with extremely high matric suction values.

Since the subgrade soil is under an impermeable asphalt surface layer, the matric suction

change in the soil under the pavement centerline is different from the soil under the

pavement shoulder. Generally, the closer the location is to the pavement centerline, the

less matric suction change in the subgrade soil is noted. Mitchells approach (Equation

(2.33)) is used to predict the horizontal matric suction profile under the pavement based

on the vertical matric suction profile. Therefore, the matric suction distribution over the


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subgrade cross section is obtained under the flexible impermeable pavement at each

steady matric suction state. Table 3.1 shows the matric suction distribution in the soil in

extremely wet condition; Table 3.2 presents the matric suction distribution in the

subgrade in extremely dry condition.

48BTable 3.1 Matric Suction Distribution in Wet Subgrade Soil

Distance from the centerline (m)Depth

(m) 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

0.15 2.5587 2.5579 2.5566 2.5546 2.5521 2.5489 2.5449 2.5400

0.30 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800

0.45 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800 2.58000.60 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800

0.75 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800 2.5800

0.90 2.5640 2.5634 2.5624 2.5610 2.5591 2.5567 2.5537 2.5500

1.05 2.5480 2.5468 2.5448 2.5420 2.5382 2.5333 2.5273 2.5200

1.17 2.5507 2.5496 2.5478 2.5451 2.5416 2.5372 2.5317 2.5250

1.32 2.5533 2.5524 2.5507 2.5483 2.5451 2.5411 2.5361 2.5300

1.47 2.5533 2.5524 2.5507 2.5483 2.5451 2.5411 2.5361 2.5300

1.59 2.5533 2.5524 2.5507 2.5483 2.5451 2.5411 2.5361 2.5300

1.74 2.5560 2.5551 2.5536 2.5515 2.5486 2.5450 2.5405 2.5350

1.89 2.5587 2.5579 2.5566 2.5546 2.5521 2.5489 2.5449 2.5400

2.01 2.5587 2.5579 2.5566 2.5546 2.5521 2.5489 2.5449 2.5400

2.16 2.5587 2.5579 2.5566 2.5546 2.5521 2.5489 2.5449 2.5400

2.28 2.5587 2.5579 2.5566 2.5546 2.5521 2.5489 2.5449 2.5400

2.43 2.5613 2.5607 2.5595 2.5578 2.5556 2.5528 2.5493 2.5450

2.55 2.5640 2.5634 2.5624 2.5610 2.5591 2.5567 2.5537 2.5500

2.70 2.5640 2.5634 2.5624 2.5610 2.5591 2.5567 2.5537 2.5500

2.85 2.5640 2.5634 2.5624 2.5610 2.5591 2.5567 2.5537 2.5500

3.00 2.5640 2.5634 2.5624 2.5610 2.5591 2.5567 2.5537 2.5500

3.15 2.5640 2.5634 2.5624 2.5610 2.5591 2.5567 2.5537 2.5500

3.30 2.5640 2.5634 2.5624 2.5610 2.5591 2.5567 2.5537 2.5500


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3.45 2.5667 2.5662 2.5653 2.5642 2.5626 2.5605 2.5580 2.5550

3.60 2.5693 2.5689 2.5683 2.5673 2.5661 2.5644 2.5624 2.5600

3.75 2.5693 2.5689 2.5683 2.5673 2.5661 2.5644 2.5624 2.5600

3.90 2.5693 2.5689 2.5683 2.5673 2.5661 2.5644 2.5624 2.5600

4.05 2.5693 2.5689 2.5683 2.5673 2.5661 2.5644 2.5624 2.5600

4.20 2.5693 2.5689 2.5683 2.5673 2.5661 2.5644 2.5624 2.5600

4.35 2.5693 2.5689 2.5683 2.5673 2.5661 2.5644 2.5624 2.5600

4.50 2.5747 2.5745 2.5741 2.5737 2.5730 2.5722 2.5712 2.5700

49BTable 3.2 Matric Suction Distribution in Dry Subgrade Soil


(m) 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

0.15 3.5795 3.6164 3.6790 3.7687 3.8878 4.0392 4.2268 4.4550

0.30 3.5315 3.5667 3.6262 3.7116 3.8250 3.9692 4.1477 4.3650

0.45 3.4836 3.5169 3.5735 3.6546 3.7623 3.8992 4.0687 4.2750

0.60 3.4383 3.4700 3.5237 3.6007 3.7030 3.8330 3.9940 4.1900

0.75 3.3983 3.4285 3.4797 3.5531 3.6507 3.7746 3.9282 4.1150

0.90 3.3583 3.3870 3.4357 3.5056 3.5983 3.7163 3.8623 4.0400

1.05 3.3183 3.3456 3.3918 3.4580 3.5460 3.6579 3.7964 3.9650

1.17 3.2810 3.3069 3.3508 3.4137 3.4972 3.6034 3.7349 3.89501.32 3.2463 3.2710 3.3127 3.3725 3.4519 3.5528 3.6778 3.8300

1.47 3.2144 3.2378 3.2775 3.3344 3.4100 3.5061 3.6251 3.7700

1.59 3.1824 3.2046 3.2423 3.2964 3.3682 3.4594 3.5724 3.7100

1.74 3.1531 3.1742 3.2101 3.2615 3.3298 3.4166 3.5241 3.6550

1.89 3.1237 3.1438 3.1778 3.2267 3.2914 3.3738 3.4758 3.6000

2.01 3.0944 3.1134 3.1456 3.1918 3.2531 3.3310 3.4275 3.5450

2.16 3.0704 3.0885 3.1192 3.1633 3.2217 3.2960 3.3880 3.5000

2.28 3.0491 3.0664 3.0958 3.1379 3.1938 3.2649 3.3529 3.4600

2.43 3.0304 3.0471 3.0753 3.1157 3.1694 3.2376 3.3221 3.4250

2.55 3.0118 3.0277 3.0548 3.0935 3.1450 3.2104 3.2914 3.3900

2.70 2.9931 3.0084 3.0342 3.0713 3.1206 3.1832 3.2607 3.3550

2.85 2.9718 2.9863 3.0108 3.0460 3.0927 3.1520 3.2255 3.3150


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3.00 2.9505 2.9642 2.9874 3.0206 3.0648 3.1209 3.1904 3.2750

3.15 2.9318 2.9448 2.9668 2.9984 3.0403 3.0937 3.1597 3.2400

3.30 2.9105 2.9227 2.9434 2.9731 3.0124 3.0625 3.1245 3.2000

3.45 2.8918 2.9034 2.9229 2.9509 2.9880 3.0353 3.0938 3.1650

3.60 2.8759 2.8868 2.9053 2.9319 2.9671 3.0119 3.0674 3.1350

3.75 2.8599 2.8702 2.8877 2.9128 2.9462 2.9886 3.0411 3.1050

3.90 2.8439 2.8536 2.8701 2.8938 2.9253 2.9652 3.0147 3.0750

4.05 2.8279 2.8370 2.8525 2.8748 2.9043 2.9419 2.9884 3.0450

4.20 2.8146 2.8232 2.8379 2.8589 2.8869 2.9224 2.9664 3.0200

4.35 2.8012 2.8094 2.8232 2.8431 2.8695 2.9030 2.9445 2.9950

4.50 2.6866 2.6906 2.6972 2.7068 2.7195 2.7357 2.7557 2.7800

The matric suction change at every location of the subgrade from the wet

condition to the dry condition is then calculated based on the matric suction distributions

at the two steady matric suction states. The logarithm of the matric suction change in the

subgrade ( ( )if hh /log10 ) is also computed and shown in Table 3.3. The subgrade soils

under the pavement centerline are assumed to have zero matric suction change. For the

field soil that is not under the pavement, the logarithm of the matric suction change is

considered uniform between two steady matric suction states because no impermeable

cover exists on the top of field soil.

50BTable 3.3 Logarithm of Matric Suction Change in Modeled Pavement Subgrade


(m) 0.25 0.75 1.25 1.75 2.25 2.75 3.25 3.75 4.00

0.15 0.0729 0.1481 0.1542 0.1635 0.1758 0.1914 0.2102 0.2322 0.2440

0.30 0.0682 0.1385 0.1442 0.1529 0.1645 0.1790 0.1966 0.2173 0.2283

0.45 0.0652 0.1325 0.1380 0.1463 0.1575 0.1716 0.1886 0.2086 0.21930.60 0.0624 0.1267 0.1320 0.1401 0.1508 0.1644 0.1808 0.2002 0.2106

0.75 0.0598 0.1216 0.1267 0.1344 0.1449 0.1580 0.1739 0.1926 0.2027

0.90 0.0586 0.1191 0.1242 0.1319 0.1422 0.1552 0.1711 0.1898 0.1998


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1.05 0.0574 0.1166 0.1216 0.1292 0.1394 0.1524 0.1681 0.1868 0.1968

1.17 0.0547 0.1111 0.1160 0.1232 0.1330 0.1455 0.1606 0.1785 0.1882

1.32 0.0521 0.1060 0.1106 0.1176 0.1270 0.1389 0.1535 0.1707 0.1801

1.47 0.0500 0.1016 0.1061 0.1128 0.1219 0.1334 0.1475 0.1642 0.1732

1.59 0.0478 0.0972 0.1015 0.1080 0.1167 0.1278 0.1414 0.1575 0.1662

1.74 0.0456 0.0927 0.0968 0.1030 0.1114 0.1220 0.1350 0.1505 0.1589

1.89 0.0433 0.0881 0.0920 0.0979 0.1059 0.1161 0.1286 0.1434 0.1515

2.01 0.0413 0.0840 0.0877 0.0934 0.1010 0.1108 0.1228 0.1370 0.1448

2.16 0.0396 0.0805 0.0841 0.0896 0.0970 0.1064 0.1180 0.1318 0.1392

2.28 0.0381 0.0775 0.0809 0.0862 0.0934 0.1025 0.1136 0.1270 0.1342

2.43 0.0365 0.0743 0.0776 0.0827 0.0896 0.0983 0.1091 0.1220 0.1290

2.55 0.0350 0.0711 0.0743 0.0792 0.0858 0.0942 0.1045 0.1169 0.12372.70 0.0336 0.0684 0.0715 0.0762 0.0825 0.0907 0.1007 0.1126 0.1191

2.85 0.0320 0.0652 0.0682 0.0727 0.0788 0.0866 0.0962 0.1077 0.1139

3.00 0.0305 0.0620 0.0649 0.0692 0.0750 0.0825 0.0916 0.1027 0.1087

3.15 0.0291 0.0592 0.0619 0.0661 0.0717 0.0788 0.0876 0.0982 0.1040

3.30 0.0275 0.0560 0.0586 0.0625 0.0678 0.0746 0.0830 0.0931 0.0986

3.45 0.0259 0.0527 0.0551 0.0588 0.0639 0.0703 0.0782 0.0878 0.0930

3.60 0.0245 0.0498 0.0521 0.0556 0.0604 0.0665 0.0740 0.0831 0.0880

3.75 0.0233 0.0473 0.0495 0.0529 0.0574 0.0632 0.0704 0.0791 0.0838

3.90 0.0220 0.0449 0.0469 0.0501 0.0544 0.0600 0.0668 0.0751 0.0796

4.05 0.0208 0.0424 0.0443 0.0474 0.0515 0.0567 0.0632 0.0711 0.0753

4.20 0.0198 0.0403 0.0422 0.0450 0.0489 0.0540 0.0602 0.0677 0.0718

4.35 0.0188 0.0382 0.0400 0.0427 0.0464 0.0512 0.0571 0.0643 0.0682

4.50 0.0091 0.0185 0.0193 0.0207 0.0225 0.0248 0.0276 0.0311 0.0330

45B

3.2.2 Model Constraints

If the matric suction change is uniform in an unconstrained elastic soil body, the

resultant swelling or shrinkage occurs in such a way as to cause a cubic element of the

soil solid to remain cubic, while experiencing changes of length on each of its sides.

Normal strains develop in each direction without inducing any normal stress. In this case,


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the matric suction change also does not produce shear strains or shear stresses. However,

if the soil body has a nonuniform suction change field, as studied in this research, or if

the soil expansion or shrinkage is prohibited from taking place freely because of

restrictions placed on the boundaries, even if the matric suction change is uniform, the

shrinkage stress will develop in the soil. Once the shrinkage tensile stress exceeds the

tensile strength of the soil, shrinkage cracks will initiate and develop in the subgrade soil.

As a result, the boundary conditions are directly related to the magnitude of the shrinkage

stresses as well as the development of the shrinkage cracks.

In the proposed pavement model shown in Figure 3.1, no constraint is assigned to

the pavement surface (edge AB ), the shoulder of the pavement (edge BC), and the

surface of the field soil (edge CD ). Boundary conditions are specified at three

boundaries: pavement centerline (edge AF), the bottom of pavement subgrade and field

soil (edge FE), and the right vertical edge of the field soil (edge DE). Because of

pavement symmetry, edge AF in the proposed model is not allowed to have horizontal

displacement. Since the model size in depth is large enough for the subgrade to assume

no significant deformation below 6 m, edge FE is specified zero displacement. At edge

DE, four types of model constraint are applied respectively in order to consider both

lateral confinement and possible shrinkage cracks in the field soil:

First, edge DE is fixed in the horizontal direction (no horizontal displacement

allowed) under the assumption that the field soil is intact without macro crack

(see Figure 3.2).

Second, no boundary condition is applied at edge DE, as shown in Figure 3.3,

which indicates that the field soil is able to deform freely at its right edge (edge

DE). This case is equivalent to the situation in which a 6-meter-long vertical

crack develops at the right edge of the field soil.


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Third, the upper 2 m of edge DE is not constrained, and the rest of this

boundary is fixed in the horizontal direction, which is illustrated in Figure 3.4.

This case simulates a 2-meter-deep crack developing from the soil surface

downward at the right edge of the field soil.

Fourth, a 2-meter-deep top-down crack is introduced at the location, which is 4

m horizontally away from and on the left side of edge DE, while edge DE is

not allowed to have horizontal displacement, as shown in Figure 3.5. This case

simulates that a shrinkage crack develops in the middle of the field soil in the

proposed model but no shrinkage crack is presented at the right edge of the field

soil.


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Figure3.2Pro

posedPavementModelwiththeFirstModelConstraint


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Figure3.3

ProposedPavementModelwith

theSecondModelConstraint


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Figure3.4ProposedPavementModelwithth

eThirdModelConstraint


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Figure3.5ProposedPavementModelwiththeFourthModelConstraint


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46B3.2.3 Simulation of Matric Suction Change

As stated in previous sections, the expansive soils not only have strains associated

with the displacement functions, but also have strains due to matric suction variations.

Assuming the subgrade soil in this research is linearly elastic and isotropic, the

volumetric change of the expansive soil is evenly distributed in transverse ( 1x ),

longitudinal ( 2x ) and vertical directions ( 3x ). In other words, let the matric suction h of

an elastic isotropic body in an arbitrary zero configuration be increased by a small

amount, all infinitesimal line elements in the volume undergo equal shrinkage since the

body is isotropic. All line elements maintain their initial directions. According to

Equation (3.1), the strain components due to the matric suction change are:

===

i

f

hh

h10332211 log

3

1 (3.2)

0213213312312 ====== (3.3)

The matric suction induced strains can be superimposed to the stress induced

strains to give:

( )

+=

i

f

h h

h

E 1033221111 log3

11

(3.4)

( )

++=

i

f

hh

h

E1033221122 log

3

11 (3.5)

( )

++=

i

f

hh

h

E1033221133 log

3

11 (3.6)

Under the plane strain assumption, 022 = . Therefore, Equations (3.4) and (3.6)

can be rearranged as follows:

( ) ( )

+=

i

f

hh

h

E10

331111 log

3111 (3.7)

( ) ( )

+=

i

f

hh

h

E10

113333 log

3

111

(3.8)


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In terms of strain components, these expressions become:

( )( )( )[ ]

+

+=

i

f

hh

hEE10331111 log

3

1

211

211

(3.9)

( )( )( )[ ]

+

+=

i

f

hh

hEE10113333 log

3

1

211

211

(3.10)

There is an analogy between the matric suction variation in the soil and the

temperature change in a solid. Considering a change in temperature ( )yxT , , the change

of length, L , of a small linear element of length L in an unconstrained body is

calculated by Equation (3.11):

LTL = (3.11)

in which is the thermal expansion coefficient. If a point is allowed to have free

expansion, the thermal strain, t , associated with the temperature change is then:

Tt = (3.12)

For the plain strain problem with 022 = , the full stress-strain relations are as follows

(Ugural, et al., 1995):

( ) ( )

+

+= T

E

331111

11 (3.13)

( ) ( )

+

+= T

E

113333

11 (3.14)

In terms of strain components, the stress-strain relations become:

( )( )( )[ ] T

EE

211

211331111

++

= (3.15)

( )( )( )[ ] T

EE

211

211113333

+

+= (3.16)

In both the matric suction variation problem and the temperature change problem,

the body has strains associated with the displacement functions as well as strains due to

other causes (moisture differential or temperature variation). The constitutive equations

are similar for the two problems. When comparing the corresponding constitutive


57/139


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46

66BFigure 3.6 Definition of Thermal Expansion Coefficient in ABAQUS (ABAQUS, 2006)

ABAQUS assumes that there is no initial thermal strain when the reference temperature

is not equal to the initial temperature. This assumption is enforced by the second term in

Equation 3.17 which represents the strain due to the difference between the initial

temperature, I , and the reference temperature, 0 .

In the pavement model proposed in this research, the initial temperature of the

subgrade is assumed to be zero, and the final equivalent temperature is the logarithm of

the matric suction change. Because the matric suction change varied from different

locations in the subgrade soil, the subgrade is partitioned into a number of grids. Each

grid is assigned a final equivalent temperature which is the logarithm of the matric

suction change (

i

f

h

h10log ) at the corresponding location shown in Table 3.1. The

thermal expansion coefficient in the simulation is

h

31 . The thermal expansion

coefficients used in this model have negative values because an increase in matric suction

results in shrinkage of the soil instead of expansion. In other words, the subgrade and


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field soil in the proposed model behave as a Negative Thermal Expansion material,

which contracts upon heating rather than expanding.

23B

3.3FINITE ELEMENT MESH

The finite element mesh distribution is designed to provide adequate accuracy

without consuming too much computational effort. The mesh size is 20 mm for the

pavement and the subgrade soil. Biased seed is assigned to the field soil in order to obtain

denser mesh in the location closer to the pavement and sparser mesh in the region farther

from the pavement.

Each element is a 4-node bilinear plane strain quadrilateral continuum element

(CPE4R). Such an element provides a first-order interpolation with reduced integration.

Reduced integration reduces running time by using a lower-order integration to form the

element stiffness. In total, 92,800 elements are generated in this model.

24B

3.4SIMULATION RESULTS AND ANALYSIS

The simulation results are represented by contour maps in terms of the

distribution of the normal stress in the transverse direction because the tensile stress

distribution determines the onset of the shrinkage cracks. The location with the largest

tensile stress is the most likely place for the initiation of a shrinkage crack. Figures 3.7,

3.8, 3.9 and 3.10 show the simulation results of the proposed model with the four

different boundary conditions, respectively. Since the stress distribution in the subgrade

soil is of the most interest, only a part of the subgrade soil and the pavement in their

deformed shape are presented in each figure in order to show the contours more clearly.

As can be seen from Figures 3.7 to 3.10, in all four cases studied with different

model constraints, the largest transverse tensile stress in the pavement subgrade, max11 ,

develops in the area close to the pavement shoulder and close to the interface of the


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subgrade and the base. The difference in the four cases is the magnitude of max11 . With

the first model constraint shown in Figure 3.2, the right edge of the field soil is not

allowed to move horizontally, andmax11

is around 0.30 MPa (Figure 3.7). When a 2-

meter-long top-down crack develops at the right edge of the field soil (the third model

constraint shown in Figure 3.4), the strain energy induced by the nonuniform matric

suction variation in the soil body is released by this crack to generate the crack surfaces.

In this case, max11 decrease to approximate 0.28 MPa (Figure 3.9). If the length of this

crack increases to 6 m (the second model constraint illustrated in Figure 3.3), which

means the right edge of the field soil is allowed to move freely in the transverse direction,

more strain energy is released, and max11 is found to be 0.22 MPa (Figure 3.8). When

the 2-meter-long crack develops closer to the pavement shoulder (4 m horizontally away

from the right edge of the field soil), which is the fourth model constraint shown in

Figure 3.5, max11 is around 0.25 MPa (Figure 3.6).

These findings indicate that whether the field soil is intact or has shrinkage

cracks, the magnitude of max11 is considerably larger than the soil tensile strength

reported in the literature (Ayad et al., 1997). According to the crack initiation criterion, a

crack will develop in the soil if the tensile stress exceeds the tensile strength of the soil.

Therefore, independent of the shrinkage cracks in the field soil, macro cracks tend to

initiate in the pavement subgrade close to the pavement shoulder and close to the

interface of base and subgrade. This matches the location of the observed longitudinal

cracks on in-service pavement sections.

After the onset of the shrinkage cracks in the pavement subgrade, fracture

mechanics theory will be used to determine whether or not the crack is stable in the next

chapter. In order to reduce calculation effort, only one type of constraint will be assigned

at the right edge of the field soil when studying the crack propagation. Of the four cases


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studied with different model constraints, the most critical case with the largest max11 is

the model with the first constraint in which the field soil is intact without any shrinkage

cracks. Consequently, in the analysis of crack propagation, the boundary conditions of

the proposed model will be as follows:

Zero horizontal displacement at edge AF in Figure 3.1;

Zero displacement at edg

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Longitudinal Cracking in Pavement - R Luo

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