Behavior of piles and pile groups under lateral load

BEHAVIOR OF PILES AND PILE
GROUPS UNDER LATERAL LOAD
us.Department
of Transportation
Federal Highway
Administration
REPROD.UC ED BY
NATIONAL TECHNICAL
INFORMATION SERVICE
us. DEPARTMENT OF COM.MERCE
SPRINGFIELD, VA. 22161
PB86-238466
Research, Development,
and Technology
Turner·Fairbank Highway
Research Center
6300 Georgetown Pike
McLean, Virginia 22101
I'leport No.
FHW AfRO-85/1 06
Final Report
March 1986
is document is available to the U.S. public through the National Technical Information Service, Springfield, Virginia 22161

FOREWORD
This report details and describes design procedures for piles
subjected to lateral loads. It will be of interest to
geotechnical and bridge engineers.
This report presents the results of the University of Texas,
research project, "Behavior of Piles and Pile Groups Under
Lateral Load." The program was conducted for the Federal
Highway Administration, Office of Engineering and Highway
Operations Research and Development, Washington, D.C., under
Interagency agreement DTFH61-84-Y-30005. This final report
covers the period of research and development from September
28, 1982, to May 1, 1983.
Sufficient copies of the report are being distributed by FHWA
BuIletin to provide a minimum of two copies to each FHWA regional
office, two copies to each FHWA division, and two copies to each
State highway agency. Direct distribution is being made to the
division offices.
Office of Engi eering
NOTICE
and Highway per at ions
Research and Development
This document is disseminated under the sponsorship of the
Department of Transportation in the interest of information
exchange. The United States Government assumes no liability
for its contents or use thereof. The contents of this report
reflect the views of the contractor, who is responsible for the
accuracy of the data presented herein. The contents do not
necessarily reflect the official policy of the Department of
Transportation. This report does not constitute a standard,
specification, or regulation.
The United States Government does not endorse products or
manufacturers. Trade or manufacturers' names appear herein
only because they are considered essential to the object of
this document.
•,~

1. Report No. 2. Government Accession No.
FH~JA/RD-85/.l06
4. Title and Subtitle
Behavior of Piles and Pile Groups Under Lateral
Load
Technical Report Documentation Page
3. Recipient's Cotalog No.
5. Report Dote
March 1986
~------~----~~--------~6. Performing Orgoni zation Code
I--=_ _~_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _--! B. Performing Organization Report No.
7. Author! s)
L. C. Reese
9. Performing Organi zation Name and Address 10. Wark Unit No. (TRAIS)
FCP 35P2-192The University of Texas at Austin
College of Engineering 11. Contract or Grant No.
DTFH61-84-Y-30005Bureau of Engineering Research
Austin~ Texas 78712 13. Type of Report and Period Covered
~~~~~~~~~~~-----.----------~
12. Sponsoring ~ilency Name and Address
Federal Highway Administration Final Report
Office of Engineering &Highway Operations Aug. 1983 - Dec. 1983
Research and Development
Washington, D.C. 20590
15. Supplementary Notes
14. Sponsoring Agency Code
CME/0158
FHWA Contract Manager: Carl D. Ea1y (HNR-30)
16. Abstract
Several methods of analysis and design of piles under lateral loading are in use.
Presumptive values that suggest allowable loads, but very conservative ones, are
included in some manuals of practice. Batter piles may be employed with an
assumption, not entirely correct, that no lateral load is taken by vertical piles.
Several rational methods, in which the equations of mechanics are satisfied, have
been proposed. The methods of Broms and of Poulos and his coworkers are reviewed.
The rational method utilizes different equations to solve the governing differential
equation along with the use of nonlinear curves to describe the soil response.
Curves showing soil resistance p as a function of pile deflection y have been
recommended for several types of soil and pile loading.
Case studies are presented where results from analysis are compared with those
from experiment. Design recommendations are made and needed research is outlined.
17. Key Words
Piles, lateral loading, analysis, design
soil response p-y curves, deflection,
bending, group effects, static loading,
cyclic loading, difference-equation
methods, nondimensiona1 curves
lB. Distribution Statement
No restrictions. This document is
available to the public through the
National Technical Information Service,
Springfield, VA 22161
19. Security Clossif. (of this report) 20. Security Classif. (of thi s page) 21. No. of Pages 22. P ri ce
Unclassified Unclassified 311
Form DOT F 1700.7 (B_72) Reproduction of completed poge authorized

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PREFACE
In 1977, the Implementation Division of the Federal Highway Adminis-
tration sponsored the writer and colleagues in the preparation of a
two-volume work entitled, "Design Manual for Drilled Shafts.1I Volume 2 of
that work presented design procedures for drilled shafts subjected to lat-
eral loads. This present volume, a manual for highway engineers, is
related to piles and drilled shafts and updates and enlarges on the 1977
work. This manual is sponsored by the Research Division of the Federal
Highway Administration.
The objectives of the manual are to present background material,
design procedures, and methods of verifying computations for deep founda-
tions under lateral load. Emphasis is placed on the use of nonlinear
curves for soil response and the use of numerical procedures to solve the
governing differential equation. An Executive Summary of this work
includes recommendations for further research and the means of carrying
out that research.
A companion work is under preparation for the Implementation Divi-
sion of FHWA and is entitled, IIHandbook on Design of Piles and Drilled
Shafts under Lateral Load. 1I A draft of that work was employed in two
two-day workshops, the first in Austin, Texas in June, 1983, and the sec-
ond in Albany, New York in July, 1983.
iii

Chapter 1.
Chapter 2.
Chapter 3.
TABLE OF CONTENTS
Introduction
Soil-Structure Interaction
Methods of Solution of Laterally Loaded Piles
Design Problems
Design Organization
Factor of Safety
References
The Differential Equation
Relation between Curvature and Bending Moment
The Differential Equation of the Elastic Curve
Derivation of the Differential Equation for the
Beam-Column
Summary
Example Exercise
References
Exercises
Soil Response
Reaction of Soil to Lateral Deflection of Pile
Use of Theory of E1 asti city to Determi ne Soil
Behavior
Skempton
Terzaghi
McClelland and Focht
Use of Soil Models to Determine Soil Behavior
Soil Models for Saturated Clay
Soil Models for Sand
Experimental Methods for Obtaining Soil Response
Curves
Soil Response from Direct Measurement
Soil Response from Experimental Moment
Curves
Nondimensiona1 Methods for Obtaining Soil
Response
Recommendations for p-y Curves for Clays
Response of Soft Clay below the Water Table
Field Experiments
iv
1
2
5
5
7
8
9
11
11
13
14
16
17
20
20
21
23
25
25
27
29
30
30
31
34
34
35
36
37
37
37

Chapter 4.
Chapter 5.
TABLE OF CONTENTS (continued)
Recommendations for Computing p-y
Curves
Recommended Soil Tests
Example Curves
Response of Stiff Clay below the Water
Table
Field Experiments
Curves
Example Curves
Response of Stiff Clay above the Water
Table
Field Experiments
Curves
Example Curves
Unified Criteria for Clays below the Water
Table
Introduction
Curves
Example Curves
Recommendations for p-y Curves for Sand
Response of Sand below the Water Table
Field Experiments
Curves
Simplified Equations
Example Curves
Response of Sand above the Water Table
Recommendations for p-y Curves for Rock
References
Exercises
Solutiorl"5 for Laterally Loaded Piles with Soil
Modulus Assumed Constant, Constant Pile Stiff-
ness, No Axial Loading
Solution of the Differential Equation
Pile of Infinite Length
Pile of Finite Length
References
Exercises
The Difference Equation Method for Solving the
Differential Equation for a Laterally Loaded
Pile
Introduction
v
Page
37
40
40
41
41
41
47
47
48
48
49
52
52
52
52
53
59
59
59
59
60
65
67
67
68
68
70
71
73
73
75
79
79
80
81
81

Relationships in Difference Form
The Gleser Method of Solution
Example Computation
Difference Equations for Case of Axial Loading
and for Changes in Bending Stiffness
Computer Program COM622
References
Exercises
81
83
84
87
92
93
94
Chapter 6. Nondimensional Method for the Analysis of Lat-
erally Loaded Piles
95
Dimensional Analysis for Elastic Piles 95
Dimensional Analysis for Rigid Piles 100
Forms of Variation of Soil Modulus with Depth 103
Solution Procedures 106
Case I - Pile Head Free to Rotate 107
Case II - Pile Head Fixed Against Rotation 110
Case III - Pile Head Restrained Against 118
Rotation
Solution of Example Problem, Assuming Es = kx 119
Solution of Example Problem, E = kxn and E = kO 125
+ k x s s
Altern~te Solution to Restrained-Head Case 128
Concluding Comment 136
References 138
Exercises 139
Chapter 7. Other Methods of Design 141
Broms Method 141
Ultimate Lateral Load for Piles in Cohesive 141
Soil
Short, Free-Head Piles in Cohesive Soil 141
Long, Free-Head Piles in Cohesive Soil 143
Influence of Pile length, Free-Head Piles 145
in Cohesive Soil
Short, Fixed-Head Piles in Cohesive Soil 145
Intermedi ate Le.nglh, Fi xed-Head Pi 1es in 145
Cohesive Soil
Long, Fixed-Head Piles in Cohesive Soil 146
Influence of Pile Length, Fixed-Head Piles 146
in Cohesive Soil
Deflection of Piles in Cohesive Soil 148
Effects of Nature of. Loadi n9 on Pi 1es in 149
Cohesive Soil
Ultimate Lateral Load for Piles in Cohe- 149
sionless Soil
Short, Free-Head Piles in Cohesionless Soil 150
vi

Chapter 8.
Page
Long, Free-Head Piles in Cohesion1ess Soil 152
Influence of Pile Length, Free-Head Piles 154
in Cohesion1ess Soil
Short, Fixed-Head Piles in Cohesion1ess 154
Soil
Intermedi ate Length, Fi xed-Head Pil es in 154
Cohesion1ess Soil
Long, Fixed-Head Piles in Cohesion1ess Soil 154
Influence of Pile Length, Fixed-Head Piles 155
in Cohesion1ess Soil
Deflection of Piles in Cohesion1ess Soil 156
Effects of Nature of Loading on Piles in 157
Cohesion1ess Soil
Poulos Method 157
Model Employed in Elastic Analysis 157
Free-Head Piles, Elastic Behavior 158
Fixed-Head Piles, Elastic Behavior 159
Effect of Local Yield of Soil Along Pile 161
Determination of Properties of Soil 162
Example Computation 166
Pressuremeter Method 167
Introduction 167
Pressuremeter Curve 167
Determining Pressuremeter Modulus 168
Development of p-y Curves 169
Example Computation 171
Method Using Charts 172
References 177
Exerci ses 179
Structural Design of Piles and Drilled Shafts
Nature of Loading
Fail ure Modes
Concepts of Design
Structural Steel
Reinforced Concrete
Timber
Design of a Structural Steel Member
Computation of Design Loads
Step-by-Step Procedure
Example Problem
Design of a Reinforced Concrete Member
Computation of Bending Stiffness
Example Problem
References
Exercises
vii
181
181
182
183
183
183
184
184
184
184
185
187
187
187
189
191
194
195

Chapter 9.
Case Studies of Single Piles Under Lateral Loading
Introduction
Parametric Studies of Piles in Clay Using Com-
puter Method
Soft Clay below Water Table
Stiff Clay below Water Table
Stiff Clay above Water Table
Unified Criteria for Stiff Clay below Water
Table
Sand
Comments on Parametric Studies
Effect of Depth of Penetration
Comments on Methods Used for Case Studies
Case Studies of Piles in Clay
Japanese Test
St. Gabriel
Southern California
Lake Austin
Sabine
Houston
Manor
Case Studies of Piles in Sand
Gi 11 Tests
Arkansas River
Florida
Mustang Island
Comments on Results of Case Studi~s
References
Exercises
Page
197
197
199
199
199
201
205
205
210
210
211
215
215
216
218
222
223
227
228
230
230
236
237
239
242
243
244
Chapter 10. Analysis of Pile Groups Under Lateral Loading 245
Introduction 245
Distribution of Load to Each Pile in a Group 245
Problem Statement 245
Loading and Movement of the Structure 246
Movement of a Pi 1e Head 248
Forces and Moments' 248
Equilibrium Equations 249
Solution Procedure 250
Example Problem 251
Behavi or of a Group of Close1y-Spaced Pil es 257
Pou1os-Focht-Koch Method 258
Single-Pile Method 266
Example Problem 266
Pou1os-Focht-Koch Solution 267
Solution Assuming Group Behaves as a 270
Single Pile
Comment on Solution of Example Problem 271
References 271
viii

Page
Chapter 11. Step-by-Step Procedure for Design 273
Appendix 1. Solution of Coefficients for a Pile of Finite 277
Length, Constant Soil Modulus, Constant Pile
Stiffness
Appendi x 2. Gl eser Method of Sol uti on of Difference 287
Equations
Appendix 3. Computer Program COM622
Input Guide for COM622
Example Problems
Listing of Program
Listing of Input for Example Problems
Output for Example Problems
Appendix 4. Nondimensional Coefficients
Appendix 5. Computer Program PMEIX
299
300
302
303
310
312
323
371
Input Guide 372
Statement of the Problem 372
Outline of the Solution 372
Procedure 373
Example 375
Application to Load-Deflection Analysis of 377
Drilled Shafts or Piles
Program Capabilities 377
Data Input 379
Printed Output 379
Other Output 379
Example Problems 383
Li sti ng of Program 384
Listing of Input for Example Problems 392
Output for Example Problems 393
Note: Appendixes are available upon request from Materials Division,
Office of Engineering and Highway Operations Research and
Development, 6300 Georgetown Pike, McLean, Virginia 22101
ix

Figure No.
1.1
1.2
1.3
1.4
1.5
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3
3.4
LIST OF FIGURES
Title
Strip footing
Model of a pile under axial load
Model of a pile under lateral load
Examples of laterally-loaded piles
Results of computations where pile penetration
is controlled by lateral loading
A stra ight beam under bendi ng moment (after
Popov, 1952)
Segment of a deflected pile
Element from beam-column
Sign conventions
Form of the resul ts obtained from a complete
solution
Section of a beam (pile) with uniform load
Graphical definition of p and y
(a) view of elevation of section of pile
(b) view A-A - earth pressure distribution pri-
or to lateral loading
(c) view A-A - earth pressure distribution
after lateral loading
Typical p-y curves
(a) family of curves
(b) characteristic shape of p-y curve
Assumed passive wedge-type failure for clay
(a) shape of wedge
(b) forces acting on wedge
Assumed lateral flow-around type of failure for
clay
(a) section through pile
(b) Mohr-Coulomb diagram
(c) forces acting on section of pile
x
Page
2
4
4
6
9
12
14
15
18
18
19
22
24
31
32

Fi gure No.
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15a
3.15b
3.16
LIST OF FIGURES (continued)
Title
Assumed passive wedge-type failure of pile in
sand
(a) general shape of wedge
(b) forces on wedge
(c) forces on pile
Assumed mode of soil failure by 1atera1 flow
around a pile in sand
(a) section through pile
(b) Mohr-Coulomb diagram representing states
of stress of soil flowing around a pile
Characteri sti c shapes of the p-y curves for
soft clay below the water table
(a) for static loading
(b) for cyclic loading (from Matlock, 1970)
Soi 1 profi 1e used for example p-y curves for
soft cl ay
Example p-y curves for soft clay below water
table, Matlock criteria, cyclic loading
Characteri stic shape of p-y curve for static
loading in stiff clay below the water table
(after Reese, Cox, Koop, 1975)
Values of constants A and As c
Characteristic shape of p-y curve for cyclic
load~ng in stiff clay below water table (after
Reese, Cox, Koop, 1975)
Soil profile used for example p-y curves for
stiff cl ay
Example p-y curves for stiff clay below the
water table, R~ese criteria, cyclic loading
Characteristic shape of p-y curve for static
loading in stiff clay above water table
Characteristic shape of p-y curve for cyclic
loading in stiff clay above water table
Example p-y curves for stiff clay above water
table, Welch criteria, cyclic loading
xi
33
35
38
41
42
44
45
46
48
49
50
51
53

Figure No.
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
4.1
4.2
5.1
5.2
6.1
6.2
6.3
6.4
6.5
Titl e
Characteristic shape of p-y curve for unified
clay criteria for static loading
Characteristic shape of p-y curve for unified
clay criteria for cyclic loading
Example p-y curves for soft clay below water
table, unified criteria, cyclic loading
Example p-y curves for stiff clay below water
table, unified criteria, cyclic loading
Characteristic shape of a family of p-y curves
for static and cyclic loading in sand
Values of coefficients A and A
c s
Nondimensional coefficient B for soil resis-
tance versus depth
Example p-y curves for sand below water table,
Reese criteria, cyclic loading
Recommended p-y curve for design of dri 11 ed
shafts in vuggy limestone
Soil response curve
Boundary conditions at top of pile
Representation of deflected pile
Method of subdividing pile
Pile deflection produced by lateral load at
groundline (Reese and Matlock, 1956)
Pile deflection produced by moment applied at
Slope of pile caused by lateral load at ground-
line (Reese and Matlock, 1956)
Slope of pile caused by moment applied at
Bending moment produced by lateral load at
xii
Page
54
58
60
61
62
63
63
67
69
73
75
82
83
108
109
111
112
113

Fi gure No.
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
, 6.15
6.16
6.17
6.18
Title
Bendi ng moment produced by moment app1i ed at
Shear produced by 1atera1 load at groundl i ne
(Reese and Matlock, 1956)
Shear produced by moment applied at groundline
(Reese and Matlock, 1956)
Deflection of pile fixed against rotation at
Plot of p-y curves for example problem, stiff
clay above water table, cyclic loading
Trial plots of soil modulus values
Interpolation for final value of relative
stiffness factor T
Deflection and moment diagrams for example
problem
Trial fitting of Es =kxn for solution of exam-
ple problem
Deflection, moment, and shear diagrams for
example problem, Es =kO + k1
x
Latera1 forces app1i ed to an offshore struc-
ture, Example Problem 2 (Matlock and Reese,
1961)
The superstructure and the pile, considered as
elastic elements of the problem (Matlock and
Reese, 1961)
Typical resistance-deflection curves predicted
for the soil at various depths (Matlock and
Reese, 1961)
Page
114
115
116
117
121
123
124
126
127
130
131
132
132
6.19 Nondimensional coefficients for lateral 134
defl ecti on of a pi 1e, assumi ng ~oi 1 modul us
proportional to depth, or E =kx L!ong pile case Zmax 5 to lQU
s
6.20
6.21
Trial plots of soil modulus values
Interpolation for final value of relative stiff-
ness factor T
xiii
136
137

Fi gure No.
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
Title
Assumed di stri buti on of soi 1 resistance for
cohesive soil
Deflection, load, shear, and moment diagrams
for a short pi 1e in cohesive soi 1 that is
unrestrained against rotation
Design curves for short piles under lateral
load in cohesive soil (after Broms)
Design curves for long piles under lateral load
in cohesive soil (after Broms)
for an i ntermedi ate-l ength pi 1e in cohesive
soil that is fixed against rotation at its top
Failure mode of a short pile in cohesionless
soil that is unrestrained against rotation
for a short pile in cohesionless soil that is
unrestrained against rotation
Design curves for long piles under lateral load
in cohesionless soil (after Broms)
Stresses acting on (a) pile; (b) soil adjacent
to pile (after Poulos)
Influence factors I for free-head pile (after
Poulos) YP
Infl uence factors I M and I for free-head
pile (after Poulos) Y sp
Influence factors I Mfor free-head pile (after
Poulos) s
Maximum bending moment for free-head pile (af-
ter Poulos)
Influence factors IYF for fixed-head pile (af-
ter Poulos)
Maximum negative bending moment for fixed-head
pile (after Poulos)
xiv
141
142
143
144
146
151
151
153
158
160
160
160
160
161
162

Fi gure No.
7.16
7.17
7.18
7.19
7.20
7.21
7.22
7.23
7.24
7.25
8.1
8.2
8.3
8.4
8.5
8.6
Title
Influence of distribution of yield pressure on
load-displacement relationship, free-head case
(after Poulos)
Degradation parameter, t (from Poulos)
Typical curve from Menard pressuremeter
Soil response curves proposed by Menard
Soil and test pile at Plancoet
Comparison of results from various analytical
methods with results from experiment at Plan-
coet .
Values of maximum bending moment in an 18-in.
diameter concrete shaft in clay (after Reese
and Allen)
Values of maximum bending moment in an 18-in.
diameter concrete shaft in clay (after Reese
and Allen)
Empirical curves showing response of driven,
precast concrete piles (Manoliu, 1976)
Empirical curves showing response of drilled
shafts (Bhushan, 1981)
Sri dge bent
Example to demonstrate the analysis of a steel
pipe
Relationship between moment and curvature for a
concrete member
Example to demonstrate the analysis of a rein-
forced concrete pile
Relationship between moment and curvature for
the example problem
Values of load to be employed in computer pro-
gram to analyze laterally loaded drilled shaft
xv
Page
163
165
168
170
172
173
174
175
176
177
181
186
189
192
192
193

Figure No.
8.7
9.1
9.2
9.3
9.4
9.S
9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13
9.14
9.1S
Title
Results from computer analysis of the drilled
shaft
Comparison between results for ±SO percent var-
iation in c for soft clay below water table
iation in £SO for soft clay below water table
iation in EI for soft clay below water table
iation in c for submerged stiff clays
iation in £SO for submerged stiff clays
iation in k for submerged stiff clays
iation in EI for submerged stiff clays
iation in c for stiff clay above water table
iation in £SO for stiff clay above water table
iation in EI for stiff clays above water table
iation in c for unified criteria
iation in £SO for unified criteria
iation in k for unified criteria
iation in EI for unified criteria
Comparison between results for ±20 percent var-
iation in I/J using sand criteria for cyclic
loading
xvi
194
200
200
201
202
202
203
203
204
204
205
206
206
207
207
208

Fi gure No.
9.16
9.17
9.18
9.19
9.20
9.21
9.22
9.23
9.24
9.25
9.26
9.27
9.28
9.29
9.30
9.31
Title
iation in r using sand criteria for cyclic
loading
iation in ks using sand criteria for cyclic
loading
iation in EI using sand criteria for cyclic
loading
Effect of depth of embedment on lateral
deflection in sand
Relationship between Es and undrained shear
strength for cohesive soil used in case studies
Relationship between k and. for cohesionless
soil
Relationship between Es and angle of internal
friction for cohesionless soil
Information for the analysis of Japanese test
Comparison of measured and computed results for
Japanese Test
Information for analysis of test at St. Gabriel
St. Gabriel Test
Information for the analysis of Southern Cali-
fornia Test Pile 2
Southern California Test Pile 2
Information for the analysis of Southern Cali-
fornia Test Pile 6
xvii
Page
208
209
209
211
213
214
215
216
217
219
219
220
220
221
222
223

Figure No.
9.32
9.33
9.34
9.35
9.36
9.37
9.38
9.39
9.40
9.41
9.42
9.43
9.44
9.45
9.46
Title
Comparison of measured and computed deflections
for Lake Austin Test
Compari son of mea sured and computed max imum
moments for Lake Austin Test
Compari son of me'asured and computed defl ection s
for Sabi ne Test
Compari son of measured and computed maximum
moments for Sabine Test
for Houston Test
(a) cyclic loading (10 cycles)
(b) static loading
moments for Houston Test
for Manor Test
(a) cyclic loading
(b) static loading
Comparison of measured and computed moments for
Manor Test
Information for the analysis of tests in
hydraul ic fi 11
Gill Test Pile 9
Gill Test Pile 10
Gill Test Pile 11
Gill Test Pile 12
Soils information for analysis of tests at
Arkansas Ri ver
for Arkansas River Test Pile 2
xviii
Page
224
225
226
227
229
230
231
232
233
234
234
235
235
237
238

Fi gure No.
9.47
9.48
9.49
9.50
9.51
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
Title
moments for Arkansas River Test Pile 2
Arkansas River Test Pile 6
Florida Test
for Mustang Island Test
(a) cyclic loading
(b) static loading
Comparison of measured and computed maximum
moments for Mustang Island Test
Typical pile-supported bent
Simp1i fi ed structure showi ng coordi nate sys-
tems and sign conventions (after Reese and Mat-
lock)
Set of pile resistance functions for a given
pile
Sketch of a pile-supported retaining wall
Interaction diagram of the reinforced concrete
pile
Axial load versus settlement for reinforced
concrete pile
Pile loading, Case 4
Interaction factor apH for free-head piles
subjected to horizontal load (Poulos, 1971)
Interaction factors apM for free-head
piles subjected to moment (Poulos, 1971)
Interaction factors a F for fixed-head pile
(Poulos, 1971) p
Infl uence factors IpH for a free-head pil e
(Poulos, 1971)
xix
238
239
240
241
242
246
247
252
253
254
254
257
260
261
262
263

Fi gure No.
10.12
10.13
10.14
10.15
10.16
10.17
A3.1
A3.2
A5.1
A5.2
A5.3
A5.4
A5.5
A5.6
Title
Influence factors IpM for a free-head pile
(Poulos, 1971)
Influence factors IpF for a fixed-head pile
(Poulos, 1971)
lIyll factor i nfl uence on computed pi 1e-head
deflection
Plan and elevation of foundation analyzed in
example problem
Graphical solution for V-factor
Bendi ng moment curve for pi 1e wi th greatest
load, example solution
Sample problems
p-y curves for 16-in. diameter pile in clay
Portion of a beam subjected to bending
(a) the elastic curve
(b) cross-section
(c) strain diagram
Beam cross-section for example problem
Stress-strain curve for concrete used by Pro-
gram PMEIX
Stress-strain curve for steel used by Program
PMEIX
Data input form for Computer Program PMEIX
Concrete column cross-sections for example
problem
xx
263
264
265
267
269
270
302
302
374
374
378
378
380
383

Table No.
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
4.1
6.1
6.2
6.3
6.4
6.5
6.6
7.1
7.2
LIST OF TABLES
Title Page
Terzaghi's recommendations for soil modulus aT 28
for laterally loaded piles in stiff clay
Terzaghi's recommendations for values of k for 29
laterally loaded piles in sand
Representative values of £50 39
Representative values of k for stiff clays 43
Representative values of £50 for stiff clays 43
Representative values of £50 55
Curve parameters for the unified criteria 56
Representative values for k 57
Representative values of k for submerged sand 64
Representative values of k for sand above water 64
table
Nondimensiona1 coefficients for p-y curves for 66
sand (after Fenske)
Table of functions for pile of infinite length 77
Moment coefficients at top of pile for
fixed-head case
Computed p-y curves
Computed deflections
Computed values of soil modulus
Computed values of soil modulus, Es =kO + k1x
Sample computations for first trial
Values of rheological factor a (after Bague1in,
et a1., 1948)
Poi nts on p-y curves deri ved from data from
pressuremeter
xxi
118
120
122
122
129
135
170
173

Table No.
9.1
9.2
10.1
10.2
10.3
A4.1
A4.2
A4.3
A4.4
A4.5
A4.6
A4.7
LIST OF TABLES (continued)
Title Page
Initial parameters for soil 198
Initial parameters for pile 198
Values of loading employed in analyses 255
Computed movements of origin of global coordi- 255
nate system
Computed movements and loads at pile heads 256
A and B coefficients for elastic piles, E =k, 325
Z = 10 s
max
A and B coeffi ci ents for e1astic pi 1es, Es = 326
kx 0.2 5, Z =10
max
A and B coefficients for elastic piles, Es = 327
kx 0.5 Z = 10, max
kx, Zmax =10
kx 2
Z = 10, max
A and B coeffi ci ents for e1astic pil es, Es = 330
kx4
Z = 10, max
a and b coefficients for rigid piles, Es =k 332
A4.8 a and b coefficients for rigid piles, E = 333
kx 0.25 S
A4.9
A4.10
A4.11
A4.12
A4.13
A4.14
a and b coefficients for rigid piles, E =kX~5 334
s
a and b coefficients for rigid piles, E =kx 335s
a and b coefficients for rigid piles, E =kx 2
336S
a and b coefficients for rigid piles, E =kx 4
337s
A and B coefficients for elastic piles, Es =kO 339
+ k1x where kO/k1T =0.1, Zmax =10
A and B coefficients for elastic piles, Es =kO 340
xxii

Table No.
A4.15
A4.16
A4.17
A4.18
A4.19
A4.20
A4.21
A4.22
A4.23
A4.24
A4.25
A4.26
A4.27
A4.28
A4.29
A4.30
Title
A and B coefficients for elastic piles, Es =kO
a and b coefficients for rigid piles, E =k +
I T
sO.
k1x where kO k1 =0.1
a and b coefficients for rigid piles, Es =kO +
k1x where kO/k1T =0.2
a and b coefficients for rigid piles, Es = kO +
A and B coeffi ci ents for e1astic pil es, Es =
kx, Zmax =10.0
A and B coefficients for elastic piles, Es =
kx, Zmax =4.0
kx, Zmax =3.5
kx, Zmax =3.0
kx, Zmax = 2.8
kx, Zmax =2.6
xx;;;
Page
341
342
343
344
346
347
348
349
350
351
353
354
355
356
357
358

Table No. Title ~~
A4.31 A and B coefficients for elastic pil es, E = 359
kx, Zmax =2.4
s
A4.32 A and B coefficients for elastic piles, E = 360
kx, Zmax =2.2
s
A4.33 Deflection coefficients Cy
for elastic pil es, 362
E = kx Z =10.0s 'max
for elastic piles, 363
E = kx Z =4.0s 'max
E = kx Z =3.5s 'max
E = kx Z =3.0s 'max
A4.37 Defl ection coefficients Cy
E = kx Z =2.8s 'max
E = kx Z =2.6s 'max
E = kx Z =2.4s 'max
E = kx Z =2.2s 'max
A5.1 Detailed input guide with definitions of vari- 381
ables
xxiv

a.
1
A
A
c
A
s
b
b.
1
LIST OF NOTATIONS
horizontal coordinate of global axis system in
pile group analysis
nondimensional coefficients, same
A-coefficient except for rigid-pile theory
coefficient used to define the shape of the
p-y curve, unified criteria for clay
as
empirical coefficient used in equations for p-y
curves for stiff cl ays below water surface,
cyclic loading
curves for stiff clays below water surface,
static loading
nondimensional coefficients in elastic-pile
theory re1ati ng to an app1i ed force Pt' for
deflection, slope, moment, shear and soil
reaction, respectively
curves for sand, cyclic loading
curves for sand, static loading
coefficients in solutions for the difference
equation method
deflection coefficient for long pile with pile
top restrained against rotation
nondimensional coefficients for piles of infi-
nite length and finite length, constant pile
stiffness, constant soil stiffness, no axial
loading
pile diameter or width of foundation (L)
vertical coordinate of global axis system in
pile group analysis (L)
nondimensional coefficients same as
B-coefficients, except for rigid-pile theory
coeffi ci ents in sol uti ons for the difference
equation method
curves for sand, cyclic loading
xxv

c
C
Cy
* * C*CI , C2, 3
DE
°p
DR
e
e
E
E
Ecp
Es
LIST OF NOTATIONS (continued)
curves for sand, static loading
nondimensiona1 coefficients in elastic-pile
theory re1ati ng to an app1i ed moment Mt for
deflection, slope, moment, shear and soil
reaction, respectively
equation method
undrained shear strength (F/L2)
average undrained strength of clay from ground
surface to depth (F/L2)
average undrained shear strength (F/L2)
coefficient related to stress level used in p-y
curves for stiff clay above water surface
coefficient to be determined by use of boundary
conditions for case of constant pile stiffness
and constant soil stiffness, no axial loading
coefficient used in equations for p-y curves for
sand
nondimensiona1 deflection coefficient assuming
E =ks c
equation method
degradation parameters for soil modulus
degradation parameters for yield pressure
rate factor
eccentricity of lateral load
Napierian base
Young1s modulus (F/L2)
term used in difference equation solution
soil modulus after cyclic loading (F/L2)
soil modulus (secant to p-y curve)(F/L2)
xxvi

Esi or Es(max)
Esm
Esp
EIc
F
v·1
G
m
initial or maximum soil modulus (F/L2)
soil modulus at node m(F/L2)
soil modulus (values suggested by Poulos) (F/L2)
pile stiffness of combined pile and jacket leg
(F/L2)
flexural rigidity of pile (F/L2)
the Menard modulus of deformation (F/L2)
parameter in Broms method for computing pi 1e
load (L)
bending stress on pile (F/L2)
coefficient used to define deterioration of soil
resistance at large deformations, unified crite-
ria for clay, static loading
hori zonta1 component of force on any II i-th" pil e
( F)
moment coefficient at top of pile for fixed-head
case
force against a pile in clay from wedge of soil
( F)
rate coefficient (limited data suggest a range
of from 0.05 - 0.3)
force against a pile in sand from a wedge of soil
( F)
verti ca1 component of force on any II i -th" pil e
( F)
deflection coefficient for fixed head pile
parameter in Broms method for computing pile
load (L)
shear modulus from pressuremeter (F/L2)
terms used in difference equation solution
X
=L ' depth coefficient in rigid-pile theory
xxvii

h
I
J
lIST OF NOTATIONS (continued)
increment length in difference-equation method
( l)
horizontal translation in global coordinate (l)
pile increment length (l)
specific depth below ground surface (l)
lateral load on pile j (F)
the lateral load on pile k (F)
total lateral load on pile group (F)
=M
moment of inertia (l4)
in fl uence coeffi cient for computing pi 1e-head
rotation for applied moment at groundline
influence coefficient for computing pile-head
rotation for applied shear at groundline
i nfl uence coeffi ci ent for computing pil e-head
deflection for a pile with fixed head
i nfl uence coeffi ci ent for computing pi 1e-head
deflection for applied moment at groundline
i nfl uence coeffi ci ent for computing pil e-head
deflection for applied shear at groundline
surface displacement influence value
elastic influence coefficient for fixed-head
pile
elastic influence coefficient for deflection
caused by horizontal load
caused by moment
caused by horizontal load
factor used in equation for ultimate soil resis-
tance near ground surface for soft clay
xxviii

J
J
m
J
x
J
y
J
1
,J
2
,J
3
,J
4
k
ks
kO' k1, k2, .. ,
kM
ka
K
Ka
KO
K
P
KR
L
LI
m
m
m
a constant having the same dimensions as the
soil modulus for rigid pile analysis (F/L2)
=Mt/Yt modulus for computing Mt from Yt (F)
=Px/x
t
modulus for computing Px from xt (F/L)
=Pt/Y
t
modulus for computing Pt from Yt (F/L)
coefficients used in difference equation sol-
ution
constant giving variation of soil modulus with
depth (F/L3
)
coefficient used in equations for p-y curves for
stiff clays below water surface, cyclic loading
(F/L3
)
initial slope of p-y curve for sand (F/L3
)
constants of soil modulus variation in E =kO +
k X+ k X2
S
1 2
slope of initial portion of Menard's soil
response curve (F/L3
)
= Mt/, spri ng stiffness of restrained pil e
head (F L)
reducti on factor used in expression for force
against pile from wedge of soil
minimum coefficient of active earth pressure
coefficient of earth pressure at rest
Rankine coefficient ~f passive pressure
pile flexibility factor
length of pile (L)
liquidity index for clay
number of piles in group
pile node number
slope used in defining portion of p-y curve for
sand
xxix

M
Mc
Mm
M
max
MPos
max
Mpos,
M+
Mt
Mt ·
1
M/St
My, Myield
M+ M-
y' y
n
n
N
N
OR
p
Pc
Pcd
PcR
Pct
Pf
bending moment (F-L)
bending moment at pile top (f-L)
bending moment at node m(F-L)
maximum bending moment in pile (F-L)
maximum positive bending moment (F-L)
positive moment (F-L)
bending moment at pile head (F-L)
bending moment at "i-th" pile head (F-L)
rotational restraint constant at pile top (F-L)
yield moment of pile (F-L)
positive yield moment and negative yield moment
of pil e (F-L)
exponent used in equations for p-y curves for
sand
exponent in Es = kxn , t/J(Z) =Zn or t/J( h) = hn
tensile force active on beam-column (F)
number of cycles of load application used in p-y
curves for stiff clay above water surface
overconsolidation ratio for clay
soil resistance (F/L)
ultimate soil resistance for pile in stiff clay
below water surface (F/L)
ultimate soil resistance at depth for pile in
stiff clay below water surface (F/L)
residual resistance on cyclic p-y curves, uni-
fied criteria for clay (F/L)
ultimate soil resistance near ground surface for
pile in stiff clay below water (F/L)
the initial volume of the cavity (L3
)
xxx

p~
p
x
pressure at the point where there is no longer a
straight-line relationship between pressure and
volume (F/L2)
total horizontal load on pile group (F)
a specific resistance on p-y curves for sand
( F/L)
1imit pressure of typical curve from Menard
pressuremeter (F/L2)
soil resistance at node m(F/L)
soil reaction measured from Menard pressuremeter
( F/L)
ultimate soil resistance for pile in sand (F/L)
ultimate soil resistance at depth for pile in
sand (F/L)
total vertical load on pile group (F)
axial load at pile head (F)
pile in sand (F/L)
lateral force at pile top (F)
ul t imate soi 1 resistance or ul t imate soi 1
reaction (FIL)
pile inc1ay (FI L)
ultimate soil resi stance at depth for pile in
cl ay (F/L)
pile in sand (F/L)
ultimate soil resistance at depth for pil e in
sand (F/L)
axial load at pile top (F)
ultimate lateral load on a pile (F)
xxxi

PI
~P/~V
q
qyS
Qn
Qv
Ql'Q2,Q3 ,Q4
rO' r
1
, r
2
, ...
R
R
t
St
Sm
S
St
SI' S2' S3
t
t
T
the distribution of the yield pressure defined
by Poulos (F/L2)
soil resistance per uni t 1ength at, a depth Z
below the ground surface (F/L)
plasticity index for clay
slope of curve between Vo and Vf
foundation pressure (F/L2)
failure stress for foundation (F/L2)
unconfined compressive strength of clay (F/L2)
limiting pile-soil interaction stress (yield
pressure) after cyclic loading (F/L2)
yield pressure for static loading (in Poulos
method) (F/L2)
normal shear stress (F/L2)
vertical shear stress (F/L2)
constants in polynomial soil modulus functions
relative stiffness factor
=EtIt , flexural rigidity at pile top (F-L2)
sensitivity of clay
slope of pile at node m
slope
rotation at groundline
coefficients for simplified equations for p-y
curves for sand
degradation parameter
pile top node
5/TI
=I l< relative stiffness factor (IlL)
xxxi i

v
v
x
• y
shear (F)
volume of cavity (l3)
vertical translation in global coordinate (l)
initial volume of the measuring cell (l3)
midpoint volume (l3)
shear at node m(F)
volume at start of the straight-line portion of
the curve and equal to the initial volume of the
cavity (l3)
shearing force parallel to y-axis (F)
1i qui d 1imit for clay
coordinate along pile, beam (l)
transition depth at intersection of equations
for computing ultimate soil resistance against a
pile in clay (l)
transition depth at intersection of equations
for computing ultimate soil resistance against a
pile in sand (l)
vertical displacement at pile head (l)
pile deflection and for y-coordinate (l)
deflection coordinate for p-y curves for stiff
clay above water surface, cyclic loading (l)
pile deflection with pile head fixed against
rotation (l)
deflection at intersection of the initial linear
portion and the curved portion of the p-y curve,
unified criteria for clay (l)
a specific deflection on p-y curves for sand (l)
pile deflection at node m(l)
xxxiii

Yt
Yu
VA' SA' MA, VA' PA
z
z
a
a
a
pFkj
a specifi c defl ecti on on p-y curves for stiff
clay below water surface, cyclic loading (l)
deflection coordinate for p-y curves for stiff
clay above water table, static loading (l)
pile top deflection (l)
components of pil e response due to an app1ied
force Pt , namely, deflection, slope, moment,
shear and soil reaction, respectively
components of pile response due to an applied
moment Mt , namely deflection, slope, moment,
shear and soil reaction, respectively
horizontal deflection of pile group (l)
depth (l)
cri t ica1 depth for pressuremeter method sug-
gested by Menard (l)
x
=T ' depth coefficient in elastic-pile theory
l
= T ' maximum value of elastic-pile theory co-
efficient
=Es , soil modulus, coefficient of subgrade re-
action (F/l2)
angle used in defining geometry of soil wedge
rotational angle in the global coordinate system
Terzaghi's soil modulus for stiff clay (F/l2)
the coefficient to get the influence of pile j
on pile k
the angle between the line joining the pile cen-
ters and the direction of loading
~ , relative stiffness fac"tor
angle used in defining geometry of soil wedge
average unit weight of soil (F/l3
)
xxxiv

E
E
C
E
X
8
a.1
A-
r
A-
z
p
p
(1
(1
x
(1
v
'[
~
~(h)
HZ)
bouyant unit weight or average unit weight used
in computing effective stress (F/L3
)
axial strain of soil
axial strain of soil
ax i a1 strain of soil correspondi ng to one-half
the maximum principal stress difference
axial strain
angle of rotation
the inclined angle between vertical line and
pile axis of the "i-th" batter pile
loading rate
reference loading rate (perhaps. static loading)
soil modulus coefficient between the ground sur-
face and critical depth
Poisson1s ratio
mean settlement of a foundation (L)
radius of curvature of elastic curve (L)
deflection of the "k-th" pile (L)
the unit reference displacement of a single pile
under a unit horizontal load, computed by using
elastic theory (L)
normal stress (F/L2)
bending stress (F/L2)
deviator stress (F/L2)
average effective stress (F/L2)
shear stress (F/L2)
angle of internal friction of sand
= Es/J, nondimensional soil modulus function
of rigid pile theory
= Es ·T4/EI, nondimensional soil modulus function
of elastic pile theory
xxxv

Behavior of piles and pile groups under lateral load

CHAPTER 1. INTRODUCTION
Laterally loaded piles are found in many structures, both onshore and
offshore. In many instances in the past, and even today, pile foundations
have been designed so that each pile takes only a nominal lateral load or
batter piles are employed. When batter piles are utilized in a structure,
the assumption is frequently made that any horizontal load is sustained by
the hori zonta1 component of the ax iall oad. The assumption that batter
piles do not deflect laterally is, of course, incorrect as will be shown
subsequently.
With increasing cost of labor and materials and with decreasing cost
of computations, situations are a~ising where it is cost effective to
employ more engi neeri ng effort. Furthermore, desi gners are fi ndi ng it
desirable to create more complex structures and severe loadings are being
encountered, such as those on offshore structures. Thus, in some cases it
is necessary to consider as well as possible the various deformations of a
structure and its foundation under a wide range of loading. Therefore,
procedures such as those given herein are needed to allow analyses of
foundations in as rational a manner as possible.
As a foundation problem, the analysis of a pile under lateral loading
is complicated by the fact that the soil reaction (resistance) is depen-
dent on the pil e movement, and the pil e movement, on the other hand, is
dependent on the soil response. Thus, the problem is one of
soil-structure interaction.
The method of solution of the problem of the laterally loaded pile
described herein (the p-y method) is being used in the United States and
abroad. Numerous references in the following chapters will illustrate the
use of the method. To illustrate the use abroad, references are cited
from Italy (Jami1kowski, 1977), France (Baguelin, et al., 1978), Britain
(George and Wood, 1976), Australia (Poulos and Davis, 1980), and Norway
(Det Norske Veritas, 1977). The method is included in Planning, Designing
and Constructing Fixed Offshore Platforms, RP2A, American Petroleum
Institute. That publication has guided the design of offshore drilling
platforms in the United States and has significantly influenced their
design elsewhere. The method is expected to be used increasingly in the
design of onshore facilities.
1

However. a number of additional developments are needed. One of the
most important is simply the acquisition of sufficient data to improve the
quality of the recommended soil-response curves. It would be desirable if
enough data were available to allow a statistical approach to the use of
the soil-response curves.
1.1 SOIL-STRUCTURE INTERACTION
The term II so il-structure interaction ll
has been used frequently in
connection with the analysis of structures that sustain seismic loads;
however, the term has relevance to loads that are short-term, repeated at
relatively low frequencies, or sustained. Reflection will show that every
problem in foundation design, if fully solved, is one o·f soil-structure
interaction. For example, the strip footing in Fig. 1.1 can be
considered. Not only is it desirable to know at what load the footing
will plunge so that an appropriate factor of safety can be employed to
prevent a soil failure, it is necessary to know the way the bearing stress
is distributed at the base of the footing so that the footing can be prop-
erly reinforced.
wall
Fi9. 1.1. Strip footill0'
In order for a bearing stress to be mobilized the footing must move
down, however slightly. It is unlikely that a uniform bearing stress will
deve lop such as the one shown. Depend i ng on the support i ng so i 1, the
bearing stress at the edge of the footing may either be lower or higher
than the average. However, for purposes of discussion the assumption is
2

made that the stress is uniform as shown. The extension of the footing
beyond the wall behaves as a short, cantilever beam and the downward move-
ment of the edge of the footing is less than that at the center.
Conceptually, then, the bearing stress should be different at the edge and
at the center of the footing to reflect the difference in downward move-
ment.
The pattern of the distribution of the bearing stress should change
with the change in applied load because the stress-deformation character-
istics of soil are nonlinear. Thus, as loading on the footing increases
there will be a nonlinear increase in the bending moment in the footing at
the edge of the wall. There is, of course, a complex state of stress in
the soil beneath the footing and a complex pattern of deformations. The
soil response at the base of the footing is more complex if the loading
has a lateral component or an eccentricity.
The problem of the strip footing is frequently trivial because suffi-
cient reinforcement can be provided for a small expense to make the foot-
ing safe against any pattern of distribution of bearing stress. The same
argument cannot be made for a mat foundation, however, where the thickness
of the mat and the amount of reinforcing steel will vary widely according
to the distribution of bearing stress. The problem of the mat foundation
is a soil-structure-interaction problem that needs additional attention.
The pile foundation is an excellent case to use in discussing
soil-structure interaction. While the material in this volume is directed
toward the pile under lateral loading, it is of interest to consider the
general behavior of a pile under axial loading as well. Figure 1.2 shows
a model of an axially loaded deep foundation. The soil has been replaced
,
with a series of mechanisms and the pile has been replaced with a stiff
spring. A study of the model will indicate the following significant
pOints about a soil-structure -interaction problem: the pile is
deformable, a movement of the pile is necessary to mobilize soil
resistance, the soil response is a nonlinear function of pile movement,
there is a limiting soil response, and if the model can be described
numeri ca lly, computations can be made to obtain the response of the
system.
The above characteristics of a pile under axial load also pertain to
a pile under lateral loading, shown in Fig. 1.3. As may be seen in the
figure, the soil is again replaced by a set of mechanisms that indicate a
3

Q Q
1 t
--=-1j1 t
1 t
=u1 t
--=1j1 t
1 r
~
1 t
-4l1 t
1 r --::rj
Fig. 1.2. Model of a pile under
ax i all oad.
•x
Fig. 1.3. Model of a pile under
lateral load.
nonlinear response to the lateral deflection of the pile. The mechanisms
indicate the soil resistance p per unit length of the pile as a function
of the pile deflection y. The p-y curves will be discussed in detail in a
l~ter chapter. If such curves are available, the computations for pile
deflection and bending moment can be ~ade readily if pile dimensions and
pile-head loading are known.
A feature that is common in the two models shown in Figs. 1.2 and 1.3
is that the soil is characterized by a set of discrete, independent mech-
anisms. This sort of modelling is not strictly correct, of.course,
.because the soil is a continuum and a deformation at any point in the con-
tinuum will cause a deformation at all other points. The theoretical dif-
ficulty of modelling the soil as indicated in Fig. 1.3 causes little
practical difficulty, as will be discussed later.
4

The methods that are used to analyze the behavior of a single pile
under lateral load can also be extended in developing approximate methods
for the analysis of a group of closely-spaced piles. This problem in
soil-structure interaction is treated in a later chapter.
An important problem in the mechanics of pile behavior is the compu-
tation of the magnitude of the loads and moments that are distributed to a
group of widely-spaced piles, including batter piles, that support a pile
cap or structure. As demonstrated later, the solution to such a problem
can be made as exactly as the behavior of the individual piles under axial
and lateral load can be computed.
1.2 METHODS OF SOLUTION OF LATERALLY LOADED PILES
The principal method of solution presented herein requires the mod-
elling of the soil by p-y curves and the computation of the pile response
by di gita1 computer. The differential equation that governs the pil e
behavior, even with nonlinear soil response, can be conveniently solved by
use of difference equations.
In addition to the computer solution, the use of nondimensional
curves has an important role in the analysis of laterally loaded piles.
Nondimensional methods can be used to demonstrate with clarity the nature
of the computer method and, furthermore, can be used to obtain a check of
the computer results.
Two other methods of analysis are presented, the methods of Broms
(1964a, 1964b, 1965) and Poulos and Davis (1980). Broms· method is ingen-
ious and is based primarily on the use of limiting values of soil resist-
ance. The method of Poulos and Davis is based on the theory of
elasticity. Both of these methods have had considerable use in practice
and the designer of a particular foundation may wish to employ one or both
of them as a check or to give additional insight into a design problem.
1.3 DESIGN PROBLEMS
Some of the applications of piles under lateral loading are shown in
Fig. 1.4. There are other examples, including high-rise buildings, sol-
dier piles in a retaining structure, well-head supports,
slope-stabilizing elements, and river crossings for pipelines.
The principal kind of loading in most of the cases is repeated or
cyclic, and sustained loading is also present. The methods presented here-
5

Transmission Tower Pier 8 Breasting Dolphin Offshore Structure
Bridge Foundation
HIGHWAY 360
Overhead Sign
Retaining Wall; Bridge Abutment
C3 ; : s"'zc:s ,. ...c
Pipe Support 1Anchorage
Fig. 1.4. Examples of laterally-loaded piles.
6

in can be utilized to analyze directly those cases where there are
repeated loadings. With regard to sustained loading, the methods yield an
excellent preliminary analysis and the geotechnical engineer must use
some computations and a good deal of judgement to complete a solution.
Concern i ng design, addi tiona1 comments about the subsurface soi 1
investigation and the inspection of construction are appropriate. In the
sections giving procedures for computing p-y curves there are suggestions
for the determination of significant soil properties. The importance of a
soil investigation of high quality cannot be over-emphasized. In partic-
ular, because piles under lateral loading derive a significant amount of
their lateral support from soils near the ground surface, the soil inves-
tigation must be especially thorough for the near-surface soils.
The inspection of the construction is important and should be done by
someone familiar with the design process. A number of things could be
done by the contractor inadvertently that could have a detrimental effect
on the performance of a pile under lateral load.
The geotechnical engineer must give attention to factors such as
expansive clay, negative friction, downslope movement, and changes in
soil properties with time. These factors and many such others are not
discussed herein.
1.4 DESIGN ORGANIZATION
The writer has observed that there unfortunately is sometimes a lim-
ited use of the methods presented in this work because of the separation
of respons i bi 1iti es. The appropri ate use of the methods requi res the
skills of geotechnical engineers and structural engineers, along with the
support of computer technicians. It is inappropriate for the geotechnical
engineer to provide data on p-y curves and not be connected further with
the design. It is also inappropriate for the geotechnical engineer to try
to perform an analysis of a pile without careful consideration of how the
pile interacts with the superstructure, the work of the structural engi-
neer. It is inappropriate for the structural engineer to proceed with a
design if there are even minor changes that affect the soil response.
Therefore, in many offices there is the need for a management deci-
sion that geotechnical engineers and structural engineers will work
closely throughout the design and construction of a project involving
7

piles under lateral loading. Such close cooperation may already be pres-
ent in many offices; it is essential in the design of laterally loaded
pil es.
1.5 FACTOR OF SAFETY
The ordi nary procedures for estab1i shi ng an appropri ate factor of
safety will apply to the design of a single pile and the pile group under
lateral loading. Many factors are to be considered, of course, including
the quality of the information on loading, the quality of the soil data
and data on other materials, the adequacy of the design methods, and the
possible result of a failure. In regard to a failure, the designer must
consider whether loss of life might result, a large monetary loss, or a
minor monetary loss.
Two aspects of a soil-structure-i nteraction problem di fferent i ate
that problem from others in foundation engineering: the methodology does
not have much experimental validation, and the problem is nonlinear. Con-
cerning the nonlinear aspects, the designer must put the factor of safety
into the load rather than into the material properties. That is, the ser-
vice load must be increased by the factor of safety and computations made
wi th the factored load. The computation s wi th the servi ce load mi ght
indicate a moderate deflection and bending stress while a small increase
in load could result in a failure. Such a result is possible because
there could be a considerable loss in soil resistance with a small
increase in deflection.
Concerning the adequacy of design methods, a study of the later sec-
tion of this work will show that the methods are rational and validated to
a certain extent. However, experimental data are limited. Thus, the
designer should make computations not only with a range of loads but with
an upper bound and with a lower bound for the soil response. These
upper-bound and lower-bound values can probably best be estab1i shed by
taking the maximum values of soil properties that can be expected and the
minimum values. Also, the effects of varying the parameters that are used
in the soil-response criteria (p-y curves) can be studied. The computa-
tions will yield insight into. the probable response of the pile-soil sys-
tem.
There are in general two types of failure: a failure of the pile
material as reflected by an excessive bending moment, and a soil failure
8

as reflected by excessive deflection of a pile. However, there could be
some applications where the limits on pile-head deflection are small and
the allowable deflection is exceeded even if the soil is still substan-
tially in the elastic range.
There are applications of the pile under lateral loading where the
pile carries little or no axial load (such as a support for an overhead
sign or a breasting dolphin) and where the pile penetration is determined
by lateral loading. Figure 1.5 shows how to deal with such a case. When a
pile is short, the deflection of the pile at the groundline can be large
because the bottom of the pile will deflect. As the pile penetration is
increased, soil resistance at the bottom of the pile will increase and the
groundline deflection will reach a limiting value where increased pene-
tration will cause no decrease in groundline deflection. Thus, the
designer will make computations for a series of pile penetrations and will
determine a penetration that will yield an appropriate factor of safety.
Groundline
Deflection
Critical Penetration
Pile Penetration
Fig. 1.5. Results of cnmputations where pile penetration
is controlled by lateral loading.
1.6 REFERENCES
American Petroleum Institute, Planning, Designing and Constructing
Fixed Offshore Platforms, API RP2A, 1979.
Baguelin, F•• Jezequel, J. F., and Shields, D. H., The Pressuremeter
and Foundation Engineering, Trans Tech Publications, 1978.
9

Broms, Bengt B., "Lateral Resistance of Piles in Cohesive Soils,"
Proceedings, American Society of Civil Engineers, Vol. 90, No. SM2, March
1964, pp. 27-63.
Broms, Bengt B., "Lateral Resistance of Piles in Cohesionless
Soils," Proceedings, American Society of Civil Engineers, Vol. 90, No.
SM3, May 1964, pp. 123-156.
Broms, Bengt B., "Design of Laterally Loaded Piles," Proceedings,
Ameri can Soci ety of Ci vi 1 Engi neers, Vo 1. 91, No. SM3, May 1965, pp.
79-99.
Det Norske Veritas, Rules for the Design, Construction, and
Inspection of Offshore Structures, Det Norske Veritas, 1977.
George, P., and Wood, D., Offshore Soil Mechanic~, Cambridge Univer-
sity Engineering Department, 1976.
Jamil kowski, M., "Des i gn of Laterally Loaded Pil es, II General Lec-
ture, International Conference on Soil Mechanics and Foundation Engineer-
ing, Tokyo, Japan, 1977.
Poulos, H. G., and Davis, E. H., Pile Foundation Analysis and Design,
Wiley, New York, 1980.
10

CHAPTER 2. THE DIFFERENTIAL EQUATION
The problem of the laterally loaded pile is similar to the
beam-on-foundation problem. The interaction between the soil and the
structure (pile or beam) must be treated quantitatively in the problem
solution. The two conditions that must be satisfied for a rational analy-
sis of the problem are: (1) each element of the structure must be in equi-
librium and (2) compatibility must be maintained between the
superstructure, foundation, and supporting soil. If the assumption is
made that compatibility between the pile and the superstructure can be
maintained by selecting appropriate boundary conditions at the top of the
pile, the remaining problem is to obtain a solution that insures equilib-
rium and compatibility of each element of the pile, taking into account
the soil response along the pile. Such a solution can be made by solving
the differential equation that describes the pile behavior.
A derivation of the differential equation for a beam or a pile under
lateral loading is presented so that the assumptions that are made can be
understood.
2.1 RELATION BETWEEN CURVATURE AND BENDING MOMENT
A segment of an initially straight beam deformed by a bending moment
is shown in Fig. 2.1 (Popov, 1952). The initially straight neutral axis,
A-B, becomes curved in a bent beam.
A fundamental assumption made in establishing the flexure formula is
that plane sections initially perpendicular to the axis of the beam remain
plane in the bent beam. The lines m-m and p-p represent two such planes.
The extensions of these lines intersect at a point 0 which is the center
for the radius of curvature p for the infinitesimal arc n-n l
.
The line s-s2' at distance ~ away from n-n l
, has been stretched due
to bending. The extension of the line, s2-s1' is determined by construct-
ing line nl
-s
l
parallel to n-s. Triangles non l
and 5
1
n
1
5
2
are 5imilar.
Therefore:
_p_ = n
n-nl 52-51
n 52-51
11
(2.1)
(2.2)

o 0
~ ~
I
I

", -----.---r-
Fig. 2.1. A straight beam under bending moment
(after Popov, 1952).
The strain E is defined as the change in length of an element divided
by the initial length of that element. Using this definition, the strain
at any point x along the beam is equal to
Substituting Eq. 2.2 into Eq. 2.3
Using Hooke's Law, the strain of the element s-s2 is:
where
€
X E
Ox =bending stress in element s-s2
E = modulus of elasticity.
Combining Eqs. 2.4 and 2.5
or 0"
X
=.i!L .p
The flexure formula for bending is:
0"
X
= J:!!L .
I
12
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)

Combining Eqs. 2.6 and 2.7
Mn _ En
and
-1- - -p-
1 = 11...
p EI
2.2 THE DIFFERENTIAL EQUATION OF THE ELASTIC CURVE
(2.8)
(2.9)
The next step in the derivation is to obtain an expression for p in
terms of x and y. The curvature of a line in analytic geometry is defined
by Eq. 2.10.
1
p
= (2.10)
In the usual cases of the bending of a beam or pile, the slope dy/dx is
very small. Therefore, the square of the slope is a negligible quantity.
Thus,
(2.11)
Finally, combining Eqs. 2.9 and 2.11
~=~ •
EI dX2
(2.12)
Eq. 2.12 is the desired differential equation.
In applying Eq. 2.12, the pile is assumed to be vertical with the
x-axis lying along the axis of the unloaded pile. The deflection of a
point on the elastic curve of the pile is given by y, Fig. 2.2. Deflection
to the right is positive. Slopes of the elastic curve at points 1 and 2
are negative while slopes at 3 and 4 are positive. However, as indicated
in the figure, the moment is positive in both instances.
Other relationships which are needed, along with those already
defined, are:
y =deflection of the elastic curve
~ = S = slope of the elastic curve
d2 v M
~ = -- where M= moment
dx 2 E1 '
13
(2.13)
(2.14)

d3
y = ..i-. , where V = shear (2.15)
dx 3 EI
~=~ , where p
dx 4 EI
= soil reaction. (2.16)
~----~------------------~y
51 < 52 . +M
2 -52
Fig. 2.2. Segment of a deflected pile.
2.3 DERIVATION OF THE DIFFERENTIAL EQUATION FOR THE BEAM-
COLUMN
In most instances the axial load on a laterally loaded pile is of
such magnitude that it has a small influence on bending moment. However,
there are occasions when it is necessary to include a term for the effect
of axial loading in the analytical process. The derivation for the dif-
ferential equation has been made by Hetenyi (1946) and is shown in the
following paragraphs.
Methods are presented later for the solution of the differential
equation. When the solution involves consideration of the axial load, it
will be necessary to employ a computer program o The program is described
14

later and offers the user an opportunity for doing stability analyses as
well as the usual computations of pile behavior. It should be noted at
this point that the interaction between behavior under lateral loading and
under axial loading can normally be uncoupled because the axial load is
usually almost constant over the portion of the pile where bending moment
is significant.
It will now be assumed that a bar on an elastic foundation is sub-
jected not only to the vertical loading, but also to the pair of horizon-
tal compressive forces Px acting in the center of gravity of the end cross-
sections of the bar.
If an infinitely small unloaded element, bounded by two verticals a
distance dx apart, is cut out of this bar (see Fig. 2.3), the equilibrium
of moments (ignoring second-order terms) leads to the equation
(2.17)
or
~+p ~-V =0.
ux x dx v
(2.18)
T~---Y
x
+
y - - - - -..
dx
1
y+dy------.P.
p =-E. y
x
Fig. 2.3. Element from beam-column.
15

Differentiating Eq. 2.18 with respect to x, the following equation is
obtained
d2M + p ~ _ dVv = 0
dx 2 X dx 2 dx •
The following identities are noted:
dV
v- = P
dx
p = -EsY •
And making the indicated substitutions, Eq. 2.19 becomes
EI ~ + p ~ + E Y = 0 .
dx4 X dx 2 5
(2.19)
(2.20)
The direction of the shearing force V is shown in Fig. 2.3. The
v
shearing force in the plane normal to the deflection line can be obtained
as
V =V cos S - P sin S.
n v x
(2.21)
Because S is usually small, cos S = 1 and sin S = .91tan S = dx' Thus, Eq.
2.22 is obtained.
v = V - P ~n v x dx
(2.22)
Vn will mostly be used in computations but Vv can be computed from Eq.
2.22 where dy/dx is equal to the rotation S.
2.4 SUMMARY
The assumptions that must be made in deri vi ng the di fferent i a1
equations are shown below:
(1) The pile is straight and has a uniform cross section.
(2) The pile has a longitudinal plane of symmetry; loads and
reactions lie in that plane.
(3) The pile material is homogeneous.
(4) The proportional limit of the pile material is not
exceeded.
(5) The modulus of elasticity of the pile material is the same
for tension and compression.
16

(6) Transverse deflections of the pile are small.
(7) The pile is not subjected to dynamic loading.
(8) Deflections due to shearing stresses are negligible.
The assumption of a uniform cross section can be eliminated by
rewriting the differential equation or by solving sets of simultaneous
differential equations. Also, most of the other assumptions can be elimi-
nated if one wishes to modify the differential equation.
The sign conventions that are employed are shown in Fig. 2.4. For
ease of understanding, the sign conventions are presented for a beam that
is oriented like a pile. A solution of the differential equation yields a
set of curves such as shown in Fig. 2.5.
Techniques for the solution of the differential equation will be dis-
cussed in a later chaoter.
2.5 EXAMPLE EXERCISE
As an example of the sol ution of the second-order differential
equation, Eq. 2.19, the problem of a simply-supported beam with uniform
loading will be considered. The desired solution is an expression for y.
Cutting a free body from the beam shown in Fig. 2.6 at some point x
along the beam and solving for the moment in the beam where it is cut, the
following expression results:
Mx = (E{-) x - (T) x.
Substituting expression for moment from Eq. 2.23 into Eq. 2.12
d
2
y =l (~_~)
~ EI 2 2 '
Integrating Eq. 2.24
~ _1- '(PLX2
E)dx - EI 4 - 6 + C1•
From symmetry, the slope is zero at the mid-point of the beam,
~ = 0 at x = 2L
dx
and this enables the constant of integration C1
to be evaluated.
expressions in Eq. 2.26 to solve Eq. 2.25
1 ( nL L2
.p. L3 )0=- .J:..=.. _ _ . - +C
EI 4 4 6 8 1•
17
(2.23)
(2.24)
(2.25)
(2.26)
Using
(2.27)

Pb
T
p y
load
DiaQram
1
Positive
load
(a)
y
Shear Moment Slope
DiaQram DiaQram DiaQram
--
~
y
-~ -~0- +
--- ..:J x
Positi ve Positive Positive
Shear Moment Slope
(b) ( c) (d)
Fig. 2.4. Sign conventions.
S=~dJt M=EI~ V=EI~
Deflection
DiaQram
y
i~
x
Positive
Deflection
(e)
Fig. 2.5. Form of the results obtained from a complete solution.
18

...
1
y
pL ~
p(Negative)-2
~.
L
~
"'
~
..
pL
2"
,
x
Fig. 2.6. Section of a beam (pile) with uniform load.
or
-~ .c1 - 24EI
(2.28)
Substituting expression for C1 into Eq. 2.25
.91. =1- (pLx2
_ ~) _ ~ "
dx EI 4 6 24EI
(2.29)
Integrating Eq. 2.29
1 (nL _x3 ~ ) nL 3
y = IT "4' 3 - 24 - t4rr . x + C2" (2.30)
To solve for C2 the condition is employed that y = 0 at x = O. Using
these boundary conditions, Eq. 2.30 becomes
1
o = IT (0 - 0) - 0 + C2; C
2 = O. (2.31)
Finally, Eq. 2.30 becomes
(2.32)
Eq. 2.32 is the desired solution.
19

2.6 REFERENCES
Hetenyi, M. ,Beams on Elastic Foundation, The University of Michigan
Press, Ann Arbor, 1946.
Popov, E. P., Mechanics of Materials, Prentice-Hall, New York, 1952,
pp. 269-276.
2.7 EXERCISES
2.1 Use the method in Section 2.5 and solve each of the following
problems of a cantilever beam: (a) a force at the free end, (b) a moment
at the free end, (c) a rotation at the free end, and (d) a deflection at
the free end.
2.2 Repeat problem 2.1 with the EI of the half of the beam at the
fixed end being twice as great as the EI of the half of the beam at the
free end.
20

CHAPTER 3. SOIL RESPONSE
As noted earlier, the soil response is characterized as a set of dis-
crete mechanisms as suggested by Winkler (1867). The discrete mechanisms
i ndi cate that the soil response at a poi nt is not dependent on pil e
deflection elsewhere; thus, a continuum is not perfectly modelled. Howev-
er, the continuum could be modelled properly if information were available
concerning the interaction effects from one soil slice to the next. Each
discrete mechanism could then represent a family of curves, with the
appropriate curve in the family at a point selected to reflect the effects
from the soil above and below that point. The present state-of-the-art
concerning the response of the soil to a deflected pile does not allow or
justify an approach more sophisticated than that suggested by the set of
discrete mechanisms. Futhermore, a small amount of unpublished exper-
imental data suggests that the soil response at a point is unaffected by
those changes in defl ected shape that can be achi eved by a1teri ng the
rotational restraint at the pile head by any practical amount.
Proceeding with the concept that the soil response can be treated by
employing a set of mechanisms, a discussion of the physical meaning of one
of these mechanisms is helpful. Figure 3.la is a view of a pile after it
has been installed and before any lateral load has been applied. The
behavior of the soil at the depth xl is to be considered. The stress dis-
tribution against the pile, before any lateral loading, is shown in Fig.
3.lb. The assumption implied by the figure is that the pile has been dri-
ven without any residual deflection and bending moment; thus, there is no
lateral force against the pile at the depth xl or elsewhere. It is
assumed that a lateral load is now applied to the pile and that it is
caused to deflect an amount Yl at the depth xl. The stress distribution
is altered, of course, and that shown in Fig. 3. lc could represent the new
distribution.
The integration of the stress distribution shown in Fig. 3.lc would
yield the force per unit length along the pile, Pl. The quantity p is
defined as the soil reaction or soil resistance. It acts in opposition to
the deflection y; hence, p and yare opposite in sign. If one were able to
predict the stress distribution for a range of deflections, the successive
integrations would yield p-values corresponding to y-values, allowing the
21

A
x
Fi g. 3.1.
Ground Surface
y
A
I (0)
Graphical definition of p and y
(a) view of elevation of section of pile
(b) view A-A - earth pressure distribution
prior to lateral loading
(c) view A-A - earth pressure distribution
after lateral loadin~
22
ee)

p-y curve to be developed for the depth xl' Similar exercises at other
depths would yield a family of p-y curves.
With regard to p-y curves, the assumptions are made (1) that there is
no shear stress at the surface of the pile parallel to its axis (the
direction of the soil resistance is perpendicular to the axis of the pile)
and (2) that any lateral resistance or moment at the base of the pile can
be accounted for by a p-y curve at the side of the pile near the base. Any
errors due to these assumptions are thought to be negligible.
3.1 REACTION OF SOIL TO LATERAL DEFLECTION OF PILE
As might be expected, from the definition of a p-y curve given in the
previous section, the soil resistance p is a nonlinear function of the
deflection y. A family of p-y curves, plotted in the appropriate quad-
rants, is shown in Fig. 3.2a. That the curves are plotted in the second
and fourth quadrants is merely an indication that the soil resistance p is
opposite in sign to the deflection y. While the p-y curves in Fig. 3.2a
are only illustrative, they are typical of many such families of curves in
that the initial stiffness and the maximum resistance increase with depth.
A typical p-y curve is shown in Fig. 3.2b; it is plotted in the first
quadrant for convenience. The curve is strongly nonlinear, changing from
an initial stiffness E . to an ultimate resistance p . As is evident, theS1 u
soil modulus E is not a constant except for a small range of deflections.
s
The three factors that have the most influence on a p-y curve are the
soil properties, the pile geometry, and the nature of loading. The corre-
lations that have been developed for predicting soil response are based on
the best estimate of the properties of the in situ soil with no adjustment
for the effects of the method of installation on soil properties. The
logic supporting this approach is that the effects of pile installation on
soil properties are principally confined to a zone of soil close to the
pile wall, while a mass of soil of several diameters from the pile is
stressed as lateral deflection occurs. There are instances, of course,
where the method of pile installation must be considered; for example, if
a pile is jetted into place, a considerable volume of soil could be
removed with a significant effect on the soil response.
The principal dimension of a pile affecting the soil response is its
diameter. All of the recommendations for developing p-y curves include a
23

Y
Y
Y
---y
X-X2
x-x!
X-X
4
( a )
'2
I
~
JE~......
a.
oj
U
C ;'
E ;'
III
~ Es'in
Q)
0: ;'
,/
'0
,/
(f)
,/
Deflection. y(jn.)
( b)
Fig. 3.2. Typical p-y curves
(a) family of curves
(b) characteristic shape
of p-y curve.
--- p
ult
term for the diameter of the pile; if the cross-section of the pile is not
circular, the width of the pile perpendicular to the direction of loading
is usually taken as the diameter.
The p-y curves are strongly responsive to the nature of the loading.
Recommendations have been developed for predicting curves for short-term
static loading and for cyclic (or repeated) loading. However, there are
no current recommendations for the cases where the loading is dynamic or
sustained. Recommendations for P7Y curves where the inertia of the soil
is considered are needed because of the desirability of developing
rati ona1 methods of analyzi ng pil e-supported structures that are sub-
jected to earthquake loadings. With regard to sustained loadings, as from
a retaining wall, it is unlikely that criteria can be developed for pre-
dicting p-y curves. The problem must be solved as a whole, taking into
account the three-dimensional consolidation that will occur as well as the
time-dependent changes in loading. From the standpoint of practice, the
engineer can probably estimate some additional amount of deflection that
will occur if he has the earth pressure distribution along a pile at the
beginning of the loading period.
24

3.2 USE OF THEORY OF ELASTICITY TO DETERMINE SOIL BEHAVIOR
The theory of elasticity is of limited use in solving the problem of
the response of soil to pile deflection because experience has shown that
the soil-response curve is linear only for small deflections of the pile.
In spite of this limitation, some useful contributions, based essentially
on the theory of e1asti ci ty, have been made. The papers of Skempton
(1951), Terzaghi (1955), and McClelland and Focht (1958) have been
selected for review.
Skempton
The author states that "simple theoretical considerations" were
employed to develop a prediction for load-settlement curves for footings.
The ,theory can be employed to obtain the p-y curve for a pile if it is
assumed that the depth is such that the curve is not affected by the free
surface of the soil, that the state of stress is the same in the horizon-
tal and vertical directions, and that the soil is isotropic.
The mean settlement of a foundation, p, of width b, on the surface of
a semi-infinite solid, based on the theory of elasticity, is given by the
following expression.
where
p = qbI
p
q =foundation pressure,
I =influence value,
P
v =Poisson's ratio of the solid, and
E =Young's moqulus of the solid.
(3.1)
With regard to Eq. 3.1, Poisson's ratio can be assumed to be equal to
1/2 for saturated clays if there is no change in water content and I can
P
be taken as TI/4 for a rigid circular footing on the surface. Furthermore,
for a rigid circular footing, the failure stress qf may be taken as equal
to 6.8c, where c is the undrained shear strength. Making the substi-
tutions indicated, and setting p equal to PI for the particular case
Skempton
01 4 q
b = E/c = qf
noted that the
(3.2)
influence value I decreases with depth below the
P
surface but the bearing capacity factor increases; therefore, as a first
approximation Eq. 3.2 is valid for any depth.
25

In an undrained compression test the axial strain is given by the
following equation.
(°1 - (3)
E: =
of.'::.
= -
E
(3.3)
where
E =Young1s modulus at the stress (01 - (3).
For saturated clays with no water content change, 'Eq. 3.3 may be
rewritten as follows.
where
2
E: = - -
E/c
(°1 - (3)
(01 - (3)f
(01 - (3)f =failure stress.
(3.4)
It may be noted by comparing Eqs. 3.2 and 3.4 that, for the same
ratio of applied stress to ultimate stress, the strain in the footing test
(or pile under lateral loading) is related to the strain in the laboratory
compression test by the following equation.
PI
- = 2E:
b
(3.5)
Skempton1s arguments based on the theory of elasticity and also on the
actual behavior of full-scale foundations led to the following conclu-
sion:
Thus, to a degree of approximation (20 percent) com-
parable with the accuracy of the assumptions, it may
be taken that Eq. 3.5 applies to a circular or any
rectangular footing.
While the analytical approach employed by Skempton involves numerous
approximations, the method has gained some acceptance because of the
experimental evidence presented by Skempton and others (Reese, et al.,
1975).
Skempton stated that the failure stress for a footing reaches a maxi-
mum value of 9c. If one takes that value of a pile in saturated clay under
lateral loading, Pu becomes 9cb. A p-y curve could be obtained, then, by
taking points from a laboratory stress-strain curve and using Eq. 3.5 to
obtain deflection and 4.5 o~b to obtain soil resistance. The procedure
would presumably be val id at depths beyond where the presence of the
ground surface woul d not reduce the soil resistance. Skempton di d not
suggest that his ideas could be used in obtaining p-y curves and no sug-
26

gestions in that regard are made here. However, the concepts employed by
Skempton are useful and of interest.
Terzaghi
While Skempton presented an analytical approach for the prediction
of curves giving the deflection of a footing as a function of bearing
stress, Terzaghi's approach is much less direct. He does call on princi-
ples of mechanics in his presentation; however, he fails to give a
step-by-step procedure for his derivations and he also fails to cite any
experi menta1 evi dence that woul d verify hi s recommendations. One woul d
suppose that Terzaghi has used the theory of e1asti ci ty freely in hi s
development of the recommended coefficients and that he had some exper-
imen~al evidence at his disposal but that he chose not to cite his refer-
ences for his own reasons. While the paper has some obvious shortcomings,
the numerical recommendations for soil response are summarized here
because of thei r hi stori ca1 importance and because the recommendations
continue to have value.
A reference to an incident in Terzaghi's later years is pertinent.
He was one of the principal speakers at the Eighth Texas Conference on
Soil Mechanics and Foundation Engineering in the early Fall of 1956 where
he read his notable paper, "Submarine Slope Failures." Professor Hudson
Matlock and his colleagues were engaged in running the first field tests
of a fully-instrumented, laterally loaded pile at Lake Austin. Terzaghi
visited the site, expressed an interest in the testing program, but had no
particular suggestions to make. His paper on subgrade reaction had just
been published and Terzaghi remarked that he was not particularly proud of
the paper and that he had only agreed to publish it at the urging of a num-
ber of his acquaintances.
Terzaghi's recommendations for the coefficient of subgrade reaction
(p-y curves) for stiff clay were based on his notion that the deforma-
tional characteristics of stiff clay are "more or less independent of
depth." Thus, he proposed, in effect, that the p-y curves should be con-
stant with depth. He further proposed that the ratio between p and y
should be constant and defined by the symbol aT. Therefore, his family of
p-y curves for stiff clay consists of a series of straight lines, all of
the same slope, passing through the origin of the coordinate system.
Terzaghi recognized, of course, that the pile could not be deflected
to an unlimited extent with a linear increase in soil reaction. He stated
27

that the linear relationship between p and y was valid for values of p
that were smaller th~n about one-half of the ultimate bearing stress.
Table 3.1 presents Terzaghi1s recommendations for stiff clay. The
units have been changed to reflect current practices. The values of ~T'
it should be noted, are independent of pile diameter.
TABLE 3.1. TERZAGHI'S RECOr¥l~lENDATIONS FOR SOIL ~10DULUS aT
FOR LATERALLY LOADED PILES IN STIFF CLAY.
Consistency of Clay Stiff Very Stiff Hard
qu' T/sq ft 1-2 2-4 > 4
~T' lb/sq in. 460-925 925-1850 1850-up
With regard to sand, Terzaghi based his recommendations on the fact
that the stiffness of the sand increases with confining stress (or with
depth). However, he recommended, as with stiff clay, that the soil
resistance should be a linear function of y. Again, as with clay he stip-
ulated that his recommended values were valid only for a soil reaction
that was no more than one-half of the maximum bearing stress.
Thus, the family of p-y curves recommended by Terzaghi for sand con-
si~ts of a series of straight lines, with the slope of the lines being
zeto at the ground surface and increasing linearly with depth. Because
Es ' the soil modulus is equal to ply
where
E =kxs
k =constant giving variation of soil modulus
wi th depth, and
x =depth below ground surface.
Table 3.2 shows recommendations for k.
28
(3.6)

TABLE 3.2. TERZAGHI'S RECOMMENDATIONS FOR VALUES OF k FOR
LATERALLY LOADED PILES IN SAND.
Relative Density of Sand
Dry or moist, k, lb/cu in.
Submerged sand, k, lb/cu in.
McClelland and Focht
Loose
3.5-10.4
2.1-6.4
Medium
13-40
8-27
Dense
51-102
32-64
The paper by these authors has considerable importance for several
reasons: it is the first paper to report experimental p-y curves from a
full-scale, instrumented, pile-load test, and it shows conclusively that
the soil modulus is not a soil property but is a functjon of depth and pile
deflection. While the paper is not strongly based on the theory of elas-
ticity, it is included in this section because it closely parallels
Skempton I s approach and because stress-strain curves from 1aboratory
tests are employed in obtaining p-y curves.
The paper recommends the performance of consolidated-undrained tri-
axial tests with the confining pressure equal to the overburden pressure.
To obtain values of the soil resistance p from the stress-strain curves,
the authors recommend the following equation:
·where
p =5.5 ball
all =deviator stress (0
1 - 0
3)
b = pil e di ameter •
(3.7)
Equation 3.7 agrees well with Skempton1s recommendations for the case
where the depth divided by the pile diameter is about three or more. As
noted earlier, Skempton proposed a factor of 4.5 instead of the 5.5 shown
in Eq. 3.7.
To obtain values of pil e defl ection y from stress-stra in curves,
McClelland and Focht propose
y =0.5 bE •
(3.8)
Skempton1s corresponding equation suggests a value of 2 rather than 0.5.
Part of the difference in these two numbers probably derives from the dif-
29

ference in the two sets of experiments. Skempton's studies were for foot-
ings that were at or near the ground surface; these footings could be
expected to have more relative deflection than a segment of pile that is
for the most part at some distance below the ground surface.
3.3 USE OF SOIL MODELS TO DETERMINE SOIL BEHAVIOR
The preceding section has presented solutions employing the theory
of elasticity, or related approaches, to obtain the response of the soil
to the lateral deflection of a pile. This section reviews the use of soil
models to obtain expressions that will indicate approximately the ulti-
mate resistance against a pile that can be developed near the ground sur-
face and at some depth below the ground surface.
Soil Models for Saturated Clay
The assumed model for estimating the ultimate soil resistance near
the ground surface is shown in Fig. 3.3 (Reese, 1958). The force Fp is
F =c bH [tan a + (I+K) cot a ]
p a c c
+ 1/2~bH2 + C H2 sec a (3.9)
a c
where
c = average undrained shear strength
a
K = a reduction factor to be multipl ied by c to yield thea
average sliding stress between the pile and the stiff
cl ay, and
~ = average unit wei ght of soil.
(the other terms are defined in the figure)
It is possible to take the partial derivative of Eq. 3.9 with respect to
the angl e a and set it equal to zero to fi nd the angl e at whi ch the
equation is minimized. However, as an approximation the angle a is taken. c
as 4~ and K is assumed equal to zero. Differentiation of the resulting
expression with respect to H yields an expression for the ultimate soil
resistance near the ground surface as follows:
(p) =2c b + ~bH + 2.83 c H •u ca a a (3.10)
It can be reasoned that, at some distance below the ground surface,
the soil must flow around the deflected pile. The model for such movement
is shown in Fig. 3.4a. If it is assumed that blocks 1, 2, 4, and 5 fail by
shear and that block 3 develops resistance by sliding, the stress condi-
tions are represented by Fig. 3.4b. By examining a free body of a section
of the pile, Fig. 3.4c, one can conclude that the ultimate soil resistance
30

(b)

( a)
Ffl x
1 H
WI
I Fp
f

fs
.

. A
yb
Ft "-.. :W F
f
~tln ~-Fpetc
~
Fig. 3.3. Assumed passive wedge-type failure for clay
(a) shape of wedge
(b) forces acting on wedge.
at depth is:
(p ) b = 11 cb • (3.11 )u c
Equations 3.10 and 3.11 are, of course, approximate but they do indi-
cate the general form- of the expressions that give the ultimate soil
resistance along the pile. The equations can be solved simultaneously to
find the depth at which the failure would change from the wedge type to
the flow-around type.
Soil Models for Sand
The soil model for computing the ultimate resistance near the ground
surface for sand is shown in Fig. 3.5a (Reese, Cox, and Koop, 1974). The
total lateral force Fpt (Fig. 3.5c) may be computed by subtracting the
active force Fa' computed using Rankine theory, from the passive force Fp'
31

-C
ar-- 4 3
----r5 - -
I 'b - -
L '------:--...:........1
I
I
I
I
--t
(7"3 I
I
I
2
---+---
--OJ
---122--
t
PILE MOVEMENT
-L~__~~~____~~~~_____
c
T
(0 )
( b)
(c)
Fig. 3.4. Assumed lateral flow-around type of failure for clay
(a) section through pile (b) Mohr-Coulomb diagram
(c) forces acting on section of pile.
computed from the model. The force Fpis computed assumi ng that the
Mohr-Coulomb failure condition is satisfied on planes ADE, BCF, and AEFB.
The directions of the forces are shown in Fig. 3.5b. No frictional force
is assumed to be acting on the face of the pile. The equation for Fpt is
as fo 11 ows.
(3.12)
32

where
Fi g. 3.5.
A
(0)
-F.P
( b)
B
DIRECTION OF
PILE MOVEMENT
PILE OF
DIAMETER b
-_... ~t--F.a
(c)
Assumed passive wedge-type failure of pile in sand
(a) general shape of wedge (b) forces on wedge
(c) forces on pile.
K =coefficient of earth pressure at resto
K =minimum coefficient of active earth pressure.a
The ultimate soil resistance near the ground surface per unit length
of the pile is obtained by differentiating Eq. 3.12.
[
K Htan ¢ sinS tanS
( ) =. H 0 + (b + H tans tanu )
Pu sa ( tan (s- ¢)cosus tan (s- ¢l s
(3.13)
33

Bowman (1958) suggested values of a from ¢/3 to ¢/2 for loose sand up to ¢s
for dense sand. The value of ~ is approximated as follows.
~ = 45 + ~ (3.14 )_
2
The model for computing the ultimate soil resistance at some distance
below the ground surface is shown in Fig. 3.6a. The stress 01' at the back
of the pile must be equal to or larger than the minimum active earth pres-
sure; if not, the soil could fail by slumping. This assumption is based
on two-dimensional behavior, of course, and is subject to some uncertain-
ty. However, the assumption should be adequate for the present purposes.
Assuming the states of stress shown in Fig. 3.6b, the ultimate soil
resistance for horizontal flow around the pile is
(p ) b =K b~H (tan8~ - 1) + K b~H tan¢ tan4~. . (3.15)usa a
As in the case for clay, Eqs. 3.14 and 3.15 are approximate but they
serve a useful purpose in indicating the form, if not the magnitude, of
the ultimate soil resistance. The two equations can be solved simultane-
ously to find the approximate depth at which the soil failure changes from
the wedge type to the flow-around type.
3.4 EXPERIMENTAL METHODS FOR OBTAINING SOIL RESPONSE
CURVES
The above paragraphs descri be methods for obta i ni ng soi 1 response
based primarily on theory. (An exception is the method of McClelland and
Focht that was based on some experimental results:) The strategy that has
been employed for obtaining design criteria is to make use of the theore-
tical methods, to obtain p-y curves from full-scale field experiments, and
to derive such empirical factors as necessary so that there is close
agreement between results from adjusted theoretical solutions and those
from experiments. Thus, an important procedure is obtaining experimental
p-y curves. 1
Soil Response from Direct Measurement
A number of attempts have been made to make direct measurement of p
and y in the field. Measuring the deflection involves the conceptually
simple process of sighting down a hollow pipe from a fixed position at
scales that have been placed at intervals along the length of the pile.
The method is cumbersome in practice and has not been very successful.
The measurement of the soil resistance directly involves the design
of an instrument that will integrate the soil stresses at a point along
34

" t /, /
0;6-- ¥ -/2), CT6
/ ,
/ J "
t~
5 PILE
-
" . /
, /
, /
-- Ii' ---OJOJ )V,
/ "-
/ . '
MOVEMENT
(a )
(b)
CT
Fig. 3.6. Assumed mode of soil failure by lateral flow around a pile
in sand
(a) section through the pile
(b) Mohr-Coulomb diagram representing states of stress
of soil flowing around a pile.I
the pile. The design of such an instrument has been proposed but none has
yet been built. Some attempts have been made to measure the soil pressure
at a few points around the exterior of a pile with the view that the soil
pressures at other points can be estimated. This method has met with lit-
tle success.
Soil Response from Experimental Moment Curves
Almost all of the successful experiments that yielded p-y curves have
involved the measurement of bending moment by the use of electrical
resistance strain gauges. The deflection can be obtained with consider-
able accuracy by two integrations of the moment curves. The deflection
35

and the slope at the groundline have to be measured accurately and it is
helpful if the pile is long enough so that there are at least two points
of zero deflection along the lower portion of the pile.
The computation of soil resistance along the 1ength of the pi 1e
involves two differentiations of a bending moment curve. Matlock (1970)
made extremely accurate measurements of bending moment and was able to do
the differentiations numerically. However, most other investigators have
fitted analytic curves through the points of experimental bending moment
and have performed the differentiations mathematically.
With families of curves showing the distribution of deflection and
soil resistance, p-y curves can be plotted. A check can be made of the
accuracy of the analyses by using the experimental p-y.curves to compute
bending-moment curves. The computed bending moments should agree closely
with those from experiment.
Nondimensional Methods for Obtaining Soil Response
Reese and Cox (1968) described a method for obtaining p-y cuves for
those instances where on..ly pi 1e-head measurements were made duri ng 1at-
eral loading. They noted that nondimensiona1 curves can be obtained for
many variations of soil modulus with depth. Equations for the soil modu-
lus involving two parameters were employed, such as shown in Eqs. 3.16 and
3.17.
Es =k1 ~ k2x
E = k xs 1
(3.16)
(3.17)
From measurement of pile-head deflection and rotation at the ground1ine,
the two parameters were computed for a given applied load and moment.
With an expression for soil modulus for a particular load, the soil
resistance and deflection along the pile were computed.
The procedure was repeated for each of the applied loadings and p-y
curves were plotted from the compute~ families of curves of deflection and
soil resistance. While the method is approximate, the p-y curves computed
in this fashion do reflect the measured behavior of the pile head. Soil
response derived from a sizeable number of such experiments can add sig-
nificantly to the existing information.
As previously indicated, the major field experiments that have led to
the development of the current criteria for p-y curves have involved the
acquisition of experimental moment curves. However, nondimensional meth-
ods of analyses have assisted in the development of p-y curves in some
instances.
36

3.5 RECOMMENDATIONS FOR p-y CURVES FOR CLAYS
Three major experimental programs were performed for piles in clays
to yield the criteria which follow. In each case the piles were subjected
to short-term static loads and to repeated (cyclic) loads. The exper-
imental program is described briefly in the paragraphs that follow, a
step-by-step procedure is given for computing the p-y curves, recommen-
dations are given for obtaining the necessary data on soil properties, and
example curves are presented.
The final portion of this section on clays presents a method that has
been developed for predicting p-y curves in clays of any shear strength.
This "unified" method is based on all of the major experiments in clay
below the water table.
As noted in the following sections, repeated loading of the clay has
a pronounced effect on the soil response, particularly when water covers
the ground surface. The loss of resistance from repeated loading is due
to two effects: the breakdown of the structure of the clay (remolding)
and scour. The remolding is a result of the repeated strains that occur
due to the deflection of the pile. The scour occurs when the pile
deflects enough to cause a gap to remain between pile and soil when the
load is removed. Water will flow into the gap and will be ejected on the
next application of load. The water in most cases will move out at a high
velocity and carry out particles of clay.
If the clay is above the water table, only the first of the two
effects will be present. Therefore, the recommendations for p-y curves
that are presented are dependent on the position of the water table.
Response of Soft Clay below the Water Table
Field Experiments. Matlock (1970) performed lateral load tests
employing a steel pipe pile that was 12.7S in. in diameter and 42 ft long.
It was driven into clays ne,ar Lake Austin that had a shear strength of
about 800 lb/sq ft. The pile was recovered, taken to Sabine Pass, Texas,
and driven into clay with a shear strength that averaged about 300 lb/sQ
ft in the significant upper zone.
Recommendations for Computing P-y Curves. The following procedure
is for short-term static loading and is illustrated by Fig. 3.7a.
1. Obtain the best possible estimate of the variation with
. depth of undrained shear strength c and submerged uni t
weight rl. Also obtain the values of ESO' the strain corre-
37

( a)
( b)
1.0
OL.~--------~------------o I 8
L
Yeo
1.00 /.------
./ For x~x,.(depth where flow
,/ . around failure gowrns)
L 0.50 -
Pu
x
0.72 x,
O~--~--------------~~---01 3 IS
Fig. 3.7. Characteristic shapes of the p-y curves
for soft clay below the water table
(a) for static loading
(b) for cyclic loading
(from r·1atlock. 1970).
sponding to one-half the maximum principal-stress differ-
ence. If no stress-strain curves are available, typical
values of £50 are given in Table 3.3.
2. Compute the ultimate soil resistance per unit length of
pile, using the smaller of the values given by equations
below.
p =[3 + ~ x +~ x] cb
u c. b
Pu =9 cb
38
(3.18)
(3.19)

TABLE 3,3. REPRESENTATIVE VALUES OF £50'
Consistency of Clay
Soft
Medium
Stiff
(Also see Tables 3.5 and 3.6)
where
0.020
0.010
0.005
r l
=average effective unit weight from ground surface
to p-y curve
x =depth from ground surface to p-y curve
c = shear strength at depth x
b =width of pile.
Matlock (1970) states that the value of J was determi ned
experimentally to be 0.5 for a soft clay and about 0.25 for
a medium clay. A value of 0.5 is frequently used for J.
The value of p is computed at each depth where a p-y curveu
is desired, based on shear strength at that depth.
3. Compute the deflection, Y50' at one-half the ultimate soil
resistance from the following equation:
Y50 =2.5 E
50b. (3.20)
4. Points describing the p-y curve are now computed from the
following relationship.
1
l = 0.5 (L) "3
Pu Y50
(3.21)
The value of p remains constant beyond y =8Y50'
The following procedure is for cyclic loading and is illustrated in
Fig. 3.7b.
1. Construct the p-y curve in the same manner as for short-term
static loading for values of p less than 0.72p .u
39

2. Solve Eqs. 3.18 and 3.19 simultaneously to find the depth,
xr ' where the transition occurs. If the unit weight and
shear strength are constant in the upper zone, then
6cbx = -:;-----:-
r (y I b + Jc)
(3.22)
If the unit weight and shear strength vary with depth, the
value of xr should be computed with the soil properties at
the depth where the p-y curve is desired.
3. If the depth to the p-y curve is greater than or equal to
x , then pis equal to O. 72p for all values of y greaterr u
than 3ySO.
4. If the depth to the p-y curve is less than xr ' then the val-
ue of p decreases from 0.72pu at y =3ySO to the value given
by the following expression at y = lSySO.
p = 0. 72pu ( :r )
The value of p remains constant beyond y =lSysO.
(3.23)
Recommended Soil Tests. For determining the various shear strengths
of the soil required in the p-y construction, Matlock (1970) recommended
the following tests in order of preference:
soil .
1. in-situ vane-shear tests with parallel sampling for soil
identification,
2. unconsolidated-undrained triaxial compression tests having
a confining stress equal to the overburden pressure with c
being defined as half the total maximum principal stress
difference,
3. miniature vane tests of samples in tubes, and
4. unconfi ned compress i on tests.
Tests must also be performed to determi ne the unit wei ght of the
Example Curves. An example set of p-y curves was computed for soft
clay for a pile with a diameter of 48 in. The soil profile that was used
is shown in Fig. 3.8. The submerged unit weight was assumed to be 20 lb/cu
ft at the mudline and 40 lb/cu ft at a depth of 80 ft and to vary linearly.
In the absence of a stress-strain curve for the soil, ESO was taken as 0.01
for the full depth of the soil profile. Theloading was assumed to be cyclic.
40

Shear Strength, (T /sq ft)
0 0.5 1.0 1.5 2.0
0
50
In -Situ
- Strength
- 100 Profile
--~
.2 150
-CJ
..-•~
• 200CL
250
Fig. 3.8. Soil profile used for example p-y
curves for soft cla~
The p-y curves were computed for the following depths below the mud-
line: 0, 1, 2, 4, 8, 12, 20, 40, and 60 ft. The plotted curves are shown
in Fig. 3.9 (curves for 0 and 1 ft too close to axis to be shown).
Response of Stiff Clay below the Water Table
Field Experiments. Reese, Cox, and Koop (1975) performed lateral
load tests employing steel-pipe piles that were 24 in. in diameter and 50
ft long. The piles were driven into stiff clay at a site near Manor,
Texas. The clay had an undrained shear strength ranging from about 1 T/sq
ft at the ground surface to about 3 T/sq ft at a depth of 12 ft.
Recommendations for Computing p-y Curves. The following procedure
is for short-term static loading and is illustrated by Fig. 3.10.
1.
2.
Obta in values for undrained soil shear strength c, soil
submerged unit weight ll, and pile diameter b.
Compute the average undrained soi 1 shear strength covera
the depth x.
pile using the smaller of the values given by the equation
below:
Pct = 2cab + y'bx + 2.83 cax (3.24)
Pcd = 11 cb. (3.25)
41

~
.5
"-.a
1500
, - - - - - - - - - - - - - - - - - - - - 6 0 ft
,------------------------------------40ft
~IOOO
q)
u
c:
c
-.!!!
III
Q)
0::
'0
(/)
,..--------------------20ft
5
Deflection, y(in.l
Fig. 3.9. Example p-y curves for soft clay below water table,
Matlock criteria, cyclic loading.
4. Choose the appropriate value of As from Fig. 3.11 for the
particular nondimensional depth.
S. Establish the initial straight-line portion of the p-y
curve:
p = (kx)y . ( 3.26)
Use the appropriate value of k or k from Table 3.4 for k.s c
6. Compute the following:
yso =£SOb. (3.27)
Use an appropriate value of £SO from results of laboratory
tests or, in the absence of laboratory tests, from Table
3.S.
7. Establish the first parabolic portion of the p-y curve,
using the following equation and obtaining p from Eqs.
c
3.24 or 3.2S.
p = 0. 5pc (L)0.5
Y50
(3.28)
Equation 3.28 should define the portion of the p-y curve
42

TABLE 3.4. REPRESENTATIVE VALUES OF k FOR STIFF CLAYS.
Average Undrained Shear Strength*
T/sq ft
0.5-1 1-2 2-4--
ks (Static) lb/cu in. 500 1000 2000
kc (Cyclic) lb/cu in. 200 400 800
* The average shear strength should be computed from the shear
strength of the soil to a depth of 5 pile diameters. It
should be defined as half the total maximum principal stress
difference in an unconsolidated undrained triaxial test.
TABLE 3.5. REPRESENTATIVE VALUES OF s50 FOR STIFF CLAYS
s50 (i n./ in. )
Average Undrained Shear Strength
0.5-1
0.007
43
T/sq ft
1-2
0.005
2-4
0.004

c
.......
..Q
-c.
[
y ]0.5 /
p=0.5pc Yeo /
/
/
,(
/ ------..
STATIC
1- - ....Y.,
[
Ii] 1.25
J!,,,... •0.055 Pc . A. Yeo
,; 0.5pc ---
u
c
o
-IJ)
IJ)
cv
a:::
o
en
Fig. 3.10.
E =- 0.0625 Pc
SS Y50
6ASY50
Deflection, y (in.)
Characteristic shape of p-y curve for static loading
in stiff clay below 'the water table
(after Reese, Cox, Koop, 1975).
from the point of the intersection with Eq. 3.26 to a point
where y is equal to AsYSO (see note in step 10).
8. Establish the second parabolic portion of the p-y curve,
p = 0. 5pc (yy )0.5 _ 0. 055pc (y ~ ~SY50) 1.25 •
50 s 50 ) (3.29)
from the point where y is equal to AsYSO to a point where y
is equal to 6AsYSO (see note in step 10).
9. Establish the next straight-line portion of the p-y curve,
44

x
b
A
o 0.2 0.4 0.6 0.8 1.0
O~----------~------~-----r----~
2
4
6
8
10
12
0
I
I I
I I
I I
I I
I I
I I
I I
I I
I I
Fig. 3.11. Values of constants As and Ac'
from the point where y is equal to 6AsY50 to a point where y
is equal to 18AsY50 (see note in step 10).
10. Establish the final straight-line portion of the p-y curve,
P = 0.5p (6A )0.5 _ 0.411p - 0.75p A (3.31)
c s c c s
or
p = pc(1.225 vrr;:;, - 0.75A
s
- 0.411) . (3.32)
from the point where y is equal to 18AsY50 and for all larg-
er values of y (see following note).
Note: The step-by-step procedure is out1i ned, and Fi g.
3.10 is drawn, as if there is an intersection between Eqs.
45

3.26 and 3.28. However, there may be no intersection of Eq.
3.26 with any of the other equations or, if no intersection
occurs, Eq. 3.26 defines the complete p-y curve.
Fig. 3.12.
Q.
o
CYCLIC
E O.085PeIe • - _ ____
Yeo
1.8yp
Deflection, y( in.)
Figo 3.12. Characteristic shape of p-y curve for cyclic loading
in stiff clay below water table
(after Reese, Cox, Koop, 1975).
1. Steps 1, 2, 3, 5, and 6 are the same as for the static case.
4. Choose the appropriate value of Ac from Fig. 3.11 for the
particular nondimensional depth.
Compute the following:
yp =4.1 A
sY50' (3.33)
7. Establish the parabolic portion of the p-y curve,
[
Y - 0.45 YP 2.5 ] .
p = Acpc 1 - - (3.34)
0.45 Yp
46

from the point of the intersection with Eq. 3.26 to where y
is equal to 0.6y (see note in step 9).
p
8. Establish the next straight-line portion of the p-y curve,
p = 0.936 A p - 0.085 P {y - 0.6y } • (3.35)
c c Y50 c· p
from the poi nt where y is equa1 to O. 6y to the poi nt where
p
y is equal to 1.8yp (see note in step 9).
9. Establish the final straight-line portion of the p-y curve,
p = 0.936 A P - 0.102 P y (3.36)
c c Y50 c P •
from the pOint where y is equal to 1.8y
p
and for all larger
values of y (see following note).
Note: The step-by-step procedure is out1i ned, and Fi g.
3.12 is drawn, as if there is an intersection between Eqs.
3.26 and 3.34. However, there may be no intersecti on of
those two equations and there may be no intersection of Eq.
3.26 with any of the other equations defining the p-y curve.
If there is no intersection, the equation should be
employed that gives the smallest value of p for any value of
y.
Recommended Soil Tests. Triaxial compression tests of the unconsol-
idated-undrained type with confining pressures conforming to the in situ
overburden pressures are recommended for determining the shear strength
of the soil. The value of £50 should be taken as the strain during the
test corresponding to the stress equal to half the maximum
total-principal-stress difference. The shear strength, c, should be
interpreted as one-half of the maximum total-stress difference. Values
obtained from the triaxial tests might be somewhat conservative but would
represent more realistic strength values than other tests. The unit
weight of the soil must be determined.
Example Curves. An example set of p-y curves was computed for stiff
clay for a pile with a diameter of 48 in. The soil profile that was used
is shown in Fig. 3.13. The submerged unit weight of the soil was assumed
to be 50 lb/cu ft for the entire depth. In the absence of a stress-strain
47

curve, ESO was taken as 0.005 for the full depth of the soil profile. The
slope of the initial portion of the p-y curve was established by assuming
a value of k of 463 1b/cu in. The loading was assumed to be cyclic.
Shear Strength.CT/sqft)
oO~____~r-______I~.O~____~~____~zo
200
In-Situ Strength Profile ~
~----~.....
250~----~------~~-L--~~----~
Fig. 3-.13. Soil profile used for example p-y
curves for stiff c1a~
line: 0, 2, 4, 8, 12, 20, 40, and 60 ft. The plotted curves are shown in
Fig. 3.14.
Response of Stiff Clay above the Water Table
Field Experiments. A lateral load test was
Houston on a drilled shaft, 36 in. in diameter.
instrumented at intervals along its
performed at a site in
A 10-in. diameter pipe,
length with e1ec-
trica1-resistance-strain gauges, was positioned along the axis of the
shaft before concrete was placed. The embedded length of the shaft was 42
ft. The average undrained shear strength of the clay in the upper 20 ft
48

........
~250
-0..
..0> rv-v"V""U~
OL~~~
C
C
-en
'en.0>
a:::
'0
en 1000
0.4 0.6 08 1.0 1.2 1.4 1.6 1.8
Deflection I Y(in.)
Fig. '3.14. Example p-y curves 'for stiff clay below the water table,
Reese criteria, cyclic loading.
was approximately 2,200 lb/sq ft. The experiments and their intepretation
are discussed in detail by Welch and Reese (1972) and Reese and Welch
(1975).
Recommendations for Computing p-y Curves. The following procedure
is for short-term static loading and is illustrated in Fig. 3.15a.
1. Obtain values for undrained shear strength c, soil unit
weight l, and pile diameter b. Also obtain the values of
£50 from stress-strain curves. If no stress-strain curves
are available, use a value from £50 of 0.010 or 0.005 as
given in Table 3.3, the larger value being more conserva-
tive.
49
2.0

-c
.......
.c
/P= Pu
-------- -~~-------------------
a.
Q)
o
c
~ " P ( Y )1/4.iii - .. 0.5 ...!L
Q) Pu Yeo
a::
'0
CJ)
16yIO
Deflection t y(in.)
Fig. 3.15a. Characteristic shape of p-y curve for static
loading in stiff clay above water table.
shaft, Pu' using the smaller of the values given by Eqs.
3.18 and 3.19. (In the use of Eq. 3.18 the shear strength
is taken as the average from the ground surface to the depth
being considered and J is taken as 0.5. The unit weight of
the soil should reflect the position of the water table.)
3. Compute the deflection, yso' at one-half the ultimate soil
resistance from Eq. 3.20.
4. Poi nts descri bi ng the p-y curve may be computed from the
relationship below .
..£... = 0.5 (...L) 1/4
Pu Y50
(3.37)
5. Beyondy= 16ySO' p is equal to Pu for all values of y.
Fig. 3.1Sb.
50

.....a.
Q)
o
c
c
-en
'in
Q)
a:
'0
Cf)
:::::::';:::::=;1--...,..1 - - - -
: : ~Yc. Y. + YIOC leigN,
I
Yc • Ys + YIO C logNa
Yc • Y. + YIO ClOON.
ISYso 16yIO 16ylO
.. + +
9.6(YIO)IOQN. 9.6(ylO) IOON1 - 96(yIOHooN.
Deflection , y (in.)
Fig. 3.15b. Characteristic shape of p-y curve for cyclic
loading in stiff clay above water table.
1. Determi ne the p-y curve for short-term static 1oadi ng by
the procedure previously given.
2. Determine the number of times the design lateral load will
be applied to the pile.
3. For several values of p/p obtain the value of C, the param-u
eter describing the effect of repeated loading on deforma-
tion, from a relationship developed by laboratory tests,
(Welch and Reese, 1972), or in the absence of tests, from
the following equation.
(3.38)
4. At the value of p corresponding to the values of p/pu
selected in step 3, compute new values of y for cyclic load-
ing from the following equation.
Yc =Ys + Y50 • C • 10gN (3.391
51

where
Yc =deflection under N-cycles of load,
Ys =defl ection under short-term static load,
ySO = defl ection under short-term static load at
one-half the ultimate resistance, and
N = number of cycles of load application.
s. The p-y curve defines the soil response after N-cycles of
load.
Recommended Soil Tests. Triaxial compression tests of the unconsol-
idated-undrained type with confining stresses equal to the overburden
pressures at the elevations from which the samples were taken are recom-
mended to determine the shear strength. The value of ESO should be taken
as the strain during the test corresponding to the stress equal to half
the maximum total principal stress difference. The undrained shear
strength, c, should be defined as one-half the maximum
total-principal-stress difference. The unit weight of the soil must also
be determi ned.
Example Curves. An example set of p-y curves was computed for stiff
clay above the water table for a pile with a diameter of 48 in. The soil
profile that was used is shown in Fig. 3.13. The unit weight of the soil
was assumed to be 112 lb/cu ft for the entire depth. In the absence of a
stress-strain curve, ESO was taken as O.OOS. Equation 3.38 was used to
compute values for the parameter C and it was assumed that there is to be
100 cycles of load application.
The p-y curves were computed for the fo 11 owi ng depths be low the
groundline: 0, 1,2, 4, 8, 12, 20, 40, and 60 ft. The plotted curves are
shown in Fig. 3.16.
Unified Criteria for Clays below the Water Table
Introduction. As was noted in the previous section, no recommen-
dations were made for ascertaining for what range of undrained shear
strength one should employ the criteria for soft clay and for what range
one should employ the criteria for stiff clay. Sullivan (1977) examined
the original experiments and developed a set of recommendations that yield
computed behavi ors in reasonably good agreement wi th the experimental
results from the Sabine tests reported by Matlock (1970) and with those
from the Manor tests reported by Reese, Cox and Koop (197S). However, as
will be seen from the fo 11 owi ng presentation, there is a need for the
52

100
~
c
:.::::: 8000
60ft.0
a.
Q)
()
c:
0
-Ul
·in
Q)
0::
'6
CJ}
6000
4000
{20ft
40ft
12 ft
8ft
4 ft
Oft
2ft
'I ft
o&------L----__~_____L______~_____L______~_____L___
o 4 8 12 16
Deflection I
20
Y(in.)
24 28
Fig. 3.16. Example p-y curves for stiff clay above water
table, Welch criteria, cyclic loading.
engineer to employ some judgement in selecting appropriate parameters for
use in the prediction equations.
Recommendations for Computing p-y Curves. The following procedure is
for short-term static loading and is illustrated in Fig. 3.17.
1. Obtain values for the undrained shear strength c, the sub-
merged unit weight ll, and the pile diameter b. Also obtain
va 1ues of ESO from stress-strain curves. If no
stress-strain curves are available, the values in Table 3.6
are provided as guidelines for selection of ESO'
2. Compute ca and 0v' for x < 12b, where
ca =average undrained shear strength,
a =average effective stress, and
v
x =depth.
3. Compute the variation of Pu with depth using the equations
below.
53

1.0
PR
- - - --------------- Pu = I for x >12b
p (Y )1/--0.5-
Pu Yeo
.&.- -'-F + (I-F) 12b for xs 12b
Pu
o~----------~~------------~------------~-----
o 10 20
Fig. 3.17. Characteristic shape of p-y curve for
unified clay criteria for static loading.
4.
a. For x < 12b, Pu is the smaller of the values computed
from the two equations below.
Pu = (3 + O. 5 ~) cb
b. For x > 12b,
Pu =9 cb.
The steps below are for a particular depth, x.
Select the coefficients, A and F, as indicated below.
(3.40)
(3.41)
(3.42)
The
coefficients A and F, determined empirically for the load
tests at Sabine and Manor, are given in Table 3.7. The
terms used in Table 3.7, not defined previously, are
defined below.
WL=liquid limit,
PI = plasticity index,
54

TABLE 3.6. REPRESENTATIVE VALUES
OF E50'
c
(lb/sq ft)
250 - 500
500 - 1000
1000 - 2000
2000 - 4000
4000 - 8000
(Also see Tables 3.3 and 3.5)
LI =liquidity index,
2
1
0.7
0.5
0.4
OR =overconsolidation ratio, and
St =sensitivity.
The recommended procedure for estimating A and F for other
clays is given below.
a. Determine as many of the following properties of the
clay as possible, c, ESO' OR' St' degree of fissuring,
ratio of residual to peak undrained shear strength, WL,
PI, and LI.
b. Compare the properties of the soil in question to the
properties of the Sabine and Manor clays listed in
Table 3.7.
c. If the properties are similar to either the Sabine or
Manor clay properties, use A and F for the similar clay.
d. If the properties are not similar to either, the en-
gineer should estimate A and F using his judgement and
Table 3.7 as guides.
S. Compute:
ySO = AESOb. (3.43)
55

TABLE 3.7. CURVE PARAMETERS FOR THE UNIFIED CRITERIA.
Clay Description A F
Sabine River 2.5 1.0
Inorganic, Intact
c = 300 lb/sq ft
£50 = 0.7%
OR =e 1
St =e 2
wL = 92
PI = 68
LI = 1
Manor 0.35 0.5
Inorganic, Very fissured
c =e 2400 1bl sq ft
£50 = 0.5%
OR > 10
St =e 1
wL = 77
PI = 60
LI = 0.2
56

6. Obtain (E) . When no other method is available Eq. 3.44s max
and Table 3.S may be used as guidelines.
(Es)max = kx. (3.44)
TABLE 3.8. REPRESENTATIVE VALUES FOR k.
c k
(lb/sq ft) (lb/cu in. )
250 - 500 30
500 - 1000 100
1000 - 2000 300
2000 - 4000 1000
4000 - 8000 3000
(Also see Table 3.4)
7. Compute the deflection at the intersection between the ini-
tial linear portion and curved portion, from the equation
below.
Sa.
8b.
(Yg can be no larger than SySO)
ForO<y<y
g
,
p =(Es)maxY'
For Yg < Y < 8y50' 1
p = 0.5p (1-)3 .u Y50
Sc. For SySO < Y < 30yso '
PR - Pu
P = Pu + 22 Y (y - 8Y50)
50
57
(3.45)
(3.46)
(3.47)
(3.48)

where
p = p (F + (l-F) _X_)R u 12 b •
(PR will be equal to or less than pu)
Bd. For y > 30Y50'
p =PRo
(3.49)
(3.50)
The following procedure is for cyclic loading and i~ illustrated in
Fig. 3.1B.
1.0
~~Rz 0.5 (I~b) for x sl2b
10 20 30
~CR =0.5 for x>12b
u
Fig. 3.18. Characteristic shape of p-y curve for unified
clay criteria for cyclic loading.
1. Repeat steps 1 through Ba for static loading.
2. Compute
3a.
x
p = 0 5 p -- < 0.5 p •
CR • u 12 b - u
For Yg < Y< Y50'
1
p=O.5p (L)3.u Y50
58
(3.51)
(3.52)

3b. For ySO < y < 20yso'
PeR - 0.5 Pu
P = 0.5 P + --'------ (y - ySO)·
u 19 Y50
3c. For y > 20ySO'
(3.53)
p =PCR· (3.54)
The procedure outlined above for both static and cyclic loading
assumes an intersection of the curves defined by Eqs. 3.46 and 3.47. If
that intersection does not occur, the p-y curve is defined by Eq. 3.46
until it intersects a portion of the curve defined by Eqs. 3.48 or 3.50
for static loading, and Eqs. 3.52 or 3.53 for cyclic loading.
Example Curves. Two example sets of p-y curves were computed using
the unified criteria; each of the sets is for a pile of 48 in. in diameter
and for cyclic loading.
Figure 3.19 shows the set of p-y curves for soft clay; the soil pro-
file used is shown in Fig. 3.8. The value of £50 was assumed to be 0.02 at
the mudline and 0.01 at a depth of 80 ft. The unit weight was assumed to
be 20 lb/cu ft at the groundline and 40 lb/cu ft at a depth of 80 ft. The
value of A was assumed to be 2.5 and the value of F was assumed to be 1.0.
The value of k for computing the maximum value of the soil modulus was
assumed to be 400,000 lb/cu ft. The p-y curves were computed for the fol-
lowing depths: 0, 1,2,4,8, 12,20, and 40 ft (curves for a and 1 ft too
close to axis to be shown).
Figure 3.20 shows the set of p-y curves for stiff clay; the soil pro-
file used is shown in Fig. 3.13. The value of £50 was assumed to be 0.006
and the unit weight of the soil was assumed to be 50 lb/cu ft. The value
of A was assumed to be 0.35 and the value of F wasassumedto be 0.5. The
value of k for computing the maximum value of the soil modulus was assumed
to be 800,000 lb/cu ft. The p-y curves were computed for the following
depths: 0, 1,2,4,8, 12, 20, and 40 ft.
3.6 RECOMMENDATIONS FOR p-y CURVES FOR SAND
As shown below, a major experimental program was conducted on the
behavior of laterally loaded piles in sand below the water table. The
results can be extended to sand above the water table.
Response of Sand below the Water Table
Field Experiments. An extensive series of tests were performed at a
site on Mustang Island, near Corpus Christi (Cox, Reese, and Grubbs,
59

~oo
-:-
.S
..... 600 40ft:e
-Q.
I)
u
c
c
- 400I/)
'enI)
It:
=a(J)
200
20ft
12 ft
20 40 60
Deflection , yOn.)
Fig. 3.19. Example p-y curves for soft clay below water
table, unified criteria, cyclic loading.
1974). Two steel pipe piles, 24 in. in diameter, were driven into sand in
a manner to simulate the driving of an open-ended pipe, and were subjected
to ldteral loading. The embedded length of the piles was 69 ft. One of
the piles was subjected to short-term loading and the other to repeated
loading.
The soil at the site was a uniformly graded, fine sand with an angle
of internal friction of 39 degrees. The submerged unit weight was 66
lb/cu ft. The water surface was maintained a few inches above the mudline
throughout the test program.
Recommendations for Computing P-y Curves. The following procedure
is for short-term static loading and for cyclic loading and is illustrated
in Fig. 3.21 (Reese, Cox, and Koop, 1974).
1. Obtain values for the angle of internal friction ~, the soil
unit weight r, and pile diameter b.
2. Make the following preliminary computations.
a = 1. Q = 45 + 1. K = 04' and K = tan2 (45 _1)
2 ' iJ 2' 0 . , a 2
(3.55)
60

-a.
CD
U
C
o
-II)
·en
CD
a::
·0
en
mo
0.5 1.0 1.5
Deflection, y(in)
Fig. 3.20. Example p-y curves for stiff clay below water
table, unified criteria, cyclic loading.
pile using the smaller of the values given by the equations
below, where x is equal to the depth below the ground sur-
face. [ Koxtan ¢ sins tans
p = yX . + (b + x tans tana)
st tan (S-~)cosa tan (S-¢)
+ Kox tanB (tan <I> sinB - tana) - Kab ] (3.56)
Psd =KaQlX (tanB~ - 1) + KoblX tan ¢ tan4~ (3.57)
For sand below the water table, the submerged unit weight II should be
used.
61

x= X4
""'t x=x3
.5
.......
:!
Q. x=x2
-II
u x=x,c:
~°iii
II
It:
°0(/)
x=o
b/60 3b/ao
Deflection • y (In.)
Fig. 3.21. Characteristic shape of a family of p-y curves
for static and cyclic loading in sand.
4. In making the computations in Step 3, find the depth xt
at
which there is an intersection at Eqs. 3056 and 3.57. Above
this depth use Eq. 3.56. Below this depth use Eq. 3.57.
5. Select a depth at which a p-y curve is desired.
6. Establish Yu as 3b/80. Compute Pu by the following
equation:
Pu =Asps or Pu =Acps' (3.58)
Use the appropriate value of A or A
s c
from Fig. 3.22 for
the particular nondimensional depth, and for either the
static or cyclic case. Use the appropriate equation
Eq. 3.56 or Eq. 3.57 by referring to the computation
4.
for p ,s
in step
7. Establish Ym as b/60. Compute Pm by the following equation:
p =B p or p =B p . (3.59)
m ssm c s
Use the appropriate value of Bs or Bc from Fig. 3.23 for
the particular nondimensional depth, and for either the
static or cylic case. Use the appropriate equation for p .
s
The two straight-line portions of the p-y curve, beyond the
point where y is equal to b/60, can now be established.
62

A
0 1.0 3.0
Ac(CYCLIC) 1.0 ,t AI (STATIC)
2.0 ~.
~
x
3.0 I0
4.0
5.0 x -b > 5.0, A= 0.88
6.0
Fig. 3.22. Values of coefficients Ac and As'
o
1.0
2.0
x 3.0b
4.0
5.0
6.0
,
Bc(CYCLIC)~
'.

It
-I
B
1.0 2.0
x
b >5.0, Be·0.55
Ss-0.50
3.0
Fig. 3.23. Nondimensional coefficient B for soil
resistance versus depth.
63

8. Establish the initial straight-line portion of the
p-y curve,
p = (kx)y.
Use the appropriate value of k from Table 3.9 or 3.10.
TABLE 3.9. REPRESENTATIVE VALUES OF k FOR SUBMERGED SAND.
Relative lJensity Loose Med i um Dense
1-----------------------------
Recommended k (lb/cu in.) 20 60 125
1------------------------1
(3.60)
TABLE 3.10. REPRESENTATIVE VALUES OF k FOR SAND ABOVE WATER TABLE.
Relative Density Loose Medium Dense
Reconmended k (1 b/cu in.) 25 90 225
9. Establish the parabolic section of the p-y curve,
- linp = C Y • (3.61)
Fit the parabola between points k ~nd mas follows:
a. Get the slope of line between points mand u by,
Pu - Pm
m = (3.62)
Yu - Ym
b. Obtain the power of the parabolic section by,
64

Pm
n =--
mYm
c. Obtain the coefficient C as follows:
C =
Pm
lin
Ym
d. Determine point k as,
= ( C)n/n-1Yk kx •
(3.63)
(3.64)
(3.65)
e. Compute appropriate number of points on the parabola by
using Eq. 3.6l.
Note: The step-by-step procedure is outlined, and Fig.
3.21 is drawn, as if there is an intersection between the
initial straight-line portion of the p-y curve and the par-
abolic portion of the curve at point k. However, in some
instances there may be no intersection with the parabola.
Equation 3.60 defines the p-y curve until there is an inter-
section with another branch of the p-y curve or if no inter-
secti on occurs, Eq. 3.60 defi nes the complete p-y curve.
The soi 1 response curves for other depths can be found
repeating the above steps for each desired depth.
Simplified Equations
In his work on the ultimate resistance of a plate in sand, Bowman
(1958) stated that the angle a ranges from ¢/2 for loose sand to ¢ for
dense sand. Reese, et al. (1974) reported that the value of a was found
f~om measurements of the contours of the wedge that formed at the ground
surface and that a from the Mustang Island tests ranged from ¢/3 for stat-
ic loading to 3¢/4 for cyclic loading. The angle ~ that further defines
the shape of the wedge of sand at the ground surface is not easy to measure
experimentally and also can be expected to vary. However, Reese, et al.
(1974) selected values of a and ~ of ¢/2 and 45 + ¢/2, respectively, in
developing correlations with experimental results from Mustang Island.
Fenske (1981) points out that Eqs. 3.56 and 3.57 can be simplified if
a is ¢/2 and ~ is 45 + ¢/2. The simplified equations are:
Pst =lb2[Sl(x/b) + S2(x/b)2]
Psd =lb2[S3(x/b)]
65
(3.66)
(3.67)

where
51 =(Kp - Ka) (3.68)
52 =(tan ~)(K tan a + K [ tan ¢ sin ~(sec a+l) - tan aJ)
p 0
53 =K2(K + K tan ¢) - K .
P P 0 a
The depth of transition xt can be found
(3.69)
(3.70)
by equating the expressions in
Eqs. 3.66 and 3.67, as follows:
xt/b =(53 - 51)/52, (3.71)
It can be seen that 51' 52' 53' and xt/b are functions only of ¢; therefore,
the values shown in Table 3.11 can be computed.
TABLE 3.11. NONDIMENSIONAL COEFFICIENTS FOR p-y CURVES
FOR SAND (after Fenske).
¢, deg. SI S2 S3 xt/b
25.0 2.05805 1. 21808 15.68459 11. 18690
26.0 2.17061 1. 33495 17.68745 11.62351
27.0 2.28742 1. 46177 . 19.95332 12.08526
28.0 2.40879 1. 59947 22.52060 12.57407
29.0 2.53509 1. 74906 25.43390 13 .09204
30.0 2.66667 1.91170 28.74513 13.64147
31.0 2.80394 2.08866 32.51489 14.22489
32.0 2.94733 2.28134 36.81400 14.84507
33.0 3.09732 2.49133 41.72552 15.50508
34.0 3.25442 2.72037 47.34702 16.20830
35.0 3.41918 2.97045 53.79347 16.95848
1
36.0 3.59222 3.24·376 61. 20067 17.75976
37.0 3.77421 3.54280 69.72952 18.61673
38.0 3.96586 3.87034 79.57113 19.53452
39.0 4.16799 4.22954 90.95327 20.51883
40.0 4.38147 4.62396 ·104.14818 21. 57604
66

Recommended Soil Tests. Triaxial compression tests are recommended
for obtaining the angle of internal friction of the sand. Confining pres-
sures should be used which are close or equal to those at the depths being
considered in the analysis. Tests must be performed to determine the unit
weight of the sand. In many instances, however, undisturbed samples of
sand cannot be obtained and the value of ¢ must be obtained from corre-
lations with static cone penetration tests or from dynamic penetration
tests.
Example Curves. An example set of p-y curves was computed for sand
below the water table for a pile with a diameter of 48 in. The sand is
assumed to have an angle of internal friction of 34° and a submerged unit
weight of 62.4 lb/cu ft. The loading was assumed to be cyclic.
line: 0, 1, 2, 4, 8, 12, and 20 ft. The plotted curves are shown in Fig.
3.24.
-
8000
,-__________ 20f
~-------------------12ft
~___-----------------8fl
0.80 1.60 2.40 3.20 4.00 4.80
Deflection, y (in.)
Fig. 3.24. Example p-y curves for sand below water table,
Reese criteria, cyclic loading.
67

Response of Sand Above the Water Table
The procedure in the previous section can be used for sand above the
water table if appropriate adjustments are made in the unit weight and
angle of internal friction of the sand. Some small-scale experiments were
performed by Parker and Reese (1971) and recommendations for p-y curves
for dry sand were developed from those experiments. The results from the
Parker and Reese experiments should be useful as check of solutions made
using results from the test program using full-scale piles.
3.7 RECOMMENDATIONS FOR p-y CURVES FOR ROCK
It is hardly surprising that not much information is available on the
behavior of piles that have been installed in rock. Some other type of
foundat ion woul d normally be used. However, a study was made of the
behavior of an instrumented drilled shaft that was installed in a vuggy
limestone in the Florida Keys (Reese and Nyman, 1978). The test was per-
formed for the purpose of gaining information for the design of founda-
tions for highway bridges in the Florida Keys.
Difficulty was encountered in obtaining properties of the intact
rock. Cores broke during excavation and penetrometer tests were mislead-
ing (because of the vugs) or could not be run. It was possible to test two
cores from the site. The small discontinuities on the outside surface of
the specimens were coated with a thin layer of gypsum cement in an effort
to minimize stress concentrations. The ends of. the specimens were cut
with a rock saw and lapped flat and parallel. The specimens were 5.88
in. in diameter and with heights of 11.88 in. for Specimen 1 and 10.44 in.
for Specimen 2. The undrained shear strength of the specimens were taken
as one-half the unconfined compressive strength and were 17.4 and 13.6
T/sq ft for Specimens 1 and 2, respectively.
The rock at the site was alsq investigated by in-situ-grout-plug
tests under the direction of Dr. John Schmertmann (1977). A 5.5 in. diam-
eter hole was drilled into the limestone, a high strength steel bar was
placed to the bottom of the hole, and a grout plug was cast over the lower
end of the bar. The bar was pulled until failure occurred and the grout
was examined to see that failure occurred at ~he interface of the grout
and limestone. Tests were performed at three borings and the following
results were obtained, in T/sq ft: depth into limestone from 2.5 to 5 ft,
23.8, 13.7, and 12.0; depth into limestone from 8 to 10 ft, 18.2, 21.7,
68

and 26.5; depth into limestone from 18 to 20 ft. 13.7 and 10.7. The aver-
age of the eight tests was 16.3 T/sq. However. the rock was stronger in
the zone where the deflections of the drilled shaft were most significant
and a shear strength of 18 T/sq ft was selected for correlation.
The drilled shaft was 48 in. in diameter and penetrated 43.7 ft into
the limestone. The overburden of fill was 14 ft thick and was cased. The
load was applied about 11.5 ft above the limestone. A maximum load of 75
tons was applied to the drilled shaft. The maximum deflection at the
point of load applicaton was 0.71 in. and at the top of the rock (bottom
of casing) it was 0.0213 in. While the curve of load versus deflection
was nonlinear, there was no indication of failure of the rock.
A single p-y curve, shown in Fig. 3.25, was proposed for the design
of piles under lateral loading in the Florida Keys. Data are insufficient
to indicate a family of curves to reflect any increased resistancewith
depth. Cyclic loading caused no measurable decrease resistance by the
rock.
1--' Perform proof test if deflection
I is in this range
I Pult =b Su - -
I
I
1 _
IEs -2000su
I
I
y1 =O.0004b
y
Assume brittle fracture
in this range
Fig. 3.25 0 Recommended p-y curve for design of drilled shaft
in vuggy limeston~
69

As shown in the figure, load tests are recommended if deflections of
the rock (and pile) are greater than 0.004b and brittle fracture is
assumed if the lateral stress (force per unit of length) against the rock
becomes greater than the diameter times the shear strength Su of the rock.
The p-y curve shown in Fig. 3.25 should be employed with considerable
caution because of the limited amount of experimental data and because of
the great variability in rock. The behavior of rock at a site could very
well be controlled not by the strength of intact specimens but by joints,
cracks, and secondary structure of the rock.
3.8 REFERENCES
Bowman, E. R., Investigation of the Lateral Resistance to Movement
of a Plate in Cohesionless Soil, Unpublished Thesis, The University of
Texas, January 1958.
Cox, W. R., Reese, L. C., and Grubbs, B. R., Field Testing of Lat-
era lly Loaded Pil es in Sand, II Proceedi ngs, Offshore Technology Confer-
ence, Paper No. 2079, Houston, Texas, May 1974.
Fenske, Carl W., personal communication, 1981.
Matlock, Hudson, Corre1ations for Design of Laterally Loaded Pil es
in Soft Clay, Paper No. OTC 1204, Proceedings, Second Annual Offshore
Technology Conference, Houston, Texas, Vol. 1, 1970, pp. 577-594.
McClelland, B. and Focht, J. A., Jr., Soil Modulus for Laterally
Loaded Piles, Transactions, American Society of Civil Engineers, Vol.
123, 1958, pp. 1049-1086.
Parker, F., Jr., and Reese, L. C., Lateral Pile - Soil Interaction
Curves for Sand, Proceedings, The International Symposium on the Engi-
neering Properties of Sea-Floor Soils and their Geophysical Identifica-
tion, The University of Washington, Seattle, Washington, July 1971.
Reese, L. C., Di scuss ion of Soi 1 Modul us for Laterally Loaded
Piles, by Bramlette McClelland and John A. Focht, Jr., Transactions,
American Society of Civil Engineers, Vol. 123, 1958, pp. 1071.
Reese, l. C. and Cox, W. R., Soil Behavior from Analysis of Tests of
Uninstrumented Piles under Lateral Loading, ASTM Special Technical Pub-
lication 444, American Society for Testing and Materials, San Francisco,
California, June 1968, pp. 161-176.
Reese, l. C., Cox, W. R., and Koop, F. D., Analysis of Laterally
Loaded Piles in Sand, Paper No. OTC 2080, Proceedings, Fifth Annual Off-
shore Technology Conference, Houston, Texas, 1974, Vol. II, pp. 473-485.
Reese, l. C., Cox, W. R. and Koop, F. D., Field Testing and Analysis
of Laterally Loaded Pil es in Stiff 'clay, II Paper No. OTC 2312, Proceedi ngs,
70

Seventh Offshore Technology Conference, Houston, Texas, Vol. 11,1975,
pp. 672-690.
Reese, L. C., and Nyman, K. J., Field Load Tests of Instrumented
Drilled Shafts at Islamorado, Florida, a report to Girdler Foundation and
Exploration Corporation, Clearwater, Florida, Bureau of Engineering
Research, The University of Texas at Austin, February 28, 1978.
Reese, L. C. and Welch, R. C., IILateral Loading of Deep Foundations
in Stiff Clay, Proceedings, American Society of Civil Engineers, Vo1.
101, No. GT7, February 1975, pp. 633-649.
Schmertmann, John H., Report on Development of a Keys Limerock Shear
Test for Drilled Shaft Design, a report to Girdler Foundation and Explo-
ration Company, Clearwater, Florida, 1977.
Skempton, A. W., liThe Bearing Capacity of Clays, Proceedings,
Building Research Congress, Division I, London, England, 1951.
Sullivan, W. R., Development and Evaluation of a Unified Method for
the Analysis of Laterally Loaded Piles in Clay, Unpublished Thesis, The
University of Texas at Austin, May 1977.
Terzaghi, Karl, IIEvaluation of Coefficients of Subgrade Reaction,1I
Geotechnique, Vol. 5, December 1955, pp. 297-326.
Winkler, E., Die Lehre von Elastizitat und Festigkeit (On Elasticity
and Fixity), Prague, 1867.
Welch, R. C. and Reese, L. C., IILaterally Loaded Behavior of Drilled
Shafts, Research Report No. 3-5-65-89, Center for Highway Research, Uni-
versity of Texas at Austin, May 1972.
3.9 EXERCISES
3.1 Given a clay with an undrained shear strength of 800 lb/sq ft a
submerged unit weight of 48 lb/cu ft and an E50 of 0.02, make necessary
computations and plot p-y curves for both static and cyclic loading for
depths of 6 ft and 12 ft. Assume the pile diameter to be 24 in. Use rec-
ommendations for p-y curves for soft clay below the water surface.
3.2 Repeat problem 1 using the Unified Criteria Method.
3.3 Repeat problem 1 for a stiff clay with undrained shear strength
of 4000 lb/sq ft, E50 of 0.005, and a dry unit weight of 115 lb/cu ft.
Plot p-y curves for both stiff clay above water surface and stiff clay
below the water surface.
3.4 Repeat problem 1 for a sand with an angle of internal friction
of 38° and a submerged unit weight of 55 lb/cu ft.
71

CHAPTER 4. SOLUTIONS FOR LATERALLY LOADED PILES WITH SOIL
MODULUS ASSUMED CONSTANT,
CONSTANT PILE STIFFNESS, NO AXIAL LOADING
4.1 SOLUTION OF THE DIFFERENTIAL EQUATION
The pile is assumed to be supported along its entire length by a con-
tinuous stratum of soil which is capable of exerting a reaction to the
pile in a direction opposite to the pile deflection. Fig. 4.1 shows that
the soil resistance p per unit of length of the pile is related to the
deflection y by the soil modulus E. For the case being considered the
s
soil modulus Es is assumed to have the same value for all points along the
pile and is defined by the constant a. Furthermore, EI is constant and
there is no axial loading.
-0-
I
..Q)
o
c:
o
-(/)
(/)
Q)
a::
o
en
-p
- =0(
y
Deflection, yOn)
Fig. 4.1. Soil response curve.
The second-order differential equation is shown in Eq. 4.1 and the
fourth-order differential equation is shown in Eq. 4.2. It should be not-
ed that the fourth-order equation is derived by differentiation, assuming
that the stiffness EI is constant.
~ = Ji..
dx 2 E1 Preceding page blank (4.1)
73

(4.2 )
Employing the basic relationship between the soil resistance p and
pile deflection y, Eq. 4.3, and employing the identity in Eq. 4.4, Eq. 4.5
is derived.
p =-a,y
4 _ a
e - 4E1
~ + 4e4y = 0
dx4
(4.3)
(4.4)
(4.5)
The parameter ~ may be defined as the relative stiffness factor; the
influence of S on the solution of Eq. 4.5 will be indicated later.
The solution of Eq. 4.5 can be easily obtained by standard tech-
niques, as shown in Eqs. 4.6 through 4.10.
(04 + 4S4)y =0
m4 + 4S4 =0
m1 = -m3
= S(1 + i)
m2
= -m4
=S(-1 + i)
(4.6)
(4.7)
(4.8)
(4.9)
y = eex(CI
cos ex + (2 sin ex) + e -ex(C3
cos ex + C
4
sin ex)
(4.10)
The coefficients C1, C2
, C3
, and C4
must be evaluated for the various
boundary conditions that are desired. The evaluation of these coeffi-
cients must involve the use of the derivatives that are shown in Eqs. 4.11
through 4.14.
gy ex - -
dx =Se (C1 cos Sx + C2 sin Sx - C1 sin Sx + C2 cos Sx)
+ se-ex (-C3
cos Sx - C
4
sin Sx - Ssin Sx + C
4
cos Sx) (4.11.)
d2 ex - -
~2 = 2S2e (C2 cos Sx - C1 sin Sx)
dx -ex - -
+2S2e (C3 sin Sx - C
4
cos Sx) (4.12)
::; =2s
3
e
e
:(xC2 :.OS SX - C1 ~in Sx - C2 .:..in Sx - C1 .:.OS Sx)
+ 2S3e e (-C
3
sin Sx + C
4
cos Sx + C
3
cos Sx + C
4
sin Sx) (4.13)
(4.14)
74

4.2 PILE OF INFINITE LENGTH
If one considers a long pile, one which can be considered to have an
infinite length, a simple set of equations can be derived. Because
- -deflections must be small for large values of x, C1 =C2 =O. This conclu-
sion is reached by examining Eq. 4.10. The term eSxincreases without lim-
it as x increases. The terms sin ~x and cos ~x oscillate between +1 and
-1; therefore, the only way that the expression for y can have a finite
- -
value is for C1 and C2 to approach zero as x becomes large. The first case
to be considered is shown in Fig. 4.2(a). The boundary conditions are
given by Eqs. 4.15 and 4.16.
d2y Mt
at x = 0, - = -
(0)
Mf
FJ 0--y
Ix
( b)
.!L.'fi--Y
Ix
(4.15)
(4.16)
(e)
/spring (tokes no shear but
restra ins pile - head
D rotation)
't .. J.---y
x
Fig. 4.2. Boundary conditions at top of pile
The use of Eq. 4.15 leads to
2EIS2(C3
(0) - C4
(1)] =
-Mt
C4 = -2E-I-s-2 (4.17)
75

The use of Eq. 4.16 leads to
2EI S3[-C3(0) + C4(1) + C3(1) + C
4
(0)] = P
t
' and
P
C + C = t
3 4 2ElS3·
(4.18)
With the determination of the coefficients, substitutions can be
made and relevant equations derived as shown below.
Timoshenko (1941) says the IIlongli pile solution is satisfactory
where aL ~ 4. Solutions will be shown later for the case where the nondi-
mensional length of the pile is less than 4.
Making use of Eqs. 4.17 and 4.18, expressions for y, S, M, V, and p
can be written and are shown in Eqs. 4.19 through 4.23.
-SXe
y = 2ElS2 [
Pt
13 cos SX + Mt (cos SX - sin
s -sx [2Pt
S2
(sin SX + cos sx)
Mt
sx]= -e E + ElS cos
s
[P SX + cos SX)]
- x t . + Mt (sinM= e S S-sln SX
V = e- Sx [Pt(cos SX - sin f3x·) - 2Mt S sin sx]
p = 2f3e- Sx [-Pt cos SX - MtS(cos f3X - sin sx)]
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
It is convenient to define some functions which make it easier to
write the above equations. These are:
Al = e-f3x(cos f3X + sin f3x)
Bl = e-Sx(cos SX - sin sx)
C
1
= e-Sxcos sx
01 = e-Sxsin SX.
Using these functions, Eqs. 4.19 through 4.23 become:
. 2PtS Mt
y = -a.- C1 + 2Els2 Bl
-2PtS2 Mt
S = a. Al - EIS C1
Pt
M= S- 01 + Mt A1
V = Pt
B1 - 2Mt
S01
p = -2Pt SC1 - 2Mt f32B1
Values for AI' B1, C1, and 01 are shown in Table 4.1.
76
(4.24)
(4.25)
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)

TABLE4.1.TABLEOFFUNCTIONSFORPILEOFINFINITELENGTH.
f3xA,B,C,0,f3xA,B,C,0,I!
01.00001.00001.00000.00002.4-0.0056-0.1282-0.06690.0613
0.10.99070.81000.90030.09032.6-0.0254-0.1019-0.06360.0383
0.20.96510.63980.80240.16272.8-0.0369-0.0777-0.05730.0204
0.30.92670.48880.70770.21893.2-0.0431-0.0383-0.0407-0.0024
0.40.87840.35640.61740.26103.6-0.0366-0.0124-0.0245-0.0121
0.50.82310.24150.53230.29084.0-0.02580.0019-0.0120-0.0139
0.60.76280.14310.45300.30994.4-0.01550.0079-0.0038-0.0117I
0.70.69970.05990.37980.31994.8-0.00750.00890.0007-0.0082
-.s0.80.6354-0.00930.31310.32235.2-0.00230.00750.0026-0.0049
-.s0.90.5712-0.06570.25270.31855.60.00050.00520.0029-0.0023
1.00.5083-0.11080.19880.30966.00.00170.00310.0024-0.0007
1.10.4476-0.14570.15100.29676.40.00180.00150.00170.0003,
1.20.3899-0.17160.10910.28076.80.00150.00040.00100.0006
1.30.3355-0.18970.07290.26267.20.0011-0.000140.000450.00060
1.40.2849-0.20110.0419'0.,24307.60.00061-0.000360.000120.00049
1.50.2384-0.20680.01580.22268.00.00028-0.00038-0.00050.00033
1.60.1959-0.2077-0.00590.20188.40.00007-0.00031-0.000120.00019
1.70.1576-0.2047-0.02350.18128.8-0.00003-0.00021-0.000120.00009
1.80.1234-0.1985-0.03760.16109.2-0.00008-0.00012-0.000100.00002
1.90.0932-0.1899-0.04840.14159.6-0.00008-0.00005-0.00007-0.00001
2.00.0667-0.1794-0.05630.123010.0-0.00006-0.00001-0.00004-0.00002
2.20.0244-0.1548-0.06520.0895
I

For a pile whose head is fixed against rotation, as shown in Fig.
4.2(b), the solution may be obtained by employing the boundary conditions
as given in Eqs. 4.33 and 4.34.
at x = 0, ~ - 0 (4.33)dx -
d3y Pt
- = (4.34)
dx 3 EI
procedures as shown above, it was found that C
3 = C
4 =
The solution for long piles is given in Eqs. 4.35 through 4.39.
PtS
y = ---;- Al (4.35 )
(4.36)
(4.37)
(4.38)
(4.39)
It is convenient frequently to have a solution for a third set of
boundary conditions, as shown in Fig. 4.2(c). The boundary conditions are
given in Eqs. 4.40 and 4.41.
at x = D,
EI~
dx2
~
dx
d
3
y =~
dx 3 EI
(4.40)
(4.41)
Employing these boundary conditions, the coefficients C3
and C4
were eval-
uated as shown in Eqs. 4.42 and 4.43. For convenience in writing, the
rotational restraint Mt/St is given the symbol k
e
•
C3
=
Pt
(2EIS + k
e
)
EI (a + 4S3ke)
(4.42)
C4
KP
t
=
EI(a + 4S3ke)
(4.43)
Equations 4.42 and 4.43 may be substituted into Eqs. 4.10 through 4.14 to
obtain the expressions for the pile response.
78

4.3 PILE OF FINITE LENGTH
A solution for the case of the pile of finite length is useful. The
following derivation is for one set of boundary conditions, as shown.
at x = a
at x = L
d3y Pt
v = P or - =-
t dx3 EI
d2v
M = a or ~ = a
dx2
d3v
V = a or =-..t- = a
dx 3
Employing Eqs. 4.44 and 4.12, Eq. 4.48 results.
M
E~ = 2 S2(C2 - C4)
Employing Eqs. 4.45 and 4.13, Eq. 4.49 is obtained.
Pt - - - -
El = 2 S3(-C1 + C2 + C3 + C4)
SL - -
(4.44)
(4.45)
(4.46)
(4.47)
(4.48)
(4.49)
a = 2~2e (C
2 cos ~L - C1 sin ~L)
+ 2~2e-S[(C3 sin ~L - C4 cos ~L) . (4.50)
a = 2~3eSL(C cos ~L - C1 sin ~L - C2 sin ~L - C1 cos ~L)
+ 2fh e-st(-C3
si n ~L + C4 cos ~L + C3
cos ~L + C4 si n ~L)
(4.51)
Equations 4.48 through 4.51 can be solved in any convenient way for
- -
the coefficients C1 through C4. A step-by-step procedure that is
straightforward is shown in the Appendix.
4.4 REFERENCES
Timoshenko, S. P., Strength of Materials, Part II, Advanced Theory
and Problems, 2nd Edition - Tenth Printing. D. Van Nostrand Company,
Inc., 1941, p. 20.
79

4.5 EXERCISES
4.1 Assume a fixed-head steel pipe pile with a 36-in. outside diam-
eter and a wall thickness of 1.0 in. a length of 70 ft, and a lateral load
of 40 kips at the mudline. Assume no axial load and that EI is constant
with depth. Assume Es = 2000 lb/sq in. and constant with depth. Compute
deflection and bending moment as a function of depth.
4.2 Assume the pile in problem 1 is free to rotate at the mudline,
find the maximum bending moment, and depth to point of maximum bending
moment for increments of load of 10 kips until a plastic hinge develops in
the stee1.
4.3 Derive expressions for pile response with pile head restrained
against rotation as shown in Fig. 4.2(c) for a long pile. Then repeat
problem 1 assuming ke =Mt/St =6 x 108
in~lb.
80

CHAPTER 5. THE DIFFERENCE EQUATION METHOD
FOR SOLVING THE DIFFERENTIAL EQUATION
FOR A LATERALLY LOADED PILE
5.1 INTRODUCTION
If the re1ation shi p between soil reaction p and defl ection y is
linear and defined by the soil modulus E as shown in Fig. 3.2(b), thes .
soil resistance per unit length of pile is equal to the modulus multiplied
by the deflection. If the soil modulus is constant with depth and if the
pile can be considered to be of infinite length, the differential equation
can be solved rather easily. The solution is
y = ei3X
(C1
cos ~x + C2
sin ~x)
+ e-·i3X (C
3 cos ~x + C4
sin ~x)
as shown in Chapter 4. As was shown, the coefficients C1, C2
, C3, and C4
can be evaluated by using the boundary conditions.
If the soil modulus has a random variation with depth, the soil
resistance is equal to some function of x multiplied by the deflection.
The solution of the differential equation can be made by writing the dif-
ferential equation in difference form. This method was suggested by Pal-
mer and Thompson (1948). A conveni ent way of sol vi ng the difference
equation has been suggested by Gleser (1953). Contributions to the gener-
al method have been made by Focht and McClelland (1955) and Howe (1955).
The differential equation is
o . (5.1 )
5.2 RELATIONSHIPS IN DIFFERENCE FORM
Figure 5.1 shows a portion of the elastic curve of a pile. Relation-
ships in difference form are as follows:
(~
dx2/
x=m
Ym- 1 - Ym+1
2h
Ym- Ym+1
h Ym-1 - 2Ym+ Ym+1- ----·h--~- -
h2
81
(5.2)
(5.3)

Y
r'~
I
h
I
I /)..;
I /; /
1 Ym+ 2 / ;
1 /1 /
Yon. I I /
/
I
1/ II Ym
I
I I I
J. Ym-I / /
h
h
I
/ / I
Ym-~ I I /
I'
//1I
I 11/
I
/11I
h
l1 ,_':1
I
x
Fig. 5.1. Representation of deflected pile,
In a similar manner
(~)
Ym-2 - 2Ym_1 + 2Ym+1
-
dx 3 2h 3
x=m
(~)
Ym-2 - 4y + 6y -m-1 m-
dx4 h4
x=m
Equation 5.5 is substituted into Eq. 5.1
- Ym+2
4Ym+1 + Ym+2
-E h4
sm
EI
(5.4)
(5.5)
(5.6)
Figure 5.2 shows the manner in which the pile is subdivided. Two
imaginary points are shown below the tip of the pile and two above the top
of the pile. Since Es is presumably known for all points along the pile,
it is possible to write t + 1 algebraic equations, similar to Eq. 5.6, for
points 0 through t. Two boundary conditions at the tip of the pile and two
82

at the top of the pile yield four additional equations, giving a total of
t + 5 simultaneous equations. When solved, these equations give the
deflection of the pile from point -2 through point t + 2. A solution can
be obtained for any number of subdivisions of the pile.
t +2
t +
t
6
5
4
;5
2
1
o
-I
-2
Fig. 5.2. Method of subdividing pile.
5.3 THE GLESER METHOD OF SOLUTION
Appendix 2 presents a detailed derivation of the method proposed by
Gleser (1953) for the solution of the simultaneous algebraic equations.
The detailed derivations that are presented are intended to provide
the reader with sufficient information on the difference-equation method
to allow for the evaluation of the method. In addition, the derivations
that are presented provide guidance in developing equations for addi-
tional sets of boundary conditions. For further guidance, a step-by-step
computation procedure is presented.
1. Compute the A-values, using Eq. A2.7.
2. Compute BO' B1, and B2 using Eqs. A2.14, A2.15, and A2.23.
3. Compute other B-values, using Eqs. A2.35 and A2.36 through
B2t+1·
83

4. Compute Ci, C'2, and C3 using Eqs. A2.47, A2.S6, and A2.S7.
These steps are applicable to all cases.
(Steps S through 11 pertain to Cases 1 and 2)
S. Compute b
1
using Eq. A2.41 and b
S
using Eq. A2.63.
6.
7.
Compute Yt ' Yt+1' and Yt+2 using Eqs. A2.42, A2.73, and A2.74.
Compute Yt - 1, Yt -2' and other y-values using Eq. A2.34 and
appropriate B-values.
8. Compute slope using
S
m
Ym- l - Ym+l
=~~---
2h
9. Compute moment using
EI
Mm = h2 (Ym-l - 2Ym + Ym+l)·
10. Compute shear using
V - EI ( 2 + )
m - 2h3 Ym-2 - Ym- l Ym+l - Ym+2 •
11. Compute soil reaction using
(S.7)
(5.8)
(5.9)
P =-E Y (5.10)
m s m·
m
(Steps 12 through 14 pertain to Case 3)
12. Compute b1 using Eq. A2.41.
13. Compute Yt ' Yt+1' and Yt+2 using Eqs. A2.89, A2.82, and A2.80.
14. Other computations proceed as from Step 7, above.
5.4 EXAMPLE COMPUTATION
An example computation is presented to illustrate the step-by-step
procedure.
Pile: 24 in. in diameter by 1 in. wall thickness;
I =4787 in.4 ; length =1200 in.
Loading: fixed-head case, Pt =60,000 lb
Soil modulus: Es =kx, k =S lb/cu in.
Number of increments: 5, h = 240 in.
Find: Yt
and Mt
Computation of A-values:
k X h4
A = - - - = 0.1155 x
m EI
84

Point x, in. Am
0 1,200 138.60
1 960 110.88
2 720 83.16
3 480 55.44
4 240 27.72
5 0 0
Computation of B-values:
2 2
BO = AO + 2 = 138.60 + 2 = 0.014225
B1 = 2BO = 0.028450
B2 =-5-+---A:-----2-B-
1
= 5 + 110.88 = (0.028450)2 = 0.008634
B3 = B2(4-B1) = 0.008634(4~0.028450) = 0.034290
B = 1
4 6 + A2 - BO - B3 (4-B1)
1
= -------------------,,------:-- = 0.011235
6 + 83.16 - 0.014225 - 0.034290(4-0.028450)
B5 = B4(4-B3) = 0.011235(4-0.034290) = 0.044555
1
= = 0.016327
6 + 55.44 - 0.008634 - 0.044555(4-0.034290)
B7 = B
6
(4-B5) = 0.016327(4-0.044555) = 0.064581
1 1
B = = --------------------
8 6 + A
4
-B
4
-B7(4-B
5
) 6 + 27.72 - 0.011235 - 0.064581(4-0.044555)
= 0.029895
B9 = B
8
(4-B
7
) = 0.029895(4-0.064581) = 0.117649
85

1
= = 0.181127
6 + 0 - 0.016327 - 0.117649(4-0.064581)
Bll = B10(4-B9) = 0.181127(4-0.117649) = 0.703199 •
Computation of C*-values:
1 1 1
Ci =- =- =-~- = 5.520988
B2t B10 0.181127
C2= CiB2t+l - 2 - B2t_2(2 - B2t_3) = CiBll - 2 - B8(2-B7)
= 5.520988(0.703199) - 2 - 0.029895(2-0.064581) = 1.824494
C3= Ci - B2t_4 - B2t_l(2-B2t_3) = Ci - B6 - B9(2 - B7)
= 5.520988 - 0.016327 - 0.117649(2-0.064581) = 5.276961 •
Computation of y-values:
2Pth3 2(60,000) (1200 )3
b1 = = -5- = 11.55128
EI (30 x 106 )(4787)
11.55128(1 + 0.29895)
= 5.276961(1+0.029895) _ 1.824494(0.117649) = 2.2719015 in.
B2t-1(Yt) 0.117649(2.2719015)
Y6 = Yt+l = 1 + B = 1 + 0.029895 = 0.260341 in.
2t-2
Y7 = Yt+2 = Ci(B2t+1)(Yt +l) - CiY~ = CiBIIY6 - CiY5
= 5.520988(0.703199)(0.260341) - 5.520988(2.2719015) = -11.571676 in.
Y4 = -B
8
Y6 + B9Y5 = - (0.029895 x 0.260341) + (0.117649 x
2.2719015) = 0.259504 in.
Y3 = -B6Y5 + B7Y4 = - (0.016327 x 2.2719015) + (0.064581 x
0.259504) = - 0.0203344 in.
86

Y2 = -B4Y4 + B5Y3
= (0.011235 x 0.259504) +(0.044555 x
- 0.0203344) - - 0.0038215 in.
Y1 = -B2Y3 + B3Y2
= (0.008634 x -0.0203344) + (0.034290 x
- 0.0038215) = 0.0000442 in.
YO = -BOY2 + B1Y1 = - (0.014225 x -0.0038215) + (0.028450 x
0.0000442) = 0.0000556 in.
Computation of shear and moment at mudline:
V5 = 30 x 10
6
(4787) [-0.0203344 - 2(0.259504) + 2(0.260341) - (-11.532405)]
2(240)3
V5 = 59,800 lbs. Checks load at mudline.
EI )M5 = h2 (Y4 - 2Y5 + Y6
M = 30 x 10
6
(4787) [0.259504 _ 2(2.2719015) + 0.260341]
5 (240)2
= -10,060,000 ino-lbs
Dividing the pile into only 5 increments leads to serious errors
because of failure to represent properly the elastic curve. Using 50
increments, the va 1ues of moment and defl ecti on at the top of the pil e
were calculated to be
Mt = - 6,870,000 in.-lb,
Yt = 0.730 in.
As may be understood, care should be used in deciding the mesh size and
the number of significant figures to employ in solving the difference
equations.
5.5 DIFFERENCE EQUATIONS FOR CASE OF AXIAL LOADING AND
FOR CHANGES IN BENDING STIFFNESS
The solution procedure as presented by Gleser serves to illustrate
the difference-equation method and the equations, for the case of a coarse
87

mesh, can be readily solved by hand. But for many practical problems it
is necessary to address the effects of axial loading and changes in bend-
ing stiffness. Thus, the following derivation is presented but not in
detail as was done in Appendix 2.
The governing differential equation was presented in Chapter 2;
renumbered here for convenience.
EI f2 + p ~ + E y = 0
dx4 X dx2 S
(5.11)
Rewriting the first term of the equation with respect to moment and set-
ting Es to K for ease in writing, we have
d2M d2 v
- + P =-...t... + KY = o.
dx2 X dx2 (5.12)
The finite difference expressions for the first two terms of Eq. 5.12
at poi nt mare
and
where
(::~)m = [ Ym-2 Rm_1 + Ym-l (-2Rm - 2Rm_1)
p (~)X dx2
m
+ y (4R + R 1 + R 1)m m m- m+
= Px (Ym-1 - 2Ym + Ym+1) ,
h2
Rm =flexural rigidity at point (m), that is,
Rm = EmIM •
(5.13)
(5.13)
(5.14)
Substituting expressions from Eqs. 5.13 and 5.14 into 5.12 results in
the differential equation in finite difference form.
Ym-2Rm-1 + Ym-1 (-2Rm_1 - 2Rm + pxh2 )
+ Ym (Rm-1 + 4Rm + Rm+1 - 2Pxh2 + Kmh4) + Ym+1
(-2Rm - 2Rm+1 + PX
h2 ) + Ym+2 Rm+1 = 0 (5.15)
88

It is noted that the axial force P which produces compression is
x
assumed to be positive. It is also noted that P acts through the axis of
x
the pile; thus, P causes no moment at the top of the pile.
x
Applying the boundary conditions to the top and bottom of the pile,
the solution to Eq. 5.15 can proceed (Gleser, 1953).
Using the previous notation, the two boundary conditions at the bot-
tom of the pile (point 0) are zero bending moment,
(6) = 0,
dx 2
o
and zero shear,
R (.cSt.) +
o dx 3 0
p s!l = O.
x dx
O
For simplicity it is assumed that
R_
1
=R
O
=R
1
·
These boundary conditions are, in finite difference form,
y-1 - 2yO + Y1 = 0,
= Y-l (2 - P() -Yl (2 - P()+ Y2,
(5.16)
(5.17)
(5.18)
(5.19)
respectively. Using these boundary conditions in finite difference form
with Eq. 5.15 where m is equal to zero, and rearranging terms, results in
the following equations.
Yo =aOY1 - bOY2'
(5.20)
2RO + 2R1 - 2Pxh2
(5.21)
(5.22)
Equation 5.15 can be expressed for all values of m other than 0 and
the top of the pile by the following relationships:
Ym = amYm+1 - bmYm+2' (5.23)
am
-2b 1R + a b R + 2R - 2b 1R + 2R 1 - P h2 (1-b 1)= m- m-1 m-2 m-1 m-1 m m- m m+ x m-
(5.24)
89

c =m
_ Rm+1
b - -- andm c 'm
R - 2a R -m-1 m-1 m-1
(5.25)
(5.26)
Three sets of boundary conditions are considered at'the top of the
pile where m=t.
1. The lateral load CPt) and the moment CMt ) are known.
2. The lateral load (Pt ) and the slope of the elastic curve (St) are
known.
3. The 1atera1 load (Pt) and the rotationa l-restra i nt constant
(Mt/St ) are known.
For convenience in establishing expressions for these boundary con-
ditions, the following constants are defined:
P h2
X
E =--
R •
t
(5.27)
(5.28)
(5.29)
(5.30)
(5.31)
The difference equations expressing the first of the boundary condi-
tions for the top of the pile are:
Rt
2hil (Yt-2 - 2Yt _1 + 2Yt +1 - Yt +2)
+ Px
2h (Yt - 1 - Yt +1) = Pt , (5.32)
:; (Yt-l - 2Yt + Yt+1) = M
t
. (5.33)
90

After some substitutions the difference equations for the deflection at
the top of the pile and at the two imaginary points above the top of the
pile are:
(5.34)
Yt+1 =
(5.35)
(5.36)
where
QI HI
GIH2
(1 - at
G1 ) 1 (5.37)= +--+
G
2
bt 'G2
Q2 J
3
at J2 J
2H2 (5.38)= +-- - --G2bt G2
GI = 2 - a
t
_
1 (5.39)
G2 = I-b
t
_
1
(5.40)
HI = -2a 1 Ea
t
_
1
- b
t
_
2 + a
t
_
1
a
t
_
2 ' and (5.41)
t-
H2 = -a
t
_
2
b
t
_
1
+ 2b
t
_
1
+ 2 + E(1 + b
t
_
1
) (5.42)
The difference equations for the second set of boundary conditions
are Eqs. 5.32 and 5.43:
Yt - I - Yt +
1 =J 1. (5.43)
The resulting difference equations for the deflections at the three
points at the top of the pile are:
Q4
Yt - Q '
3
(5.44)
=
at
_
1
Yt - J1 , andYt +l G4
(5.45)
at Yt +l - Y
Yt +2 = t
b
t
(5.46)
where
H
2
a
t
_
1
a
t
a
t
_
1
1
Q3 = HI + +-
G4 b
t
G4 bt
(5.47)
91

J
1
H2 J
1at
Q
4
= J
3 + - - - -- , and
G
4
b
t
G
4
(5.48)
G
4
= 1 + b
t
_
1
, (5.49)
and the other constants are as previously defined.
The difference equations for the third set of boundary conditions are
Eqs. 5.32 and 5.50:
Yt -1 - 2Yt + Yt +1
---....::.-------- = J4•
Yt-1 - Yt +1
(5.50)
The resulting diference equations for the deflections at the three
points at the top of the pile are:
at 1
H + H2H3 - -- H +--1 bt 3 bt
(5•51 )
Yt +1
= Yt (G1
+ J
4
a
t
_
11 = H3Yt ' and
G2
+ J
4G
4
(5.52)
Yt +2 = ~t ( atyt+1 - Yt ) , (5.53)
where
G
1
+ J
4
a
t
_
1H = ~--~;...-:::;--=-
3 G2 + J
4G4
(5.54)
The other constants have been previously defined.
The di fferent i a1 equation can be rev; sed and di fference equations
can be written to deal with distributed loads, applied shear and moment at
any point along the pile, and inertial effects. A wide variety of prob-
lems can be addressed with the resulting nonlinear beam-column problem.
Matlock (1956, 1957, 1958, 1962, 1964) has given attention to problem sol-
utions using an approach similar to that presented herein.
5.6 COMPUTER PROGRAM COM622
A computer program, COM622, has been written for the beam-column
equations presented in Eqs. 5.12 through 5.54. A listing of the program,
input forms, sample input, and sample output are in Appendix 3.
92

Some comments pertaining to the use of the program are presented in
comment cards in the listing. The following comments may also be useful.
Limitations of the program:
• The pile is assumed to remain elastic;
• The ability to analyze group effects is not included;
• Information on response of complex soil profiles, e.g., layered
soils, is extremely limited at present;
• Time effects, such as those associated with sustained or cyclic
loading, are not directly included but can be taken into account
indirectly by adjusting p-y curves in some appropriate manner.
The prudent engineer should treat results of this program as an aid
in the overall process of engineering analysis and design, not as the sole
basis for design nor as the final word on how a laterally loaded deep
foundation will perform.
Computer program COM622 requires that p-y curves be computed accord-
ing to the procedures presented in Chapter 3; the tables of p versus y
then are used as input to COM622. Computer Program COM624 has been writ-
ten in which the criteria for generating p-y curves are subroutines of the
program and the engineer only has to specify soil properties, pile geom-
etry, and the kind of loading (static or cyclic). COM624 is being dis-
tributed only to a user l
s group in order that the program can be
maintained. An individual or firm who wishes information on the program
and user's group can obtain it by writing to: The Geotechnical Engineer-
ing Center, The University of Texas at Austin, College of Engineering,
'Cockre1l Hall 6.2, Austin, Texas 78712.
5.7 REFERENCES
Focht, John A. ,Jr., and McClelland, Bramlette, Analysis of Laterally
Loaded Piles by Difference Equation Solution, presented at Spring meet-
ing of the Texas Secti~n of American Society of Civil Engineers, Corpus
Christi, Texas, April 1955. Published later in three parts in The Texas
Engineer publication of the Texas Section of American Society of
Engi neers, 1955.
G1eser, Sol M., Lateral Load Tests on Vertical Fixed-Head and
Free-Head Pil es, II Symposium on Lateral Load Tests on Pil es, Ameri can Soci-
ety Testing Materials Special Training Publication No. 154, pp. 75-101,
1953.
Howe, R. J., A Numeri ca1 Method for Predi cti ng the Behavi or of Lat-
era lly Loaded Pil i ng, II TS Memorandum 9, Shell Oil Company, Houston, Texas,
1955.
93

Matlock, Hudson, Correlations for Design of Laterally Loaded Piles
in Soft Clay, a report to Shell Development Company, Houston, Texas, Sep-
tember 15, 1962, 71 pp.
Matlock, Hudson, and Haliburton, T. Allan, A Program for
Finite-Element Solution of Beam-Columns on Nonlinear Supports, a report
to the California Company, Shell Development Company, June, 1964, 171 pp.
Matlock, Hudson, and Ripperger, E. A., Measurement of Soil Pressure
on a La,tera lly Loaded Pi 1e, Proceedi ngs, Ameri can Soci ety for Testing
Materials, Vol. 58, pp. 1245-1259, Boston, Massachusetts, 1958.
Matlock, Hudson, Ripperger, E. A., and Fitzgibbon, Don P., Static
and Cyclic Lateral Loading of an Instrumented Pile, a report to Shell Oil
Company, Austin, Texas, 1956.
Matlock, Hudson, Ripperger, E. A., and Reese, L. C., Recommen-
dations Pertaining to the Design of Laterally Loaded Piles, a report to
Shell Oil Company, Austin, Texas, 1957.
Palmer, A. L., and Thompson, James B., Horizontal Pressures on Pile
Foundations, Proceedings, Third International Conference on Soil Mechan-
ics and Foundation Engineering, Rotterdam, Vol. 5, pp. 156-161, 1948.
5.8 EXERCISES
5.1 Derive equations for the case where the boundary conditions at
the pile head are Pt and Mt/St .
5.2 Derive equations for the case where the boundary conditions at
the pile head are Pt and St'
5.3 Solve the following example by hand computations using the dif-
ference equations and compare y and M with similar values for thet max
long pile case,
Pile:
using equations in Chapter 4.
Loading:
Soil :
Number of
24 in. in diameter by 1 in. wall thickness,
length =1200 in.
free-head case, Pt =40,000 lb
Stiff clay, constant modulus E =800 lb/sq in.
s
increments =6
(Note: the solutions should be identical except for the effect of length
and the effect of inaccuracy due to the crude mesh size.)
5.4 Repeat Problem 3 using COM622 and compare results.
5.5 Repeat problems 4.1 and 4.2 in Chapter 4 using COM622. Check
and compare results. Vary increment lengths and study errors.
94

CHAPTER 6. NONDIMENSIONAL METHOD FOR THE
ANALYSIS OF LATERALLY LOADED PILES
A nondimensional method for the analysis of laterally loaded piles
was presented in Chapter 4 for the case where the soil modulus is con-
stant. That solution has limited usefulness because almost never would
the value of soil modulus be constant over the pile length. There is some
value, however, in having nondimensional solutions for other variations
of soil modulus with depth. Some such solutions are developed in this
section.
While nondimensional solutions can be developed for problems where a
number of parameters are involved, a limited approach has been selected.
Pile stiffness is assumed to be constant and the effect of axial loading
is ignored. But nondimensional curves are developed, as shown below, for
a number of variations of soil modulus with depth and for piles of various
lengths (Matlock and Reese, 1962).*
Considering the nonlinearity of p-y relations at various depths, Es
is a function of both x and y. Therefore, the form of the E -versus-depth
s
relationship also will change if the loading is changed. However, it may
be assumed temporarily (subject to adjustment of Es values by successive
trial) that the soil modulus is some function of x only, or that
Es = Es(x). (6.1)
For solution of the problem, the elastic curve y(x) of the pile must
be determined, together with various derivatives that are of interest.
Ihe derivatives yield values of slope, moment, shear, and soil reaction as
functions of depth.
6.1 DIMENSIONAL ANALYSIS FOR ELASTIC PILES
The principles of dimensional analysis may be used to establish the
form of nondimensional relations for the laterally loaded pile. With the
use of model theory the necessary relations will be determined between a
prototype having any given set of dimensions, and a similar model for
which solutions may be available.
*Note: The derivations that are presented follow closely the referenced
paper.
95

For very long piles, the length L loses significance because the
deflection may be nearly zero for much of the length of the pile. It is
convenient to introduce some characteristic length as a substitute. A
linear dimension T is therefore included in the quantities to be consid-
ered. The specific definition of T will vary with the form of the func-
tion for soil modulus versus depth. However, it will be seen later that,
for eac~ definition used, T expresses a relation between the stiffness of
the soil and the flexural stiffness of the pile and is called the rel a-
tive stiffness factor.
For the case of an applied shear Pt and moment Mt the solution for
deflections of the elastic curve will include the relative stiffness fac-
tor and be expressed as
y = y(x, T, L, Es' EI, Pt
, Mt
). (6.2)
Other boundary values can be substituted for Pt
and Mt .
If the assumption of elastic behavior is introduced for the pile, and
if deflections remain small relative to the pile dimensions, the principle
of superposition may be employed. Thus, the effects of an imposed lateral
load Pt
and imposed moment Mt may be considered separately. If YA repres-
ents the deflection due to the lateral load P
t and
caused by the moment Mt , the total deflection is
if YB is the deflection
(6.3)y=YA+YB'
It is the ratios of YA
generalized solutions
to Pt and of YB to Mt which are sought in reaching
for the elastic pile. The solutions may be
expressed for Case A as
YA
p = fA(x. T, L, Es' EI) ,
t
and for Case B as
(6.4)
YB
~ = fB(x, T, L, Est EI) , (6.5)
t
where fA and fB represent two different functions of the same terms. In
each case there are six terms and two dimensions (force and length).
There are therefore four independent nondimensional groups which can be
formed. The arrangements chosen are, for Case A,
x
T
L
T
E T4
s
EI
(6.6)
96

and for Case B,
YB EI
t·1 T2
t
x
T
L
T EI
(6.7)
To satisfy conditions of similarity, each of these groups must be equal
for both model and prototype, as shown below.
x x
--.E.=-.!!!. (6.8)
T Tp m
L Lm
--.E.=_ (6.9)
Tp Tm
E T 4 E T 4
sp p sm m
=
(6.10)
EI EIp m
YA EI YA EI
P P m m= (6.11)
Pt
T 3 Pt
T 3
P m
p m
YB EI YB EIp m
p =
m (6.12)
M T 2 Mt
T 2
tp P mm
A group of nondimensional parameters may be defined which will have
the same numerical value for any model and its prototype. These are shown
below.
Depth Coefficient, Z - x
- T (6.13)
Maximum Depth Coefficient, z L
max =T (6.14)
E T4
Soil Modulus Function, /(z) = _s_
EI
(6.15)
A
YAEl
=--
Y P T3
t
(6.16)Case A Deflection Coefficient,
Case B Deflection Coefficient, B
YB EI
---
Y ~T2
(6.17)
97

Thus, from definitions 6.13 through 6.17, for (1) similar soil-pile stiff-
nesses, (2) similar positions along the piles, and (3) similar pile
lengths (unless lengths are very great and need not be considered), the
solution of the problem can be expressed from Eq. 6.3 and from Eqs. 6.16
and 6.17, as
= [L] A + [Mt T2 ]Y EI Y EI By. (6.18)
By the same type of reasoning other forms of the solution can be
expressed as shown below.
Slope, = [Pt T2] A + [~]. B
EI S EI S
(6.19)
Moment, (6.20)
Shea r, (6.21)
Soil Reaction, p = PA + PB = (6.22)
A particular set of A and B coefficients must be obtained as func-
tions of the depth parameter, Z, by a solution. of a particular model.
However, the above expressions are independent of the characteristics of
the model except that elastic behavior and small deflections are assumed.
The parameter T is still an undefined characteristic length and the vari-
ation of Es with depth, or the corres~onding form of ¢(Z), has not been
specifi ed.
While the relations derived above are applicable to step-tapered
piles which are frequently used in construction, it is necessary that
structural similarity be maintained between the mathematical model and
the prototype.
From beam theory, as presented earlier, the basic equation for an
elastic beam is
EI f2 = p.
dx4
98
(6.23)

Introducing the definition of p = -E y, the equation for a beam on an
s
elastic foundation, or for a laterally loaded pile, is
4 E
!1 + 2. y = O.
dx4 EI
(6.24)
Where an applied lateral load Pt and an applied moment Mt are consid-
ered separately according to principle of superposition, the equation
becomes, for Case A,
d
4
YA Es
dx4 + EI YA = 0, (6.25)
and for Case B,
(6.26)
Substituting the definitions of nondimensional parameters contained in
Eqs. 6.13 through 6.17, a nondimensional differential equation can be
written for Case A as
d4A
~ + ~(Z)A = 0,
dZ Y
(6.27)
and for Case B as
d4B
~ + ¢ (Z)B = O.
dZ Y
(6.28)
To produce a particular set of nondimensional A and B coefficients,
it is necessary (1) to specify ~(Z), including a convenient definition of
of the relative stiffness factor T, and (2) to solve the differential
equations (6.27 and 6.28). The resulting A and B coefficients may then be
us?d, with Eqs. 6.18 through 6.22, to compute deflection, slope, moment,
shear, and soil reaction for any pile problem which is similar to the case
for which nondimensional solutions have been obtained.
Based on the boundary conditions Pt and Mt and the resulting A and B
coefficients, relations have been derived so that problems may be solved
for cases in which other boundary conditions are known. As shown later,
nondimensional relationships have been derived so that almost any con-
ceivable structure-soil-pile problem can be solved.
To obtain the A and B coefficients that are needed to make solutions
with the nondimensional method, Eqs. 6.27 and 6.28 can be solved by use of
99

di fference equations. Coeffi ci ents for other types of boundary condi-
tions can be obtained in a similar manner.
6.2 DIMENSIONAL ANALYSIS FOR RIGID PILES
Piles or posts having relatively shallow embedment are frequently
encountered in practice. Such piles behave essentially as rigid members,
and the difference-equation method used in the elastic-theory solutions
may become inaccurate because of the small successive differences which
are involved. For such cases, a simpler theory is applicable, in which
the pile is considered to be a rigid member (Matlock and Reese, 1962).
Although computations are simpler for the rigid pile than for the
elastic pile, it is still convenient to use generalized. solutions and to
consider separately the effects of applied lateral load and applied
moment.
In the derivation of the equations for the rigid pile it is conven-
ient to include an additional term J that is later given particular defi-
nitions. The definitions of J depend on the form of the variation of the
soil modulus with depth. For the present, J is simply a constant having
the same dimensions (force x length-2
) as the soil modulus E .. s
For either Case A (M
t
= 0) or Case B (Pt = 0) there are a total of six
factors to be considered. For Case A,
YA =YA (x,L,Es,J,Pt ), (6.29)
and for Case B,
(6.30)
In each trial computation in an actual design problem, the soil is consid-
ered to be elastic. Thus, for either Case A or Case B, it is the ratio of
deflection to loading which is sought in reaching generalized solutions.
This reduces the number of nondimensional groups to three. For Case A
these are
yAJL x E
s (6.:n)
Pt
' L ' J
,
and for Case B,
y JL2 X E
B s (6.32)
Mt
' L J
For similarity between a prototype and a computed model, each nondimen-
sional group may be defined as a dimensionless parameter. These are as
shown below.
100

Depth Coefficient,
Soil Modulus Function,
Case A Defl ection
Coefficient,
Case B Deflection
Coeffi ci ent,
By superposition, the total deflection is
Y=YA +Y
B
x
h = I
jJ(h)
a
y
J
From reasoning similar to the above, other forms of the
be expressed as showh below.
Slope, S = SA + SB = [:Lt2] as +LML] bs
Moment, M= MA + MB = [PtL] am + [Mt] bm
Shear, V = VA + VB = [pt ] av + [ MLtJ bv
Soi 1 Reacti on, p = PA + PB = [~]a +[Mt]bL P L2 P
(6.33)
(6.34)
(6.35)
(6.36)
(6.37)
(6.38)
solution can
(6.39)
(6.40)
(6.41)
(6.42)
For any given problem the slope (S =dy/dx) is a constant and all higher
derivatives of yare zero. The last three expressions are related to the
first two through the relation between soil reaction and pile deflection,
E =-p (6.43)
s y
or, in terms of the nondimensional coefficients,
-a -b
jJ(h) = ~ = ~
a b
(6.44)
y Y
101

The above dimensional analysis will apply to any form of the soil modulus
functions Es or ¢(h). The soil modulus constant J is to be defined subse-
quently.
The nondimensional soil modulus function ¢(h) is equivalent to the
corresponding function ¢(Z) used with the elastic-pile theory except that
¢(h) is related to the length of the pile rather than to a relative stiff-
ness between the pile and the soil.
For any given ¢(h), there exists a single set of nondimen-
sional-coefficient curves (for deflection, slope, moment, shear, and soil
reaction). Design problems may be solved by essentially the same proce-
dures as for the elastic-pile case. The choice of which theory to use is
aided by comparing the results of nondimensional solutions obtained by the
two methods.
The equation for deflection y of a rigid pile is
y=Yt+Sx (6.45)
where Yt is the deflection at x = 0 and S is the constant slope of the
pile. The soil reaction is
p = -EsYt - Es S x. (6.46)
By statics, the equation for shear is
V= Pt + J ~ pdx. (6.47)
Substituting the expression for p in Eq. 6.46 into Eq. 6.47 yields
V= Pt - Yt JoXEsdX - S J~ xEsdx.
The equation for moment is
x
M= Mt + Vx - J 0 xpdx,
or,
(6.48)
(6.49)
M= M + Vx + y J x xE + S J x X2E dx. (6.50)
t t 0 s 0 s
The shear and moment are zero at the bottom of the pile. Thus, the
following equations may be written from Eqs. 6.48 and 6.50 so that Yt and
S may be evaluated by simultaneous solution.
P
t
= Yt J LE dx + S J LxE dx
0L s O L s
Mt = -Yt J 0 xEsdx - S J0 x2Esdx
The values obtained for Yt and S are then
and 6.50 to complete the solution.
(6.51)
(6.52)
substituted into Eqs. 6.48
As in the procedure used in the elastic-pile theory, unit values may
be introduced into the solution to obtain numerically correct values of
the nondimensional coefficients defined in Eqs. 6.33 through 6.36. This
amounts to determining the nondimensional coefficients from the results
102

of a numerically convenient model having unit values of L, P
t
, and M
t
.
Coefficients in Es(x) are chosen to agree with those in the soil modulus
function ¢(h), and J is thus made equal to unity.
6.3 FORMS OF VARIATION OF SOIL MODULUS WITH DEPTH
In solving problems of laterally loaded piles by using nondimen-
sional methods, the constants in the expressions describing the variation
of soil modulus Es with depth x are adjusted by trial until reasonable
compatibility is obtained. The selected form of the soil modulus with
depth should be kept as simple as possible so that a minimum number of
constants needs to be adjusted.
Two general forms are a power form,
E = kxn
s
and a polynomial form,
Es =ko + k1x + k2x2
•
The form Es = kx is seen to be a special case of either of these.
similar to Eq. 6.53 has been suggested previously (Palmer and
1954).
(6.53)
(6.54)
A form
Brown,
The relative stiffness factor T of the elastic-pile theory and the
soil modulus constant J of the rigid-pile theory must be defined for each
form of the soil modulus-depth relation. While T and J may be defined in
any way, it is convenient to select definitions that will simplify the
corresponding nondimensional functions.
From the elastic-pile theory, Eq. 6.15 defining the nondimensional
function for soil modulus is
E T4
¢(Z) =_5_
EI
If the form E =k xn is substituted in Eq. 6.15, the result is
s
() k n 4
¢Z =ETxT. (6.55)
For the elastic-pile case, it is convenient to define the relative stiff-
ness factor T by the following expression.
Tn+4 = I!.
k
Substituting this definition into Eq. 6.55 gives
x
n
T4
. [-Tx ] n.¢(Z) = Tn+4 =
103
(6.56)
(6.57)

Because x / T = Z, the general nondimensional function for soil modulus is
~(Z)=Zn. (6.58)
The above expression contains only one arbitrary constant, the power n.
Therefore, for each value of n which may be selected, one complete set of
independent, nondimensional solutions may be obtained from solution of
Eqs. 6.27 and 6.28. For relatively short, elastic piles, separate compu-
tations must be made for each Z considered.
max
Appendix 4 contains a number of tables for making solutions using
nondimensional methods. Tables A4.1 through A4.6 are for elastic piles
and for the case where E = kx n. One of the tables is for the case where n
s
= 0, or Es is a constant, and another table is for the case where n = 1, or
Es = kx. All of the tables are for the case where the nondimensional
length of the pile is 10, that is, the pile acts as a long pile. A cover
page is placed with the tables to provide a summary of the important
equations and to show the general shape of the functions. Five additional
sets of nondimensional coefficients are referenced subsequently; the same
general format was employed in presenting each set of tables.
From the ri gi d-pil e theory the function for soil modul us has been
defined by Eq. 6.34 as ~(h) = E /J. If the soil modulus constant J is now
s
defined as
J = kL n (6.59),
the corresponding general nondimensional function for soil modulus is
n
p(h) = ~ (6.60)
kL
n
or, since h = x / L,
~(h)=hn. (6.61)
Only one set of nondimensional curves will be needed for each selected
value of n, regardless of the length L.
Nondimensional coefficients for rigid piles for the case where E =
s
kx
n
are presented in Tables A4.7 through A4.12 in Appendix 4.
When a polynomi ali s used to express the form of the soil modul us
variation with depth, the relative stiffness factor T, or the soil modulus
constant J, may be defined to simplify only one of the terms in the poly-
nom; a1•
For the elastic-pile case, introducing the polynomial form into Eq.
6.15 gives
104

kT4 k T5
o 1
=--+--
EI EI
x
- +
T
k T6
2
EI
(6.62)
To simplify the second term, as an example, T may be defined by the fol-
lowing expression.
T5 = II.
k1
The resulting soil modulus function is
where
t(l)=ro +l+r
2
P
_ ko 1
ro -kl T
k2
r 2 = kl [T)
, and
For the rigid pile theory, from Eq. 6.34,
ko k1 x k2x2
p(h) = - + - + - - ••••
J J J
Again to simplify the second term, J is defined by
J = k L
1
and
or
t(h) = ro + h + r2 h2
where h = x / Land
k
ro =_0 ,and
k1L
_ k2L
r 2 - -k- .
1
(6.63)
(6.64)
(6.65)
(6.66)
(6.67)
(6.68)
(6.69)
(6.70)
(6.71)
(6.72)
A separate set of nondimensional curves would be needed for each
desired combination of r-constants. Because of the complexity which oth-
erwise would result, it does not appear reasonable to vary more than one
constant and such forms as those following appear to be about as compli-
cated as should be considered.
t(l) =r + l
o
t(l) = r + Po
(6.73)
(6.74)
While it would be permissable for some of the r-constants to have negative
values, care must be taken that p does not become negative.
105

Nondimensional coefficients for elastic piles for the case where E =
s
ko + k1x are presented in Tables A4.13 through A4.18. Nondimensional
coefficients for rigid piles for the case where Es = ko + k1x are pre-
sented in Tables A4.19 through A4.24.
Because of the utility of employing the relatively simple variation
of Es with depth, Es =kx, tables are included in Appendix 4 for that form
of variation of E. The coefficients are presented in Tables A4.25
s
through A4.40. The next section gives some details on solutions with Es =
kx; in that section the nondimensional coefficients for Es =kx are shown
as curves.
6.4 SOLUTION PROCEDURES
Reese and Matlock (1956) presented several arguments for the use of
E = kx as a viable variation in the soil modulus with depth. McClellands
and Focht (1958) made
test, and as noted in
as defining the early
use of E = kxs
in analyzing the results of a field
Chapter 3, E = kx is recommended in some instances
s
part of p-y curves. Further, an examination of the
recommended families of p-y curves reveals that the ultimate resistance is
always lower at the ground surface for soils with constant shear strength
with depth. That fact, coupled with the fact that computed deflections
are larger near the ground surface leads to the idea of an increasing soil
modulus with depth (but does not necessarily suggest a zero modulus at the
ground surface for clays).
A number of authors have suggested the use of Es =kx and the nondi-
mensional curves presented herein in solving the problem of the pile under
lateral loading (Department of Navy, 1971; George and Wood, 1976; Poulos
and Davis, 1980).
As was shown earlier, solutions are available for more complicated
variations in E with depth than E =kx; however, the use of E =kx is as s s
favorable choice at least for the initial computations.
Prior to initiating the solution procedures, it is desirable to reit-
erate the limitations of the nondimensional method: the effect of axial
load on bending cannot be investigated, and the pile stiffness must be
assumed to be constant. Of course, all of the limitations imposed during
the derivation of the differential equation, e.g., no shear distortion and
small deflections, are also present. The solution procedure is described
below for three sets of boundary conditions at the top of the pile:
106

1) pile head free to rotate, 2) pile head fixed against rotation, and 3)
pile head restrained against rotation (Reese and Matlock, 1956).
Case I - Pile Head Free to Rotate
1. Construct p-y curves at various depths by procedures recom-
mended in Chapter 3, wi th the spaci ng between p-y curves
being closer near the ground surface than near the bottom of
the pil e.
2. Assume a value of T, the relative stiffness factor. The
relative stiffness factor is given as:
5
T = IEI/k (6.75)
where
EI = flexural rigidity of pile, and
k =constant relating the secant modulus of soil
reaction to depth (Es =kx).
3. Compute the depth coefficient Z ,as follows:max
x
= max
zmax T
(6.76)
4. Compute the deflection y at each depth along the pile where
a p-y curve is available by using the following equation:
P T3 M T2
Y =Ay -tr-+ By -tr-where
Ay =deflection coefficient, found in Fig. 6.1,
Pt = shear at top of pile,
T = relative stiffness factor,
B =deflection coefficient, found in Fig. 6.2,y
M
t = moment at top of pi 1e, and
EI = flexural rigidity of pile.
(6.77)
The particular curves to be employed in getting the Ay and
B coefficients depend on the value of Z computed in Stepy max
3.
5. From a p-y curve, select the value of soil resistance p that
corresponds to the pile deflction value y at the depth of
the p-y curve. Repeat this procedure for every p-y curve
that is available.
107

5.0
:
4.0
~
3.0 f-
-
«
K~,:
.
~
~
-c
.~
2.0u
--~
U
c
0
f-
- 1.0u
~
-CI.I
0
0.0
-1.0 f-
-2.0
0.0
I I
.-
.-
.-
.-
~
~
.~,
~.
.-
~. ~~
~~
-. .- .- [)810, 4.......
3
Zmax= 2
.
o-
J
1.0 2.0 3.0 4.0 5.0
Depth Coefficient, Z
3
Y=A (Pt T ) x =Z(T)
AYE I
X
where T=(El/k)
1/15
Fig. 6.1. Pile deflection produced by lateral load
at groundline (Reese and Matlock, 1956).
108

5.0~
4.0 ~
'.
'.
0 3.01-'·.
CD
~
-c
.!!
.~
-~2.01-
o
u
-u
CD
-CD
o
l'
1
j)~'
o,.
.....
-I.0-
-201
0.0
1
'.
'.'.
~, .
:--.. '.
~~
Zmox= 2
1
1.0
~
1 1
.-
-
-
-
4,5810
----3
,-
I I 1
2.0 3.0 4.0 5.0
'Ii =8 (MtT~ x=Z(T)B 'I E I oj
115
where T =(E Ilk)
Fig. 6.2. Pile deflection produced by moment applied
109

6. Compute a secant modul us of soil reaction Es using Es =
-ply. Plot the Es values versus depth.
7. From the Es vs. depth plot in Step 6, compute the constant k
which relates E to depth (k = E Ix). Give more weight tos s
the Es values near the ground surface.
8. Compute a value of the relative stiffness factor T from the
value of k found in Step 7. Compare this value of T to the
va1ue of T assumed inStep 2. Repeat Steps 2 through 8
using the new value of T each time until the assumed value
of T equals the calculated value of T.
9. When the iterative procedure has been completed, the values
of deflection along the pile are known from Step 4 of the
final iteration. Values of soil reaction may be computed
from the basic expression: p = -E y. Values of slope,
s
moment, and shear along the pile can be found by using the
following equations:
and
P T2 MtT
S=A _t_+B
s EI s EI
M
V=Ap +B -1v t v T
(6.78)
(6.79)
The appropriate coefficients to be used in the above
equations may be obtained from Figs. 6.3 through 6.8.
Case II - Pile Head Fixed Against Rotation. Case II may be used to
obtain a solution for the case where the superstructure translates under
load but does not rotate and where the superstructure is very, very stiff
in relation to the pile.
1. Perform Steps 1, 2, and 3 of the solution procedure as for
free-head piles, Case I.
2. Compute the deflection y at each depth along the pile where
a p-y curve is available by using the following equation:
P T3
YF = Fy -iI- . (6.80)
The defl ection coeffi ci ents Fy may be found by enteri ng
Fig. 6.9 with the appropriate value of z .max
110

't
-c:Q)
u
--!
u
Q)
Co
0
u;
0.0
·0.5f-
-1.0 ~
r-
~.' ,/
-1.5
-
-2.0
-2.5 f-
-3.0 -
.....
-3.5
0.0
I
j
/'/'/I. /
~. /
~ l/
i/./'. ~
/
/
Zmox =2
.......
I
1.0
I I
5
/
~.-. 10
i·
1.1# ..7.-
---r .
ji 10-
3
/
//
-
..
.-
.........
i i I
2.0 3.0 4.0 5.0
Depth Coefficient. Z
x
S=A(PtT
3
) x=Z(T)
'A S E I
I/~
where T =(El/k)
Fig. 6.3. Slope of pile caused by lateral load
at groundline (Reese and Matlock, 1956~
111

0.0 I
I .,
.- _.-:...- . 5
- --
6al~ 10
A' ... ~
---
~
I' 3
./
~
A~/
'-f,
t/
-0.5
,,/
fo-
/1
-
/',1
~
-1.0
)~l
~:I -m-1.5
.
-c
Q)
u
--8 -2.0fo- .' .-
u
Q)
Q.
o
(/)
-2.5fo-
-3.0 fo-:
-3.5
0.0
:
J
1.0
Zmox = 2
.-
'-
I I
2.0 3.0 4.0 5.0
S =8 (MtT) x =Z(T)
e s E I
x where T =(E II k)
115
Fig. 6.4. Slope of pile caused by moment applied at
groundline (Reese and Matlock, 1956).
112

:IE
«~
-c:
.!.!
.J
--Q)
0
u
-c:Q)
a~
, ,
I.0- .-
o.8- .-
r '-
o.
1/' '
i6r-
/1
-
:/....

o.
'.
I
4- / , -,

o.2)
~'.
~
~
.-
j
ic
,f '.
':~Zmax =2 . 3' 4'.~~ ~
0 -
rto-
o.
o.21- -
o.4 I I
0.0 1.0 2.0 3.0 4.0 5.0
MA=At.1·( PtT) x =Z(T)
x I/~
where T =(EI Ik)
Fig. 6.5. Bending moment produced by lateral load at
113

::::E
(JJ
-c::
.!!:!
.J
--Q)
0
u
-c::
Q)
E
0
:i
1.0
0.8 I-
0.6I-
0.4~
0.2 I-
0.0
0.2 -
0.4
0.0
~
I I
-
~.....~
~.
.~
.-
'.~
.~ .-
',
~.
'.,~
--
l
'.
, .~
. .-
1.,

,~.
Zmox =2 3~~' .4 ~,..-
r::..- - - -1--
-
I I i i
LO 2.0 3.0 4.0 5.0
Depth Coeffici ent, Z
MB=BM(Mt ) x =Z(T)
x where T =(Ell k)
1/5
Fig. 6.6. Bending moment produced by moment applied at
114

I I
I.6- .-
I.2- .-
o.S[:'.l -
u O.4f-
--8u
~
o
~ o.(J)
G
-0.4....
-0 .8-
2-0.1
0.0
~
'. ..
.-
..
~
.. ',. 4/~
i'
3,
'. ~
, I~.p
'.~ ;
I
V~.
~
,
I.A
.- : ........ ..:;:0:'[i....
~
: ,
- .,,-
Zmax :: 2
-
i ,
1.0 2.0 3.0 4.0 5.0
VA=Ay(Pt x =Z(T)
x
I'll
where T =(EI Ik)
Fig. 6.7. Shear produced by lateral load at
115

I I I -T
0.4f- -
0.2 f-o -
5
i- -::::: .......
0.0 ~p
r I
~I
4 ~.. I

II)
i-O.2
cu
u
;;:
-cu-
8
... -0.4
o
cu
55
-0.6
-O.S
-
-
f-o
-
-0.4
0.0
..~ I
7~ I
~ I
-
' // ~

~
r'. ~
~ v;/ -
~'.
:/3
: !/
..- .-'.
: :
'.
j Zmax =2'.
:
.-
:
..'
I I I I
1.0 2.0 3.0 4.0 5.0
Depth Coefficient. Z
)( VB Bv~~t) X =Z(T)
where T =(E 1/ k)
1/5
Fig. 6.8. Shear produced by moment applied at
116

1.0
0.8
0.6
--c:
Q)
:5;! 0.4
--8u
.~
~ 0.2
;::
Q)
o
0.0
-0.2
r-, .I- .,
~.
~.
,I-
I-
I-
f-
....
-04
0.0
I T I
-
['.
y.
~
I
I
'. I
_~.
.-
l~

~ .-~
.
5 alo~.
~
~
.-
.•..,'. II.
, .-
Zmox=2
~ 10,

..:::--
:-. 5
4.......·.....

3 .-
j I I I
1.0 2.0 3.0 4.0 5.0
x
y =F ( Pt T 3) X=Z (T)
F Y E I
115
where T =(El/k)
Fig. 6.9. Deflection of pile fixed against rotation
117

3. The solution proceeds in steps similar to those of Steps 5
through 8 for the free-head case.
4. Compute the moment at the top of the pile M
t
from the fol-
lowing equation:
Mt = Fmt Pt T
The value of Fmt may be found by entering Table 6.1 with the
appropriate value of zmax.
TABLE 6.1. MOMENT COEFFICIENTS AT TOP OF PILE
FOR FIXED-HEAD CASE.
Zmax F
mt
2 -1.06
3 -0.97
4 -0.93
5 and above -0.93
5. Compute values of slope, moment, shear, and soil reaction
along the pile by following the procedure in Step 9 for the
free-head pil e.
Case III - Pile Head Restrained Against Rotation. Case III may be
used to obtain a solution for the case where the superstructure translates
under load and where pile-head rotation is restrained.
1. Perform Steps 1, 2, and 3 of the sol ution procedure for
free- head piles, Case I.
2. Obtain the value of the spring stiffness ke of the pile
superstructure system. The spring stiffness is defined as
follows:
where
Mt =moment at top of pile, and
St =slope at top of pile.
118
(6.83)

3. Compute the slope at the top of the pile St as follows:
Pt T2 MtT
St = Ast ~ + Bst EJ: (6.84)
where
Ast =slope coefficient, found in Fig. 6.3, and
Bst =slope coefficient, found in Fig. 6.4.
4. Solve Eqs. 6.83 and 6.84 for the moment at the top of the
pil e Mt .
5. Perform Steps 4 through 9 of the sol uti on procedure for
free-head piles, Case I.
This completes the solution of the laterally loaded pile problem for
three sets of boundary conditions. The solution gives values of
deflection, slope, moment, shear, and soil reaction as a function of
depth. To illustrate the solution procedures, an example is presented.
6.5 SOLUTION OF EXAMPLE PROBLEM, ASSUMING Es = kx
Find the deflection, moment and shear as a function of depth along a
pile that is free to rotate and is subjected to a horizontal force and a
moment. The p-y curves are to be constructed at 0, 2, 4, 8, 12, 16, and 24
ft. The soil is a stiff clay above the water table. Other data for the
problem are shown below.
Pt =35,000 lbs
Mt =3.02 x 10 7
in.-lbs
L =60 ft
b =2 ft
EI =7.39 x 10 10
lb-sq in.
c =1,000 lb/sq ft
~ =110 lb/cu ft
N=1,000 cycles
The solution will proceed in the step-by-step manner as described for
Case I.
1. Construct p-y curves.
- Assume £50 =0.01 in the absence of stress-strain curves.
- Compute p as the smaller of the values from Eqs. 3.18 and
u
3.19 for depths of 0, 24, 48, 96, 144, 192, and 288 in.
- Compute Y50 from Eq. 3.20 and compute points on the p-y
curves for short-term static loading using Eq. 3.37.
119

- Compute y values for cyclic loading by use of Eq. 3.39 The
results of the computations are shown in Table 6.2 and in
Fig. 6.10.
TABLE 6.2. COMPUTED p-y CURVES*.
Depth, in. 0 24 48 96 144 192 288
Ystatic Ycyclic p, lb/in.
0.000 0.000 0 0 0 0 0 0 0
0.001 0.003 51 63 75 99 123 147 152
0.015 0.04 100 123 147 195 243 291 299
0.24 0.67 199 247 294 390 485 580 596
0.60 1.68 250 310 370 490 610 730 750
1. 24 3.48 300 372 444 588 731 875 899
2.50 7.00 357 443 529 700 872 1043 1072
5.00 14.00 425 527 629 833 1036 1240 1274
9.60 26.88 500 620 740 980 1220 1460 1500
*p-y cyclic curves are plotted in Fig. 6.10
2. Assume T: T =125 in. The corresponding k is 2.42 cu in.
x 60(12)
3, Compute z • z =~ = = 5.76. max' max T 125
4. Compute the deflection y at depths of 0, 2, 4, 8, 12, 16,
and 24 ft using Eq. 6.77. (Use Figs. 6.1 and 6.2; the com-
putations are tabulated in Table 6.3.)
5. From the set of p-y curves (Fig. 6.10 and Table 6.2), the
values of p are selected corresponding to the y-values com-
puted in Step 4. (See tabulation in Step 6.)
6. Compute the E value at each depth (see Table 6.4).
s
120

':'
C
........
.c
-0.
..Q)
u
c
c
iii
en
Q)
a::
0
(f)
X=
1500
1000
Deflect ion, y (in.)
288n.
192 il.
144in.
96 in.
48 in.
24 in.
oin.
Fig. 6.10. Plot of p-y curves for example problem, stiff clay
above water table, cyclic loading.
7. A plot of E vs. depth is shown in Fig. 6.11. The k values
is:
500
k = Es/x = 303 = 1.65 lblcu in.
8. Compute T:
T = 5/E1/k = ij7.39 x 10
10
= 136' 1 126 .1.65 In. In.
121

TABLE 6.3. COMPUTED DEFLECTIONS.
x, in. z Ay YA' in. By YB' in. . *y, In.
0 0 2.40 2.22 1.62 10.34 12.56
24 0.19 2.07 1. 91 1.29 8.24 10.15
48 0.38 1. 78 1.65 1.02 6.51 8.16
96 0.77 1. 23 1.14 0.58 3.70 4.84
144 1.15 0.76 0.70 0.25 1.60 2.30
192 1. 54 0.38 0.35 0.05 0.32 0.67
288 2.30
P T3 MT2
*y t t (6.77)= Ay-rr- + By -rr-
TABLE 6.4. COMPUTED VALUES OF SOIL MODULUS.
x, in. y, in. p, lb/in. Es ' 1b/sq in.
0 12.56 411 33
24 10.15 481 47
48 8.16 546 67
96 4.84 631 130
144 2.30 652 283
192 0.67 580 866
288
122

.c
)(
E • lb/SQ in.
o 100 200 400 500 600
o~~----~------~------~------~~------~----~
100
200
300
400
+0
'+
- - - 0 Trial I
--- + Trial 2
-

o
+
-
-
-
-
-
-

500~------L-------L-------L-------~------L-----~
Fi g. 6.11. Trial plots of soil modulus values (The first trial
corresponds to computations in Table 6.4.).
This completes the first iteration of tpe solution procedure. Before
proceeding to the next iteration, the results thus far will be examined
for guidance with regard to further computations.
It is evident from Fig. 6.11 that E = kx is not a good represen-
s
tation of the variation of the soil modulus with depth. A straight line
passing through the origin does not fit the plotted points. At this point
it could be desirable to use the nondimensiona1 solutions based on a power
or polynomina1 function (these solutions are presented later in this chap-
ter). However, the solution will proceed by use of the nondimensiona1
123

, !
curves based on Es = kx in order to gain an approximate idea of the final
design.
The solution has not been found because the k that was tried is not
equal to the k that was obtained. Rather than making the next trial with a
T of 136 in. (the value obtained in the first trial); it is preferable to
select a larger value in order to speed the convergence; so a value of 145
in. is selected for the next trial. The selected value of T for the second
trial corresponds to a k of 1.15 lb/cu in.
The computation of values of soil modulus for the second trial are
not shown but proceed as shown in Tables 6.3 and 6.4. The plot of the val-
ues of soil modulus for Trial 2 is shown in Fig. 6.11. The value of k was
found to be 1.20 lb/cu in., leading to a value of T of 144 in. The values
of T that were tried and those obtained for each of the trials are shown in
Fig. 6.12. As may be seen in the figure, convergence was achieved with a
value of T of 143 in.
150
O+~i2
140 Trial 1
~
' Final T =143 in.
,......,+
130
130 140 150
Fig. 6.12. Interpolation for final value of relative
stiffness factor ~
124

9. Compute the values of moment and shear using Eqs. 6.79 and
6.80 (see Fig. 6.13). Also shown in Fig. 6.13 are plots of
the moment and shear diagrams from a computer solution of
the example problem.
As may be seen, excellent agreement is found between the computer
solutions and the nondimensional solutions. This good agreement may be
fortuitous; however, it is not unusual to get reasonably good agreement
between solutions by computer and those by the nondimensional procedure.
The example is presented to illustrate the computational procedure
and not as an exercise in design. As noted earlier, computations should
be performed for a number of loads in making a design and parametric stu-
dies are desirable.
6.6 SOLUTION OF EXAMPLE PROBLEM, Es = kx
n
AND Es = kO + kl x
As shown in Fig. 6.11, the computed values of Es vs. x failed to pass
through the origin but, except for that, seemed possibly to follow a para-
n
bolic curve. Therefore, it was decided to try the use of E =kx and to
s
see if a better fit to the computer solution could be obtained.
The first step was to use T of 143 in. and to compute the Es values at
points where p-y curves were available. These points are plotted in Fig.
6.14. The next step was to find the best fit of a parabola through those
points (shown as circles in Fig. 6.14). The analytical fitting of a para-
bola proved unproductive because, in spite of appearances, the points from
the Es = kx solution do not follow the parabolic equation.
The next step was to make some trial fits with selected values of k
and n. This latter procedure is perhaps preferable to analytical fitting
because the engineer can insure a better fit near the groundline where
soil resistance has more influence on pile behavior. Two trials were made
with different values of k and n and the results are shown in Fig. 6.14.
The values in Trial 2 were thought to be preferable.
Tables A4.1 through A4.6 were then consulted and it was found that a
table was not available for a value of n of 1.2. Tables were available for
n = 1 (E = kx) and n = 2.s
of k that would yield the
It was decided to use n =2 and to find values
best fit to E from p-y curves. An alternate
s
procedure would have been to use double interpolation; that is, interpo-
lating with z as an argument using tables with n = 1 and n = 2, and then
interpolating again with n as the argument. This latter procedure did not
125

-15
I
-15
I
-30
V o
o
o
o
o
o
Y(in.)
-10 -5 0 5 10
I I I I I
V (lb x 104)
-10 -5 0 5 10
I I I I I
-20
M(in.-Ib x 104 )
-10 o 10 20
o
o
Depth, in.
Fig. 6.13. Deflection and moment diagrams
for example problem.
126
15
I
15
I
30

.c:
)(
o 100 200
200
300
Es ' Ib/sQ in.
300 400 500
o E =kx (T =I. 43 )
+ Trial I : k=0.7 t n= 1.1
6 Tria12: k =05, n =1.2
o
+
seem attractive. Of course, the computer program could be used to develop
additional nondimensional coefficients; however, it seems undesirable to
have large numbers of curves or tables.
Several trials were made with the table for n =2 and it soon became
apparent that the use of Es =kx 2
was not yielding a good fit at all to the
early part of the points for E versus x from the p-y curves. The computed
s
curves for n = 2 are relatively quite steep near the origin; thus, the
value of E at x =0 (whlch is almost constant) could not be fitted and the
s
fitting was deviating further and further from other values near the
n
groundl i ne. Therefore, the use of E = kx was abandoned for the examp1es
problem.
The procedure continues with the use of Es = kO + k1x. A trial was
made by fitting a straight line, not passing through the origin, through
points shown for Es = kx in Fig. 6.14. The value selected for kO was 20
lb/sq in. and for k1 was 0.93 lb/cu in. With these values the following
127

computations were made:
T = EI 1/5 = 151.37 in.
~
kO
0.142--
kIT
Z = 4.75.
max
The long pil e sol ution coul d be used and values were taken from Tables
A4.13 and A4.14. The double interpolation for values of Ay and By is
shown in Table 6.5. The values of E shown in Table 6.5 agree well withs
the values computed using E = kx with a T of 143 in. Therefore, it was
s
decided to make no further trials.
Tables A4.13 and A4.14 were used to interpolate values of the coeffi-
cients for shear and moment. Computations were made and curves of com-
puted deflection, moment, and shear are shown in Fig. 6.15. Also shown in
the figure are similar curves obtained from the computer program.
An examination of the figure shows that the more complicated form of
soil modulus with depth gave little or no improvement. The curves for
deflection, moment, and shear as computed by Es = kO + k1x are very close
to those computed with Es =kx.
6. 7 ALTERNATE SOLUTION TO RESTRAINED-HEAD CASE
Earlier in this chapter a procedure was presented for dealing with
the problem where the pile head is attached elastically to the superstruc-
ture. Thre is an alternate solution to the restrained-heqd problem, as
will be shown in this section. The solution is presented in connection
with an example (Matlock and Reese, 1961). The solution is developed for
the case where Es = kx. Solutions for other variations of Es with x are
possible but the desirability for these other solutions is questionable.
A typical offshore structure is shown in Fig. 6.16. While an off-
shore structure is used as an exmple, the method applies equally well to a
bridge bent. The specific problem considered is that of solving for the
bending moments in the portion of the structural system which lies beneath
the soil surface. In erecting such a structure, a prefabricated weld-
ed-pipe framework or jacket is set in place on the ocean bottom and pipe
piles are driven through the vertical members of the jacket.
128

......
N
~
x
in.
0
24
48
96
144
192
-~
k1T
ZA
Y
0.0002.151
0.1591.918
0.3171.690
0.6341.258
0.9510.880
1.2680.569
--------
TABLE6.5.COMPUTEDVALUESOFSOILMODULUS,Es=kO+k
1
x.
0.10.20.142
ByABABYAYB
YYYYin.in.
1.4681.9301.3482.058l.4W3.3813.28
1.2181.7171.1091.8341.0823.0110.13
0.9931.5080.8951.6560.9522.728.91
0.6161.1140.5401.1980.5841.975.47
0.3330.7710.2770.8340.3091.372.89
0.1350.4910.0970.5360.1190.881.11
--
YPEs
in.lb/in.lb/in.
16.6644026
13.1451739
11.6359551
7.4470895
4.26762179
L99755379.I
----
---

Y (in)
-15 -10
-r 0 5 10
I I I I I
V(Ib x104 )
-15 -10 -5 0 5 10
I I I I
M (in.-Ib x104)
I I
-30 -20 -10 0 10 20
•
v •
•
•
•
•
•
•
•o
700
Depth, in.
Fig. 6.15. Deflection, moment, and shear diagrams for
example problem, Es = kO + k1x.
130
15
I
15
I
30

Resultant wind
and wave force
---o!.~
=600,OOOlb
+
mean sea level
t t f
Fig. 6.16. Lateral forces applied to an offshore structure,
Example Problem 2 (Matlock and Reese, 1961).
The elastic elements of the problem are described in Fig. 6.17. The
annular space between the pile and the jacket column is assumed to be
grouted so that the two members will bend as a composite section. This is
,
frequently, but not always, done in actual practice.
The elastic angular restraint provided by the portion of the struc-
ture above the soil may be analyzed by determining the moment required to
produce a unit value of rotation at the connection. This value, and the
imposed lateral load, constitute the boundary conditions for this partic-
ular problem. For the example problem, for each pile the elastic angular
restraint Mt/St is equal to 6.176 x 109
in.-lb/radian and the lateral load
Pt is equal to 150,000 lb.
The force-deformation characteristics of the soil are described bya
set of predicted p-y curves, as shown in Fig. 6.18.
131

Annular space orouted
h =240'
140
~
Ipof pile
=7375 in.4 -0.0.=30
-
E of steel =30 X 106
Ib/sq, in.
Fig. 6.18. Typical resistance-
deflection curves pre-
dicted for the soil at
various depths (Matlock
and Reese, 1961).
Fig. 6.17. The superstructure and
the pile, considered as
elastic elements of the
problem (Matlock and
Reese, 1961).
:'? -500
-.c
Q. -400I~~J-~+----J..-.::::::i===R
,,u
g -~O~~~~~----~--~--~~...III
'Vi
CD
a:
-IOOI-I+-#--+-,.,....F=----!-----+-----+--l
I 2
Deflection, y (in)
132

As shown earlier in this chapter, the deflection y for the case where
Es = kx is
P T3 M T2
y=A --!-+B _t_
Y EI Y EI
(6.77)
where EI is the flexural rigidity of the pile and where T is the relative
stiffness factor, defined by
T5 =rrk • (6.75)
It is convenient to define an additional set of nondimensional deflection
coefficients by rearranging Eq. 6.77 as follows:
P T3
t
Y = Cy EI '
where, at any depth coefficient Z,
M
C A +_tB.
Y = Y PtT y
(6.85)
(6.86)
Depending on the angular restraint provided by the structure, values
of Mt/PtT will range from zero for the pinned-end case to -0.93 for the
case where the structure prevents any rotation of the pile ahead. Values
of Cy are given by the curves in Fig. 6.19. Tables for C
y are also
included in Appendix 4 and are Tables A4.33 through A4.40.
To begin the solution of the example problem it is necessary to
assume, temporarily at least, that the form of soil modulus variation E =
s
kx will be a satisfactory approximation of the actual final Es variation.
Also, available nondimensional solutions are limited to a pile of constant
bending stiffness. For the example hand solution, the pile stiffness will
be assumed equal to that of the combined pile and jacket leg.
The slope at the top of the pile is
Pt
T2 MtT
S = A - - + Bs EI '
t St EIe t e
(6.87)
where the subscript t indicates values at Z = O. The relation between M
t
and St from Fig. 6.17 is
S = h M
t
• (6.88)
t 3.5 EIe
133

Deflection Coefficient, Cy
o0 0.5 1.0 1.5 2.0
0.5
N
~
-c
Q)
u
1.0
--Q)
0
U
..s:::.
-Q.
Q)
c 1.5
2.0
Fi g. 6.19.
1
Ii / J I L L L V / V
I
'I I V L / V V V
~ I~j !j
/
Mi 05~~yV L L
V VL'
/1/ / V ~oLlLL- fo-
PiIT -'
I / I / /1/ /VV:~V./
/ 11/ V vI /I/V~~
F'Ixe~ ~leac --..., 1//VIIIIv//:~~V
lea ~e I~/VIV/ ~ ~ ~
/~'Ii~~~~ +M
Iff~~~V I+P ,..-t y
t-
nl(.2
j.rt//, ~~rFree-Head Case
/;,~~1-~ (M=O)
~~
~ I
~J
1/
IJ
:'rl
Zmax=IO
Nondimensional coefficients for lateral deflection of a pile,
assuming soil modulus proportional to depth, or Es = kx,
long pile case Zmax 5 to 10.
Combining Equations 6.87 and 6.88, and rearranging,
Mt As T -1.623T -T
2.5
t- = = =
h
~~~ + 1.750T
(6.89).
PtT 42.25 + 1.078 T1:5 - B T• St
Because the relative stiffness factor T depends on the coefficient of soil
modulus variation k and this quantity in turn depends on nonlinear soil
resistance characteristics, as noted earlier the solution must proceed by
a process of repeated trial and adjustment of values of T (or k) until the
deflection and resistance patterns of the piles are made to agree as
closely as possible with the resistance-deflection (p-y) relations previ-
134

ously estimated for the soil and shown in Fig. 6.18. Also, as noted ear-
lier, even though the final set of soil moduli (Es =-ply) may not vary in
a perfectly linear fashion with depth, proper fitting of Es =kx will usu-
ally produce satisfactory solutions.
For the fi rst tri a1, T wi 11 be assumed equal to 200 in. From Eq.
6.89 the corresponding value of Mt/PtT is -0.776. For this value of
Mt/PtT, values of Cy are interpolated from Fig. 6.19 and are given in
Table 6.6 at depths corresponding to the positions of the several p-y
curves of Fig. 6.18. Values of deflection yare then computed at each
depth. By reference to Fig. 6.18, values of soil resistance pare
obtained, and soil modulus values E are computed.
s
TABLE 6.6. SAMPLE COMPUTATIONS FOR FIRST TRIAL.
-,
Depth Depth Deflection Deflection Soil Soil
Coefficient Coefficient Resistance Modulus
x Z C Y P Esy
from P T3 fromx Fig. 6.16 C t Fig. 6.15 =.:.£=
f =
~y Y
in. in. lb/in. lb/sq in.
30 0.15 1.13 3.20 -132 41
60 0.30 1. 06 3.00 -285 95
90 0.45 0.99 2.81 -420 149
150 0.75 0.82 2.32 -578 249
240 1. 20 0.57 1. 62 -675 416
Values of soil modulus from the first trial are plotted versus depth
as shown in Fig. 6.20. A straight line through the origin is fitted to the
points, with more weight being given to points at depths less than x ~
0.5T than at greater depths. For this straight line, the coefficient of
soil modulus variation resulting from the first trial is computed as
135

Soil Modulus, E (Ib/sq in,)
200 600 700 800
50
100
-..c
0. 150Q)
o
Final Trial
k = 1.95 Ib/cu In,
200
First Trial
250L-------------~--L-------~~
Fig. 6.20. Trial plots of soil modulus values. The first
trial corresponds to computations in Table 6.5.
E
k =~ = 1.6 lb/cu in.
x
The correspondng value of the relative stiffness factor is
T 5~ 194 in.
(obtained) =vf~--k-- =
(6.90)
(6.91)
If the value of T(obtained) were equal to the value of TCtried)' the proc-
ess would have been completed. To facilitate additional estimating and to
reach closure with a minimum of trials, a plot of T-values is used, as
shown in Fig. 6.21. Two trials will usually allow interpolation for the
final value of T. A final set of computations for Es values is then made
as a check.
6.8 CONCLUDING COMMENT
While the nondimensional methods described above are satisfactory
for many problems, most laterally-loaded piles can be analyzed efficient-
ly by means of a computer program such as COM622.
136

'C
Q)
c:
·0
1;
o
I-
225
200
175
;.~
-..
l'*/i$:~
:)
~+--Trial 1
o
O
'Final T = 185 in.
Trial 2
~
/+
150~--------~--------~~------~
150 175 200 225
Ttried
Fig. 6.21. Interpolation for final value
of relative stiffness factor ~
As stated previ ous ly, the program uses successive di ffer-
ence-equation computations based on repeated reference to the p-y curves
to determine at increments along the pile the values of soil modulus. The
procedure insures both compatibility and equilibrium for the soil, the
pile, and the superstructure. Some of the advantages of using the comput-
er program are given below.
1. Step changes in the fl exura1 stiffness of the pil e may be
introduced at any depth.
2. The pile length may be changed as desired.
3. The boundary conditions at the top of the pile may be speci-
fied as the lateral load and a) the moment, b) slope, and c)
the rotational spring constant moment/slope). In addition,
an axial load may be specified.
However, the nondimensional method should be employed on almost
every occasion as a check of the computer solution or to give preliminary
design information.
137

6.9 REFERENCES
George, P. and Wood, D., Offshore Soil Mechanics, Cambridge Univer-
sity Engineering Department, 1976.
Matlock, Hudson, and Reese, L. C., IIFoundation Analysis of Offshore
Pil e-Supported Structure, II Proceedi ngs, Fifth Internati ona 1 Conference,
International Society of Soil Mechanics and Foundation Engineering, Par-
is, Vol. 2, 1961, p. 91.
Matlock, Hudson, and Reese, L. C. IIGeneralized Solutions for Lat-
erally Loaded Piles,1I Transactions, American Society of Civil Engineers,
Vol. 127, Part 1, 1962, p. 1220-1251.
McClelland, Bramlette, and Focht, John A., Jr., IISoil Modulus for
Laterally Loaded Piles,1I Transactions, American Society of Civil Engi-
neers, Vol. 123, p. 1049, New York, 1958.
Palmer, L. A., and Brown, P. P., IIPiles Subjected to Lateral Thrust
Part II - Analysis of Pressure, Deflection, Moment, and Shear by the Meth-
od of Difference Equations,1I Supplement of Symposium on Lateral Load Tests
on Piles, ASTM Special Technical Publication, No. 154-A, pp. 22-44, 1954.
John Wiley and Sons, 1980.
Reese, Lymon C., and Matlock, Hudson, IINondimensional Solutions for
Laterally Loaded Piles with Soil Modulus Assumed Proportional to Depth,1I
Proceedings, Eighth Texas Conference on Soil Modulus and Foundation Engi-
neering, Special Publication No. 29, Bureau of Engineering Research, The
University of Texas, Austin, September 1956.
U. S. Department of the Navy, Design Manual, Soil Mechanics, Founda-
tions, and Earth Structures, NAVFAC DM-7, 1971.
138

6.10 EXERCISES
1. The pile shown below is subjected to cyclic loading. The pile
head is restrained against rotation. Analyze the pile response by:
(a) Computer program, and
(b) Hand calculation using nondimensional Cy curves.
Soi 1 condi ti ons:
Soft saturated clay deposit with the following properties:
C
u
= 500 lb/sq ft
y' = 45 lb/cu ft
E:50 = 0.01
E = kx
s
k = 300 lb/cu in.
(c) Repeat the problem with pile head
(i) free to rotate
(ii) fixed against rotation
axial load =10,000 Ib
F; =20,000 Ib J.M.. = 6.11 x 107in;lb- _ _ _a S
180
r720'
El
l
=3.14x 1010 Ib/sq in.
EI2
= 2.2 x 10
10
fb/sq in.
10=16
2. Rework the problem given in the example in 6.5 with the applied
moment equal to zero.
139

3. Rework the problem given in the example in 6.5 with the load and
moment decreased to 15, 45, and 75 percent of the values used in the exam-
ple.
4. Develop p-y curves for a sand below water, ¢ =36°, ~I =50 lb/cu
ft, and cyclic loading. Find the groundline load versus deflection up to
the point where the maximum steel stress is 30 ksi, keeping the applied
lateral load and moment at the same ratio as given in the example.
5. Rework the problem given in 6.7 assuming no jacket-leg extension
(use EI of pile alone) and compare results with computer solution.
6. Use the computer and develop nondimensional curves for Es = kx
n
where n is equal to 1.2, 1.4, 1.6, and 1.8.
140

CHAPTER 7. OTHER METHODS OF DESIGN
Three methods for the design of piles under lateral loading are
reviewed that differ from the p-y method previously presented. However,
there are common features in all of the methods.
7.1 BROMS METHOD
The method was presented in three papers published in 1964 and 1965
(Broms, 1964a, 1964b, 1965). As shown in the following paragraphs, a pile
can be designed to sustain a lateral load by solving some simple equations
or by referring to charts and graphs.
Ultimate Lateral Load for Piles in Cohesive Soil.
Broms adopted a distribution of soil
resistance, as shown in Fig. 7.1, that
allows the ultimate lateral load to be com-
puted by equations of static equilibrium.
The elimination of soil resistance for the
top 1.5 diameters of the pile is a result
of lower resistance in that zone because a
wedge of soil can move up and out when the
pile is deflected. The selection of nine
times the undrained shear strength times
the pile diameter as the ultimate soil
resistance, regardless of depth, is based
on calculations with the soil flowing from
the front to the back of the pile.
Short, Free-Head Piles in Cohesive
Soil. For short piles that are unres-
trained against rotation, the patterns that
Fi g. 7.1.
1.5b
I 9cb I
Assumed distri-
bution of soil
resistance for
cohesive soil.
were selected for behavior are shown in Fig. 7.2. The following equation
results from the integration of the upper part of the shear diagram to the
point of zero shear (the point of maximum moment)
MPoS =Pee + 1.5b + f) - 9cbf2/2.
max
But the point where shear is zero is
Therefore,
f = P/9 cb.
MPoS =Pee + 1.5b + O.5f).
max
141
(7.1 )
(7.2)
(7.3)

Fig. 7.2. Deflection, load, shear, and moment diagrams
for a short pile in cohesive soil that is
unrestrained against rotation.
Integration of the lower portion of the shear diagram yields
MPoS =2. 25cbg2 •
max
It may be seen that
L = (1. 5b + f + g).
(7.4 )
(7.5)
Equations 7.2 through 7.5 may be solved for the load Pult that will pro-
duce a soil failure. After obtaining a value of Pult the maximum moment
can be computed and compared with the moment capacity of the pile. An
appropriate factor of safety should be employed.
As an example of the use of the equations, assume the following:
b =1 ft (Assume 12-in. 0.0. steel pipe by 0.75 in. wall,
1= 421 in.4), e = 2 ft, L = 8 ft, and c = 1kip/sq ft.
Equations 7.2 through 7.5 are solved simultaneously and the following
quadratic equation is obtained.
p2
+ 243P - 3422 =0
Pult =13.4 kips
Substituting into Eq. 7.3 yields the maximum moment.
M = 13.4(2 + 1.5 + 0.744) = 57.0 ft-k
max
142

Assuming no axial load, the maximum stress is
fb = (57.0)(12)(6)/421 = 9.7 kips/sq in.
The computed maximum stress is tolerable for a steel pipe, especially when
a factor of safety is applied to Pult ' The computations, then, show that
the short pile would fail due to a soil failure.
Broms presented a convenient set of curves for solving the problem of
the short-pile (see Fig. 7.3). Entering the curves with L/b of 8 and e/b
of 2, one obtains a value of Pult of 13.5 kips, which agrees with the
results computed above.
60~---------r----~----~--~
Q)
(J
40c::
0
-fn
.-en
cu
300:::
0
r-
cu
2
-0
~
Q)
-0
e
-::::
0
0 4 16 20
Embedment Length, Lib
Fig. 7.3. Design curves for short piles under
lateral load in cohesive soil
(after Broms).
Long, Free-Head Piles in Cohesive Soil. As the pile in cohesive soil
with the unrestrained head becomes longer, failure will occur with the
formation of a plastic hinge at a depth of I.S0b + f. Equation 7.3 can
143

then be used directly to solve for the ultimate lateral load that can be
app1ied. The shape of the pi 1e under load wi 11 be di fferent than that
shown in Fig. 7.2 but the equations of mechanics for the upper portion of
the pile remain unchanged.
A plastic hinge will develop when the yield stress of the steel is
attained over the entire cross-section. For the pile that is used in the
example, the yield moment is 317 ft-k if the yield strength of the steel
is selected as 40,000 lbfsq in.
Substituting into Eq. 7.3
Pult
317 = Pult (2 + 1.5 + ~)
Pult =50.3 kips.
Broms presented a set of curves for solving the problem of the long
pile (see Fig. 7.4). Entering the curves with a value of Mfcb 3
of 317,y
one obtains a value of Pult of about 50 kips.
~
~ 6
-Q.3
... 4
~
u
c
o
-en
en
~
a:=
-o..~
-o
..J
~
-o
e
-
Free-Headed
1:o
4 6 10 20 40 60 100 200 400 600
Yield Moment, My /cb3
Fig. 7.4. Design curves for lonq piles under lateral
load in cohesive soil (after Broms).
*Note: The length of the pile for which these curves are val id must be
ascertained (see text).
144

Influence of Pile Length, Free-Head Piles in Cohesive Soil. Consid-
eration may need to be given to the pile length at which the pile ceases to
be a short pile. The value of the yield moment may be computed from the
pile geometry and material properties and used with Eqs. 7.2 through 7.5
to solve for a critical length. Longer piles will fail by yielding. Or a
particular solution may start with use of the short-pile equations; if the
resulting moment is larger than the yield moment, the long-pile equations
must be used.
For the example problem, the length at which the short-pile equations
cease to be valid may be found by substituting a value of P
ult
of 50.3 kips
into Eq. 7.2 and solving for f and substituting a value of Mmax of 317 ft-k
into Eq. 7.4 and solving for g. Equation 7.5 can then be solved for L.
The value of L was found to be 19.0 ft. Thus, for the example problem the
value of Pult increases from zero to 50.3 kips as the lengt~ of the pile
increases from 1.5 ft to 19.0 ft, and above a length of 19.0 ft the value
of Pult remains constant at 50.3 kips.
Short, Fixed-Head Piles in Cohesive Soil. For a pile that is fixed
against rotation at its top, the mode of failure depends on the length of
the pile. For a short pile, failure consists of a horizontal movement of
the pile through the soil with the full soil resistance developing over
the length of the pile except for the top one and one-half pile diameters,
where it is expressly eliminated. A simple equation can be written for
this mode of failure, based on force equilibrium.
Pult =9cb(L - 1.5b) (7.6)
Intermediate Length, Fixed-Head Piles in Cohesive Soil. As the pile
becomes longer, an intermediate length is reached such that a plastic
hinge develops at the top of the pile. Rotation at the top of the pile
will occur and a point of zero deflection will exist somewhere along the
length of the pile. Figure 7.5 presents the diagrams of mechanics for the
case of the restrained pile of intermediate length.
The equation for moment equilibrium for the point where the shear is
zero (where the positive moment is maximum) is:
MPoS =P(1.5b + f) - f(9cb)(f/2) - M .max y
Substituting a value for f,
MPos =P(1.5b + 0.5f) - M .max y (7.7)
145

p
Myr---r-----,- --1
II
I
1
1
1
...... ' H• I I
1 1 I
--....,j... /1'. ~/' •
I I 1.5b---I '--Jl;~~~I I
I I
I I
II I
I
I
I I
I I
I I
I I
I
L~'-
9
9cb
4.5cbg
MPOS
max
Fig. 7.5. Deflection, load, shear, and moment diagrams for
an intermediate-length pile in cohesive soil that
is fixed against rotation at its top.
Employing the shear diagram for the lower portion of the pile,
Pos
M =2.25cbg2
• (7.8)max
The other equations that are needed to solve for Pu1t are:
L =1.5b + f + g (7 .g)
and
f = Pig cb. (7.10)
Equations 7.7 through 7.10 can be solved for the behavi or of the
restrained pile of intermediate length.
Long, Fixed-Head Piles in Cohesive Soil. The mechanics for a long
pile that is restrained at its top is similar to that shown in Fig. 7.5
except that a plastic hinge develops at the point of the maximum positive
moment. Thus, the MPoS in Eq. 7.7 becomes M and the following equationmax y
results:
2M
P = Y
(1.5b + 0.5f)
(7.11)
Equations 7.10 and 7.11 can be solved to obtain Pu1t for the long pile.
Influence of Pile Length, Fixed-Head Piles in Cohesive Soil. The
example problem will be solved for the pile lengths where the pile goes
146

from one mode of behavior to another. Starting with the short pile, an
equation can be wri tten for moment equi 1i bri urn for the case where the
yield moment has developed at the top of the pile and where the moment at
its bottom is zero. Referring to Fig. 7.5, but with the soil resistance
only on the right-hand side of the pile, taking moments about the bottom
of the pile yields the following equation.
PL - 9 cb(L - 1.5b)(L - 1.5b)/2 - My = 0
Summing forces in the horizontal direction yield the next equation.
P - 9 cb(L - 1.5b) = 0 (same as Eq. 7.6)
The simultaneous solution of the two equations yields the desired
expression.
Pult =My/(0.5L + 0.75b)
Equations 7.6 and 7.12 can be solved simultaneously for Pult.and
follows:
from Eq. 7.6, Pult =9(L - 1.5)
from Eq. 7.12, Pult =317/(0.5L + 0.75)
then L =8.53 ft and Pult =63.2 k.
(7.12)
for L, as
For the determination of the length where the behavior changes from
that of the pile of intermediate length to that of a long pile, Eqs. 7.7
through 7.10 can be used with M set equal to My' as follows:max
from Eq. 7.7. P 1t = (2)(317)
u 1.5 + O.Sf
from Eq. 7.8, g = (23
.VS)0.5 = 11.87 ft
from Eq. 7.9, L =1.5 + f + g
from Eq. 7.10, f =Pult/9
then L =23.83 ft and Pult =94.2 k.
In summary, for the example problem the value of Pu1t increases from
zero to 63.2 kips as the length of the pile increases from 1.5 ft to 8.5
ft, increases from 63.2 kips to 94.2 kips as the length increases from 8.5
ft to 23.8 ft, and above a length of 23.8 ft the value of Pu1t remains con-
stant at 94.2 kips.
In his presentation, Broms showed a curve in Fig. 7.3 for the short
pile that was restrained against rotation at its top. That curve·is omit-
ted here because the computation can be made so readily with Eq. 7.6.
Broms' curve for the long pile that is fixed against rotation at its top
is retained in Fig. 7.4 but a note is added to insure proper use of the
147

curve. For the example problem, a value of 93 kips was obtained for Pu1t '
which agrees well with the computed value. No curves are presented for
the pile of intermediate length.
Deflection of Piles in Cohesive Soil.
Broms suggested that for cohesive soils the assumption of a coeffi-
cient of subgrade reaction that is constant with depth can be used with
good results for predicting the lateral deflection at the ground1ine. He
further suggests that the coefficient of subgrade reaction u should be
taken as the average over a depth of 0.8~L, where
ex 0.25
B = (4EI) (4.4)*
where a =soil modulus (subgrade reaction)
EI =pile stiffness.
Broms presented equations and curves for computing the deflection at the
ground1ine. His presentation follows the procedure presented in Chapter 4
and the methods in that chapter are recommended here for computing
deflection.
With regard to values of the coefficient of subgrade reaction, Broms
used work of himself and Vesic (1961a, 1961b) for selection of values,
depending on the unconfined compressive strength of the soil. The writer
believes that the values suggested by Terzaghi (1955) yield results that
are compatible with other assumptions; thus, values shown in Table 3.1
are recommended.
Broms suggested that the use of a constant for the coefficient of
subgrade reaction is valid only for a load of one-half to one-third of the
ultimate lateral capacity of a pile.
For the example problem, the long pile in cohesive soil that is
restrained against rotation at its top will be considered. A value of
Pu1t of 94.2 kips was computed. A working load of 35 kips is selected for
an example computation and, using Table 3.1, a value of a is selected as
50 1b/sq in. The value of ~ is
( )
0.25
B - 50
(4)(30 x 106 )(421)
1 .
=-- In.
178
*Broms' notation has been changed to agree with that in Chapter 4.
148

The value of ~L must be equal to or greater than 4 for the pile to act as a
long pile; therefore, the length must be at least 60 feet. The deflection
at the top of the pile may be computed from Eq. 4.35, using a value from
Table 4.1.
PtB (35,000)(1.0)
Yt = ~ Alt = (178)(50) = 3.9 in. (4.35)
Had the pile been shorter than 60 ft, values of the deflection coefficient
can be obtained from other tables in Chapter 4.
The further use of Broms I recommendation to compute the groundl i ne
deflection of piles in cohesive soils will be demonstrated in Chapter 8
where case studies are presented.
Effects of Nature of Loading on Piles in Cohesive Soil.
The values of soil modulus presented by Terzaghi are apparently for
short-term 1oadi ng. Terzaghi di d not di scuss dynami c 1oadi ng or the
effects of repeated loading. Also, because Terzaghi's coefficients were
for overconsolidated clays only, the effects of sustained loading would
probably be minimal. Because the nature of the loading is so important in
regard to pile response, some of Broms ' remarks are presented here.
Broms suggested that the increase in the deflection of a pile under
lateral loading due to consolidation can be assumed to be the same as
would take place with time for spread footings and rafts founded on the
ground surface or at some distance below the ground surface. Broms sug-
gested that test data for footings on stiff clay indicate that the coeffi-
cient of subgrade reaction to be used for long-time lateral deflections
should be taken as 1/2 to 1/4 of the initial coefficient of subgrade
reaction. The value of the coefficient of subgrade reaction for normally
consolidated clay should be 1/4 to 1/6 of the initial value.
Broms suggested that repetitive loads cause a gradual decrease in the
shear strength of the soil located in the immediate vicinity of a pile.
He stated that unpublished data indicate that repetitive loading can
decrease the ultimate lateral resistance of the soil to about one-half its
initial value.
Ultimate Lateral Load for Piles in Cohesionless Soil.
As for the case of cohesive soil, two failure modes were considered;
a soil failure and a failure of the pil e by the formation of a p1asti c
hinge. With regard to a soil failure in cohesionless soil, Broms assumed
that the ultimate lateral resistance is equal to three times the Rankine
149

passive pressure. Thus, at a depth Z below the ground surface the soil
resistance per unit of length P
z
can be obtained from the following
equations.
Pz = 3blZKp
Kp = tan2
(45 +~)
l = unit wei ght of soil
Kp = Rankine coefficient of passive pressure
~ = angle of internal friction of soil
(7.13)
(7.14 )
Short, Free-Head Piles in Cohesionless Soil. For short piles that
are unrestrained against rotation, a soil failure will occur. The curve
showing soil reaction as a function of depth is shaped approximately as
shown in Fig. 7.6. The use of Ma as an applied moment at the top of the
pile follows the procedure adopted by Broms. If both P and Ma are acting,
the result would be merely to increase the magnitude of e. It is unlikely
in practice that Ma alone would be applied.
The patterns that were selected for behavior are shown in Fig. 7.7.
Failure takes place when the pile rotates such that the ultimate soil
resistance develops from the ground surface to the center of rotation.
The high values of soil resistance that develop at the toe of the pile are
replaced by a concentrated load as shown in Fig. 7.7.
The following equation results after taking moments about the bottom
of the pile.
Solving for P when Ma is equal to zero,
ybL3K
p = 2(e + ~)
(7.15)
(7.16)
And solving for Ma when P is equal to zero,
M =0.5lbL3 K (7.17)a p •
Equations 7.15 through 7.17 can be solved for the load or moment, or a
combination of the two, that will cause a soil failure. The maximum
moment will then be found, at the depth f below the ground surface, and
compared with the moment capacity of the pile. An appropriate factor of
safety should be used. The distance f can be computed by solving for the
point where the shear is equal to zero.
(7.18)
150

p ~Mo
e
L
17
1 I
--I ~~________~__
1 I
1 I
I I
I 1
I I
I I
I I
Fig. 7.6. Failure mode of a short pile in cohesionless soil
that is unrestrained against rotation.
p
e
L
H3YbLK
Fig. 7.7. Deflection, load, shear, and moment diagrams for
a short pile in cohesionless soil that is
unrestrained against rotation.
151

Solving Eq. 7.18 for an expression for f
f = 0.816(P/lbK )0.5 •
P
(7.19)
The maximum positive bending moment can then be computed by referring to
Fi g. 7.7.
K ybf3
MPoS = P(e + f) - P + M
max 2 a
Or, by substituting expression for Eq. 7.18 into the above equation, the
following expression is obtained for maximum moment.
MPoS =Pee + f) - Pf/3 + M (7.20)
max a
As an example of the use of the equations, the pile used previously
is considered. The angle of internal friction of the sand is assumed to
be 34 degrees and the unit weight is assumed to be 55 pounds per cubic foot
(the water table is assumed to be above the ground surface). Assume Ma is
equal to zero. Equations 7.14 and 7.16 yield the following:
K =tan2
( 45 + 34) =3 54
P 2 '
P = (0.055)(1)(8)3(3.54) 4 98 k'
ult 2(2 + 8) =. lpS.
The distance f can be computed by solving Eq. 7.19.
f - ( 4.98 ) 0.5 - 4 13 ft
- (1.5)(0.055)(1)(3.54) -.
The maximum positive bending moment can be found using Eq. 7.20.
Mmax =(4.98)(2 + 4.13) - (4.98)(4.13)/3 + 0 =23.7 ft-k
Assuming no axial load, the maximum bending stress fb is
fb =(23.7)(12)(6)/421 =4.05kip/sq in.
The computed maximum stress is undoubtedly tolerable, especially when a
factor of safety is used to reduc~' Pul t' Broms presented curves fo·r the
solution of the case where a short, unrestrained pile undergoes a soil
failure; however, Eqs. 7.15 and 7.18 are so elementary that such curves
are unnecessary.
Long, Free-Head Piles in Cohesionless Soil. As the pile in cohesion-
less soil with the unrestrained head becomes longer, failure will occur
with the formation of a plastic hinge in the pile at the depth f below the
ground surface. It is assumed that the ultimate soil resistance develops
from the ground surface to the point of the plastic hinge. Also, the
shear is zero at the point of maximum moment. The value of f can be
152

obtained from Eq. 7.19 as shown above. The maximum positive moment can
then be computed and Eq. 7.20 is obtained as before. Assuming that M isa
equal to zero, an expression can be developed for Pult as follows:
M
P
ult
= y
e + 0.544[P
ult
/(ybK
p
)]O.S
{7 .21)
For the example problem, Eq. 7.21 can be solved, as follows:
P = .317 = 34.36 kips.
ult 2 + 0.544[P
ult
/{(O.055)(I)(3.54)}]O.S
Broms presented a set of curves for solving the problem of the long
pile in cohesionless soils (see Fig. 7.8). Entering the curves with a
value of M/b4~K of 1628, one obtains a value of P lt of about 35 kips.y p u
The logarithmic scales are somewhat difficult to read and it may be desira-
ble to make a solution using Eq. 7.21. Equations 7.19 and 7.20 must be
used in any case if a moment is applied at the top of the pile.
Q.
~
)...
~~ IOOO~-------r--------r--------r--------r-------~
a..::l
oJ
u
c
o+-
VI
VI
Q)
cr
o~
Q)
+-
o
....J
Q)
+-
o
E
+-
100
10 ~•.•....
--H-
b
1.0 10.0 100.0 1000.0 10000.0
Yield Moment My/ b4
Y Kp
Fig. 7.8. Design curves for long piles under lateral
load in cohesionless soil (after Broms).
*Note: The length of pile for which this curve is valid must be ascer-
tained (see text).
153

Influence of Pile Length, Free-Head Piles in Cohesionless Soil.
There may be a need to solve for the pile length where there is a change in
behavior from the short-pile case to the long-pile case. As for the case
of the pile in cohesive soils, the yield moment may be used with Eqs. 7.15
through 7.17 to solve for the critical length of the pile. Alternatively,
the short-pile equations would then be compared with the yield moment. If
the yield moment is less, the long-pile equations must be used.
For the example problem, the value of P
ult
of 34.36 kips is substi-
tuted into Eq. 7.16 and a value of L of 19.7 ft is computed. Thus, for the
pile that is unrestrained against rotation the value of P
ult
increases
from zero when L is zero to a value of 34.36 kips when L is 19.7 ft. For
larger values of L, the value of P
ult
remains constant at 34.36 kips.
Short, Fixed-Head Piles in Cohesionless Soil. For a pile that is
fixed against rotation at its top, as for cohesive soils, the mode of fai-
lure for a pile in cohesionless soil depends on the length of the pile.
For a short pile, the mode of failure will be a horizontal movement of the
pile through the soil, with the ultimate soil resistance developing over
the full length of the pile. The equation for static equilibrium in the
horizontal direction leads to a simple expression.
Pult = 1.5lL2
bKp
(7 .22)
Intermediate Length, Fixed-Head Piles in Cohesionless Soil. As the
pile becomes longer, an intermediate length is reached such that a plastic
hinge develops at the top of the pile. Rotation at the top of the pile
wi11 occur, and a point of zero deflection will exist somewhere along the
length of the pile. The assumed soil resistance will be the same as shown
in Fig. 7.7. Taking moments about the toe of the pile leads to the follow-
ing equation for the ultimate load.
P lt = M/L + O.5lbL2
Ku y p
(7 .23)
Equation 7.23 can be solved to obtain P
ult
for the pile of intermediate
length.
Long, Fixed-Head Piles in Cohesionless Soil. As the length of the
pile increases more, the mode of behavior will be that of a long pile. A
plastic hinge will form at the top of the pile where there is a negative
bending moment and at some depth f where there is a positive bending
moment. The shear at depth f is zero and the ultimate soil resistance is
as shown in Fig. 7.7. The value of f may be determined from Eq. 7.19 but
that equation is re-numbered and presented here for convenience.
154

f =0.816(P/~bK )0.5
P
(7 .24)
Taking moments at point f leads to the following equation for the ultimate
lateral load on a long pile that is fixed against rotation at its top.
M+ + M-
P = -----....:y~---y~---
ult e + 0.544[P
ult
/ybK
p
)]0.5
(7.25)
Equations 7.24 and 7.25 can be solved to obtain P
ult
for the long pile.
Influence of Pile Length, Fixed-Head Piles in Cohesionless Soil. The
example problem will be solved for'the pile lengths where the pile goes
from one mode of behavior to another. An equation can be written for the
case where the yield moment has developed at the top of the short pile.
The ~quation is:
P lt =M/L + 0.5~bL2K • (7.26)
u y p
Equations 7.23 and 7.26 are, of course, identical but the repetition is
for clarity. Equations 7.22 and 7.26 can be solved for Pult and for L, as
follows:
from Eq. 7.22, P
ult
=0.292L2
from Eq. 7.26, P
ult
=317/L + 0.09735L2
then L =11.77 ft and Pult = 40.4 kips.
For the determination of the length where the behavior changes from
that of a pile of intermediate length to that of a long pile, the value of
P
ult
from Eq. 7.23 may be set equal to that in Eq. 7.25. It is assumed
that the pile has the same yield moment over its entire length in this
example.
from Eq. 7.23, P
ult
= O.09735L2 + 3t7
from Eq. 7.25, Pult = 317)0 5
2 + 0.544(Pult/0~1947 .
then L =20.5 ft and Pult =56.4 kjps
In summary, for the example problem the value of Pult increases from
zero to 40.4 kips as the length of the pile increases from zero to 11.77
ft, increases from 40.4 kips to 56.4 kips as the length increases from
11.77 ft to 20.5 ft, and above 20.5 ft the value of Pult remains constant
at 56.4 kips.
In his presentation, Broms showed curves for short piles that were
unrestrained against rotation at their top. Those curves are omitted
because the equations for those cases are so easy to solve. Broms' curve
for the long pile that is fixed against rotation at its top is retained in
155

Fig. 7.8 but a note is added to
example problem, a value of 68
poorly with the computed value.
ensure proper use of the curve. For the
kips was obtained for Pult ' which agrees
The difficulty probably lies in the ina-
bility to read the logarithmic scales accurately. No curves are presented
for the pile of intermediate length with fixed head.
Deflection of Piles in Cohesionless Soil.
Broms noted that Terzaghi (1955) has shown that the' coeffi ci ent of
lateral subgrade reaction for a cohesionless soil can be assumed to
increase approximately linearly with depth (see discussion of Terzaghi's
recommendations in Chapter 3). As noted earlier, and using the formu-
lations of this work, Terzaghi recommends the following equation for the
soil modul us.
E =kxs
(7.27)
Table 3.2 presents Terzaghi's recommendations for values of k. Broms sug-
gested that Terzaghi's values can be used only for computing deflections
up to the working load and that the measured deflections are usually larg-
er than the computed ones except for piles that are placed with the aid of
jetting.
Broms presented equations and curves for use in computing the lateral
deflection of a pile; however, the methods presented in Chapter 6 are con-
sidered to be appropriate.
As an example problem, the long pile in cohesionless soil that is
restrained against rotation at its top is considered. The value of Pult
waS computed to be 56.4 kips and a working load of 20 kips is selected.
Using Table 3.2, a value of k of 10 lb/cu in. is selected. The groundline
deflection may be computed from the following equations.
P
t
T3
YF=F
YEr
T =(EIIk)o.2
(6.81)
(6.75)
Obtaining a value of Fyt from Fig. 6.9 of 0.93 and substituting into the
above equations, T is found to be equal to 66.1 in. and the deflection at
the groundline is found to be 0.43 in. For the solution to be valid, the
length of the pile should be at least 5T or 27.5 ft. Had the pile been
shorter, other values of Fyt could have been obtained from Fig. 6.9.
The further use of Broms ' recommendations to compute the groundline
deflection of piles in cohesionless soils will be demonstrated in Chapter
8 where case studies are presented.
156

Effects of Nature of Loading on Piles in Cohesionless Soil.
Broms noted that piles installed in cohesionless soil will experi-
ence the majority of the lateral deflection under the initial application
of the load. There will be only a small amount of creep under sustained
loads.
Repetitive loading and vibration, on the other hand, can cause sig-
nificant additional deflection, especially if the relative density of the
cohesionless soil is low. Broms noted that work of Prakash (1962) shows
that the lateral deflection of a pile group in sand increased to twice the
initial deflection after 40 cycles of load. The increase in deflection
corresponds to a decrease in the soil modulus to one-third its initial
value.
For piles subjected to repeated loading, Broms recommended for cohe-
sionless soils of low relative density that the soil modulus be decreased
to one-fourth its initial value and that the value of the soil modulus be
decreased to one-half its initial value for soils of high relative densi-
ty. He suggested that these recommendations be used with caution because
of the scarcity of experimental data.
7.2 POULOS METHOD
Several authors have proposed methods for the analysis of laterally
loaded piles where the equations of elasticity have been used to develop
interaction equations. Poulos and his co-workers at the University of
Sydney have been especially active (Poulos, 1971a; Poulos, 1971b; Poulos,
1973; Poulos and Davis, 1980, Poulos, 1982) and the presentation herein is
based principally on his work (1971a).
Model Employed in Elastic Analysis.
Poulos (1971a) assumed the pile to be a thin, rectangular, vertical
strip of width b, length L, and constant stiffness EI. The possible hori-
zontal shear stresses developed between the soil and the sides of the pile
were not taken into account. The pile was divided into n+l elements, as
shown in Fig. 7.9, with all elements being of an equal length h, except
those at the top and at the tip of the pile, which are of length h/2. Each
element was acted upon by a uniform, horizontal stress q which was assumed
to be constant across the width of the pile.
The soil was assumed to be an ideal, homogeneous, isotropic, linear,
elastic material of semi-infinite dimensions. The soil was assumed to
157

M
2
L
-Q.::::J_
-n n
n+1 n+1
Fig. 7.9. Stresses acting on (a) pile;
(b) soil adjacent to pile
(after Poulos)
have a Young's modulus of E P and a Poisson's ratio of v . The soil prop-s s
erties were assumed to be unaffected by the presence of the pile. As will
be noted later, Poulos presented some discussion of cases where Esp was
not constant with depth.
Poulos used the Mindlin equation for horizontal displacement due to
horizontal load within a semi-infinite mass to compute soil displacement.
Beam theory was used to compute pil e di sp1acements. The soil and pil e
displacements are evaluated and equated at the element centers except that
displacements are computed at the top and at the. tip of the pile. Poulos
sub-divided the pile into 21 elements.
Free-Head Piles, Elastic Behavio~
The behavior of a pile under lateral load was expressed by Poulos in
terms of non-dimensional influence factors. For a free-head pile, one
.with no rotational restraint at the groundline, the equation for horizon-
tal displacement at the groundline.is1Eq. 7.28.
* (7.28)
*Some of the notation employed by Poulos has been changed to agree with
notation previously used herein; however, it was necessary to retain some
of the Poulos notation even though there will be more than one symbol for
the same parameter. The reader should consult the section on Notation for
a list of symbols and their meanings.
158

where
Yt
Pt
Esp
l
= groundline deflection
= shear at groundline
= soil modulus (values suggested by Poulos are given later)
= pile length
= influence coefficient for computing pile-head deflection
for applied shear at groundline
I = influence coefficient for computing pile-head deflectionyM
for applied moment at groundline
Figures 7.10 and 7.11 give values of Iyp and IyM ' respectively, as a func-
tion of KR, lib, and vs' Poulos defined KR as the pile flexibility factor
and its value may be computed by use of Eq. 7.29.
where
where
EI
K = - -
R E l4
sP
(7.29)
EI = pile stiffness
The equation for rotation of the pile head at the groundline is
Pt
Mt
St = Isp E l2 + IsM E l3
sP sP
(7.30)
St = rotation at groundline
Isp = influence coefficient for computing pile-head rotation
for applied shear at groundline
IsM = influence coefficient for computing pile-head rotation
for applied moment at groundline
Figures 7.11 and 7.12 give values of Isp and IsM' respectively.
The maximum moment in a free-head pile subjected to a lateral load is
shown in Fig. 7.13 as a function of KR, P
t
, L, and lib. Poulos suggested
that the maximum moment typically occurs at a depth of between O.ll and
0.4L below the groundline. The lower depths are associated with stiffer
piles.
Fixed-Head Piles, Elastic Behavio~
For a pile that is fixed against rotation at the groundline, such as
a pile that is built into a rigid concrete mat, the deflection at the
groundline may be computed by using the following equation.
P
t
Yt = IyF Espl (7.31)
159

Iyp 5
2
IL-~__~~~~~~~--~
10-6 10-5 104
103
102
10
1
I 10
KR
Fig. 7.1Q Influence factors Iyp
for free-head pile
(after Poulos).
~:; 0.5
10
10
Fig. 7.12. Influence factors 1st,1
for free-head pile
(a fter Poulos).
IyM a
M
POS
max
160
I~~~~--~~~~--~~
10
6
10-5 10
4
10
3
10
2
10
1
10
KR
Fig. 7.11. Influence factors
IyM and Isp for free-
head pile tafter Poulos).
0.16,----r----,-----,r-----r----,----,
0.14
0.12 Values of Lib
0.1
0.OO~_5---L-4---L~--~--~--1--~~
10 10 10 10 10
KR
Fig. 7.13. Maximum bending moment
for free-head pile
(a fter Poulos).

where
Yt = groundline deflection
I = influence coefficient for computing pile-head headyF
deflection for a pile with fixed-head
Figure 7.14 gives values of IyF '
5
2
Vs =0.5
LIb
10
Fig. 7.14. Influence factors IyF for fixed-head pile
(after Poulos).
The bending moment at the top of a fixed-head pile is giver. by Fig.
7.15 as a function of KR, Pt , L, and Lib.
Effect of Local Yield of Soil Along Pile.
Poulos noted that elastic analysis showed that very high values of
soil pressure q developed near the top of a pile. An approximate analysis
was made by modifying the elastic analysis. Yielding of soil was assumed
at each element until all elements had yielded. It was assumed that the
horizontal displacements of the soil at the elements where elastic condi-
tions prevail can be computed by elastic theory using the known pressures
at elements where soil has yielded.
161

Mneg
max
P.t L
-0.6
-0.5
Us =0.5
-0.4
-0.3 Values of Lib
-0.2
-0.1
0.0 -6
10 10
4
10
3
10
2
10' 10
KR
Fig. 7.15. Maximum negative bending moment for
fixed-head pile (after Poulos).
Furthermore, because of the complexity of the analysis, only six ele-
ments were employed rather than the 21 elements used in the previous sol-
utions. Poulos presented curves for the free-head case only, as shown in
Fig. 7.16, that allow the groundline deflection to be computed in terms of
parameters previously employed and in terms of the distribution of the
yield pressure Py' Poulos noted that the free-head case is more severe
than the fixed-head case because the deflection of the free-head pile at
the groundline would be more than for the fixed-head pile for the same
load; hence, the computations using elastic analysis would show the soil
pressures to be greater for the free-head case. He further noted that the
information shown in Fig. 7.16 should be employed with caution because of
the assumptions that had to be made in developing the figure.
Determination of Properties of Soil
Poulos suggested that the best method for obtaining Esp is to carry
out a field loading test and to compute Esp from measured groundline
deflections.
162

0.018
Case R Distribution
0.016
-o-Py
0.014 ..,.-.
/-'
, / ' 2 0.67Py
0.012 / 2 -[SPyPt /
I. 33 Py
P. L2 0.010
/ ,,-3y / ...--- 0.167 Py
I
./ ...... 4
O.ooa ./ ./
-UPy/ ./ :3
1./b=25
0.006 /~/ Vs =0.5 1.633 Py
II K.=O.OI
~y0.004 /1 4
II Horizontal Load
, Only 2 Y
P=Id Py
Es
0.00
0 2 4 6 8 10 12
I
Fig. 7.16. Influence of distribution of yield pressure
on load-displacement relationship, free-head
case (after Poulos).
To provide a rough guide, values of Esp were back-figured from data
collected by Broms (March 1964; May 1964). The guidelines that were given
are shown below. For cohesive soils:
Esp =15c to 95c
where
c = undrained shear strength of clay.
For all cases considered, the average value of soil modulus was
Esp =40c.
For cohesionless soils the following table was presented. It was noted
that the soil modulus for sand is not constant with depth so that the use
of the values in the table below is questionable.
163

Soil Density Range of E P'
lb/sq in~
Average E P'
lb/sq in~
Loose 130-300 250
Medium 300-600 500
Dense 600-1,400 1,000
Poulos stated that for a cohesive soil with a un iform c the value of
the yield pressure Py theoretically increases from 2c at the groundline to
a va 1ue of ll.4lc at a depth of about three pil e di ameters below the
groundl i ne. He adopted the Broms I recommendation of 9c for design pur-
poses. With regard to cohesionless soil, Poulos suggested the use of a
triangular distribution of yield pressure with the yield pressure being
equal to three times the Rankine passive pressure, a suggestion made ori-
ginally by Broms.
Poulos noted in the discussion of the proposed method that the creep
of the soil at higher load levels can cause a discrepancy between results
from analysis and those from experiment.
Poulos (1982) gave an extended discussion of the behavior of a single
pile due to cycling the lateral load. He identified two effects: the
structural IIshakedownll
of the pile-soil system in which permanent defor-
mations accumulate with increasing load cycles with no changes in the
pile-soil properties, and a decrease in strength and stiffness of the soil
due to cyclic loading. His paper dealt mainly with the degradation of the
soil due to cyclic loading.
Poulos defi ned degradation parameters for soil modul us DE and for
yield pressure Dp as shown
DE = Ecp/Esp
Dp = qy/qyS
where
by Eqs. 7.32 and 7.33, respectively.
= soil modulus after cyclic loading
soil modulus for static loading
(7.32)
(7.33 )
= limiting pile-soil interaction stress (yield pressure)
after cyclic loading
qys =yield pressure for static loading
Poulos noted that a limited amount of data are available on degradation
factors and he suggested the use of data summarized by Idriss, et al.
164

(1978). Poulos prepared Fig. 7.17 from the Idriss data, with EC redefin-
ing the cyclic strain and E redefining a representative value of cyclic
cr
strain. The value of E can be varied to influence the cyclic degrada-cr
tion. The parameter t is defined by Eq. 7.34.
where
N = number of cycles
--- 0.3 .....----,,.----r----r------,
L.
Q)
-Q)
E
~ 0.2
o
a.
c
o
:;: 0.1
o
'0
oL.
01
Q) 0.0 ~---'----'----'---......
o 0.0 0.1 0.2 0.3 0.4
Cyl ic Strain Ratio, (Ec/Ecr)
Fig. 7.17. Degradation parameter t,
(from Poulos).
(7.34)
The effect of the rate of loading on the degradation was also consid-
ered. The degradation factors DE and Dp were multiplied by the rate fac-
tor DR that is defined in Eq. 7.35.
where
(7.35)
Fp = rate coefficient (limited data suggest a range of from
0.05 to 0.3)
Ar = reference loading rate (perhaps static loading)
A = loading rate
The computation procedure is initiated by selecting values of soil
modulus and yield pressure for each element and a distribution of dis-
placement is computed. The cyclic displacements, number of cycles, and
rate of loading are used to establish degradation factors that can be used
165

in the next cycle. The procedure is continued until convergence is
achieved. Poulos indicated that a computer program, not presented in his
paper, has been written to perform the analysis.
The presentation outlined above is insufficient to allow for the com-
putation of the behavior under cyclic loading of a given pile in a given
soil profile; however, the discussion does serve to illustrate the nature
of the problem.
Example Computation.
The pile and soil employed in the Broms method will be used in solv-
ing an example problem. The clay with an undrained shear strength of 1.0
kip/sq ft would have a value of Esp of approximately 40 kip/sq ft. The
pile length is assumed to be 40 ft; thus, the value of KR can be compu-
ted as shown below.
K = (30 x 106
)(421)(144) = 8.57 x 10-4
R (4 X 104)(480)4
From Fig. 7.13
M~~~ =(0.06)(Pt L).
Assuming that the maximum moment is 317 ft-k, the Pt at failure of the
pile is 132 kips. If it is assumed that a safe load of 50 kips can be
applied to the pile, the groundline deflection may be computed by use of
Eq. 7.28. The value of l yp was obtained from Fig. 7.10.
lyt = l yp EspL
(8.5)(50)= = 0.27 ft = 3.2 in.
(40)( 40)
The computed values seem large compared to values obtained from the method
of Broms. The error probably is in assuming that the elastic method can
be used to compute the bending moment in the pile at the failure condi-
tion.
An alternate procedure is to assume that the maximum deflection is
limited to 1.0 in. Substituting this value into Eq. 7.28 yields a lateral
load as shown below.
_ YtEspL (1.0/12)(40)(40)
Pt - I = 8 5 = 15.7 kips
yP •
The bending moment corresponding to the load of 15.7 kips is found to be
37.7 ft-k which yields a bending stress of 6.45 kip/sq in.
166

7.3 PRESSUREMETER METHOD
The use of results from pressuremeter tests to design piles under
lateral loading has been given attention in technical literature (Gambin,
1963; Baguelin and Jezequel, 1972; Baguelin, et al., 1978; Briaud, et al.,
1982). The method produces p-y curves that can be used with a computer
program to obtain pile response.
Introduction.
The pressuremeter as a design tool was developed by Menard (1956) and
there is a considerable body of literature on the device. No attempt is
made herein to present the developments that have led to the physical
devi ces that are in use and the means of performi ng tests with these
devices or to present the various theories that have been proposed for
interpreting the pressuremeter test.
The self-boring pressuremeter ·has been developed (Baguelin and
Jezequel, 1973) but the method outlined is based on the use of the stand-
ard Menard device. The rules for the prediction of p-y curves are empir-
ical to a large extent but a number of field experiments have been carried
out for the purpose of checking the validity of the method (Baguelin, et
al., 1978, p. 312).
Pressuremeter Curve.
A typical curve from the Menard pressuremeter is shown in Fig. 7.18.
The large volume change for a small increase in pressure at the early part
of the curve results from the drilling of an oversized hole prior to plac-
ing the pressuremeter probe. The next portion of the curve reflects a
linear relationship between pressure and volume changes. Then, the curve
becomes nonlinear and a limiting pressure is indicated. The following
definitions relate to the shape of the pressuremeter curve.
v = volume at start of the straight-line portion of the curve
o
and equal to the initial volume of the cavity
P
f
= pressure at the point where there is no longer a
straight-line relationship between pressure and volume
v = initial volume of the measuring cell
c
p2, = limit pressure (to be determined at point where initial
volume has doubled; thus, p2, is the the value of pressure
at the point where the volume of fluid is equal to 2vo +
v. Note that the volume of fl ui din the cell when the
c
167

..Q,)
...:::J
en
en
Q,)
...a..
- - - - - - - - - - - - - - -
3
Volume, L
Fig. 7.18. Typical curve from Menard pressuremete~
soil has been pushed back to its original position is v +
c
v but the volume indicator would register only v .o 0
Determining Pressuremeter Modulus
The assumption is made that the soil is elastic between Vo and vf
where the pressuremeter curve is a straight line. The following equation
gives the relationship between the shear modulus and the slope of the
straight-line portion of the pressuremeter curve (Baguelin, et al., 1978,
p. 153).
where
G
M
= v ~
!:,.v
G
M
= shear modulus from pressuremeter
v =volume of cavity
!:,.p/l1v = slope of curve between v0 and vf
(7.36)
The volume of the cavity changes between Vo and vf and the volume vm at
the midpoint is used.
168

In order to obtain the Young's modulus E from the shear modulus G,
the expression from mechanics is employed.
E
G = 2(1 + v) (7.37)
where
v =Poisson's ratio
The value of Poisson's ratio can vary widely for a soil but Menard chose a
value of 0.33 (Baguelin, et al., 1978, p. 154). Employing this value of
v, the following equation results.
where
E = 2.66V ~
M m f:,v
EM = the Menard modulus of deformation
vm = midpoint volume
Development of p-y Curves
(7.38)
The shape of the curves giving the soil response for a pile under
lateral loading is shown in Fig. 7.19. In the upper curve in Fig. 7.19,
the values of Pf and Pk', may be taken di rectly from the pressuremeter
curve. The curve can be drawn, then, by computing a value of k using onem
of the following two equations.
where
1- = 9~ Bo(;- 2.6S)a + ~ b b 0.6 mkM m 0 6Em '
1 b 4(2.65)a + 3a
18
b 0.6 m
Bo = a reference width, usually 60 cm (2 ft)
(7.39)
(7.40)
kM = slope of initial portion of Menard's soil response curve
(see Fig. 7.19)
b = pile diameter
EM = the Menard modulus of deformation
a = rheological factor (see Table 7.1)
As shown in Fig. 7.19, the ordinates for the curve where the depth l
is zero is one-half of those where the depth is equal to lc. The depth lc
is defined as the critical depth and is suggested by Menard to be about 2 b
for cohesive soils and about 4 b for granular soils. If there is a pile
169

B Pu=11~ - --- - - - - - -- - - - ---~--.....;.....;.;......---
Z2! Zc
~ = 1112
Z=Q
Fig. 7.19. Soil response curves proposed by Menard.
TABLE 7.1. VALUES OF RHEOLOGICAL FACTOR a
(after Baguelin, et al., 1948).
Soil Type Peat Clay Silt Sand
Over-consolidated 1 2/3 1/2
Norma lly
Consolidated 1 2/3 1/2 1/3
Weathered and/or
Remolded 1/2 1/2 1/3
170
Sand and
Gravel
1/3
1/4
1/4

cap, the critical depth is zero. Between the ground surface and the crit-
ical depth, the soil modulus should be AZkMgiven by Eq. 7.41.
1 + (Z/Z )
AZ = 2 c (7.41)
The Menard curves for a pile under lateral loading can be developed
from a pressuremeter curve (Fig. 7.18) and by use of Eqs. 7.38 through
7.41. The final step in the development of p-y curves as employed herein
is to convert the Menard curve. As noted in the above development, the
following conversions are necessary.
(7 .42)
and
E . =kMbSl
(7 .43)
With Eqs. 7.42 and 7.43, a family of p-y curves can be developed from the
Menard curves.
Example Computation
As an example of the use of the Menard approach to the analysis of
piles under lateral loading, a report by Baguelin and Jezequel (1971) is
employed. A test was performed on a stiff pile at a test site at Plancoet.
A profile of the pile and soil is shown in Fig. 7.20. The following soil
properties, based on information in the report, were used in analyses.
silt: total unit weight, 17.1 kN/cu m
submerged unit weight, 7.26 kN/cu m
£50' 0.020*
a (see Table 7.1),0.5
undrained shear strength, 35 kN/sq m*
sand: submerged unit weight, 7.35 kN/cu m
a, 0.33
undrained shear strength, 29 kN/sq m*
*Note: the values of €5B and shear strength are not used, of course, in
getting p-y curves from ressuremeter resul ts but are needed for use in
other methods. Shear strength for sand and for silt should more properly
be given in terms of an angle of internal friction; however, values of
undrained shear strength were given in the report and used in making com-
putations using methods other than the pressuremeter.
171

F;
0.6m
I 1.2m
Depth,m
I 0.8m..
~ --~--.
I I =0.003875 m4
2
I E1 =772210 kN-m
2
Silt ~ ...
3 I,
..
I..
I..
Sand
5 ::
... .,.
6 :.:.:
b = 949mm
fig. 7.20. Soil and test pile at P1anco~t.
The first three columns in Table 7.2 give results from the pressure-
meter tests performed at Plancoet and the other columns in the table show
the development of the p-y curves. The curves were employed to compute
the behavior of the pile under lateral loading. The results of the compu-
tations of deflection at the groundline are shown in Fig. 7.21, along with
the measured values. Also shown in Fig. 7.21 are results from other meth-
ods of analysis. The fact that the method of analysis employing COM624
gave the best agreement between analysis and experiment could, of course,
be fortuitous.
Chapter 9 presents comparisons of results from analysis with results
from a number of experiments. The compari son of the resul ts from the
pressuremeter are placed here because pressuremeter tests were not per-
formed at any of the other sites where experiments are studied.
7.4 METHOD USING CHARTS
A method of design could be used in which preliminary designs or
designs for small-scale projects could be made by reference to charts or
172

TABLE 7.2. POINTS ON p-y CURVES DERIVED FROM DATA FROM PRESSUREMETER
Depth Pf PQ, EM kM AZ AZkM Pl=Pf b Yl P2=pQ,b Y2
Z
bars bars bars bars bars kN/m m kN/m mm /cm /cm
0.5 0.85 1.44 20 0.57 0.632 0.36 80.8 0.023 136.8 0.056
1.0 0.95 1. 54 10.3 0.29 0.763 0.22 90.3 0.043 146.3 0.097
1.5 1. 05 2.07 13 0.37 0.895 0.33 99.8 0.032 196.7 0.094
2.0 1.15 1.68 9.4 0.27 1.0 0.27 109.3 0.042 159.6 0.081
3.0 1. 35 2.98 13.7 0.39 1.0 0.39 128.3 0.035 283.7 0.119
4.0 1. 55 3.23 18 0.51 1.0 0.51 147.3 0.030 306.9 0.096
5.0 1. 75 3.48 26.5 0.75 1.0 0.75 166.3 0.023 330.6 0.069
100
.......
z
~ 80
-0
0
0
-1 60
0
L-
a
+-
0
-1 40
Groundline Deflection (mm)
Fig. 7.21. Comparison of results from various analytical methods
with results from experiment at Plancoet.
173

diagrams. A design office could perform a number of analyses using a com-
puter program with assumptions being made about pile size and materials,
soil properties, boundary conditions, and nature of loading. The results
would then be analyzed and diagrams made for convenient use.
Examples of the kinds of charts that can be developed are shown in
Figs. 7.22 and 7.23. As noted in the figures, the charts were produced
for a reinforced-concrete pile with a diameter of 18 in. The soil has a
constant shear strength with depth and is below the water surface. The
loading is cyclic. The difference between the two charts is that Fig.
7.22 is for a soil with an undrained shear strength of 1000 lb/sq ft and
Fig. 7.23 is for a strength of 2000 lb/sq ft. It is of interest to note
that the limiting condition in Fig. 7.22 is deflection. The maximum
deflection has arbitrarily been set at 10 percent of the diameter of the
shaft or 1.8 in. On the other hand, the limiting condition in Fig. 7.23 is
bending moment with the ultimate moment on the pile being reached before
the limiting deflection.
10rr------~--~-,~-----,------~------~------_r------_r------~
Ultimate moment capacity of shaft .. 950 in.- kips
Reinforcement percentage. • I 'Yo
:-... Flexural ri9idity. EI • 3.5 x 109
Ib-sq in.
8 ' ' Shoff Len9th • 40 ft
Soil unit wei9ht. • 57.5 Ib/cu ft
Undrained shear stren9th •c =1000 Ib/Sq ff
..... 1nltal k .. 500lb/cuin.
6
• 50 .. 0.006
0..1- 4
2
-150 o 150 300 450 600 750
Mr ( in.- kips)
Fig. 7.22. Values of maximum bending moment in an 18-in. diameter
concrete shaft in clay (after Reese and Allen).
174
900

II)
Q.
:i:
.
a:-
15~--~--~~----T-------'-------'-------~-------r-------r------~
9
6
~300
Shaft properties are the same as those in FiQ. 2.32
Soil unit weioht, = 62.51b/tt
Undrained shear strenoth, • 2000 Ib 1ft
...~ Inital k • 600 Ib/ln.
,~ • • 0.0045
'~ (p- y criteria are for stiff
cloy below water surface)
-150 o
MT , in.- kips
Fig. 7.23. Values of maximum bending moment in an
18-in. diameter concrete shaft in clay
(after Reese and Allenl
900
If the moment and shear at the top of the pile (drilled shaft,
drilled pier, caisson) are known, it is a simple matter to enter a curve
and arrive at the maximum bending moment in the pile. The difficulty
arises with the specifics: pile dimensions, shear strength and other soil
properties, and nature of loading. An extremely large number of charts
would be necessary if a chart is at hand for conditions that are approxi-
mately equal to those at a site.
An office could encounter designs that are similar enough in nature
to each other that it would be justified to develop a series of charts.
Even that approach might be questionable if the office has convenient
access to a digital computer of appropriate power.
Simplified methods of design can be more useful if based on results
of field load tests on piles of a certain type. Two proposals of this sort
will be presented.
175

Manoliu (1976) examined the results from 27 different sites of 160
field tests of precast concrete piles that were installed by driving. The
soils ranged from sands to gravels to soft clays. Results were plotted
and empirical curves were fitted through the plotted points. There was
considerable scatter but Manoliu suggested that the curves shown in Fig.
7.24 can be used for preliminary design. For a given design, a groundline
deflection Yt can be selected and the value of k obtained from Fig. 7.24.
Then the relative stiffness factor T can be computed from Eq. 6.75. The
lateral load can then be computed depending on pile-head restraint, using
methods presented in Sect. 6.4. Manoliu indicated that the piles being
tested genera lly coul d be defi ned as long with a Zmax greater than 4.
Bhushan, et al. (1981) reported results on full-scale tests of drilled
shafts that were installed in sand. The results were analyzed and Fig.
7.25 was prepared. The fi gure can be used for pre1imi na ry designs of
40
35
30
gravelsj gravelly sands
E 25 clean sands j very stiff clays()
::::I medium sands; stiff clays()
20........
Z fine sands, silty sands
- soft si It y and clayey soils.:.:
15
10
5
o--~~--------~--~~--~--~
00.2 0.5 1.0 1.5 2.0 2.5 3.0
Yt' em
Fig. 7.24. Empirical curves showing response of driven,
precast concrete piles (Manoliu, 1976).
176

0 35
50
40
c· 80
:::s(,)
, 120..0
-.JI:
160
200
240
0 0.02 0.04 0.06 0.08 0.10
Ytl b
Fig. 7.25. Empirical curves showing response of
drilled shafts (Bhushan, 1981 )
drilled shafts in sand in a manner similar to that suggested by Manoliu.
The curves in Fig. 7.25 are for sands above the water table. For sands
below the water table, Bhushan suggests that the values of k from Fig.
7.25 be halved. Bhushan did not restrict his curves to long piles; if the
drilled shafts are short, the appropriate values of the nondimensional
deflection coefficients should be employed.
7.5 REFERENCES
Baguelin, F., and Jezequel, J. F., Etude experimentaJe de founda-
tions profonde rigides sollicitees horizontalement, Report 71-B-687,
Laboratoire Central de Ponts et Chausees, Paris, May 1971.
Baguelin, F., and Jezequel, J. F., Etude experimentale du comporte-
ment de pieux sollicites horizontalement, Annales de l'Institute Tech-
nique du Batiment et des Travaux Publics, Supplement au No. 297, Serie
SF/I, September 1972, pp. 153-204.
177

Baguelin, F., and Jezequel, J. F., Le pressiometre autoforeur,
Annales de l'Institut Technique du Batiment et des Travaux Publics, Sup-
plement No. 307-308, Serie Sols et Foundations, No. 97, July-August 1973,
pp. 133-160.
Baguelin, F., Jezequel, J. F., and Shields, D. H., The Pressuremeter
and Foundation Engineering, Trans Tech Publications, Clausthal, Germany,
1978.
Bhushan, Kul, Lee, L. J., and Grime, D. B., Lateral Load Tests of
Drilled Piers in Sand, Drilled Piers and Caissons, American Society of
Civil Engineers, 1981, pp. 114-130.
Briaud, Jean-Louis, Smith, T. D., and Meyer, B., Design of Laterally
Loaded Piles Using Pressuremeter Test Results , Proceedings, Symposium on
the Pressuremeter and its Marine Applications, Paris, France, April
19-20, 1982.
Broms, Bengt B., Lateral Resistance of Piles in Cohesive Soils,
Proceedings, American Society of Civil Engineers, Vol. 90, No. SM2, March
1964a, pp. 27-63.
Broms, Bengt B., II Latera1 Resistance of Pil es in Cohesion 1ess
Soil s, II Proceedi ngs, Ameri can Soci ety of Ci vil Engi neers, VoL 90, No.
SM3, May 1964b, pp. 123-156.
Broms, Bengt B., Design of Laterally Loaded Piles, Proceedings,
American Society of Civil Engineers, Vol. 91, No. SM3, May 1965, pp.
79-99.
Gambin, M., Calcul du tassement d'une foundation profonde en fonc-
tion des resultats pressiometriques, Sols-Soils, Vol. II, No.7, Decem-
ber 1963, pp. 11-31.
Idriss, 1. M., Dobry, R., and Singh, R. D., Nonlinear Behavior of
Soft Cl ays Duri ng Cycl i c Loadi ng, II Proceedi ngs, Ameri can Soci ety of Ci vi 1
Engineers, Vol. 104, No. GTI2, 1978, pp. 1427-1447.
Manoliu, 1., Lateral Bearing Capacity of Precast Driven Pile, Pro-
ceedi ngs, Si xth European Conference on Soi 1 Mechani cs and Foundation
Engineering, Vienna, 1976, pp. 515-518.
Menard, L. F., An Apparatus for Measuring the Strength of Soils in
Place, M.S. Thesis, The University of Illinois, 1956.
Poulos, H. G., Behavior of Laterally Loaded Piles: I-Single Piles,
Proceedings, American Society of Civil Engineers, Vol. 97, No. SM5, May
1971a, pp. 711-731.
Poulos, H. G., Behavi or of Latera lly Loaded Pil es: II-Pil e Groups, II
Proceedings, American Society of Civil Engineers, Vol. 97, No. SM5, May
1971b, pp. 733-751.
178

Poulos, H. G., IILoad-Deflection Prediction for Laterally Loaded
Piles,1I Austrian Geomechanics Journal, Vol. G3, No.1, 1973, pp. 1-8.
Wiley, New York, 1980.
Poulos, H. G., IISingle Pile Response to Cyclic Lateral Load, Pro-
ceedings, American Society of Civil Engineers, Vol. 108, No. GT3, March
1982, pp. 355-375.
Prakash, S., IIBehavior of Pile Groups Subjected to Lateral Loads,1I
Unpublished Dissertation, University of Illinois, 1962.
Reese, L. C., and Allen, J. D., Drilled Shaft Manual, Vol. II, Struc-
tural Analysis and Design for Lateral Loading, I.P.77-21, U. S. Department
of Transportation, Washington, D. C., July 1977.
Terzaghi, K., IIEvaluation of Coefficients of Subgrade Reaction,1I
Geotechnique, Vol. V, 1955, pp. 297-326.
Vesic, A. S., IIBending of Beams Resting on Isotropic Elastic Solid,1I
Proceedings, American Society of Civil Engineers, Vol. 87, No. SM2, April
1961a, pp. 35-53.
Vesic, A. S., Beams of Elastic Subgrade and the Winkler's Hypoth-
esi s, II Proceedings, Fi fth Internationa1 Conference on Soi 1 Mechani cs and
Foundation Engineering, Vol. 1, 1961b, Paris, France, pp. 845-850.
7.6 EXERCISES
7.1 Use the pile shown in the example for the Broms method and solve
for the ultimate capacity of the pile as a function of depth using the
soil profile. Use Broms' equations with an e of 3 ft.
7.2 Use the data in problem 1 and a long pile and select a service
load at one-third the ultimate capacity of the pile. Compute the ground-
line deflection, using Broms and Poulos methods.
7.3 Repeat problem 1 using the soil profile in Fig. 3.13.
7.4 Repeat problem 1 using a sand with an angle of internal friction
of 340
and a submerged unit weight of 62.4 lb/cu ft.
179

CHAPTER 8. STRUCTURAL DESIGN OF PILES AND DRILLED SHAFTS
8.1 NATURE OF LOADING
A pile or drilled shaft that supports a bridge, a bridge abutment, or
a retaining wall will normally be subjected to an axial load, a lateral
load, and a moment. An example is a bridge bent shown in Fig. 8.1.
I ~ Trud
0 II IL 0
I I II
)---
'iJ
-::
Current
-- ....
.'7/0 , ,
~
- --
Fig. 8.1. Bridge bent.
The live loads from trucks and other vehicles as well as the dead load of
the structure must be sustained by the deep foundations. In addition to
the vertical loads from traffic there can be transverse loads due to wind
and current. Also, the vehicles may apply longitudinal loads because of
braking. In short, there can be a complicated system of forces applied to
a deep foundation used in a bridge or used in other structures found in
transportation facilities. Thus, a deep foundation at the groundline will
be subjected to an axial load, a lateral load, and a moment. Because the
pile response may be nonlinear, iterations between the piles and the
Preceding page blank 181

superstructure may be necessary in order to satisfy the conditions of
equilibrium and continuity. That is, the loadings on a pile may have to
be estimated and the pile-head movements computed. With loads and move-
ments, the stiffness of the pile can be found; these stiffnesses can then
be employed in the analysis of the superstructure and the process repeated
until convergence is achieved.
It is possible that the shear and moment that act on'a pile head do
not lie in the same plane. However, the assumption is implicit herein
that all forces on a pile head do lie in the same plane; or, if not, that
the pile behavior can be found by superposition.
8.2 FAILURE MODES
The most common failure mode in a pile or drilled shaft is excessive
stress. The computation of stress from an axial load is straightforward
and the procedures described herein can be employed to obtain the maximum
bending moment. The computation requires a knowledge of the bending
stiffness of the pile and of the ultimate moment the pile can sustain.
For steel piles, the stiffness and ultimate moment can be found in tables;
for drilled shafts with steel reinforcement, the ultimate moment can be
computed by Computer Program PMEIX (see Appendix 5). The stiffness, con-
sidering a cracked section, may also be computed by PMEIX or may be based
on the gross moment of inertia of the concrete section.
Excessive deflection of a pile or drilled shaft may also constitute
faflure. The excessive deflection, which may be computed by procedures
described herein, may be due to a soil failure where the bottom of the
pile has moved laterally. Or excessive deflection may be a result of too
much elastic deformation of the pile itself.
A deep foundation may also fail by buckling. Buckling is unlikely,
however, if there is a pile cap at or near the ground surface. Soil that is
weak can usually provide sufficient lateral restraint that buckling is not
a problem.
The illustration in Fig. 8.1 shows that the deep foundation is con-
t i nuous to the pil e cap. There is no specifi c di scuss i on of buck1i ng
included herein; however, Computer Program COM622 described earlier can
be used to analyze the pile-column. The design loads (service loads times
the factor of safety) are applied at the top of the unsupported
pile-column, then the axial load is increased incrementally until the lat-
182

eral deflection becomes excessive, and at that axial load the pile is con-
sidered to have buckled. The procedure is also applicable to a pile in
weak soil with little or no unsupported length.
8.3 CONCEPTS OF DESIGN
Structural Steel
Two design philosophies of steel are in current use. The working
stress method has been in principal use and is still used by many design-
ers. According to this philosophy, a structural member is designed such
that stresses computed under the action of IIworking or service loads do
not exceed some predetermined values. These allowable stresses are given
by the AISC specifications (American Institute of Steel Construction,
1978) or by the AASHTO specifications (Standard Specifications for High-
way Bri dges, Ameri can Associ ati on of State Hi ghway and Transportation
Officials, 1977).
The other design philosophy is generally referred to as limit states
design and more recently as the load-and-resistance-factor design (LFRD).
In this approach the service loads are factored and the strength at an
ultimate limit state is checked to ensure that the factored loads can be
achieved.
Because working-stress design is still strong in its popularity and
the principal method specified by both AISC and AASHTO, it is recommended
for use in the analysis of members of structural steel.
Reinforced Concrete
In general, most present-day d~sign of reinforced concrete is being
done using ultimate strength concepts. The design of a member is designed
based on the ultimate strength of the member; the m~thod is similar to the
LFRD method for structural steel. In the ultimate strength approach, as
in the LFRD method for steel, the service loads are factored. The'com-
puted ultimate strength of the member is reduced by a capacity-reduction
factor. If the factored loads lead to a required capacity that is less
than the computed value of reduced ultimate capacity of the section, the
design is adequate. In general, the Building Code Requirements for Rein-
forced Concrete of ACI (American Concrete Institute, 1977) or AASHTO
(1977) are used in design.
183

Timber
If a pile or drilled shaft is to be made with timber, a work-
ing-stress method is recommended as outlined in the Timber Construction
Manual of the American Institute of Timber Construction (1974). Similar
to structural steel, the allowable stresses prescribed in a timber spec-
ification would ensure an adequate factor of safety against failure. The
stresses in the timber pile at service load levels would'be compared to
the allowable levels to ensure an adequate design.
8.4 DESIGN OF A STRUCTURAL STEEL MEMBER
For the working-stress method of structural steel design, the design
loads are based on in-service conditions. The actual dead weight of the
as-built structure, the calculated values of live load such as truck load-
ing on a bridge, and the effects of other loading such as wind are used.
Frequently the design of a steel pil e is controlled by the soil
resistance to axial loading. The pile is then checked to see that the
steel is not over-stressed. Sometimes, however, the design may be con-
trolled by the stresses in the steel. In this latter case, the
step-by-step procedure outlined below may be employed.
1. Determine the working or in-service loads acting on the
member.
2. Select a member using previous experience or preliminary
analysis.
3. Determine the EI value (stiffness) of the structural steel
section for use in the analysis of the member.
4. Using the in-service loads and the EI of the pile, obtain
the maximum shear and moment in the member. A computer pro-
gram or another method may be used.
5. For the calculated maximum moment and shear, determine the
stresses in the section.
6. Compare the actual stresses in the member to the allowable
stresses. If the actual stresses are slightly less than the
allowable stresses, the member is adequate. If the member
is over-stressed or if it is grossly under-stressed, a new
selection is made and the process is repeated.
184

The equations from the AISC specifications are listed here for con-
venience. To check the combined stress state for strength:
(8.1)
where
f = axial stress from service loadsa
fb = bending stress from service loads
Fy =yield stress of steel
Fbx = allowable stress in bending
(F
bx
determined from AISC section 1.5.1.4).
Equation 8.1 has been written to indicate bending about only one axis; it
is assumed that there will be no biaxial bending. To check the shear
stress:
where
F = 0.4 F
v y
(8.2)
Fv =allowable shear stress.
7. When a member of appropriate size is found, check for other
failure modes such as buckling as outlined earlier.
In the example computations presented in earlier chapters, it was
shown that the bending stress in a pile is maximum at or near the ground
surface and decreases rapidly with depth. If a pipe pile is employed, a
pipe with a thicker wall may be selected for the top several feet and a
section with a thinner wall used below. If a structural shape is to be
employed, it may be possible to add plates in the top few feet to with-
stand the bending stresses and achieve overall economy.
Example Problem
The data for an example problem are shown in Fig. 8.2. It is desired
to determine if the steel pipe has adequate strength for the given condi-
tions. Previous analyses are assumed to have shown that deflection does
not control nor is buckling of concern.
Because there is bending about one axis only, AISC Eq. 1.6-1b b'ecomes
The computation of the stresses to substitute into Eq. 8.1 is as follows:
185

~
'VII
Fy = 36kips/sq in.
E = 29,000kips/sqin.
I = 5,438 in4
b = 25 in.
Px = 750Kips
~Mt
~
Moment Shear
Mmax= 5000 in.-kips Vmax = 250
Fig. 8.2. Example to demonstrate the analysis of a steel pipe.
A = n(ro
2 - ri 2) = n[(12.5)2 - (11.5)2] = 75.4 sq in.
fa = 750/75.4 = 9.95 kips/sq in.
f = Mc = (5000)(12.5) = 11 49 k' / .
b I 5438 • lpS sq In.
(see AISC Section 1.5.1.4.1, paragraph 7)
Fb = 0.66Fy if ~ 3i
OO
Y
where d =depth of section and t =thickness of wall.
£!. = ~~ = 25
t 1
3300 = 3300 = 92
F 36y
25 92 therefore, Fb =0.66 Fy
Fbx =0.66(36) =23.8 kips/sq in.
Substitution into Eq. 8.1 yields the following:
~i~~ + 1i;.~9 = 0.461 + 0.483 = 0.943 1.0.
Therefore, the section is satisfactory for combined axial load and flex-
ure.
186

To check shearing stresses, AISC Specification 1.5.1.2.1 is con-
sulted.
F =0.40 Fv y
The computation of the stresses to use in checking the adequacy of the
section in resisting shear is as follows:
Fv = 0.4 Fy =(0.4)(36) =14.4 kips/sq in.
Av =area resisting shear
~(0.5)(Agross) = 37.7 sq in.
fa = Vmax/Av =250/37.7 =6.63 kips/sq in.
6.63 14.4.
Therefore, the section is satisfactory with regard to shearing stresses.
8.5 DESIGN OF A REINFORCED CONCRETE MEMBER
Because design of reinforced concrete is currently done using ulti-
mate strength concepts, the loads on a structure at failure are above ser-
vice load levels and therefore factored loads are used. The load factors
to be used depend on the type of load and the governing concrete code. The
loads which need to be considered include dead load, live load, wind load,
earthquake load, thermal load, creep and shrinkage effects, earth pres-
sure; each has a specified load factor. The magnitude of the load factor
for each type of load depends on which specification has been adopted. If
the pile or drilled shaft is for a bridge structure, the governing code is
probably AASHTO (1977). If the foundation is for a building, the ACI Spe-
cification (1977) has probably been adopted by the local building authori-
ties. Both AASHTO and ACI require the member selected to have adequate
strength for different loading combinations. The design of the member is
based on the greatest required strength.
Computation of Bending Stiffness
A value of bending stiffness EI must be determined in order to com-
pute the behavior of a pile or drilled shaft under lateral loading. Con-
crete has a low tensile strength and the assumption made is that the
concrete cracks if there are tensile stresses. Thus, the EI of a rein-
forced concrete section wi 11 depend on the magni tude of the bendi ng
moment. Some i nvesti gators have suggested that the effective bendi ng
stiffness lies between that of the gross concrete section and that of the
fully cracked section. Because the magnitude of bending stiffness does
187

not have a large effect on deflection, moment, and shear that are computed
in the analysis under lateral load, some investigators prefer to use the
stiffness of the gross concrete section. However, in the example that
follows the stiffness of the cracked section is used.
Because the bending stiffness of a reinforced-concrete section
depends on the axial load and bending moment, it follows that the EI var-
ies along the length of a pile. If the magnitude of the EI were critical
to the solution of the bending moment, a computer program for the computa-
tion of bending stiffness could be incorporated as a subroutine into the
computer program for the analysis of a laterally loaded pile. In such a
case, another level of iteration would be required to compute the behavior
of a reinforced-concrete pile. The use of a variable stiffness in the
analysis of a laterally loaded pile appears undesirable at present in view
of other uncertainties of greater importance.
In any case, however, a computer program is needed in order to com-
pute the ultimate moment that can be sustained by a reinforced concrete
section. The program can also compute the bending stiffness. Computer
Program PMEIX is described in Appendix 5.
Program PMEIX produces data giving moment versus curvature for a giv-
en level of axial load. The M-¢ curve varies for different levels of
axial load as is shown in the following figure. The value of the ultimate
moment that can be sustained may be taken directly from such a figure as
Fig. 8.3.
The EI which may be used in analyses is the slope of the M-¢ curve
after the section has cracked for the known level of axial load.
llM
(EI)effective = ll¢
In using a program like PMEIX, which generates a curve for a given axial
load like one of those shown in Fig. 8.3, the axial load should be the fac-
tored axial load divided by the applicable capacity reduction factor, ¢.
The reason is that the strength of the concrete section will be checked at
its ultimate strength which is at the level of the factored loads.
In lieu of a more accurate analysis such as that given by Program
PMEIX, ACI equations (10-9) and (10-10) can be used to detemine an EI val-
ue. However, these equations can be very conservative.
Creep of concrete under sustained loads also has an effect on stiff-
ness. As stiffness decreases, deflections increase producing sec-
188

-c
Q)
E
o
:E
P
,------......x P. = 0.3
, 0
16M
I
Curvature, cp
x ~ = 0.0
o
P =applied axial load
Po =strength under pure
axial load
Fig. 8.3. Relationship between moment and curvature
for a concrete member.
ond-order or P-de lta effects. To take thi s into account, ACI (Sec.
10.11.5.2) recommends reducing the effective EI value by the term (1 + ~d)
where ~d is the ratio of the maximum factored dead load moment to the max-
imum factored total load moment.
(
EI) - (EI)effective
creep 1 + Bd
~d is always positive and has a maximum value of 1. Because for the pile
problem these moments are not known until an analysis is done, which in
turn depends on the EI value used, taking a value of ~d = 1 would produce
the smallest EI.
The following is a step-by-step procedure which can be used in the
analysis and design of a reinforced concrete pile or drilled shaft.
1. Determine the loads acting on the structure and then use the
appropriate load factors. The following load factors for
gravity loads and capacity-reduction factor for a spirally
reinforced shaft are specified by the American Concrete In-
stitute.
Load factor for dead load is 1.4,
Load factor for live load is 1.7, and
Capacity-reduction factor ~ is 0.75.
189

2. By previous experience or preliminary analysis select a
member size and reinforcement.
3. Take the factored axial load, divided by the appropriate
capacity-reduction factor, and run Program PMEIX to deter-
mine a stiffness (EI) value at the level of axial load to be
used in the shaft analysis program.
4. Run computer program COM622 using the factored loads which
have also been di vi ded by the appropri ate capaci-
ty-reduction factor. The shaft analysis program will yield
the maximum shear and moment in the member.
5. From the earlier run of PMEIX the ultimate moment capacity
may be determined. Multiply this nominal ultimate moment
by the appropri ate capaci ty-reducti on factor to get the
allowable ultimate moment.
6. Compare the allowable ultimate moment to the maximum moment
computed in the analysis of the pile under lateral loading.
If the allowable ultimate moment is equal to or slightly
greater than the maximum moment from the analysis, the sec-
tion is adequate. If the section is under-sized or grossly
over-sized, a new section is selected and the analysis is
repeated. Also, check the shear capacity of the section.
In performing steps 5 and 6, the relevant equations from the code are
to be employed. The equations from the ACI speci!ications are listed here
for convenience.
where
where
M = ~M
u N
M = allowable ultimate momentu
~ = capacity-reduction factor
MN = nominal ultimatelmoment.
Vu =~(Vc + Vs)
v = shear capacity of a member
u
Vc = shear contribution of the concrete
V = shear contri butfon of the reinforcing
s steel
~ = capacity-reduction factor for shear (0.85).
190
(8.3)
(8.4)
(8.5)

where
Nu = factored axial load
A = gross area of concreteg
b
w
= diameter of circular section
d =distance from extreme compression fiber to centroid
of tensile reinforcement.
7. The final step is to check for other failure modes such as
buckling when a member of appropriate size is found. The
procedure for checking for buckling was outlined earlier.
As noted in the discussion of the structural steel member, the steel
rei nforcement ina rei nforced concrete pi 1e may be needed on ly in the
upper portion of the pile. The designer may be able to achieve consider-
abl e economy by matchi ng the requi red rei nforcement to the computed
stresses.
Example Problem
The data for an example problem are shown in Fig. 8.4. It is desired
to determine if the reinforced concrete section has adequate strength for
the given conditions. Previous analyses are assumed to have shown that
deflection does not control nor is buckling of concern. Load factors and
capacity-reduction factor given earlier are used to compute loadings.
Pux =1.4(350 ) + 1.7(150 ) =745 kips
Mut = 1.4(lxl06
) + 1.7(0.5x106
) = 2.25 x 106
in.-lb
Put = 1.4(25k) + 1.7(10k) = 52 kips
The bending stiffness EI is determined by using Computer Program PMEIX
(see Appendix 5). The axial load to be used in PMEIX was found as follnws:
Pux 745
P
N
= - = - = 993 kips.
x ¢ 0.75
Using PMEIX at an axial load level of 993 kips, the moment-curvature (M-¢)
relationship shown in Fig. 8.5 was determined. Figure 8.5 was analyzed
and the following bending stiffness was obtained.
(EI) = llM ~ 0.67 x 10
7
in.-lb = 8.4 x 1010 lb-sq in.
effective ll¢ 0.80 x 10- 4 rad/in.
Taking into consideration creep yields:
(EI)effective 8.4 x 1010
EI = = 4.2 x 1010 1b- sq in.
1 + Bd 1 + 1
The factored loads divided by the capacity-reduction factor resulted in
the moment and shear shown in Fig. 8.6.
191

r7'.• •.'
30 . #8
Given:
As-totol = 12.0 (0.79sq in.) =9.48sq in.
f~ = 4,000 Ib/sq in.
fy = 60,000 Ib/sq in.
Es = 29 x 106 1b/ sq in.
Ec = 57.000~ = 3,605,000 Ib/sq in.
Cover = 3 to center of bar
Spiral Column
Assume Bd = I
Service Loads Given:
p. = ~DL + PxLL = 350 kips + 150 kips
M t = MtDL + MtLL = Ix 106in.- lb 6
+ 0.5 x 10 in.-Ib
~ = ~DL + PtLL = 25kips + 10kips
Fig. 8.4. Example to demonstrate the analysis of a
reinforced concrete pile.
------- 1.5..0
T.
M-cp for P =973 kipsc
0
)( __ ---0
1.0 ....--- ----,Jf---- -----
~ Ultimate Nominal CapacityI~
I
-c
tQ)
E 0.5 6M0 I
,
Z
1.0 2.0 3.0
Curvatur~, cp (1x 10 • rod/ in.)
Fig. 8.5. Relationship between moment and curvature
for the example problem.
192

6
2.25 x 10 = 3 x Id in.-/b
0.75
52 kips - 69k'
0.75 - IpS
Fig. 8.6. Values of load to be employed in computer program
to analyze laterally loaded drilled shaft.
The bending stiffness of 4.2 x 10 10
sq in.-lb was employed, along
with the loading shown in Fig. 8.6, and Computer Program COM622 was
employed. The details of the computer analysis are not shown here but the
significant results are presented in Fig. 8.7. The allowable ultimate
values are now compared to those from analysis. From the previous M-¢
curve, and using engineering judgement, the ultimate nominal moment
capacity for the given level of axial load is 0.95 x 10 7
in.-lb. The rea-
son the maximum value of about 1.1 x 107
in.-lb was not chosen is that at
this value the EI value is much less, yet the shaft analysis program used
an EI value in the initial range. The 0.95 x 10 7
in.-lb value represents
-the point on the M-¢ curve where the EI value begins to be drastically
reduced. The allowable ultimate moment Mu is equal to ¢MN. Thus,
Mu = 0.75(0.95 X 10 7
) = 7.1 X 106
in.-lb.
The maximum moment was found from computer analysis (Fig. 8.7) to be 6 x
106
in.-lb; therefore M M and the section is satisfactory for bend-u max
ing.
To check the shear, Eqs. 8.5 and 8.4 are used.
V = 2 [1 + 745,000 ] y4000 (30)(24) = 1.4 x 10slb = 140 kips
c (2000)(707)
Vu =0.85 Vc =118 kips
The maximum shear found from the computer analysis was 75 kips; therefore,
the section is adequate for shear because V V . There is no need tou max
compute the contribution to shear of the steel reinforcement.
193

6
Mmox= 6 x 10 in·lb Vmox=75kips
Fig. 8.7. Results from computer analysis of the drilled shaft.
The computed maximum moment at ultimate is about 15% less than the·
allowable maximum moment, and the capacity of the section in shear is more
than adequat~. Therefore, the designer might consider a reduction in the
size of the section, in the amount of reinforcing steel, or in the
strength of the concrete. Any change in the section would have to consid-
er a number of factors other than the strength of the section.
8.6 REFERENCES
Manual of Steel Construction, American Institute of Steel Con-
struction, Eighth Edition, Chicago, Illinois, 1978.
Standard Specifications for Highway Bridges, American Association of
State Highway and Transportation Officials, Twelth Edition, 1977.
BUilding Code Requirements and Commentary for Reinforced Concrete
CACI 318-77), American Concrete Institute, Detroit, Michigan, 1977.
Timber Construction Manual, American Institute of Timber Con-
struction, Wiley, New York, 1974.
194

8.7 EXERCISES
8.1 Select a steel structural shape to replace the pipe pile in the
example. Assume no change in the moment and shear obtained from the com-
puter program.
8.2 Reduce the amount of steel in the reinforced concrete section
and re-work the example. Assume the moment and shear obtained from the
computer program to be unchanged but re-run PMEIX to obtain the moment
versus curvature relationship for the new section.
195

CHAPTER 9. CASE STUDIES OF SINGLE PILES UNDER
LATERAL LOADING
9.1 INTRODUCTION
For single piles under lateral loading, a comparison of results from
analyses with results from experiments is useful to provide an understand-
ing concerning the quality of the analytical methods. The presumption is
made that the experiments reveal the true behavi or of a speci fi c pi 1e
under lateral loading at a specific site. Emphasis in the analyses will
be placed on the computer method described in detail herein; however, the
Broms and Poulos methods presented in Chapter 7 will also be employed.
Separate sections in this chapter are presented for piles in sand and
for piles in clay. The response of the soil is so different in these two
instances that separate comparisons are desirable.
Several cases are selected for study. In order to perform the ana-
lyses it is necessary that information be available on pile dimensions and
properti es of the pil e materi a1, on the engi neeri ng properties of the
soil, on the magnitude of load and its method of application, and on the
response of the pile to loading. There are a limited number of cases in
the technical literature where the above information is available.
Pri or to presenting the compari sons between results from ana lyses
ind results from experiments, it is of interest to present the results of
some parametric studies. Only the computer method is employed in these
parametric studies.
The principal aim of the parametric studies is to investigate the
influence on pile response of various parameters. Most of the parameters
that were investigated i nvo1ved soil properties, but some studi es were
aimed at investigating the influence of the bending stiffness EI of a
pile. Studies were made of the four methods of predicting soil behavior
for clay and of the single method of predicting the response of sand. The
initial parameters selected for the soil are shown in Table 9.1 and for
the pile are shown in Table 9.2. As shown in the following paragraphs,
the effects of varying some of these parameters are investigated.
The general procedure employed in these studies follows that used by
Meyer (1979). The parameters varied for the clay criteria are: c, £50'
k, and EI; the parameters varied for the sand criteria are: ~, r, k, and
EI. Cyclic loading was employed in all cases because that is the condi-
Preceding page blank 197

TABLE 9.1. INITIAL PARAMETERS FOR SOIL.
Soil Soft Clay Stiff Clay Stiff Clay Uniti ed Sand
Properties below W. T. below W. T. above W.T. Criteria
c(lbjsq in. ) 6.0 60 15 6.0
Sso 0.02 0.002 0.005 0.01
~ 35°
y(lbjcu ft) 45 45 110 45 45 to 75
k(lbjcU in.) 500 200 100 60
TABLE 9.2. INITIAL PARAMETERS FOR PILE.
b (in.) 16
EI (lb-sq in.) 3.13 x 1010
L (ft) 75
tion most often encountered in practice. The pile head was assumed to be
free to rotate.
The percentage change of the input parameters was computed as fol-
lows:
t:,.% = New value - initial value 100. (9.1 )
Initial value
The change of percentage of the input parameters was based on the sensi-
tivity of the results to the change. A change of ±50% was used in several
instances.
198

A study was made also concerning the depth of embedment. As noted
earlier, short piles fail because of soil failure and the depth of embed-
ment is critical in such instances. Computations show that a small
increase in the depth of embedment can cause a significant increase in the
lateral capacity.
9.2 PARAMETRIC STUDIES OF PILES IN CLAY USING COMPUTER
METHOD
Soft Clay below Water Table
The results of analyses using the Matlock (1970) criteria are shown
in Figs. 9.1 through 9.3. Figure 9.1 shows the effects on maximum bending
moment and deflection of varying the undrained shear strength from 432
lb/sq ft to 1296 lb/sq ft. As might have been expected, the maximum bend-
ing moment and the deflection show almost a linear variation with the
undrained shear strength.
Figure 9.2 shows the effects on maximum bending moment and deflection
of varying ESO from 0.01 to 0.03. The changes in bending moment and
deflection are relatively small.
Fi gure 9.3 shows the effects on maximum bendi ng moment and on
deflection of varying the EI of the 16-in.-diameter pile from 1.S7 x 10 10
lb-sq in. to 4.70 x 10 10
lb-sq in. The effect on the bending moment is
negligible to small but the effect on the deflection is significant. A
tripling of the EI results in about a SO% decrease in deflection.
Stiff Clay below Water Table
The results of analyses using the Reese, Cox, Koop (197S) criteria
are shown in Figs. 9.4 through 9.7. Figure 9.4 shows the effects on maxi-
mum bendi ng moment of varyi ng the undrained shear strength from 4,320
lb/sq ft to 12,960 lb/sq ft. At first glance it is surprising that the
bending moment and deflection were not affected more by the change in
shear strength; however, the loads are relatively small. As the bending
moment approaches its ultimate, more differences in the curves in Fig. 9.4
would develop.
Figure 9.S shows the effects on pile response of changing ESO from
0.001 to 0.003. The changes in bending moment and deflection are rela-
tively small except at the loads of larger magnitude.
Figure 9.6 shows the effects on pile response of changing the initial
values of k (where the initial portion of a p-y curve is defined by Es =
199

30~------~----~------~~ 30~------w---~--~------~
J)
Cl.
~20
-0
C
o
-.J
c...Q)
-c
-.J
/
/
/
·_·--50%
0%
----+50%
OL-------~------~------~
2
Maximum Moment, in:-Ib x 106
-C
C
.3
,/'
/
. -·--50%
0%
--- +50 %
O~------~--------~------~
2
Groundline Deflection, in.
Fig. 9.1. Comparison between results for ±SO percent variation in
c for soft clay below water table.
30 30~------~~~~~------~
J)
Cl.
~ 20 .920
~
0 -
C 0
o C
~ 0
C
...Q)
-C
-.J
·_·--50%
- 0%
---+50%
OL-------~------~------~
2
Maximum Moment, in:-Ib x 106
...J
·_·--50%
0%
---+50 %
O~------~------~~------~
2
Groundline Deflection I in.
Fig. 9.2. Comparison between results for ±SO percent variation in
ESO for soft clay below water table.
200

30~------.-------~~----~ 30~------or.~~---,~------,
0
o
o
...J
010
...Q)
-o
...J
·_·--50%
0%
---+50%
2
Maximum Moment, In.-Ib x 106
3
~20
.::II!
. -·--50%
0%
---+50%
2
Groundllne Deflection, in.
Fig. 9.3. Comparison between results for ±50 percent variation
in EI for soft clay below water table.
kx). The value of k was changed from 250 lb/sq in. to 750 lb/sq in. As
may be seen the lower value of k caused a significant increase in
deflection and bending moment, especially at the higher loads.
Figure 9.7 shows the effects on pile response of changing the pile
stiffness EI from 1.57 x 10 10
lb-sq in. to 4.70 x 10 10
lb-sq in. The
effects on both bending moment and deflection became significant at the
higher loads.
Stiff Clay above Water Table
The results of analyses using the Reese and Welch (1975) criteria are
shown in Figs. 9.8 through 9.10. One hundred cycles of loading were
employed in the studies. Figure 9.8 shows the effects on maximum bending
moment and groundline deflection of changing the undrained shear strength
from 1080 lb/sq ft to 2160 lb/sq ft. At the larger loads there are signif-
icant increases in both deflection and bending moment.
Figure 9.9 shows the effects of changing e50 from 0.0025 to 0.0075.
The effects are relatively small for the full range of loading.
Figure 9.10 shows the effects of changing the bending stiffness of
the pile EI from 1.57 x 10 10
lb-sq in. to 4.70 x 10 10
lb-sq in. The
effects on the maximum bending were negligible but the effects on deflec-
tion were considerable.
201

40~------T---~--~------~
I I
'300-
..:
40
,30
0-
..:
-0
g20·
....J
·_·--50%
0%
---+50%
2 3
Maximum Moment, in:·lb x 106
o....
-
-
~ 10-o
..J
·_·--50% _
0%
- --+50%
OL-____~I--------~I------~
0 1 2 3
Fig. 9.4. Comparison between results for ±SO percent variation
in c for submerged stiff clays.
.:--
/
·_·--50%
0%
---+50%
2
40
/'
~/30
' 10-
..:
I-0
020
.3 I
0....Q)
10 -
-0
..J
/ . ...-r
/
I
-
-
·_·--50% -
---- 0%
- - -+50%
OL-------~~------~I------~
3 0 2 3
Maximum Moment I in:-Ib x 106 Groundline Deflection, in.
Fig. 9.S. Comparison between results for ±SO percent variation
in sso for submerged stiff clays.
202

40~------r-~~--~~------' 40rn~~--r-------~------,
en 30
Q.
o...Q)
-o
-l
. _. --50%
0%
- --+50%
o...
~IO
.5
·_·--50%
0%
---+50%
o~------~------~------~ O~------~------~------~
o 2 3 0 2
Groundllne Deflection, In.Maximum Moment, In.-Ib x 106
Fig. 9.6. Comparison between results for ±50 percent variation in
k for submerged stiff clays.
30
--0
020o
-l
40 , 1 . __
I .--........
I .r
I
30N/
~ I.:.:
I
0
-
-
·_·--50%
... ·_·--50% -~IO
0% 0 0%-l
---+50% ---+50%
0 00
I I
0 I' 2 3 2
Maximum Moment, in.-Ib x 106 Groundline Deflection, in.
EI for submerged stiff clays.
203
3

40
/' 40
/'
,.,.
/' /
30 /
/
30 /'
/
/'
/
20 .
/
/
·_.--50% / ·_·--50%
0%
/
0%
---+50% --- +50%
2 3 00 2
Maximum Moment, in.-Ib x 106 Groundline Deflection, in.
c for stiff clay above water table.
40
.,1/,
40
//Ifo/'
30
.'/ j(.~ 30
.~ /f
20
/// 20 /1/
11/
IfIi!
10 !II ·_·--50% .- ·--50%
0% 0%
---+50% ---+50%
0 0
0 2 3 0 2
Maximum Moment, in:-Ib x 106 Groundline Deflection, in.
E50 for stiff clay above water table.
204
,..
:3
3

40~------~----~-r-------. 40~------~----~-r------~
o~
~IO
o
...l
. -·--50%
0%
---+50%
2 3
Maximum Moment, in;-Ib x 106
/
·_·--50%
0%
--- +50%
O~------~--------~------~
o 0.5 1.0 1.5
in E1 for stiff clays above water table.
Unified Criteria for Stiff Clay below Water Table
The results of analyses usi ng the Sull ivan, Reese, Fenske (1980)
method are shown in Figs. 9.11 through 9.14. Figure 9.11 shows the
effects on bending moment of changing the undrained shear strength from
432 lb/sq ft to 1296 lb/sq ft. The effects on both maximum bending moment
and deflection are severe.
Figure 9.12 shows the effects of changing E50 from 0.005 to 0.015.
The effects are negligible at the smaller loads but become significant as
the load increases.
slope
1b/sq
used.
Figure 9.13 shows the effects of varying the value of k (the initial
of the p-y curves is obtained from E =kx) from 100 lb/sq in. to 300s
in. The effects are negligible for the range of loading that was
Figure 9.14 shows the effects of changing the EI of the pile from
1.57 x 10 10
lb-sq in. to 4.70 x 10 10
lb-sq in. The effects are signif-
icant at the higher loads on both maximum bending moment and groundline
deflection.
Sand
The results of analyses using the Reese, Cox, Koop (1974) method are
shown in Figs. 9.15 through 9.18. Figure 9.15 shows the effects of chang-
205

20 20
-
/
/
.e.15 / ........ ~15 -.-.x:
/
/' .x: --.........
0
I
,/
.0
/
.310
h/
0 0...
'I
...Q)
·_·--50% Q)
·_·--50%
- -0
5 0% 0 5 0%...J ...J
--- +50% ---+50%
0 0
0 2 :3 0 2
Maximum Moment, in:-Ib x 106 Groundline Deflection, in.
in c for unified criteria.
20~------~~~--~~----~
15
~IO
o
...J
o...
2 5
.3
. _. --50%
0%
- - - +50%
o
.310
o...Q)
o 5...J
. -·--50%
0%
--- +50%
:3
O~------~-------'-------~
2 :3
Maximum Moment, In;-lbxl06
o 2
in E50 for unified criteria.
206

20~------~~~--~------~
~15
c...Q)
C 5
....J
·_·--50%
0%
-- - +50%
2 3
Maximum Moment I in.- Ib x 106
20~--~~~------~------,
15
10
5 - -·--50%
0%
---+50%
00 2 3
Groundline Deflect ion, in.
in k for unified criteria.
rJ) 15
Q.
c...
~ 5c
....J
- ----50%
0%
- - - +50%
2 3
Maximum Moment I in:-Ib x 106
5
-.-
·----50%
0%
- - - +50%
2 3
Groundline Deflection I In.
in EIfor unified criteria_
207

60~------~-------r~--~~ 60~----~~~----~~------~
en
~40
~
~
CI
o
..J
~20Q)
-CI
...J
·_·--20%
0%
---+20%
2 4 6
Maximum Moment, in.-Ib x 106
~
~
CI
o
..J
/
/
. -·--20%
0%
---+20%
2 4
Groundline Deflection, In.
in ¢ using sand criteria for cyclic loading.
6
60 60~------~-r~--'-------~
~ ~
1:J ~
CI CI
o 0
..J ..J
~ 20Q)
§ . -·--20%
0%
---+20%
2 4 6
Maximum Moment, in;-Ib x 106
2
. -·--20%
0%
---+20%
4
in y using sand criteria for cyclic loading,
208
6

fIJ
Q.
60~------~-------r----~~ 60~------.-ur----,-------~
~40 40
-,::J
c
o
-'
-c
....J
fIJ
Q.
·----50% - -·--50%
0% 0%
- - - +50% ---+50%
0 0
0 2 4 6 0 2 4
Maximum Moment, in:-Ib x 106 Groundline Deflection, In.
in ks using sand criteria for cyclic loading.
6
60 60~----~r-~----~~--~
/
~40
/
40
-c
....J
-----50%
0%
-'-- +50%
O~------~-------L------~
o 2 4 6
20
.. - ---50%
0%
---+50%
2 4 6
in E1 using sand criteria for cyclic loading.
209

ing the angle of internal friction ¢ from 28 to 42. As may be seen, the
effects are significant for the full range of loading.
Figure 9.16 shows the effects of changing the submerged unit weight
II from 48 lb/cu ft to 73 lb/cu ft (about the maximum range that could be
expected in practice). The effects are relatively small for the full
range of loading.
Fi gure 9.17 shows the effects of changi ng the value of k , that
s
establishes the initial slope of the p-y curves, from 30 lb/sq in. to 90
lb/sq in. As may be seen, the effects are negligible on both the maximum
bending moment and the groundline deflection.
Figure 9.18 shows the effects of changing the bending stiffness EI of
the pile from 1.57 x 1010
lb-sq in. to 4.70 x 10 10
lb-sq in. The effects
are negligible on the deflection and significant on the groundline
deflection.
Comments on Parametric Studies
The curves shown in Figs. 9.1 through 9.18 provide some guidance on
the influence of various parameters. While there is a considerable amount
of guidance to be gained by a designer from a study of the curves, the
curves are specific in that the pile diameter (16 in.), method of loading
(cyclic), and pile head condition (free to rotate) remained unchanged
throughout the study. The designer is encouraged to perform parametric
studies of a similar sort for the particular problem that is encountered.
Effect of Depth of Penetration
A further study is of interest regarding the pile selected for the
parametric studies. Figure 9.19 shows the results of studies where the
penetration of the pile is gradually reduced. As may be seen, the ground-
line deflection (and other aspects of pile response) is unaffected with
increased penetration beyond a critical length. However, as the pene-
tration becomes less than the critical length, the deflection undergoes a
sharp increase, indicating a soil failure. For the free-head pile in
sand, the critical depth is about 18 feet for a lateral load of 10 kips and
perhaps 24 feet for a lateral load of 30 kips.
In practice, the designer should usually make certain that the pene-
tration is below the critical depth. An increase of a few feet of pile
length in some instances can ensure a much more favorable response of a
pile under lateral load.
210

C
c:
o
1.0..-------.--------------.
0.8
10 kips
12 24 36 48 60
Fig. 9.19.
Depth of Embedment, ft
Effect of depth of embedment
on lateral deflection in sand
9.3 COMMENTS ON METHODS USED FOR CASE STUDIES
The technical literature was consulted and a number of articles or
reports were found where experimental results were given on lateral load
tests. The literature search was not exhaustive; however, the analyses of
the tests that are presented should be useful in giving additional under-
standing of the methods of analysis. No attempt was made to select only
those tests that appeared to correlate better with analysis; the criterion
for selection was that sufficient information was available to allow a
test to be analyzed. However, the tests were included from which recomm-
endations for p-y curves were developed because those tests were well doc-
umented (Matlock, 1970; Reese, et al., 1974; Reese, et al., 1975; Reese
and Welch, 1975).
No parametric studies were done to ensure the best fit of the analyt-
ical methods to the experiments with one exception. Where the pil~ fail-
ure came as a result of a soil failure, some computer analyses were done
with piles of different penetrations. As noted earlier in this chapter,
when piles are short the penetration is a critical parameter.
211

The computations by computer were made by using the appropriate p-y
criteria. The ultimate moment of a steel member was computed by assuming
that the yield stress of the steel was developed across the entire sec-
tion. The ultimate moment of a reinforced concrete member was computed by
using Computer Program PMEIX. The ultimate load by computer was found at
the point where the computed maximum moment in the pile was equal to the
ultimate moment. As noted above, there were a few cases where the piles
failed by excessive deflection. Also, in some of the cases where the
.u1timate moment is used to control the ultimate load, the deflection could
be considered to be excessive. It should be noted, however, that the
ultimate lateral load must be divided by an appropriate factor of safety
to obtain the service load. It is the deflection at the service load that
controls the design.
The computations by the method of Broms were made as outlined in
Chapter 7. An indication is given in each case as to whether or ~ot the
ultimate load was due to a soil failure or to a pile (material) failure.
The computations by the method of Poulos, outlined in Chapter 7, yields
only a linear relationship between load and deflection as does the Broms
method for deflection. Therefore, on the figures giving comparisons
between theory and experiment, the Broms and Poulos curves for deflection
are stopped well below the ultimate load.
Some additional explanation beyond that presented in Chapter 7 is
needed about the procedures employed in making computations with the Broms
and Poulos methods. For some of the case histories considered herein, the
soil deposit was not homogeneous as assumed in the analysis. Therefore,
certain assumptions were used to evaluate an equivalent property repre-
sentative of the non-homogeneous soil deposit.
In order to assess a single value for a soil parameter in a layered
deposit, the parameters were averaged for a depth of five pile diameters.
A depth of five pile diameters was used because the uppermost soils influ-
ence the pile behavior significantly. This depth was used for determining
strength parameters (c, ~) and unit weights (~). These parameters were
used to determine the maximum lateral pile capacity according to the Broms
method and horizontal deflections according to the methods of both Poulos
and Broms. Lateral deflections were computed at the ground1ine consider-
ing the effects of both lateral load and moment.
212

In addition, specific relationships between parameters representing
soil strength and parameters representing soil modulus had to be assumed.
These relationships are discussed below.
For cohesive soils, values of the soil modulus, a, for use in the
Broms method were selected based on the shear strength of the soil. Shown
in Fig. 9.20 is the relationship proposed by Terzaghi (1955), and pre-
sented in Table 3.1 of this text.
c
.- 300r-----------.---------~---------~~---------~----------.---------~
a-
lii
-
.0
en
:I
~ 200
o
~
o
-c
o 100N
...o
Z
Fig. 9.20.
1000
Suggested by Terzaghi
relationship used for
computation of deflections
2000 3000 4000 5000
Undrained Shear Strength t Ib/sqft
6000
Relationship between E and undrained shear strength
for cohesive soil usedsin case studies.
For a pile in cohesive soil, the lateral deflection was calculated
using a beam-on-an-elastic-foundation equation. This equation is as fol-
lows (Scott, 1981):
= 2PtS sinh SL cosh SL - sin SL cos SL
Yt a sinh 2 SL - sin2 SL
2M S2 sinh2 SL + sin2 SL
+
t
a sinh2 SL - sin2 SL (9.2)
213

where
Yt = deflection at groundline
P
t = shear at groundline
Mt = moment at groundline
~ = (al4EI) 0.2 5
EI = bending stiffness of pile
a = soil modulus
L = pile length.
For a pile in cohesionless material, a relationship between the angle
of internal friction, ¢, and the constant giving a variation in soil modu-
lus with depth, k, was needed for the Broms computations. Terzaghi (1955)
recommended values of k for different relative densities of the sand as
presented in Table 3.2; however, Terzaghi's relationship is inconvenient
due to inaccuracies in determining in-situ relative densities. There-
fore, the correlation shown in Fig. 9.21 was used to obtain the relation-
ship between ¢ and k.
III
::l
::l
~
0
::E
0
(/)
c:
c
0
-0
...0

c:
::l
u
.......
.0
-~
.c:
.-Co
Q)
0
.c:
-~
100
50
0
25
Above Water Table
30 35 40
,.1..0
Angle of Internal Friction, ~
45
Fig. 9.21. Relationship between k and ¢ for cohesionless soil.
50
Using the appropriate figure to determine either a or k, the Broms
deflections were computed according to the methods outlined in Chapter 7.
214

Values of elastic modulus, Es ' for cohesive soil for the Poulos com-
putations were taken as equal to 40 times the undrained shear strength.
Values of elastic modulus for cohesionless soils were originally pre-
sented by Poulos as functions of relative density; however, as previously
done for Terzaghi's relationship, values of E were related to ¢ as shown
s
in Fig. 9.22 .
.c
cr
~ 1000
.0
In
LLJ
In
:::J
500:::J
~
0
:E
o
en o ~------~--------~------~--------~------~
20 25 30 35
Angle of Internal Friction t
Fig. 9.22. Relationship between Es and angle of
internal friction for cohesionless soil.
9.4 CASE STUDIES OF PILES IN CLAY
Japanese Test
45
The results of short-term tests of free-head p~pe piles under lateral
load were reported by the Japanese Committee of Research for Piles Sub-
jected to Earthquake (1965). The results of Test Pile 3 will be dis-
cussed. The test pile, shown in Fig. 9.23, was installed by jacking the
closed-ended pile into place.
The soil at the site was a soft, medium to highly plastic, silty clay
with a high sensitivity. The undrained shear strength for the deposit,
shown in 9.23, was obtained from undrained triaxial tests. The strains at
failure were generally less than 5 percent, and the specimens failed by
britt1e fracturi ng. The E 50 values were obtained from the reported
stress-strain curves.
215

17 ft.
c:J
Ground Surface
G.W.T.
Pile Properties
EI =2.52 x 109 Ib-sq in.
b= 12 in.
My = 6 1ft - kips
Cross-section
of Pile
Soil Properties
De p th c E'50 y k
(ft) ( I b Isq ft ) (% ) (I b leu ft ) (I b / cu in.)
0 570 31 65
17 900 31 65
Fig. 9.23. Information for the analysis of Japanese test.
0.125 in.
12 in.
A comparison of the measured and computed results is presented in
Fig. 9.24. The figure shows that the ultimate lateral load computed by
the. Broms method is higher than that from the computer. The load versus
deflection curves from the Broms and Poulos methods are quite conserva-
tive; the computer gave a load-deflection curve that was in reasonably
good agreement with the experiment. Reasonably good agreement was also
obtained between computed and measured maximum moment.
St. Gabriel
A load test was performed on a free-head, 10 in., concrete-filled,
pipe pile near St. Gabriel, Louisiana (Capazzoli, 1968). The loading was
short-term. The test piles were driven vertically to a depth of 115 ft.
The test setup and pile properties are shown in Fig. 9.25.
The soil at the site was a soft to medium, intact, silty clay. The
natural moisture content of the clay varied from 35 to 46 percent in the
upper 10 ft of soil. The undrained shear strengths, shown in Fig. 9.25,
were obtained from triaxial tests.
216

20~------------------~-------------------r------~
til
Q.
~ 15
-a..
0
g 10
....I
o...CD
-o
...J
Pult Broms (pile failure)
Pult Computer
--
_ - Poulos
-
Groundline
o Measured
Broms
2
Deflection I Yt I in.
20~------~------~------~--------r-------~------~
til
.e- 15
~
0..-
0
o 10o
....I
o...CD
-.3 5
M = 73 2 X 104 ft-kipsult .
Computer
o Measured
OL-------~--------~------~----~--~__----~--------~
o 20 40 60 80 100
Maximum Moment I ft- kips
Fig. 9.24. Comparison of measured and computed results
for Japanese test.
217
120

A comparison of the measured and computed results is shown in Fig.
9.26. The ultimate lateral load computed by the Broms method is higher
than that from the computer. The in i t ia1 slope of the
1oad-versus-defl ecti on curve from the computer agrees we11 wi th the
experiment. The deflection curve from the computer is somewhat conserva-
tive at the higher loads. The Poulos and Broms deflection curves are
slightly conservative to very conservative, respectively.
Southern California
Bhushan, et a1. (1978) reported the resul ts of 1atera1 load tests
performed on drilled shafts for a transmission line. Cyclic loading was
not used. The results of three tests performed at two sites will be dis-
cussed. The three piles were straight-sided and reinforced with 3 percent
steel. The lateral loads were applied incrementally, and each increment
was held constant for at least 40 minutes.
At both sites, the soils were silty and sandy clays of low to medium
plasticity. The liquid limit was between 30 and S8 and the plasticity
index was between IS and 20. The natural water content was at or below the
plastic limit, indicating that the soil was heavily overconsolidated.
The values of undrained shear strength and ESO were obtained from
undrained triaxial tests of intact samples. The authors reported a great
deal of scatter in the results of the tests used to define the undrained
strength, c. The large amount of scatter in c is common for desiccated,
heavily overconsolidated soils. In the following analyses, the average c
and ESO values reported by the authors for each test site were used.
The data used in the analysis of Test Pile 2 are shown in Fig. 9.27.
A comparison of the measured and computed results is shown in Fig. 9.28.
As seen in the figure, the computed values of Pult from the Broms method
and from the computer are in rea'scnable agreement but are perhaps less
than the value that would have been obtained by experiment had the pile
been loaded to collapse. Because the pile was short, computations were
performed by computer for the IS-ft 1ength and for an 18-ft 1ength. The
18-ft length resulted in a higher load but the load-deflection curves for
smaller loads were identical.
The deflection curve for Test Pile 2 from the computer is in good
agreement with the experiment at lighter loads but is conservative at lar-
ger loads. The deflection curves from the Broms and Poulos methods are
conservative.
218

1ft
Ground Surface
6ft
-L--...Jj~G.;;...'W.T.
Pile Properties
lOin.
Pile Cross section
115 f t
E I =3 8 x I 0 9 I b - sq in.
b =10 In.
VI 20
c.
~
-0..
'C
C
o
....J
c 10
...I)
c
....J
My= 116 ft:-kips
Soil Properties
Depth =0-15ft
c =600 Ib/sq ft
E 50 = 1
%
Y = 110 Ib/cuft
Fig. 9.25. Information for analysis of test at St. Gabriel.
Broms
o 2 4 6 8 10
Groundline Deflection, Yt ,in.
Fig. 9.26. Comparison of measured and computed results for
St. Gabriel Test.
219
12

0.75 ft
p t ----'-~_I
15 ft
Dial Gage
Ground Surface
Pile Properties
EI =8.2 x lOll Ib-sq in.
b=48 in.
My = 4385 ft- kips
Soil Properties
Depth = 0 15 ft
c = 5500 Ib /sq ft
E
50 =0.94%
y. =130 Ib / CU ft
48 in.
Pile Cross section
total area of rebars
= 3.0% of gross area
Fig. 9.27. Information for the analysis of Southern
California Test Pile 2.
600~------------~--------------r------------~----------------~------------~------------~
soil failures L =18 ft
- 400t:J L =15ft
o
o
.J
o
Puit O
o
~ 200 0
o
.J
/ Poulos
./
,/ Broms
o Measured
,/
O~~~==~~--------L----L--------~--~
o 2 4 6 8 10
for Southern California Test Pile 2.
220
12

The data used in the analyses of Test Piles 6 and 8 are shown in Fig.
9.29. As noted, Test Pile 6 was 48 in. in diameter and Test Pile 8 was 24
in. in di ameter. A compari son of the measured and computed resul ts for
Test Pile 6 is shown in Fig. 9.30. A soil failure was computed by the
Broms method and by the computer. The values from both methods of analy-
sis, even when the pile was increased in length to 20 ft in the computer
analysis, are conservative .
.75 ft
15.5 ft
Test Pile
Number
6
8
Ground Surface
~~~/
Pile Properties
EI b
(I b- sq in.) ( in. )
8 2 x lOll 48
52xl0 10 24
5011 Properties
Depth=0-16ft
c = 4750 Ib/sq ft
€ 50 = 0.72 %
Y = I 3 0 IbI cu ft
241n.
Pile Cross section
total area of rebars
= 3.0 % of gross area
My
(f t-kips)
4385
503
Fig. 9.29. Information for the analysis of Southern California
Tes t Pi 1e 6.
The deflection curve for Test Pile 6 from the computer is in good
agreement with the experiment at lighter loads but is conservative at lar-
ger loads. The deflection curves from Broms and Poulos are conservative.
221

600r-------r-------r-------r-----~
soil failure
r:C 400
0 Pult
g ~~~------------~~~~--------~~
...J
c~
Q)
-c
...J
0
0
o Measured
..... Poulos
.......... -Broms
---
2 :3 4
Groundline Deflection, Yt ' in.
A comparison of the measured and computed results for Test Pile 8 is
shown in Fig. 9.31. Comparatively, the results for this test are similar
to those of the other two in this series. A pile failure was computed, in
contrast with soil failures for the other two tests, but again the com-
puted results are generally conservative with respect to the experiment.
It is of interest to note that if a factor of safety had been used to
reduce the computed Pult to obtain a service load, there would be good to
excellent agreement between computed and measured deflections.
Lake Austin
The test program consisted of both short-term and cyclic tests of a
tree-head pile (Matlock, 1970). Water was kept above the ground surface
for the entire test program. The pl1e was 12.75 in. in diameter, had a
penetration of 42 ft, and was instrumented to measure moment along its
length. The load was applied at a few inches above the mudline. The bend-
ing stiffness EI of the pile was 10.9 x 109
lb-sq in.
The clay at the site was slightly overconsolidated by desiccation and
was slightly fissured. The shear strength was measured with a vane and
averaged 800 lb/sq ft. The £50 was 0.012 and the submerged unit weight
was 50 lb/cu ft.
222

300r-------,--------r-------r------~
III
c..
-~200
-~
o
o
...J
~ 100
Q)
-.5
0
0
o
o
o
Computer (pile fai lure)
Broms (pile fai lure)
__ Broms o Measured
2 4 6 8
Figure 9.32 shows a comparison of measured and computed deflections.
The results from cyclic loading are shown in Fig. 9.32a and from static
loading in Fig. 9.32b. Also shown by a note in Fig. 9.32b are the computed
ultimate loads by the Broms and computer methods. It can be seen that the
experimental loading was stopped well below the ultimate capacity of the
piles. The instrumented piles were employed at another site.
There is reasonable agreement between the deflection curves from the
experiment and from the computer at the lower loads, but a considerable
deviation at the higher loads. The Poulos curve fell close to the exper-
imental curve for static loading but the Broms curve was conservative.
A comparison of the measured and computed maximum moments for the
Lake Austin pile are shown in Fig. 9.33. The comparison for static load-
ing is good but the computer under-predicts the maximum moments for cyclic
loading.
Sabine
The piles used in the Lake Austin tests were pulled and re-driven at
Sabine (Matlock, 1970). Static and cyclic loads were applied with the
pile head free to rotate and with the pile head restrained against rota-
tion. The restrained-head tests are not discussed herein. The point of
application of the load was 12 in. above the mudline. Water was kept
above the mudline throughout the testing program.
223

30
(I)
Q.
Pult Computer
~
-Q. 20
-~
0
00
..J
0
0
0~
CD
-0
..J
o Measured
0
0 2 3
Groundline Deflection, Yt ,In.
Pu1t = 5 I kips Broms
Pu1t = 44 kips Computer
30~------~------~----~~
0:-20
~
o
o
....J
-o
....J 1
--_Broms
o Measured
o~------~------~------~
o 2
(a) cycli c
loading
(b) static
loading
Fig~ 9.32. Comparison of measured and computed
deflections for Lake Austin Test.
224

28~----~----~------r-----~
24
a.. 20
-'t:I
C
o
...J 16
c...Q)
-~ 12
8
4
0
0 4
Computed
Static -
0 Measured I
D. Measured I
8 12
Static
Cyclic
Maximum Moment I in.-Ib x 105
16
Fig. 9.33. Comparison of measured and computed
maximum moments for Lake Austin Test.
The clay at the site was a slightly overconsolidated marine deposit.
The undrained shear strength was about 300 lb/sq ft and the submerged unit
weight was 35 lb/cu ft. The strain E50 at one-half the compressive
strength was 0.007.
A comparison of measured and computed deflections is shown in Fig.
9.34. The results from cyclic loading are shown in Fig. 9.34a and from
static loading in Fig. 9.34b. The computed ultimate loads from the Broms
method and from the computer are shown in Fig. 9.34b. There is excellent
agreement between the experimental and the computed deflections for stat-
ic loading. The deflections for the cylic case are under-predicted by the
computer. The deflection curves from the Broms and Poulos methods, shown
in Fig. 9.34b, are conservative.
225

15
_ 10
0
o
o
....I.
o...Q)
-o
....I
(/)
a.
oX
~
'0
0
0
....I
0
...Q)
-0
....I
0
0
20
15
10
0
0
Fi g.
Pult
Pult
9.34.
o
o
2
Groundline Deflection I
= 32 kips Broms
= 30 kips Computer
Computer
,Broms
,
,,
2
Groundline Deflection I
Comparison of measured
deflections for Sabine
226
o o
o Measured
:3 4
Yt lin.
0 Measured
:3 4
Yt I in.
and computed
Test.
(a) cyclic
loading
(b) static
loading

Figure 9.35 shows a comparison of the measured and computed maximum
moments for the Sabine tests. In both instances, the comparisons are
exce11 ent.
f/J
Co
.¥:
0..-
'0
c
0
..J
c...Q)
-c
..J
Houston
20~------~------~------~------~
15
Computed
Static
10
5
0 Measured Cyclic
Do Measured Static
0
0 5 10 15 20
Maximum Moment, in:lb x 10~
maximum moments for Sabine Test.
A drilled shaft was tested at a site in Houston (Reese and Welch,
1975). The diameter of the pile was 30 in. and the penetration was 42 ft.
The foundation was instrumented along its length for the measurement of
bending moment. Difficulty was encountered in getting the excavation
filled with concrete because of the close spacing of the reinforcing steel
and there was a cavity near the top of the shaft. Field measurements
indicated that the bending stiffness EI of the pile was about 1.47 x 1011
lb-sq in.
227

The pile head was free to rotate and a combined static-cylic loading
program was employed. A given load was applied, measurements were taken,
and the load was removed and re-applied a number of times. A total of 20
cycles of loading was applied at each load increment.
The soil profile consisted of 28 ft of stiff to very stiff red clay,
2 ft of interspersed silt and clay layers, and very stiff silty clay to a
depth o{ 42 ft. The water table was at a depth of 18 ft. The undrained
shear strength was determined by triaxial tests and was found to vary
widely in the top 20 ft. A value of 2.2 kips/sq ft was selected for ana-
lyses. The average value of ESO was 0.005, and the unit weight of the clay
was 120 lb/cu ft.
A comparison of measured and computed deflections is presented in
Fig. 9.36. The results from cyclic loading (10 cycles) are shown in Fig.
9.36a and from static loading in Fig. 9.36b. The computed ultimate loads
from the Broms method and from the computer method are shown in Fi g.
9.36b. There is good to excellent agreement between the experimental
deflection curves and the results from the computer. The Broms method
appears to under-estimate the ultimate load. The deflection curves from
the Poulos and Broms methods are conservative.
Figure 9.37 shows the comparisons between the measured and computed
maximum moments for the Houston test. The agreement is good to excellent.
Manor
Tests were performed on steel pi pe pil es that were 25.25 in. in
diameter and with a penetration of 49 ft. The piles were instrumented
along their length for the measurement of bending moment. The pile heads
were unrestrained against rotation and both static and cy~lic tests were
performed.
An excavation about 5 ft deep was made at the site and water was
ponded for several weeks prior to and during the testing. The soil was a
stiff, fissured clay. The undrained shear strength was measured by triax-
ial tests; there was much scatter in the results but in ~eneral the shear
strength increased rapidly with depth. The following depths and undrained
shear strengths were used in the analyses (feet and kips/sq ft, respec-
tively): 0, 0.4; 1.0, 1.6; 13.0, 7.0; 21.0, 7.0. The average value of E50
at the site was 0.005 and the submerged unit weight of the clay was 65
lb/cu ft.
228

120
II) o~
..ill:
0..-
80
Pult Broms
~
0
0
...J
0
....Q)
-0
...J
0
0.0
,.,.,.
0.4
,. Poulos,.
0.8
o Measured
1.2
Groundline Deflection, Yt tin.
1.4
deflections for Houston Test.
229
(b) static
loading

en
Q.
~
e:-
-,:,
0
0
....J
0
~
CD
-0
....J
100r---~----~----~~--~----~--~-'~--~
80
60
40
20
Computed Cyclic
(10 cycles)
o Measured Static
t::. Measured Cyclic
o L-____~____~____~____~____~~____~__~
o 2 :3 4 5 6 7
Fig. 9.37. Comparison of measured and computed maximum moments
for Houston Test.
Figure 9.38 shows the comparison between the measured and computed
deflections. The results from cyclic loading are shown in Fig. 9.38a and
from static loading in 9.38b. The computed ultimate loads from the Broms
method and from the computer method are indicated in Fig. 9.38b. There is
generally good agreement between the experimental deflection curves and
those from the computer. The deflection curves from the Broms and Poulos
methods are conservative.
maximum moments for the Manor test. The agreement is good to excellent.
9.5 CASE STUDIES OF PILES IN SAND
Gill Tests
Gill (1969) reported the results of four lateral load tests performed
on pipe piles. The piles were of different stiffnesses and were all
embedded to a sufficient depth so that they behaved as flexible members.
The pil e heads were free to rotate duri ng testing and the loads were
applied statically.
230

120r-------~------~------~--------
~ 80
C
o
o
...J
o 40'-
CI)
-o
...J
o o
o
o
o Measured
O~------~------~------~------~
(a) Cyclic
loading
0.0 0.4 0.8 1.2 1.6
Groundllne Deflection, Yt , in.
160r-------~------~------~------~
Pult Broms 290 kips
Pu1t Computer 142 kips
o
~ 120
:¥
-a..
~
C
800
0
..J
0
'-
CI)
-0
...J
o Measured
O~------~--------~------~--------~
0.0 0.4 0.8 1.2 1.6
deflections for Manor Test.
231
(b) static
loading

160~------~------~------~
tn
c.. 120
~
0..-
'0 CY
0
80 Computed 0
f:y-I
Static
// Computed
0 ~ Cyclic...Q)
-0
...J
40
0 Measured Static
6 Measured Cyclic
00 40 80 120
Fig. 9.39. Comparison of measured and computed moments
for Manor Test.
The soil at the site was mainly an old hydraulic fill which had been
placed in the 1940's (Gill, 1969). A compacted granular surface had
reportedly been placed over the hydraulic fill. This compacted surface
could account for the high blow count of 58 blows/ft at a depth of 2 ft,
shown in Fig. 9.40. Below 2 ft, the blow count decreased rapidly until it
reached 16 blows/ft at a depth of 4.5 ft. No information concerning the
SPT resistance of the material was given below 4.5 ft, and it was assumed
that the relative density was constant below that depth.
The data shown in Fig. 9.40 were used in the analyses and the result-
ing curves are shown in Figs. 9.41 through 9.44. As may be seen in the
figures, the initial slopes of the curves from the computer, from Broms,
and from Poulos are in good agreement and also agree reasonably well with
the initial slopes of the experimental curves. The ultimate capacities
obtained from the computer and from Broms are in reasonable agreement. At
the larger loads the deflection curves from the computer agree well with
the experiments or are somewhat conservative.
232

GWT
Depth
o 2
2 4.5
4.5 32
Boring
Depth Blow Count
(blows/ft)
Ground
Line
cp
41
40
38
( ft )
......
• eO : '
.. ..·0 .....
.::~:.:~:. 58
::.. e.; ~
.. e ......
. o· .. 0
.. , .......
... :... 16
5 ... ' 0-
'.' :,:
, .: ::::., '
, SP·.
: S' ..
Uniform Medium
Sand
10
15
.... ,- ..
· . ... : ..• r
.... ' ... ,':• J • ..
.. .. .. .. ~
.. . . . . eO
.......
• .. 'Of ..
,.. .
.0 t.
·':' ........
-.. ;.:,
o .. ..
' . . . t-
O .. I ...,
., .. ,' ..

Soil Properties
y k
(Ib/cu tt) (lb/cu in.)
125 275
115 175
60 110
Ko
0.4
0.4
0.4
t
1.67 ft
L
Length
(ft)
Pile Width
IT
Pile Properties
Test Pile b EI
Number ( in.) (lb-sqin,)
P9 4,75 2.17 x 108
PIO 8.62 2.17 x 109
PII 12.75 7.46 x 109
PI2 16,00 1.69 x 1010
Fig. 9.40. Information for the analysis of tests in hydraulic fill,
233
L
( ft)
18
24
30
30

VI
c.
.:.:
0:-
'0
0
0
..J
C...CD
-0
..J
VI
Co
....::
-c-
'0
0
0
..J
0
...CD
-c
..J
5
Computer
4 Broms ( pile failure)
3
0 Measured
Poulos
2
Broms
2
Groundline Deflection I Yt , in.
Gi 11 Test Pil e 9.
20
Pult Computer
15
O Measured
Poulos
10 Broms
0
5
3
o~------------------~--------------------~------~
o 2
Groundline Deflection I Yt I in.
Gill Test Pile 10.
234

0
CI
40~--------------~~--------------~
Pult Computer
.Pult Broms (pile failure)
o 20-...J
CI
...CD
-CI
..J
Measured
o------------------~----------------~
o 2
for Gill Test Pile 11.
60
Pult Com uter
U)
Q.
~
~
0
30CI
0
..J
CI
-
CD
- o MeasuredCI
..J
0
0 2
Groundllne Deflection, Yt ' in.
for Gill Test Pile 12.
235

Arkansas River
A number of lateral load tests were performed for the Corps of Engi-
neers by Fugro and Associates at a site on the Arkansas River near Pine
Bluff, Arkansas (Alizadeh and Davisson, 1970). Test Piles 2 was a pipe
pile with a 16 in. outside diameter and modified by welding four steel
channels (4 x 7.25) at 90 degrees apart around the exterior of the pipe.
It was installed by driving. The bending stiffness of the pile was 2.44 x
1010
lb-sq in. and it had a penetration of 53 ft. The ultimate moment
capacity My of the pile section was computed to be 778 ft-kips.
The pile head was free to rotate during testing and the lateral load
was applied 0.1 ft above the groundline. Static loading was used in the
testing program. The pile was instrumented along its length for the meas-
urement of bending moment.
The soil conditions at the site are shown in Fig. 9.45. As may be
noted, the water table was near the ground surface. The soil of primary
importance with regard to lateral loading is the sand, classified as SP by
the Unified method; the sand extends from the ground surface to a depth of
about 22 ft.
Figure 9.46 shows the comparison between the measured and computed
deflections for Test Pile 2. The computer predicts a higher Pult than
does Broms. The deflections from Broms and Poulos are larger than the
measured deflections and the deflection curve from the computer is stiffer
than either of these at lower loads. There is some indication that the
sand near the ground surface was denser than assumed, which may account
for some of the lack of agreement between the computer and the experiment.
The compari son between the measured and computed maximum bendi ng
moments is shown in Fig. 9.47. As may be seen, the computed results are
somewhat conservative with regard to the measured values.
Test Pile 6 at the Arkansas River site was a 14BP73 steel bearing
pipe that was driven into place. The pile had a bending stiffness EI of
2.15 x 10 10
lb-sq in. and a width of 14 in. Its ultimate-moment capacity
was computed to be 397 ft-kips. The penetration of the pile was 40 ft.
A comparison of the measured and computed results for Test Pile 6 is
shown in Fig. 9.48. The results are similar to those for Test Pile 2 as
shown in Fig. 9.46.
236

Bottom of Excavation
Depth Blow Count
( ft ) ( blow / ft )
~ 0 : .~. ~------~~~r---
G.W.T. -~--......: ......•.t-----
::•. :.: - II
..~ '.;,..... ,.......:':!;... - ..
.:.:. : -14
10-!'SP;'
·ot.' :' ~
. ':j _.' - 35
',:} .-:/
,.. ' ',
:';'.....,-.. - 25
Soil Properties
Depth y k
'. , -'J
.. .... :..
20 - :.:.~ -23
cp( ft) (lb/cu ft) (Ib/cu in.)
... , ~,
0-3 38 115 160
,SM': ~27
:; .::. -21
:'SP.,.. ' 49
:.SM ':30- ' .. -34
::.: SP':::
~'.:. ;:i
.. '
'. SM 'J-43
., .
..........,' '.
3-20 39 63
20-60 40 63
Fig. 9.45. Soils information for analysis of tests
at Arkansas Rive~
Florida
90
90
Ko
1.0
1.0
1.0
A load test was performed by the Florida Power and Light Company
(Davis, 1977) on a 56-in. 0.0. steel tube that was vibrated to a depth of
26 ft. The interior of the tube was filled with concrete and a utility
pole was embedded in the upper part of the tube so that the load could be
applied at 51 ft above the groundline. The bending stiffness of the pile
was computed to be 1.77 x 1012 lb-sq in. in the top 4 ft and 8.8 x 1011
lb-sq in. below that. The ultimate-moment capacity of the pile was com-
puted to be 4,630 ft-kips. The loads were applied statically.
The soil profile consisted of 13 ft of medium dense sand overlying
stiff to very stiff sandy, silty clay. The water table was at a depth of 2
ft. The angle of internal friction of the sand was estimated at 38° and
237

120
Pult Computer (120 kips)
90 Ftlt Broms (pile failure)
.,.Q.
..
a:-
600
0
0..J
e
.!
0
30..J
o Measured
o~------~--------~--------~--------~------~
o 0.5 1.0 1.5 2.0
Groundllne Deflection, Yt' in.
Fig. 9.46. Comparison of measured and computed deflections
for Arkansas River Test Pile 2.
120--------------------~----------------~
.,. 90
.9-
M
..
Computer
..
8 60
..J
~ 0
•-o
..J
o Measured
5 10
Fig. 9.47. Comparison of measured and computed maximum moments
238
2.5

the undrained shear strength of the clay was estimated at 2.5 kips/sq ft.
The total unit weight of the soil was 115 lb/cu ft and the submerged unit
weight was 60 lb/cu ft.
tr
Pult Computer (74 Kips)
60 Pult Broms (Pile Failure)
.,. 40
-ag
o
..J
g
..•-g
..J
/
/
/
/
/
0.5
/
/
/
Poulos
1.0
o
1.5
Groundline Deflection, Yt I in.
Measured
2.0
2.5
Figure 9.49 gives the comparison between the measured and computed
results. The agreement between measured deflections and those from com-
puter are in excellent agreement. The deflection curves from Broms and
Poulos are conservative. The same value of of ultimate load (84 kips) was
obtained from the Broms method and from the computer.
Mustang Island
Tests were performed at Mustang Island near Corpus Christi, Texas, on
two 24-in. 0.0. pipe piles that were instrumented along their lengths for
the measurement of bending moment (Reese, et al., 1974). One of the piles
was subjected to static loads and the other to cyclic loads. The pile
heads were free to rotate and the loads were applied one foot above the
groundline. The penetration of the piles was 69 ft.
239

The soil at the site consisted of clean fine sand to silty fine sand.
An excavation was made at the site to a depth of about 5 ft and the natural
water table was above the testing surface. A program of in situ tests and
laboratory soil tests was conducted at the site. The angle of internal
friction was found to be 390
and the submerged unit was 66 lb/cu ft.
In
Q.
~
60~----~------r------r------'
50
Pult Computer 84 kips
Pult Broms 84 kips
- 40
~
0
0
0
30-l
0
~
CD
...0
-l 20
o Measured
O~-----L------L-----~----~
0.0 0.25 0.50 0.75 1.00
Groundllne Deflection, Yt' In.
results for Fl or; da Test.
Figure 9.50 shows the comparison betweeen the measured and computed
deflections. The results from cyclic loading are shown in Fig. 9.50a and
from static loading in 9.50b. The computed ultimate loads from the Broms
method and from the computer method are indicated in Fig. 9.50b. There is
exce11 ent agreement between the experi menta1 defl ecti on curves and the
ones from the computer. The deflection curves from the Broms and Poulos
methods are slightly conservative.
240

60r-----~----~----,_----,_----1r----_r----_,
50
- 40
--
--0
o
.3 30
o
-
CD
-o 20....J
0
0.0
60
50
UI
0..
.Jill:
40
--
--0
0
0 30
....J
0
-
CD
- 200
....J
10
0
0
Fi g. 9.50.
Computer
o Measured
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Groundline Deflection, Yt , in.
Pult Broms 92 kips
Pult Computer 102 kips
o Measured
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Groundline Deflection, Yt , in.
for Mustang Island Test.
241
(a) cyclic
loadincJ
( b) static
loadin9

maximum moments for the Mustang Island Test. The agreement is excellent.
70~----~----~----~----~----~
60
~
,-
50
~In
Q.
,-.:.t:
-40 ,c{a..
Computed
/
0 /
c Static / Computed0
-l 30 cf Cyclic
/
c /... /Q)
- y!c
20.....J
10 o Measured I Static
6 Measured I Cyclic
0
0 2 3 4 5
Maximum Moment, in:lbx 106
Fig. 9.51. Comparison of measured and computed maximum
moments for Mustang Island Test.
9.6 COMMENTS ON RESULTS OF CASE STUDIES
The compari sons in the precedi ng sections show agreements between
experiment and analysis that range from fair to excellent. At the present
time it cannot be stated with certainty that the fair agreement is due to
deficiencies in the analytical methods or to poor data from the exper-
iments. It can be stated, however, that an excellent soil investigation
is mandatory when design of piles under lateral loading is to be done. As
noted earlier, the soils near the ground surface need careful attention.
The construction method also needs careful control.
242

The computer method appears to be more versatile. Even in those cas-
es where agreement between the results from the computer and those from
experiment, the use of a factor of safety in the range of those normally
used for foundation design would lead to an acceptable solution.
The reader can examine the cases that are presented and reach a deci-
sion about the factor of safety that should be employed in any particular
design. On important jobs, of course, the design should be confirmed with
a full-scale field load test. The test should be performed in such a way
that the experimental results can be analyzed with the computer. That is,
the careful measurement of load, pile-head deflection, and pile-head
rotation will allow the soil response to be ascertained (Reese and Cox,
1968). The soil response so obtained can then be used in the design for
different pile-head conditions or for piles of different sizes, with
appropriate adjustment.
9.7 REFERENCES
Alizadeh, M., and Davisson, M. T., Lateral Load Tests on Piles -
Arkansas River Project, Journal of the Soil Mechanics and Foundation
Division, American Society of Civil Engineers, Vol. 96, SM5, September
1970, pp. 1583-1604.
Bhushan, K., Haley, S. C., and Fong, P. T., Lateral Load Tests on
Drilled Piers in Stiff Clays, Preprint 3248, American Society of Civil
Engineering Spring Convention and Exhibit, April 1978, 28 pp.
Capozzoli, L., Test Pile Program at St. Gabriel, Louisiana, Louis
J. Capozzoli and Associates, July 1968.
Committee of Piles Subjected to Earthquake, Architectural Institute
of Japan, Lateral Bearing Capacity and Dynamic Behavior of Pile Founda-
tion (Loading Test of Single and Grouped Piles), May 1965, pp. 1-69 (in
Japanese).
Davis, L. H., Tubular Steel Foundation, Test Report RD-1517, Flo-
rida Power and Light Company, Miami, Florida, 1977.
Gill, H. L., Soil-Pile Interaction Under Lateral Loading, Confer-
ence on In-Situ Testing of Soils and Rocks, London, England, 1969, pp.
141-147.
Matlock, Hudson, Correlations for Design of Laterally Loaded Piles
in Soft Clay, Paper No. OTC 1204, Proceedings, Second Annual 'Offshore
Technology Conference, Houston, Texas, 1970, Vol. 1, pp. 577-594.
Meyer, Barry J., Analysis of Single Piles under Lateral Loading,
Masters Thesis, University of Texas, Austin, Texas, December 1979.
243

Reese, Lymon C., and Cox, W. R., Soil Behavior from Analysis of
Tests of Uninstrumented Piles Under Lateral Loading, Special Technical
Publication 444, American Society for Testing and Materials, Vol. 123,
1968, pp. 161-176.
Reese, Lymon C., Cox, W. R., and Koop, F. D., Analysis of Laterally
Loaded Pil es in Sand, II Paper No. OTC 2080, Proceedi ngs, Si xth Annua 1 Off-
shore Technology Conference, Houston, Texas, 1974, Vol. 2, pp. 473-483.
Reese, Lymon C., Cox, W. R., and Koop, F. D., Field Testing and
Analysis of Laterally Loaded Piles in Stiff Clay, Paper No. OTC 2312,
Proceedings, Seventh Annual Offshore Technology Conference, Houston, Tex-
as, 1975, Vol. 2, pp. 671-690 ..
Reese, Lymon C. and Welch, R. C., Lateral Loading of Deep Founda-
tions in Stiff Clay, Journal of the Geotechnical Engineering Division,
American Society of Civil Engineers, Vol. 101, No. GT7, Proc. Paper 11456,
July 1975, pp. 633-649.
Scott, Ronald F., Foundation Analysis, Prentice-Hall, Englewood
Cl i ffs, New Jersey, 1981.
Sullivan, W. Randall, Reese, Lymon C., and Fenske, Carl W., Unified
Method for Ana lysi s of Laterally Loaded Pil es in Cl ay, II Proceedi ngs, Num-
ericsl Mehtods in Offshore Piling, Institution of Civil Engineers, Lon-
don, England, May 1979, pp. 107-118.
Terzaghi, Karl, Evaluation of Coefficients of Subgrade Reaction,
Geotechnigue, Vol. 5, December 1955, pp. 297-326.
Terzaghi, Karl, and Peck, Ralph B., Soil Mechanics in Engineering
Practice, John Wiley and Sons, Inc., New York, 1948.
9.8 EXERCISES
9.1 Do a parameter study for a 30-in. 0.0. pile in sand.
9.2 Do a parameter study for a 30-in. 0.0. pile in clay.
9.3 Find in the technical literature or in company files the results
of a field test of a pile under lateral loading. Analyze the test accord-
ing to the procedures employed in this chapter.
244

CHAPTER 10. ANALYSIS OF PILE GROUPS UNDER LATERAL LOADING
10.1 INTRODUCTION
There are two general problems in the analysis of pile groups: the
computation of the loads coming to each pile in the group, and the deter-
mination of the efficiency of a group of closely-spaced piles. Each of
these problems will be discussed in the following sections.
The methods that are presented are applicable to a pile group that is
symmetrical about the line of action of the lateral load. That is, there
is no twisting of the pile group so that no pile is subjected to torsion.
Therefore, each pile in the group can undergo two translations and a rota-
tion~ However, the method that is presented for obtaining the distrib-
ution of loading to each pile can be extended to the general case where
each pile can undergo three translations and three rotations (Reese, et
a1., 1970; OINei11, et a1., 1977; Bryant, 1977).
In all of the analyses presented in this section, the assumption is
made that the soil does not act against the pile cap. In many instances,
of course, the pile cap is cast against the soil. However, it is possible
that soil can settle away from the cap and that the piles will sustain the
full load. Thus, it is conservative and perhaps logical to assume that
the pile cap is ineffective in carrying any load.
If the piles that support a structure are spaced far enough apart
that the stress transfer between them is minimal and if the loading is
shear only, the methods presented earlier in this work can be employed.
Kuthy, et a1. (1977) present an excellent treatment of this latter prob-
lem.
10.2 DISTRIBUTION OF LOAD TO EACH PILE IN A GROUP
The derivation of the equations presented in this section is based on
the assumption that the piles are spaced far enough apart that there is no
loss of efficiency; thus, the distribution of stress and deformation from
a given pile to other piles in the group need not be considered. However,
the method that is derived can be used with a group of closely-spaced
piles but another level of iteration will be required.
Problem Statement
The problem to be solved is shown in Fig. 10.1. Three piles support-
ing a pile cap are shown. The piles may be of any size and placed on any
245

batter and may have any penetration below the groundline. The bent may be
supported by any number of piles but, as noted earlier, the piles are
assumed to be placed far enough apart that each is 100% efficient.

TB
Fig. 10.1. Typical pile-supported bent.
The soil and loading may have any characteristics for which the
response of a single pile may be computed.
Loading and Movement of the Structure
The derivation of the necessary equations in general form proceeds
conveniently from consideration of a simplified structure such as that
shown in Fig. 10.2 (Reese and Matlock, 1966; Reese, 1966). The sign con-
ventions for the loading and for the geometry are shown. A global coordi-
nate system, a-b, is established with reference to the structure. A
coordinate system, x-y, is established for each of the piles. For conven-
ience in deriving the equilibrium equations for solution of the problem,
the a-b axes are located so that all of the coordinates of the pile heads
are positive.
246

b
M(+)
b,(+)
a (+)
~ (+)
/x,
b
~h
(+)
U~---r---;:Il7:vT.(+~)~------I:s(+)
Fig. 10.2. Simplified structure showing coordinate systems
and sign conventions (after Reese and Matlockt
247
(a )
( b)

The soil is not shown, but as shown in Fig. 10.2b, it is desirable to
replace the piles with a set of springs (mechanisms) that represent the
interaction between the piles and the supporting soil.
Movement of a Pile Head
If the global coordinate system translates horizontally ~h and ver-
tically ~v and if the coordinate system rotates through the angle a , the
s
movement of the head of each of the piles can be readily f6und. The angle
as is assumed to be small in the derivation.
The movement of a pile head x
t
in the direction of the axis of the
pil e is
xt=(~h+ba)sin8+(~v+aa)cos8. (10.1)s s
The movement of a pile head Yt transverse to the direction of the axis of
the pile (the lateral deflection) is
Yt = (~h + bas) cos 8 - (~v + aas) sin 8. (10.2)
The assumption is made in deriving Eqs. 10.1 and 10.2 that the pile heads
have the same relative positions in space before and after loading.
Forces and Moments
The movements computed by Eqs. 10.1 and 10.2 will generate forces and
moments at the pil e head. The assumption is made that curves can be
developed, usually nonlinear, that give the relationship between
pile-head movement and pile-head forces. A secant to a curve is obtained
at the point of deflection and called the modulus of pile-head resistance.
The values of the moduli, so obtained, can then be used, as shown below,
to 'compute the components of movement of the structure. If the values of
the moduli that were selected were incorrect, iterations are made until
convergence is obtained.
Using sign conventions established for the single pile under lateral
loading, the lateral force Pt at the pile head may be defined as follows:
Pt = JyYt. (10.3)
If there is some rotational restraint at the pile-head, the moment is
Mt = -JmYt. (10.4)
The moduli Jy and Jm are not single-valued functions of pile-head trans-
lation but are functions also of the rotation a of the structure.
s
If it is assumed that a compressive load causes a positive deflection
along the pile axis, the axial force P may be defined as follows:
x
Px=Jxxt . (10.5)
It is usually assumed that P
x
is a single-value function of xt .
248

The forces at the pile head defined in Eqs. 10.3 through 10.5 may now
be resolved into vertical and horizontal components of force on the struc-
ture, as follows:
Fv =-(Pxcos 8 - Ptsin 8) , and
Fh =-(Pxsin 8 + Ptcos 8)
(10.6)
(10.7)
The moment on the structure is
Ms =Jmyt'
Equilibrium Equations
(10.8)
The equilibrium equations can now be written, as follows:
P + EF =a (10.9)v v. '1
Ph + EFh. =0, and (10.10)
1
M + EM + Ea. F + Eb. Fh =O. (10.11 )S. 1 V. 1 .
1 1 1
The subscript i refers to values from any i-th pile. Using Eqs. 10.1
through 10.8, Eqs. 10.9 through 10.11 may be written in terms of the
structural movements.
L
+ [a. (J cos 2 e. + J sin2 e.)1 X. 1 y. 1
1 1
+ b. (J - J )sin e. cos e.] a
1 x. y. 1 1 S
1 1
I[(J -x.1
J )sin e. COS e.]~v +y. 1 1
1
COS e.1
= P
V
[J sin2
x.1
+ [a.(J - J )sin e.
1 Xi Yi 1
+ b.(J sin2 e. + J1 X. 1 y.
1 1
cos2
ei )las I= P
h
(10.12)
ei
+ J cos 2
ei] lIh
Yi
(10.13)
sin e. + a.(J cos 2 e. + J sin 2 e.) + b.(J - J )
1 1 Xi 1 Yi 1 1 Xi Yi
sin ei cos ei ]lIv + [-J cos e. + a.(J - J )sin e. cos e.
mi 1 1 Xi Yi 1 1
+ b.(J cos 2 e. + J cos 2 e.)]lIh + (J (a. sin e. - b. cos e.)
1 Xi 1 Yi 1 mi 1 1 1 1
+ (J cos 2 e. + J sin2 e.)a. 2 + (J sin2 e. + J cos 2 e.)b. 2
x. 1 y. 1 1 Xi 1 Yi 1 1
1 1
+ 2(J - J )(sin e. cos eo)aob} I = M (10.14)
x. y. 1 1 1 1 S
1 1
249

Several of the terms in the above equation occur a number of times;
it is convenient to define five terms as shown below.
A. =J cos2
9. + J sin 2
9. (10.15)1 x. 1 Yi 1
1
B. = (J - J )
1 Xi Yi
sin 9i cos 9i (10.16)
C. =J sin 2
9. + J cos2
9. (10.17)
1 Xi 1 Yi 1
D. =J sin 9. (10.18)1 m. 1
1
E. =-J cos 9. (10.19)1 m. 1
1
Equations 10.12 through 10.14 can be simpl Hied by use of the above
expressions. Equations 10.20 through 10.22 are the final form of the
equilibrium equations.
L'IV [L: A. ]
1
+ lIh[L: B.]
1
+ a [L:
S
lIV[L: B. ]
1
+ lIh[L: C.]
1
+ a [L:
S
lIV[L: D. + L: a.A. + L: b.B.]
1 1 1 1 1
+ a [L: a.D. + L: a. 2
S 1 1
+ L: 2a.b.B.] = M1 1 1
1
A. + L:
1
a.A. + L: b. B.] = P (10.20)1 1 1 1 V
a.B. + L: b. C.] = Ph (10.21)
1 1 1 1
+ lIh[L: E. + L: a.B. + L: b. C.]
1 1 1 1 1
b.E. + L: b. 2 C.
1 1 1 1
(10.22)
The equilibrium equations can be solved in any convenient way.
Because of the number of operations required, it is usually convenient to
use a digital computer (Awoshika, 1971; Lam, 1982).
Solution Procedure
1. Select a coordinate center and find the horizontal compo-
nent, the verti ca1 component, and the moment through and
about that point.
2. Compute by some procedure (Reese, 1964; Coyle and Reese,
1966; Coyle and Sulaiman, 1967; Kraft, et al., 1981) a curve
showing axial load versus axial deflection for each pile in
the group. An alternate procedure is to use the resul ts
from a field load test. A typical curve is shown in Fig.
10.3a.
3. Use procedures presented earlier in this work and compute
curves showi ng 1atera1 load as a function of 1atera1
250

deflection and moment as a function of lateral deflection.
In making these computations, attention must be paid to the
effect of structural rotation on the boundary conditions at
each pile head. Typical curves are shown in Figs. 10.3b and
10.3c.
4. Trial values of J x' Jy
' and J m are estimated for each pile
in the structure.
5. Equations 10.15 through 10.22 are solved for values of 6V,
6h,andas ·
6. Pile-head movements are computed and new values of J x' Jy
'
and J
m are obtained for each pile. Curves such as those
shown in Fig. 10.3 are employed or a computer solution
yields directly the values of the moduli.
7. Equations 10.15 through 10.22 are solved again for new val-
ues of 6V, 6h, and as.
8. Iteration is continued until the computed values of the
structural movements agree, within a given tolerance, with
the values from the previous computation.
9. The loads and moments at each pile head can then be used to
compute the stresses along the length of each pile.
Example Problem
Figure 10.4 shows a pile-supported retaining wall with the piles
spaced 8 ft apart. The piles are 14-in. in outside diameter with 4 No.7
reinforcing-steel bars spaced equally. The centers of the bars are on an
8-in. circle. The yield strength of the reinforcing steel is 60 kips/sq
in. and the compressive strength of the concrete is 2.67 kips/sq in. The
length of the piles is 40 ft.
The backfill is a free-draining, granular soil without any fine par-
ticles. The surface of the backfill is treated to facilitate runoff and
weep holes are provided so that water will not collect behind the wall.
The forces PI' P2, Ps ' and P
w
were computed as follows: 21.4, 4.6,
18.4, and 22.5 kips, respectively. The resolution of the loads at the
origin of the global coordinate system resulted in the following service
loads: P
v
= 46 kips, Ph = 21 kips, and M= 40 ft-kips (some rounding was
done) .
The moment of inertia of the gross section of the pile was used in
the analysis. The bending stiffness E1 of the piles was computed to be
251

p,.1
Mt·
(_)1
PX'
J - 1
X'---
1 Xt.1
Xt·1
y
(a)
(b)
(c )
Fig. 10.3. Set of pile resistance functions
for a given pile.
252

HI varies a
I
J12'
~
Origin of global
coordi nate system
1.5'
Ps Pw
1.75'
2'
Fig. 10.4. Sketch of a pile-supported retaining wall.
5.56 X 109
lb-sq in. Computer Program PMEIX was run and an interaction
diagram for the pile was obtained. That diagram is shown in Fig. 10.5.
A field load test was performed at the site and the ultimate axial
capacity of a pile was found to be 176 kips. An analysis was made'to
develop a curve showing axial load versus settlement. The curve is shown
in Fig. 10.6.
The subsurface soils at the site consist of silty clay. The water
content in the top 10 ft averaged 20% and below 10 ft it averaged 44%. The
water table was reported to be at a depth of 10 ft. There was a consider-
able range in the undrained shear strength of the clay and an average val-
ue of 3 kips/sq ft was used in the analysis. A value of the submerged unit
weight of 46 lb/cu ft was employed and the value of s50 was estimated to
be 0.005.
253

400
...
0 200o
o
...J
o
x
«
....,,,

•
,,,
/
/
/..
•I
00 5 10
Fig. 10.5.
.=~
-C
QJ
0.2EQJ
--QJ
II)
0.3
Ultimate Moment, in.-kips x 102
Interaction diagram of the reinforced
oncrete pile.
Axial Load, kips
100 150
Fig. 10.6. Axial load versus settlement
for reinforced concrete pile.
In making the computations, the'assumptions were made that all of the
load was carried by the piles with none of the load taken by passive earth
pressure or by the base of the footing. It was further assumed that the
pile heads were free to rotate. As noted earlier, the factor of safety
254

must be in the loading. Therefore, the loadings shown in Table 10.1 were
used in the preliminary computations.
TABLE 10.1. VALUES OF LOADING EMPLOYED IN ANALYSES.
Case Loads, kips moment, ft-kips Comment
P Phv
1 46 21 40 service load
2 69 31.5 60 1.5 times service load
3 92 42 80 2 times service load
4 115 52.5 100 2.5 times service load
Table 10.2 shows the movements of the origin of the global coordinate
system when Eqs. 10.20 through 10.22 were solved simultaneously. The
loadings were such that the pile response was almost linear so only a
small number of iterations werE required to achieve convergence. The com-
puted pile-head movements, loads, and moments are shown in Table 10.3.
TABLE 10.2. COMPUTED MOVEMENTS OF ORIGIN OF GLOBAL COORDINATE SYSTEM.
Case Vertical Movement ~v Horizontal Movement ~h Rotation a
inches inches rad
1 0.004 0.08 9 x 10-5
2 0.005 0.12 1.4 x 10-4
3 0.008 0.16 1.6 x 10-4
4 0.017 0.203 8.4 x 10-4
255

TABLE 10.3. COMPUTED MOVEMENTS AND LOADS AT PILE HEADS.
C Pile 1 Pil e 2
a P Pt Mmax Yt Px
P
t Mmaxxt Yt x
xt
s
e in. in. kips kips in.-kips in. in. kips kips in.-kips
1 .0.005 0.08 9.7 6.0 148 0.02 0.077 38.9 5.8 143
2 0.008 0.12 14.5 9.0 222 0.03 0.116 58.3 8.6 215
3 0.011 0.162 19.3 12.1 298 0.04 0.156 77.7 11. 5 288
4 0.013 0.203 24.2 15.2 373 0.06 0.194 97.2 14.3 360
The computed loading on the piles is shown in Fig. 10.7 for Case 4.
The following check is made to see that the equilibrium equations are sat-
isfied.
IF = 24.2 + 97.2 cos 14° - 14.3 sin 14°
v
= 24.2 + 94.3 - 3.5 = 114.9 kips OK
IFh = 15.2 + 14.3 cos 14° - 97.2 sin 14°
= 15.2 + 13.9 + 23.6 = 52.7 kips OK
IM = -(24.2)(1.5) + (97.2 cos 14° )(1.5)
- (14.3 sin 14° )(1.5)
= -36.3 + 141.4 - 5.2 =99.9 ft-kips OK
Thus, the retaining wall is in equilibrium. A further check can be made
to see that the conditions of compatibility are satisfied. One check can
be made at once. Referring to Fig. 10.6, an axial load of 97.2 kips
results in an axial deflection of about 0.054 in., a value in reasonable
agreement with the value in Table 1~.3. Further checks on compatibility
can be made by using the pile-head loadings and Computer Program COM622 to
see if the computed deflections under lateral load are consistent with the
values tabulated in Table 10.3.
No firm conclusions can be made concerning the adequacy of-the par-
.
ticular design without further study: If the assumptions made in perform-
ing the analyses are appropriate, the results of the analyses show the
foundation to be capable of supporting the load. As a matter of fact, the
piles could probably support a wall of greater height.
256

•
24.2
,
Fig. 10.7. Pile loading, Case 4.
With a multiplier of 2.5 times the service loads, the batter pile
takes an axial load of 97 kips which is well short of the ultimate capaci-
ty of the pile as shown in Fig. 10.6. Figure 10.5 shows that Pile 1 should
be able to sustain an ultimate moment of about 700 in-kips and under Case
4 (2.5 times the service load) the actual moment is about 373 in.-kips.
The example problem illustrates the use of the procedure. The exam-
ple shows further that an the appropriate solution of a design problem
probably should involve a number of computations in which the important
parameters in the problem are varied through a range that represents the
reasonable uncertainty .
10.3 BEHAVIOR OF A GROUP OF CLOSELY-SPACED PILES
The analysis of a group of closely-spaced piles under lateral loading
can be undertaken by a number of methods, two of which will be presented
herein. The most obvious problem is to take into account the effect of a
single pile on others in the group. Approximate solutions have been
developed (Poulos, 1971b; Banerjee and Davies, 1979) by use of the
equations of elasticity; however, as demonstrated in Chapter 8, the
assumption of a linear response of the soil-pile system is inadequate to
deal with many problems of a pile under lateral loading.
The first method that will be presented uses a combination of the
elastic method with the p-y method. The second method is the single-pile
approach to group analysis.
257

The principal difficulty in achieving an acceptable solution to the
closely-spaced group is that there have been few full-scale load tests of
such groups. Such tests are expensive and difficult to carry out. There-
fore, the methods that are presented here must only be used in consider-
ation of the assumptions that underlie each of the methods.
Pou10s-Focht-Koch Method
The, following equation was developed by Poulos (1971a, 1971b) to
obtain the deflection and load on each of the piles in a group, assuming
the soil to act elastically.
where
m
Pk = PF .I (HJ.apFkJ· + Hk)
J=l
(10.23)
H.
J
Hk
=deflection of the k-th pile
=the unit reference displacement of a single pile
under a unit horizontal load, computed by using
elastic theory
=lateral load on pile j
=the coefficent to get the influence of pile
j on pile k in computing the deflection p
(the subscript F pertains to the fixed-head case
and is used here for convenience; there are also
influence coefficients as shown later where shear
is applied, ~pHkj' and where moment is applied,
~pMkj )
Hk = lateral load on pile k
m = number of pi 1es in group.
If the total load on the group is H
G
, then
m
Ij=l
H =
G
(10.24)
If the piles are connected to a cap such that each of the piles Js
caused to deflect an equal amount, the deflection Pk i.s equal to YG' th~
deflection of the group. If there are m piles in the group, m + 1
equations can be formulated using Eqs. 10.23 and 10.24 and solved for the
group deflection and the load H on each pile in the group.
258

In order to write the equations it is necessary to have the influence
coefficients. Poulos has supplied a family of curves for the ~-values
with the curves based on a Poisson's ratio of 0.5. The curves must be
entered with values of Lib, sib, ~, and K
R
, where L is pile length, b is
pile diameter, s is center to center spacing, ~ is the angle between the
line through the two piles in question and the line giving the direction
of the loading, and KR is defined by the following equation.
where
EI
KR - -E-L-4
s
E =modulus of elasticity of pile material
I = moment of inertia of pile
E =soil moduluss
(10.25)
Figures 10.8 and 10.9 present Poulos curves for free-head piles that are
subjected to shear and to moment, respectively. Figure 10.10 presents the
Poulos curve for a fixed-head pile.
Focht and Koch (1973) have proposed modifications of the Poulos meth-
od. They suggested a revision of Eq. 10.23 as follows:
(10.26)
where
R = relative stiffness factor.
The relative stiffness factor is the ratio of the mudline deflection
of a single pile computed by the p-y curve approach, y , to the deflection
s
p computed by the Poulos method that assumes elastic soil. In both
instances, the lateral load on the single pile is the total lateral load
on the pile group divided by the number of piles.
Equation 10.27 is the Poulos equation for the deflection of a single
pile with free head (Eq. 10.27 is identical to Eq. 7.28 except for differ-
ence in symbols).
p = I _H_ + I _M_
pF E L pM E L2
S s
(10.27)
The influence coefficients IpH and IpM may be obtained from Figs. 10.11
and 10.12, respectively.
259

I.O~~--r--r--~~--~--~--~~
.9 values of LIb
.8
.7
.6
Zls= 0.5
/3= 0°
/3 =90°
K R= I()~
:x:
0:4. .5

.3
.2
.1
0

'i5, ...........10......... .......
--I.0 ......-+---+---+---+---I----+O---t----+==--I
.9
.8
.7
.6
values of LIb
Zls= 0.5
{3 =0°
/3 =9(j
:x:
0:4. .5
.4
.3
.2
.1
O.O~~~~~~~~--~--~~~~
o 2 3 4 5
..2 .15 .1 .05 0.0
SIb bls
Fig. 10.8. Interaction factors cxpH for free-head
piles subjected to horizontal load
(Poulos, 1971).
260

·7
.6
:I: .5
-e-
0
.4
~
:i: .3
Q...
0
.2
.1
1.0
.9
.8
.7
:I:
.6~
tI
~ .5
:i:
.4Q...
0
.3
.2
•1
0.00
Fig. 10.9.
of Lib
liS =0.5
/3 = 0°
--- /3 = 90°
-5
K R = 10
values of Lib
liS =0.5
°/3=0
--- /3 = 90°
K R=0.1
-2 3 4 5
.2 .15 .1 .05 0.0
51 b b/s
Interaction factors a for free-head piles
subjected to moment pM (Poulos, 1971t
261

1.0
.9
values of LIb
.8 Zls =0.5
.7
{3= 0°
--- {3= 90°
.6 -5
KR=IO
u...
.5a..
tj
.4
.3
.2
1.0
.9 values of LIb
ZlS = 0.5
.8
{3 =0°
.7 --- {3 =90°
.6 K R=0.1
u...
~~Oo
.5
'
a.. , ........
t:l
, ........ 0
.4 -....... ........ :- 25-
•:3
..........
10'.
.2
.1
0.0
0 2 :3 4 5
.2 .15 .1 .05 0
Sib bls
Fig. 10.10. Interaction factors a
pF for fixed-head pile
(Poulos, 1971).
262

I values of LIb 1Is= 0.5
20
/0
5
2
O~~~--L-__L-__~__~__~~
10-6 10-5 Icr4 10-3 10-2 10-1 10
KR
Fig. 10.11 .. Influence factors IpH for a free-head pile
(Poulos, 1971).
1000
values of LIb 1Is= 0.5
:I:
100-.....
ca
::
~
.....
10
I~~__~~~~__L-~~__~~
10-6 10-5
10-4
10-3 10-2 10-1 10
KR
Fig. 10.12. Influence factors IpM for a free-head pile
(Poulos, 1971).
263

Equation 10.28 is the Poulos equation for the deflection of a single
pile with fixed head.
H
p = I
pF E L
s
The influence coefficients IpF may be obtained from Fig. 10.13.
5 0 ~-r---r--r---r---r---r------.
values of Lib
20
10
5
2
I~_I...--_L..---'_---L_---L_---L-===
10-
6
10-
5
16
4
16
3
16
2
10-
1
10
KR
Fig. 10.13. Influence factors IpF for a fixed-head pile
(Poulos, 1971).
(10.28 )
The curves that are presented in Figs. 10.11 through 10.13 are
entered with values of KR and Lib. (Figures 10.11 through 10.13 are the
same as Figs. 7.10, 7.11, and 7.14, respectively, but are repeated here
for convenience.) A value of Poisson's ratio of 0.5 was used in develop-
ing all three curves.
It is important in using the Poulos equations to obtain a value of
the soil modulus Es that is as accurate as possible. It is generally
agreed that the best method for determining Es is to perform a field load-
ing test. However, such tests are not practical in many instances for
both economic and practical reasons. In the absence of such tests, some
264

correlations of E with the undrained shear strength can be used as a
s
rough guide. The stress-strain data obtained from triaxial tests can fur-
nish representative Es values that are better than those obtained from
correlations with the undrained shear strength. Focht and Koch state that
Es should be selected from available stress-strain-test results using a
low stress level in the soil. They suggest that the value should be at
least as great as the secant modulus corresponding to a stress equal to 50
percent of the strength and probably as great as the initial tangent modu-
lus indicated by most laboratory tests.
Using Eqs. 10.24 and 10.26, a set of simultaneous equations is formu-
lated and solved for the group deflection YG and the lateral load on each
pile in the group. The pile with the greatest load is selected for analy-
sis by employing modified p-y curves. The p-values are modified by
employing a multiplication factor to reflect the IIshadowingll effect of
closely-spaced piles. The y-values are modified by multiplying all of the
deflections in the p-y curves by a V-factor of 2, 3, 4 and so on. The
deflection of the single pile is computed with the modified p-y curves and
the V-factor is found that gives agreement between the single-pile
deflection so computed and the deflection of the pile group. With this
appropriate V-factor, the pile behavior can be computed with the modified
p-y curves, completing the solution. Figure 10.14 presents the form of
the solution that employs the V-factor .
...
2u
~ deflection
- from modified Poulus - ply
ona lysis
Pile -head Deflection
Fig. 10.14. Y factor influence on computed pile-head deflection.
265

Single-Pile Method
The single-pile method of analysis is based on the assumption that
the soil contained between the piles moves with the group. Thus, the pile
group with the contained soil can be treated as a single pile of large
diameter.
The first step in the use of the single-pile method is to select the
group to be analyzed and to ascertain the loading. A plan view of the
piles at the groundline is prepared and the minimum length is found for a
line that encloses the group. This length is considered to be the circum-
ference of a pile of large diameter; thus, the length is divided by ~ to
obtain the diameter of the imaginary pile.
The next step is to determine the stiffness of the group. For a lat-
eral load passing through the tops of the piles, the stiffness of the
group is taken as the sum of the stiffness of the individual piles. Thus,
it is assumed that the deflection at the pile top is the same for each pile
in the group and, further, that the deflected shape of each pile is iden-
tical. Some judgement must be used if the piles in the group have differ-
ent lengths.
An analysis is made for the imaginary pile, taking into account the
nature of the loading and the boundary conditions at the pile head. The
shear and moment for the imaginary large-sized pile is shared by the indi-
vidual piles according to the ratio of the lateral stiffness of the indi-
vidual pile to that of the group.
The shear, moment, pile-head deflection, and pile-head rotation
yield a unique solution for each pile in the group. As a final step, it is
necessary to compare the single-pile solution to that of the group. It
could possibly occur that the piles in the group could have an efficiency
greater than one, in which case the single-pile solutions would control.
Example Problem
A sketch of an example problem is shown in Fig. 10.15. It is assumed
that steel piles are embedded in a reinforced concrete mat in such a way
that the pile heads do not rotate. The piles are 14HP89 by 40 ft long and
placed so that bending is about the strong axis. The moment of inertia is
904 in.4 and the modulus of elasticity is 30 x 106
lb/sq in. The width of
the section is 14.7 in. and the depth is 13.83 in.
266

7 4 I
H H H
3'
8 5 2
H H H
3'
9 6 3
H H H
Fig. 10.15. Plan and elevation of foundation
analyzed in example problem.
The soil is assumed to be a sand with an angle of internal friction
of 34 degrees, the unit weight is 114 lb/cu ft, and the assumed Poulos
soil modulus is 3000 lb/sq ft.
Pou los- Focht- Koch Solution
Computation of value of R:
The lateral load is 450/9 or 50 kips
Computation of p using Eqs. 10.25 and 10.28
K = (30 x 106
)(904) = 1.70 x 10-4
R (3000)(480)4
L/b = 480/14.7 =32.7
IpF =7.8 (from Fig. 10.13)
(50,000)
p = (7.8) (3000)(480) = 0.27 in.
Y
t
= 0.35 in. (from COM622 solution)
R = 0.35/0.27 =1.29.
267

Computation of Poulos deflection under unit load:
PF = 0.27/50,000 = 0.54 x 10- 5
in./lb.
Determination of a-values:
(Because a number of the a-values are identical, it is convenient to
obtain these identical values together rather than to determine the values
pile-by-pile as they are used.)
al -2 = a2- l =a2-3 =a3- 2 =a4- 5 =a5- 4 =a5_ 6 =a6- 5 =
a7-8 =a8- 7 =a8- 9 =a9_ 8
~ =90Q
, sib =(3)(12)/14.7 =2.45, a =0.35
(from Fig. 10.10)
(Note: the a-value was obtained by straight-line
interpolation)
a1~3 =a 3- 1 =a4-6 =a 6- 4 =a7- 9 =a9- 7
~ =90Q
, sib =4.9, a =0.23
al - 4 = a4- 1 =a4- 7 =a7- 4 =a2- 5 =a5- 2 =a5- 8 =a8- 5 =
a3- 6 =a 6- 3 =a6-9 =a 9- 6
~ =0°, sib =2.45, a =0.53
al - 7 =a7- 1 =a2-8 =a8- 2 =a3- 9 =a9- 3
~ =0°, sib =4.9, a =0.40
al - 5 = a5- l =a 2-4 =a4- 2 =a2- 6 =a6- 2 =a3- 5 =a5- 3
a4-8 = a8- 4 =a5-7 =a7- 5 =a5- 9 =a9- 5 =a6- 8 =a8- 6
~ =45°, sib =3.46, a =0.38
al - 6 =a 6- 1 =a3-4 =a4- 3 =a4- 9 =a9- 4 =a6-7 =a7- 6
~ =63°, sib =5.47, a =0.25
al - 8 =a 8- 1 =a2-7 =a 7- 2 =a2- 9 =a9- 2 =a8- 3 =a3-8
~ =26.6°, sib =5.47, a =0.28
al - 9 = a9- l =a3- 7 =a 7- 3
~ =45Q
, sib =6.93, a = 0.22
Simultaneous equations:
Piles 1,3,7,9 have identical equations:
PI =P3 =P7 =P9
=PF [RH1 + (al - 3 + al - 7 + al - 9)Hl
+ (al - 2 + al - 8)H2 + (al - 4 + al - 6)H4
+ a
l - 5H5]
=0.54 x 10 -S[1.29H
1
+ (0.23 + 0.40 + 0.22) HI
+ (0.35 + 0.28)H
2
+ (0.53 + 0.25)H4 + 0.38 H5]
=0.54 x 10 -S[2.14H
l
+ 0.63H
2
+ 0.78 H4 + 0.38H5] (A)
268

Piles 2 and g have identical equations:
Substitutions as shown above yield the following equation:
P2 = Pg = 0.54 x 1O-s[1.26H1 + 1.69H2 + 0.76H4 + 0.53H5J (B)
Piles 4 and 6 have identical equations:
P4 = P6 = 0.54 x 1O-S[1.56H1 + 0.76H2 + 1. 52H4 + 0.35H5J (C)
Pil e 5:
P5 = 0.54 x 10 -S[1.52H1 + 1.06H2 + 0.70H4 + 1.29H5J (0)
Summation of loads:
4Hl + 2H2 + 2H4 + H5 = 450 kips (E)
Solving equations A, B, C, 0, E simultaneously, noting that:
YG = PI = P2 = P3 = P4 = P5 = P6 = P7 = Pg = P9
HI = 61.9 kips, H2 = 49.1 kips, H4 = 40.9 kips, H5 = 22.5 kips
YG=0.91in.
Using a computer program to solve for the moment curve in Piles
1, 3, 7, 9, the piles with the heaviest load (Sullivan and
Reese, 1990):
plotted.
--:-
c:
--~
The V-factors of 2, 4, and 5 were employed and Fig. 10.16 was
1.5,.-----T------r----,------,
1.0
0.91
0.5
0.0'-----...1.------------'------
o 2 4 4.8
Y - factor
6
Fig. 10.16. Graphical solution for Y-facto~
269
8

As noted, a V-factor of 4.8 yielded a deflection of 0.91. The
bending moment curve corresponding to a deflecton of 0.91 is shown in Fig.
10.17. The maximum bending stress is computed as follows:
c
..c
Mcf = - =
I
(4.4 x 10
3
)(6.915) = 33.7 kips/sq in.
904
Or-------,-----~~------~r_------~------~
80
160
a. 240
Q)
o
320
400
480~------~------~------~-------A~____~
-800 -600 -400 -200 o 200
Moment, in.-Ibs x 10
6
Fig. 10.17. Bending moment curve for pile with greatest load,
example solution.
Solution Assuming Group Behaves as a Single Pile
The computer progY'am was run with a pile diameter of 109 in. and a
moment of inertia of 8136 in.4 (9 times 904). The results were as fol-
lows:
270

Yt = 1. 04 in.
Mt = M = 3.53 X 10 7
in.-lb for group
max
= 3.92 x 106
in.-lb for single pile
Bending stress =31.9 k/sq in.
The deflection and stress are in reasonable agreement with the previous
solution.
Comment on Solution of Example Problem
For the example problem, good agreement was found between the two
methods of analysis. However, that good agreement could be fortuitous.
As noted earlier, there is a need for more full-scale load tests on pile
groups under lateral loading. If the size of the construction project
justifies, consideration should be given to a field test program prior to
making final designs.
The methods demonstrated herein are recommended for preliminary stu-
dies. Studies should be done to investigate the effects on the results of
changing the values of the input parameters through a range consistent
with reasonable expectations. If final designs are made on the basis of
results from these methods, an appropriate factor of safety should be
carefully considered.
10.4 REFERENCES
Awoshika, Katsuyuki, and Reese, Lymon C., Analysis of Foundation
with Wi de ly Spaced Batter Pil es, II Research Report 117-3F, Project
3-5-68-117, Center for Highway Research, University of Texas at Austin,
February 1971.
Banerjee, P. K., and Davies, T. G., Analysis of Some Reported Case
Hi stori es of Laterally Loaded Pil e Groups, II Proceedi ngs, Numeri ca1 Meth-
ods in Offshore Piling, The Institution of Civil Engineers, London, May
1979, pp. 101-108.
Bryant, L. M., Three-Dimensional Analysis of Framed Structures with
Nonlinear Pile Foundations, Unpublished Dissertation, The University of
Texas at Austin, 95 pages, December. 1977.
Coyle, H. M., and Reese, L. C., IILoad Transfer for AXially Loaded
Piles in Clay,1I Proceedings, American Society of Civil Engineers, Vol. 92,
No. SM2, March 1966, pp. 1-26.
Coyle, H. M., and Sulaiman, I. H., IISkin Friction for Steel Piles in
Sand,1 Proceedi ngs, Ameri can Soci ety for Ci vi 1 Engi neers, Vol. 93, No.
SM6, November 1967, pp. 261-278.
271

Focht, J. A., Jr., and Koch, K. J., Rational Analysis of the Lateral
Performance of Offshore Pile Groups, Proceedings, Fifth Annual Offshore
Technology Conference, Paper No. 1896, Vol. II, Houston, 1973, pp.
701-708.
Kraft, L. M., Jr., Ray, R. P., and Kagawa, T., Theoretical t-z
Curves, Proceedi ngs, Ameri can Soci ety of Ci vil Engi neers, VoL 107, No.
GT11, November 1981, pp. 1543-1561.
Kuthy, R. A., Ungerer, R. P., Renfrew, W. W., Hiss, J. G. F., Jr.,
and Rizzuto, 1. F., Lateral Load Capacity of Vertical Pile Groups,
Research Report 47, Engineering Research and Development Bureau, New York
State Department of Transportation, Albany, April 1977,37 pages.
Lam, Philip, Computer Program for Analysis of Widely Spaced Batter
Piles, Unpublished Thesis, The University of Texas at Austin, August
1981.
O'Neill, M. W., Ghazzaly, o. 1., and Ha, H. B., Analysis of
Three-Dimensional Pile Groups with Nonlinear Soil Response and
Pil e-Soil-Pil e Interaction, Proceedi ngs, Offshore Techno logy Conference,
Houston, Texas, Vol. II, Paper No. 2838, pp. 245-256, 1977.
Poulos, H. G., Behavior of Laterally Loaded Piles: I - Single
Pil es, Proceedi ngs, Ameri can Soci ety of Ci vil Engi neers, Vol. 97, No.
SM5, May 1971a, pp. 711-731.
Poulos, H. G., Behavior of Laterally Loaded Piles: II - Pile
Groups, Proceedings, American Society of Civil Engineers, Vol. 97, No.
SM5, May 1971b, pp. 733-751.
Reese, Lymon C., Load versus Settlement for an Axially Loaded Pile,
Proceedi ngs, Part II, Symposi um on Beari ng Capacity of Pil es, Central
Bui.1ding Research Institute, Roorkee, February 1964, pp. 18-38.
Reese, Lymon C., Ana lys is of a Bri dge Foundation Supported by Batter
Piles, Proceedings, Fourth Annual Symposium on Engineering Geology and
Soil Engineering, Moscow, Idaho, April 1966, pp. 61-73.
Reese, Lymon C., and Matlock, Hudson, Behavior of a Two-Dimensional
Pile Group Under Inclined and Eccentric Loading, Proceedings, Offshore
Exploration Conference, Long Beach, California, 1966, pp. 123-140.
Reese, Lymon C., O'Neill, M. W., and Smith, R. E., Generalized Anal-
ysis of Pile Foundations, Proceedings, American Society of Civil Engi-
neers, Vol. 96, No. SM1, pp. 235-250, January 1970.
Sullivan, W. Randall, and Reese, Lymon C., Documentation of Comput-
er Program COM624, Geotechnical Engineering Center, Department of Civil
Engineering, University of Texas at Austin, February 1980.
272

CHAPTER 11. STEP-BY-STEP PROCEDURE FOR DESIGN
The material presented in the preceding chapters is designed for self-
study. An engineer can begin with Chapter 1 with a reading schedule and
read directly through the manual. The worked examples can be checked and
some of the exercises can be worked out.
If such a self-study program is elected, the guidelines that follow
will serve to confirm the design procedures that are suggested. The pro-
cedures that are shown below may be modified to agree with the particular
preferences of the designer. However, if some time has passed since the
self-study or if the designer elects merely to scan much of the material
that has been presented, the step-by-step procedures may prove to be use-
ful.
1. A structural engineer and a geotechnical engineer should
work together to establish the general nature of the prob-
lem and to cooperate as solutions are developed (Chapter
1) .
2. The nature, magnitude, and direction of the various load-
ings on the pile should be found. The service load and
design load, both axial and lateral, should be established
(Chapter 1). Where there is uncertainty about the magni-
tude of the load, an upper-bound value and a lower-bound
value may be used.
3. A pile should be selected for analysis (if not already done
from considerations of axial loading) and the pile-head
conditions should be considered, whether fixed, free, or
partially restrained (Chapters 2, 4, 5, 6, and 7). The
Broms method (Chapter 7) could prove useful if lateral load
only controls design.
4. A check should be made to see whether or not any of the
assumptions made in deriving the differential equation are
violated (Chapter 2); if so, a change in the pile make-up
should be considered.
5. All of the i nformati on on the soil properties at the site
should be analyzed and a soil profile should be selected for
design (Chapter 3). If there is uncertainty about the soil
properties, upper-bound values and lower-bound values may
273

be selected. The importance of the various soil parameters
may be understood by reference to the first part of Chapter
9.
6. Having at hand the loading, the pile geometry, and a soil
profil e, p-y curves can be developed accordi ng to proce-
dures in Chapter 3. Or if Computer Program COM624 is avail-
able, the program may be used directly in design or the soil
subrouti nes in the program may be used to generate p-y
curves, a desirable step to give a better understanding of
the method of computation.
7. Computer Program PMEIX (Appendix 5) can be used to obtain
the bending stiffness of a reinforced concrete pile. The
bending stiffness of other kinds of piles can be obtained
from handbooks or from the elementary principles of mechan-
i cs.
8. Computer Program PMEIX (Appendix 5) can be used to compute
the bending moment capacity of a reinforced concrete pile.
The moment capacity of other kinds of piles can be obtained
from handbooks or from elementary principles of mechanics.
The case studies of piles in clay and in sand in Chapter 9
can provide guidance in performing Steps 7 and 8.
9. The engineer who has not made many computations or is making
only infrequent computations should use the procedures in
Chapter 6 to gain familiarity with the design process and to
gain some insight into the particular design being done.
10. The input for Computer Program COM622 or Computer Program
COM624 can be prepared. The best estimate of the important
parameters can be made for small jobs and a series of sol-
utions performed with increasing loads until a soil failure
(excessive deflection) or a pile failure (excessive bending
moment) is found. Only in rare cases should a soil failure
be allowed, because a small increase in the length of a pile
can lead to a sharp increase in ability to carry load.
11. If the job is large and if time allows for small jobs,
parameter studies can be made using the upper-bound values
and lower-bound values of soil properties (and possibly
other variables) to gain additional insight into the possi-
274

ble range in the response of the pile. The computer sol-
utions are inexpensive and especially so if results can be
read directly from the screen of a terminal.
12. As a part of using the computer program as outlined in Steps
11 and 12, checks must be made to see that an appropriate
increment length and a suitable closure tolerance are being
employed (Chapter 5). The number of significant figures
used in the internal computations of the computer must also
be satisfactory (Appendix 3).
13. From the computer solutions in Steps 10 and 11, the pile
being analyzed can be judged to be satisfactory or a new
pi 1e can be selected and the steps repeated. A specifi c
design will result, yield pile diameter (or projected
width), length, and bending capacity along the length of
the pile. Further, information can be gained as to the most
favorable way to connect the pile head to the superstruc-
ture.
14. In performing the computer solutions as outlined in Steps
10 and 11, the engineer should make use of the methods pre-
sented in Chapters 4, 6, and 7 to make approximate solutions
and to make checks on computer runs so that confidence is
developed in the solution techniques.
15. If the piles occur in groups, the methods shown in Chapter
10 should be employed. In some cases it is not necessary to
make the detail ed analyses presented in Chapter 10, but
judgement can be used in making any necessary design mod-
ifications.
16. The methods presented in Chapter 8 should be employed to see
that the pi 1es that are selected using the concepts of
soil-structure interaction are adequate from the standpoint
of structural behavior.
17. A design office may have the same type of pile, say a steel
bearing pile, to design in many instances in the same type
of soil profile, say a sand with a high water table. A com-
puter program, COM622 or COM624, can be used to develop
design charts to allow preliminary designs, or final
275

designs in some cases, to be made rapidly. Such design
charts are demonstrated in Chapter 7.
'U.S. GOVERNMENT PRINTING OFFICE, 1986--491-810/20705
276

FEDERALLY COORDINATED PROGRAM (FCP) OF HIGHWAY RESEARCH,
DEVELOPMENT, AND TECHNOLOGY
The Offices of Research, Development, and
Technology (RDT) of the Federal Highway
Administration (FHWA) are responsible for a broad
research, development, and technology transfer pro-
gram. This program is accomplished using numerous
methods of funding and management. The efforts
include work done in-house by RDT staff, con-
tracts using administrative funds, and a Federal-aid
program conducted by or through State highway or
transportation agencies, which include the Highway
Planning and Research (HPR) program, the Na-
tional Cooperative Highway Research Program
(NCHRP) managed by the Transportation Research
Board, and the one-half of one percent training pro-
gram conducted by the National Highway Institute.
The FCP is a carefully selected group of projects,
separated into broad categories, formulated to use
research, development, and technology transfer
resources to obtain solutions to urgent national
highway problems.
The diagonal double stripe on the cover of this report
represents a highway. It is color-coded to identify
the FCP category to which the report's subject per-
tains. A red stripe indicates category 1, dark blue
for category 2, light blue for category 3, brown for
category 4, gray for category 5, and green for
category 9.
FCP Category Descriptions
1. Highway Design and Operation for Safety
Safety RDT addresses problems associated
with the responsibilities of the FHWA under the
Highway Safety Act. It includes investigation of
appropriate design standards, roadside hard-
ware, traffic control devices, and collection or
analysis of physical and scientific data for the
formulation of improved safety regulations to
better protect all motorists, bicycles, and
pedestrians.
2. Traffic Control and Management
Traffic RDT is concerned with increasing the
operational efficiency of existing highways by
advancing technology and balancing the
demand-capacity relationship through traffic
management techniques such as bus and carpool
preferential treatment, coordinated signal tim-
ing, motorist information, and rerouting of
traffic.
3. Highway Operations
This category addresses preserving the Nation's
highways, natural resources, and community
attributes. It includes activities in physical
maintenance, traffic services for maintenance
zoning, management of human resources and
equipment, and identification of highway
elements that affect the quality of the human en-
vironment. The goals of projects within this
category are to maximize operational efficiency
and safety to the traveling public while conserv-
ing resources and reducing adverse highway and
traffic impacts through protections and enhance-
ment of environmental features.
4. Pavement Design, Construction, and
Management
Pavement RDT is concerned with pavement
design and rehabilititation methods and pro-
cedures, construction technology, recycled
highway materials, improved pavement binders,
and improved pavement management. The goals
will emphasize improvements to highway
performance over the network's life cycle, thus
extending maintenance-free operation and max-
imizing benefits. Specific areas of effort will in-
clude material characterizations, pavement
damage predictions, methods to minimize local
pavement defects, quality control specifications,
long-term pavement monitoring, and life cycle
cost analyses.
5. Structural Design and Hydraulics
Structural RDT is concerned with furthering the
latest technological advances in structural and
hydraulic designs, fabrication processes, and con-
struction techniques to provide safe, efficient
highway structures at reasonable costs. This
category deals with bridge superstructures, earth
structures. foundations. culverts, river
mechanics, and hydraulics. In addition, it in-
cludes material aspects of structures (metal and
concrete) along with their protection from cor-
rosive or degrading environments.
9. RDT Management and Coordination
Activities in this category include fundamental
work for new concepts and system character-
ization before the investigation reaches a point
where it is incorporated within other categories
of the FCP. Concepts on the feasibility of new
technology for highway safety are included in this
category. RDT reports not within other FCP
projects will be published as Category 9 projects.
,
I

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