ARIZONA DEPARTMENT OF TRANSPORTATION
REPORT NUMBER: FHWAAZ99483
OPTIMIZATION OF DRILLED
SHAFT GROUP SPACING
Interim Report
Prepared by:
Kenneth D. Walsh
William N. Houston
Sandra L. Houston
Daniel N. Frechette
Brent Douthitt
Arizona State University
Civil Engineering Department
Tempe, Arizona 85287
November 1999
Prepared for:
Arizona Department of Transportation
206 South 17th Avenue
Phoenix, Arizona 85007
in cooperation with
U.S. Department of Transportation
Federal Highway Administration
The contents of this report reflect the views of the authors who
are responsible for the facts and the accuracy of the data
presented herein. The contents do not necessarily reflect the
official views or policies of the Arizona Department of
Transportation or the Federal Highways Administration. This
report does not constitute a standard, specification, or
regulation. Trade or manufacturer's names which may appear
herein are cited only because they are considered essential to
the objectives of the report. The U.S. Government and the State
of Arizona do not endorse products or manufacturers.
Technical Report Documentation Page
1 . Report No. 2. Government Accession No.
FHWAAZ99483
4. Title and Subtitle
INTERIM REPORT FOR PROJECT NO. T991300060
Optimization of Drilled Shaft Group Spacing
3. Recipient's Catalog No.
5. Report Date
November, 1999
6. Performing Organization Code
7. Author 8. Performing Organization Report No.
Kenneth D. Walsh, William N. Houston, Sandra L. Houston, Daniel
N. Frechette, and Brent Douthitt
9. Performing Organization Name and Address 10. Work Unit No.
ARIZONA STATE UNIVERSITY
MAIL CODE 0204 11. Contract or Grant No.
TEMPE, ARIZONA 852870204 SPR483
12. Sponsoring Agency Name and Address 13.Type of Report & Period Covered
ARIZONA DEPARTMENT OF TRANSPORTATION FINAL INTERIM
206 S. 17TH AVENUE
PHOENIX, ARIZONA 85007 14. Sponsoring Agency Code
1 5. Supplementary Notes
16. Abstract
The report presents a summary of findings from an assessment of the technical literature, experience of
engineers, and unpublished reports on lateral loads on pile groups. Specific interest is adopted in the
design methods for drilled shafts, in particular of drilled shafts installed under similar conditions to those
common in Arizona. These conditions were determined through a file search of asbuilt drawings for
ADOT abutments supported on drilled shafts. Nine design methods were identified and are summarized
herein. Based on a survey of practice, the most important of these appear to be the group reduction
factor, the modulus of subgrade reaction reduction, and the pmultiplier. Each of these methods was
compared to the other. A number of states, including Arizona, were found to interpret the factors
presented in AASHTO 4.6.5.6.1.4 as group reduction factors, however, an evaluation of the apparent
source documents indicates that they were intended for use as modulus of subgrade reaction reduction
factors. Factors available in the literature for all methods were found not to represent Arizona soil or
structural conditions well, so a series of finite element models and field load tests are recommended.
Appendices B, C, D, E, F, G, H, I are available from the Arizona Transportation
Research Center upon request.
17. Key Words
Pile Groups, Lateral Loads, Pile Foundations,
Drilled Shafts Design Methods, Soil Models,
Evaluation of Practice, Abutments, Condition
Assessment
1 9. Security Classification 20. Security Classification
Unclassified Unclassified
18. Distribution Statement
21. No. of Pages 22. Price
312
APPROXIMATE CONVERSIONS TO SI UNIT S
Symbol When You Know Multlply By To Find Symbol
In
It
yd
ml
oz
lb
T
fl oz
gal
fl I
yd'
Inches
leet
yards
mlles
square Inches
square leet
square yards
square mllee
acres
ounces
pounds
short tons (2000 lb)
fluid ounces
gallons
cubic reet
cubic yards
LENGTH
2.54
0,3048
0.914
1.81
AREA
8,452
0,0929
0,838
2.59
0.396
MASS (weight)
28,35
0.454
0,907
VOLUME
29.67
3.785
0,0328
0.786
.
canllmatars
meters
meters
kllometara
centimeters squared
meters squared
meters squared
kllometers squared
hectares
grams
kllograms
m egagrams
mllllmeters
Hiers
meters cubed
meters cubed
Nole! Volumea greater than 1000 l shall be shown In m •.
Fahrenheit
temperature
TEMPERATURE (exact)
5/9 (arter
subtracting 32)
Celslus
temperature
These factors conrorm to the requirement or FHWA Order 5190, 1A
• SI la the symbol for the International System of Measurements
cm
m
m
km
Icm
m•
m'
km 1
ha
g
kg
Mg
ml
L
m'
m•
Symbol
mm
m
yd
km
mm
m•
yd'
ha
g
kg
Mg
ml
L
m'
m'
APPROXIMATI! CONVl!RSIONS TO 81 UNITS
When You Know
mllllmetere
meters
meters
kllometers
mllllmeters squared
meters squared
kllometers squared
hectares (10,000 m 1 )
Multlply By
LENGTH
0.039
3.28
1.09
0.821
ARl!A
0,0018
10,784
0.39
2.53
MASS (weight)
grams
kllograma
megagrams (1000 kg)
mllllmeters
Hiers
meters cubed
meters cubed
0.0353
2.205
1,103
VOLUME
0,034
0.284
35,315
1.308
To Find
lnchee
feet
yards
mllas
square Inches
square feet
square mlles
acres
ounces
pounds
short tons
fluld ounces
gallons
cubic feet
cubic yards
TEMPERATURE (exact)
Celslus
temperature
9/5 (then
add 32)
Fahrenheit
temperature
' 32 98,8 212 ° F
•40 ° F O 140 80 120 180 200 J I I I I I I I d1 I I I I I I I I II I I I I I
I I I I I I I I t I I I I I I
·40 ° C 20 0 20 40 80 80 100 ° 0
37
symbol
In
ft
yd
ml
ln1
ft I
ml'
80
oz
lb
T
fl oz
gal
n•
yd'
OF
TABLE OF CONTENTS
Executive Sumary .......................................................................................................... i
1.0 Introduction ............................................................................................................. l
2. 0 Sumary of Literature and Practice ......................................................................... 3
2.1 Literature Review ....................................................................................................................... 3
2.1.1 Methods of Analysis .......................................................................................................... 3
2.1.2 Single Pile Problem ........................................................................................................... 3
2.2 Description of Analytical Approaches ....................................................................................... 11
2.2.1 Elastic Analysis ............................................................................................................... 12
2.2.2 Hybrid Analysis ............................................................................................................... 14
2.2.3 Group Reduction Factor Design ....................................................................................... 16
2.2.4 Coeficient of Lateral Subgrade Reaction Reduction ........................................................ 30
2.2.5 Pmultiplier Design ......................................................................................................... 35
2.2.6 LoadandResistance Factor Design (LRFD) Procedure ................................................... ·40
2.2.7 Group Amplification Procedure ....................................................................................... 47
2.2.8 Strain Wedge (SW) Method ............................................................................................. 55
2.2. 9 Finite Element Modeling (FEM) ...................................................................................... 56
2.3 Comparison of Analytical Approaches ...................................................................................... 58
2.3.1 Definitions ...................................................................................................................... 58
2.3.2 Boundary Conditions ....................................................................................................... 59
2.3.3 Problems for Analysis ...................................................................................................... 62
2.3.4 Group Reduction Factor Design ....................................................................................... 62
2.3.5 Coeficient of Lateral Subgrade Reaction Reduction ........................................................ 63
2.3.6 Pmultipliers ................................................................................................................... 68
2.4 Finite Element Modeling .......................................................................................................... 73
2.4.1 Elements ......................................................................................................................... 73
2.4.2 Material Behavior ............................................................................................................ 75
2.4.3 Computer Program Requirements .................................................................................... 81
2.4.4 Computer Programs Investigated ..................................................................................... 84
2.5 Sumary ofPractice. ................................................................................................................ 94
2.5.1 State Departments of Transportation ............................................................................... 94
2.5.2 Maricopa County Area Consultants ................................................................................. 97
3.0 Sumary ofHistoric Use ..................................................................................... 101
3 .1 Project Identification and Plan Review ..................................................................................... 101
3.1.1 Drilled ShaftLength ....................................................................................................... 101
3 .1. 2 Drilled Shaft Diameter ................................................................................................... 102
3.1.3 Drilled Shaft Group Geometry ........................................................................................ 103
3.1.4 Drilled Shaft Group Spacing .......................................................................................... 105
3.1.5 Soil Conditions ............................................................................................................... 106
3.2 Performance of Structures ........................................................................................................ 107
3.2.1 Progress Report on Abutment Structure Inspections by WTI.. ......................................... 107
4.0 Data Gaps ............................................................................................................ 109
4.1 Drilled ShaftLength ................................................................................................................ 109
4.2 Drilled Shaft Diameter ............................................................................................................ 110
4.3 Drilled Shaft Group Geometty ................................................................................................. 110
4.4 Drilled Shaft Group Spacing .................................................................................................... 112
4.5 Soil Conditions ........................................................................................................................ 112
4.6 Boundary Conditions ............................................................................................................... 113
5. 0 Summary and Recommendations for Finishing Study ........................................... 115
5.1 Sumary ................................................................................................................................ 115
5 .2 Recommendations for Further Study ........................................................................................ 118
Table 2.1: Pile and Soil Properties for Example ............................................................... 8
Table 2.2: Reduction Factors for Group Reduction Factor Design ................................. 16
Table 2.3: Definitions of Group Reduction Factors ........................................................ 18
Table 2.4: Comparison ofModulus ofSubgrade Terminology ........................................ 24
Table 2.5: Range in TValues for a Pile of Very Low El. ................................................ 25
Table 2.6: Range in TValues for Comon Prototype Piles ............................................ 25
Table 2. 7: Reduction Factor, R vs. Pile Spacing ............................................................. 32
Table 2.8: Pmultipliers vs. Row Position ...................................................................... 38
Table 2.9: Summary of Inherent Soil Variability and Measurement Variability Index (after
Phoon et al. 1995) ................................................................................................. 46
Table 2.10: Moment Coeficients Am and Bm (after Matlock and Reese 1961) ................ 52
Table 2.11: Minimum Length Criterion (after Duncan et al., 1994) ................................ 55
Table 2.12: Eficiency versus Group Reduction Factor Design ....................................... 63
Table 2.13: Results of Problem 1 using DM7.2 ............................................................. 64
Table 2.14: Efect of Slenderness Ratio using DM7.2 (Problem 2) ................................ 65
Table 2.15: Free Head using DM7.2 (Problem 3) .......................................................... 66
Table 2.16: FixedHead using DM7.2 (Problem 3) ........................................................ 66
Table 2.17: Results of Problem 1 using pmultipliers ...................................................... 70
Table 2.18: Efect of Slenderness Ratio using pmultipliers (Problem 2) ......................... 70
Table 2.19: Free Head using pmultipliers (Problem 3) ................................................... 71
Table 2.20: Fixed Head using pmultipliers (Problem 3) ................................................. 72
Table 2.21 Description of Reasons for Rejecting Programs ............................................ 84
Table 2.22 Programs Rejected ....................................................................................... 85
Table 2.23: Design Procedures for State DOTs .............................................................. 95
Table 2.24: Allowable Lateral Deflection for State DOTs .............................................. 97
Table 2.25: Survey Results for Geotechnical Consultants ............................................... 98
Table 2.26: Survey Results for Structural Consultants .................................................... 99
Table 4 .1: Boundary Conditions for Pile Caps in Literature .......................................... 114
Figure 2.1: Beam on Elastic Foundation (Symbols defined on Page 6) ............................. 4
Figure 2.2: Element from BeamColumn (After Hetenyi, 1946) ........................................ 5
Figure 2.3: Example py curve ......................................................................................... 7
Figure 2.4: Example LoadDeflection Curve .................................................................... 8
Figure 2.5: Influence Factors IpF Fixed Head Pile (From Poulos 1971a) ........................ 11
Figure 2.6: Increased Deflection of Pile due to Elastic Interaction of Piles in the Group . 13
Figure 2.7: 10 Pile Group used in Example Analysis ...................................................... 14
Figure 2.8: Efect of Basis ofDefinition ......................................................................... 20
Figure 2.9: Efect ofFixity Condition ............................................................................ 21
Figure 2.10: Group Reduction Factor versus deflection for FullScale Tests in Clay ....... 22
Figure 2.11: Group Reduction Factor versus Deflection for FullScale Tests in Sand ..... 22
Figure 2.12: Group reduction factor versus Deflection for Model Pile in Sand (From
Sarsby, 1985) ......................................................................................................... 23
Figure 2.13: Pile Group in Very Soft Material and Lateral Load Remaining in Pile with
Depth .................................................................................................................... 26
Figure 2.14: Pile Group in Uniform Material and Accompanying Load Remaining in Pile
with Depth ............................................................................................................. 26
Figure 2.15: Efect of Type of Study for Fixed/Fixed Group reduction factor ................. 27
Figure 2.16: Efect ofRelative Stifness of Pile and Soil ................................................. 28
Figure 2.17: Prototype Scale Results ............................................................................. 29
Figure 2.18: Influence values for modulus of subgrade reaction reduction procedure for
pile fixed against rotation (from DM7.2 1986) ....................................................... 33
Figure 2.19: Comparison ofpy curves for a Single Shaft and a Shaft in a Group at a
Depth of 5 ft . ......................................................................................................... 34
Figure 2.20: Loaddeflection curve for pmultiplier example ........................................ :37
Figure 2.21: Example of pmultiplier .............................................................................. 39
Figure 2.22: Identification of Rows with respect to Applied Load .................................. 39
Figure 2.23: LoadDeflection Curves (a) Sand; (b) Clay (After Duncan et al. 1994) ...... . 49
Figure 2.24: Moment Deflection Curves: (a) Sand; (b) Clay (After Duncan et al. 1994). 50
Figure 2.25: Nonlinear Superposition ofDeflection due to Load and Moment: (a) Step 1;
(b) Step 2; (c) Step 3; (d) Step 4; (e) Step 5; (t) Step 6 (From Duncan et al , 1994)51
Figure 2.26: LoadMoment Curves: (a) Clay; (b) Sand .................................................. 51
Figure 2.27: Strain Wedge (From Ashour et al., 1998) .................................................. 56
Figure 2.28: Failure Modes for Free, Laterally Loaded Piles: (a) Flexible (b) Rigid (After
Broms, 1965) ......................................................................................................... 61
Figure 2.29: Failure Modes for Fixed Laterally Loaded Piles: (a) Flexible; (b)
Intermediate; (c) Rigid (After Broms, 1965). .......................................................... 61
Figure 2.30: Relationship between Eficiency and Reduction Factor using DM7.2
(Problems 1 and 2) ................................................................................................. 65
Figure 2.31: Results of Problem 3 using DM7.2 ........................................................... 67
Figure 2.32: Eficiency versus Deflection for Coeficient of Lateral Subgrade Reaction
Reduction using COM624P for Sand ..................................................................... 68
Figure 2.33: Efect of Pile Slenderness Ratio using pmultipliers (Problem 2) ................. 71
Figure 2.34: Results of Pile Head Fixity using pmultipliers (Problem 3) ........................ 72
Figure 2.35 Laterally Loaded Pile with Gap and Slip Regions ........................................ 74
Figure 2.36 StressStrain Curves from Triaxial Tests ..................................................... 77
Figure 2.37 Yield Surfaces in the Deviatoric Plane ......................................................... 79
Figure 2. 3 8 Yield Surface in the Meridional Plane with Hardening ................................. 81
Figure 2.39: Yield surface with cap in meridional plane .................................................. 82
Figure 3 .1: Drilled Shaft Length .................................................................................. 102
Figure 3 .2: Drilled Shaft Diameter ............................................................................... 103
Figure 3.3: Pile Layout; (a) InLine, (b) Staggered ....................................................... 103
Figure 3.4: Staggered versus InLine Geometry ........................................................... 104
Figure 3. 5: Number of Shafts in Group ........................................................................ 104
Figure 3.6: Normalized Diagonal Spacing ofDrilled Shaft Groups in Arizona .............. 105
Figure 3.7: Normalized RowRow Spacing ofDrilled Shaft Groups in Arizona ............ 106
Figure 3.8: Percentage of Projects Identified versus various soil conditions .................. 106
Figure 4 .1 : Drilled Shaft Length .................................................................................. 109
Figure 4.2: Drilled Shaft Diameter ............................................................................... 110
Figure 4. 3: Staggered versus InLine Geometry ........................................................... 111
Figure 4 .4: Number of Shafts in Group ........................................................................ 111
Figure 4.5: Normalized Diagonal Spacing ofDrilled Shaft Groups in Arizona .............. 112
Figure 4.6: Soil conditions at testing site locations ....................................................... 113
Figure 4. 7: Depth to Water Level ................................................................................ 113
APPENDICES:
Appendix A
AppendixB
Appendix C
AppendixD
List of References Cited ....................................................................... A1
Elastic Analysis Method: Figures and Example .....................................B 1
Hybrid Method Example ....................................................................... C1
Tabular Summary of Literature Used in this Report ..............................D 1
AppendixE
AppendixF
Appendix G
AppendixH
Appendix I
Coeficient ofLateral Subgrade Modulus Charts ................................... E1
Department of Transportation and Maricopa County
Consultant Protocols .............................................................................F 1
Table of Plan Review Information ......................................................... G1
Department of Transportation Contacts ................................................H 1
Western Technologies Inc. Report ......................................................... I1
Executive Summary
The objective of this report was to develop a deeper understanding of laterally loaded pile group behavior.
This understanding was to be developed through an extensive literature review and interviews with
practitioners in the consulting and Department of Transportation (DOT) communities and prominent
researchers. Information gathered form this phase was augmented with an evaluation of the conditions of
use of laterally loaded pile groups in Arizona transportation projects. The project consist of 10 Tasks,
namely:
Task 1: Kickof Meeting
Task 2: Information Review
Task 3: Define Curent Usage
Task 4: Evaluate Analytical Approaches
Task 5: Interim Report
Task 6: Analytical Analysis
Task 7: Design Field Load Tests
Task 8: Reports
Task 9: Executive Presentation
Task 10:Technical Presentation
This report is submitted in accordance with the agreed scope of work for Task 5, and is intended to
sumarize the results of our eforts on Tasks 14. The report specifically covers the objectives of
sumarizing the state of the practice in laterally loaded pile group design and analysis, describing in detail
analytical approaches and describing common uses of pile groups under lateral load in Arizona. Based on
these results, recommendations are provided for completing the analysis and design of field testing portions
of this study, Tasks 6 and 7.
Summary of Literature and Practice
This chapter consisted of a number of subtasks. The first subtask involved reviewing information from a
number of sources: technical literature, unpublished reports, ongoing research, and experience of engineers.
The second subtask involved determining how common analytical methods compare with each other for
Arizona conditions. The remaining subtask involved determining what constitutive models might be
appropriate for finite element modeling and selecting an appropriate code.
Information Review
Technical Literature
The review of the technical literature included both results of load tests and experiments on groups of
laterally loaded piles and methods of analysis. The results of the load tests are too numerous to explain
here, however, the general trend was that fullscale tests showed less interaction between piles within the
group than model tests. There were 9 various methods of analysis for laterally loaded pile groups found in
the literature. These methods were as follows:
1. Elastic analysis
2. Hybrid analysis
3. Group reduction factor design
4. Coeficient of lateral subgrade reaction reduction
5. pmultiplier design
6. Load and resistance factor design (LRFD)
7. Group amplification procedure
8. Strain wedge method
9. Finite element modeling
Of these procedures the most commonly used are the group reduction factor method, coefficient of lateral
subgrade reaction reduction, and pmultiplier design. The group reduction factor method involves
developing a loaddeflection curve for a single pile. The engineer then selects an allowable deflection and
then enters the curve and atains the allowable load. This load is then multiplied by the group reduction
factor to obtain the allowable load for a pile in the pile group. The coeficient of lateral subgrade reaction
reduction method reduces the coeficient of lateral subgrade reaction, which is then used in the design
calculations for the pile group. The pmultiplier approach uses a pmultiplier that is aplied to the paxis of
the py curve. The analysis is then performed with the new py curve and a new loaddeflection curve is
developed.
Experience of Engineers
In an efort to tap the experience of engineers both locally and nationally a survey was conducted on each
level. There were three areas of emphasis for the surveys: pile group design procedure, deflection criteria,
and how are driled shafts designed. As stated previously, there were two geographical areas of interest for
the survey: local and nationaly.
Locally an efort was made to contact both geotechnical and structural engineers who have been involved
in the design of laterally loaded pile groups. From the surveys it was clear that Maricopa County area
consultants follow the ADOT recomendation, which is to use the AASHTO recommendations, and the
reduction factor is applied to the capacity of the single pile. The static deflection is usually kept less than 1
[in]. The comon way to design laterally loaded piles is to use py based computer programs, COM624
and LPIT.,E.
On a national scale all the departments of transportation were contacted and a survey conducted. When
contacting the various state DOTs both structural and geotechnical engineers were contacted depending on
whom was the most knowledgeable about the design procedure. The survey results from the state DOTs
for design procedure are presented below:
Analysis AASHTO Pmultiplier Elastic
Method 15 12 1
How Factors Capacity Soil Prop. Don't Know
Applied
12 3 0
The results of allowable deflections for the various state D0Ts at both static and seismic levels are
presented below:
Static Seismic
< l" > l" Not Sure Varies <2" >2" Not Sure
Not
Varies
Considered
19 3 2 21 7 2 7 17 12
The comon way to design laterally loaded piles is to use py based computer programs, COM624 and
LPil.,E.
Unpublished Reports and Ongoing Research
An efort was made by the researchers to find unpublished reports on lateraly loaded pile groups. The
main groups contacted were Maricopa County engineers, state department of transportation engineers
ii
around the country, and prominent researchers in the field of deep foundations. These inquiries pointed the
researchers towards very few unpublished reports.
With regards to ongoing research the researchers were able to obtain a copy of the interim report of
NCHRP 2 49 titled Static and Dynamic Lateral Loading of Pile Groups (O'Neill et al., 1997). The report
provided some insight, however, there were no results presented in the report.
During the technical literature review the procedure outlined in theMSHTO 1996 Standard Specifications
for Highway Bridges was reviewed. As ADOT and others have interpreted AASHTO, the specification for
groups oflaterally loaded drilled shafts (Sec. 4.6.5.6.1.4) states that the "efects of group action for inline
CTC (centertocenter)< 8B amy be considered using ....t he artio of lateral resistance ofs afht in group to
single shaft." There is no other guidance on how to apply these ratios to the group action problem. Many
have interpreted the word "resistance" in its common usage elsewhere in ASHTO as implying force, and
accordingly developed an understanding that the values given for lateral load resistance are ratios of forces.
The source in AASHTO is the Canadian Geotechnical Society (1985). This was interpreted to mean the
Canadian Foundation Engineering Manual (CFEM) which is published by the Canadian Geotechnical
Society. This reference recommends applying the reduction factors to the coeficient oflateral subgrade
reaction rather than reducing the lateral force capacity of the single pile. This method was further
substantiated when examining the origin of the CFEM recommendations, in Davisson (1970). It is clear
that the source referenced in the AASHTO specifications has a diferent interpretation than that in use by
most states.
Comparison of Analytical Methods
The second subtask, comparing analytical approaches, was performed for a typical drilled shaft group
under Arizona Conditions. The analytical approaches that were compared were the group reduction
method, coeficient of lateral subgrade reaction reduction method and pmultiplier method. The
comparison of these 3 methods to Arizona conditions is very dependent on the pile slenderness ratio (LIT).
There are three categories ofL/I': rigid, (L/f < 2); intermediate, (2 <LIT< 4); and flexible (L/I' >4). The
results are dificult to sumarize but the general trends of eficiency are presented below.
Pile Slenderness Group Reduction Coeficient of Lateral Pmultipliers
Factor Subgrade Reaction
Fixed1 1 Free Fixed Free Fixed Free
Rigid Same* Same Same Same Same Same
Intermediate Same Same Ef. > Coef. Ef. > Coef. Ef. > pmult Same
Flexible Same Same Ef. > Coef. Ef. > Coef. Ef. > pmult. Ef. > pmult
1 Head fitxty condi.t.i on
* Same means that the eficiency was equal to the reduction factor used in the procedure
For example a 42 in drilled shaft, 30 ft long was used at a cemented silt site, which is categorized as an
intermediate pile. Using a coeficient of lateral subgrade reaction reduction of 0.25 results in an eficiency
of O. 4 2. The entries in the table represent a comparison of the reduction factor used in the procedure to the
overall eficiency of the group, where eficiency is defined as the ratio of loads used to compare the load in
the pile group divided by the load in a single pile times the number of piles in the group, at the same
deflection. In equation form eficiency is as follows:
£ =
(
pgroup
)
=
(
pgroup/pile
)
P.mgle X
N
I'. P.mgle I'.
iii
where:
E = eficiency
P group = the load carried by the pile group
PsingJe = the load carried by the single pile
P group/pile
= the average load carried by each pile in the group
N = number of piles in the group
A = the deflection at which the loads were compared
Finite Element Modeling
The final subtask was selection of constitutive models and programs for analysis of a laterally loaded
driled shaft group. In an efort to select an appropriate finite element code 96 programs were examined.
The criteria for these programs were element types, material models, operating system, preprocessing,
mesh generation, and solution methods. From the search of finite element programs three programs were
selected for use: ABAQAS, MSC/NASTRAN, and ADINA. These three programs were selected as
possibilities with the emphasis being on using ABAQAS.
There are several commercially available programs, such as GROUP and FLPIER that are available to
solve the laterally loaded pile group problem. However, these programs and others like them will not be
studied by the researchers. The reason being that these programs are based on empiricism. They assume
the behavior of the soil through py curves. In order to perform a detailed analysis of the problem it is
necessary to be able to analyze from scratch. This means without any preconceived notion of the solution.
It is important to note that the FEM work is mainly research work and the subsequent recommendations
will not require the use of FEM.
Summary of Historic Use
The objective of this task was to determine typical applications for which laterally loaded drilled shaft
groups are use in Arizona and to develop an understanding of the performance of these drilled shaft groups
in the field. The results of the first subtask are presented first. It was determined that the primary use for
drilled shaft groups in Arizona was in bridge abutments. The ADOT files were searched and the plans
reviewed for abutments founded on driled shafts. There were 120 abutments that were found to be located
on groups of driled shafts. From the plan review the diameter, length, group geometry, centertocenter
spacing, and soil type were obtained. The following table provides the ranges of values that represent
Arizona conditions.
Parameter
Length
Diameter
Centertocenter spacing
Group Geometry
Inline vs. Staggered
Number of piles in group
Soil Type
Pile Cap
In Contact with the Soil (Cap friction)
Below Soil Surface (Passive Resistance)
Maximum Ran e
3140 ft
36 in
33.5 [SID]
Staggered
1120
Sandyclay w/ cementation and SGC
Almost All
Almost Al
The second subtask was determining how groups of driled shafts are performing in the field. Field visits
and inspections were conducted by WTI. To date WTI has inspected 53 abutments. The results of the site
iv
visits were that there is no significant movement or damage of the abutments, although development of
such movements would be a function of construction sequencing, time, and other variables.
Data Gaps
The soil conditions and drilled shaft geometry commonly used in Arizona were compared to those
identified in the literature. Eight projects were identified in the literature for which 11 load tests were
performed. The ranges in Arizona conditions that were not well represented in the literature are presented
in the following table.
Parameter
Length
Diameter
Centertocenter spacing
Group Geometry
Inline vs. Staggered
Number of piles in group
Soil Type
Pile Cap
In Contact with the Soil (Cap friction)
Below Soil Surface (Passive Resistance)
Data Ga s
4150 ft
30 and 36 in
3.54 [SID]
Staggered
> 10 shafts
Clayeysilt w/ cementation and SGC
2 cases in literature
1 case I literature
These data gaps are of interest because if it is thought that a fullscale load test is necessary than it should
be designed to fill in as many gaps as possible. If a test was to be conducted the most important aspect to
test would be the case of stagger due the absence of any data in the drilled shaft literature.
Conclusions
From the development of the interim report there were three important conclusions. The first conclusion
deals with the AASHTO Specifications (1996) for closely spaced laterally loaded drilled shafts. As
previously stated, the AASHTO specifications state that the "efects of group action for inline CTC
( centertocenter) < 8B may be considered using .... the ratio of lateral resistance of shaft in group to single
shaft." Many states including Arizona implement the AASHTO specifications as load reduction factors.
The source cited in the AASHTO specifications (CGS, 1985) and Davisson (1970) recomend using the
reduction factors to reduce the modulus of subgrade reaction. It is clear that the sources referenced in the
AASHTO specifications have a diferent interpretation than most states that are using the ASHTO
specifications.
The second conclusion is that field load tests should be conducted to fill in the data gaps. Based on the
comparison of the literature data and the data from Arizona uses it is clear that some data gaps exist,
specifically with regards to soil type. Arizona conditions have a large percentage of sites where the soil
conditions are predominantly either sand, gravel, and cobbles, or cemented finegrained material. To fill in
these data gaps it is necessary to conduct a fullscale field load test Such tests are beyond the scope of this
work. However, the design of a field load test will be undertaken as part of this project.
The third conclusion is that a finite element analysis should be conducted in the second phase of the
project. This conclusion is based on the fact that curently there are no recomendations that fit Arizona
conditions and that fullscale load tests are too costly to perform parametric studies. In an efort to develop
group reduction factors for Arizona conditions a FE analysis is to be conducted. The model will first be
calibrated to actual fullscale tests for conditions that best match Arizona conditions. Once the model has
been calibrated the analysis will be expanded to general conditions which are representative of Arizona
transportation applications.
V
The presentation of the final report will be done in two parts. Part 1 will include the research results along
with any necessary background infonation on the laterally loaded pile group problem. Part 2 of the final
report will be written specifically for use by practitioners. It will provide practitioners with stepbystep
guidelines as to how to implement the results of the curent research into practice with examples. This
document will be used in all technical presentations. If apropriate the procedure will separate the efect of
the pile and pile cap, and the format will allow the use of COM624P or other programs. The design
procedure will also include deformation (deflection), criteria
vi
1. 0 Introduction
This report is submitted to the Arizona Department of Transportation in accordance with
the requirements of Project Number T991300060. This project was started in March of
1999, with the objective of developing a deeper understanding of laterally loaded pile
group behavior. This understanding was to be developed through literature review and
interviews with practitioners in the consulting and DOT communities and prominent
researchers. This information was to be augmented with an evaluation of the conditions of
use of laterally loaded pile groups in Arizona transportation projects. The project consists
of 10 Tasks, namely:
Task 1:
Task 2:
Task 3:
Task 4:
Task 5:
Task 6:
Task 7:
Task 8:
Task 9:
Task 10:
Kickof Meeting
Information Review
Define Current Usage
Evaluate Analytical Approaches
Interim Report
Analytical Analysis
Design Field Load Tests
Reports
Executive Presentation
Technical Presentation
This report is submitted in accordance with the agreed scope of work for Task 5, and is
intended to summarize the results of our eforts on Tasks 14. The report specifically
addresses the objectives of summarizing the state of the practice in laterally loaded pile
group design and analysis, describing in detail analytical approaches (including the finite
element approach), and describing comon uses of pile groups under lateral load in
Arizona. Based on these results, recommendations are provided for completing the
analysis and design of field testing portions of this study, Tasks 6 and 7.
The report consists of a number of chapters. Following this introductory chapter is
Chapter 2, Information Review. This chapter includes a review of the archival literature
and a number of directed reports relating to laterally loaded pile group behavior and
design procedures. A summary of a number of analytical methods, including example
calculations, is provided. These methods are then normalized for comparison using typical
Arizona soil conditions and pile geometries. An important analytical method, the finite
element approach, is summarized, along with our findings regarding potential programs
for use in Task 6. Finally, a summary of the state of the practice oflaterally loaded pile
group design procedures in Arizona and in other state DOTs is provided.
Chapter 3, Sumary of Historic Use, provides a description of the conditions of use of
laterally loaded pile groups in Arizona transportation applications. This information was
obtained by review of ADOT files to identify relevant projects, which were found to
number 120. Project plans were reviewed to obtain geometric characteristics, and then
the projects were visited to assess their current conditions. Where possible, the design
methods used for these projects were identified. The soil and drilled shaft conditions
1
common in Arizona are compared to those identified in the literature in Chapter 4, Data
Gaps. This process of comparison will allow for more focused field and analytical analysis
in the balance of this study, without duplicating those results already available in the
literature.
Finally, the report is summarized in Chapter 5, Summary and Recommendations.
Recommendations for completing Tasks 6 and 7 are found here.
The researchers would like to acknowledge the Technical Advisory Committee for this
project, chaired by Mr. Frank McCullagh of ADOT. Other members include Mr. Doug
Alexander, Mr. Gene Hansen, Mr. J.J. Liu, and Mr. Shafi Hasan of ADOT, Mr. Kamel
Alqalam ofFHWA, and Mr. Dan Heller of TY Lin. In addition, we are grateful to the
Steering Comittee members, including Mr. Keith Dahlen of AGRA Earth &
Environmental; Mr. Kenneth Ricker of Ricker, Atkinson, McBee and Associates; Mr.
Randolph Marwig of Western Technologies; Inc., Mr. Robert Turton ofHDR; and Mr ..
Dwaine Sergent of Kleinfelder.
The researchers are also grateful to the staf of ADOT records, specifically Mr. Sunil
Athalye, Ms. Wendy Fields and Ms. Marcia Hurley who helped the researchers with
finding projects and pulling files. We are also grateful to Professor Michael O'Neill of the
University of Houston, Professor Kyle Rollins of Brigham Young University, Professor
Dan Brown of Auburn University, and Professor Gary Norris of the University of Nevada,
Reno, who provided useful comments and suggestions and agreed to be interviewed for
this work.
2
2.0 Summary of Literature and Practice
2.1 Literature Review
This section contains a review of the available literature relating to drilled shaft and driven
pile group performance under lateral load. The objectives of this section are to present a
summary of methods of analysis commonly undertaken, to present values for empirical
factors commonly used in those methods, and to compare those methods from a design
standpoint. A number of variables are important to the laterally loaded pile group
problem, and these variables and their efects will be summarized. Finally, a historical
context for research related to the question will be presented.
2.1.1 Methods of Analysis
The prediction of the response to lateral load of a single pile is somewhat dificult. The
problem is complicated dramatically by the combination of piles into a group. Overlapping
stresses from one pile influence the response of the soil ahead of other piles. Soilstructure
interaction becomes pronounced in some cases, but not in all cases. Due to the
complexities of the problem, simplifications are usually adopted which transform the
problem to some function of the single pile problem. What follows will be a sumary of
the methods that have been encountered in the literature. More emphasis will be given to
those methods believed to be commonly used by DOTs or consultants based on survey
data to be presented later in this report.
2.1.2 Single Pile Problem
The group problem is often addressed by reference or scaling based on the single pile
response. In general, there are 4 potential solution methods for the single pile problem:
1. py curve analysis
2. Elastic analysis
3. Field load testing
4. Modulus of subgrade reaction
2.1.2.1 py Curve Analysis
The use of py curves in obtaining a solution to the problem of a single pile under lateral
load is fairly straightforward analysis. The solution is primarily based upon beam on
elastic foundation (BEF) theory by Hentenyi (1946), in which the foundation is modeled
as a Winkler medium. In a Winkler medium, the pressure in the foundation is proportional
at every point to the deflection occurring at that point, and is independent of the pressure
or deflections occurring elsewhere in the foundation. This relationship between pressure
and deflection implies a lack of continuity in the foundation. The behavior of the
3
foundation is as if it is composed of rows of closely spaced but independent springs. The
diferential equation is derived below using the following assumptions:
1. The pile is straight with uniform crosssection ofhomogenous material, and is
described by its stifuess EI.
2. Pile has longitudinal plane of symmetry
3. Pile stays in the elastic range
4. Pile properties are the same in tension and compression
5. Pile is only subjected to static loads
6. Deformation due to shearing stresses are small
7. Transverse deformations of the pile are small
N

X  p = ky

X
y
Figure 2.1: Beam on Elastic Foundation (Symbols defined on Page 6)
Hetenyi (1946) gave the derivation of the diferential equation for the beamcolumn on a
Winkler foundation. He assumed that a beam on an elastic foundation is subjected to
horizontal loading and vertical compressive forces N acting at the centroid of the end
crosssection of the beam Figure 2.1.
If an infinitely small element, bounded by two horizontals a distance dx apart, is cut out of
this bar (see Figure 2.2), the equilibrium moments (ignoring second order terms) leads to
the equation
L MA =( M +c/M) M  NdyQvdx = 0 Equation 2.1
or
Equation 2.2
4
Diferentiating Equation 2.2 with respect to x, the following equation is obtained
Prior to continuing it is important to note the following identities:
where:
dQV =p
dx
b = width of the beam
ko = coeficient of lateral sub grade reaction
y = deflection
k = bko = modulus of subgrade reaction
Substituting in the appropriate identities; Equation 2.3 becomes:
Ef
d4y +N
d
zy +ky=O
dx4 dx2
X __Y_ ___ N
dx
y+dy
X
y
N
Figure 2.2: Element from Beam.Column (After Hetenyi, 1946)
Equation 2.3
Equation 2.4
5
The direction of the shearing force is shown in Figure 2.2. The shearing force in the plane
normal to the deflection line can be obtained from the following:
Qn = Qv
cos0N sin0 Equation 2.5
Since 0 is usually small, one can assume the small angle relationships: cos 0 = I and sin 0
= tan 0 = dy/dx. Then Equation 2.5 becomes:
Equation 2.6
A summary of equations that are used in analyzing piles under lateral load are:
where:
d 3y dy
EI+N=Q
dx3 dx V
dy
=0
dx
p = soil reaction per unit length
Q = shear in the pile
M = bending moment in the pile
N = axial load
0 = slope of the elastic curve defined by the axis of the pile
Equation 2. 7
Equation 2.8
Equation 2.9
Equation 2.10
The py curve analysis solves the beam on elastic foundation (Equations 2.72.10) using a
Winkler medium as the soil model. Py curves describe the soil as a nonlinear spring to
characterize the forcedeformation characteristics of the soil. The secant of the py curve
is equal to the soil modulus, Es. The modulus is essentially only a computation device
(used to solve Equation 2.7) which varies with depth and pile deflection, and is not a
unique soil property. An example of a family of py curve is shown in Figure 2.3. The py
curves relate soil resistance (p) to pile deflection (y) at various depths below the ground
surface. In general the py curves are nonlinear and depend on several parameters of the
pile and soil. These parameters include the diameter of the pile, depth along the pile, and
the shear strength and unit weight of the soil.
6
:c 700 ++++++:=""""++1
.9! 600 ++=At+:'++++1
·!.9a
o 500 ++jj:::;;;;,,qttj
C
i
;::, 400 +V:l,"l++tt;
300 ++ 71'+=:l""""++tti UI ·; Cl)
200
rr:;.1r=t=2Ef==:+===9:=rr1
c: 100 +l,:;,,,"""l""'+!++tt ;
0+'+'+'+'1
0 1 2 3
y, Lateral Deflection of Pile [in]
Figure 2.3: Example py curve
The curves may be developed from site specific soil test data (see 2.1.2.2) or from
relationships developed from fullscale field load tests reported in the literature.
4
The secant of the py curve, Es, is used as input into Equation 2.7 (in terms ofp) which is
then solved for y at various depths, x. This process is repeated until compatibility between
the pile and soil is obtained. The analysis is then repeated for several load cases to obtain
a loaddeflection curve for the single pile. Reese and Matlock (1956) developed a hand
solution using a py analysis based on Es linearly increasing with depth. Matlock and
Reese (1960) later improved upon this method when they developed a model that could
handle a fixed form of variation of Es with depth. Matlock and Reese (1961) again
improved the py model developing a procedure that can handle arbitrary variations of
modulus with depth. The hand solution for this procedure is iterative and very lengthy.
However, the method has been programmed and is widely available in a number of
programs. The most widely used are COM624 and LPILE.
An example is performed to provide insight into the procedure. The pile that is used for
this example will be used in all further examples, so that comparisons between methods
can be made. The characteristics of the pile and the soil were taken from a laterally loaded
pile group at the intersection of Warner Road and the Price Freeway and are presented in
Table 2.1.
7
Table 2.1: Pile and Soil Properties for Example
• Diameter (D) = 42 in • Length (L) = 30 ft
• Modulus of Elasticity (Ep) = 4(10)6 psi • Undrained Shear Strength (Su)= 1 ksf
• Moment of Inertia (Ip)= 1.53(10)5 in4
• Relative Stiflhess Factor (T) = 134 in
• Coeficient of Variation of Sub grade Reaction (nh)= 13. 9 pci
• Soil Modulus (E1 ) = 2800 psi
An analysis was performed on the single pile described previously. The analysis was
performed using the computer program COM624 using py curves generated using the
procedure outlined by Matlock (1970). The loaddeflection curve in Figure 2.4 is the
result of the single pile analysis. From this curve a deflection of 0.38 in at a horizontal
load of 100 [kips] is obtained.
0 I'''''+''''+''''+'''.,
0 1
2.1.2.2 Field Load Testing
2
Deflection [in]
Figure 2.4: Example LoadDeflection Curve
3 4
The most straightforward approach to single pile design is to perform a field load test. A
field load test would consist of installing a pile at the site with adequate instrumentation to
measure the load and deflection. Once the data is obtained a loaddeflection curve can be
drawn for use in design. The engineer would enter the loaddeflection curve with either a
specified load or an allowable deflection and read of the corresponding deflection or load
respectively. If the test is done at full scale, the results can be used directly in design.
8
To allow for the translation of the results to any scale, the engineer needs to instrument
the pile with strain gauges prior to loading to measure the bending moment. This allows
the engineer to determine the magnitude of the moment along the length of the pile.
Graphical diferentiation of the moment curve produces a shear diagram, which can be
diferentiated again to produce a soil reaction curve (or py curve). The soil reaction
curve could then be used to estimate the behavior of other piles of diferent sizes as
described in section 2.1.2 .1. To obtain a complete curve of deflection with depth the
engineer can double integrate the moment curve. This allows the designer to determine
the point of fixity of the pile.
As an alternative to the installation of strain gauges, it is possible to simply measure the
load and deflection of the pile head, as described above, and use trial and error to find a
set of py curves which produces a good match to the measured loaddeflection curve.
Then COM624 can be used to calculate the shear and moment for design.
It is important to note that when a field load test is performed it is still necessary to
perform a geotechnical investigation to determine the spatial heterogeneity of the site. If
the site is relatively uniform, a single test may be suficient to characterize the site.
2.1.2.3 Elastic Analysis
Another approach to the single pile problem is to model the foundation as an elastic
continuum rather than isolated springs. An elastic continuum is an ideal, elastic,
homogeneous, isotropic mass having constant elastic parameters E and Vs (where Eis the
modulus of elasticity of the soil and Vs is Poisson's ratio of the soil). It represents the case
of complete continuity of the foundation. The most widely used solution for this
foundation type for laterally loaded piles was done by Poulos (1971a). The displacements
within the soil were evaluated using Mindlin's equation for horizontal displacement due to
a horizontal load within a semiinfinite mass. In this model the soil is represented as an
ideal elastic half space. Since it is an elastic continuum, possible local yielding between
the piles and the soil is not taken into account. The pile used in Poulos' (1971a) analysis is
assumed to be a thin rectangular vertical strip of width d, length L, and constant :flexibility
Erfp  The development of the relevant equations is lengthy and will not be repeated here.
Conveniently, Poulos developed a hand solution to the single pile problem using charts of
influence factors. Equations and influence factors were developed to solve for the
displacement, moment, and rotation of a single pile for both free and fixed head
conditions.
To illustrate Poulos' procedure, an example is presented of a fixed head pile subjected to
horizontal load. The remaining cases are fully described in Poulos (1971a) and will not be
discussed here. The pile and soil properties were presented in Table 2.1. The equation for
the deflection of a fixed head single pile at the ground surface is as follows:
I H
p pF EL Equation 2.11
s
9
where:
H = Applied horizontal load
Es
= Soil Modulus
L = Length of the pile
IpF = Displacement influence factor for fixedhead pile subjected to lateral load,
where:
IpF = function of KR, pile flexibility factor and LID, pile slenderness:
. 2
K   Equation 2.1 R

E L4
s
where:
Ep = Young's Modulus of the pile
Ip = Moment of Inertia of the pile
From the example pile
(4xl0 6 _
lb
2 )(1.53xl05 in4)
KR = m
4 = 1.30xl 02
( 2800 i:' )((30 fte n ))
L =
30 ft = 8 6
42 in(!)
D
12 in
Taking LID = 8.6 10, because Poulos' charts begin with an LID ratio of 10, Figure 2. 5
reveals a value of I pF
= 2.2. Applying a horizontal load (H) of 100 [kips] results in a
displacement, p, given by:
lOO ki s(
lOOO lb
p )
1 kip
p = 2.2  ,
( 2soo i:' )(3oft(1n ))
= 0.22 in
Note that this is substantially less than the previous estimate using py curves, 0.38 in,
indicating the importance of the nonlinear efects.
10
50,
20
Values of Lid
10
5
2.2 1
2
103
KR
1 10
Figure 2.5: Influence Factors lp F" Fixed Head Pile (From Poulos 1971a)
2.1.2.4 Modulus of Subgrade Reaction
The modulus of subgrade reaction is another method to solve the single pile problem.
However, it is dificult to separate it from the pile group method, so it will be presented in
that section (2.2.4).
2.2 Description of Analytical Approaches
There are a number of analytical methods used for the problem of a group of laterally
loaded piles. The list of analytical methods encountered during the literature search
includes the following:
1. Elastic analysis
2. Hybrid analysis
3. Group reduction factor design
4. Pmultiplier design
5. Modulus of sub grade reaction reduction
6. LoadandResistance Factor Design philosophy (LRFD)
7. Group Amplification Procedure
8. Strain Wedge (SW) Method
9. Finite Element Modeling (FEM)
11
These methods will be described individually so as to develop an understanding of each
method. Once this is accomplished, we will present a comparison, where possible, of the
methods and their results.
2.2.1 Elastic Analysis
The elastic analysis method has been applied to pile group behavior by Poulos (1971b).
Solutions are available for the following three cases:
1. A freehead pile group with all displacements equal
2. A freehead pile group in which equal horizontal load and or moment is applied to
each pile in the group.
3. A fixedhead pile group in which all piles displace the same amount
Of these three cases, the third is the most related to Arizona conditions and is the only one
described here. The remaining two cases are fully described in Poulos (1971b).
Approximate solutions for displacement and rotation have been obtained by assuming that
the principle of superposition holds. In other words, the increase in displacement of a pile
due to all the surrounding piles can be calculated by summing the increase in displacement
due to each pile in turn (Equation 2.13).
Equation 2.13
where:
a2 ,a3 ,a4 are the values of the interaction factors for pile 1 due to piles 2, 3, and 4.
Poulos provides figures of interaction factors, which are presented in Appendix B. The
interaction factor is a function of spacing between piles, angle between the piles or
departure angle (P)(Figure 2.6), and KR, (Equation 2.12). Once the interaction factors
have been determined the displacement of the group can be calculated. The ratio of the
displacement of the group, pg, to the displacement, p1, of a single pile carrying the same
load as a pile in the group is:
p
1.=l+a Equation 2.14
P1
Figure 2.6 also shows the influence of adjacent piles on deflection. In Figure 2.6 a1 is the
deflection due to the load on pile 1, a2 is the additional deflection of pile 1 due to the load
on pile 2 and so on.
12
p
Pile 1 Pile 2
Pile 3 Q 0 Pile 4
Figure 2.6: Increased Deflection of Pile due to Elastic Interaction of Piles in the Group
The general theory is now applied to the specific case of a fixed head pile group in which
all the piles have the same deflection. The equation to determine the displacement of any
pile k in the group with a fixed head subject to a horizontal load by superposition is:
where:
Equation 2.15
PF = the displacement of a single fixedhead pile due to a unit horizontal load
Hj = the load on pile j
CX.pFkj = the value of CX.pF for two piles coresponding to the spacing between piles k
and j and the angle B between the direction ofloading and the line joining
the centers of the piles k and j.
The theoretical value for the unit reference displacement, PF , for a single pile was
previously described in the single pile section. The only diference from the single pile
analysis is that the value of His 1 (i.e. load applied is 1 to obtain PF .
A 10 pile group was analyzed having the same pile properties as the one analyzed for the
single pile and having a group geometry as shown in Figure 2.7. For an average
horizontal load per pile of 100 [kips] the resulting deflection is 0.67 in. In order to obtain
a group reduction factor based on load, it is necessary to determine the amount ofload the
group can resist at a deflection level of 0.22 in (the deflection at a load of 100 kips from
elastic analysis). At a deflection level of 0.22 in the load carried by the group was 328
[kips] or 32.8 [kips/pile]. This results in a reduction factor of approximately 33%. The
step by step procedure is lengthy and, therefore, presented in Appendix B.
13
Load
+
D =42 in
f 0
to
0 0
0 0
(a) Plan View
Load
.,
(b) Elevation View
Figure 2.7: 10 Pile Group used in Example Analysis
The previous example was performed using the hand procedure as described in Poulos
(1971b). This method has been added to the programs PIGLET (Randolph, 1980) and
DEFPIG (Poulos, 1980). Some modifications that were made to the method include
eliminating the need to assume uniform elastic properties. However, the method still
doesn't consider the nonlinear behavior of the soil.
2.2.1.1 Advantages and Disadvantages
The primary advantage of the elastic method is the ease in understanding the theory behind
the method and the ability to follow a hand calculation. The elastic formulation does not
lend itself to consideration of nonlinear behavior so that it is less appropriate for higher
deflection levels, and there is some doubt as to whether pilesoil interaction is finally
accounted for, given that measured reduction factors for fullscale field tests greatly
exceed values calculated with this method.
2.2.2 Hybrid Analysis
The hybrid analysis combines the theory of soil nonlinearity for a single pile analysis along
with an elastic interaction analysis. The most widely used hybrid analysis was developed
by Focht and Koch (1973). Simply stated, the analytical method is a solution of:
where:
Ya =ys+ Yg
YG = total group deflection at load P per pile
Ys = single pile deflection at load P per pile
yg = additional deflection due to group efects
Equation 2.16
14
The theory behind this method is that the displacement of an individual pile subjected to
lateral load is large enough to create plastic strain. Due to the presence of plastic strain, Ys
is calculated using a nonlinear py curve method. The additional displacement of a single
pile due to the interaction with other piles in the group is assumed to be much smaller, and
is therefore calculated using an elastic analysis. The group efects are calculated using an
equation similar to Poulos' (1971b) approach. The only modification is replacing the
understood value of unity preceding the shear load, Hk, acting on pile k by a relative
stifness factor, R, defined as Ys divided by p. Here Ys is the displacement of the single pile
using nonlinear or py analysis, and p is the displacement of a single pile using elastic
analysis.
Equation 2.17
Equation 2.17 states that the total deflection of pile k is the "plastic" deformation, Ys, plus
the integrated elastic efect of loads on all other piles in the group, which has been defined
as yg .
An example calculation was performed using the same pile group used in the elastic
analysis (Figure 2.7). For an average lateral load per pile of 100 [kips], the resulting
deflection is O. 84 in. In order to obtain group eficiency based on load it is necessary to
determine the amount ofload the group can resist at a deflection level of0.38 in
(deflection of a single pile with a 100 [kip] load using nonlinear analysis). At a deflection
level of 0.38 in the load carried by the group was 454 [kips] or 45.4 [kips/pile]. This
results in a group reduction factor of (45.4/100), or approximately 45%. Note that this
value is derived for a higher deflection than that used for the elastic analysis in the
previous section. As for the elastic approach, the step by step hand procedure is lengthy
and is presented in Appendix C.
2.2.2.1 Advantages and Disadvantages
The hybrid analysis is an improvement over the elastic method, in that it accounts for the
nonlinearity of the soil stressstrain relationship when analyzing a single pile. However, it
does not account for nonlinear soil efects when dealing with the pile groups. The hand
procedure is long and time consuming, and to the authors' knowledge a commercial
program for this procedure is not available. Although the reduction factor of 45%
calculated with this method approaches the lower bound oflabmeasured reduction factors
for small scale model tests (Figure 2.15), it is still far below the reduction factors
measured for full scale pile groups in the field.
15
2.2.3 Group Reduction Factor Design
The group reduction factor design method requires, as input, a loaddeflection curve for
the single pile. The most common method of obtaining a loaddeflection curve is to use py
curve analysis. The engineer enters the loaddeflection curve so developed at an
acceptable level of deflection to obtain the single pile capacity. The single pile capacity is
then multiplied by a reduction factor to obtain the lateral load capacity of each pile in the
group at the deflection under consideration. In the literature, this reduction factor is often
called eficiency. Many engineers and regulatory agencies appear to apply the factors
specified by ASHTO as group reduction factors.
A brief example will serve to illustrate the group reduction factor design procedure.
Assume a single pile analysis has been performed using a py curve approach. Based on
this analysis, a single pile loaddeflection curve is produced, and at a load of 100 [kips] the
expected lateral deflection is 0.38 in (Figure 2.4). The spacing of the piles in the group is
three diameters (3D), therefore, a group reduction factor of 0.25 was used following one
interpretation of the ASHTO specifications. Using these parameters, the expected
horizontal capacity of each pile in the group, at a deflection of O .3 8 in is:
Pperpile = ( P.mglepiie)E = (100 kips)(0.25) = 25 kips
Until relatively recently, it was common in Arizona practice to use a group reduction
factor of 1. 0 for a spacing of at least 3D. With the arrival of the most recent ASHTO
specifications, these values were dramatically reduced (Table 2.2). It is important to note
that the factors presented in the AASHTO specifications maybe applied in another maner
discussed later. An interim policy was adopted based on a briefliterature study (Walsh et
al., 1998).
Table 2.2: Reduction Factors for Group Reduction Factor Design
Ratio ofLateral
Resistance of
Center to Center Shaft in Group to
Shaft Spacing for Historical
InLineLoading Practice
8B 1.00
6B 1.00
4B 1.00
3B 1.00
2.2.3 .1 Advantages and Disadvantages
Single Shaft
(AASHTO)
1.00
0.70
0.40
0.25
Interim Policy
Cap
above Cap in Contact
soil w/ soil
1.00 1.00
0.84 0.92
0.68 0.84
0.60 0.80
The advantages of the group reduction factor design method are its simplicity and ease of
application to design. The major disadvantage is that the group reduction factor is an
average value for the entire group, and thus does not account for variations from row to
16
row. Most recommendations provide only a single set of recommendations. Therefore,
these factors are applied to a range of conditions and deflection levels. The reduction
factors are based primarily on lab and field measurements, rather than theory. However, a
limited number of finite element analyses have produced results generally consistent with
field measurements. The procedure reduces the pile stifuess along with the soil stifness,
even though the pile stifuess is not afected by group interaction.
2.2.3 .2 Definitions of Group Reduction Factors
Aside from the values presented in Table 2.2, other reduction factor values can be found in
the literature. However, one of the most dificult challenges to overcome when one
assesses group reduction factors in the literature is the lack of a widely accepted
definition. Group reduction factors were found based on the following definitions:
• Computing the ratio of the average ultimate load on the piles in a pile group to the
ultimate load of a single pile of the same size as each of the piles in the group.
• Computing the ratio of the load resisted by a pile in a pile group to the load resisted by
a single pile of the same size as each of the piles in the group at some level of
deflection, this deflection being the same for both the group and the single pile.
• Computing the ratio of the deflection of a single pile to the deflection of a pile group
composed of the same size piles and at the same average load per pile.
Thus, there are diferent schools of thought regarding how the group reduction factor
could be defined, and broadly these schools could be classed as either based on ultimate
load, based on load, or based on deflection. It is the authors' belief that definitions based
on the ultimate load are not generally applicable in the urban systems of central Arizona,
as the deflection of the pile cap is typically limited in design to some low value (in the
range of0.51.0 in, or around 3 percent of the pile diameter for common drilled shaft
sizes). However, such a definition may be implemented in parts of the state where seismic
design may be more important. To the extent that linear elastic conditions exist in the soil,
which is often assumed at low stress levels, the choice of evaluating the eficiency based
on deflection or load should make no diference as the ratio should be more or less
equivalent for either definition.
The issue is clouded even more by the structural boundary condition at the top of each
pile. The pile head is idealized as either free to rotate at the top (free head pile) or fixed
against rotation at the top (fixed head pile). For the most part, in field and model testing
the single pile is a free head pile, because of the dificulty involved in fixing the top of a
single pile. However, it is very common to assume that pile groups are conected by an
efectively rigid pile cap and therefore cannot rotate. In analytical studies, it is no
particular problem to enforce a fixed head boundary condition in either a single pile or a
pile group, and this is often done in computational solutions. So, one finds rapidly that
there are a number of potential definitions of eficiency, based on whether one chooses to
17
consider loads or deflections, and based on how one chooses to fix the top of the pile. The
possible definitions are summarized in Table 2.3.
Table 2.3: Definitions of Group Reduction Factors
Single Pile Free Single Pile Fixed Single Pile Free
Head, Group Piles Head, Group Piles Head, Group Piles
Fixed Head Fixed Head Free Head
Definition Based on
( Efi>
½,
)
•
( Efix/jJ
6
Ratio of Deflections ( Efrt/2 )
free D.
Definition Based on
( Elm½,)
P (E%)P
Ratio ofLoads ( Efi>
½
)
free p
Given that so many bases for defining the group reduction factor exist, the problem that
arises is that of knowing which definition to use for a given application. The solution to
this problem is to use the definition which is most appropriate to the conditions of the
design process. For purposes of design in Arizona, most piles are fixed into rigid caps so
that rotation of the top is unlikely. Therefore, design based on a fixed group is probably
best. The free/fix condition will probably only be useful when a freehead single pile load
test is performed. Even in this case, it would be possible to use trial and error with COM
624 to find py curves which produce a match with the measurements from the free head
pile load test. Then the fixed head single pile response could be computed and a definition
of the reduction factor based on the fix/fix condition could be implemented.
Another issue is whether the definition should be based on load ratios or deflection ratios.
In practice this selection would depend on how the designer intended to limit deflection.
Based on the survey information to be presented later in this report, the most common
approach in Arizona is to design based on a py curve developed from curves in the
literature and scaled according to the observed soil properties as measured in laboratory
tests and/or the experience of the geotechnical designer. This py curve is then used to
obtain a loaddeflection curve for the single pile with a fixed head using one of the
available computer programs. Then, the load at the acceptable pile group deflection is
picked from the single pile loaddeflection curve, and this load is multiplied by the group
reduction factor to obtain the allowable lateral load capacity of each pile in the group.
Given this procedure, the most interesting reduction factors from the literature would be
those which correspond to (E.rix1.ruJ p.
So, while in concept the group reduction factor appears to be relatively simple to apply, it
has been found that the value one would use could potentially be related to a number of
factors. First among these are the issues presented above, which relate directly to the
definition of the group reduction factor. Additional dependencies exist, however. The
18
group reduction factor is related as well to the deflection level, the soil type, the pile cap
boundary conditions, and the relative stiflhess of the pile and the soil. These dependencies
will be treated individually in the following sections.
2.2.3 .3 Impact of Definition Chosen
Lateral load reduction factors can be found which relate single pile response to group
response in many ways, including deflectionbased factors and loadbased factors,
including freehead or fixedhead conditions, including diferent boundary conditions for
the pile group, and including diferent strain levels. An attempt was made to report group
reduction factors solely in terms of the response of the single pile compared to the average
response of the pile group, and only at strain levels expected in the prototype foundations
for fullheight abutment walls, on the order of 3% of the pile diameter. This meant
entering the tables or figures of the papers reviewed to calculate group reduction factors,
which in many cases were diferent from those reported by the authors of those papers. A
complete tabulation of all group reduction factors found or calculated with the
appropriate conditions and a bibliographic citation can be found in Appendix D.
In an efort to observe any diferences that might exist, the data were separated by
definition and by fixity at the ground surface. Figure 2.8 presents group reduction factors
calculated as a ratio ofloads at a deflection near 3% of the pile diameter and calculated as
a ratio of deflections at a pile load able to create around 3% deflection, versus pile
spacing. The data in Figure 2.8were derived primarily from actual pile load tests on single
piles and pile groups. Most of the data points come from smallscale model tests and a
small number of the data points come from analytical studies. At first glance, one might
assume that the data points plotting above one are eroneous, given that group reduction
factors should theoretically be one or less. However, these data points correspond to full
scale load tests where the pile cap for the group was in contact with the soil and the single
pile had no cap. Thus difering boundary conditions explain these data points.
The solid symbols in Figure 2.8 were developed from loadbased definitions, the open
symbols arise from deflectionbased definitions. Furthermore, the tests with the single pile
fixed and the group pile fixed (fix/fix) have been separated from the tests with both the
single pile and the group free (free/free). While deflectionbased reduction factors are
generally slightly lower than loadbased reduction factors, the diferences do not appear to
be extremely significant. Note from Figure 2.8 that the open symbols tend to be the
lowest points in both graphs at most spacings shown, and the closed symbols tend to be
the highest points. However, there are a number of exceptions, and the body of the data
in these two figures show very similar trends.
19
2 2
I Load
(a) Free/Free
I Load
(b) Fix/Fix
. 0 o Deflection o Deflection ....
1.5 0 st 1.5
LL cu 0
C:
LL
•
0 C:
ts 0
:::, 1 t)
"'C 1
I
:::,
Q) "'C
0:: Q)
C.
0::
•
:::, C. 0
0 :,
e C> 0.5 c, 0.5
O+++, 0 +++,
0 5 10 15 0 5 10 15
Spacing (D) Spacing (D)
Figure 2.8: Effect of Basis of Definition
It is also apparent from Figure 2.8 that there are only minor differences between the
free/free and fixed/fixed reduction factors. At any given spacing, the fixed/fixed results
tend to have a slightly lower lowend, and the free/free results tend to have a slightly
higher highend, but the trends are again very similar. This result is not entirely unexpected
as, in the free/free and fixed/fixed cases, the same rotational constraints at the pile head
exist for the single pile and each pile in the group, so that the efect of a change in
rotational constraint would appear in both the numerator and the denominator in the
group reduction factor calculation. However, there are very significant changes introduced
when the rotational constraint is diferent for the single pile and the piles in the pile group.
Figure 2.9 shows the free/free and fixed/fixed results together with the free/fix load tests 
these latter being those cases reported in which the single pile was tested free head, and
the pile group was tested in a fixed head condition. Again, solid symbols are used for
eficiencies based on a ratio of load, and open symbols for eficiencies based on a ratio of
deflection.
The circular symbols in Figure 2.9 represent the free/fixed case, and represent in large part
the increase in pile stifihess which arises when the rotational constraint changes from the
freehead condition of the single pile to the fixedhead condition of the piles in the group,
and hence the group reduction factor reported is very large. In the authors' opinion, this
group reduction factor is only useful for the case when a lateral load test on a single pile is
actually performed as part of the design process, and in fact all of the points shown result
from exactly that sort of process. Given that lateral load testing of single piles is relatively
rare in engineering practice, and because it is more appropriate to consider a consistent set
of reduction factors, henceforth the test results corresponding to the free/fixed state will
be reported only after conversion to a fixed/fixed state. This process requires a conversion
factor for the diference in fixity and henceforth the tests shown in the free/fixed state will
be reported only after conversion to a fixed/fixed state. This process requires a conversion
20
factor for the difference in fixity of the single pile. In some cases, the appropriate factor
was reported in the literature (e.g. Matlock and Foo's (1976) discussion of Kim and
Brungraber (1976) contains the appropriate factor for that case). In other cases, a
COM624 analysis was performed on the single pile, using the py curves given or assumed
py curves, until the pile head loaddeflection curve was matched very well for the freehead
condition. This model was then rerun with the single pile changed to the fixed head
case, allowing the ratio of fixedhead response to freehead response to be calculated. The
exact value of this ratio is a function of the pile and soil properties, but seems to range
from about 2 to about 2.4 for typical pile sizes.
3.5
3

0
•
2.5
·

•
0 2
u
:J
•
0
1.5 a:
a.
:J
e 1
C)
0.5
•
l
!
h el
t'i
C a,.
L er
•
0
0 2
•
II
l'"I
. a
n
4
•
•
0
a II

I.A.
•
I I
6
Spacing (D)
••
H
j I
8
Figure 2.9: Efect of Fixity Condition
2.2.3.4 Deflection Level and Soil Type
• Free/FreeP
a Free/FreeD
• Fix/FixP .
l Fix/FixD
• Free/FixP .
o Free/FixD
0
II l
'
10 12
As stated previously, the most interesting group reduction factor is that based on a ratio of
loads. If a ratio ofloads is used then they must be taken at a certain deflection level. The
deflection level one chooses to determine group reduction factor has a large impact on the
answer one obtains. The results from a few load tests reported in the literature have been
analyzed to determine the efect of deflection on group reduction factor. Examples of
these results are shown in Figure 2.10, Figure 2.11, and Figure 2.12.
21
0.90 ·································································· ···················· .. ·········""·························· ..................................................................... .
 Rollins (1998)
. 0.85 0  Brown (1988)
ca
u. 0.80
'•
0
t5
,: 0.75
"C
Q. 0.70
::,
0
C, 0.65
0.60
0 0.05 0 .1 0.15
MD
Figure 2.10: Group Reduction Factor versus deflection for FullScale Tests in Clay
1 .5  ....................................... ································,,,
1 .4 ;:,,:.,, +,
I.
§ 1.3 +·+._ lfl
co
u. 1.2
C:
0
n 1.1
::J
 Townsend (1997)
·Ismael (1990)
•Vanderpool(1996)
1.0
a:
c. 0.9 ++f++;
:J
0.8 r====t=======t=====t,i
0.7 +=··''l'"CC..."" ·=·=·== +=..• +
0.6 1"'""'''.+... _ ....;
0 0.02 0.04 0.06 0.08 0.1
MD
Figure 2.11: Group Reduction Factor versus Deflection for FullScale Tests in Sand
22
....
0
(.)
ro
u.
C:
0
n
::I
"O
CJ)
a::
a.
::I
e
C)
....
0
(.)
ro
u.
C:
0
(.)
::I
"O
CJ)
a::
a.
::I
e
C)
1 . 1
1
0.9
0.8
0.7
0.6
0
0.9
0.8
0.7
0.6
0.5
0.4
0
120 4D h80 120 16D I
0.5 1
l4Piles I
0.5 1
1 .5
ID
1.5
ID
2
2
2.5
2.5
3
3
Figure 2.12: Group reduction factor versus Deflection for Model Pile in Sand (From Sarsby, 1985)
For full scale tests it appears that for clay soils (Figure 2.10) the group reduction factor
has a tendency to decrease with increasing deflection. For sand (Figure 2 .11) the trend is
less clear. For small scale tests in sand the results of Sarsby (1985) are presented (Figure
2.12). At first glance it appears that the group reduction factor increases and then
decreases at high levels of deflection. However, it is important to note that the model
piles used in the study were 6 [mm] in diameter. Therefore, at all deflection levels up to
about 50% of the diameter the group reduction factor increased.
23
2.2.3 .5 Relative Stiflhess of Pile and Soil
The most commonly used expression for relative stiflhess of the pile compared to the soil
is T, where:
Equation 2.18
Here, E is the modulus of elasticity of the pile material and / is the moment of inertia of
the cross section of the pile. The parameter nh is the coeficient of variation of sub grade
reaction. The modulus of subgrade reaction, k, is related to nh and depth, z, by
Equation 2.19
where k is a function of the pile width (as shown on page 5). Note that, although Tis a 
parameter that is widely used to quantify relative stiflhess, nh is actually the rate of
increase in soil stiflhess with depth rather than the soil stiflhess itself Nonetheless, n1
tends to be higher for stifer, stronger materials, particularly granular materials, and thus T
provides a general indication of relative stifthess.
Prior to discussing the design procedure it is important to have an understanding of what
the variables used in the procedure mean. This is very important for the coeficient of
lateral subgrade reaction because diferent researchers and manuals use diferent notation.
Table 2.4 compares the terminology for the modulus of subgrade reaction used in a
number of important studies.
Property of
Interest
Modulus of
Subgrade
Reaction
Coeficient
of Variation
of Subgrade
Reaction·
Relative
Stifness (T)
Table 2.4: Comparison of Modulus of Subgrade Terminology
Matlock&
Reese
(1961)
NIA
k
Prakash
(1962)
k
,W,
T=Vk
Davisson
(1970)
k
Canadian
Found Eng.
Manual
(1985)
NAVFAC
(1986)
NIA
f
Many of the researchers used diferent names when identifying this parameter. For the remainder of this
report the names in the first column of the table and the variables as defined by Davisson will be used.
24
For typical conditions in fullheight abutments in Arizona (3 foot diameter concrete shaft,
soil properties on the order of ¢=32°, c =500 [psf], relatively high density), one can
compute a T value in the range of 90100 inches. Model piles comonly reported in the
literature typically have Tvalues which are much lower. It should be pointed out that if
EI is low enough, T can be low for almost any soil one might encounter. This is illustrated
in Table 2.5 below for a hypothetical 1/2 in diameter model pile made of aluminum.
Table 2.5: Range in TValues for a Pile of Very Low El
EI lb in2 T in
1 30679 7.9
60 30679 3.5
The values of T shown in Table 2.5 are typical of the values corresponding to almost all of
the model tests reported in the literature. For example, a very important set of model tests
:frequently cited is that of Prakash (1962). For Prakash's tests, the 0.5 in O.D. aluminum
tubes used for piles and the dense sand produced a Tvalue of2.85 in. Table 2.6 shows the
range in T values for actual prototype piles that one might encounter in practice. These
data show that for the full range of soil types, T ranges :from about 3 6 in to about 3 50 in.
Table 2.6: Range in TValues for Common Prototype Piles
Pile Types
1 ft dia. 10.75 in 8 ft dia.
f = nh reinf. 3 ft dia. 5 ft dia. dia. Steel, Steel, 2 in
[tons/ft3 ] Soil Type and Condition cone. reinf cone. reinf. cone. 3/8 in wall wall
3 very soft cohesive (Su
= 300 fpsfl) 65 157 236 68 355
5 very loose sand (Dr
= 25%) 59 142 213 61 321
12 med.  stif cohesive (Su
= 1000 49 119 179 51 269
r psfj)
22 med. Dense sand (Dr
= 50%), 44 105 158 45 238
stif  v. stif cohesive (Su
= 2000·
fpsfJ)
55 Very dense sand (Dr
= 93%), 36 88 132 38 198
v. stif cohesive (Su
= 4000 [psfj)
Using the relative stiflhess factor, T, one can develop an appreciation for the impact of
soilpile stifness on the response. Consider a hypothetical condition in which a pile group
is founded in very soft material of near zero shear strength, as shown in Figure 2.13. A
horizontal load applied near the top of this group would create a deflection pattern similar
to a cantilever beam, and the deflection would be mostly a function of the pile properties.
This is because essentially no loads would be transferred to the upper soil of near zero
resistance, each pile would deflect identically and carry the same horizontal load  and
also the same load as a single pile deflected to the same shape. Thus, one would expect
the pile to behave in a linear elastic fashion, and the group reduction factor to be very near
one whether the group reduction factor is evaluated :from load or deflection. The efect of
pilesoil interaction would be minimal. In this case, T would be very high, as EI would be a
25
large number compared to a very small nh, provided nh for the upper very soft material
were used. The important aspect of this example is that lateral load is not transferred to
the soil quickly. As long as the lateral load remaining in the pile is high, the pile or group
of piles behaves like cantilever beams and the eficiency is high as described above.
··
I
•I> · .................................................... .
. ___ .T _i
f
f / Very soft material,
/ e.g. slury
Stif material
Lateral Load in Pile
Slow decrease with
depth leads to high
group reduction factor
(close to 1)
Figure 2.13: Pile Group in Very Soft Material and Lateral Load Remaining in Pile with Depth
At the other extreme, consider a very flexible pile in a very stif uniform soil. In this case,
the deflection would be heavily controlled by the soil behavior, because lateral load is
quickly transferred to the soil and cantilever action is relatively less important, Figure
2.14. Overlapping stress fields lead to low group reduction factors. The loadbased group
reduction factor would be expected to be diferent from the deflectionbased group
reduction factor. The deflection would be a function of the stress state in the soil, and the
stress in front of any one pile in the group will be increased by the stress distribution from
other piles, tending to increase the deflection. In other words, loads would be transferred
quickly to the soil due to the low flexural stifness of the pile, stress superposition would
afect all the piles in a group, and group reduction factor could be well below 1. In this
case, T would be relatively low, as El would be a smaller number relative to nh.
............................ 
Stif Material
Lateral Load in Pile
Rapid decrease with depth
leads to lower group
reduction factor
Figure 2.14: Pile Group in Uniform Material and Accompanying Load Remaining in Pile with Depth
In fact, one can observe that there is a diference in the group reduction factors reported in
the literature for small scale (model, low1) pile groups and prototype scale (fullscale)
pile groups (Figure 2.15).
26
1.6
)l( Model
1.4 • •
•Full Scale
• O FE/Analytical .... 0 1.2 •
u. 1
0.8 :J
"C )K
0.6
::J
e 0.4
i
0.2
0
0 2 4 6 8 10 12 14
Spacing (D)
Figure 2.15: Effect of Type of Study for Fixed/Fixed Group reduction factor
The model tests typically have the lowest T values, and full scale results correspond to
very high T values. Finite element studies may be performed with a larger simulated T than
the model tests, but nonlinear constitutive models are not often used, leading to
somewhat higher degrees of stress overlap in most cases than might be expected in the
field. Clearly, the lowest group reduction factors across the range of Tvalues are for the
model tests, and the highest are for the fullscale tests, with finite element tests somewhere
in between.
27
The importance of Tis made even more clear if one separates those results for which T of
the test is in the neighborhood of a realistic prototype pile foundation (to be referred to as
"high T," with a threshold of about 3 5 inches), from those for which T of the test has a
value which is unrealistically low given the dimensions and materials used in prototype
piles (to be referred to as "low T," values less than 13 inches). Clearly from Figure 2.16
there have been many more tests conducted with very low T than with a T which would be
reasonable for prototype pile dimensions, no doubt because of cost.
1.6
1.4
1.2
0
1
C:
0 0.8
::,
'"C 0.6
g. 0.4
0.2
0
0 2
High T
LowT
4 6 8 10
Spacing (D)
Figure 2.16: Effect of Relative Stifness of Pile and Soil
o LowT
• High T
12 14
The highTtests form a significantly higher band than the lowTtests. At a spacing of3D,
the middle of the lowTrange is approximately 0.65, while the middle of the highTis
approximately 1.0.
There are two important conclusions from these comparisons. First, the values of group
reduction factor should be higher for piles of high Tthan for piles oflow T. Second, in
order to get reduction factors which are appropriate for fullscale field piles, it is necessary
to use tests or analyses in which the T values of the test match the T values of the
prototype piles.
2.2.3.6 Pile Cap Boundary Conditions
In the case of a buried pile cap, which is common for a fullheight abutment in Arizona
with drilled shafts to provide for added lateral resistance, the pile cap would be fairly stif
28
The soil which is in contact with the cap provides substantial resistance to rotation and
lateral movement. These boundary conditions are typical for the abutment pile cap, but
are not present for the single pile, which is very important because it accounts for the fact
that essentially all of the fullscale lateral load tests on pile groups where abutment
boundary conditions are matched show reduction factors above, and often well above, 1.0.
The solid symbols on Figure 2.17 come from tests with pile cap boundary conditions
similar to the prototype piles for fullheight abutments, in that the cap was in contact with
the soil surface. In the tests represented by the open symbols, the pile cap boundary
conditions were unlike the prototype, in that the cap was not in contact with the soil
surface. Clearly, this factor is extremely important; when the prototype boundary
conditions are matched, a group reduction factor above one often results, even at a
spacing of3D.
1.6
1.4
0
t5 1.2
L
C 1
0
g 0.8
0.6
a.
5 0.4
CD 0.2
0
"

0
•
•
•

c,
0 C)
i
0
I
2
.
o
0
I
4
Spacing (D)
•
80
Figure 2.17: Prototype Scale Results
2.2.3.7 Use of ASHTO Guidelines
u
•Abutment BC
OQther BC
6 8
During the technical literature review the procedure outlined in the ASHTO 1996
Standard Specifications for Highway Bridges was reviewed. As ADOT and others have
interpreted AASHTO, the specification for groups oflaterally loaded drilled shafts (Sec.
4.6.5.6.1.4) states that the "efects of group action for inline CTC (centertocenter)< 8B
may be considered using .... the ratio of lateral resistance of shaft in group to single shaft."
There is no other guidance on how to apply these ratios to the group action problem.
Many have interpreted the word "resistance" in its common usage elsewhere in AASHTO
as implying force, and accordingly developed an understanding that the values given for
lateral load resistance are ratios of forces. The source in ASHTO is the Canadian
29
Geotechnical Society (1985). This was interpreted to mean the Canadian Foundation
Engineering Manual (CFEM) which is published by the Canadian Geotechnical Society.
This reference recomends applying the reduction factors to the coeficient of lateral
sub grade reaction rather than reducing the lateral force capacity of the single pile. This
method was further substantiated when examining the origin of the CFEM
recommendations, in Davisson (1970). Davisson stated that the efective value ofk (kef)
for a pile group is less than that for a single pile. He provided the following relationships
between spacing and kef:
@3D, ketr = 25%k
@8D,ketr = 100%k
Equation 2.20
These relationships clearly show that Davisson intended the reduction factors to be applied
to the modulus of subgrade reaction and not the single pile capacity. These
recommendations have been used in Foundations and Earth Structures (1986). It is clear
that the source referenced in the ASHTO specifications has a diferent interpretation
than that in use by most states.
2.2.4 Coeficient of Lateral Sub grade Reaction Reduction
The theory of subgrade reaction is another solution of a foundation element under lateral
load. The coeficient oflateral subgrade reaction k is related to Es in Equation 2.21. It
models the soil as a Winkler foundation, which is then used to solve the beam on elastic
foundation. The coeficient oflateral subgrade reaction reduction procedure reduces the
stifihess of the soil, but the pile stifness is unchanged
k _ Es
o
D
Equation 2.21
where:
D = the pile diameter
The following procedure is used for calculating the lateral load capacity as stated in
Foundations and Earth Structures (1986) also referred to as DM 7.2. The design
procedure includes design for a single pile and reduction factors for pile groups. The
procedure defined in DM 7 .2 assumes that the lateral load does not exceed about onethird
of the ultimate lateral capacity. The lateral load analysis is dependent on two criteria:
the soil conditions and the loading conditions.
2.2.4.1 Soil Conditions
The soil conditions insitu are modeled by the coeficient of lateral sub grade reaction, k.
The value of k is a function of the soil type. DM 7.2 classifies soils into two categories: 1)
granular soil and normally to slightly overconsolidated cohesive soil and 2) heavily
overconsolidated cohesive soils
30
2.2.4.1.1 Granular Soil and Normally to Slightly Overconsolidated Cohesive Soil
Soils that fit into this category have insitu ko values that increase linearly with depth. The
formula used to define k0 is:
k = f xz
o D Equation 2.22
where:
ko = coeficient of lateral subgrade reaction [tons/:ft:3]
f = coeficient of variation of lateral sub grade reaction [ tons/ft3 ]
z = depth ft
D = width/diameter of loaded area ft
Selection off is dependent on whether the soil is finegrained or coarsegrained. The
value off can be obtained from Figure El (Appendix E) using the unconfined compressive
strength, Qu, or the relative density of the soil, Dr.
2.2.4.1.2 Heavily Overconsolidated Cohesive Soils
When heavily overconsolidated cohesive soils are encountered, it is comon to assume
that the coeficient of lateral sub grade reaction is constant with depth, and defined within
the limits presented below:
35xc<k0 <70xc Equation 2.23
where:
ko = coeficient of lateral subgrade reaction [tons/ft3 ]
c = undrained shear strength
Soils fitting this description are analyzed using the analysis of beams on elastic foundation
directly.
2.2.4.2 Boundary Conditions
There are three principal boundary conditions that were considered in the DM 7.2
procedure. The boundary conditions are: 1) pile with a flexible cap or hinged end
condition (free); 2) pile with a rigid cap fixed against rotation at the ground surface; 3)
pile with rigid cap above the ground surface. These principal loading conditions are
illustrated with design procedures in Figure E2. The procedure for a fixed head pile
subject to a horizontal load is presented because it most resembles Arizona conditions.
The remaining procedures with accompanying figures are presented in Appendix E. The
equation for the lateral deflection for a fixed head pile subjected to lateral load is presented
for later use (Equation 2.24). The variables are defined in Figure 2.18.
31
o =F (
PT3
) p 8 EI
Equation 2.24
The design of pile groups must take into account the interaction of one pile with the other
piles in the group. The recommendation in DM 7.2 is that group action should be
considered when the spacing between piles is less than 8 pile diameters in the direction of
loading. Group action is evaluated by applying a reduction factor, R, to the coeficient of
lateral subgrade reaction, k, in the direction ofloading (Table 2. 7).
Table 2.7: Reduction Factor, R vs. Pile Spacing
Pile spacing in
direction of loading
D = pile diameter
8D
6D
4D
30
2.2.4.3 Application ofR in Design
Subgrade reaction
reduction factor
[R]
1.00
0.70
0.40
0.25
The application of the reduction factor is not entirely straightforward. The definition of ko
was presented in Equation 2.22. It is clear that for a given pile/shaft geometry at a given
depth, both D and z remain constant. Therefore, a reduction in k is equivalent to a
reduction inf One then multiplies/for the soil at the site by R to obtain a reduced value
of/for the group,fgroup (Equation 2.25).
fg roup = .Jingle X R Equation 2.25
To obtain the lateral capacity of a pile in the group, Tgroup must be calculated using/group in
Equation 2.18, and then Equation 2.24 is solved for P using the data in Table 2.1 and
Figure 2.18 and Tg orup· The capacity oft he group is equal to the capacity obtained for a
pile in the group with reduced k0, multiplied by the number of piles in the group, N
(Equation 2.26).
pgroup = N X educed single Equation 2.26
2.2.4.4 Advantages and Disadvantages
The DM7.2 procedure, like the pmultiplier procedure, accounts for group efects by
reducing only the soil stifness, in this case through the modulus of subgrade reaction. A
minor disadvantage of this method is that it is only applicable up to onethird of ultimate
lateral capacity because the method assumes a linear relationship between load and
deflection. The limitation on lateral capacity would be a problem when large loads
32
associated with extreme events such as earthquakes, floods, or ship impact need to be
considered .
I J .. I J
p r":P
v z
1L
.
... ,
. . .. .

.,. .... PT3 .
. p"fyjl ...
"
5 , .,.
.,;.' ,
l
. . i
,: 4 l
4
l 5:
10
,,
.
0.2 0
£ I
DEFLECTION C0£FFICIENT Us·> :,· .
FOR APPU£0 LATEJIAL U>AD (P) .,d :."". ·:· ··
. L  =.2
T . ••···
.. i . • .
;,j
I.,.
·  .oil
,
..
I l . :. .f.!" ;.·
!i ,. IO.,. " ;.•
,.'!"
.· .,,,,,,
.. i
l:il ,
. ...
15" . . ' _ DEFJNmONS
. P=UJERAL FCR<EAPPUED fl.E
H= VERTICAL DISTAt«::EBETWEEN PAN> SURFACE
M• PH•MOMENTON PiLE APi.JED .i GROut,IOsutFACE
Z="DEPTH. 8El.OW GROND,(TO POINT 10 SE CHECKED)
Gf(Z)SOILMODULUS ·OF ELASTICITY
f = COEflCl9CT'OF VARIATION OF LATERAL. SUBGRADE.
REACTION. (SE FIGURE 9 .. )
Li: LEIWTH OF PiL£ BEi.OW GROUND SURFACE
T = RELATIVE STIFRES FACTOR
I£ :a: ·MODMOMUENTSW O FOF IENRTE$ALTIACI FOTYR OLFE· ,ACOSEL S EINOT
Bp,M1>1Vp=DEFL.ml'tQN,MOMENT, a SHEAR AT.ANY DEPTH
Z OUE l0 FORCE P.
.i
"
.,·




1
,
1<1m,Mm,Vm:OEFLECTIOH,11Ef8'Sf£ARAT1tNYDEPTH" z DUE,OMOl,8TM,. I l I I·. I I I .
0.2 0.4 O.&
DEFLECTION COEFICIENT FB
0.8 1..0
Figure 2.18: Influence values for modulus of subgrade reaction reduction procedure for pile fixed against rotation
(from DM7.2 1986)
An example of the modulus of sub grade reaction multiplier method is presented using the
soil and pile properties from Table 2.1. From the applied load, P = 100 [kips], Figure 2 .18
gives Fil = 1 .05. Solving Equation 2.24 using these properties results in (Bp)singie = 0.42 in.
o = F ( PT
3
) = 1.0 5((100 kips)(134 in)
3
) = 0.42 in P O EP IP (6.22xl0 n lbin 2 )
If the piles are spaced at 3D, the reduction factor R = 0.25 (Table 2.7) is applied to fsmg1e,
!group = fsingle x R = (13.9 pci) x 0.25 = 3.5 pci
Then, T group is calculated using fgroup :
Using Tgroup results in a new value of Fil = 1.10 from Figure 2.18. Equation 2.24 is used to
solve for P at Ogroup
= Osingle below:
33
P = g 1:.l! =   = 42 kips
o
(
E I
J 0.42 in (
( 6.22xl 011 lb in2 )J F8 T3 1.1 (1 77 in)3
The result is a lateral capacity of 42 [kips] per pile in the group at a deflection of 0.42 in.
An alternative approach for using the coeficient of lateral sub grade reaction reduction
could be performed using the computer program COM624P. This alternative approach
reduces the coeficient of lateral soil reaction, k, by the recommended reduction factor as
in DM7.2. The modified value ofk is then used by the computer to develop the py
curves internally. COM624P can than be run to obtain a load deflection curve with the
reduced k py curves. This procedure is simplified compared with the DM7 .2 procedure.
This procedure is also easier to implement due to the fact that many engineers are already
using the COM624P or similar program.
There are significant drawbacks to this procedure. First, this method is only applicable to
sands. The internally generated py curves for other soil types are not a factor ofk.
Therefore, a modification of the k value for a clay material doesn't efect the development
of py curves resulting in no efect on the loaddisplacement curve of the shaft. Secondly,
the k values for sands are only used to predict the initial slope of the py curve. This
means that only the initial portion of the py curve is afected by the reduction in k. After
relatively small deflections, the single pile and the group pile py curves are the same
(Figure 2.19).
a'.
2500 ...,.,=======
Group Shaft
 • Single Shaft
2000 ++++'.
1500 +++++;!
J.
 I
1000 ++V=="'+=tlfi
/1/
500
/ I
0 11,, 0 0.1 0.2 0.3 0.4 0.5
y[in]
Figure 2.19: Comparison of py curves for a Single Shaft and a Shaft in a Group at a Depth of 5 ft.
34
2.2.5 Pmultiplier Design
Another method used for the design of pile groups is the pmultiplier method, which is
commonly attributed to Brown (1988). The pmultiplier method is similar to the modulus
of sub grade reaction procedure except that the paxis of the py curve is reduced instead
of the modulus of sub grade reaction, k,. The py curve approach can be related to the
modulus of subgrade reaction using the soil modulus, Es. The equation relating the py
curve to Es is as follows:
E =
p Equation 2.27
From the Equation 2.21 and Equation 2.27 it is clear that a reduction in p of the py curve
is analogous to a reduction in k. The main diference between the two procedures is that
E. is independent of deflection for the modulus of sub grade reaction procedure while the··
pmultiplier method allow Es to vary with deflection.
The following procedure is used for calculating the lateral load capacity as stated in
Design and Construction of Driven Pile Foundations. GROUP and FLPJER are widely
available and commonly used computer programs that employ the pmultiplier method to
analyze a group of laterally loaded piles. Instead of modifying the load response of the
single pile, the pmultiplier method involves modifying the py curve obtained for a single
isolated pile by multiplying the pvalues in the py curve by the pmultiplier, denoted as Pm ·
An example is presented in Figure 2.21. It is comon to apply diferent pmultipliers to
each row of piles/shafts in the group as a function of its position within the group, with
respect to the loading conditions (Figure 2.22). There are a number of results and
recommendations available for values of Pm (Table 2. 8). It is, of course, advised that the
engineer use pmultipliers from the case most similar to the design under consideration
and in this context the lack of drilled shaft testing is significant for Arizona usage. The pmultiplier
design approach can be summarized by the following steps (FHW A 1996):
Step 1) Develop py curves for a single isolated pile
There are three ways a py curve can be developed: 1) an instrumented lateral pile
load test; 2) based upon published correlations in the literature with soil properties;
and 3) based on insitu test data. LPILE and COM624P combine options 2 and 3
to develop py curves internally.
Step 2)Develop loaddeflection and loadmoment data
For each row in the pile group, generate a separate loaddeflection curve using the
py curve from Step 1, with p at each y multiplied by the appropriate value of Pm
for that row.
Step 3)Develop load deflection curve for pile group
Enter the loaddeflection curve for each row at a given deflection and obtain the
lateral resistance for that row at that deflection. Sum the lateral resistance for all
the rows together to obtain the lateral resistance of the group, and plot against the
35
deflection assumed. (This procedure is only applicable if all rows in the group
have the same number of piles, otherwise the procedure should be done on a pile
by pile basis.) Repeat the procedure for several deflections to obtain a loaddeflection
curve for the pile group.
Step 4) Group lateral capacity
To obtain the lateral capacity of the pile group enter the group loaddeflection
curve at an acceptable level of deflection for the design.
Step S)Evaluate pile structural acceptability.
Once the deflection and loading criteria have been established, it is necessary to
evaluate the structural adequacy of the piles. First, plot the maximum bending
moment, from the computer analysis, versus deflection for each row of piles.
Determine the maximum bending moment and the resulting stress for a single pile
in each row. Check if the maximum pile stress is less than the pile yield stress, if_
not, the design must be modified.
Step 6) Refine evaluation
Refine the pile group response evaluation taking into account the superstructuresubstructure
interaction.
Programs that solve only the single pile problem, such as LPILE or COM624P will be
used for Steps 1 through 3 only, and the balance will be completed by hand. Computer
programs such as GROUP and FLPIER complete Steps 1 through 3 internally, with the
remaining steps done by the engineer. These programs have default pmultipliers that can
be modified by the engineer to site specific conditions using engineering judgement.
36
800
 • pm = 0.8
pm = 0.4
.....
600
. 
....

.
 .
.
.
= ..
"O 400
cu ....
0
,,.
,,..



 


200


I' ,/
. ./"
/ /
. /
/ /
, 1/
0
0 5 10 15 20 25
Deflection [mm]
Figure 2.20: Loaddeflection curve for pmultiplier example
An example to illustrate the procedure is as follows. The single pile is the same as used
previously for the group reduction method. At a load of 100 [kips] the expected lateral
deflection is 0.38 in. The pile group is composed often piles in two rows (Figure 2.7).
Figure 2.20 shows the loaddeflection curves for the pile group, a single pile with a pmultiplier
equal to 0.8, and a single pile with a pmultiplier equal to 0.4. These pmultipliers
are used for the first and second row of the pile group using the FHW A (1996)
recommendations. The average loaddeflection curve for each pile in the group is
obtained by adding five times the first row plus 5 times the second row (there are five piles
in each row) then dividing by ten. From Figure 2.20 the average load capacity for each
pile in the group is equal to 67.5 [kips) at a deflection of 0.38 in (9.65 [mm]).
37
Table 2.8: Prnultipliers vs. Row Position
Reference:
Pile Properties
Lead
2nd Row 3n1 Row
Trailing Last
Soil Prooerties Row Rows Row
Rollins (1998) Clayey
Driven steel pipe pile
filled with concrete. 0.60 0.40 0.40  
silt (CLML, ML);
D=0.324 [m]
Ruestra and Townsend
Jetted/driven 0.76 [m]
(1997) Loose fine
square prestressed 0.80 0.70 0.30 0.30 
sand (SP) Dr :o 20
concrete pile
40%
Brown et al. (1988)
Driven 0.272 [m] OD
Clean medium sand
steel pipe pile filled 0.80 0.40 0.30  
(SP) Dr :o 50% sand
with grout
placed after driving
Brown et al. (1987) Stif
Driven 0.272 [m] OD
clay (CL to CH); 0.70 0.60 0.50
steel pipe pile filled  
overconsolidated by
with grout
0.70 0.50 0.40
dessication
Meimon et al. (1986) Driven 0.284 x 0.270
Silty clay (CL); Su = [m] steel Hpile with
0.90 0.50   
500 [psf]; f =3842°; side plates to form a
c' = 0 box section
Mc Vay et al. (1995)
Driven openended
medium dense sand 0.80 0.40 0.30  
Dr :o 55%, Centrifuge
pipe pile 0.43 [m]
Mc Vay et al. (1995)
Driven openended
loose medium sand 0.65 0.45 0.35  
Dr :o 33%, Centrifuge
pipe pile 0.43 (m]
Driven open ended
Pinto et al. (1997) sand aluminum pipe pile;
0.80 0.45 0.30 
Dr :o 55%, Centrifuge D = 0.43 [m]; L = 
13.3 [m]
Driven open ended
Pinto et al. (1997) sand aluminum pipe pile;
0.65 0.45 0.35 
Dr :o 33%, Centrifuge D = 0.43 [m]; L = 
13.3 [m]
Zhang (1999) uniform
Bored concrete shaft
low to medium dense
D = 1.5 [m]
0.50 0.40 0.30  
sandy silt, f =3 5
Zhang (1999) uniform Driven 0.8 [m] square
low to medium dense prestressed concrete 0.90 0.70 0.50 0.40 
sandy silt, f =35 piles
GROUP (1996) Recommendation * ** 
FLPIER (1996) Recommendation 0.80 0.40 0.30 0.20 0.30
FHWA (1996) Recommendation 0.80 0.40 0.30 0.30 
( )0.2579
*Leading Row Pm = 0.7309
( )0.3251
**Trailing Rows Pm = 0.5791
38
p
Load
single pile
group pile
y L]
Figure 2.21: Example of pmultiplier
Trailing
Row
0
0
0
Middle
Row
0
0
0
Leading
Row
0
0
0
Figure 2.22: Identification of Rows with respect to Applied Load
2.2.5.1 Advantages and Disadvantages
In the pmultiplier method, the soil stifness is reduced by the pmultiplier; the pile stifness
is unchanged. The pmultiplier accounts for the fact that not all the piles resist the same
amount ofload for a group in which all piles deflect equally, so one can choose to design
for the worst case pile. However, as pointed out earlier, not all structural designers will
take advantage of this result. The method allows the soil modulus, Es, to vary
independently with depth. The major disadvantage of the pmultiplier method is that the
39
same pmultiplier is used for the entire length of the pile. This is a disadvantage because
the stress overlap, and thus the degree of influence of surrounding piles, is a function of
pile deflection, and the pile deflection varies along the length of the pile (Norris, 1994).
Furthermore, nearly all pmultiplier recomendations have been developed from tests on
driven piles (Table 2.8), and there is some evidence that the behavior of groups of drilled
shafts is somewhat diferent (Zhang, 1999).
2.2.6 LoadandResistance Factor Design (LRFD) Procedure
LRFD is a relatively new design procedure with regards to geotechnical engineering.
Therefore a complete and thorough background on the method is presented.
2.2.6.1 History
Commonly design in the U.S. has been performed using the allowable stress design (ASD)
method, in which a factor of safety is used to take into account all uncertainty in loads and
material resistance. At the end of the 1980s, a new specification was under development
in which the uncertainty in load(s) and material resistance(s) were to be represented by
factors. Uncertainties in load would be represented by load factors, which generally
would have a value greater than one. The uncertainties in material resistance would be
represented by resistance factors, which generally have a value less than one. A relevant
procedure based on this approach was approved by ASSHTO in 1994. Adoption is
scheduled in the near future.
2.2.6.1.1 Alowable Stress Design (ASD)
As previously mentioned the ASD method is the common design method in the U.S .. The
general design procedure for ASD can be represented by Equation 2.28 as follows:
where:
.!1
> "f.Q
FS  applied
Rn = Nominal (ultimate) resistance
FS = Factor of safety
"f.Qapplied = Summation of all applied forces
Equation 2.28
In general the factor of safety is qualitative and not directly related to the uncertainties
associated with the load or resistance. The engineer determines the factor of safety based
upon his/her experience and the current state of practice.
40
2.2.6.1.2 Load and Resistance Factor Design (LRFD)
The LRFD method is a quantitative procedure. The procedure attempts to quantify the
risks and uncertainties associated with the safety of a system in mathematical terms, in a
reliabilitybased design method. The basic LRFD equation is as follows:
Equation 2.29
where:
Y1 = Statisticallybased load factor, generally greater than one
Q1 =Load
Rn = Nominal (ultimate) resistance
cl> = Statisticallybased resistance factor, generally less than one.
2.2.6.2 Limit States
In the LRFD method the safety of a structure is evaluated at various limit states. A limit
state is defined as a condition beyond which a structural component, such as a foundation
or other bridge component, ceases to fulfill the function for which it was designed. The
limit states which must be evaluated in the AASHTO LRFD specification (AASHTO,
1997a) method are:
• Service Limit State
• Strength Limit State
• Extreme Limit State
Fatigue Limit State
Each limit state is explained briefly below. It is important to note that several load
combinations define each limit state. Therefore, several load cases must be checked at
each limit state.
2.2.6.2.1 Service Limit State
Service limit state refers to the structural performance of the structure under service load
conditions. Evaluation of a structure at this state is performed to determine if the bridge
components under regular service conditions exceed restrictions on stress, deflection, and
crack widths.
2.2.6.2.2 Strength Limit State
The strength limit state refers to providing suficient strength or resistance to counteract
the applied loads for the statistically significant load combinations that a bridge is expected
to experience in its design life. The strength limit state is reached when either partial or
total collapse of the structure occurs.
41
2.2.6.2.3 Extreme Limit State
The extreme limit state refers to the structural survival of a bridge during extreme events.
An extreme event doesn't always occur during the life of the structure. There is some
probability associated with it happening during the life of the structure. The events that
are considered extreme events include earthquakes, floods, or collision by vessel, vehicle,
or ice flow. Due to the classification of these events as extreme, the probability of these
events occurring simultaneously is extremely low, therefore, these events are analyzed
separately. However, there is an exception to this rule. Flooding, due to the possibility
that it can occur in combination or as a result of other extreme events, is often considered
in conjunction with other extreme events.
2.2.6.2.4 Fatigue Limit State
The fatigue limit state is an analysis of the structure due to repeated loads. A set of
restrictions is placed on the stress range that would be caused by repeated loading of a
design truck.
2.2.6.3 Load and Resistance Factors
The LRFD method uses both load and resistance factors in the design process. The actual
values of these factors vary depending on the limit state under consideration. However,
even though the values are diferent, the criteria that are considered in determining their
values do not change.
2.2.6.3.1 Resistance Factors
The resistance factor, ¢,for a particular limit state must account for uncertainties in
(FHWA, 1998):
•
•
•
•
•
Material properties (variability)
Reliability of the equations used to predict resistance, i.e. amount of scatter in
the data.
Quality of construction workmanship. Normally a structure is designed
assuming that good construction practices will be used.
Extent of soil exploration, i.e. amount of knowledge the designer has about the
site. The more detailed the soil exploration the better the designer is able to
account for the heterogeneity of the site.
Consequence of failure. As the importance of a structure increases then the
consequence of failure also increases.
42
2.2.6.3.2 Load Factors
The load factor, y;, chosen for a particular load type must consider uncertainties in
(FHWA, 1998):
• Magnitude of loads. As the certainty of the designer about the estimated loads
on the structure decreases the load factor increases.
• Arrangement (positions) of loads. The arrangement of loads is very important.
There are some cases where an increase in the load will help the stability of the
structure. In this case a load factor close to 1 will be chosen.
• Possible combinations of loads. The combination of loads is taken into
account as to the probability of certain loads acting in combination with one
another or separately.
2.2.6.4 General LRFD Equation
The general LRFD equation is composed of three components: load modifiers, load
factors, and resistance factors. The previously presented equation for LRFD (Equation
2.29) assumes that the load modifier, 'f/;, was equal to 1 and was not included. The
complete equation is presented in Equation 2.30.
Equation 2.30
2.2.6.4.1 Load Modifiers
In the ASHTO LRFD specification, each factored load is adjusted by a load modifier, 'f/;,
to account for the combined efects of ductility, 'f/n, redundancy, 'f/R, and operational
importance, 'f/1. There are two limiting conditions for 'f/; that must be satisfied.
Loads for which a maximum value of y; is appropriate:
Equation 2.31
For loads for which a minimum value of y; is appropriate:
1
'I'/; = s 1.00
'f/D X 'f/R X 'f/1
Equation 2.32
There are primarily only two limit states that consider the load modifier: strength and
service. For design the following guidelines are presented for selection of the load
modifier. The design guidelines vary depending on the limit state under consideration.
Designing at the strength limit state, values of 'f/; range as follows (FHW A, 1998):
• Ductility  'f/D
• 'f/D:; 1.05 for nonductile components and conections
• 'f/D = 1. 00 for conventional designs and details
43
• 1JD?: 0.95 for components and connections for which additional ductility
enhancing measures are specified
• Redundancy  1]R
• 1]R 1.05 for nonredundant components and connections
• 1]R = 1. 00 for conventional levels of redundancy
• 1]R?: 0.95 for exceptional levels of redundancy
• Operational Importance  1]1
• 1]1 1.05 for important structures
• 1]1 = l. 00 for typical structures
• 1]1 ?: 0. 95 for relatively less important structures
The criteria used to determine the operational importance of a structure should be base
on social, survival and/or security or defense requirements.
For design at the service limit state, 1]v = 1]R = 1]1 = 1.0.
2.2.6.4.2 Load Factors
The primary loads of concern to a geotechnical engineer are earth loads, surcharge, and
downdrag. These loads can be classified into two main categories: permanent and
transient.
2. 2. 6. 4. 2.1 Permanent Loads
Permanent loads are defined as those loads that are considered to be acting on the
structure at all times. The following is a brief list of permanent loads (FHW A, 1998):
• Dead
•
•
•
• structural components
• utilities
• vertical pressure from earth
Downdrag
Lateral earth pressure
Earth surcharge loading
2.2.6.4.2.2 Transient Loads
Transient loads are loads experienced by the structure only some of the time. The
following is a brief list of transient loads that are or may be considered during design
(FHW A, 1998):
• Loads associated with vehicles
• Pedestrian live load
44
• Live load surcharge due to trafic loads on backfill
• Water load and stream pressure force
• Wind load
• Friction force
• Force efects due to superimposed deformations, i.e. temperature change, settlement,
and creep or shrinkage
• Earthquake
• Collision due to ice, vehicle , or vessel
2.2. 6.4.2.3 Load Factors and Load Combinations
Most substructure designs require the evaluation of foundation and structure performance
at the Strength I and Service I limit states. These limit states are chosen because they are
analogous to evaluations of ultimate capacity and deformation behavior in ASD,
respectively. A table ofload factors can be found in Load and Resistance Factor Design
(LRFD) for Highway Bridge Substructures (FHW A, 1998).
2.2.6.4.3 Resistance Factors
The resistance factors recommended by the ASHTO LRFD specifications are not related
in a precise manner to variables such as the number of borings, exploration depth, or
boring spacing, but instead reflect the standard of care representative of each data set
considered. Guidelines for exploration and testing programs are based on three criteria:
FHWA (1998) recommendations, the availability of information from previous
explorations in the vicinity of the site; and engineering experience and judgement.
A simple quantitative measure of the variability, or dispersion, of data or an engineering
property is the coeficient of variation, COV, which is the ratio of the standard deviation
to the mean of the data set. The greater the value ofCOV the less reliable the data and
the lower the COV the greater the resistance factor. For any given COV the site
characteristics are obviously quantified with greater certainty by obtaining more samples
and doing more testing ..
It is important to remember that while optimizing the exploration program one should also
consider the reliability of the diferent methods available for engineering property
assessment of soil and rock. There are three primary sources of error which contribute to
the uncertainty of material resistance estimates (FHWA, 1998):
• Inherent spatial variability represented by the uncertainty in using point
measurements compared to measurements reflecting a larger volumetric extent
• Measurement error due to equipment and testing procedures
• Model error reflected by the uncertainty of the predictive method
The LRFD manual (1994) provides many tables for this endeavor. Table 2.9 is presented
as an example to the reader.
45
Table 2.9: Summary of Inherent Soil Variability and Measurement Variability Index (after Phoon et al. 1995)
Various Soil
Inherent Soil Measurement ASTM
Properties
Soil Type Variability Variability Precision
MeanCOV MeanCOV Estimate COV
Wn Finegrained 0.18 0.08 
W1 Finegrained 0.18 0.07 0.05
Wn Finegrained 0.16 0.1 0.17
PI Finegrained 0.29 0.24 
LL Clav, Silt 0.74  
'Y Finegrained 0.09 0.01 
Yd Finegrained 0.07  
Sand 0.19  
Dr
Sand 0.61  
The resistance factors presented in the LRFD manual are not absolute. The engineer can
modify resistance factors presented in LRFD when unusual or highly variable soil and rock
conditions are encountered, or when very uniform or well defined soil and rock conditions
exist. Any modification of resistance factors would be similar to modifying the factor of
safety in ASD when critical situations or geologic conditions are present.
2.2.6.5 Design of Laterally Loaded Pile
During the design of a laterally loaded drilled shaft group using the LRFD method, two
failure criteria are checked: structural failure of the shaft and excessive deflection of the
shaft at ground line. Passive failure is another potential failure mode, but is not considered
because failure of this type generally occurs at relatively excessive deflection, which
exceeds tolerable movements.
Structural failure of the shaft is checked after determining that there are not excessive
deflections at the ground line due to the applied load. The structural integrity of the shaft
is checked based on the design of reinforced concrete. This is beyond the scope of this
research project and is not presented here.
Section 10.8.2.4 in LRFD provides the criteria for designing and checking deflection for a
laterally loaded drilled shaft. This section states that the lateral displacement of a single
pile and a pile group can be calculated using a py analysis.
There is no mention of group capacity for laterally loaded drilled shafts in LRFD.
However, a procedure is outlined in the driven pile section