Analysis of Laterally Loaded Long or
Intermediate Drilled Shafts of Small or
Large Diameter in Layered Soil
Final Report
Report CA04- 0252
December 2008
Division of Research
& Innovation
Analysis of Laterally Loaded Long or Intermediate Drilled Shafts
of Small or Large Diameter in Layered Soils
Final Report
Report No. CA04- 0252
December 2008
Prepared By:
Department of Civil and Environmental Engineering
University of Nevada, Reno
Reno, NV 89557
Prepared For:
California Department of Transportation
Engineering Services Center
1801 30th Street
Sacramento, CA 95816
California Department of Transportation
Division of Research and Innovation, MS- 83
1227 O Street
Sacramento, CA 95814
DISCLAIMER STATEMENT
This document is disseminated in the interest of information exchange. The contents of this report
reflect the views of the authors who are responsible for the facts and accuracy of the data presented
herein. The contents do not necessarily reflect the official views or policies of the State of California
or the Federal Highway Administration. This publication does not constitute a standard,
specification or regulation. This report does not constitute an endorsement by the Department of
any product described herein.
STATE OF CALIFORNIA DEPARTMENT OF TRANSPORTATION
TECHNICAL REPORT DOCUMENTATION PAGE
TR0003 ( REV. 10/ 98)
1. REPORT NUMBER
CA04- 0252
2. GOVERNMENT ASSOCIATION NUMBER
3. RECIPIENT’S CATALOG NUMBER
4. TITLE AND SUBTITLE
Analysis of Laterally Loaded Long or Intermediate Drilled Shafts of Small or
Large Diameter in Layered Soil
5. REPORT DATE
December, 2008
6. PERFORMING ORGANIZATION CODE
7. AUTHOR( S)
Mohamed Ashour, Gary Norris, Sherif Elfass
8. PERFORMING ORGANIZATION REPORT NO.
UNR / CCEER 01- 02
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Department of Civil & Environmental Engineering
University of Nevada
Reno, NV 89557- 0152
10. WORK UNIT NUMBER
11. CONTRACT OR GRANT NUMBER
DRI Research Task No. 0252
Contract No. 59A0348
12. SPONSORING AGENCY AND ADDRESS
California Department of Transportation
Engineering Services Center
1801 30th Street
Sacramento, CA 95816
California Department of Transportation
Division of Research and Innovation, MS- 83
1227 O Street
Sacramento, CA 95814
13. TYPE OF REPORT AND PERIOD COVERED
Final Report
14. SPONSORING AGENCY CODE
913
15. SUPPLEMENTAL NOTES
This report may also be referenced as report UNR / CCEER 01- 02 published by the University of Nevada, Reno.
16. ABSTRACT
Strain wedge ( SW) model formulation has been used, in previous work, to evaluate the response of a single pile or a group of piles ( including its
pile cap) in layered soils to lateral loading. The SW model approach provides appropriate prediction for the behavior of an isolated pile and pile
group under lateral static loading in layered soil ( sand and/ or clay). The SW model analysis covers the entire range of soil strain or pile deflection
that may be encountered in practice. The method allows development of p- y curves for the single pile based on soil- pile interaction by considering
the effect of both soil and pile properties ( i. e. pile size, shape, bending stiffness, and pile head fixity condition) on the nature of the p- y curve.
This study has extended the capability of the SW model in order to predict the response of laterally loaded large diameter shafts considering 1)
the influence of shaft type ( long, intermediate or short) on the lateral shaft response; 2) the nonlinear behavior of shaft material ( steel and/ or
concrete) and its effect on the soil- shaft- interaction; 3) developing ( partial or complete) liquefaction in the surrounding soil profile based on far and
near- field induced pore water pressure; and 4) vertical side shear resistance along the shaft wall that has a significant contribution to the lateral
shaft response.
The incorporation of the nonlinear behavior of shaft material, soil liquefaction and vertical side shear resistance has a significant influence on the
nature of the calculated p- y curves and the associated t- z curves. Contrary to the traditional Matlock- Reese p- y curve that was established for
small diameter long ( slender) piles and does not account for soil liquefaction and the variation in the shaft bending stiffness, the current approach
for large diameter shafts can provide the p- y curve based on varying liquefaction conditions, vertical and horizontal shear resistance along the
shaft, and the degradation in shaft flexural stiffness. In addition, the technique presented allows the classification and the analysis of the shaft as
long, intermediate or short based on soil- shaft interaction.
17. KEY WORDS
Laterally Loaded Deep Foundations, Drilled Shafts,
Strain Wedge Model, Layered Soils, Nonlinear Behavior
of Shaft Material, Liquefaction, Vertical Side Shear
18. DISTRIBUTION STATEMENT
No restrictions. This document is available to the public
through the National Technical Information Service,
Springfield, VA 22161
19. SECURITY CLASSIFICATION ( of this report)
Unclassified
20. NUMBER OF PAGES
219 Pages
21. PRICE
Reproduction of completed page authorized
ANALYSIS OF LATERALLY LOADED LONG OR INTERMEDIATE
DRILLED SHAFTS OF SMALL OR LARGE
DIAMETER IN LAYERED SOIL
( FINAL)
CCEER 01- 02
Prepared by:
Mohamed Ashour
Research Assistant Professor
Gary Norris
Professor of Civil Engineering
and
Sherif Elfass
Research Assistant Professor
University of Nevada, Reno
Department of Civil Engineering
Prepared for:
State of California
Department of Transportation
Contract No. 59A0348
June 2004
i
ACKNOWLEDGMENTS
The authors would like to thank Caltrans for its financial support of this project. The authors would also
like to acknowledge Dr. Saad El- Azazy, Mr. Anoosh Shamsabadi, Dr. Abbas Abghari, Mr. Angel
Perez- Copo, Mr Steve McBride, Mr. Bob Tanaka and Mr. Tom Schatz for their support and guidance
as the Caltrans monitors for this project.
ii
DISCLAIMER
The contents of this report reflect the views of the authors who are responsible for the facts and
accuracy of the data presented herein. The contents do not necessarily reflect the official views or
policies of the State of California or the Federal Highway Administration. This report does not
constitute standard specifications, or regulations.
iii
ABSTRACT
Strain wedge ( SW) model formulation has been used, in previous work, to evaluate the response of a single
pile or a group of piles ( including its pile cap) in layered soils to lateral loading. The SW model approach
provides appropriate prediction for the behavior of an isolated pile and pile group under lateral static loading
in layered soil ( sand and/ or clay). The SW model analysis covers the entire range of soil strain or pile
deflection that may be encountered in practice. The method allows development of p- y curves for the single
pile based on soil- pile interaction by considering the effect of both soil and pile properties ( i. e. pile size,
shape, bending stiffness, and pile head fixity condition) on the nature of the p- y curve.
This study has extended the capability of the SW model in order to predict the response of laterally loaded
large diameter shafts considering 1) the influence of shaft type ( long, intermediate or short) on the lateral
shaft response; 2) the nonlinear behavior of shaft material ( steel and/ or concrete) and its effect on the soil-shaft-
interaction; 3) developing ( partial or complete) liquefaction in the surrounding soil profile based on far-and
near- field induced porewater pressure; and 4) vertical side shear resistance along the shaft wall that has
a significant contribution to the lateral shaft response.
The incorporation of the nonlinear behavior of shaft material, soil liquefaction and vertical side shear
resistance has a significant influence on the nature of the calculated p- y curves and the associated t- z curves.
Contrary to the traditional Matlock- Reese p- y curve that was established for small diameter long ( slender)
piles and does not account for soil liquefaction and the variation in the shaft bending stiffness, the current
approach for large diameter shafts can provide the p- y curve based on varying liquefaction conditions,
vertical and horizontal shear resistance along the shaft, and the degradation in shaft flexural stiffness. In
addition, the technique presented allows the classification and the analysis of the shaft as long, intermediate
or short based on soil- shaft interaction.
iv
TABLE OF CONTENTS
CHAPTER 1
INTRODUCTION.................................................................................................... 1- 1
CHAPTER 2
SHAFT CLASSIFIFACTION AND CHARACTRIZATION ............................ 2- 1
2.1 SHAFT CLASSIFICATION.......................................................................... 2- 1
2.2 FOUNDATION STIFFNESS MODELING.................................................. 2- 2
2.3 LARGE DIAMETER SHAFT....................................................................... 2- 2
CHAPTER 3
VERTICAL SIDE SHEAR AND TIP RESIATNCES OF
LARGE DIAMTER SHAFTS IN CLAY............................................................... 3- 1
3.1 INTRODUCTION ......................................................................................... 3- 1
3.2 LOAD TRANSFER AND PILE SETTLEMENT......................................... 3- 2
3.3 DEVELOPED t- z CURVE RELATIONSHIP .............................................. 3- 5
3.3.1 Ramberg- Osgood Model for Clay...................................................... 3- 7
3.4 PILE TIP ( SHAFT BASE) RESISTANCE IN CLAY.................................. 3- 8
3.5 PROCEDURE VALIDATION...................................................................... 3- 9
3.5.1 Comparison with the Seed- Reese t- z Curve in
Soft Clay ( California Test)................................................................. 3- 9
3.6 SUMMARY................................................................................................... 3- 11
CHAPTER 4
VERTICAL SIDE SHEAR AND TIP RESIATNCES OF
LARGE DIAMTER SHAFTS IN SAND............................................................... 4- 1
4.1 INTRODUCTION ......................................................................................... 4- 1
4.2 PILE TIP ( SHAFT BASE) RESISTANCE AND .........................................
SETTLEMENT ( QT – zT) IN SAND............................................................. 4- 1
v
4.2.1 Pile Tip Settlement............................................................................. 4- 5
4.3 LOAD TRANSFER ALONG THE PILE/ SHAFT
SIDE ( VERTICAL SIDE SHEAR)................................................................ 4- 6
4.3.1 Method of Slices for Calculating the Shear Deformation
and Vertical Displacement in Cohesionless Soil................................ 4- 6
4.3.2 Ramberg- Osgood Model for Sand ..................................................... 4- 11
4.3.3 Procedure Steps to Assess Load Transfer and Pile
Settlement in Sand ( t- z Curve)........................................................... 4- 12
4.4 PROCEDURE VALIDATION...................................................................... 4- 15
4.5 SUMMARY................................................................................................... 4- 16
CHAPTER 5
MODELING LATERALLY LOADED LARGE DIAMTER
SHAFTS USING THE SW MODEL...................................................................... 5- 1
5.1 INTRODUCTION ......................................................................................... 5- 1
5.2 THE THEORETICAL BASIS OF STRAIN
WEDGE MODEL CHARACTERIZATION................................................. 5- 2
5.3 SOIL PASSIVE WEDGE CONFIGURATION............................................ 5- 3
5.4 STRAIN WEDGE MODEL IN LAYERED SOIL........................................ 5- 5
5.5 SOIL STRESS- STRAIN RELATIONSHIP.................................................. 5- 7
5.5.1 Horizontal Stress Level ( SL).............................................................. 5- 9
5.6 SHEAR STRESS ALONG THE PILE SIDES ( SLt)..................................... 5- 11
5.6.1 Pile Side Shear in Sand ...................................................................... 5- 11
5.6.2 Pile Side Shear Stress in Clay............................................................ 5- 11
5.7 SOIL PROPERTY CHARACTERIZATION IN
THE STRAIN WEDGE MODEL.................................................................. 5- 12
5.7.1 Properties Employed for Sand Soil .................................................... 5- 13
5.7.2 The Properties Employed for Normally Consolidated Clay.............. 5- 14
5.8 SOIL- PILE INTERACTION IN THE STRAIN WEDGE MODEL............. 5- 16
5.9 PILE HEAD DEFLECTION ......................................................................... 5- 19
5.10 ULTIMATE RESISTANCE CRITERIA IN
vi
STRAIN WEDGE MODEL........................................................................... 5- 20
5.10.1 Ultimate Resistance Criterion of Sand Soil........................................ 5- 20
5.10.2 Ultimate Resistance Criterion of Clay Soil........................................ 5- 21
5.11 VERTICAL SIDE SHEAR RESISTANCE .................................................. 5- 22
5.12 SHAFT BASE RESISTANCE ...................................................................... 5- 22
5.13 STABILITY ANALYSIS IN THE STRAIN WEDGE MODEL .................. 5- 24
5.13.1 Local Stability of a Soil Sublayer in the Strain Wedge Model.......... 5- 24
5.13.2 Global Stability in the Strain Wedge Model...................................... 5- 24
5.14 SUMMARY................................................................................................... 5- 25
CHAPTER 6
SHAFTS IN LIQUEFIABLE SOILS ..................................................................... 6- 1
6.1 INTRODUCTION ......................................................................................... 6- 1
6.2 METHOD OF ANALYSIS............................................................................ 6- 3
6.2.1 Free- Field Excess Pore Water Pressure, uxs, ff ............................................... 6- 4
6.2.2 Near- Field Excess Pore Water Pressure, uxs, nf .............................. 6- 5
6.3 CASE STUDIES ............................................................................................ 6- 12
6.3.1 Post- Liquefaction Response of Completely
Liquefied Nevada Sand ...................................................................... 6- 12
6.3.2 Post- Liquefaction Response of Completely
Liquefied Ione Sand ........................................................................... 6- 13
6.3.3 Post- liquefaction Response of Partially and Completely
Liquefied Fraser River Sand .............................................................. 6- 13
6.4 UNDRAINED STRAIN WEDGE MODEL FOR LIQUEFIED SAND....... 6- 13
6.5 SOIL- PILE INTERACTION IN THE SW MODEL UNDER
UNDRAINED CONDITIONS ...................................................................... 6- 16
6.6 SUMMARY................................................................................................... 6- 17
CHAPTER 7
FAILURE CRITERIA OF SHAFT MATERIALS .............................................. 7- 1
7.1 INTRODUCTION ......................................................................................... 7- 1
vii
7.2 COMBINATION OF MATERIAL MODELING WITH
THE STRAIN WEDGE MODEL.................................................................. 7- 3
7.2.1 Material Modeling of Concrete Strength and Failure Criteria ........... 7- 4
7.2.2 Material Modeling of Steel Strength.................................................. 7- 7
7.3 MOMENT- CURVATURE ( M- F ) RELATIONSHIP................................... 7- 9
7.4 ANALYSIS PROCEDURE ........................................................................... 7- 10
7.4.1 Steel Shaft .......................................................................................... 7- 10
7.4.2 Reinforced Concrete Shaft ................................................................. 7- 15
7.4.3 Concrete Shaft with Steel Case ( Cast in Steel Shell, CISS)............... 7- 18
7.4.4 Reinforced Concrete Shaft with Steel Case
( Cast in Steel Shell, CISS) ................................................................. 7- 20
7.5 SUMMARY ......................................................................... 7- 21
CHAPTER 8
VERIFICATIONS WITH FIELD LOAD TESTS ................................................ 8- 1
8.1 INPUT DATA................................................................................................ 8- 1
8.1.1 Shaft Properties.................................................................................. 8- 1
8.1.2 Soil Properties.................................................................................... 8- 1
8.1.3 Liquefaction analysis ( for saturated sand) ......................................... 8- 2
8.1.4 Loads ( shear force, moment and axial load)...................................... 8- 2
8.1.5 Earthquake Excitation ( Liquefaction) ................................................ 8- 2
8.2 LAS VEGAS FIELD TEST ( SHORT SHAFT)............................................. 8- 3
8.3 SOUTHERN CALIFORNIA FIELD TEST ( SHORT SHAFT).................... 8- 5
8.4 TREASURE ISLAND FULL- SCALE LOAD TEST
ON PILE IN LIQUEFIED SOIL ............................................................ 8- 6
8.5 COOPER RIVER BRIDGE TEST AT THE MOUNT
PLEASANT SITE, SOUTH CAROLINA SITE............................................ 8- 8
8.6 UNIVERSITY OF CALIFORNIA, LOS ANGELES ( UCLA)
FULL- SCALE LOAD TEST ON LARGE DIAMETER SHAFT................. 8- 11
8.7 FULL- SCALE LOAD TEST ON A BORED PILE IN LAYERED
SAND AND CLAY SOIL.............................................................................. 8- 13
viii
8.8 SUMMARY................................................................................................... 8- 13
APPENDIX I
REFERENCES
1- 1
CHAP TER 1
INTRODUCTION
The problem of a laterally loaded large diameter shafts has been under investigation and
research for the last decade. At present, the p- y method developed by Matlock ( 1970)
and Reese ( 1977) for slender piles is the most commonly used procedure for the analysis
of laterally loaded piles/ shafts. The confidence in this method is derived from the fact
that the p- y curves employed have been obtained ( back calculated) from a few full- scale
field tests. Many researchers since have attempted to improve the performance of the p- y
method by evaluating the p- y curve based on the results of the pressuremeter test or
dilatometer test.
The main drawback with the p- y approach is that p- y curves are not unique. Instead the p-y
relationships for a given soil can be significantly influenced by pile properties and soil
continuity and are not properly considered in the p- y approach. In addition, the p- y curve
has been used with large diameter long/ intermediate/ short shafts, which is a compromise.
The SW model proposed by Norris ( 1986) analyzes the response of laterally loaded piles
based on a representative soil- pile interaction that incorporates pile and soil properties
( Ashour et al. 1998). The SW model does not require p- y curves as input but instead
predicts the p- y curve at any point along the deflected part of the loaded pile using a
laterally loaded soil- pile interaction model. The effect of pile properties and surrounding
soil profile on the nature of the p- y curve has been presented by Ashour and Norris
( 2000). However, the current SW model still lack the incorporation of the vertical side
shear resistance that has growing effect on the lateral response of large diameter
piles/ shafts. In addition, many of the large diameter shafts could be designed as long
shafts and in reality they behave as intermediate shafts. Compared to the long shaft
characteristics, the intermediate shaft should maintain softer response. It is customary to
use the traditional p- y curves for the analysis of all types of piles/ shafts
( short/ intermediate/ long) which carries significant comprise.
1- 2
The lateral response of piles/ shafts in liquefied soil using the p- y method is based on the
use of traditional p- y curve shape for soft clay corresponding to the undrained residual
strength ( Sr) of liquefied sand. Typically Sr is estimated using the standard penetration
test ( SPT) corrected blowcount, ( N1) 60, versus residual strength developed by Seed and
Harder ( 1990). For a given ( N1) 60 value, the estimated values of Sr associated with the
lower and upper bounds of this relationship vary considerably. Even if a reasonable
estimate of Sr is made, the use of Sr with the clay curve shape does not correctly reflect
the level of strain in a liquefied dilative sandy material. The p- y relationship for a
liquefied soil should be representative of a realistic undrained stress- strain relationship of
the soil in the soil- pile interaction model for developing or liquefied soil. Because the
traditional p- y curve approach is based on static field load tests, it has been adapted to the
liquefaction condition by using the soft clay p- y shapes with liquefied sand strength
values.
In the last several years, the SW model has been improved and modified through a
number of research phases with Caltrans to accommodate:
· a laterally loaded pile with different head conditions that is embedded in multiple soil
layers ( report to Caltrans, Ashour et al. 1996)
· nonlinear modeling of pile materials ( report to Caltrans, Ashour and Norris 2001);
· pile in liquefiable soil ( report to Caltrans, Ashour and Norris 2000); and
· pile group with or without cap ( report to Caltrans, Ashour and Norris 1999)
The current report focuses on the analysis of large diameter shafts under lateral loading
and the additional influential parameters, such as vertical side shear resistance, compared
to piles. It also addresses the case of complete liquefaction and how the completely
liquefied soil rebuilds significant resistance due to its dilative nature after losing its whole
strength. The assessment of the t- z curve along the length of shaft and its effect on the
shaft lateral response is one of the contributions addressed in this report
1- 3
The classification of the shaft type whether it behaves as short, intermediate or long shaft
has a crucial effect on the analysis implemented. The mechanism of shaft deformation
and soil reaction is governed by shaft type ( geometry, stiffness and head conditions) as
presented in Chapter 2.
The assessment of the vertical side shear due to the shaft vertical movement induced by
either axial or lateral loading is presented in Chapter 3 and 4. New approach for the
prediction of the t- z curve in sand and clay is also presented. Since the lateral resistance
of the shaft base has growing effect on the short/ intermediate shaft lateral response, a
methodology to evaluate the shaft base resistance in clay/ sand is also presented in
Chapters 3 and 4.
The SW model relates one- dimensional BEF analysis ( p- y response) to a three-dimensional
soil pile interaction response. Because of this relation, the SW model is also
capable of determining the maximum moment and developing p- y curves for a pile under
consideration since the pile load and deflection at any depth along the pile can be
determined. The SW model has been upgraded to deal with short, intermediate and long
shafts using varying mechanism. The degradation in pile/ shaft bending stiffness and the
effect of vertical side shear resistance are also integrated in the assessed p- y curve. A
detailed summary of the theory incorporated into the SW model is presented in Chapter
5.
Soil ( complete and partial) liquefaction and the variation in soil resistance around the
shaft due to the lateral load from the superstructure are presented in Chapter 6. Based on
the results obtained from the Treasure Island field test ( sponsored by Caltrans), it is
obvious that none of the current techniques used to analyze piles/ shafts in liquefied soils
reflects the actual behavior of shafts under developing liquefaction. New approach is
presented in Chapter 6 to assess the behavior of liquefied soil and will be incorporated in
the SW model analysis as seen in Chapter 8.
1- 4
The nonlinear behavior of shaft material ( steel and concrete) is a major issue in the
analysis of large diameter shafts. Such nonlinear behavior of shaft material should be
reflected on the nature of the p- y curve and the formation of a plastic hinge as presented
in Chapter 7.
Several case studies are presented in this study to exhibit the capability of the SW model
and how the shaft classification, shaft material modeling ( steel and/ or concrete) and soil
liquefaction can be all implemented in the SW model analysis. Comparisons with field
results and other techniques also are presented in Chapter 8.
2- 1
CHAP TER 2
CLASSIFICATION AND CHARACTERIZATION OF
LARGE DIAMETER SHAFTS
2.1 SHAFT CLASSIFICATION
The lateral load analysis procedures differ for short, intermediate and long shafts. The short,
intermediate and long shaft classifications are based on shaft properties ( i. e. length, diameter and
bending stiffness) and the soil conditions described as follows. A shaft is considered “ short” so
long as it maintains a lateral deflection pattern close to a straight line. A shaft classified as
“ intermediate” under a given combination of applied loads and soil conditions may respond as a
“ short” shaft for the same soil profile for a different combination of applied loads and degraded
soil properties ( e. g. a result of soil liquefaction).
The shaft is defined as “ long” when L/ T / 4. L is the shaft length below ground surface and T
is the relative stiffness defined as T = ( EI/ f) 0.2 where f is the coefficient of subgrade reaction
( F/ L3). The computer Shaft treats the given shaft as a short shaft. The value of relative stiffness,
T, varies with EI and f. For a short shaft, the bending stiffness ( EI) in the analysis could have a
fixed value ( linear elastic). The coefficient of subgrade reaction, f, varies with level of deflection
and decreases with increasing lateral load. The chart ( Fig. 2- 1) attributable to Terzaghi ( DM 7.2,
NAVFAC 1982) and modified by Norris ( 1986) provides average values of f as a function of soil
properties only ( independent of pile shape, EI, head fixity, etc).
The shaft behaves as an “ intermediate” shaft when [ 4 > ( L/ T) > 2]. When an intermediate shaft
is analyzed as a long shaft it results in overestimated lateral response. It should be noted that the
classification of the shaft type in the present study ( i. e. evaluation of its relative stiffness, T) is
based on the initial bending stiffness of the shaft and an average of the coefficient of subgrade
reaction ( f) including the free- field liquefaction effect.
2- 2
The shaft classification for the same shaft my change according to the level loading and the
conditions ( e. g. liquefied or non- liquefied) of the surrounding soils. In addition, shaft stiffness
also varies with level of loading and the induced bending moment along the shaft. Therefore, the
criterion mentioned above is not accurate and does not reflect the actual type of shaft with the
progressive state of loading. For example, a shaft could behave as a long shaft under static
loading and then respond as an intermediate shaft under developing liquefaction. Such response
is due to the changing conditions of the surrounding soil. The analysis carried out in this study
changes according to the type of shafts.
2.2 FOUNDATION STIFFNESS MATRIX
The structural engineer targets the shaft- head stiffness ( at the base of the column) in 6 degrees of
freedom as seen in Figs. 2- 2 through and 2- 4. In reality, the bending stiffness ( EI) of the cross
section varies with moment. In order to deal with an equivalent linear elastic behavior, a
constant reduced bending stiffness ( EIr) for the shaft cross section can be used to account for the
effect of the cracked concrete section under applied loads. However, it is very difficult to
identify the appropriate reduction ratio for the shaft stiffness at a particular level of loading. The
technique presented in this report allows the assessment of the displacement and rotational
stiffness based on the varying bending stiffness of the shaft loaded. Such nonlinear modeling of
shaft material reflects a realistic representation for the shaft behavior according to the level of
loading, and the nonlinear response of shaft material and the surrounding soil The structural
engineer can also replace the nonlinear shaft- head stiffnesses shown in Figs. 2- 3 and 2- 4 by
using the shaft foundation and the p- y curve resulting from the presented technique along with
the superstructure ( complete solution) to model the superstructure- soil- shaft behavior as shown
in Fig 2- 6.
2.3 LARGE DIAMETER SHAFT
The computer programs LPILE/ COM624P have been developed using lateral load tests
performed on long slender piles. The Vertical Shear Resistance ( Vv) acting along the pile or
shaft perimeter has no significant influence on the lateral response of shafts and piles of
diameters less than 3 feet. However, Vv contributes significantly to the capacity of large
diameter shafts. The shaft analysis presented in this report accounts for the Horizontal and
2- 3
Vertical Shear Resistance ( Vh and Vv) acting along the sides of large diameter shafts in addition
to base resistance ( Fig. 2- 7). The t- z curve for soil ( sand, clay, c- j soil and rock) is evaluated
and employed in the analysis to account for the vertical shear resistance.
It should be noted herein that there are basic differences between the traditional p- y curves used
with LPILE/ COM624P and the Strain Wedge ( SW) model technique employed in the current
Shaft analysis.
· The traditional p- y relationships used in LPILE/ COM624P do not account for the vertical
side shear ( Vv) acting along the sides of large diameter shafts because these relationships
were developed for piles with small diameters where side shear is not significant.
· The traditional p- y relationships used in LPILE/ COM624P were developed for long piles
and not for intermediate/ short shafts or piles. The p- y relationships for long piles are
stiffer than those of short piles/ shafts and their direct use in the analysis of short shafts is
not realistic.
· The traditional p- y relationships for sand used in LPILE/ COM624P are multiplied,
without any explanation, by an empirical correction factor of 1.55 ( Morrison and Reese,
1986)
· The bending stiffness of the pile/ shaft has a marked effect on the nature of the resulting
p- y curve relationship. The traditional p- y relationships used in LPILE/ COM624P do not
consider this effect. That is, the traditional p- y relationships used in LPILE/ COM624P
were developed for piles with diameters less than 3 feet that have much lower values of
bending stiffness ( EI) than the large diameter shafts.
· The traditional p- y curves for sand, developed about 30 years ago, is based on a static
load test of a 2- ft diameter long steel pipe pile. They do not consider soil liquefaction.
· The traditional p- y curves have no direct link with the stress- strain relationship of the
soil. Therefore, it is not feasible to incorporate the actual stress- strain behavior of
liquefied soil in the traditional p- y curve formula.
· The traditional p- y curve cannot account for the varying pore water pressure in liquefied
soil. It can only consider the pore water pressure ratio ( ru) in the free field ( away from
the shaft) by reducing the effective unit weight of soil by a ratio equal to ru. Because of
2- 4
this limitation, the traditional p- y curve, even after modification via ru, is incapable of
modeling the increase in pore water pressure around the shaft from the added
superstructure loading.
Fig. 2- 1 f vs. qu for Fine Grained Soil and f vs. Dr for Coarse Grained Soils
2- 5
Fig. 2- 2 Bridge Shaft Foundation and Its Global Axes
Single Shaft
2- 6
Fig. 2- 3 Foundation Stiffnesses for a Single Shaft
P2
K22
K11
K66
P1
M3
Y
Y
X X
P2
K22
K33
K44
P3
M1
Y
Y
Z Z
A) Loading in the X- X Direction
( Axis 1)
B) Loading in the Z- Z Direction
( Axis 3)
Single Shaft
2- 7
Fig. 2- 4 Foundation Springs at the Base of a Bridge Column
in The X- X Direction.
Y
X X
Z
Z
Y
Foundation Springs in
the Longitudinal Direction
K11
K66 K22
Column Nodes
2- 8
Fig. 2- 6 Superstructure- Shaft- Soil Modeling as a Beam on Elastic Foundation ( BEF)
y
p
p
p
p
y
y
y
p
y
( Es) 1
( Es) 2
( Es) 3
( Es) 4
( Es) 5
Ph
Pa
2- 9
Fig. 2- 7 Configuration of a Large Diameter Shaft
y
p
Soil- Shaft Horizontal Resistance
Po
M o
V t
F t
Vv
Vv
FP
FP
FP
Vh
Vh
Vh
Mt
Z
T
Soil- Shaft Side Shear Resistance
Po
Mo
PV PV
FP
FP
FP
F v
F v
F v
F t
Mt
V t
M R
M R
M R
3- 1
CHAP TER 3
VERTICAL SIDE SHAER AND PILE POINT TIP RESISTANCE OF
A PILE / SHAFT IN CLAY
3.1 INTRODUCTION
The primary focus of this chapter is the evaluation of the vertical side shear induced by
the vertical displacement accompanying the deflection of a laterally loaded shaft. The
prediction of the vertical side shear of a laterally loaded shaft is not feasible unless a
relationship between the vertical shaft displacement and the associated shear resistance is
first established. The most common means to date is the t- z curve method proposed by
Seed and Reese ( 1957). The associated curves were developed using experimental data
from the vane shear test to represent the relationship between the induced shear stress
( due to load transfer) and vertical movement ( z) along the side of the pile shaft ( Fig. 3- 1).
Other procedures are available to generate the t- z curve along the pile shaft ( Coyle and
Reese 1966; Grosch and Reese 1980; Holmquist and Matlock 1976 etc.). Most of these
procedures are empirical and based on field and experimental data. Others are based on
theoretical concepts such as the methods presented by Randolph and Worth ( 1978), Kraft
et al. ( 1981) in addition to the numerical techniques adopted by Poulos and Davis ( 1968),
Butterfield and Banerjee ( 1971), and the finite element method.
It should be noted that any developed t- z relationship is a function of the pile/ shaft and
soil properties ( such as shaft diameter, cross section shape and material, axial stiffness,
method of installation and clay shear stress- strain- strength). This requires the
incorporation of as many soil and pile properties as useful and practical in the suggested
analysis.
Coyle and Reese ( 1966) presented an analytical method to assess the load transfer
relationship for piles in clay. The method is addressed in this chapter and requires the
use of a t- z curve such as those curves suggested by Seed and Reese ( 1957), and Coyle
and Reese ( 1966) shown in Figs. 3- 1 and 3- 2. However, the t- z curve presented by Seed
3- 2
and Reese ( 1957) is based on the vane shear test, and the t- z curve developed by Coyle
and Reese ( 1966) is based on data obtained from a number of pile load tests from the
field ( Fig. 3- 2).
The current chapter presents a procedure for evaluating the change in the axial load with
depth for piles in clay called “ friction” piles since most of the axial load is carried by the
shaft ( as opposed to the pile point). The load transfer mechanism presented by Coyle and
Reese ( 1966) is used in the proposed analysis in association with the t- z curve developed
herein. In fact, the axially loaded pile analysis is just a means to develop the nonlinear t-z
curves for clay that will be used later to assess the vertical side shear resistance of a
laterally loaded large diameter shaft undergoing vertical movement at its edges as it
rotates from vertical.
3.2 LOAD TRANSFER AND PILE SETTLEMENT
In order to construct the load transfer and pile- head movement in clay under vertical load,
the t- z curve for that particular soil should be assessed. The load transferred from shaft
skin to the surrounding clay soil is a function of the diameter and the surface roughness
of the shaft, clay properties ( cohesion, type of consolidation and level of disturbance) in
addition to the shaft base resistance. The development of a representative procedure
allows the assessment of the t- z curve in soil ( sand and/ or clay) that leads to the
prediction of a nonlinear vertical load- settlement response at the shaft head. Such a
relationship provides the mobilized shaft- head settlement under axial load and the ration
of load displacement or vertical pile head stiffness.
The procedure developed by Coyle and Reese ( 1966) to assess the load- settlement curve
is employed in this section. However, such a procedure requires knowledge of the t- z
curves ( theoretical or experimental) that represent the load transfer to the surrounding soil
at a particular depth for the pile movement ( z).
The following steps present the procedure that is employed to assess the load transfer and
pile movement in clay soil:
3- 3
1. Based on Skempton assumptions ( 1951), assume a small shaft base resistance, qP
( small percentage of qnet = 9 C).
qP = 9 Cm = 9 C SL = SL qnet ( 3- 1)
QP = qP Abase = SL qnet Abase ( 3- 2)
C is equal to the clay undrained shear strength, Su. Abase is the area of the pile tip
( shaft base).
2. Using the SL evaluated above and the stress- strain relationship presented in
Chapter 5 [ Norris ( 1986) and Ashour et al. ( 1998)], compute the induced axial
( deviatoric) soil strain, e P and the shaft base displacement, zP
zP = e P B ( 3- 3)
where B the diameter of the shaft base. See Section 3- 3 for more details.
3. Divide the pile length into segments equal in length ( hs). Take the load QB at the
base of the bottom segment as ( QP) and movement at its base ( zB) equal to ( zP).
Estimate a midpoint movement for the bottom segment ( segment 4 as seen in Fig.
3- 3). For the first trial, the midpoint movement can be assumed equal to the shaft
base movement.
4. Calculate the elastic axial deformation of the bottom half of this segment,
base
B s
EA
Q h / 2
z elastic = ( 3- 4)
The total movement of the midpoint in the bottom segment ( segment 4) is equal to
T elastic z = z + z ( 3- 5)
5. Based on the soil properties of the surrounding soil ( Su and e 50), use a Ramberg-
Osgood formula ( Eqn. 3- 6) to characterize the backbone response ( Richart 1975).
3- 4
ú ú
û
ù
ê ê
ë
é
÷ ÷ ø
ö
ç ç è
æ
= = +
- 1
1
R
r r ult ult z
z
t
t
b
t
t
g
g
( 3- 6)
z = total midpoint movement of a pile/ shaft segment
g = average shear strain in soil adjacent to the shaft segment
t = average shear stress in soil adjacent to the shaft segment
g r is the reference strain, as shown in Fig. 3- 4, and equals to Gi / t ult
zr = shaft segment movement associated to g r
e 50 = axial strain at SL = 0.5 ( i. e. s d = Su). e 50 can be obtained from the chart
provided in Chapter 5 using the value of Su.
b and R- 1 are the fitting parameters of the a Ramberg- Osgood model given in
Eqn. 3- 7. These parameters are evaluated in section 3.2.1.
6. Using Eqn. 3- 6 which is rewritten in the form of Eqn. 3- 7, the average shear stress
level ( SLt = t / t ult) in clay around the shaft segment can be obtained iteratively based
on movement z evaluated in Eqn. 3- 5.
[ ( ) 1 ] 1 = = + - R
t t
r r
SL SL
z
z b
g
g
( Solved for SLt) ( 3- 7)
7. Shear stress at clay- shaft contact surface is then calculated, i. e.
t = SLt t ult or t = SLt a C ( 3- 8)
where a is the ratio of CA/ C that expresses the variation in the cohesion of the
disturbed clay ( CA) due to pile installation and freeze, as seen in Fig. 3- 5 ( DM7.2
, 1986). It should be noted that the drop in soil cohesion is accompanied by a
drop in the initial shear modulus ( Gi) of the clay
3- 5
8. The axial load carried by the shaft segment in skin friction / adhesion ( Qs) is
expressed as
Qs = p B Hs t ( 3- 9)
9. Calculate the total axial load ( Qi) carried at the top of the bottom segment ( i = 4).
Qi = Qs + QB ( 3- 10)
10. Determine the elastic deformation in the bottom half of the bottom segment
assuming a linear variation of the load distribution along the segment.
Qmid = ( Qi + QB) / 2 ( 3- 11)
8EA
( Q 3 Q ) H
/
2
Q
z mid i B s
elastic
+
= ÷ ø
ö ç è
æ +
= H EA
Q
s
B ( 3- 12)
11. Compute the new midpoint movement of the bottom segment.
z = zP + zelastic ( 3- 13)
12. Compare the z value calculated from step 11 with the previously evaluated
estimated movement of the midpoint from step 4 and check the tolerance.
13. Repeat steps 4 through 12 using the new values of z and Qmid until convergence is
achieved
14. Calculate the movement at the top of the segment i= 4 as
AE
Q Q H
z z i B s
i B 2
+
= +
15. The load at the base ( QB) of segment i = 3 is taken equal to Q4 ( i. e. Qi+ 1) while zB
of segment 3 is taken equal to z4 and steps 4- 13 are repeated until convergence for
segment 3 is obtained. This procedure is repeated for successive segments going
up until reaching the top of the pile where pile head load Q is Q1 and pile top
3- 6
movement d is z1
. Based on presented procedure, a set of pile- head load-settlement
coordinate values ( Q - d ) can be obtained on coordinate pair for each
assumed value of QT. As a result the load transferred to the soil along the length
of the pile can be calculated for any load increment.
16. Knowing the shear stress ( t ) and the associated displacement at each depth ( i. e.
the midpoint of the pile segment), points on the t- z curve can be assessed at each
new load.
3.3 DEVELOPED t- z CURVE RELATIONSHIP
For a given displacement ( z), the mobilized shear stress ( t ) at the shaft- soil interface can
be expressed as a function of the ultimate shear strength ( t ult) via the shear stress level
( SLt).
SLt = t / t ult ( 3- 14)
The shear displacement of the soil around the pile decreases with increasing distance
from the pile wall ( Fig. 3- 6). Based on a model study ( Robinsky and Morrison 1964) of
the soil displacement pattern adjacent to a vertically loaded pile, it has been estimated
( Norris, 1986) that the average shear strain, g , within a zone of B/ 2 wide adjacent to the
pile accounts for 75% of the shear displacement, z, as shown in Fig. 3- 7. A linear shear
strain, g , in the influenced zone ( B/ 2) can be expressed as
B
z
B
z 1.5
/ 2
0.75 g = = ( 3- 15)
Therefore,
1.5
B
z
= g ( 3- 16)
As seen in Fig. 3- 7 and because z is directly related to g based on shaft diameter ( Eqn. 3-
16), note that
3- 7
f f z
z
g
g 50 = 50 ( 3- 17)
where z50 and g 50 are the shaft displacement and the associated shear strain in the soil at
SLt = 0.5 ( i. e. t = 0.5 t ult). zf and g f are the shaft displacement and the associated shear
strain at failure where SLt = 1.0 ( i. e. t = t ult). Therefore, the variation in the shear strain
( g ) occurs in concert with the variation in shaft displacement z ( Fig. 3- 4). It should be
noted that soil shear modulus ( G) exhibits its lowest value next to the pile skin and
increases with distance away from the pile to reach it is maximum value ( Gi) at g and z @
0 ( Fig. 3- 6). Contrary to the shear modulus, the vertical displacement ( z) and the shear
strain ( g ) reach their maximum value in the soil adjacent to the pile face and decrease
with increasing radial distance from the pile.
3.3.1 Ramberg- Osgood Model for Clay
With the above mentioned transformation of the t- z curves to t - g curves, a Ramberg-
Osgood model represented by Eqn. 3- 6 can be used to characterize the t- z curve.
ú ú
û
ù
ê ê
ë
é
÷ ÷ ø
ö
ç ç è
æ
= = +
- 1
1
R
r r ult ult z
z
t
t
b
t
t
g
g
( 3- 18)
At t / t ult = 1 then
= - 1
r
f
g
g
b ( 3- 19)
At t / t ult = 0.5 and g = g 50, then
log ( 0.5)
1
2 1
log
log ( 0.5)
2 1
log
1
50 50
÷ ÷ ÷ ÷ ÷
ø
ö
ç ç ç ç ç
è
æ
-
-
=
÷ ÷ ÷ ÷
ø
ö
ç ç ç ç
è
æ -
- = r
f
r r
R
g
g
g
g
b
g
g
( 3- 20)
3- 8
The initial shear modulus ( Gi) and the shear modulus ( G50) at SL = 0.5 can be determined
via their direct relationship with the normal stress- strain relationship and Poisson’s ratio
( n )
2( 1 ) 3
G i
i i E E
=
+
=
n
n for clay = 0.5 ( 3- 21)
and
50
50 50
50 2 ( 1 ) 3 3
G
n e
u E E S
= =
+
= ( 3- 22)
As seen in Fig. 3- 4,
i
ult
i
u
r G G
S t
g = = ( 3- 23)
50
50
0.5
G
S g = u ( 3- 24)
The shear strain at failure ( g f) is determined in terms of the normal strain at failure ( e f),
i. e.
( 1 ) 1.5
f f
f
e
n
e
g =
+
= ( 3- 25)
The normal stress- strain relationship of clay ( s d - e ) is assessed based on the procedure
presented in Chapter 5 that utilizes e 50 and Su of clay. The initial Young’s modulus of
clay ( Ei) is determined at a very small value of the normal strain ( e ) or stress level ( SL).
In the same fashion, e f is evaluated at SL = 1 or the normal strength s df = 2Su.
3.4 PILE TIP ( SHAFT BASE) RESISTANCE IN CLAY
In regard to the pile tip resistance ( QT – zT) response, the concept of Skempton’s
characterization ( 1951) is used as follows,
3- 9
T net base base Q = q A = 9 C A
where clay cohesion, C, represents the undrained shear strength, Su. The stress level ( SL
= s d / s df) in clay is proportional to the pressure level ( PL = q/ qnet). Different from the
strain- deflection relationship established by Skempton ( 1951) for strip footing ( y50 = 2.5
e 50 B), the vertical soil strain ( e 1) beneath the base of the shaft is expressed as
E E E
1 2 3
1
s
n
s
n
s
e
D
+
D
+
D
=
for s 2 = s 3 and n = 0.5, then
E E
1 3 3
1 ( 1 2 )
s
n
s s
e
D
+ -
D - D
=
E E
d s s s
e
D
=
D - D
= 1 3
1
Therefore, for a constant Young’s modulus ( E) with depth, the strain or e 1 profile has the
same shape as the elastic ( D s 1 - D s 3) variation or Schmertmann’s Iz factor ( Schmertmann
1970, Schmertmann et al. 1979 and Norris 1986). Taking e 1 at depth B/ 2 below the shaft
base ( the peak of the Iz curve), the shaft base displacement ( zT) is a function of the area of
the triangular variation ( Fig. 3- 9), or
z B T = e ( 3- 26)
Dealing with different values for the pile tip resistance, the associated deviatoric stress ( e )
and base movement ( a function of strain, e ) can be determined ( given the stress- strain, s d
- e relationship of the clay immediately below pile tip) in order to construct the pile point
load- point displacement curve.
3.5 PROCEDURE VALIDATION
3.5.1 Comparison with the Seed- Reese t- z Curve in Soft Clay ( California Test)
The test reported by Seed and Reese ( 1957) was conducted in the San Francisco Bay area
of California. As shown in Fig. 3- 10, the soil conditions at that site consisted of 4 ft of
3- 10
fill, 5 ft of sandy clay, and around 21 ft of organic soft clay “ bay mud”. The water table
was approximately 4 ft below ground.
Several 6- in.- diameter pipe piles ( 20 to 22 ft long) were driven into the above soil profile.
The pipe pile had a coned tip and maximum load of 6000 lb. The top 9 ft of the
nonhemogeneous soil was cased leaving an embedment in clay of 13 ft.
A number of disturbed and undisturbed unconfined compression tests were conducted to
determine the unconfined compressive strength of clay ( Fig. 3- 11). Seven loading tests
were performed on the same pile at different periods of time that ranged from 3 hours to
33 days. As shown in Fig. 3- 12, the ultimate bearing capacity of the clay reached a stable
and constant value ( 6200 lb) by the time of the seventh test. As a result, Coyle and Reese
( 1966) considered the results of the seventh load test as representative for stable load
transfer- pile movement response.
Coyle and Reese ( 1966) used the data obtained from the current field test conducted by
Seed and Reese ( 1957) to compute the values of the load transfer response and pile
movement at different depths as seen in Fig. 3- 13. Figure 3- 14 exhibits an equivalent set
of the t- z curves at the same depths that are constructed by using the procedure presented
herein and based on the undrained compressive strength of clay that is described by the
dashed line shown in Fig. 3- 11. The good agreement between the experimental and
predicted t- z curves can be seen in the comparison presented in Fig. 3- 15. Such
agreement speaks to capability of the technique presented. The predicted t- z curve at the
deepest two points ( 20 and 22 feet below ground) and seen in Fig. 3- 15 can be improved
by a slight increase in the undrained compressive strength utilized.
The good agreement between the predicted and experimental t- z curves resulted in an
excellent assessment for load distribution ( due to shear resistance) along the pile. Fig. 3-
16 shows the assessed load distribution and tip resistance that are based on the procedure
presented and induced in 1000- lb axial load increments up to an axial load of 6000 lb. A
3- 11
comparison between the measured and predicted load distributions along the pile is
shown in Fig. 3- 17.
The measured pile head load- settlement curves under seven cases of axial loads are
shown in Fig. 3- 18. The loading tests were performed at different periods of time after
driving the pile. As mentioned earlier, the seventh test ( after 33 days of driving the pile)
is considered for the validation of the procedure presented. Reasonable agreement can be
observed between the predicted and measured pile head load- settlement curve ( Fig. 3-
18).
It should be noted that Seed and Reese ( 1957) established a procedure that allows the
assessment of the pile load- settlement curve and the distribution of the pile skin
resistance based on the data collected from vane shear test shown in Fig. 3- 1. In addition,
some assumptions should be made for the point load movement in order to get good
agreement with the actual pile response. Seed and Reese ( 1957) presented explanation for
the lack of agreement between their calculated and measured data. The undrained
compressive strength collected using the vane shear test was the major source of that
disagreement.
3.6 SUMMARY
The procedure to evaluate the t- z and load- settlement curves for a pile in clay presented
here is based on elastic theory and Ramberg- Osgood characterization of the stress- strain
behavior of soil. This procedure allows the assessment of the mobilized resistance of the
pile using the developed t- z curve and the pile point load- displacement relationship. The
results obtained in comparison with the field data show the capability and the flexible
nature of the suggested technique. Based on the comparison study presented in this
chapter, the good agreement between the measured and predicted load transfer along the
pile, pile movement, pile- head settlement and pile tip resistance shows the consistency of
the technique’s assumptions. The findings in this chapter will be employed in Chapter 5
to evaluate the vertical side shear resistance induced by the lateral deflection of a large
diameter shaft and its contribution to the lateral resistance of the shaft.
3- 12
Fig. 3- 1 Shear Resistance vs. Movement Determined by the Vane Shear Test
( Seed and Reese 1957)
Fig. 3- 2 Ratio of Load Transfer to Soil Shear Strength Vs. Pile Movement
for a Number of Field Tests ( Coyle and Reese 1966)
3- 13
Fig. 3- 3 Modeling Axially Loaded Pile Divided into Segments
V v1 V v1
( z mid ) 1
Q = Q1
V v2 V v2
Q 2
V v3 V v3
Q 3
V v4 V v4
Q 4
QP
z = z 1
z P
L
Q1
Q3
Q4
QP
zP
h
h
h
h
Segment
1
Segment
2
Segment
3
Segment
4
Segment
1
Segment
2
Segment
3
Segment
4
=
( EA/ h ) 1
( EA/ h) 2
( EA/ h) 3
z = z 2
( zmid) 2
( z mid ) 3
( z mid ) 4
z = z3
z = z4
( z mid ) 4
z = z4
z = z3
Q = Q1
z = z 1
( z mid ) 1
z = z 2
( z mid ) 2
( zmid) 3
3- 14
Fig. 3- 4 Basic ( Normal or Shear) Stress- Strain Curve
Fig. 3- 5 Changes in Clay Cohesion Adjacent to the Pile Due to Pile Installation
( DM7.2 1986)
e e or g r or g r
s ult or t ult
s or t
E i or G i
e 50 or g 50
s 50 or t 50
3- 15
Fig. 3- 6 Soil Layer Deformations Around Axially Loaded Pile
Fig. 3- 7 Idealized Relationship Between Shear Strain in Soil ( g )
and Pile Displacement ( Z) ( Norris, 1986)
B
B/ 2
0.75 z
g
z
Q o
Q T
Sheared soil layers
3- 16
Fig. 3- 8 Soil Shear Resistance Vs. Shear Strain ( g ) or Pile Movement ( z)
Fig. 3- 9 Schmertmann Strain Distribution Below Foundation Base
( after Norris, 1986)
z or g
t ult t
z 50 or g 50
t 50
z f or g f
e   = s d / E
B/ 2
B
2B
D
3- 17
Fig. 3- 10 Driven Pile and Soil Profile for Fig. 3- 11 Results of Soil Tests for the Undrained Shear Strength
California Test ( Seed and Reese 1957) of the Bay Mud in California Test ( Seed and Reese 1957)
3- 18
Fig. 3- 12 Variation of the Clay Bearing Capacity with Time
( California Test, Seed and Reese 1957)
Fig. 3- 13 Measured Load Transfer ( t ) – Pile Movement ( z) Curve for California Test
( Coyle and Reese 1966)
3- 19
Fig. 3- 14 Predicted Load Transfer ( t ) – Pile Movement ( z) Curve for
California Test Using the Suggested Procedure
Fig. 3- 15 Comparison Between Measured and Predicted Load Transfer ( t ) – Pile
Movement ( z) Curve for the California Test
0 0.02 0.04 0.06 0.08 0.1
Pile Movement, z, in
0
100
200
300
400
500
Load Transfer, t, lb/ ft2
Measured
Predicted
3- 20
Fig. 3- 16 Predicted Load Distribution along Fig. 3- 17 Comparison of Measured and Predicted Load Distribution
the Pile in California Test along the Pile in California Test ( Seed and Reese 1957)
0 2000 4000 6000 8000
Pile- Head Axial Load, lb
24
22
20
18
16
14
12
10
8
Depth, ft
3- 21
Fig. 3- 18 Pile- Head Load- Settlement Curves for Seven Loading Tests at Different Time
Periods for the California Test in Comparison with the Predicted Results
Predicted
4- 1
CHAP TER 4
VERTICAL SIDE SHAER AND POINT RESISTANCE OF
PILE/ SHAFT IN SAND
4.1 INTRODUCTION
The friction pile in cohesionless soil gains its support from the pile tip resistance and the transfer
of load via the pile wall along its length. It has been suggested that the load transferred by skin
friction pile can be neglected which is not always the case. The load transferred via the pile wall
depends on the diameter and length of the pile, the surface roughness, and soil properties. It
should also be mentioned that both pile point and skin resistances are interdependent.
The assessment of the mobilized load transfer of a pile in sand depends on the success in
developing a representative t- z relationship. This can be achieved via empirical relationships
( Kraft et al. 1981) or numerical methods ( Randolph and Worth, 1978). The semi- empirical
procedure presented in this chapter employs the stress- strain relationship of sand and findings
from experimental tests. The t- z curve obtained based on the current study will be used in
Chapter 5 to account for the vertical side shear resistance that develops with the laterally loaded
large diameter shafts.
The method of slices presented in this chapter reflects the analytical portion of this technique that
allows the assessment of the attenuating shear stress/ strain and vertical displacement within the
vicinity of the driven pile. As a result, the load transfer and the t- z curve can be assessed using a
combination between the tip and side resistances of the pile.
PILE POINT ( SHAFT BASE) RESISTANCE AND SETTLEMENT
( QP – zP) IN SAND
It is evident that the associated pile tip resistance manipulates the side resistance of the pile shaft.
As presented in the analysis procedure, the pile tip resistance should be assumed at the first step.
As a result, the shear resistance and displacement of the upper segments of the pile can be
4- 2
computed based on the assumed pile tip movement. This indicates the need for a practical
technique that allows the assessment of the pile tip load- displacement relationship under a
mobilized or developing state. Most of the available techniques provide the ultimate pile tip
resistance that is independent of the specified settlement. In other words, the pile tip settlement
at the ultimate tip resistance is a function of the pile diameter ( e. g. 5 to 10% of pile tip diameter).
Thereafter, a hyperbolic curve is used to describe the load- settlement curve based on the
estimated ultimate resistance and settlement of the pile tip.
Elfass ( 2001) developed an approach that allows the assessment of the mobilized pile tip
resistance in sand and the accompanying settlement over the whole range of soil strain up to and
beyond soil failure. In association with the pile side shear resistance technique presented in
Section 4- 2, the approach established by Elfass ( 2001) will be employed in the current study to
compute the pile tip load- settlement in sand.
The failure mechanism developed by Elfass ( 2001) assumes four failure zones represented by
four Mohr circles as shown in Fig. 4.1. This mechanism yields the bearing capacity ( q) and its
relationship with the deviatoric stress ( s d) of the last ( fourth Mohr circle) as shown in Fig 4- 2. .
s d = 0.6 q ( 4- 1)
The pile tip resistance ( QP ) is given as,
base
d
P base Q q A A
0.6
= = s ( 4- 2)
where Abase is the cross sectional area of the pile tip ( shaft base).
As seen in Fig. 4- 1, the Mohr Columb strength envelope is nonlinear and requires the evaluation
of the secant angle of the fourth circle ( j IV) tangent to the curvilinear envelope. The angle of the
secant line tangent to first circle ( j I) at effective overburden pressure can be obtained from the
field blow data count ( SPT test) or a laboratory triaxial test at approximately 1 tsf ( 100 kPa)
confining pressure. Due to the increase in the confining pressure ( s 3 ) from one circle to the
4- 3
next, the friction angle ( j ) decreases from j I at ( s 3 ) I to j IV at IV ( s 3 ) based on the following
Bolton ( 1986) relationship modified by Elfass ( 2001) ( Fig. 4- 3)
peak diff j = j + j min ( 4- 3)
( ) 1
3
2 tan 45 / 2
3 3 10 ln 3
2
-
ï þ
ï ý ü
ï î
ï í ì
ú û
ù
ê ë
é
÷ ÷ ø
ö
ç ç è
j = = - æ + + j s diff R R I D ( 4- 4)
s 3 is in kPa. j min is the lowest friction angle that j may reach at high confining pressure, as
shown in Fig. 4- 4 and Dr is inputted as its decimal value.
Knowing the sand relative density ( Dr) and the associated friction angle under the original
confining pressure ( s 3 = s vo ) , the reduction in the friction angle ( D j ) due to the increase of the
confining pressure from s vo to IV ( s 3 ) can be evaluated based on Eqns. 4- 3 and 4- 4, as
described in the following steps:
1. Based on Eqn. 4- 4, calculate ( j diff) I at the original confining pressure ( s 3 = s vo )
( )
1
3
2 tan 45 / 2
( ) 3 10 ln
2
-
ï þ
ï ý ü
ï î
ï í ì
ú ú û
ù
ê ê ë
é
÷ ÷ ø ö
ç ç è
= - æ + + vo
I
diff I R j D j s ( 4- 5)
2. Assume a value for the deviatoric stress ( s d) of the fourth circle ( Fig. 4- 2). As a result,
0.6
q d s
= ( 4- 6)
IV vo q d vo q ( s 3 ) = s + - s = s + 0.4 ( 4- 7)
3. Assume a reduction ( D j = 3 or 4 degrees) in the sand friction angle at ( s 3 = s vo )
due to the increase in the confining pressure from s vo to IV ( s 3 ) , as seen in Fig. 4- 4.
Therefore,
j IV = j I - D j ( 4- 8)
4- 4
4. As presented by Elfass ( 2001) and shown in Fig. 4- 4, j changes in a linear pattern with
the logarithmic increase of s 3 . The friction angle j IV associated with the confining
pressure IV ( s 3 ) can be calculated as
vo
IV
IV I s
s
j j j
( )
log
3 = - D ( 4- 9)
5. According to the computed friction angle ( j IV), use Eqn. 4- 4 to evaluate ( j diff) IV.
( )
( ) 1
3
2 tan 45 / 2
( ) 3 10 ln 3
2
-
ï þ
ï ý ü
ï î
ï í ì
ú ú û
ù
ê ê ë
é
÷ ÷ ø
ö
ç ç è
æ + +
= - IV
IV
diff VI R D s
j
j ( 4- 10)
6. Having the values of ( j diff) I and ( j diff) IV, a revised value for D j can be obtained.
D j = ( j diff) I - ( j diff) IV ( 4- 11)
7. Compare the value of D j obtained in step 6 with the assumed D j in step3. If they are
different, take the new value and repeat the steps 3 through 7 until the value of j IV
converges and the difference in D j reached is within the targeted tolerance.
8. Using the calculated values of j I and j IV, the deviatoric stress at failure can be expressed
as
( ) ( tan 2 ( 45 / 2 ) 1 )
= 3 + - df IV IV s s j ( 4- 12)
9. The current stress level ( SL) in soil ( Zone 4 below pile tip) is evaluated as
( )
( ) d df
df
d
IV
SL m s SL s
s
s
j
j
= =
+ -
+ -
= ;
tan 45 / 2 1
tan 45 / 2 1
2
2
( 4- 13)
where
÷ ÷
ø
ö
ç ç
è
æ
+
= -
( ) / 2
/ 2
sin
3
1
IV d
d
m s s
j s ( 4- 14)
4- 5
4.2.1 Pile Tip Settlement
As presented in Chapter 3 with clay soil, the pile tip displacement in sand can be determined
based on the drained stress- strain relationship presented in Chapter 5 ( Norris 1986 and Ashour et
al. 1998). The soil strain ( e ) below the pile tip is evaluated according to the following equations:
Corresponding to a triaxial test at a given confining pressure ( s 3 ) at a deviator stress ( s d) and
stress level ( SL) as given by Eqns. 4- 12 through 4- 14.
e
l
e 50
3.707
SL e SL
= ( 4- 15)
The value 3.707 and l represent the fitting parameters of the power function relationship, and e 50
symbolizes the soil strain at 50 percent stress level. l is equal to 3.19 for SL less than 0.5 and l
decreases linearly with SL from 3.19 at 0.5 to 2.14 at SL equal to 0.8.
Equation 4- 16 represents the final loading zone which extends from 80 percent to 100 percent
stress level. The following equation is used to assess the strain ( e ) in this range:
( ) SL 0.80
m + q
100
SL = 0.2 + ³
ú ú û
ù
ê ê ë
é
exp ln ;
e
e
( 4- 16)
where m= 59.0 and q= 95.4 e 50 are the required values of the fitting parameters.
The two relationships mentioned above are developed based on unpublished experimental results
( Norris 1977).
For a constant Young’s modulus ( E) with depth, the strain or e 1 profile has the same shape as the
elastic ( D s 1 - D s 3) variation or Schmertmann’s Iz factor ( Schmertmann 1970, Schmertmann et al.
1979 and Norris 1986). Taking e 1 at depth B/ 2 below the shaft base ( the peak of the Iz curve),
the shaft base displacement ( zP) is a function of the area of the triangular variation ( Fig. 3- 9).
4- 6
z B P = e ( 4- 17)
where B is the diameter of the pile point ( shaft base). Dealing with different values for pile tip
resistance ( Eqn. 4- 2), the associated deviatoric stress ( Eqn. 4- 1), stress level ( Eqn. 4- 13) and
principal strain ( e ) ( Eqns. 4- 15 and 4- 16) can be used to assess base movement in order to
construct the pile tip load- settlement ( QP – zP) curve.
4.2 LOAD TRANSFER ALONG THE PILE/ SHAFT SIDE
( VERTICAL SIDE SHEAR)
4.3.1 Method of Slices for Calculating the Shear Deformation and
Vertical Displacement in Cohesionless Soil
The methodology presented in this chapter is called the method of slices. The soil around the
pile/ shaft is modeled as soil horizontal slices that deform vertically as shown in Fig. 4- 5. The
shear stress/ strain caused by the shaft settlement ( z) at a particular depth gradually decreases
along the radial distance ( r) from the pile wall. As seen in Fig. 4- 6, the shear stress ( t ) and strain
( g ) experience their largest values ( t max and g max) just at the contact surface between the shaft and
the adjacent sand. Due to the shear resistance of sand, the induced shear stress/ strain decreases
to zero and large radial distance ( r).
Randolph and Worth ( 1978) and Kraft et al. ( 1981) assume the shear stress decreases with
distance such that t r = t oro in which t o is the shear stress ( t max) at the pile wall ( ro); and t is the
shear stress angular ring at distance r. However, Randolph and Worth ( 1978) argued this
assumption and indicated that the shear stress decreases rapidly with the distance r. Based on
this assumption, Terzaghi ( 1943) showed a more decreasing parabolic pattern ( similar to the one
shown in Fig. 4- 7) for the horizontal variation of the shear stress caused by the axially loaded
sheet pile embedded in a homogenous mass of soil. Robinsky and Morrison ( 1964) performed
experimental tests on model piles embedded in sand that exhibited the parabolic deflection
pattern seen in Fig. 4- 7. The following relationship describes the attenuation in the shear stress
( t ) in soil with the distance r for such a parabolic pattern.
4- 7
2
2
r
r o
o
=
t
t
( 4- 18)
In order to understand the slice method, the stress- strain conditions of a small soil element at the
contact surface with the pile shaft is analyzed. Figure 4- 8 shows the induced shear stress on the
soil- pile contact surface.
The lateral earth pressure coefficient ( K) varies, with the radial distance, from 1 at the pile wall
( due to pile installation) to K = Ko = 1 – sin j in the free- field where the z- movement- induced
shear stress ( t ) reaches zero. Therefore, the horizontal effective stress at the pile wall after
installation ( prior to loading of the pile) just equals the vertical effective overburden, s vo ( i. e.
lateral earth pressure coefficient K = 1). It should be noted that t o represents the t max induced at
the pile wall. Accordingly, a Mohr circle with a center at s vo and a diameter of 2 t o ( t max = t o)
develops at r = ro, as shown in Fig. 4- 8. With radial distance from the pile, the horizontal normal
stress ( s h) and the deviator stress ( s d) continue to drop from s vo and 2 t max at ro to s vo ( 1 - sin j )
and ( 1 ) s vo - Ko or s vo sin j in the far- field ( where t due to z is 0). The corresponding shear
strain ( g = g max) causes a major normal strain e 1,
e 1 = ( 1 + n ) g ( 4- 19)
In addition, the shear modulus ( G) is related to the Young’s modulus ( E) at the given effective
confining pressure ( s 3 ) and normal strain ( e 1), i. e.
2( 1 + n )
= E
G ( 4- 20)
The method of slices described in Fig. 4- 10, is based on the shear stress variation concepts
presented above. The proposed method of slices provides the radius of the soil ring ( radial
distance, r) over which the induced shear stress diminishes, as shown in Fig. 4- 7.
4- 8
As shown in Fig. 4- 11 for soil ring 1, the horizontal stress ( s h) on the soil- pile interface ( inner
surface of the first soil slice) is equal to s vo . At the same time, the horizontal stress ( s h) on the
outer surface is expressed as
s h = s vo - D t ( 4- 21)
The horizontal ( radial and tangential) equilibrium is based on the ring action for the whole ring
of soil ( 2 p r) around the pile. The vertical equilibrium is also conducted on a full ring of soil.
The vertical equilibrium of the first soil ring ( slice) adjacent to the pile wall is expressed by the
following equations:
å = 0 y F ( 4- 22)
cos cos 0 1 R - R - D T - W = B B T T j j ( 4- 23)
Therefore,
cos cos 0 1 R - R - D T - W = B B T T j j ( 4- 24)
and
W R R T B B T T = cos j - cos j - D 1 ( 4- 25)
where D T represents the reduction in the vertical shear force along the radial width ( D r) of the
horizontal soil ring.
The following steps explain the implementation of the method of slices:
1. Divide the pile length into a number of segments that are equal in length ( Hs). Note that
the effective stress ( s vo ) ( i. e. the initial confining stress) increases with depth for each
pile segment.
2. Assume a shear stress developed at the soil- pile interface ( r = ro) equal to that at soil
failure or t ult. It should be noted that there might be a slip condition ( e. g. t limit = K s vo
tan d ) at the soil pile interface that limits to a value t limit less than t ult.
4- 9
3. Determine the developing confining pressure s 3 due to t max ( Fig. 4- 11)
s 3 = o s vo = 1 - sin j K ( 4- 26)
where j the friction angle at failure.
4. Increase the radial distance ( r) from ro to r1 by a small incremental amount ( D r). As a
result, the vertical shear stress on the face of the slice at r1 will drop to t 1 as expressed in
Eqn. 4- 21.
5. The horizontal stress ( s h) on the vertical face of the soil slice decreases with the
attenuating shear stress ( t ) as shown in Fig. 4- 9 until it reaches the value of s 3 given in
Eqn. 4- 26. The Mohr circles shown in Fig. 4 describe the decrease in horizontal stress
( s h) and the mobilized friction angle ( j m) in association to the attenuation in the shear
stress ( t ) ( and the vertical shear force, T, on a vertical unit length) acting on the vertical
face of the soil ring, i. e.
D T1 = T0 – T1 = 2 p ( ro t o - r1 t 1) ( 4- 27)
( r - r )
cos
R = 2
o
2
1
T
1
T p
j
s h ( 4- 28)
( r - r )
cos
R = 2
o
2
1
B
B p
j
s vo ( 4- 29)
It should be noted that s vo is the effective stress at the middle of the slice which is used
as an average effective stress for the whole slice ( i. e. with More circle). The angles j T
and j B at the top and bottom of the first soil ring, respectively, are determined as follows,
vo
o
B s
t
j = sin - 1 ( 4- 30)
1
sin 1 1 t t t
s t
t
j D = -
- D
= -
o
vo
T where ( 4- 31)
4- 10
j B equals j T of the next slice ( soil ring 2) where t 1 and t 2 are the vertical shear stresses at
radii r1 and r2, respectively ( Fig. 4- 12).
6. Based on the induced shear stress ( t o) on the inner face of the current soil ring ( first ring)
and its Mohr circle, calculate the associated shear strain ( g ) that develop over the width
( D r) of the current soil ring. For each horizontal soil slice i ( soil ring with a width D r) and
based on the induced shear stress ( t ) as seen in Fig. 4- 10, the normal strain and stress ( e
and s d), and n will be evaluated. Thereafter, determine the associating shear strain g i and
vertical displacement zi as follows,
n
e
g
+
=
1
i
i ( 4- 32)
where
n = 0.1 + 0.4 SLi
i i i z = g D r ( 4- 33)
7. Repeat steps 1 through 6 for larger values of r ( i. e. an additional soil ring) and calculate zi
for each soil slice ( ring) until the induced vertical shear stress approaches zero at r = rf.
8. Assess the total vertical displacement at the soil- pile contact ( t = t max or t o) as follows,
å =
=
=
t 0
t t o
f i z z ( 4- 34)
zf represents the elastic vertical displacement at failure at the soil- pile contact that is
needed to construct the Ramberg- Osgood model in the next sections.
It would be noticed that the soil ring is always in horizontal equilibrium. For example, the
horizontal equilibrium for the first ring of soil can be expressed as
å = 0 x F
4- 11
sin sin 0 1 + - - = o T T B B E R j E R j ( 4- 35)
where,
o vo o s E = s 2 p r H ( 4- 37)
E v r Hs 1 1 = s 2 p ( 4- 38)
s v varies from s vo at the sand- pile contact surface to s vo ( 1 - sin j ) at rf where the induced
shear stress ( t ) = 0, as shown in Fig. 4- 7.
4.3.2 Ramberg- Osgood Model for Sand
As presented in Chapter 3 with the clay soil, Ramberg- Osgood model represented by Eqn. 4- 39
can be used to characterize the t- z curve.
ú ú
û
ù
ê ê
ë
é
÷ ÷ ø
ö
ç ç è
æ
= = +
- 1
1
R
r r ult ult z
z
t
t
b
t
t
g
g
( 4- 39)
At t / t ult = 1 then
= - 1
r
f
g
g
b ( 4- 40)
At t / t ult = 0.5 and g = g 50, then
log ( 0.5)
1
2 1
log
log ( 0.5)
2 1
log
1
50 50
÷ ÷ ÷ ÷ ÷
ø
ö
ç ç ç ç ç
è
æ
-
-
=
÷ ÷ ÷ ÷
ø
ö
ç ç ç ç
è
æ -
- = r
f
r r
R
g
g
g
g
b
g
g
( 4- 41)
The initial shear modulus ( Gi) at a very low SL and the shear modulus ( G50) at SL = 0.5 can be
determined via their direct relationship with the normal stress- strain relationship and Poisson’s
ration ( n )
4- 12
2( 1 ) 2.2
G i
i i E E
=
+
=
n
n for sand = 0.1 ( 4- 42)
and
50
50 50
50 3
/ 2
2 ( 1 ) 3
G
e
s
n
E E df
= =
+
= ( 4- 43)
Therefore,
i
df
i
ult
r G G
t s / 2
g = = ( 4- 44)
The Poisson’s ratio ( n ) for sand varies 0.1 to 0.5 with the increasing values of SL as follows,
n = 0.1 + 0.4 SL ( 4- 45)
The shear strain at failure ( g f) is determined in terms of the normal strain at failure ( e f).
( 1 ) 1.5
f f
f
e
n
e
g =
+
= ( 4- 46)
The normal stress- strain relationship of sand ( s d - e ) is assessed based on the procedure
presented in Chapter 5. The initial Young’s modulus of clay ( Ei) is determined at a very small
value of the normal strain ( e ) or stress level ( SL). In the same fashion, e f is evaluated at SL = 1
or the normal strength s df. By knowing the values of g r, g 50 and g f, the constants b and R of the
Ramberg- Osgood model shown in Eqn. 4- 39 can be evaluated.
The Ramberg- Osgood model given in Eqn. 4- 39 allows the assessment of the elastic vertical
displacement that occurs at the soil- pile contact surface based on zf obtained in Section 4- 3- 1.
Equation 4- 39 can be rewritten as follows,
ú ú
û
ù
ê ê
ë
é
÷ ÷ ø
ö
ç ç è
æ
= +
- 1
1
R
r ult ult z
z
t
t
b
t
t
( 4- 47)
where,
4- 13
f
r
r f
f
r
f
r i e z z
z
z
g
g
g
g
= . . = ( 4- 48)
4.3.3 Procedure Steps to Assess Load Transfer and Pile Settlement
in Sand ( t- z Curve)
The assessment of the load transfer and associated settlement of a pile embedded in sand requires
the employment of t- z curve for that particular soil. The load transferred from pile shaft to the
surrounding sand is a function of the diameter and the surface roughness of the pile skin and
sand properties ( effective unit weight, friction angle, relative density and confining pressure) in
addition to the pile tip resistance. The development of a representative procedure allows the
assessment of the t- z curve in soil ( sand and/ or clay) that leads to the prediction of a nonlinear
load- settlement curve at the pile/ shaft head. Such a relationship provides the mobilized pile- head
settlement under axial load and vertical shear resistance.
A new procedure is developed in this chapter to assess pile/ shaft skin resistance in sand in a
mobilized fashion. The proposed procedure provides the deformation in sand around the pile in
the radial zone affected by the pile movement ( Fig. 4- 1). At the same time, the horizontal
degradation ( attenuation) of the shear stress away from the pile is evaluated by the suggested
analysis. As a result, the varying shear stress/ strain, shear modulus and deformation in the radial
distance away from the pile can be predicted based on reasonable assumptions.
The presented t- z curve is developed according to the induced displacement along the pile. The
following steps present the procedure that is employed to assess the load transfer and pile
movement in sand soil:
1. Based on the approach presented in Section 4- 2 for the pile tip resistance, assume a small
pile tip resistance, QP as given in Eqns ( 4- 1 and 4- 2)
4- 14
2. Using the SL evaluated above and the stress- strain relationship presented in Eqns. 4- 13
through 4- 16, compute the induced axial ( deviatoric) soil strain, e P and the shaft base
displacement, zP = e P B. B is the diameter of the shaft base.
3. Divide the pile length into segments equal in length ( hs). Take the load QB at the base of
the bottom segment as ( QP) and movement at its base ( zB) equal to ( zP). Estimate a
midpoint movement for the bottom segment ( segment 4 as seen in Fig. 4- 13). For the
first trial, the midpoint movement can be assumed equal to the shaft base movement.
4. Calculate the elastic axial deformation of the bottom half of this segment,
base
B s
EA
Q h / 2
z elastic = ( 4- 49)
The total movement of the midpoint in the bottom segment ( segment 4) is equal to
T elastic z = z + z ( 4- 50)
5. Based on the soil properties of the surrounding sand, use a Ramberg- Osgood formula to
characterize the backbone response ( Richart 1975).
ú ú û
ù
ê ê
ë
é
÷ ÷ ø
ö
ç ç è
æ
= = +
- 1
1
R
r r ult ult z
z
t
t
b
t
t
g
g
( 4- 51)
z = total midpoint movement of a pile/ shaft segment
g = average shear strain in soil adjacent to the shaft segment
t = average shear stress in soil adjacent to the shaft segment
g r is the reference strain, as shown in Fig. 3- 4, and given by Eqn. 4- 44
zr = shaft segment movement associated to g r
e 50 = axial strain at SL = 0.5. e 50 can be obtained from the chart provided in Chapter 5.
4- 15
b and R- 1 are the fitting parameters of the Ramberg- Osgood model given in Eqn. 4- 52.
These parameters are evaluated in section 4.2.1.
6. Using Eqn. 4- 51 which is rewritten in the form of Eqn. 4- 52, the average shear stress
level ( SLt) in sand around the shaft segment can be obtained iteratively based on
movement z evaluated in Eqn. 4- 50.
[ ( ) 1 ] 1 = = + - R
t t
r r
SL SL
z
z b
g
g
( Solved for SLt) ( 4- 52)
7. Shear stress at soil- shaft contact surface is then calculated, i. e.
t = SL s df/ 2 ( 4- 53)
8. The axial load carried by the shaft segment in skin friction / adhesion ( Qs) is
expressed as
Qs = p B hs t ( 4- 54)
9. Calculate the total axial load ( Qi) carried at the top of the bottom segment ( i = 4).
Qi = Qs + QB ( 4- 55)
10. Determine the elastic deformation in the bottom half of the bottom segment
assuming a linear variation of the load distribution along the segment.
Qmid = ( Qi + QB) / 2 ( 4- 56)
8EA
( Q 3 Q ) h
/
2
Q
z mid i B s
elastic
+ = ÷ ø
ö ç è æ + = EA h
Q
s
B ( 4- 57)
11. Compute the new midpoint movement of the bottom segment.
4- 16
z = zP + zelastic ( 4- 58)
12. Compare the z value calculated from step 11 with the previously evaluated estimated
movement of the midpoint from step 4 and check the tolerance.
14. Repeat steps 4 through 12 using the new values of z and Qmid until convergence is
achieved
15. Calculate the movement at the top of the segment i= 4 as
AE
Q Q h
z z i B s
i B 2
= + +
16. The load at the base ( QB) of segment i = 3 is taken equal to Q4 ( i. e. Qi+ 1) while zB of
segment 3 is taken equal to z4 and steps 4- 13 are repeated until convergence for segment
3 is obtained. This procedure is repeated for successive segments going up until reaching
the top of the pile where pile head load Q is Q1 and pile top movement d is z1. Based on
presented procedure, a set of pile- head load- settlement coordinate values ( Q - d ) can be
obtained on coordinate pair for each assumed value of QT. As a result the load
transferred to the soil along the length of the pile can be calculated for any load
increment.
17. Knowing the shear stress ( t ) and the associated displacement at each depth ( i. e. the
midpoint of the pile segment), points on the t- z curve can be assessed at each new load.
4.4 PROCEDURE VALIDATION
As reported by Vesic ( 1970), an 18- inch diameter steel pipe pile with 0.5- inch- thick walls was
driven and tested in five stages. The bottom section has a 2- in thick flat steel plate at the base of
the pile. Tests with this pile were performed at driving depths of 10, 20, 30, 40 and 50 ft.
Figure, 4- 14 shows the results of the standard penetration tests ( SPT) at different locations at the
test site. Figure 4- 15 the particle size distribution curves of two different types of sands. The
4- 17
fine sand curves in this figure refer to the material found mostly at the top 5 ft of the soil profile.
It should be noted that the frictions angles shown in Table 4- 1 is a little bit relatively high
compared to the associated ( N1) 60.
Table 4- 1 – Suggested Soil Data for Current Soil Profile
Soil
layer #
Soil type Thickness
( ft)
g ( pcf) ( N1) 60 f ( deg.) e 50**
1 Sand 10 110 9 30 0.009
2 Sand 10 60 15 32 0.007
3 Sand 10 60 19 35 .006
4 Sand 10 66 24 39 .004
5 Sand 10 66 32 42 0.003
Figure 4- 16 exhibits a comparison between the measured and computed data at the depths 20, 40
and 50 ft below ground. Good agreement between the measured and computed axial pile load
can be seen in Fig. 4- 16.
4.5 SUMMARY
This Chapter presents a procedure that allows the assessment of the t- z and load- settlement
curves for a pile in sand. The methodology employed is based on the elastic theory, stress- strain
relationship, and the method of slices for the vertical equilibrium. The results obtained
incorporate the pile tip and side resistance in a mobilized fashion. The results obtained in
comparison with the field data show the capability of the suggested technique. The findings of
this chapter will be employed in Chapter 5 to evaluate the vertical side shear resistance induced
by the lateral deflection of a large diameter shaft and its contribution to the lateral resistance of
the shaft.
4- 18
Fig. 4- 1 Failure Mechanism of Sand Around Pile Tip ( Elfass, 2001)
Fig. 4- 2 Relationship Between Bearing Capacity ( qnet) of Pile Tip in Sand and the Deviatoric
Stress ( s d) ( after Elfass, 2001)
( s 3) IV ( s 1) IV= qult
s vo
j I j IV
qnet
s d = 0.6q net
Zone IV
Zone I s
t
4- 19
Fig. 4- 3 Degradation in the Secant Friction Angles of Circles Tangent to a Curvilinear
Envelope of Sand Due to the Increase in the Confining Pressure ( Elfass, 2001)
Fig. 4- 4 Changes of Friction Angle ( j ) with the Confining Pressure
j
j Peak
j diff
j min
s 3
Confining Pr essure, s 3
j
D j
j min
1 10 100 s 3 ( log)
4- 20
Fig. 4- 5 Soil Deformation in the Vicinity of Axially Loaded Pile.
Q o
Q T
Sheared soil layers
4- 21
Fig. 4- 6 Shear Stress/ Strain at Soil- Pile Interface.
Fig. 4- 7 Shear and Displacement Attenuation with the Radial Distance from the Pile Wall.
t o
t n
Shaft
r o r n
r n + m
Displacement, z
z max
Distance
t n + m
z n
Z n + m
Shear Stress, t
Shaft
s vo
K s vo
t
g
Shaft
s 3
t
s 1
4- 22
Fig. 4- 8 Growth of Shear Stress at the Soil- Pile Contact Surface ( Pile Wall)
Due to Pile Movement
Fig. 4- 9 Mohr Circles that Represent the Radial Attenuation of Shear and Normal Stresses
For a Given Displacement z at the Pile Wall
s vo
( s h ) 1
t 1
s vo ( 1 - sin j )
= s h
j
j 1
s
t
( s h ) 2
j 2
t 2
t max
Shaft
s vo
K s vo
t
g
t o = t max
s vo = K s vo Normal Stress ( s )
Shear Stress ( t )
j m
j
( s 3) 1 ( s 1) 1
K = 1
due to installation
4- 23
Fig. 4- 10 Soil Rings Around the Pile and the Applying forces on Each Soil Ring ( Slice)
Wn
R
RT
Tn
T
En
E
j B
j T
B. Cross Section in the Soil
C. Cross Section ( Slice) in Soil
hs
r
r
A) Soil Rings around
Slic
e
Sli
ce
Sli
ce T
Hs
Sli
ce
z
4- 24
Fig. 4- 11 Forces and Stresses Applied on the Soil Ring ( Slice) Number 1
( s h) 1 s t vmo
ax=
t o
t 1
s vo ( 1 - sin j )
= s h
j = ( j B) o
D t ( j T) o
  s
t
W1
( RB) 1
( RT) 1
To
T1
Eo
E1
( j B ) 1
( j T) 1
t = T/ h
4- 25
Fig. 4- 12 Forces and Stresses Applied on the Soil Ring ( Slice) Number 2
s vo
( s h ) 1
t 1
s vo ( 1 - sin j )
= s h
j
D t ( j B ) 1 = ( j T ) 2
s
t
( s h ) 2
( j B ) 2
t 2
W 2
( R B ) 2
( R T ) 2
T 1
T 2
E 1
E 2
( j B ) 2
( j T ) 2
4- 26
Fig. 4- 13 Modeling Axially Loaded Pile Divided into Segments
V v1 V v1
( z mid ) 1
Q = Q1
V v2 V v2
Q 2
V v3 V v3
Q 3
V v4 V v4
Q 4
QP
z = z 1
z P
L
Q1
Q3
Q4
QP
zP
h
h
h
h
Segment
1
Segment
2
Segment
3
Segment
4
Segment
1
Segment
2
Segment
3
Segment
4
=
( EA/ h ) 1
( EA/ h) 2
( EA/ h) 3
z = z 2
( zmid) 2
( z mid ) 3
( z mid ) 4
z = z3
z = z4
( z mid ) 4
z = z4
z = z3
Q = Q1
z = z 1
( z mid ) 1
z = z 2
( z mid ) 2
( zmid) 3
4- 27
Fig. 4- 14 Results of the Standard Penetration Tests ( SPT) at
Different Locations ( Vesic, 1970)
Fig. 4- 15 Particle Size Distribution of Sands at Test Site
4- 28
Fig. 4- 16 A Comparison Between Measured and Computed Axial Pile
Load at Different Depths ( After Vesic, 1970)
0 200 400 600 800
Axial Pile Load, Q, Kips
20
10
0
Depth ( ft)
0 200 400 600 800
Computed
Measured
0 200 400 600 800
Axial Pile Load, Q, Kips
40
30
20
10
0
Depth ( ft)
0 200 400 600 800
Computed
Measured
0 200 400 600 800
Axial Pile Load, Q, Kips
50
40
30
20
10
0
Depth ( ft)
0 200 400 600 800
Computed
Measured
5- 1
CHAP TER 5
LATERAL LOADING OF A SHAFT IN LAYERED SOIL
USING THE STRAIN WEDGE MODEL
5.1 INTRODUCTION
The strain wedge ( SW) model is an approach that has been developed to predict the response of a
flexible pile under lateral loading ( Norris 1986, Ashour et al. 1996 and Ashour et al. 1998). The
main concept associated with the SW model is that traditional one- dimensional Beam on Elastic
Foundation ( BEF) pile response parameters can be characterized in terms of three- dimensional
soil- pile interaction behavior. The SW model was initially established to analyze a free- head
pile embedded in one type of uniform soil ( sand or clay). However, the SW model has been
improved and modified through additional research to accommodate a laterally loaded pile
embedded in multiple soil layers ( sand and clay). The SW model has been further modified to
include the effect of pile head conditions on soil- pile behavior. The main objective behind the
development of the SW model is to solve the BEF problem of a laterally loaded pile based on the
envisioned soil- pile interaction and its dependence on both soil and pile properties.
The problem of a laterally loaded pile in layered soil has been solved by Reese ( 1977) as a BEF
based on modeling the soil response by p- y curves. However, as mentioned by Reese ( 1983), the
nonlinear p- y curve employed does not account for soil continuity and pile properties such as
pile stiffness, pile cross- section shape and pile head conditions.
The SW model was initially developed to assess the response of a laterally loaded long ( slender)
pile ( diameter < 3 ft). As a result, the effect of the vertical side shear ( Vv) along the side of a
large diameter shaft should be integrated in the SW model analysis to account for such a
significant parameter in the analysis of large diameter shafts ( Fig. 5- 1). In addition, the
characterization of the intermediate and short shafts should be incorporated in the SW model
analysis to cover broader aspects of the shaft/ pile analysis.
5- 2
5.2 THE THEORETICAL BASIS OF STRAIN WEDGE MODEL
CHARACTERIZATION
The SW model parameters are related to an envisioned three- dimensional passive wedge of soil
developing in front of the pile. The basic purpose of the SW model is to relate stress- strain-strength
behavior of the soil in the wedge to one- dimensional BEF parameters. The SW model
is, therefore, able to provide a theoretical link between the more complex three- dimensional soil-pile
interaction and the simpler one- dimensional BEF characterization. The previously noted
correlation between the SW model response and BEF characterization reflects the following
interdependence:
· the horizontal soil strain ( e ) in the developing passive wedge in front of the pile to the
deflection pattern ( y versus depth, x) of the pile;
· the horizontal soil stress change ( D s h) in the developing passive wedge to the soil- pile
reaction ( p) associated with BEF behavior; and
· the nonlinear variation in the Young's modulus ( E = D s h/ e ) of the soil to the nonlinear
variation in the modulus of soil subgrade reaction ( Es = p/ y) associated with BEF
characterization.
The analytical relations presented above reflect soil- pile interaction response characterized by
the SW model that will be illustrated later. The reason for linking the SW model to BEF analysis
is to allow the appropriate selection of BEF parameters to solve the following fourth- order
ordinary differential equation to proceed.
where MR is the resisting bending moment per unit length induced along the shaft length ( x) due
to the vertical side shear ( VV) ( Fig. 5- 1). The closed form solution of the basic form of the above
equation has been obtained by Matlock and Reese ( 1961) for the case of uniform soil. In order
to appreciate the SW model’s enhancement of BEF analysis, one should first consider the
governing analytical formulations related to the passive wedge in front of the shaft, the soil’s
= 0
d x
d M
+
d x
+ E ( x) y + P d y
d x
EI d y R
2
2
4 s x
4
÷ ÷
ø
ö
ç ç
è
æ
÷ ÷ ø ö
ç ç è
æ
÷ ÷ ø
ö
ç ç è
æ
2
2
( 5- 1)
5- 3
stress- strain and the vertical side shear ( t- z curve) formulations, and the related soil- pile
interaction.
5.3 SOIL PASSIVE WEDGE CONFIGURATION
The SW model represents the mobilized passive wedge in front of the pile which is characterized
by base angles, j m and b m, the current passive wedge depth h, and the spread of the wedge fan
angle, j m ( the mobilized friction angle of soil). The horizontal stress change at the passive
wedge face, D s h, and side shear, t , act as shown in Fig. 5- 2. One of the main assumptions
associated with the SW model is that the deflection pattern of the pile is taken to be linear over
the controlling depth of the soil near the pile top resulting in a linearized deflection angle, d , as
seen in Fig. 5- 3.
The SW model makes the analysis simpler because forces ( F1) on the opposite faces cancel, but
the real zone of stress is like the dashed outline shown in Fig. 5- 4b which includes side shear
influence ( ô) on the shape of the strained zone. However, the ô perpendicular to the face of the
pile is still considered in the SW model analysis. As seen in Fig. 5- 4c, the horizontal equilibrium
in the SW wedge model is based on the concepts of the conventional triaxial test. The soil at the
face of the passive wedge is represented by a soil sample in the conventional triaxial test where
s vo ( i. e. K = 1) and the horizontal stress change, Äóh, ( from pile loading) are the confining and
deviatoric stresses in the triaxial test, respectively.
The relationship between the actual ( closed form solution) and linearized deflection patterns of
long pile/ shaft has been established by Norris ( 1986) ( h/ Xo = 0.69). As seen in 5- 5, the
relationship ( h/ Xo) between the actual and linearized deflection for the short shaft is equal to 1,
and varies for the intermediate shafts from 0.69 at ( L/ T = 4) to 1 at ( L/ T = 2). As presented in
Chapter 2, L is the embedded length of the shaft and T is the initial relative shaft stiffness.
It should be noted that the idea of the change in the full passive wedge ( mobilized passive wedge
at different levels of deflection) employed in the SW model has been shown experimentally by
Hughes and Goldsmith ( 1978) and previously established by Rowe ( 1956).
5- 4
Changes in the shape and depth of the upper passive wedge, along with changes in the state of
loading and shaft/ pile deflection, occur with change in the uniform strain ( e ) in the developing
passive wedge. As seen in Fig. 5- 6, two mobilized ( tip to tip) passive wedges are developed in
soil in front of the short shaft. Because of the shaft straight- line deflection pattern with a
deflection angle d , the uniform soil strain ( e ) will be the same in both ( i. e. upper and lower)
passive wedges.
As shown in Figs. 5- 5 and 5- 6, the deflection pattern is no longer a straight line for the
intermediate shaft, and the lower passive wedge has a curved shape that is similar to the
deflection pattern. Accordingly, the soil strain ( e x) at depth x below the zero crossing will not be
uniform and will be evaluated in an iterative method based on the associated deflection at that
depth ( Fig. 5- 6c)
The lateral response of the short shaft is governed by both ( upper and lower) developed passive
wedges ( Fig. 5- 6). However, with the intermediate shaft, less soil strain ( i. e. stress on soil)
develops in the lower passive soil wedge ( the inverted wedge below the point of zero crossing)
compared to the upper one ( Fig. 5- 6). The non- uniform soil strain ( e x) in the lower passive soil
wedge ( Fig. 5- 6c) becomes much smaller compared to the strain in the upper soil wedge when
the shaft deflection approaches the deflection pattern of the long shaft. Since the lateral
deflection of the long pile/ shaft below the zero crossing is always very small, the associated soil
strain and developing passive wedge will be very small as well. Consequently, the developing
upper passive soil wedge ( and uniform strain therein) dominates the lateral response of the long
pile/ shaft; hence the adopted name “ strain wedge” ( SW).
As seen in Figs. 5- 3 and 5- 6, the configuration of the wedge at any instant of load and,
therefore, base angle
mobilized friction angle, j m, and wedge depth, h, is given by the following equation:
or its complement
2
= 45 - m
m
j
Q ( 5- 2)
5- 5
The width, BC , of the wedge face at any depth is
b j m m BC = D + ( h - x) 2 tan tan ( 5- 4)
where x denotes the depth below the top of the studied passive wedge, and D symbolizes the
width of the pile cross- section. It should be noted that the SW model is based upon an effective
stress analysis of both sand and clay soils. As a result, the mobilized fanning angle, j m, is not
zero in clay soil as assumed by Reese ( 1958, 1983).
The above equations are applied to the upper and lower passive wedges in the case of short and
intermediate shafts where x for any point on the lower passive wedge ( Fig. 5- 6c) is measured
downward from the zero crossing and replaces the term ( h - x) in Eqn. 5- 4. Therefore,
( / ) / ( )
d
d
e e d e x
x x = y x = ( 5- 5)
where e and d are the uniform soil strain and linearized shaft deflection angle of the upper
passive wedge, respectively. yx and d x are the shaft deflection and secant deflection angle at
depth x below the zero crossing ( Fig. 5- 6c).
5.4 STRAIN WEDGE MODEL IN LAYERED SOIL
The SW model can handle the problem of multiple soil layers of different types. The approach
employed, which is called the multi- sublayer technique, is based upon dividing the soil profile
and the loaded pile into sublayers and segments of constant thickness, respectively, as shown in
Fig. 5- 7. Each sublayer of soil is considered to behave as a uniform soil and have its own
properties according to the sublayer location and soil type. In addition, the multi- sublayer
technique depends on the deflection pattern of the embedded pile being continuous regardless of
the variation of soil types. However, the depth, h, of the deflected portion of the pile is
controlled by the stability analysis of the pile under the conditions of soil- pile interaction. The
effects of the soil and pile properties are associated with the soil reaction along the pile by the
2
= 45 + m
m
b j ( 5- 3)
5- 6
Young's modulus of the soil, the stress level in the soil, the pile deflection, and the modulus of
subgrade reaction between the pile segment and each soil sublayer. To account for the
interaction between the soil and the pile, the deflected part of the pile is considered to respond as
a continuous beam loaded with different short segments of uniform load and supported by
nonlinear elastic supports along soil sublayers, as shown in Fig. 5- 8. At the same time, the point
of zero deflection ( Xo in Fig. 5- 5) for a pile in a particular layered soil varies according to the
applied load and the soil strain level.
The SW model in layered soil provides a means for distinguishing layers of different soil types
as well as sublayers within each layer where conditions ( e 50, SL, j m) vary even though the soil
and its properties ( ` g , e or Dr, j , etc.) remain the same. As shown in Fig. 5- 9 , there may be
different soil layers and a transition in wedge shape from one layer to the next, with all
components of the compound wedge having in common the same depth h. In fact, there may be
a continuous change over a given sublayer; but the values of stress level ( SL) and mobilized
friction angle ( j m) at the middle of each sublayer of height, Hi, are treated as the values for the
entire sublayer.
As shown in Fig. 5- 9, the geometry of the compound passive wedge depends on the properties
and the number of soil types in the soil profile, and the global equilibrium between the soil layers
and the loaded pile. An iterative process is performed to satisfy the equilibrium between the
mobilized geometry of the passive wedge of the layered soil and the deflected pattern of the pile
for any level of loading.
While the shape of the wedge in any soil layer depends upon the properties of that layer and,
therefore, satisfies the nature of a Winkler foundation of independent “ soil” springs in BEF
analysis, realize that there is forced interdependence given that all components of the compound
wedge have the same depth ( h) in common. Therefore, the mobilized depth ( h) of the compound
wedge at any time is a function of the various soils ( and their stress levels), the bending stiffness
( EI), and head fixity conditions ( fixed, free, or other) of the pile. In fact, the developing depth of
the compound wedge can be thought of as a retaining wall of changing height, h. Therefore, the
resultant “ soil” reaction, p, from any soil layer is really a “ soil- pile” reaction that depends upon
5- 7
the neighboring soil layers and the pile properties as they, in turn, influence the current depth, h.
In other words, the p- y response of a given soil layer is not unique. The governing equations of
the mobilized passive wedge shape are applied within each one- or two- foot sublayer i ( of a
given soil layer I) and can be written as follows:
( ) ( )
2
= 45 - m i
m i
j
Q ( 5- 6)
( ) ( )
2
= 45 + m i
m i
j
b ( 5- 7)
( BC ) i = D + ( h - xi ) 2 ( m ) i ( m ) i tan b tan j ( 5- 8)
where h symbolizes the entire depth of the compound passive wedge in front of the pile and xi
represents the depth from the top of the pile or compound passive wedge to the middle of the
sublayer under consideration. Equations 5- 6 through 5- 8 are applied at the middle of each
sublayer. In the case of short and intermediate shafts, xi is measured downward from the point of
zero crossing and replaces the term ( h - xi) in Eqn 5- 8, as shown in Fig. 5- 6, for analysis of the
lower wedge.
5.5 SOIL STRESS- STRAIN RELATIONSHIP
The horizontal strain ( e ) in the soil in the passive wedge in front of the pile is the predominant
parameter in the SW model; hence, the name “ strain wedge”. Consequently, the horizontal stress
change ( D s h) is constant across the width of the rectangle BCLM ( of face width BC of the
passive wedge ), as shown in Fig. 5- 4. The stress- strain relationship is defined based on the
results of the isotropically consolidated drained ( sand) or undrained ( clay) triaxial test. These
properties are summarized as follows:
· The major principle stress change ( D s h) in the wedge is in the direction of pile
movement, and it is equivalent to the deviatoric stress in the triaxial test as shown in Fig.
5- 4 ( assuming that the horizontal direction in the field is taken as the axial direction in the
triaxial test).
· The vertical stress change ( D s v) and the perpendicular horizontal stress change ( D s ph)
equal zero, corresponding to the standard triaxial compression test where deviatoric stress
5- 8
is increased while confining pressure remains constant.
· The initial horizontal effective stress is taken as
s ho s vo s vo = K =
where K= 1 due to pile installation effects. Therefore, the isotropic confining pressure in
the triaxial test is taken as the vertical effective stress ( ` s vo) at the associated depth.
· The horizontal stress change in the direction of pile movement is related to the current
level of horizontal strain ( e ) and the associated Young's modulus in the soil, as are the
deviatoric stress and the axial strain, to the secant Young’s modulus ( E = D s h/ e ) in the
triaxial test.
· Both the vertical strain ( e v ) and the horizontal strain perpendicular to pile movement ( e ph)
are equal and are given as
e v = e ph = - n e
where n is the Poisson’s ratio of the soil.
It can be demonstrated from a Mohr’s circle of soil strain, as shown in Fig. 5- 10, that shear
strain, g , is defined as
( v ) Q m ( ) Q m 1 + 2
2
1
- 2 =
2
1
=
2
g e e sin e n sin
( 5- 9)
The corresponding stress level ( SL) in sand ( see Fig. 5- 11) is
( )
( 45 + ) - 1
45 + - 1
SL = = 2
m
2
hf
h
tan / 2
tan / 2
j
j
s
s
D
D
( 5- 10)
where the horizontal stress change at failure ( or the deviatoric stress at failure in the triaxial test)
is
ú û
ù
ê ë
é
÷ ø
ö
ç è
D æ - 1
2
= 2 45 +
hf vo
j
s s tan ( 5- 11)
5- 9
In clay,
; = 2 S
SL = hf u
hf
h s
s
s D
D
D
( 5.12)
where Su represents the undrained shear strength which may vary with depth. Determination of
the values of SL and j m in clay requires the involvement of an effective stress analysis which is
presented later in this chapter.
The relationships above show clearly that the passive wedge response and configuration change
with the change of the mobilized friction angle ( j m) or stress level ( SL) in the soil. Such
behavior provides the flexibility and the accuracy for the strain wedge model to accommodate
both small and large strain cases. The above equations are applied for each soil sublayer along
the shaft in order to evaluate the varying stress level in the soil and the geometry of the passive
wedges.
A power function stress- strain relationship is employed in SW model analysis for both sand and
clay soils. It reflects the nonlinear variation in stress level ( SL) with axial strain ( e ) for the
condition of constant confining pressure. To be applicable over the entire range of soil strain, it
takes on a form that varies in stages as shown in Fig. 5- 12. The advantage of this technique is
that it allows the three stages of horizontal stress, described in the next section, to occur
simultaneously in different sublayers within the passive wedge.
5.5.1 Horizontal Stress Level ( SL)
Stage I ( e £ e 50% )
The relationship between stress level and strain at each sublayer ( i) in the first stage is assessed
using the following equation,
( ) ( - 3.707 SL )
SL = i
50 i
i
i exp
e
l e ( 5.13)
where 3.707 and l ( l = 3.19) represent the fitting parameters of the power function relationship,
and e 50 symbolizes the soil strain at 50 percent stress level at the associated confining pressure.
5- 10
Stage II ( e 50% £ e £ e 80 % )
In the second stage of the stress- strain relationship, Eqn. 5.13 is still applicable. However, the
value of the fitting parameter l is taken to vary in a linear manner with SL from 3.19 at the 50
percent stress level to 2.14 at the 80 percent stress level as shown in Fig. 5- 12b.
Stage III ( e ³ e 80% )
This stage represents the final loading zone which extends from 80 percent to 100 percent stress
level. The following equation is used to assess the stress- strain relationship in this range,
( ) SL 0.80
m + q
100
SL = 0.2 + i
i i
i
i ³ ú û
ù
ê ë
é
exp ln ;
e
e
( 5- 14)
where m= 59.0 and q= 95.4 e 50 are the required values of the fitting parameters.
The three stages mentioned above are developed based on unpublished experimental results
( Norris 1977). In addition, the continuity of the stress- strain relationship is maintained along the
SL- e curve at the merging points between the mentioned stages.
As shown in Fig. 5- 13, if e 50 of the soil is constant with depth ( x), then, for a given horizontal
strain ( e ), SL from Eqns 5- 13 or 5- 14 will be constant with x. On the other hand, since strength,
D s hf, varies with depth ( e. g., see Eqns. 5- 11 and 5- 12), D s h (= SL D s hf ) will vary in a like
fashion. However, e 50 is affected by confining pressure ( ` s vo) in sand and Su in clay. Therefore,
SL for a given e will vary somewhat with depth.
The Young’s modulus of the soil from both the shear loading phase of the triaxial test and the
strain wedge model is
( ) ( )
e
s
e
s
SL
=
E = h i i hf i
i
D D
( 5.15)
5- 11
It can be seen from the previous equations that stress level, strain and Young's modulus at each
sublayer ( i) depend on each other, which results in the need for an iterative solution technique to
satisfy the equilibrium between the three variables.
5.6 SHEAR STRESS ALONG THE PILE SIDES ( SLt)
Shear stress ( t ) along the pile sides in the SW model ( see Fig. 5- 4) is defined according to the
soil type ( sand or clay).
5.6.1 Pile Side Shear in Sand
In the case of sand, the shear stress along the pile sides depends on the effective stress ( s vo) at the
depth in question and the mobilized angle of friction between the sand and the pile ( j s). The
mobilized side shear depends on the stress level and is given by the following equation,
= ( ) ( ) ; where ( ) = 2 ( ) i vo i s i s i m i t s tan j tan j tan j ( 5- 16)
In Eqn. 5- 16, note that mobilized side shear angle, tan j s, is taken to develop at twice the rate of
the mobilized friction angle ( tan j m) in the mobilized wedge. Of course, j s is limited to the fully
developed friction angle ( j ) of the soil.
5.6.2 Pile Side Shear Stress in Clay
The shear stress along the pile sides in clay depends on the clay’s undrained shear strength. The
stress level of shear along the pile sides ( SLt) differs from that in the wedge in front of the pile.
The side shear stress level is function of the shear movement, equal to the pile deflection ( y) at
depth x from the ground surface. This implies a connection between the stress level ( SL) in the
wedge and the pile side shear stress level ( SLt). Using the Coyle- Reese ( 1966) “ t- z” shear stress
transfer curves ( Fig. 5- 14), values for SLt can be determined. The shear stress transfer curves
represent the relationship between the shear stress level experienced by a one- foot diameter pile
embedded in clay with a peak undrained strength, Su, and side resistance, t ult ( equal to z times
the adhesional strength a Su), for shear movement, y. The shear stress load transfer curves of
Coyle- Reese can be normalized by dividing curve A ( 0 < x < 3 m) by z = 0.53, curve B ( 3 < x <
5- 12
6 m) by z = 0.85, and curve C ( x > 6 m) by z = 1.0. These three values of normalization ( 0.53,
0.85, 1.0) represent the peaks of the curves A, B, and C, respectively, in Fig. 5- 15a. Figure 5-
15b shows the resultant normalized curves. Knowing pile deflection ( y), one can assess the
value of the mobilized pile side shear stress ( t ) as
= ( SL ) ( ) t i t i t ult i ( 5- 17)
where
( ) = S ) t ult i z ( a u i ( 5- 18)
and a indicates the adhesion value after Tomlinson ( 1957).
The normalized shear stress load transfer curves can be represented by the following equations.
For the normalized curves A ( x < 3 m) and B ( 3 < x < 6 m),
SL = 12.9 y D - 40.5 y D 2 2
t ( 5- 19)
For the normalized curve C ( x > 6 m)
SL = 32.3 y D - 255 y D 2 2
t ( 5- 20)
where y is in cm and D in m.
From the discussion above, it is obvious that SLt varies nonlinearly with the pile deflection, y, at
a given soil depth, x. Also, SLt changes nonlinearly with soil depth for a given value of soil
displacement/ strain ( see Fig. 5- 15). These concepts are employed in each sublayer of clay.
5.7 SOIL PROPERTY CHARACTERIZATION IN THE STRAIN WEDGE MODEL
One of the main advantages of the SW model approach is the simplicity of the required soil
properties necessary to analyze the problem of a laterally loaded pile. The properties required
represent the basic and the most common properties of soil, such as the effective unit weight and
the angle of internal friction or undrained strength.
5- 13
The soil profile is divided into one or two foot sublayers, and each sublayer is treated as an
independent entity with its own properties. In this fashion, the variation in soil properties or
response ( such as e 50 and j in the case of sand, or Su and ` j in the case of clay) at each sublayer
of soil can be explored. It is obvious that soil properties should not be averaged at the midheight
of the passive wedge in front of the pile for a uniform soil profile ( as in the earlier work of Norris
1986), or averaged for all sublayers of a single uniform soil layer of a multiple layer soil profile.
5.7.1 Properties Employed for Sand Soil
· Effective unit weight ( total above water table, buoyant below), ` g
· Void ratio, e, or relative density, Dr
· Angle of internal friction, j
· Soil strain at 50% stress level, e 50
While standard subsurface exploration techniques and available correlations may be used to
evaluate or estimate ` g , e or Dr, and j , some guidance may be required to assess e 50.
The e 50 represents the axial strain ( e 1 ) at a stress level equal to 50 percent in the e 1- SL
relationship that would result from a standard drained ( CD) triaxial test. The confining
( consolidation) pressure for such tests should reflect the effective overburden pressure ( ` s vo) at
the depth ( x) of interest. The e 50 changes from one sand to another and also changes with density
state. In order to obtain e 50 for a particular sand, one can use the group of curves shown in Fig.
5- 16 ( Norris 1986) which show a variation based upon the uniformity coefficient, Cu, and void
ratio, e. These curves have been assessed from sand samples tested with “ frictionless” ends in
CD tests at a confining pressure equal to 42.5 kPa ( Norris 1977). Since the confining pressure
changes with soil depth, e 50, as obtained from Fig. 5- 16, should be modified to match the existing
pressure as follows:
( ) ( ) ÷ ø
ö ç è
æ
42.5
= ( ) vo i
0.2
50 i 50 42.5
e e s ( 5- 21)
5- 14
( ) ( ) ú û
ù
ê ë
é
÷ ø
ö
ç è
D æ - 1
2
= 2 45 + i
hf i vo i
j
s s tan ( 5- 22)
where ` s vo should be in kPa.
5.7.2 The Properties Employed for Clay
· Effective unit weight ` g
· Plasticity index, PI
· Effective angle of friction, ` j
· Undrained shear strength, Su
· Soil strain at 50% stress level, e 50
Plasticity index, PI, and undrained shear strength, Su, are considered the governing properties
because the effective angle of internal friction, ` j , can be estimated from the PI based on Fig. 5-
17. The e 50 from an undrained triaxial test ( UU at depth x or CU with s 3 = ` s vo) can be
estimated based on Su as indicated in Fig. 5- 18.
An effective stress ( ES) analysis is employed with clay soil as well as with sand soil. The reason
behind using the ES analysis with clay, which includes the development of excess porewater
pressure with undrained loading, is to define the three- dimensional strain wedge geometry based
upon the more appropriate effective stress friction angle, ` j . The relationship between the
normally consolidated clay undrained shear strength, Su, and ` s vo is taken as
u s vo S = 0.33 ( 5- 23)
assuming that Su is the equivalent undrained standard triaxial test strength. The effective stress
analysis relies upon the evaluation of the developing excess porewater pressure based upon
Skempton's equation ( 1954), i. e.
u = B [ + A ( - ) ] D D s 3 u D s 1 D s 3 ( 5- 24)
5- 15
where B equals 1 for saturated soil. Accordingly,
u = + A ( - ) D D s 3 u D s 1 D s 3 ( 5- 25)
Note that D s 3 = 0 both in the shear phase of the triaxial test and in the strain wedge. Therefore,
the mobilized excess porewater pressure is
u s 1 D u = A D ( 5- 26)
where D s 1 represents the deviatoric stress change in the triaxial test and D s h in the field, i. e.
u s h D u = A D ( 5- 27)
Therefore, using the previous relationships, the Skempton equation can be rewritten for any
sublayer ( i) as follows:
( u ) = ( A ) SL ( ) = ( A ) SL 2 ( S ) i u i i hf i u i i u i D D s ( 5- 28)
The initial value of parameter Au is 0.333 and occurs at very small strain for elastic soil response.
In addition, the value of parameter Auf that occurs at failure at any sublayer ( i) is given by the
following relationship
÷ ÷ ø
ö
ç ç
è
æ
1
-
( )
1 ( S )
1 +
2
1
( A ) =
vo i i
u i
uf i s sin j
/
( 5- 29)
after Wu ( 1966) as indicated in Fig. 5- 19.
In Eqn. 5.29, ` j symbolizes the effective stress angle of internal friction; and, based on Eqn. 5-
23, Su/ ` s vo equals 0.33. However, Au is taken to change with stress level in a linear fashion as
( A ) = 0.333 + SL [ ( A ) - 0.333 ] u i i uf i ( 5- 30)
5- 16
By evaluating the value of Au, one can effectively calculate the excess porewater pressure, and
then can determine the value of the effective horizontal stress, ( s –
vo + D s h - D u), and the effective
confining pressure, ( s –
vo - D u) at each sublayer, as shown in Fig. 5- 19. Note that the mobilized
effective stress friction angle, j –
m, can be obtained from the following relationship.
( )
( - u )
+ - u
=
2
( )
45 +
vo i
2 m i vo h i
D
D D
÷ ÷ ø
ö
ç ç è
æ
s
j s s
tan ( 5- 31)
The targeted values of ` j mi and SLi in a clay sublayer and at a particular level of strain ( e ) can be
obtained by using an iterative solution that includes Eqns 5- 12 through 5- 14, and 5- 28 through 5-
31.
5.8 SOIL- PILE INTERACTION IN THE STRAIN WEDGE MODEL
The strain wedge model relies on calculating the modulus of subgrade reaction, E s
, which
reflects the soil- pile interaction at any level of soil strain during pile loading. Es also represents
the secant slope at any point on the p- y curve, i. e.
y
p
Es = ( 5- 32)
Note that p represents the force per unit length of the pile or the BEF soil- pile reaction, and y
symbolizes the pile deflection at that soil depth. In the SW model, Es is related to the soil’s
Young's modulus, E, by two linking parameters, A and y s. It should be mentioned here that the
SW model establishes its own Es from the Young's modulus of the strained soil, and therefore,
one can assess the p- y curve using the strain wedge model analysis. Therefore, Es should first be
calculated using the strain wedge model analysis to identify the p and y values.
Corresponding to the horizontal slice ( a soil sublayer) of the passive wedge at depth x ( see Figs.
5- 2 and 5- 4), the horizontal equilibrium of horizontal and shear stresses is expressed as
p = ( ) BC S + 2 D S i h i i 1 i 2 D s t ( 5- 33)
5- 17
where S1 and S2 equal to 0.75 and 0.5, respectively, for a circular pile cross section, and equal to
1.0 each for a square pile ( Briaud et al. 1984). Alternatively, one can write the above equation as
follows:
( ) ( )
2 S
+
D
= BC S
p D
A =
h i
i 1 i 2
h i
i
i s
t
D s D
/
( 5- 34)
where A symbolizes the ratio between the equivalent pile face stress, p/ D, and the horizontal
stress change, D s h, in the soil. ( In essence, it is the multiplier that, when taken times the
horizontal stress change, gives the equivalent face stress.) From a different perspective, it
represents a normalized width ( that includes side shear and shape effects) that, when multiplied
by D s h yields p/ D. By combining the equations of the passive wedge geometry and the stress
level with the above relationship, one finds that
( ) ( ) ( ) ( )
( ) in sand
2 S
+
D
h - x 2
A = S 1 +
h i
i m m i 2 vo i s i
i 1 s
b j s f
D ÷ ÷ ø
ö
ç ç è
æ tan tan tan
( 5- 35)
( ) ( ) ( ) in clay
SL
+ S SL
D
h - x 2
A = S 1 +
i
i m m i 2 t i
1 i ÷ ÷
ø
ö
ç ç
è
æ tan b tan j
( 5- 36)
Here the parameter A is a function of pile and wedge dimensions, applied stresses, and soil
properties. However, given that D s h = E e in Eqn. 2.33,
p = A D ( s ) = A D E e i i h i i i D ( 5- 37)
For the upper passive wedge, e represents the uniform soil strain and is replaced by e x for soil
sublayers of the lower passive wedge. The second linking parameter, Y s, relates the soil strain in
the SW model to the linearized pile deflection angle, d . Referring to the normalized pile
deflection shape shown in Figs. 5- 3 and 5- 5
2
=
d g ( 5- 38)
5- 18
2 Q m
2
=
2
max sin g g
( 5- 39)
and
( )
2
1 +
=
2
-
=
2
v g e e n e max ( 5- 40)
where g denotes the shear strain in the developing passive wedge. Using Eqns. 5- 39 and 5.40,
Eqn. 5- 38 can be rewritten as
( )
2
1 + 2
d = e n sin Q m ( 5- 41)
Based on Eqn. 5- 41, the relationship between e and d can expressed as
d
e
Y = ( 5- 42)
or
( ) Q
Y
m 1 + 2
2
=
n sin
( 5- 43)
The parameter y varies with the Poisson's ratio of the soil and the soil's mobilized angle of
internal friction ( j m) and the mobilized passive wedge angle ( Q m).
Poisson's ratio for sand can vary from 0.1 at a very small strain to 0.5 or lager ( due to dilatancy)
at failure, while the base angle, Q m, can vary between 45o ( for j m = 0 at e = 0) and 25o ( for, say,
j m = 40o at failure), respectively. For this range in variation for n and j m, the parameter Y for
sand varies between 1.81 and 1.74 with an average value of 1.77. In clay soil, Poisson's ratio is
assumed to be 0.5 ( undrained behavior) and the value of the passive wedge base angle, Q m, can
vary between 45o ( for j m = 0 at e = 0) and 32.5o ( for, say, ` j m = 25o at failure). Therefore, the
value of the parameter y will vary from 1.47 to 1.33, with an average value of 1.4.
It is clear from the equations above that employing the multi- sublayer technique greatly
influences the values of soil- pile interaction as characterized by the parameter, A i
, which is
5- 19
affected by the changing effective stress and soil strength from one sublayer to another. The
final form of the modulus of subgrade reaction can be expressed as
( ) ( ) ( ) D E
h - x
= A
h - x
= A D E
y
p
E = i
i
i
i
i i
i
i
s i Y
d
e
( 5- 44)
It should be mentioned that the SW model develops its own set of non- unique p- y curves which
are function of both soil and pile properties, and are affected by soil continuity ( layering) as
presented by Ashour et al. ( 1996). For the lower passive wedge, ( h – xi) will be replaced by xi
that is measured downward from the point of zero crossing ( Fig. 5- 6).
5.9 PILE HEAD DEFLECTION
As mentioned previously, the deflection pattern of the pile in the SW model is continuous and
linear. Based on this concept, pile deflection can be assessed using a simplified technique which
provides an estimation for the linearized pile deflection, especially yo at the pile head. By using
the multi- sublayer technique, the deflection of the pile can be calculated starting with the base of
the mobilized passive wedge and moving upward along the pile, accumulating the deflection
values at each sublayer as shown in the following relationships and Fig. 5- 20.
s
i i i i y = H = H
Y
e
d ( 5- 45)
y = y i = 1 to n o i S ( 5- 46)
where the y s value changes according to the soil type ( sand or clay), and H i
indicates the
thickness of sublayer i and n symbolizes the current number of sublayers in the mobilized
passive wedge.
The main point of interest is the pile head deflection which is a function of not only the soil
strain but also of the depth of the compound passive wedge that varies with soil and pile
properties and the level of soil strain.
5.10 ULTIMATE RESISTANCE CRITERIA IN STRAIN WEDGE MODEL
5- 20
The mobilized passive wedge in front of a laterally loaded pile is limited by certain constraint
criteria in the SW model analysis. Those criteria differ from one soil to another and are applied
to each sublayer. Ultimate resistance criteria govern the shape and the load capacity of the
wedge in any sublayer in SW model analysis. The progressive development of the ultimate
resistance with depth is difficult to implement without employing the multi- sublayer technique.
5.10.1 Ultimate Resistance Criterion of Sand Soil
The mobilization of the passive wedge in sand soil depends on the horizontal stress level, SL,
and the pile side shear resistance, t . The side shear stress is a function of the mobilized side
shear friction angle, j s, as mentioned previously, and reaches its ultimate value ( j s = j ) earlier
than the mobilized friction angle, j m, in the wedge ( i. e. SLt ³ SL). This causes a decrease in the
rate of growth of sand resistance and the fanning of the passive wedge as characterized by the
second term in Eqns 5- 33 and 5- 35, respectively.
Once the stress level in the soil of a sublayer of the wedge reaches unity ( SLi = 1), the stress
change and wedge fan angle in that sublayer cease to grow. However, the width BC of the face
of the wedge can continue to increase as long as e ( and, therefore, h in Eqn. 5- 8) increases.
Consequently, soil- pile resistance, p, will continue to grow more slowly until a condition of
initial soil failure ( SLi = 1) develops in that sublayer. At this instance, p = pult where pult in sand,
given as
( p ) = ( ) BC S + 2 ( ) D S ult i hf i i 1 f i 2 D s t ( 5.47)
pult is “ a temporary” ultimate condition, i. e. the fanning angle of the sublayer is fixed and equal
to j i, but the depth of the passive wedge and, hence, BC continue to grow. The formulation
above reflects that the near- surface “ failure” wedge does not stop growing when all such
sublayers reach their ultimate resistance at SL = 1 because the value of h at this time is not
limited. Additional load applied at the pile head will merely cause the point at zero deflection
and, therefore, h to move down the pile. More soil at full strength ( SL = 1) will be mobilized to
5- 21
the deepening wedge as BC, therefore, pult will increase until either flow around failure or a
plastic hinge in the pile occurs.
Recognize that flow around failure occurs in any sublayer when it is easier for the sand at that
depth to flow around the pile in a local bearing capacity failure than for additional sand to be
brought to failure and added to the already developed wedge. However, the value at which flow
failure occurs [ Ai = ( Ault) i , ( pult) i = ( D s hf) i ( Ault) i D] in sand is so large that it is not discussed
here. Alternatively, a plastic hinge can develop in the pile when the pile material reaches its
ultimate resistance at a time when SLi £ 1 and Ai < ( Ault) i. In this case, h becomes fixed, and BCi
and pi will be limited when SLi becomes equal to 1.
5.10.2 Ultimate Resistance Criterion of Clay Soil
The situation in clay soil differs from that in sand and is given by Gowda ( 1991) as a function of
the undrained strength ( Su) i of the clay sublayer.
( p ) = 10 ( S ) D S + 2 ( S ) D S ult i u i 1 u i 2 ( 5- 48)
Consequently,
( )
( )
( )
( )
( ) = 5 S + S
D 2 S
p
=
D
p
A = 1 2
u i
ult i
hf i
ult i
ult i D s
( 5- 49)
Ault indicates the limited development of the sublayer wedge geometry for eventual development
of flow around failure ( SLi = 1) and, consequently, the maximum fanning angle in that sublayer
becomes fixed, possibly at a value j m £ ` j . If a plastic hinge develops in the pile at SLi less
than 1, then h will be limited, but BC, and pi will continue to grow until Ai is equal to Ault or pi is
equal to ( pult) i.
5- 22
5.11 VERTICAL SIDE SHEAR RESISTANCE
As seen in Fig. 5- 21, the vertical side shear stress distribution around the shaft cross section is
assumed to follow a cosine function. It is assumed that there is no contact ( active pressure) on
the backside of the shaft due to the lateral deflection. The peak ( q) of side shear stress develops
at angle q = 0 and decreases to zero at angle q = 90o. The total vertical side shear force ( Vv)
induced along a unit length of the shaft is expressed as
V q r d q r Dq v = ò = = / 2
0
/ 2
0
2 p cos 2 ( sin ) p q q q ( 5- 50)
and the induced moment ( Mx- x) per unit length of the shaft is given as
8
( sin 2
2
1
( cos 2 1)
( cos2 1)
2
1
2
2 ( cos ) ( cos ) 2 cos
/ 2 2
0
2
/ 2
0