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Drilled shafts under combined axial and lateral loads

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Title:
Drilled shafts under combined axial and lateral loads
Creator:
Gonzalez, Cesar
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Language:
English
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xv, 128 leaves : ; 28 cm.

Subjects

Subjects / Keywords:
Shafts (Excavations) ( lcsh )
Axial loads ( lcsh )
Lateral loads ( lcsh )
Boring ( lcsh )
Axial loads ( fast )
Boring ( fast )
Lateral loads ( fast )
Shafts (Excavations) ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (M.S.)--University of Colorado Denver, 2010.
Bibliography:
Includes bibliographical references (leaves 124-128).
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Cesar Gonzalez.

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|University of Colorado Denver
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Auraria Library
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Resource Identifier:
656366336 ( OCLC )
ocn656366336

Full Text
DRILLED SHAFTS UNDER COMBINED AXIAL AND LATERAL LOADS
by
Cesar Gonzalez
B.S., University of Guadalajara, Mexico, 2002
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
2010


This thesis for the Master of Science degree
by
Cesar Gonzalez
has been approved
by

^ Dale


Gonzalez, Cesar (M.S., Civil Engineering)
Drilled Shafts under Combined Axial and Lateral Loads
Thesis directed by Professor Nien-Yin Chang
ABSTRACT
This manuscript presents the numerical analyses of a drilled shaft foundation under
single and combined axial compressive and lateral loads. In order to evaluate the
soil-structure interaction effects on a drilled shaft foundation system, under single or
combined static monotonic loads, the analyses were performed using a three-
dimensional finite difference numerical code. Construction of drilled shaft
foundations for bridge piers, abutments, retaining walls, and buildings are examples
of engineering structures than can undergo the combined loading effect during their
lifespan. To better understand the behavior of the soil-structure physical system to be
modeled, the initial chapters cover an outline of past investigations of deep
foundations and more thoroughly the current state of design of drilled shafts under
various geologic conditions; such as cohesive and cohesionless soils, and cohesive
and cohesionless intermediate geomaterial. This is followed by a summary of the
mathematical numerical method utilized, the software application, and an overview
on the numerical modeling procedure of a drilled shaft foundation system. The latter
chapters include the calibration of the research model and the subject matter of this
thesis which is the combined load evaluation. The numerical analysis model was
calibrated using actual full-scale field load tests in various foundation materials, such
as cohesive and cohesionless soils, and in cohesive intermediate geomaterial. For the
combined loads research model a stiff normally consolidated clay was assumed for
the geologic profile with a drilled shaft aspect ratio of 10:1. The drilled shaft capacity-
under combined loading is dependent on the loading sequence. The results presented
in this research document are quantitative estimates of the potential range of axial and
lateral load induced displacements in a drilled shaft numerical model.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed

Nien-Yin Chang


DEDICATION
I dedicate this thesis to my parents, for their support and patience, and for helping
build a strong foundation in my life.


ACKNOWLEDGEMENT
I would like to give a very special thanks to my friend, professor, academic advisor,
and mentor Dr. Nien-Yin Chang. I am grateful for the opportunity of learning from
you every weekday for two years in the geotechnical laboratory and for the financial
assistance throughout that time. I truly appreciate your words of encouragement and
your overall patience with me throughout these many years.
I will also like to thank the members of my graduate committee, professors Shing-
Chun Trever Wang and Brian T. Brady, and the members of the soil-structure
interaction group, Kevin Lee, Russel Cox, Mohammad Abu-Hassan, and Jan Chang,
for helping me improve on my knowledge of numerical analysis and become a better
geotechnical engineer.


CONTENTS
Figures..................................................................x
Tables..................................................................xv
Chapter
1. INTRODUCTION.........................................................1
1.1 Research Overview....................................................1
1.2 Purpose..............................................................2
2. LITERATURE REVIEW....................................................3
2.1 Introduction.........................................................3
2.2 Deep Foundations under Axial Load....................................4
2.3 Deep Foundations under Lateral Load.................................12
2.4 Deep Foundations under Combined Loads...............................19
3. DESIGN PRINCIPLES OF DRILLED SHAFTS.................................21
3.1 Introduction........................................................21
3.2 Drilled Shafts under Axial Load.....................................22
3.2.1 Drilled Shafts in Cohesive Soil..................................27
3.2.1.1 Shaft Resistance in Cohesive Soil...............................27
3.2.1.2 Base Resistance in Cohesive Soil................................31
3.2.2 Drilled Shafts in Cohesionless Soil.............................32
vi


3.2.2.1 Shaft Resistance in Cohesionless Soil...............................32
3.2.2.2 Base Resistance in Cohesionless Soil................................34
3.2.3 Drilled Shafts in Intermediate Geomaterial.............................35
3.2.3.1 Shaft Resistance in Intermediate Geomaterial........................35
3.2.3.2 Base Resistance in Intermediate Geomaterial.........................38
3.2.4 Settlement of Drilled Shafts...........................................38
3.3 Drilled Shafts under Lateral Load........................................40
3.3.1 Soil-Structure Interaction.............................................40
3.3.2 Boundary Conditions....................................................44
3.3.3 Methods for Lateral Load Design........................................45
3.3.3.1 Ultimate Rigid Resistance...........................................46
3.3.3.2 Subgrade Reaction Approach..........................................47
3.3.3.2.1 P-Y Method.........................................................49
3.3.3.2.2 Characteristic Load Method.........................................50
33.3.3 Elastic Continuum Theory............................................56
4. FINITE DIFFERENCE METHOD..................................................57
4.1 Introduction...........................................................57
4.2 Fast Lagrangian Analysis of Continua in Three Dimensions (FLAC 3D)......59
4.2.1 Overview...............................................................59
4.2.2 Mathematical Model.....................................................59
4.2.3 Numerical Formulation..................................................63
vii


4.2.4 Numerical Implementation
64
5. NUMERICAL MODEL OF A DRILLED SHAFT FOUNDATION........................66
5.1 Introduction.......................................................66
5.2 Mesh Generation and Interfaces.....................................66
5.3 Boundary Conditions................................................69
5.4 Initial Static Stress State........................................70
5.5 Loading............................................................72
6. MODEL CALIBRATION WITH FIELD LOAD TESTS..............................74
6.1 Introduction.......................................................74
6.2 Drilled Shafts under Axial Load....................................76
6.2.1 23rd Street Viaduct Field Load Tests, Denver......................78
6.2.1.1 Shaft Wall Resistance Field Load Test (T-l).....................79
6.2.1.2 Base Resistance Field Load Test (T-3)...........................84
6.2.2 1-270 and 1-76 Field Tests, Denver...............................87
6.2.2.1 Base Resistance Field Load Test (T3B)...........................91
6.3 Drilled Shafts under Lateral Load..................................91
6.3.1 1-270 and 1-76 Field Load Test, Denver...........................93
6.3.1.1 Lateral Load Field Test (T3B)...................................93
6.3.2 NGES-UH Field Load Test, Houston..................................95
7. SINGLE AND COMBINED LOADS EVALUATION.................................101
viii
7.1 Introduction
101


7.2 Research Drilled Shaft Numerical Model..............................102
7.3 Single Load Analysis................................................108
7.3.1 Axial Load.........................................................108
7.3.2 Lateral Load.......................................................110
7.3.3 Single Load Results................................................112
7.4 Combined Loads Analysis.............................................112
7.4.1 Axial then Lateral Load............................................113
7.4.2 Lateral then Axial Load............................................115
8. CONCLUSIONS...........................................................117
Appendix
A. Constitutive Models in FLAC 3D........................................118
References...............................................................124
IX


LIST OF FIGURES
Figure
2.1 Drilled shaft..................................................................4
3.1 Axial load resistance of a drilled shaft......................................23
3.2 Axial resistance of a drilled shaft by layers.................................24
3.3 Bearing capacity factors......................................................25
3.4 Measured a values from field tests compared with empirical functions..........28
3.5 Frictional shaft resistance of a drilled shaft in cohesionless soil...........33
3.6 Shaft horizontal stresses and critical depth in cohesionless soils............34
3.7 Values of a\%m as a function of unconfmed compressive strength................36
3.8 Values of Fc as a function of concrete slump and depth........................37
3.9 Behavior of a drilled shaft under lateral load................................41
3.10 Soil reaction due to lateral load, profile and plan section view..............41
3.11 Typical soil resistance as a function of lateral deflection for two types
of soil stiffness............................................................42
3.12 Complete soil-structure results for a typical lateral load design.............43
3.13 Soil-structure sign conventions...............................................43
3.14 Top boundary conditions: a) free, and b) fixed................................45
3.15 Free boundary condition for: a) long, and b) short drilled shaft..............46
3.16 Fixed boundary condition for: a) long, b) intermediate, and c) short
drilled shaft................................................................47
x


3.17 Drilled shaft model forp-y method.....................................50
3.18 Load-Deflection curves: a) clay, and b) sand..........................52
3.19 Moment-Deflection curves: a) clay, and b) sand........................52
3.20 Lateral load applied above ground surface.............................53
3.21 Load-Moment curves: a) clay, and b) sand..............................54
3.22 Coefficients Am and Bm................................................55
4.1 Approximation of a continuous domain by an array of discrete points....58
4.2 Example of a FLAC 3D numerical model...................................60
4.3 Basic explicit calculation cycle.......................................60
4.4 Sign convention for positive stress components.........................61
4.5 Mechanical pressure: a) positive, b) negative..........................61
4.6 Tetrahedron............................................................64
4.7 An 8-node element with 2 overlays of 5 tetrahedrons in each overlay....65
5.1 Procedure summary for the static load analysis of a drilled shaft...67
5.2 Mesh of a drilled shaft foundation.....................................67
5.3 Types of elements used to shape the model grid.........................68
5.4 Mesh and interfaces for drilled shaft foundation.......................69
5.5 Two dimensional representation of roller boundary conditions........70
5.6 Mohr-Coulomb yield criteria............................................72
5.7 Plane of symmetry for research drilled shaft...........................73
6.1 Calibration process for a numerical model..............................75
xi


6.2 Modified Burland Triangle..................................................76
6.3 Typical axial compressive field load test setup............................77
6.4 Typical axial load-displacement curves.....................................77
6.5 Location for 23rd Street Viaduct test site.................................78
6.6 Construction of a drilled shaft in competent foundation material...........79
6.7 Geologic, construction, and model profile for the shaft wall resistance
T-l field test............................................................80
6.8 Numerical model for shaft wall resistance T-l field test...................80
6.9 Shear stress distribution for shaft wall resistance T-l field test at the
end of loading............................................................83
6.10 Load-displacement curves for shaft wall resistance T-l field test.........83
6.11 Geologic, construction, and model profile for the base resistance T-3
field test................................................................84
6.12 Numerical model for base resistance T-3 field test........................85
6.13 Vertical normal stress distribution for base resistance T-3 field test at
the end of loading........................................................86
6.14 Load-displacement curves for base resistance T-3 field test...............86
6.15 Location for 1-270 and 1-76 test site.....................................87
6.16 Geologic, construction, and model profile for T3B field tests.............88
6.17 Numerical model for T3B field tests.......................................89
6.18 Construction of a drilled shaft in caving soils with temporary casing.....89
6.19 Load-displacement curves for base resistance T3B field test...............91
6.20 Typical lateral field load test setup.....................................92
xii


6.21 Typical lateral load-displacement curves...................................92
6.22 Normal lateral stress distribution for lateral T3B field test at the end
of loading.................................................................93
6.23 Contours of x-displacement for lateral T3B field test at the end of loading.94
6.24 Load-displacement curves for lateral T3B field test........................94
6.25 Location for NGES-UH test site.............................................95
6.26 Geologic, construction, and model profile for Shaft 6 field test...........96
6.27 Numerical model for Shaft 6 field test.....................................97
6.28 Construction of a drilled shaft in caving soils............................97
6.29 Contours of x-displacement for Shaft 6 field test at the end of loading....99
6.30 Load-displacement curves for Shaft 6 field test...........................100
7.1 Typical combined axial compressive and lateral field load test setup.......102
7.2 Geologic, construction, and model profile for research drilled shaft.......103
7.3 Research drilled shaft numerical model.....................................103
7.4 Undrained shear strength with depth........................................104
7.5 Contours of undrained shear strength.......................................106
7.6 In-situ vertical normal stress distribution................................106
7.7 In-situ lateral normal stress distribution.................................107
7.8 Shear stress distribution in top 8.0 meters at the end of axial loading....108
7.9 Shear stress distribution in bottom 2.0 meters of shaft at the end of
axial loading.............................................................109
7.10 Vertical normal stress distribution at the end of axial loading...........109
xiii


7.11 Contours of vertical displacement at the end of axial loading...........110
7.12 Normal lateral stress distribution at the end of lateral loading........Ill
7.13 Contours of x-displacement at the end of lateral loading................Ill
7.14 Contours of z-displacement at the end of lateral loading................112
7.15 Load-displacement curves for single loads...............................113
7.16 Load-displacement curves for single lateral and combined
(axial-lateral) loads...................................................114
7.17 Load-displacement curves for single axial and combined
(lateral-axial) loads...................................................116
A.l Mohr-Coulomb and Tresca yield surfaces in principal stress space.........119
A.2 Mohr-Coulomb failure criteria............................................120
A.3 Distribution of representative areas to interface nodes..................121
A.4 Components of the bonded interface constitutive model....................122
xiv


LIST OF TABLES
Table
3.1 Empirical base coefficients for settlement....................................40
3.2 Minimum length of embedment for Characteristic Load Method....................56
6.1 Material properties for 23rd Street Viaduct Field Tests, Denver...............82
6.2 Material properties for 1-270 and 1-76 Field Tests, Denver....................90
6.3 Material properties for NGES-UH Field Test, Houston...........................98
7.1 Material properties for research drilled shaft model.........................105
xv


1. INTRODUCTION
1.1 Research Overview
This thesis presents the numerical analyses of a drilled shaft foundation under
combined axial and lateral loads. The behavior of a reinforced concrete drilled shaft
under a single type of load is compared to the load-displacement effect caused by the
applied combination of two types of loads, axial compressive and lateral. The loads
are applied in a static monotonic direction to the top of the drilled shafts.
A comprehensive review of research on loading of deep foundations over the past 50
years and the current state of design of drilled shaft foundations under axial or lateral
load is presented in early chapters. Since the design of a drilled shaft will vary
according to the subsurface profile conditions, the foundation geologic material was
divided for design purposes into cohesive and cohesionless soils, and cohesive and
cohesionless intermediate geomaterials. This was essential in order to develop a
better understanding of this multifaceted soil-structure interaction physical system.
The analyses were performed using a three-dimensional finite difference numerical
software called FLAC 3D. A summary of the mathematical numerical method
utilized and the software application is provided in this thesis. The chosen numerical
computer code is considered adequate to analyze the engineering mechanics of
geologic material.
An outline on the numerical modeling procedure of a drilled shaft foundation system
is also included in the literature. The numerical analysis model was calibrated using
actual full-scale field load tests in various foundation materials; such as, cohesive and
cohesionless soils, and in cohesive intermediate geomaterial. For the combined loads
research model, a stiff normally consolidated clay was assumed for the geologic
profile with a drilled shaft aspect ratio of 10:1.
The evaluation presents approximate values of the potential range of axial and lateral
induced displacements due to loading. For this research thesis, all units are presented
using the metric system (kilogram, meter, and second) unless otherwise noted in
parenthesis.
1


1.2 Purpose
It was determined that there is not sufficient research information on the effects
caused by the application of combined axial compressive and lateral loads to drilled
shafts foundations. A number of new and existing engineering structures that utilize
drilled shafts as their construction option are faced with the need to design this type
of foundation system considering that combined axial and lateral loads will be applied
at some point during the lifespan of the structure. Structures such as buildings,
transmission towers, and highway structures like bridge piers, abutments, retaining
walls, and overhead signs, can be considered to experience a combined load effect
throughout their design lifetime.
Therefore, a three-dimensional numerical analysis was considered an acceptable
method to enhance the knowledge of the soil-structure interaction behavior of this
complex physical foundation system.
2


2. LITERATURE REVIEW
2.1 Introduction
The objective of the literature review is to find available information on the general
behavior of drilled shafts under combined (axial and lateral) loads. Since the
performance is dependent on the combined application of two types of loads, it is also
necessary to evaluate the effect of these loads separately. Although a multitude of
studies exist on the subject of deep foundations under axial or lateral load, there are
only a limited number of published papers on the behavior of deep foundations under
combined loads.
This chapter will include a summary of reviewed articles that are relevant to the scope
of this thesis. Existing literature was also reviewed to establish the current state of
knowledge with regard to the design of single drilled shafts under axial or lateral load
and will be presented in the following chapter.
Foundations are the lowest part of a structure capable of adequately supporting and
transmitting the structural loads to the underlying subsurface. The type of foundation
to be used will depend on the engineering properties of the soil or rock encountered
beneath the structure. Shallow foundations are used when the surficial soil stratum is
capable of supporting the structural loads; otherwise, deep foundations are used to
transmit the loads at greater depths. Deep foundations can be separated according to
their installation method into two main groups: driven piles and drilled shafts.
Drilled shafts are concrete column-type elements, constructed by drilling into the
earth (soil and/or rock) a hole with a diameter usually greater than 75 centimeters and
placing fluid concrete in the excavation with or without steel reinforcement as shown
in Figure 2.1. In cohesive soils, drilled shafts may have enlarged bases (bell-shape)
by a construction technique known as under-reaming. Drilled shafts are also known
as bored piles, drilled piers, caissons, and augered cast-in-place piles. However,
drilled shaft is currently the designation with general acceptance for bored
foundations.
Drilled shafts are greatly used to support buildings, highway structures (overhead
signs, retaining walls, bridge piers and abutments), transmission towers, and other
engineering structures were large axial loads and lateral resistance are major factors.
Drilled shafts are also suitable for loads that result from environmental factors such as
3


wind, ice, waves, water current, scour, earthquakes, and also vessel collision and
explosive blasts.
Axial Load
ntttttt
Base Resistance
Figure 2.1 Drilled shaft (from Federal Highway Administration website)
2.2 Deep Foundations under Axial Load
Meyerhof and Murdock (1953)
Presented the results obtained from a series of axial load field tests on bored piles
(drilled shafts) and a couple of concrete driven piles in London clay at two different
test sites [1]'. Laboratory soil investigations and post-loading pile-soil examinations
1 Numbers in brackets indicate references listed in the last section of this thesis.
4


were also performed. Both sites consisted of hard to stiff fissured London clay,
where the undisturbed shear strength of the clay increased rapidly with depth.
The shear strength of the clay was reduced by up to one-half of the maximum when
the undrained triaxial compression tests were made several days after sampling, even
under high lateral pressures exceeding those due to overburden; this was due to
gradual opening of the fissures. The residual shear strength on clay tests immediately
after sampling were 50 percent less than the maximum strength. Fully remolded clay
tests at natural water content resulted in shear strengths slightly higher than tests
performed immediately after sampling by eliminating the clays natural fissures.
Fully softened clay (in free water for 3 days) with water contents 5 to 8 percent higher
than the natural water content resulted in very low, but uniform shear strengths
throughout the entire depth at both sites.
Adhesion, side resistance between the clay soil and the bored pile concrete, for stiff
and hard clays is usually less than the shear strength of the soil itself. Hence,
laboratory adhesion tests were performed in a shear box using a coarse stone of
similar texture and density as concrete on the lower box frame and undisturbed clay
on the upper frame. Adhesion increased with applied normal pressure; for large
pressures, adhesion approached the shearing strength of the clay. Also, adhesion was
correlated to an equivalent coefficient of friction between the two contacts of
approximately 0.8 for a dry stone and 0.4 for a wet stone. Residual adhesion ranged
from 0.5 to 0.8 times the maximum adhesion value.
Water from a concrete mix tends to migrate towards the clay adjacent to the borehole,
causing the clay near the pile to soften (local softening). Two water/cement mix
ratios (0.2 and 0.4) were used for the bored piles to evaluate the effect on side
resistance. It was found that the water content of clay increased rapidly within 5
centimeters (2 in.) from the shaft for mix ratios of 0.4, while remaining more or less
constant at distances 5 centimeters (2 in.) or more away from the shaft. For ratios of
0.2 (crushing failure near surface) the water content of the clay remained the same
near or with distance away from the shaft. The top-down loading system consisted of
a pile anchored frame, the test piles were loaded using a hydraulic jack. The rate of
loading was fast enough to ignore any significant consolidation without increasing the
shear strength due to a rapid load rate.
The average adhesion for bored piles can be estimated from the fully softened shear
strength of the clay. While for driven piles, the average adhesion can be taken from
the clays fully remolded shear strength. The undisturbed shear strength of the clay is
used when calculating the end bearing resistance. Driven piles carried about double
5


the load of that carried from bored piles, hence the importance of avoiding soil
softening in stiff clays.
Skempton (1959)
Axial load test results on bored piles embedded in London clay from ten different test
sites were evaluated [2], The bored piles ranged from 2.44 meters (8 ft) deep and
0.25 meters (10 in.) diameter to 27.43 meters (90 ft) deep and 0.91 meters (36 in.)
diameter. An extensive analysis was performed on the relation between the side
resistance of the drilled shaft with the clays water content near the borehole wall, as
well as the clays increase in strength with depth.
The adhesion coefficient (a), which is equal to the ratio of average adhesion strength
between the clay and the pile shaft (side resistance) to the average undisturbed shear
strength of the clay within the embedded length of the pile, was found to average
about 0.45. Values of the adhesion coefficient ranged from 0.6 to 0.3 depending on
the workmanship and soil conditions. The adhesion coefficient will always be less
than unity. Adhesion should be ignored in the zone of seasonal moisture change.
The adhesion strength is less than the strength of the clay due to softening of the clay
near the shaft wall, primarily because the clay absorbs water from the concrete and in
some cases due to the drilling operations. When drilling below the water table, water
will tend to flow out of the clay towards the open borehole. The adhesion strength
was found to be about 80 percent of the shear strength of the clay adjacent to the
concrete (softened clay). An increase in 1 percent in water content can cause a 20
percent reduction in the adhesion strength. Typical increases in the water content
range between 3 to 6 percent, within 5.0 centimeters from the shaft contact surface,
which can result in softened strengths 30 percent of the original clays strength.
The shear strength of the undisturbed clay was found to increase approximately
linearly with depth, while the adhesion coefficient was found to slightly decrease
linearly with depth. The side shear resistance which develops in the clay alongside
the shaft is essentially restricted to the narrow softened zone. The side resistance is
mobilized at an early stage in the loading, and was considered to be fully mobilized
after a settlement of 1.0 centimeter.
Coyle and Reese (1966)
Presented an analytical method to determine the load transfer mechanism of axially
loaded driven steel pipe piles embedded in clay soil [3], Reeses (1964) analytical
method for predicting load-settlement curves was employed; however, improvement
6


was emphasized on the correlations between soil properties and pile behavior. The
pile is divided into various segments to calculate the axial movement of each segment
due to the load transferred on to the surrounding soil. Pile load distribution and
movement at different depths is obtained by instrumentation of a test pile, while the
soil shear strength is acquired by field (vane shear) and laboratory (unconfined
compressive strength) tests.
Field test data from three different driven steel pile tests in clay were used to obtain a
family of curves that relate the ratio of load transfer to soil shear strength versus pile
movement. The types of piles were pipe, tapered, and rectangular, located in the San
Francisco Bay area, Omaha, and Melbourne (Australia), respectively. Ten triaxial
tests were performed on small model piles in clay for the purpose of finding load
transfer values versus pile movement. Fligher load transfer values were obtained for
rough model piles and for piles embedded in clays with lower water contents.
Field and laboratory results confirm the behavior of soft clays near the pile-soil
interface of driven piles where the loss of shear strength is due to remolding of the
clay caused by the pile driving. However, with time, the shear strength will increase
due to pore pressure dissipation. For most cases the comparison of the analytical
method with data from field tests had a good correlation.
Poulos and Davis (1968)
Evaluated the performance of single incompressible piles and piers (drilled shafts)
under axial load using elastic theory [4], based on Mindlins (1936) elastic equations.
For the pier to be considered incompressible, the surrounding soil must be assumed
soft compared to the pier; therefore, soft clay was used in the analyses. The pier
length is divided into uniform elements (segments) to determine shear stress and
displacements along the piers cylindrical sections. Three scenarios of shear stress
along the soil-pier interface were considered; however, the most acceptable scenario
was considered to be uniformly distributed shear stress along the circumference of the
soil-pier interface.
Various pier lengths to diameter (L/D) ratios were considered to account for the effect
of pier dimensions on side shear distribution, displacement, and base load. Rough
and smooth pier surfaces and variation of the soils Poissons ratio were also
evaluated. Adhesion strengths between the pier and the clay, and the clays
undrained shear strength were considered for the load-settlement behavior. Solutions
were presented for piers embedded in a semi-infinite soil mass and a finite layer.
7


It was concluded that in undrained conditions approximately 90 percent of the total
settlement will occur immediately after loading. Load transfer to the base of a pier
was small, even for a L/D ratio of two the base receives about 25 percent of the total
load. Belled base piers, which increase the load taken by the base, in soft clay are
practical for relatively short piles with a L/D ratio less than 25. For increasingly
higher L/D ratios, belled base piers will perform as a straight sided pier (without
enlarged base).
Mattes and Poulos (1969)
This paper analyzes the behavior of compressible piers [5] and is considered a
continuation of the research on incompressible piers performed by Poulos and Davis
(1968). For the pier to be considered compressible, the surrounding soil must be
considered as stiff. Elasticity theory is used to determine the behavior of soil and
pier. The pier length is divided into equal segments, where uniform shear stress is
developed around the circumference segment. Elastic modulus (Es) and Poissons
ratio (vs) of the soil are assumed constant throughout the surrounding soil mass. The
piers compressibility is evaluated by changing the pier stiffness (Aids) and comparing
it to an incompressible pier (Aids = )-
The effect of pier length to diameter (L/D) ratio, variation of Poissons ratio, belled
base option, and pier stiffness were considered for settlement behavior. The
analytical method was compared to the results obtained from a full-scale concrete pier
load test in London clay (Whitaker and Cooke, 1966) with an average value of L/D of
15, obtaining good agreement. It was concluded that with increasing compressibility
of the pier the shear stress will increase close to the ground surface and reduce near
the base of the pier. Hence, the reduction in pier stiffness will increase the settlement
at the top of the pier while reducing the settlement at the base of the pier.
Ellison et al. (1971)
Presented a two-dimensional finite element analysis of bored piles embedded in stiff
clay emphasizing on the load-deformation mechanism [6]. Five instrumented load
tests on bored piles in London Clay (Whitaker and Cooke, 1966) were used for
calibration of the model and comparison of the results. The pile dimensions varied
from 9.3 to 15.2 meters in length, and 0.64 to 0.94 meters in diameter.
The soil modeling is considered in two parts, a linear and a nonlinear portion. Special
attention is placed on the adhesion resistance between pile and soil, and on the elastic
compression of the pile. It was found that the maximum adhesion at any location
along the interface is dependant on the soil shear strength near that area. Tension
8


cracks developed in the soil near the edge of the base of the pile. Good agreement
was obtained between the load-settlement curves of the finite element results and
those of the field test.
ONeill and Reese (1972)
Evaluated the behavior of four full-scale axially loaded drilled shafts in the Beaumont
Clay formation (Houston, Texas) [7]. Load transfer mechanisms of the instrumented
drilled shafts embedded in stiff clay were investigated. Soil parameters were
established for side shear and end bearing. Field explorations revealed six different
soil layers to a depth of 18 meters. The Texas Quick Load Test Method was used
creating an undrained loading scenario for the saturated clays. Shear strengths were
obtained from unconsolidated-undrained (UU) triaxial compression tests.
Moisture migration from the concrete into the soil was examined to determine side
shear strength resistance. Undrained direct shear tests were performed on in-situ soil
samples and on mortar cast against in-situ soil specimens obtained near the pile-soil
interface. The adhesion coefficient (a) from each test was calculated as the ratio of
maximum shearing resistance of the mortar-soil sample to the undrained shear
strength of the in-situ soil samples. Plasticity index seemed to influence the value of
a along the soil layers. Compared to Tomlinsons (1969) aavg values for driven piles
in clay, the three drilled shafts constructed in dry resulted in slightly higher a values,
where as the drilled shafts constructed using slurry resulted in slightly lower values.
For all four tests at small displacements, the side resistance governed over base
resistance; however, this effect was reduced after the peak side resistance was
reached with additional displacement. The peak side resistance was reached for all
drilled shafts at displacements between 5.0 to 10.0 millimeters and the residual side
resistance was half of the peak. The use of drilling slurry was found to reduce side
shear stress.
Post-load inspection of the drilled shafts encountered increased moisture content near
the base; this effect was considered to account for some of the reduction in load
transfer near the base. It was concluded that base failure was reached at settlements
of 3 to 6 percent of the base diameter and that a value of 9 for the bearing capacity
factor Nc seemed reasonable when cohesion is obtained from UU triaxial tests.
Inspection showed shear planes at about 3.2 millimeters from the soil-concrete
interface, indicating side shear failure along the soil instead of the interface. Due to
seasonal moisture variation, the shear resistance of the top 1.5 meters of clay soil was
omitted.
9


Kulhawy and Jackson (1989)
The undrained side resistance of drilled shafts was evaluated based on 106 axial field
load tests (41 in compression and 65 in uplift) in cohesive soils; all of which were
conducted on straight-sided drilled shafts [8], The total stress or alpha method (a),
and the effective stress or beta method (/?) were used to evaluate the undrained side
resistance. Since the alpha method was originally developed for driven piles in clay
(Tomlinson, 1957), a new correlation of this method is presented specifically for
drilled shafts.
When local load test are available for calibration purposes, the total stress method can
be used with simplicity. The effective stress method requires more input parameters,
but it is more fundamentally rational. Also, the interrelationship of both methods was
examined resulting in equations containing more readily accessible parameters. The
design criteria for both methods will be explained in the next chapter.
Originally the adhesion coefficient (a) was correlated directly to the undrained shear
strength (cu), subsequent work suggested a correlation with the ratio of undrained
shear strength to the effective overburden stress (cr'v). The best correlation obtained
from the data base was related to the ratio of the atmospheric pressure (pa) to the
undrained shear strength. The new total stress method correlation should produce
slightly conservative results.
Reese and ONeill (1989)
A new design method for axially loaded drilled shafts is presented based on the
analysis of 41 load tests in overconsolidated cohesive and cohesionless soil [9]. The
design equations are limited to the range of conditions within the data base; i.e.
diameters ranging from 0.52 to 1.20 meters, lengths from 4.7 to 30.5 meters, and
other soil conditions. A design procedure for drilled shafts in rock is also presented.
The method allows for a non-homogeneous soil profile, which is divided into various
individual homogeneous layers of cohesive soil, cohesionless soil, or rock. The load-
settlement characteristics are also presented assuming an incompressible drilled shaft.
The method requires subsurface characterization; i.e. undrained triaxial test of
cohesive soil, standard penetration test (SPT) for cohesionless soil, and recovery of
rock cores.
For cohesive soils the top 1.5 meters is omitted from the calculation of shaft
resistance (a 0) due to seasonal moisture change. This consideration tends to make
the adhesion coefficient approach a constant value (a = 0.55) which does not vary as
a function of undrained shear strength. The constant value of a with depth may also
10


be due in part to the disturbance of the clay at the future shaft-soil interface from the
mechanical excavation tool. In cohesionless soils, regardless of the in-situ conditions,
the construction process tends to cause the interface friction and unit weight of the
soil at the shaft-soil interface to converge to a near constant value. Therefore, the
shaft resistance for cohesionless soils is affected mainly by the lateral effective stress
between the soil and the drilled shaft.
Results from the design method (calculated) fall within 25 percent of the measured
ultimate capacity values for cohesive soils; mixed subsurface profiles would be
slightly more conservative, while for cohesionless soils very conservative. The
method estimates accurate to conservative values of the capacity and load-settlement
characteristics of drilled shafts.
Turner et al. (1993)
This paper presents the results of 13 axial field load tests on drilled shafts in Upper
Cretaceous shales [10]. Three of these tests were performed in the Denver Blue
Formation, which is the most commonly geologic unit encountered in the Denver
metropolitan area. The Denver Blue Formation consists mostly of weakly cemented
claystone with some siltstone and sandstone layers. The measured side resistance of
the drilled shafts was correlated to the compressive strength of the supporting rock.
The measured load results were compared to published methods that predict the side
resistance of drilled shafts in weak rock.
A comparison of the measured side resistance to Denvers local practice was also
evaluated. Denver and its vicinity have based the capacity of drilled shafts on the
standard penetration test (SPT) number of blows (N-value) since the 1950s; where
the end bearing capacity is taken as N/2 in kips/ft2, and the side resistance as N/20 or
10 percent of the end bearing. This relationship is used for drilled shafts supporting
an axial load less than 4.45 MN (mega Newton). To date, no significant failures have
occurred due to the use of this local practice.
The ultimate side resistance of drilled shafts in Cretaceous shales of the region with
compressive strengths less than 0.6 MPa (mega Pascal) should be evaluated using
Kulhawys 1989 a method; for strengths greater than 0.6 MPa Horvaths 1982
method should be used.
Hassan and ONeill (1997)
Investigated the side load-transfer mechanisms of axially loaded drilled shafts
socketed into cohesive intermediate geomaterials (IGM) [11]. Two-dimensional


axisymmetric elastic-plastic finite element analysis was used. IGMs are considered
materials between very stiff soil and very soft rock, a gray zone between soil and
rock. Emphasis is made on the interface simulation of the socket, where a sinusoidal
interface roughness similar to that observed in the field was modeled. Two types of
sockets were modeled, rough and smooth interfaces.
The cohesive IGMs were classified as those geomaterials with unconfmed
compressive strengths (qu) in the range of 0.5 to 5.0 MPa (mega pascal). Drucker-
Prager elastic-plastic parameters were used to model the cohesive IGM behavior.
The adhesive bond between concrete and argillaceous IGM is considered zero for this
study (nonporous IGM), given that interface shear tests performed by Hassan (1994)
revealed that no cement paste penetrated into the clay shale. Research performed on
Eagle Ford clay shale, which is considered a typical cohesive IGM, was used as a
reference for this parametric study.
It was found that within the elastic portion of the load-settlement curve the unit side
resistance (f) increases with increasingly normal stress (crn). Also, the ultimate unit
side resistance (/max) approaches the value of the shear strength of the cohesive IGM;
therefore,^/max can be taken as qJ2 for rough sockets. For smooth sockets/max is
described in terms of the adhesion coefficient (a). For side load-transfer in rough
sockets asperities failed due to shearing beneath its bases followed by minor sliding
and consequently gap formation below the asperities. Smearing of the socket,
disintegration of the wall socket creating a soil-like material, may occur in some
cohesive IGMs reducing the side load transfer significantly and therefore should be
treated as a smooth socket in the design process.
2.3 Deep Foundations under Lateral Load
McClelland and Focht (1956)
This research focused on estimating the soil modulus (Es) based on results from full-
scale driven pile lateral load tests and consolidated-undrained (CU) triaxial tests on
undisturbed clay samples [12]. Correlations were based on soil reaction pile
deflection (p-y) curves from the pile tests and stress strain (a-s) curves from
laboratory tests. The lateral load test was conducted in 1952 on a 61 centimeter (24
in.) diameter pipe pile driven 22.86 meters (75 ft) below ground line, embedded in
normally consolidated marine clay located off the coast of Louisiana. The difference
equation method was employed to compute laterally loaded pile deflections, bending
moments, shear forces, and soil reactions.
12


Soil shear strengths were also determined from unconfined compression (qa),
remolded unconfined compression, and field vane shear tests. However, the
consolidated-undrained (CU) triaxial test exhibited stress-strain characteristics similar
to those obtained from the pile load-deflection field test. The beam theory equation
was modified to define the soil modulus as the ratio of the soil reaction to the pile
deflection at the same point (depth). The soil modulus varied widely with depth and
pile deflection; however, for a single applied load the soil modulus increased almost
linearly with depth (22 strain gages installed).
Matlock and Reese (1960)
Proposed computation methods and equations for solving non-linear load-
deformation characteristics of the soil for laterally loaded piles, for elastic and rigid
piles supported in an elastic medium [13]. Their approach is a special case of a beam
on elastic foundation, accounting for boundary conditions and the non-linear
properties of in-situ soils. Several iterations of the elastic theory were performed
until a satisfactory soil-structure response was obtained.
The soil modulus along the depth of the pile was adjusted independently for each
consecutive trial until adequate compatibility was reached among the predicted
behavior of the soil and the load-deflection of the pile. The soil modulus is
introduced as a function of both depth and a constant of soil modulus reaction (ks),
and can also be obtained from p-y curves by the slope of a secant line drawn from the
origin to any point along the curve. Two methods were used to calculate the soil
modulus, power and polynomial, where the linear form is considered a special case of
the previous two. A series of derivative equations were used in order to obtain, as a
function of depth, values of pile slope, moment, shear, and soil reaction.
Davisson and Gill (1963)
Studied the results of a laterally loaded pile in a two-layer soil profile [14]. Subgrade
reaction theory was employed to analyze the soil modulus variation with depth and/or
with pile deflection. The first concept will create a fourth order linear homogeneous
differential equation, while the latter will produce a nonlinear differential equation.
The relative thickness and modulus of the soil layers were varied and analyzed for a
complete range of values.
One of the problem statements addresses the need of an analytical method to account
for a subgrade soil modulus that varies together with pile deflection and soil depth.
Suggesting that this issue may be solved by iteration of elastic methods with
adjustment of the secant modulus of the pile load-deformation curve until field
13


compatibility is obtained. It was shown that the surface layer has great influence on
pile deflection. The seasonal moisture may reduce or increase the soil stiffness of the
upper layer; therefore, the ultimate resistance may vary with the season.
Broms (1964a and 1964b)
These journal articles recommended design methods to calculate the ultimate lateral
resistance, lateral deflections, and maximum bending moments of piles under
working loads. These methods were evaluated utilizing the theory of subgrade
reaction. The 1964a article [15] refers to piles embedded in saturated cohesive soils,
while the 1964b [16] deals with cohesionless soils. Possible failure mechanisms for
both free-headed and restrained (fixed-headed) piles under lateral loads are evaluated.
For free-headed piles, failure occurs when the applied lateral load in the pile creates a
maximum bending moment that exceeds the yielding moment of the pile section
creating a plastic hinge, typical of long piles; and when the lateral earth pressure
caused by the pile exceeds the lateral resistance of the surrounding soil and the pile
moves as a rigid unit through the soil, typical of short piles.
For restrained piles, failure may occur under three circumstances. First case is when
two plastic hinges form along the length of the pile due to maximum positive and
negative bending moments in the pile exceeding the yield moment of the pile section,
typical of long piles. Second, when one plastic hinge is formed and subsequently the
lateral earth pressure caused by the applied load to the pile exceeds the lateral
resistance of the surrounding soil and the pile rotates around a point below ground
level, typical of intermediate length piles. Third, when the lateral earth pressure
caused by the pile exceeds the lateral resistance of the surrounding soil and the pile
rotates as a rigid unit, typical of short piles.
The soil and pile are assumed to behave elastically under working loads, which are
considered in the range of one-half to one-third the ultimate lateral soil resistance.
Methods at working loads are used to compute lateral deflections, soil reactions, and
the distribution of bending moments and shear forces based on the theory of subgrade
reaction. The coefficient of subgrade reaction (kb) is assumed constant with depth for
cohesive soils and linearly increasing with depth for cohesionless soils. Different
equations were used to obtain the coefficient of subgrade reaction for short and long
piles. Results from the suggested methods were compared with measured field test
data and were considered adequate for cohesive soils; however, for cohesionless soils,
the measured values exceeded the calculated by approximately 50 percent. Ultimate
lateral resistance graphs and charts are presented for the different soil types, pile head
boundary conditions, and failure modes.
14


Spillers and Stoll (1964)
Analyzed the soil-pile response of a laterally loaded pile by simplified constitutive
equations [17]. The soil mass was considered to be initially a continuous elastic
medium and subsequently added more realistic properties to consider it as an elastic-
plastic half space (iterative solution), while the pile remained elastic. Some of the
limitations of the Winkler model and the subgrade reaction theory were mentioned.
The elastic soil concept presents high stresses near the surface similar to Winklers
uniform spring constant, which is not practical; hence, plastic yielding is added to
improve the concept. Similar nonlinear curve results were observed with those
obtained from full-scale field tests, where progressive yielding with depth is obtained.
Broms (1965)
Presented a summary of his previous two papers of 1964 for the design of laterally
loaded piles in cohesive and cohesionless soils [18]. Over-load and under-strength
factors were introduced. The ultimate lateral soil reaction was assumed equal to nine
times the undrained shear strength for cohesive soils, while considered three times the
passive Rankine earth pressure for cohesionless soils. For short piles, the ultimate
lateral resistance was found to be controlled by depth of penetration, while
independent of the pile section yield moment and vice versa for long piles.
Poulos (1971)
Evaluated the performance of piles subjected to lateral load and bending moment,
assuming the surrounding soil to be elastically homogeneous, isotropic, and semi-
infinite [19]. Soil modulus and Poissons ratio were assumed constant with depth.
This elastic theoretical assumption was considered particularly satisfactory for
cohesive soils when compared to observed pile behavior. Results from several cases
from this method are compared to those obtained from the subgrade reaction theory
and significant differences were observed. The basis of this method is considered
similar to that by Spillers and Stoll (1964) with some enhanced assumptions.
Two boundary conditions of the pile head were considered, free and restrained, while
the pile was assumed as a thin rectangular strip of constant flexibility. It was
observed that the soil modulus is more accurate if obtained from a full-scale pile load
test. The most important variables affecting the performance of the pile were said to
be the length-to-diameter (L/D) ratio and the pile flexibility factor. Pile displacement,
moment, and rotation due to local yielding of the surrounding soil were evaluated.
Furthermore, the accuracy of the method increases with more number of element
divisions (segments) along the pile.
15


Yegian and Wright (1973)
Presented a two dimensional nonlinear finite element model to develop a soil
resistance-pile displacement relationship (p-y curves) for laterally loaded single and
group piles in soft saturated clays [20], The stress-strain properties of the saturated
clay were defined by a nonlinear hyperbolic expression up to the point of full
mobilization of the shear strength and perfectly plastic beyond this point.
Emphasis is made on the soil-pile interaction, where the maximum shear resistance
(soil-pile adhesion) was considered directly proportional to the undrained shear
strength of the clay. The p-y relationships obtained from the finite element model
were compared to that of Matlocks (1970) procedure with some minor discrepancies
found between the two methods.
Reese and Welch (1975)
Proposed a method for soil-structure interaction of laterally loaded deep foundations
in stiff clays subjected to static or cyclic loading [21]. Experimental p-y curves were
obtained by testing a laterally loaded instrumented drilled shaft with a diameter of
76 centimeters (30 in.) and a total length of 13.4 meters (44 ft) located in Houston,
Texas. Theoretically, a difference equation method was used to satisfy the conditions
of equilibrium and compatibility of the nonlinear p-y response.
Values of bending moment, deflection, and rotation were measured along the length
of the drilled shaft for different number of applied loads. Stress-strain conditions of
undisturbed soil samples were obtained by unconsolidated-undrained (UU) triaxial
compression tests, where the applied confining pressure replicated the in-situ
overburden pressure. Results included deflection, slope, moment, shear, and soil
reaction, all in function of depth.
Randolph (1981)
Reported the results of a parametric study of laterally loaded single and group flexible
piles [22]. Simple and practical algebraic expressions were utilized to best-fit the
results obtained from finite element analyses. Equations of maximum bending
moment along the pile, as well as deflection and rotation at ground level were
illustrated. The soil profile was assumed an elastic continuum and the soil modulus
was represented as either homogeneous or varying linearly with depth.
The critical length of a pile was evaluated and solutions for piles beyond this length
are based on Hetenyis (1946) approach, where the pile is considered to behave
infinitely long. Single pile results are extended for group piles using interaction
16


factors between piles. This method was compared to data obtained from two lateral
load field tests, McClelland and Focht (1956) one pile, and Gill and Demars (1970)
four single piles. The result comparison was deemed rather well for the various soil
profiles; with the exception of a 20 percent under estimation of the maximum bending
moment for the 1956 load test and a maximum 20 percent discrepancy of the pile
stiffness value for the 1970 test.
Davies and Budhu (1986)
Analyzed the non-linear response of single piles under lateral load embedded in
homogeneous heavily overconsolidated clay [23], Under small strain the soil is
considered linear elastic and plastic past the yield level; hence, the stress-strain
characteristics of the soil are modeled as elastic-perfectly plastic. The major soil
parameters for this model are the undrained shear strength and the modulus of
elasticity. The bearing capacity factor (Nc) is used to estimate the bearing stress in
the front face of the pile, the adhesion coefficient is considered for estimating shear
stress along the sides of the pile, while tension stress in the back face of the pile is
also taken into account.
Mindlins (1936) elasticity equations are used as the basis to solve soil deformation.
The pile length is divided into segments and Bemoulli-Euler beam theory is utilized
to obtain pile displacements. The capabilities and limitations of the p-y method were
evaluated. The equations presented in this paper were utilized in an illustrative
example and for result comparison on a case history (Reese et al., 1975), where good
agreement was obtained. This method was considered useful for common practice.
Wang and Reese (1993)
This document presented a soil-structure computer program (COM624P) based on
iterative differential equations to solve the behavior of piles under lateral load,
including the mutual dependency of pile deflection (y) and soil reaction {p) [24], The
nonlinear response of the soil is dependent on both pile deflection and depth of soil,
where the pile is assumed to behave as a beam-column. The soil reaction is obtained
for a considered depth by integrating the increased unit stresses caused by the pile
deflection on that section depth; hence, obtaining an unbalance soil force which acts
opposite to the deflection. Finite difference techniques are used to solve the
differential equations for selected points along the pile.
Four different types of lateral loading are considered: short-term static (field load
test), cyclic (e.g. wind, waves), sustained static (time-dependent), and dynamic
(vibrations). Selected boundary conditions at the top of the pile satisfy the conditions
17


of equilibrium and compatibility. The influence of the pile depth on ground surface
deflection is analyzed under critical penetration. The stratum of soil from the ground
surface to within a few pile diameters in depth is considered of great importance since
it provides the major lateral support for the pile. The soil response is modeled as a
system of discrete springs along the depth of the pile (points). When computing the
ultimate moment and flexural rigidity of the pile it is considered to behave nonlinear
with respect to bending moment.
Soil properties, pile geometry, and type of loading are considered to have the most
effect on a p-y curve. The method of installation is not taken into consideration for
adjustment of the soil properties since beyond the pile wall an area of soil, several
times the pile diameter, is stressed under lateral loading. Recommendations for
computing p-y curves are suggested for clays (soft and stiff) and sands (loose and
dense) above and below the water table, including vuggy limestone, layered soil, and
sloping ground. Field load tests were used as a reference for computing p-y curves;
however, field and laboratory soil tests are recommended.
Computer results along the length of a pile include deflection, rotation, bending
moment, shear, and soil resistance. It is also recommended that the solutions be
verified for accuracy for any computer output.
Ashour et al. (1998)
Considered the use of the strain wedge model to predict the response of a pile under
lateral load in a layered soil profile [25], The pile is assumed a flexible one-
dimensional beam on elastic foundation, while three-dimensional nonlinear soil-pile
interaction parameters are taken into account. Soil behavior is obtained by triaxial
test stress-strain characteristics of undisturbed samples, both sands and clays are
considered. The soil response in front of the pile is represented by a passive three-
dimensional wedge which is eventually mobilized. Initial research of the strain
wedge model was presented by Norris in 1986.
The soil profile is divided into sublayers of constant thickness, where each sublayer
can have its own soil properties. The horizontal subgrade reaction modulus (kh) is
necessary for every layer during the pile loading in order to reflect proper nonlinear
soil-pile interaction. Horizontal strain (s) formed in the passive wedge is estimated
from triaxial test stress-strain relationships, where the major stress change is in the
direction of the pile movement and is considered similar to the triaxial deviatoric
stress. To solve this method, it requires basic soil parameters which can be easily
obtained from standard subsurface investigations and correlations techniques. It is
concluded that this model is based on accepted and known soil mechanics principles.
18


2.4 Deep Foundations under Combined Loads
Trochanis et al. (1991)
Investigated the three-dimensional soil-pile interaction under static and cyclic loading
using finite element analysis (FEA), for both single and group piles [26], The
nonlinear soil behavior was studied under axial, lateral, and combined axial-lateral
pile loads. The influence of slippage and separation at the soil-pile interface are
regarded as key factors in the piles overall response under axial and lateral loads,
respectively. The pile was considered as an elastic material, while the soil (clay or
sand) was modeled as a Drucker-Prager elastic-plastic continuum.
An axial field load test performed on a square 30 centimeter concrete pile in Mexico
Citys soft clay was used for comparison and validation of the finite element model.
Elastic theory results from Poulos and Davis (1980) were also used as a comparison
for purely elastic modeling of the FEA. Within the parametric study only single piles
were investigated under single and combined loads, while pile groups (two piles)
were subjected to either type of load. Pile-soil interface was modeled under two
scenarios, allowing slippage and restricting it (bonded). The effects of pile width and
slenderness ratio (length to diameter) were also studied for lateral load response.
Compared to the field test data, the numerical results of elastic soil models tend to
overestimate the realistic soil-pile interaction; hence, the importance of considering
nonlinear elastic-plastic soil models. It is observed that the axial load capacity may
increase under combined loading due to the increased shear resistance in the leading
face of the pile. However, the lateral load-deflection curve seems unchanged under
combined loading compared to that of lateral load alone.
Phillips and Lehane (2004)
Presented the performance of a full-scale driven concrete pile subjected to combined
loads, axial and lateral, and of a second pile (reaction pile) under lateral load only
[27]. Some interpretation of the field data obtained from the combined load test was
necessary due to a movement restraint in the horizontal direction caused by the axial
loading mechanism. The 35 centimeter square and 10 meter long reinforced concrete
piles were embedded mostly in estuarine clayey-silt while the tips rested on a medium
dense sand layer.
The axial load test was performed 24 hours prior to conducting the combined load
test. The maximum axial load of 168 kN (kilo Newton) was maintained until the end
of the combined load test which had a maximum applied lateral load of about 60 kN.
A second loading scenario was applied to the same piles, where the axial load was
19


reduced to 133 kN and the maximum lateral load applied was about 90 kN. The
horizontal movement of the combined load pile (0.5 cm) was much less than that of
the lateral load (single load) pile (2.5 cm) for the first loading. A similar trend was
observed under the second loading scenario up until the maximum load, where the
displacements are identical.
The axial loading mechanism provided a degree of restraint which reduced the lateral
load applied to the combined load pile. The instrumentation employed allowed to
quantify the restraint and to properly interpret the results obtained from the field tests.
20


3. DESIGN PRINCIPLES OF DRILLED SHAFTS
3.1 Introduction
This chapter is concerned with the current state of practice for the design of single
drilled shafts (groups will not be considered) under static axial or lateral load. As
mentioned in the previous chapter, drilled shafts are used to support a variety of
engineering structures. The loads applied to the drilled shafts due to these structures
are generally axial, acting in the direction of gravity; however, uplift (axial),
horizontal (lateral), bending moment, and torsion loads may also be present.
A structurally sound drilled shaft will function properly only if the supporting soil is
adequate for the loading conditions. An overstressed soil can result in excessive
displacement of the soil which can cause damage to the structure being supported.
Therefore, when designing drilled shafts, it is important to analyze them as a
combined soil-structure system. In addition to the effects of dead and live loads,
drilled shafts should be designed to withstand the worst conditions expected
throughout the lifetime of the structure.
Drilled shafts can be designed from full-scale field load test results, empirical
methods, analytical techniques, and numerical methods. It is recommended to
calibrate one design method against another. Field tests provide load capacity and
displacement data, which are reliable for similar subsurface conditions within the site.
Empirical methods relate new design calculations with those of existing field load-
displacement databanks for similar soil conditions. Analytical techniques are based
on soil mechanics principles and require information of the subsurface condition, soil
properties, and the drilled shaft. Numerical methods have an advantage over
empirical and analytical methods by taking into account the continuity of the soil
mass.
Two of the most commonly used analytical techniques are the allowable stress
method and the limit state method [28], The allowable stress method ensures that the
drilled shaft load is transmitted to the supporting stratum without causing shear
failure to the soil. This is achieved by applying safety factors to the maximum load
that would cause failure in the bearing soil. The limit state method focuses on
ultimate loads, which result in excessive displacement, causing collapse or disruption
of the functionality of the structure.
21


Structurally, drilled shafts are considered long and slender column-type elements that
may buckle under axial load. However, drilled shafts have enough lateral support
along their length from the embedding soil, so typically there is no concern about
buckling. The following analytical methods will assume that the drilled shafts have
an adequate structural design to carry the design axial and lateral loads.
3.2 Drilled Shafts under Axial Load
The total ultimate axial downward resistance (Axjai) of a drilled shaft is based mainly
on the soil properties, dimensions of the drilled shaft, location of the water table, and
construction method. The total axial load that can be supported by the drilled shaft is
provided by the sum of the shaft wall (side friction or adhesion) resistance (Ahaft) and
the base (end bearing) resistance (Aase), as shown in Figure 3.1 and described below:
(Axial C^shaft (Aase (3.1)
Ahaft =/'As = f-p-L = f (71 A) L (3.2)
A>ase ~ tfb Ab A' A> 4) (3.3)
Where,
/= average unit area skin friction or adhesion between soil and drilled shaft
surface, or shearing strength of the soil zone immediately adjacent to the
drilled shaft surface
As = drilled shaft surface in contact with soil along the embedded shaft length
p = perimeter of the drilled shaft
L = embedded length of the drilled shaft (below ground surface), L > 3 A
A = diameter of the drilled shaft (straight segment)
Ab = bearing area of the base of the drilled shaft
A = diameter of the base of the drilled shaft, A = A for straight drilled
shafts, A < 3-A
22


axial
*
z'A/7
t L
r
Figure 3.1 Axial load resistance of a drilled shaft
If the properties or conditions of the soil change along the depth of the drilled shaft
(length of embedment), the shaft resistance (£9Shaft) should be calculated by dividing
the shaft length into layers (segments) [9], as shown in Figure 3.2 and expressed by:
C?shaft=X f'PAL\ (3.4)
i=i
Where,
f = unit side shearing resistance in layer i
ALt = length of layer i
For each layer, average properties should be used acting at the middle of the layer.
23


£?axial
I
Figure 3.2 Axial resistance of a drilled shaft by layers (from Das, 2005)
The end bearing resistance ( failure of the supporting soil immediately below and adjacent to the base of the
drilled shaft. The ultimate bearing capacity ( resistance (qu\t ~ q\>), is expressed by Terzaghis (1943) general equation for circular
shallow foundations:
quh=l.3-c-Nc + y'z-L-Nq + 0.3 Dh y\ Ny (3.5)
Where,
c = average cohesion of the soil within two shaft diameter below the base of
the drilled shaft
L = depth of the base of the drilled shaft below ground surface
Db = diameter of the base of the drilled shaft
y'z = effective unit weight of soil above the base of the drilled shaft
y 'b = effective unit weight of soil below the base of the drilled shaft
24


Nc, Ny, Aq = soil bearing capacity factors that depend on the value of the angle
of internal friction, , as shown in Figure 3.3
Angle ol internal friction,
0 Nc
0 5.14
5 6.5
10 8.3
15 140
20 14.8
25 20.7
30 30.1
32 35.5
34 42.2
36 50.6
38 61.4
40 75.3
42 93.7
44 118.4
46 152.1
48 199.3
50 266.9
Nq Ny
1.0 0.0
1.6 0.5
2.5 1.2
3.9 2.6
64 5.4
10.7 10.8
18.4 22.4
23.2 30.2
29.4 41.1
37.7 56.3
48.9 78.0
64.2 109.4
85.4 155.6
115.3 224.6
158.5 330.4
222.3 496.0
319.1 762.9
Figure 3.3 Bearing capacity factors (from McCarthy, 2002)
The first term of the equation represents the contribution of the shear strength of the
soil, the second term represents the effect of the surcharge pressure, and the third term
represents the bearing resistance resulting from the weight of the soil. The weight of
the soiled removed (drilled out) is usually assumed to be equal to the weight of the
25


drilled shaft (concrete and reinforcement). The following expressions are used to
determine the bearing capacity factors [29]:
For For 0, Ac = (Aq 1) co\(j) (3.6)
Ar = 2 (Aq + 1) tan^ (3.7)
Aq = tan2f45 + 1 (en tan*) V (3.8)
Where,
e = 2.7183, base of natural logarithms
With the exception of relatively short drilled shafts, the term containing Ay is
negligible since the base diameter is generally small compared to the length of the
drilled shaft (L > 3-Db), especially for straight drilled shafts. Therefore, Equation 3.5
can be reduced to the following expression:
qb=\.3-c-Nc + r'z-L-Nt (3.9)
The maximum axial load ((9axiai) represents the load that can be applied to a drilled
shaft before failure. A factor of safety (FS) is applied to ensure that the design or
allowable load is less than the value which would cause failure, such that:
0all=%L (3.10)
FS
Information from full-scale axial load tests on drilled shafts show that with gradual
loading, initially, the load is supported by the side resistance starting at the top and
progressively moving downward. Therefore, as additional load is applied, a greater
portion is supported by the base resistance. Before the ultimate load is applied to the
drilled shaft, a serviceability limit (settlement) may be attained.
There are three generally used methods to construct a drilled shaft: dry, wet, and
casing [30], The dry method is used for soils that will not cave into the excavation
26


and that are above the water table. The wet method uses drilling slurry (usually
bentonite mix) to fill the borehole and keep it open; the slurry is later displaced by the
fluid concrete. The casing method is used for caving soils above the water table; the
casing may be left in place or removed at the end of construction before the concrete
sets.
3.2.1 Drilled Shafts in Cohesive Soil
The total ultimate axial downward resistance {Qax\a\) for clays and plastic silts is
limited by settlement or shear failure considerations. For saturated cohesive soils
with low permeability subjected to a rapidly applied load, the resistance should be
calculated using the undrained shear strength (cu) of the undisturbed clay (0=0, total
stress). For effective stress analysis the shear resistance should be calculated using
the effective friction angle (0').
3.2.1.1 Shaft Resistance in Cohesive Soil
The shaft resistance mobilizes at small displacement; it takes only a few millimeters
of displacement to reach its ultimate resistance. Failure occurs by sliding of the
drilled shaft compared to its position with the soil contact before loading (slip plane
offset).
The average unit area skin friction or adhesion if) develops between the soil and the
drilled shaft surface. For saturated undrained conditions (0= 0), the adhesion
resistance can be obtained by the following expression:
f=a-cu (3.11)
Where,
/= unit adhesion resistance for undrained conditions
cu = cohesive undrained shear strength of undisturbed clay
a = adhesion coefficient that relates unit adhesion resistance to cohesion
The above equation is also known as the alpha or total stress method. Adequate
values of a are obtained from field load test results. Therefore, the adhesion
coefficient will depend on the type of clay encountered and the method of
constructing the drilled shaft (dry, wet, and casing).
27


The following values are recommended for a:
a = 0.3 to 1.0, for cu < 250 kPa
a = 0, for the top 1.5 m and bottom one diameter (Ds) of the drilled shaft; for
bell-shaped drilled shafts a = 0 from the base to one diameter above
the top of the bell
Due to seasonal moisture change (shrinkage), the top 1.5 m of soil below the ground
surface is omitted from side resistance calculations. The bottom one diameter is
excluded due to uncertain soil disturbance from the construction process. For bell-
shaped drilled shafts there is a likelihood that a gap will develop between the soil and
the surface of the bell due to settlement.
Several investigators have reported calculated a values obtained from instrumented
field load tests [31], shown in Figure 3.4.
(kPa)
0 50 100 150 200
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Undrained Shear Strength, su (lb/ft2)
Figure 3.4 Measured a values from field tests compared with
empirical functions (from Coduto, 2001)
28


Kulhawy and Jackson (1989) reported a best fit correlation for abased on the field
test results of 106 drilled shafts [8]:
a= 0.21 +0.25
vs
< 1
(3-12)
Where,
pa = atmospheric pressure = 101.3 kPa
From Stewart and Kulhawy (1981), a values can be related to the plasticity index (PI) of a cohesive soil for dry construction with 15 < PI < 80: For normally consolidated:
a =0.9-(0.004 PI) For slightly over-consolidated (OCR < 2): (3.13)
a =0.9-(0.01 PI) For over-consolidated: (3.14)
a =0.7-(0.01 PI) and, (3.15)
o i ii (3.16)
OCR (CFV. max) (Ov. cur) (3.17)
Where,
LL = liquid limit
PL = plastic limit
OCR = overconsolidated ratio
crv max = maximum vertical stress in its geological history
crv, cur= current vertical stress
For wet construction, the a values obtained from Equations 3.13 to 3.15 should be
reduced by one-half to two-thirds.
29


Construction techniques will have an effect on the shafts interface between the
concrete and the soil. Under normal construction conditions, the side shearing
resistance along the length of the shaft will typically develop in a soil zone near, but
not at, the interface. During construction of a drilled shaft a thin wall of clay
immediately adjacent to the shaft (approximately 2.5 centimeters) will be remolded
during boring.
When fissured clay is encountered in the subsurface, caution should be performed
when obtaining the shear strength of a small specimen in the laboratory. If a sample
obtained from the field does not contain fissures, this would result in higher shear
strength than that of the in-situ clay. Also, fissures may cause a higher effect in a
laboratory sample than in the in-situ clay as a whole.
The undrained shear strength can be reduced up to one-third in situations where the
clay has been softened or remolded and therefore, a reduction in the side resistance.
Softening may take place due to pore water seeping from the surrounding clay
towards the shaft or if the clay absorbs water from the wet concrete.
The in-situ soil along the wall of the shaft will develop lateral stress relief due to
excavation and lateral expansion of the soil. The surface of the bored soil shaft is
usually rough and undulating, and as such, it is believed to act as a drainage path.
Therefore, the average unit area skin friction if) can also be expressed in terms of
effective stress for drained conditions (c ~ 0), as follows:
/= cr'h tan (/> = Ksw cr'v tan (/>' (3.18)
cr\ = y' z (3.19)
Ksw tan <)>' = P (3.20)
with, KSVi < K0
Where,
/= unit skin friction resistance for drained conditions
cr'h = effective horizontal stress
(j)' = effective friction angle of the clayey soil
Ksw = coefficient of earth pressure normal to the shaft wall
cr'y = effective vertical stress
y' = effective unit weight of the overburden soil
z = depth below the ground surface
/? = coefficient that relates earth pressure to effective friction angle
30


K0 = coefficient of earth pressure at rest
And the following expressions are considered for an infinitesimal in-situ sample:
For normally consolidated soils,
K0 = 1 sin (/>'
(3.21)
For over-consolidated soils,
K0 ~ (1 sin (f>') (OCR)0 8 sin^' (3.22)
This alternative procedure is also known as the effective stress or beta method. Since
failure is assumed to take place in the remolded soil near the drilled shaft surface, the
cohesion intercept for the remolded clay is considered to be zero. Values of (3 for
normally consolidated clays typically range between 0.25 and 0.40, but values for
overconsolidated clays are significantly higher.
3.2.1.2 Base Resistance in Cohesive Soil
The end bearing resistance requires a larger deformation for full mobilization than the
shaft resistance; therefore, the base can undergo several millimeters of displacement
before its ultimate resistance is reached. Failure usually occurs by punching shear of
the soil below the base of the drilled shaft.
The end bearing resistance of the soil at the base of the drilled shaft (qb) for saturated
undrained conditions { = 0, 7Vq = 1), can be expressed for all L > 3 Db as follows:
qb=cu-Nc (3.23)
Where,
cu = cohesive undrained shear strength
Nc = bearing capacity factor
From ONeill and Reese (1999) [32]:
Nc = 9.0, for 96 kPa < cu < 250 kPa
Nc = 8.0, for cu = 48 kPa
31


Nc = 6.5, for cu = 24 kPa
For intermediate values perform linear interpolation. The bearing capacity factor Nc
will depend on the average undrained shear strength (cu) of the soil within two
diameters of the base of the drilled shaft.
3.2.2 Drilled Shafts in Cohesionless Soil
For granular soils (sands and gravel) and non-plastic silts, the total ultimate axial
downward resistance (OaXiai) is limited only by settlement considerations. Due to the
free draining nature of granular soils the shaft and base resistance should be
calculated using an effective stress analysis {(/>').
3.2.2.1 Shaft Resistance in Cohesionless Soil
The average unit area skin friction if) along the length of the drilled shaft for drained
conditions (Figure 3.5), assuming c = 0, can be expressed by:
f- cr'h tan 5= Ksv/ cr 'v tan £ (3.24)
P = A'sw tan S (3.25)
Where,
/= unit skin friction resistance
cr'h = effective horizontal stress
5= friction angle between the drilled shaft and the cohesionless soil
KSVi = coefficient of earth pressure normal to the shaft wall
cr'v = effective vertical stress
P = coefficient that relates earth pressure to interface friction angle
The following values are recommended for S:
for dry construction, 0.8(/)'< 8 < '
for wet construction, 0.5- 32


Similar to cohesive soils, the shaft resistance values for S are obtained from field
load test results. Drilling slurry from wet construction may remain on the shaft wall
forming a film plug (cake) and thus reducing the frictional resistance of the shaft.
Direction of
movement due
to axial load
Figure 3.5 Frictional shaft resistance of a drilled shaft in
cohesionless soil (from McCarthy, 2002)
Based on uncorrected blow count (N6o) data from the standard penetration test (SPT),
the following f3 values are recommended:
For 15 < N^o < 50 ,
Sands:
(3= 1.5 -(0.245 -z05) (3.26)
with. 0.25 < fi < 1.2 /< 200 kPa
Gravel:
(3= 2.0-(0.15 -z075) (3.27)
with. 0.25 < f3< 1.8
For N60 < 15 ,
Sands and gravels:
P={N60/15) [1.5 (0.245 z05)] (3.28)
33


Unlike cohesive soils, there are no exclusion zones for determining side resistance in
cohesionless soils. Information from field load tests in cohesionless soils indicates
that the effective vertical stress increases linearly with depth to a limit called the
critical depth (Lc). Below this critical depth the effective vertical stress remains
basically constant or increases at a very low rate as shown in Figure 3.6. Consider the
following limits for critical depth:
(loose) 15-DS < Lc < 20-Ds * (dense)
With increasingly higher values of the effective friction angle, the location of the
critical depth will be deeper below the ground surface.
Figure 3.6 Shaft horizontal stresses and critical depth in cohesionless soils
3.2.2.2 Base Resistance in Cohesionless Soil
The end bearing resistance of a cohesionless soil at the base of the drilled shaft (qb)
for drained conditions, assuming c = 0, can be expressed by:
qb = r'z'L-Nq (3.29)
The average drained shear strength (') of the soil within two diameters of the base of
the drilled shaft will be used to determine the bearing capacity factor Nq, and Nc if
cohesion is considered. Shear failure at the base of a drilled shaft in cohesionless soil
34


will only occur in very loose material, serviceability or possible differential
settlement is the ruling design criteria.
Based on N6o values averaged within two diameters of the base, the end bearing
resistance can be obtained by:
3.2.3 Drilled Shafts in Intermediate Geomaterial
Intermediate geomaterial (IGM) is a relatively new classification for earth materials,
having shear strengths that range between the upper limit of a soil and the lower limit
of a rock. They can also be considered as a transitional material between, and
including, a heavily over-consolidated or very stiff soil, and a soft or weathered rock.
Limits used to define an IGM based on cu and N6o are:
For cohesive IGM,
250 kPa < cu < 2.5 MPa
For cohesionless IGM,
N60 > 50
For IGM and soils, a hybrid method can be used to obtain the total ultimate axial
downward resistance ((9axiaj) of a drilled shaft; such that the side resistance is obtained
using an effective stress method and the base resistance is calculated using a total
stress method.
3.2.3.1 Shaft Resistance in Intermediate Geomaterial
The soil shaft wall will be assumed to be relatively smooth, such that the borehole
wall is roughened naturally by the drilling process without leaving smeared soil
(cuttings). For cohesive IGM the unit side resistance can be obtained by:
For Ngo < 50 ,
qb 57.5 N6o
with qb in kPa, and qb < 2.9 MPa
(3.30)
(3.31)
35


Where,
ajgm = factor obtained from Figure 3.7, not equal to that of cohesive soil
(p\gm = factor that accounts for the presence of open joints
qu = unconfined compressive strength of IGM
Figure 3.7 was developed with the following assumptions: the interface friction angle
($m) is 30 and that a vertical displacement of 2.5 cm is required before the full side
resistance can be reached [32], The IGM elastic modulus (E\gm) is obtained from
undisturbed samples. If the interface friction is considered to be different than 30,
then a should be modified as follows:
igm = [ffigm (from figure)] [tan $t (from layer) / tan 30] (3.32)
qu (MPa)
Figure 3.7 Values of jgm as a function of unconfmed compressive
strength (from ONeill and Reese, 1999)
The pressure imparted by the fluid concrete at the middle of an IGM layer is:
on = Fc-yc-z (3.33)
Where,
36


Fc = empirical factor which depends on the concrete slump (Figure 3.8)
yc = unit weight of the fluid concrete
The value of crn is considered constant after a depth (z) of 12 meters. An empirical
factor of 0.65 in the above equation can be used for a concrete slump of 17.5
centimeters and a depth of 12 meters or greater.
Slump (mm)
Figure 3.8 Values of Fc as a function of concrete slump
and depth (from ONeill et al., 1996)
The factor ^,gm for closed joints is equal to 1.0, 0.85, and 0.6 for rock quality
designations (RQD) of 100, 70, and 50 percent, respectively. A complete set of
values for closed and opened joints can be obtained from ONeill and Reese (1999).
The unit side resistance for cohesionless IGM, with 50 < N6o < 100, can be obtained
by Equation 3.18, and for convenience is included here:
/= cr'h tan = Ksw ' (3.18)
37


3.2.3.2 Base Resistance in Intermediate Geomaterial
For cohesive IGM, with RQD = 100 percent, the base resistance can be obtained by:
qh = 2.5 qu
(3.34)
Where,
qu = averaged unconfined compressive strength of IGM within two diameters
of the base
For cohesive IGM, with closed joints and 70 < RQD < 100, the base resistance is
expressed as:
gb = 4.83 ( qu)51 (3.35)
where, qn> 0.5 MPa
From Mayne and Harris (1993), the base resistance for cohesionless IGM soils with
Ngo > 50, is calculated by:
qb = 0.59 [ N60 ]8 cr'v (3.36)
If the supporting material immediately below the base of a drilled shaft is an IGM or
stiff soil and the soil along the length of the shaft is relatively weak, then the total
axial load resistance should be based solely on the base resistance.
3.2.4 Settlement of Drilled Shafts
The axial load applied to the top of a drilled shaft will result in compressive and shear
stresses developing in the supporting soil mass. The new stress conditions in the soil
due to the axial load may result in soil compression (decrease void ratio) and volume
distortion (increase shear strain). These effects will cause settlement (vertical
displacement) of the drilled shaft foundation. Settlement may also be caused by
secondary conditions that are particular to a site such as consolidation and creep.
The total axial settlement for a drilled shaft (St) based on the semi-empirical method
by Vesic (1977) [33] is calculated as follows:
St Sds + Sb+ -Ss
(3.37)
38


Where,
Sds = settlement of drilled shaft due to shaft axial compression
St, = settlement of the base due to transmitted load
5S = settlement of the base due to transmitted load along the shaft side wall
The settlement due to axial compression of the drilled shaft can be calculated by
assuming that the drilled shaft behaves elastically, such that:
$*=(&,+ Q,)-1V (3-38)
^ds ^ds
Where,
Qbi = load transmitted by the drilled shaft to the base
OsW = factor related to the distribution of the load along the shaft wall
Qs{ = load transmitted by the drilled shaft to the side wall
^4ds = cross section area of the drilled shaft
£ds = modulus of elasticity of the composite drilled shaft material (concrete
and steel rebar)
Settlement due to axial compression should be calculated assuming that Qsi QShafi,
since the side resistance will be fully mobilized before the base resistance. The load
distribution factor ow will depend on the distribution of the unit side resistance along
the shaft wall. For normal distribution the value is usually assumed to be 0.5, and up
to 0.67 for certain extreme cases.
The settlement of the drilled shaft due to the transferred axial load to the base is:
=
cb gb,
Db- (3.39)
Where,
Cb = empirical base coefficient dependant on soil type (Table 3.1)
The transmitted load along the shaft side wall will cause settlement of the base as
follows:
5S
Cs Qs
L-qb
(3.40)
39


Cs = [0.93 + 0.16(1/A)05]
(3.41)
Where,
Cs = empirical shaft coefficient
Table 3.1 Empirical base coefficients for settlement
Soil cb
Clay (stiff to soft) 0.03 to 0.06
Silt (dense to loose) 0.09 to 0.12
Sand (dense to loose) 0.09 to 0.18
These coefficients are adequate for long-term settlement, where the end bearing layer
extends ten diameters (10-A) below the base and the soil stiffness is constant or
increases with depth. If a stratum (layer) of higher stiffness (IGM or rock) is
encountered at less than ten diameters, the coefficient values are reduced slightly. If
the stiffer layer is at five diameters below the base, A is reduced to 88 percent of the
value obtained from Equation 3.39 or if it is at one diameter below, it is reduced to 51
percent (0.51 £),)
3.3 Drilled Shafts under Lateral Load
Drilled shafts frequently support lateral loads (Q\at) in addition to axial loads (Axial)-
The structural integrity and serviceability (displacement) are the most important
factors for the design of lateral loads. The lateral deflection (y) of a drilled shaft for a
given lateral load will depend on the stiffness of the soil and the drilled shaft
(reinforced concrete), type of boundary condition (free or fixed) at the top of the
drilled shaft, and length of the drilled shaft (Figure 3.9).
3.3.1 Soil-Structure Interaction
The pressure transmitted by the drilled shaft along its length, due to a lateral load, is
not uniform and will mobilize passive pressure in the supporting soil. The soil
surrounding the shaft undergoes lateral deformation when the pressure transmitted is
40


greater than the resistance. The soil reaction will be acting in the direction opposing
the deflection of the drilled shaft (Figure 3.10).
Q\a.t
Figure 3.9 Behavior of a drilled shaft under lateral load (from Broms,
1964a. and Matlock and Reese, 1960, respectively)
Figure 3.10 Soil reaction due to lateral load, profile and
plan section view (from Coduto, 2001)
41


There is an interdependent correlation between drilled shaft deflection (y) and the soil
reaction {p) as shown in Figure 3.11, and also known asp-y curves. This soil-
structure interaction behavior must be in equilibrium for the design methods of lateral
loads.
Figure 3.11 Typical soil resistance as a function of lateral deflection
for two types of soil stiffness (from Coduto, 2001)
The top of the drilled shaft will be considered at the ground surface elevation, unless
expressed otherwise. When the lateral load is applied at some distance above the
ground surface (e), it will generate shear force (V) and bending moment (M) at the
ground surface level. These loads can cause deflection (y) and tilt (S, slope) of the
drilled shaft.
All of these factors are interdependent with the soil reaction (p) and the drilled shafts
flexural stiffness (-EdsTds). Figure 3.12 illustrates the variation of these factors with
depth which must be considered for the structural design of a drilled shaft and Figure
3.13 shows the sign conventions.
42


Loading Deflection Slope
? s
Moment Shear Soil Reaction
M V P
Figure 3.12 Complete soil-structure results for a typical lateral
load design (from Matlock and Reese, 1960)
Figure 3.13 Soil-structure sign conventions (from Matlock and Reese, 1960)
43


Based on principles of mechanics of materials and from basic beam-column concepts,
the following relations apply to a drilled shaft (beam-column) under shear lateral
force and moment, which will vary with depth as shown above in Figure 3.12:
5
dy
dz
i # i-. , d2y dS
M Ei% 1 ds 2 Eds 7ds
dz dz
v = ea-i.
d3y dM
ds ^ ds 3 j
dz dz
n-E /
P ^ds ds t 4 ,
dz dz
(3.42)
(3.43)
(3.44)
(3.45)
Where,
£ds = modulus of elasticity of the drilled shaft (reinforced-concrete)
/ds = moment of inertia of the drilled shaft
£dsfds = flexural stiffness of the drilled shaft
z = depth from ground surface
3.3.2 Boundary Conditions
The type of boundary condition at the ground surface and length of a drilled shaft
play an important role in the behavior of a laterally loaded drilled shaft. The top
boundary is the connection between the engineering structure above ground surface
and the drilled shaft. There are two basic types of boundary conditions: free and
fixed.
Free boundary condition will allow the top of the drilled shaft to move freely under
applied shear and moment loads, as shown in Figure 3.14a. For the case when both V
and Mare applied, the unknown values at the ground surface will be the S andy.
Fixed boundary condition will restrict the top of the drilled shaft from rotating, but it
is allowed to move laterally, as shown in Figure 3.14b. A foundation slab or group
cap is representative of this type of boundary condition. For this case V and S are
known at the ground surface, but not M and y.
44


Q\a
Q\at
H r7
/ /
f <"j v&znwp w i i V
/ ! / / t
a)
ri
/ /
/ /
/ i
/
/
/
b)
Figure 3.14 Top boundary conditions: a) free, and
b) fixed (from Broms, 1964a)
For design purposes, the above boundary conditions are adequate; however, drilled
shafts will actually have top boundaries that are somewhere in between that of free
and fixed conditions [31]. When only Mis applied to the top of a drilled shaft, which
is not common, there is no deflection (y = 0) at the ground surface; for this scenario,
all the parameters (V, M S, and_y) are known at the ground surface.
Drilled shafts will transfer most of the load within eight diameters of the ground
surface; hence, the importance of considering environmental effects, such as seasonal
moisture changes and scour.
3.3.3 Methods for Lateral Load Design
As mentioned in the beginning of this chapter, there are several methods used for
designing drilled shafts: field load test, empirical, analytical, and numerical methods.
The design methods need to determine the minimum diameter, steel reinforcement,
and concrete strength necessary to resist the shear force and bending moment that will
be imposed on the drilled shaft; including the depth of embedment necessary to safely
transfer the loads to the supporting soil without excessive displacement. When a
significant axial load is applied to the top of a drilled shaft, it must be considered for
the design.
Most analytical methods for lateral load design can be linked to three main
categories: 1 2 3
1. Ultimate rigid resistance
2. Subgrade reaction approach
3. Elastic continuum theory
45


3.3.3.1 Ultimate Rigid Resistance
Several researchers have used the ultimate resistance of the soil and drilled shaft to
calculate the maximum lateral load that can be applied to the top of a drilled shaft.
This approach assumes that the drilled shaft is perfectly rigid and disregards any
flexural bending of the drilled shaft.
For length considerations, a drilled shaft can be classified as short or long for free
boundary condition (Figure 3.15) and as short, intermediate, and long for fixed
conditions (Figure 3.16). Contrary to a long drilled shaft, a short drilled shaft does
not have sufficient embedment to fix the base against rotation. The length
classification is based on the stiffness of the drilled shaft and the soil, some
approximate limits will be provided at the end of Section 3.3.3.2.2.
Figure 3.15 Free boundary condition for: a) long, and
b) short drilled shaft (from Broms, 1964a)
The maximum lateral load that can be applied to a short drilled shaft will be
controlled by the soil resistance, since the soils ultimate resistance will be obtained
before the maximum bending capacity of the drilled shaft. For long drilled shafts the
maximum lateral load is controlled by the bending capacity of the drilled shaft, since
it will be reached before the ultimate resistance of the soil; and V, M, S, and y at the
base are zero. A plastic hinge will develop at the location of the yield bending
moment, My, when the flexural stiffness is exceeded.
46


£?lat Q\zS. Q\a\
Figure 3.16 Fixed boundary condition for: a) long, b) intermediate,
and c) short drilled shaft (from Broms, 1964a)
Broms developed a method (1964 1965) that evaluates the ultimate lateral
resistance within the limits of serviceability and was originally created for use in the
design of driven piles [14, 15, and 16]. The method simplifies the soil resistance
along the length of the pile from which the ultimate lateral load, Q\at, and maximum
bending moment, Mmax, are obtained. The solution process is based on the type of
soil, boundary condition, and length of pile. This method calculates the lateral
deflections based on the subgrade reaction approach.
3.3.3.2 Subgrade Reaction Approach
The concept of subgrade reaction was first introduced by Winkler in 1867 through his
research of a classical horizontal beam on an elastic foundation. The horizontal beam
(footing) is considered to rest on top of a bed of elastic springs (foundation soil). The
soil response (reaction, p) for a given load will depend on the deflection of the beam
iy) and the modulus of the soil, as follows:
p = ks-y (3.46)
Where,
ks = soil reaction modulus
47


The soil reaction modulus is considered constant throughout the foundation soil.
Based on the basic beam-column concepts noted in section 3.3.1, and rewriting
Equation 3.45 for convenience:
(3.45)
From Equations 3.45 and 3.46, the basic differential equation is obtained:
(3.47)
The classical closed form solution of this equation accounts for the flexural stiffness
of the beam; it is a non-rigid soil-structure interaction analysis. Subgrade reaction
theory does not account for continuity of the soil mass.
In 1946, Hetenyi presented his theory of a beam on elastic foundation applied to civil
engineering with closed form solutions [34], The published work is based on the
assumption that the deflections at any section along the beam will create proportional
soil reactions. The concept of a vertical beam-column embedded in the foundation
soil and loads applied at different points along the beam was evaluated. From this, a
beam-column with applied axial and lateral loads results in the following differential
equation:
kh = coefficient of horizontal subgrade reaction
From Rankines theory (1857), the passive earth pressure will increase with depth.
For a deep foundation under lateral load, using the constant soil modulus, k$, is
inadequate and it is therefore replaced with kh which increases proportional to depth.
Terzaghi, in 1955 presented the factors that influence the values of the coefficients of
subgrade reaction in sands and stiff clays, and some numerical values were
recommended [35].
(3.48)
Where,
pp = y' -z- Kp
(3.49)
48


(3.50)
1 + sin^
Kn =
p 1-sin^
tan

45+ -
2)
Ka Where,
pp = lateral mobilized passive earth pressure
Kp = coefficient of passive lateral earth pressure
Ka = coefficient of active lateral earth pressure
In the late 1950s, several researchers began to design methods for laterally loaded
driven piles centered on offshore oil platforms. The results from instrumented full-
scale lateral load tests from this work were used to develop a new empirical method
based on the observed behavior between the soil reaction (p) and the deflection (y) of
the pile, called the p-y method. The beam-column differential equation (Equation
3.48) is used to solve for different values of pile and soil stiffness along the length.
The characteristic load method developed by Duncan et al. in 1994 is a simple
approximate method for the analysis of drilled shafts under lateral load [36], Based
on the results of nonlinear p-y analyses, this method presents the solutions in the form
of correlations among dimensionless variables.
An alternative method is the strain wedge model, which considers the failure of a
deep foundation by overcoming the passive resistance of a wedge of soil (Norris
1986). In addition to basic soil properties, this design method requires the coefficient
of horizontal subgrade reaction for every layer in the model to properly reflect the
nonlinear soil-structure interaction [25],
3.3.3.2.1 P-Y Method
This two-dimensional method divides a drilled shaft into several linear segments
connected by a node, and the soil along the shaft is modeled as a series of discrete
independent nonlinear springs connected at the nodes (Figure 3.17). The beam-
column differential equation is formulated in finite difference terms at each node
along the drilled shaft.
The p-y relationship for a given soil is obtained from data acquired directly from a
laterally loaded instrumented field test. The p-y curve (Figure 3.11) for a particular
soil will be adequate only for similar type of loading (static, dynamic, duration) and
49


drilled shaft properties such as stiffness, cross-section shape and dimension, length,
and boundary condition (free or fixed). The interdependency of the soil reaction and
deflection requires iterative techniques to solve for V, M, S, and y parameters.
Computers make solving the fourth-order differential beam-column equation easier.
Numerical methods are the preferred alternatives for complex designs.
M
--->Y
Nonlinear
Springs
Nodes -
I
VWI
AAA/i
*wvI
Figure 3.17 Drilled shaft model for p-y method
The p-y curves account for the soil resistance in the front and back side of the drilled
shaft and for side shear resistance (Figure 3.10), unlike other approaches which only
account for the front side resistance in direction opposite to the lateral load.
3.3.3.2.2 Characteristic Load Method
This method is based on the results of a series of parametric analyses performed on
piles and drilled shafts with free and fixed top boundary conditions in clay and sand
using the nonlinear p-y method (Evans and Duncan 1982); therefore, the results of the
characteristic load method approximate those attained from p-y analyses [36],
50


This simple method can be used to calculate ground surface deflections due to lateral
load or moment applied at the ground surface for free, fixed, and flagpole (partial
embedment) top conditions; as well as determine the value of the maximum moment
and its location in the drilled shaft.
The dimensionless correlations are obtained by dividing the lateral load by a
characteristic (normalized) load, Qc, and by dividing the bending moment by a
characteristic moment, Mc, given by:
For clay,
Qc = 7.34-ZV (£*/?,)
f \ 0.68
V ^ds ^1 J
Mc = 3.86-ZY (E&Ri)
/ \ 0.46
c
V ^dv ^1 J
For sand,
Qc=\.57-Ds2(Eds-Rl)
'r-D.-P-K,'
, £Js ,
0.57

Mz = 1.33-Ds (Eds-R,)

r Ds- 0' k,
V -^ds ^1 J
0.40
(3.51)
(3.52)
(3.53)
(3.54)
Where,
£ds = modulus of elasticity of drilled shaft
Ri = moment of inertia ratio, ratio of drilled shaft section to that of a solid
circular section, /ds/7circuiar- (R\= 1, for uncracked drilled shaft)
As mentioned previously, a drilled shaft will transmit most of the lateral load within
eight diameters of the ground surface; therefore, the soil properties should be
averaged within this range and used in the above equations. If the cross-section of a
drilled shaft cracks the moment of inertia of the concrete portion reduces by about 40
to 50 percent, while the steel reinforcement moment of inertia remains unchanged.
When a lateral load is applied at the ground surface (Qg) without moment, the
deflection of a drilled shaft at ground surface (yg) can be obtained from Figure 3.18
51


based on the soil type and top boundary condition. When only a moment is applied at
the ground surface (Mg) the deflection yg can be obtained from Figure 3.19.
0.045
Oi
0.030
03
^ 0.015
-a
c3
o
0.000
0.00 0.05 0.10 0.15
Deflection Ratio, yg/Z)s
0.015
o
Oi
0.010
O)
"3
& 0.005
T3
03
O
0.000
0.00 0.05 0.10 0.15
Deflection Ratio, yg /Ds
Figure 3.18 Load-Deflection curves: a) clay, and
b) sand (from Duncan et al., 1994)
Figure 3.19 Moment-Deflection curves: a) clay, and
b) sand (from Duncan et al., 1994)
However, when both a lateral shear and a moment exist at the ground surface due to a
lateral load applied some distance above the ground surface (e) as shown in Figure
52


3.20, a nonlinear superposition procedure is required to obtain the deflection as
follows:
1. Calculate the deflection due only to a lateral load (ygq) from Figure 3.18,
and due only to a moment (ygm) from Figure 3.19.
2. From the Figure 3.18 obtain a value of Qg that results in the same
deflection asygm, symbolized by Qgm; and from Figure 3.19, a value of M%
that results in the same deflection as _ygq, denoted as Mm.
3. Calculate the ground surface deflection due to Q% + Q%m, represented by
_ygqm; and that due to Mg + Mgq, denoted byygmq.
4. Finally, calculate the total ground surface displacement using the
following expression:
Tg.total 0.5 (ygcjm ^ygmq) (3.55)
Figure 3.20 Lateral load applied above ground surface
(from Duncan et al., 1994)
The maximum moment due only to an applied lateral load at the ground surface can
be obtained from Figure 3.21. For a fixed top boundary, the maximum moment is
located at the ground surface.
53


0.000 0.005 0.010 0.015 0.000 0.005 0.010 0.015
Moment Ratio, Mg /Mc Moment Ratio, Mg /Mc
Figure 3.21 Load-Moment curves: a) clay, and
b) sand (from Duncan et al., 1994)
When both a lateral shear and a moment are applied at the ground surface, the
maximum moment (Mmax) will be located at some depth below the ground surface.
The value and location of the maximum moment are obtained as follows:
1. Calculate _yg,totai as described above, then determine the characteristic
length (Lj) from the following equation by trial and error:
Tg.total
2-43 Q r3 1.62-Mg ,2
-------1L, +---------L L,
E l E l
^ds ''ds ^ds -'ds
(3.56)
2. The bending moment of the drilled shaft at a depth z below the ground
surface can be obtained by:
Mz = (Am-Qg-LT) + (Bm-Mg) (3.57)
Where,
Am, Bm = dimensionless moment coefficients obtained from Figure 3.22
From Equation 3.57 and Figure 3.22, the maximum moment due to the ground
surface lateral shear load will be at 2 = 13-Lj, based on the maximum value of Am;
and the maximum moment due to the ground surface bending moment will occur at z
= 0, where the value of Bm is a maximum. When both shear and moment are applied
at the ground surface, the location of the maximum moment will vary between the
54


limits of 0 < z < 1.3-It. The maximum moment value is obtained by trial using
different values of z in Equation 3.57.
Am or Bm
Figure 3.22 Coefficients Am and Bm (from Matlock and Reese, 1961)
The main limitation of the characteristic load method is the length of embedment of
the drilled shaft; such that, it is necessary for the drilled shaft to be long enough to
avoid lateral displacement of the base. Table 3.2 presents minimum lengths of
embedment for different soil conditions and drilled shaft diameters. For shorter
lengths, less than the values presented, the drilled shaft will actually deflect more and
the moments will be less than those obtained from this method.
55


Table 3.2 Minimum length of embedment for Characteristic Load Method
Soil Type Minimum Embedment Length (A)
Cu y' Ds-(j)' Kp
Clay 1 X 105 6
Clay 3 X 105 10
Clay 1 X 106 14
Clay 3 X 106 18
Sand 1 X 104 8
Sand 4X 104 11
Sand 2 X 105 14
3.3.3.3 Elastic Continuum Theory
Elastic theory considers the soil mass as an elastic continuum and applies boundary
integral equations to solve the response of laterally loaded deep foundation.
Mindlins (1936) elastic equations are the basis for the elastic theory used in laterally
loaded deep foundations [37], Mindlin presented the solution for a concentrated force
applied in the interior of a semi-infinite homogeneous isotropic solid.
This method is recommended for drilled shaft deformation at small strains levels
since it does not account for structural yielding. Soil yielding, such as bearing, shear,
and tension failures, was introduced to this approach by Budhu and Davies (1988).
Another limitation is that it under estimates the pile maximum bending moment.
56


4. FINITE DIFFERENCE METHOD
4.1 Introduction
Mechanics of materials focuses on the set of physical laws governing and
mathematically describing the behavior of material under applied loads (strain and
motion). Numerical analysis uses methods from many areas of mathematics,
particularly calculus and linear algebra, to represent a physical system
mathematically.
For this thesis, the physical system to be modeled numerically is a drilled shaft
foundation. The system is modeled as a continuous medium that is capable of
calculating nonuniform stress distributions and obtains the stress at any point of the
model [38], Engineering theories of elasticity and plasticity are adequate for the
analysis of stress and deformation of this engineering research problem.
The finite difference method (FDM) is a mathematical numerical method used for
solving differential equations by approximation. The FDM can be used to solve both
ordinary differential equations (ODE) and partial differential equations (PDE), that is,
differential equations that have only one and more than one independent variable,
respectively. The FDM have been used since the early 1900s for two-dimensional
systems and even earlier for one-dimensional systems.
Mathematically the FDM uses finite difference approximations to replace derivatives
in the governing differential equation resulting in a finite difference equation (FDE);
solving the FDE provides an approximate solution to the differential equation. Linear
and nonlinear differential equations are transformed into linear and nonlinear
equation systems, respectively, in order to obtain the approximate solutions.
The FDM transforms a continuous domain (function) problem into a system with a
discrete set of points on a grid (grid-points) within the domain as shown in Figure 4.1;
this procedure is known as discretization. Instead of obtaining a continuous solution
throughout the domain, approximations are found at these isolated points. The
purpose of discretization is to change a differential equation of infinite dimension to
one of finite dimension; where the difference operator is a finite part of the infinite
Taylor series of the differential operator. The error between the true solution and the
approximate solution is called the discretization error or truncation error.
57


Continuous onedimsnsional
domain
D
ri-l ri

la) Discrete replacement ot
one-dimensional domain
Continuous two-dimensional
domain
It) Discrete replacement ot
two-dimensional domain
Figure 4.1 Approximation of a continuous domain by an
array of discrete points (from Crandall, 1986)
The governing equations and boundary conditions for the continuous domain can also
be reduced physically to obtain the discrete equations [39], Lumped physical
parameters of the continuous system are provided for the discrete model; where the
governing equations are obtained from physical laws such as strain and motion.
Boundary condition information provides a unique solution to the system of
differential equations [40],
Explicit and implicit methods are numerical methods that are used to solve time-
variable differential equations (both ODE and PDE). These methods are commonly
used to solve physical processes in computer analysis. Explicit methods use a
forward difference to solve an equation at a later time based on the equation at the
current time. The implicit methods use a backward difference to solve the equation at
both the current and later time. Due to this, the implicit methods require additional
computations; however, in many problems explicit methods require small time steps
(At) and therefore the computation time may be greater.
58


4.2 Fast Lagrangian Analysis of Continua
in Three Dimensions (FLAC 3D)
4.2.1 Overview
FLAC 3D is an explicit finite difference software program that evaluates the
engineering mechanics of a continuous three-dimensional numerical model [41]. The
computer code numerically simulates the behavior of geotechnical and geologic
materials as they reach equilibrium or steady plastic flow.
Materials are represented by polyhedral elements within a three-dimensional grid that
is adjusted by the user to fit the shape of the object to be modeled, which for this
thesis is a drilled shaft foundation. Figure 4.2 illustrates a general example of a
FLAC 3D numerical model with basic terminology.
Each element behaves according to prescribed linear or nonlinear stress-strain laws in
response to applied forces and boundary conditions. The material can yield and flow,
and the grid can deform (large-strain mode) and move with the material it represents.
A mixed discretization technique is used to allow the element more volumetric
flexibility (Marti and Cundall 1982).
The Lagrangian formulation, which combines momentum and energy conservation,
characterizes a point in the model by the vector components x u v, and dv,/dt, (/' = 1,
2, 3) for position, displacement, velocity and acceleration, respectively. The general
explicit calculation sequence used in FLAC 3D is shown in Figure 4.3. The equations
of motion are used first to derive new velocities and displacements from stresses and
forces; then, strain rates are derived from velocities, and new stresses from strain
rates. It takes one timestep for every cycle around the loop.
The program has eleven built-in material models in addition to an interface (slip-
plane) model. The response of the numerical model depends on FLAC 3Ds
mathematical model and on the numerical formulation described below, which is
obtained directly from the Theory and Background manual of FLAC 3D [41].
4.2.2 Mathematical Model
The behavior of the model is based on mechanical (stress) and kinematic (motion)
principles and the use of constitutive equations. The resulting mathematical
expression is a set of partial differential equations which are to be solved for
particular geometries and properties, given specific boundary and initial conditions.
59


Figure 4.2 Example of a FLAC 3D numerical model (from Itasca, 2002)
new
velocities and
displacements
Equilibrium Equation
(Equation ot Motion)
new
stresses
or forces
Stress / Strain Relation
(Constitutive Equation)
Figure 4.3 Basic explicit calculation cycle (from Itasca, 2002)
60


The sign conventions for stresses are positive for tension and negative for
compression. Shear stresses are considered positive as illustrated in Figure 4.4; shear
strains follow the same convention. Strains are positive for extension and negative
for compression. Mechanical pressure is considered positive when it acts normal to
and in the direction towards the surface of a body (push); and negative in the
direction away from the surface of a body (pull), as shown in Figure 4.5. Fluid pore
pressure is positive in compression and negative in tension.
Z
Figure 4.4 Sign convention for positive stress components (from Itasca, 2002)
Figure 4.5 Mechanical pressure: a) positive, b) negative (from Itasca, 2002)
61


The state of stress at a given point of the model is characterized by the symmetric
stress tensor cry. The traction vector [t] on a face with unit normal [n\ is given by
Cauchys equation:
The particles of the model move with velocity [v], In an infinitesimal time dt, the
model experiences an infinitesimal strain determined by the translations v, dt, and the
corresponding components of the strain-rate tensor are defined as
£,4(v"'+v'') (42)
where partial derivatives are taken with respect to components of the current position
vector [x].
The volume of an element, in addition to the rate of deformation, experiences an
instantaneous rigid-body displacement determined by the translation velocity [v] and
a rotation with angular velocity,
co
jk
(4.3)
where eyk is the permutation symbol, and [co\ is the rate of rotation tensor whose
components are
%=|k,-vJ (4.4)
From the momentum principle, use of the continuum approach yields Cauchys
motion equation.
, d\\
where p is the mass per unit volume of the element, [b\ is the body force per unit
mass, and d[v\/dt is the material derivative of the velocity. For the case of static
equilibrium of the model the acceleration d[v]ldt is zero, therefore Equation 4.5
reduces to the partial differential equation of equilibrium:
cr,/.;+M=0 (4-6)
62


The boundary conditions consist of imposed boundary tractions (see Equation 4.1)
and/or velocities (to induce given displacements). Body forces may be included too
and the initial stress state of the model needs to be specified.
The motion equations, Equation 4.5 together with Equation 4.2 for strain-rates,
constitute nine equations for fifteen unknowns. The unknowns are the 6 + 6
components of the stress- and strain-rate tensors, and the three components of the
velocity vector. Six additional relations are provided by the constitutive equations
that define the nature of the particular material under consideration.
4.2.3 Numerical Formulation
The solution in FLAC 3D is characterized by the following three techniques:
1. finite difference technique, the first-order space and time derivatives of a variable
are approximated by finite differences, assuming linear variations of the variable
over finite space and time intervals, respectively;
2. discrete-model technique, the continuous model is replaced by a discrete
equivalent model; one in which all forces (applied and interactive) are
concentrated at the nodes of a three-dimensional mesh used in the model; and
3. dynamic-solution technique, the inertial terms in the equations of motion are used
as numerical means to reach the equilibrium state of the modeled system.
From these techniques the laws of motion for the continuum are transformed into
discrete forms of Newtons law at the nodes. The resulting system of ordinary
differential equations is then solved numerically using an explicit finite difference
approach in time.
The spatial derivatives involved in the derivation of the equivalent model are those
shown in the equation of strain rates in term of velocities. In order to define the
velocity variations and corresponding space intervals, the model is discretized into
constant strain-rate elements of tetrahedral shape (Figure 4.6) whose vertices are the
nodes of the mesh mentioned previously.
63


node 4
Figure 4.6 Tetrahedron (from Itasca. 2002)
4.2.4 Numerical Implementation
In FLAC 3D, the general discretization of the model into elements (zones) is
performed by the user. Each element is then discretized automatically by the code
into sets of tetrahedrons. For example, an eight-node element can be discretized into
a maximum of two different configurations of five tetrahedrons, which correspond to
overlays 1 and 2 in Figure 4.7. The user can decide to perform the evaluation using
one overlay or a combination of two overlays.
A process called mixed discretization is used in FLAC 3D (Marti and Cundall,
1982). The principle of the mixed discretization technique is to allow more
volumetric flexibility to the element by proper adjustment of the first invariant of the
tetrahedron strain-rate tensor.
The boundary conditions of the model consist of surface tractions, concentrated loads,
and displacements. Also, body forces and initial stress conditions can be applied.
Concentrated loads are specified at given surface nodes, and imposed boundary
displacements are prescribed in terms of nodal velocities. Body forces and surface
tractions are transformed internally into a set of statically equivalent nodal forces.
This represents the initial state of the numerical system.
64


3
5
Figure 4.7 An 8-node element with 2 overlays of 5 tetrahedrons
in each overlay (from Itasca, 2002)
FLAC 3D uses an explicit time-marching finite difference solution formulation; for
every time step, the calculation sequence can be summarized as follows:
1. New strain rates are derived from nodal velocities.
2. Constitutive equations are used to calculate new stresses from the strain rates and
stresses at the previous time.
3. The equations of motion are invoked to derive new nodal velocities and
displacements from stresses and forces.
This sequence is repeated at every time step, and the maximum out-of-balance force
in the model is monitored. This force will either approach zero, indicating that the
system is reaching an equilibrium state, or it will approach a constant, nonzero value,
indicating that a portion or all of the model material is at steady-state plastic flow.
65


5. NUMERICAL MODEL OF A DRILLED
SHAFT FOUNDATION
5.1 Introduction
A drilled shaft foundation is the physical system that is numerically modeled for this
research thesis. The system is modeled as a medium capable of calculating the stress
at any discrete point of the model grid (grid point). Discretization of an infinite
dimension physical system to a finite dimension numerical model is the first step to
mathematically solving the physical problem.
The load-displacement response of a single drilled shaft foundation system is
evaluated for loading in the axial and lateral directions. The user of the numerical
code should understand the behavior of the physical system being modeled and have
an idea of what the results might look like before starting the numerical analysis.
When the results do not conform to the users expectation either there is an error in
the model input or the knowledge of the systems physical behavior is incomplete.
Unlike typical stability programs using limit equilibrium methods, with FLAC 3D the
user must determine when stability or failure of the model is achieved. Several time-
history markers can be checked when attempting to make this interpretation,
including: unbalanced force, grid point displacements, and element yield indicators.
Plots of these time-histories give valuable insight into the performance of a numerical
model and whether or not the model is approaching equilibrium.
The results from a numerical model need to be reasonable, consistent and similar to
those from other methods in order for them to be reliable. The following sections
describe the process used to complete the analyses of a drilled shaft under axial and
lateral loads. The procedure for the static load numerical analysis of a drilled shaft
foundation is summarized in Figure 5.1.
5.2 Mesh Generation and Interfaces
The materials of the physical system are represented within the numerical model by
polyhedral elements, also known as zones within FLAC 3D, which form a three-
dimensional grid that is shaped by the user to conform to the geometry of the physical
66


structure being modeled (drilled shaft foundation). Figure 5.2 illustrates an example
of a three-dimensional drilled shaft foundation model grid (mesh section).
Figure 5.1 Procedure summary for the static load analysis of a drilled shaft
Figure 5.2 Mesh of a drilled shaft foundation
67


Always start with a simple coarse mesh that captures the dominant behavior of the
physical system being modeled (keep it simple). After the initial model is stable and
the performance (physics) of the numerical model is understood, reinforced by
knowledge with other design methods or field experiments, should the mesh be
adjusted (if necessary) into a more complex and finer mesh.
The FLAC 3D code has several types of elements that can be used to shape the model
grid. For this analysis, cylinder and radial cylinder elements were used (Figure 5.3).
The cylinder elements (cylindrical mesh) with six grid points were used specifically
for shaping the concrete drilled shaft, while the radial cylinder elements (radially
graded mesh around cylindrical shape element) with ten grid points were used only
for the soil foundation.
Figure 5.3 Types of elements used to shape the model grid (from Itasca, 2002)
Three dimensional coordinates are assigned to each grid point of the elements in
order to model the grid in real space. To improve the calculation accuracy avoid
element aspect ratios (ratio between its largest and smallest dimension) higher than
ten (elongated element) and preferably stay below five. Each element is divided into
sub-zones by the mixed discretization technique described in chapter four.
The total axial resistance of a drilled shaft is provided by the shaft wall and base
resistance. Therefore, the shaft wall resistance is modeled by placing an interface
between the concrete drilled shaft perimeter and the soil shaft wall; a second interface
is used between the drilled shaft base and the bottom of the soil shaft (Figure 5.4). To
I
a) Cylinder
b) Radial Cylinder
68


properly install the interfaces, first the grids that represent the concrete drilled shaft
and the soil foundation are created separately, then the interfaces are attached to the
soil shaft, and finally the drilled shaft grid is moved into the soil shaft.
a) Soil shaft with interface at the top b) Bottom of soil shaft with interface
Figure 5.4 Mesh and interfaces for drilled shaft foundation
5.3 Boundary Conditions
From a structural view, boundary conditions are supports used to restrain the structure
against relative rigid body motion, such as displacement and/or rotation. For this
research, the boundaries conditions are applied at the edges (outer limits) of the
numerical model. This periphery boundary is necessary to limit the number of
elements in the model. Internal boundaries, such as holes or impermeable layers, are
not used for this evaluation. Applied loads are a type of boundary condition that can
affect the symmetry conditions of the numerical model, which is explained in the
loading section.
The type of boundary conditions used for this numerical model are free and roller
boundaries, as shown in Figure 5.5. The top of the model, ground surface at z = 0, is
a free surface (free movement is allowed in all three directions). The base of the
model is fixed only in the z-direction, and roller boundaries are imposed on the sides
of the model restraining displacement perpendicular to the z-axis. The rollers rest on
a grounded/fixed surface which does not move.
69


Figure 5.5 Two dimensional representation of roller boundary conditions
The periphery boundary conditions need to be placed as far away from the area of
high stresses and strains as possible, sufficient to avoid any influence on the results.
Some preliminary runs might be necessary before obtaining an adequate location for
the boundaries. For example, the lateral boundaries should extend at least ten times
that of the diameter of the drilled shaft on either side of its vertical axis; while the
bottom boundary should be placed at no less than twice the length of the drilled shaft
from the ground surface.
5.4 Initial Static Stress State
Initial elastic material properties were assigned to both groups; the soil foundation is
changed to a Mohr-Coulomb constitutive model later in the analysis, while the
concrete drilled shaft is kept as an elastic material throughout the full analysis. The
gravitational acceleration of 9.81 m/sec2 is applied acting in the negative z-direction.
j
The mass density of the material (mass/volume, kg/m ) was specified based on the
groundwater table, in which moist mass density was assigned to material above the
water table and saturated mass density to material below the groundwater table (pore
water pressure > 0). Pore pressures are a function of depth below the water table, if
applicable, and remain unchanged during the analysis; this is a mechanical non-
coupled analysis, where the calculation of groundwater flow is switched off.
70


The shear modulus (G) of the material was determined based on the material elastic
modulus (E), obtained from laboratory testing, and the Poissons ratio (v). When in-
situ geologic wave velocity information is available, the shear modulus (Gmax) can be
calculated using the material mass density (/) and shear wave velocity (Vs) as shown
in Equation 5.2 (for metric units).
E (5.1)
2(\ + v)
= r K2 (5.2)
The bulk modulus (K) of the material can then be calculated using the elastic or shear
modulus and the Poisson's ratio:
3(1 2 -v)
K 2-G_.(l-nQ
3-0-2v)
(5.3)
(5.4)
An initial stress state under gravity load using elastic material properties was obtained
at this stage before applying the soils drained strengths. A preliminary stress field in
all three directions, in-situ earth stresses (crxx ,cryy ,cr/z), was placed at this stage in
order to accelerate the run time needed to reach equilibrium.
Once finished with establishing the elastic static stress field, a static analysis was
performed using Mohr-Coulomb drained shear strengths (cohesion and friction) for
the soil foundation (in-situ stress state). For the shaft wall and base interfaces,
friction and cohesion (adhesion) properties represent the resistance between the
concrete drilled shaft perimeter and the soil foundation.
Each material element will behave according to the prescribed linear or nonlinear
stress-strain law in response to applied forces and boundary conditions (Appendix A).
The constitutive model selected for each material is simply a mathematical
formulation (constitutive equations) designed to approximate the physical observed
behavior of a real material.
The FLAC 3D Mohr-Coulomb constitutive model, used for the soil foundation,
utilizes an elastic-perfectly plastic behavior. Therefore, the materials are elastic until
71


normal and shear stresses create a Mohr circle tangent to the defined Mohr-Coulomb
yield surface, at which time the material becomes perfectly plastic (Figure 5.6).
Deformations occur until the shear and normal stresses reduce, Mohr circle falls
below the failure plane, as a result of the deformations and accompanied stress
redistribution.
Figure 5.6 Mohr-Coulomb yield criteria
5.5 Loading
The axial and lateral concentrated loads are specified at given surface nodes of the
concrete drilled shaft. History displacement markers are placed throughout several
key points of the drilled shaft foundation, such as the top and bottom of the concrete
drilled shaft and the soil shaft, prior to applying the loads in order to monitor the
response of the model with increased loading.
For this research, due to the loads, a vertical plane of symmetry exists through the x-
coordinate axis as shown in Figure 5.7, which enables the user to model half of the
physical medium without affecting the results. If the analyses were only to consider
an axial load, which is not the case, the numerical model would be reduced to one-
quarter that of the physical system due to two vertical planes of symmetry.
Therefore, a load value 0axia] / 2 and Q\a{ / 2 is applied to the top of the concrete drilled
shaft and the displacement results are a direct effect of the full Qax\a\ and Q\aX loads.
72


Figure 5.7 Plane of symmetry for research drilled shaft
The symmetry line mirrors the grid shape and loads of one side of the model to the
other, or in other words, by rotating the body of Figure 5.7b by 180 degrees about the
symmetry line it will reproduce the exact same numerical model. For simplicity the
models coordinate axes origin (x,y, and z = 0) is located at the intersection of the
ground surface and the drilled shafts vertical axis. The z-axis is oriented along the
drilled shaft axis, the upward vertical direction is positive (Figure 5.5).
The loads are applied linearly over thousands of time steps, from zero to the design
load, to avoid numerical oscillation (ringing/noise) in the model due to the large
contrast in stiffness between the concrete drilled shaft and the soil. Due to the
monotonic nature of the axial and lateral loading the computer code [41] recommends
the use of combined damping to effectively remove kinetic energy from the
numerical model.
After the preliminary analysis using the simple mesh is stable and the results are
reasonable, should discretization of the mesh be finer in the regions where high
gradients of strain and/or stress are expected. Examples of such areas are: comers in
contact with materials of different stiffness, the vicinity of concentrated loads, and
structures with abrupt changes in cross section area (thickness) or material stiffness.
A finer mesh in these areas increases the accuracy; however, it also increases the
duration of the analysis.
73


6. MODEL CALIBRATION WITH FIELD LOAD TESTS
6.1 Introduction
In order to accept the validity of a numerical model and the accuracy of its results, the
model needs to be compared as closely as possible to the real physical system it is
intended to represent. Therefore, this research drilled shaft foundation model is
calibrated using actual full-scale field load tests, with loads applied in the axial
(compressive) and lateral directions. The flow chart shown in Figure 6.1 is an
example of a calibration process that can be used for a drilled shaft model.
A field load test calculates the amount of weight that can be supported by a drilled
shaft foundation and the results can be correlated to design a foundation system under
similar geologic conditions. Since a subsurface profile can vary within a few steps of
a drill hole, it is important that the field load test is placed in a location that is
geologically representative of the area where the actual foundation system is to be
constructed.
A sound engineering analysis of a drilled shaft foundation system using numerical
modeling and/or field load testing should always include an adequate geologic field
exploration and geotechnical laboratory testing program. A numerical parametric
study should be performed when the field and laboratory information is incomplete
and there is concern on the adequacy of the material parameters to be used in the
numerical analysis.
Numerical analysis (modeling) is a key component of geotechnical engineering, as is
the geologic profile, the soil (IGM, rock) behavior, and empiricism. John Burland
(1987) presented the connection between these four fundamental components and
illustrated their relationship with a diagram of a triangle. Since then, Burlands soil
mechanics triangle has been modified into the diagram shown in Figure 6.2 [42],
The load-displacement results obtained from the field load tests enhance the
understanding of the performance of a drilled shaft foundation and help validate the
various methods used to analyze drilled shafts, including numerical modeling.
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Figure 6.1 Calibration process for a numerical model (from Itasca, 2002)
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Genesis / geology
analytical modeling
Figure 6.2 Modified Burland Triangle (from Krahn & Barbour, 2006)
6.2 Drilled Shafts under Axial Load
Field load tests can correlate the load carrying capacity of a drilled shaft with its
displacement, also known as settlement for axial compressive loads. The loads are
usually applied in increments to the top of the drilled shaft and the corresponding
displacement is recorded after each increment. The ultimate load of the drilled shaft
is obtained from the load-displacement plot and the design load is based on the
allowable displacement (settlement) for the foundation system.
Field load tests are usually continued until a predetermined settlement is obtained; for
example, when the displacement reaches 10 percent of the drilled shaft base diameter
or when the ultimate load has been surpassed by a certain percentage based on the
load-displacement curve. A typical axial compressive field load test setup is shown
in Figure 6.3 (Federal Highway Administration, FHWA), where the load is applied by
a hydraulic jack to the top of the test drilled shaft.
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Figure 6.3 Typical axial compressive field load test setup (from FHWA website)
Additional information on axial compressive field load test procedures can be found
on ASTM (American Society for Testing and Materials) standard D 1143-07.
Characteristic load-displacement plots are shown in Figure 6.4; curve A is a typical
behavior of drilled shafts for which the shaft wall resistance ((7shaft) is the dominant
factor of the total axial downward resistance (Oaxiai) and curve B is typical of drilled
shafts for which the base resistance ((?base) is the controlling factor.
Load
Figure 6.4 Typical axial load-displacement curves
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6.2.1 23rd Street Viaduct Field Load Tests, Denver
The load-displacement response of two individual drilled shafts (not a group) with
compressive axial loading is evaluated using numerical modeling. Two field load
tests performed on this site in 1992 are used for this calibration section [43], The test
site is located in downtown Denver, Colorado, on the comer of 23rd Street and Blake
Street (Figure 6.5).
Figure 6.5 Location for 23rd Street Viaduct test site
Reinforced concrete was used to form both drilled shafts. The geologic profile
consists of clay fill from the ground surface (z = 0) to a depth of 0.9 meters,
weathered claystone from 0.9 to 2.7 meters, and claystone Denver Blue Formation
was found in the exploratory drill hole from 2.7 meters to the bottom of the hole at
9.0 meters. The groundwater table (GWT) was encountered at one meter below the
ground surface. Figure 6.6 shows the construction procedure for a drilled shaft in
competent foundation material (non collapsible), which is the case for these two field
load tests.
The clay fill layer is composed of sandy clay with gravel and of low plasticity
s(CL). The weathered claystone is silty, of medium plasticity and medium
hardness. The claystone is considered a hard IGM and of medium plasticity. The
liquid limit (LL) and the plasticity index (PI) of the claystone ranged between 39 to
51, and 13 to 18, respectively. The rock quality designation (RQD) for the claystone
was about 94 with core recovery ranging from 93 to 100 percent.
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n
g
r,
£
A
i
Drilthe shaft to
the designed deplh
Cleancutaccumulatec Positioning the Concrete placed
water and bose material reinforcement cage
Figure 6.6 Construction of a drilled shaft in competent
foundation material (from FHWA website)
6.2.1.1 Shaft Wall Resistance Field Load Test (T-l)
The purpose of this field load test was to isolate the shaft wall resistance of the drilled
shaft in order to obtain a better understanding of the total axial capacity. To achieve
this, the depth of the shaft boring was extended an additional 0.3 meters and extruded
polystyrene (EPS) foam of such height was placed at the bottom of the shaft. A
geologic profile with the drilled shaft construction geometry can be shown in Figure
6.7, and was used to create the numerical model shown in Figure 6.8. As mentioned
in the previous chapter, a vertical plane of symmetry exists along the x-axis for all of
the research models included in this thesis.
The diameter of the drilled shaft (Ds) is 0.8 meters with a length (L) of 7.4 meters. A
y
maximum axial load of 2,340 kN (kilo Newtons, 1 N = 1 kg-m/sec ) was applied
resulting in a vertical displacement of 5.5 centimeters at the top of the drilled shaft.
The lateral boundaries of the model extend ten times the diameter of the drilled shaft
to 8.0 meters away from the drilled shaft vertical axis (z-axis), with exception of the
vertical plane of symmetry boundary. The base boundary is located at 12.0 meters
from the ground surface, since the grid below the base of the drilled shaft is not
considered an influence zone for this test case. A total of 2,328 elements and 2,906
grid points were used in the model.
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&>
axial
4-u
1.8 m
I
9.3 m
0.9 m
Figure 6.7 Geologic, construction, and model profile
for the shaft wall resistance T-l field test
Block Group
Hjclay
wclaystone
claystone
__drilled_shaft
pps
Figure 6.8 Numerical model for shaft wall resistance T-l field test
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The material properties used in the numerical model were obtained from field and
laboratory tests results and are shown in Table 6.1. The diameter of the borehole was
overdrilled from the ground surface down to a depth of 4.5 meters, 1.8 meters into the
claystone, to avoid any shaft wall resistance above this elevation. Corrugated metal
pipe with an inside diameter of 0.8 meters was used in the overdrilled shaft in order to
form the drilled shaft. A socket of 2.9 meters in length is formed between the drilled
shaft and the claystone.
Two interfaces were used along the shaft wall, one from the ground surface to a depth
of 4.5 meters with no shaft wall resistance and the second from 4.5 meters to a depth
of 7.4 meters with appropriate resistance. Since the undrained shear strength (cu) of
the claystone is greater than 250 kPa (kilo Pascals, 1 Pa = 1 N/m2) it is considered to
be a cohesive intermediate geomaterial (IGM); and as such, the shaft wall resistance
of the socket should be calculated using Equation 3.31, included here for
convenience:
f ^igm ^igm tfu
(3.31)
Where,
aigm = 0.15, obtained from Equation 3.32
(P\%m = 1 -0, for an RQD of 94
qu= 1.0 MPa, unconfined compressive strength of IGM
(j\nt = 22, due to smooth shaft wall surface and perched groundwater table
Therefore,
/= 150 kPa
Figure 6.9 shows the shear stress distribution in the model after the full axial loading
of 2,340 kN is applied. The difference in color scheme on both sides of the shaft wall
is due to the numerical code sign convention, the x-axis origin (x = 0) is along the
vertical axis of the drilled shaft (z = 0); however, the absolute shear stress values are
identical on both sides.
The load-displacement data, at the top of the drilled shaft, obtained from the field
load test and the results from the numerical analysis are shown in Figure 6.10.
Except for the first centimeter of displacement, the numerical model results and the
field load test data are similar.
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Table 6.1 Material properties for 23rd Street Viaduct Field Tests, Denver
Material Mass Density (kg/m3) Elastic Modulus (MPa) Poissons Ratio Shear Modulus (MPa) Bulk Modulus (MPa) Shear Strength k0
c' (kPa) r (degrees)
Reinforced Concrete 2.400 36,000 0.20 15.000 20,000 - - -
Clay Fill s(CL) 1,650 25 0.35 9.26 27.77 25 20 0.5
Weathered Claystone 1,950 250 0.31 95.42 219.3 300 0 1.0
Claystone Denver Blue 2.050 350 0.29 135.66 277.77 500 0 1.0
Extruded Polystyrene 30 7.5 0.25 3.0 5.0 200 0 -


Contour of SXZ
Magfac 0.000e+000
Gradient Calculation
1-1 .5470e+OO5 to -1 5000e+005
-1 .50000+005 to -1 .2500e-*-005
-1 .2500e+005 to -1 .00000+005
-1 .00000+005 to -7.50000+004
-7.50000+004 to -5.00000+004
-5.00000+004 to -2.5000e+004
-2.50000+004 to 0.0000e+000
00000e+000to 2 .5000e+004
I 2.5000+004 to 5.00000+004
_ 5.00000+004 to 7.5000e+004
7.5000e+004 to 1.00000+005
1 1 .00000+005 to 1 .2500e+005
1 .25000+005 to 1 .50000+005
1 .50000+005 to 1 .5405e+005
Interval 2 5e+QQ4
Figure 6.9 Shear stress (in Pascals) distribution for shaft wall
resistance T-l field test at the end of loading
T-l Load-Displacement
Figure 6.10 Load-displacement curves for shaft wall resistance T-l field test
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The discrepancy of the first centimeter displacement range can be due to any one or a
combination of reasons: a) the 2.5 centimeter lateral movement of the bottom of the
drilled shaft field test, b) the perched groundwater causing water to seep into the
socket area and reducing the interface resistance between the claystone and the
concrete, c) it usually takes a couple of centimeters of displacement before the full
shaft wall resistance is fully mobilized, etc. These observations indicate the
importance of the empiricism factor in the Burland triangle illustration above.
6.2.1.2 Base Resistance Field Load Test (T-3)
This field load test includes resistance of the shaft wall and the base (dominant). The
drilled shaft construction geometry with the geologic profile is shown in Figure 6.11,
and was used to generate the model shown in Figure 6.12. The diameter of the drilled
shaft is 0.8 meters with a length of 5.7 meters. A maximum axial load of 4,288 kN
resulted in a vertical displacement of 6.0 centimeters at the top of the drilled shaft.
The lateral and base boundaries of this model are identical to the previous one. A
total of 2,184 elements and 2,657 grid points were used for this model.
12.0 m
Clay Fill
Weathered Claystone
Claystone
Coaxial
L
^GWT £l.0m

L = 5.7 m
D = 0.8 m
16.0 m
1
1.8 m
-I-
9.3 m
0.9 m
----J
Figure 6.11 Geologic, construction, and model profile
for the base resistance T-3 field test
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Block Group
R| clay
wclaystone
~ claystone
drilled shaft
Figure 6.12
EPS material was not used in this field load test and the material properties used are
those shown in Table 6.1. The diameter of the borehole was overdrilled from the
ground surface to a depth of 4.6 meters, 1.9 meters into the claystone, to eliminate
any shaft wall resistance above this depth. Corrugated metal pipe was used in the
overdrilled shaft. Two interfaces were used along the shaft wall, one from the ground
surface to a depth of 4.6 meters with no shaft wall resistance, and the second from 4.6
meters to a depth of 5.7 meters (socket of 1.1 meters) with resistance properties
identical to those used in the previous model.
Figure 6.13 shows the vertical normal stress distribution in the model after the full
axial loading of 4,288 kN is applied (compression stress is negative). The load-
displacement data, at the top of the drilled shaft, obtained from the field load test and
the results from the numerical analysis are shown in Figure 6.14. The numerical
model results match closely with the field load test data.
The difference of displacement between the of 0.2 to 1.8 centimeter range is due to
any one or a combination of the following factors: a) immediate settlement of the
drilled shaft upon loading from disturbed remnant material in the bottom of the
borehole, b) expansion of the claystone into the borehole before placement of the
concrete, c) perched groundwater softening the claystone near the borehole, etc.
85