Technical Manual for LPile 2016
(Using Data Format Version 9)
A Program for the Analysis of Deep Foundations Under Lateral Loading
by
William M. Isenhower, Ph.D., P.E.
Shin-Tower Wang, Ph.D., P.E.
L. Gonzalo Vasquez, Ph.D., P.E.
January 14, 2016
Copyright © 2016 by Ensoft, Inc.
All rights reserved.
This book or any part thereof may not be reproduced in any form without the written permission
of Ensoft, Inc.
Date of Last Revision: January 14, 2016
Table of Contents
List of Figures ............................................................................................................................... vii
List of Tables ............................................................................................................................... xiii
Chapter 1 Introduction .................................................................................................................... 1
1-1 Compatible Designs .............................................................................................................. 1
1-2 Principles of Design.............................................................................................................. 1
1-2-1 Introduction ................................................................................................................... 1
1-2-2 Modelling of Nonlinear Response of Soil ..................................................................... 2
1-2-3 Limit States ................................................................................................................... 2
1-2-4 Step-by-Step Procedure ................................................................................................. 2
1-2-5 Suggestions for the Designing Engineer ....................................................................... 3
1-3 Modeling a Pile Foundation ................................................................................................. 5
1-3-1 Introduction ................................................................................................................... 5
1-3-2 Example Model of Individual Pile with Axial and Lateral Loading ............................. 7
1-3-3 Computation of Foundation Stiffness ........................................................................... 8
1-3-4 Concluding Comments .................................................................................................. 9
1-4 Organization of Technical Manual ....................................................................................... 9
Chapter 2 Solution for Pile Response to Lateral Loading ............................................................ 11
2-1 Introduction ........................................................................................................................ 11
2-1-1 Influence of Pile Installation and Loading on Soil Characteristics ............................. 11
2-1-2 Models Used in Analyses of Laterally Loaded Single Piles ....................................... 14
2-1-3 Computational Approach for Single Piles ................................................................... 21
2-1-4 Pile Buckling Analysis ................................................................................................ 22
2-1-5 Analysis of Critical Pile Length .................................................................................. 23
2-1-6 Occurrences of Lateral Loads on Piles ........................................................................ 24
2-2 Derivation of Differential Equation for the Beam-Column and Methods of Solution ....... 30
2-2-1 Derivation of the Differential Equation ...................................................................... 30
2-2-2 Solution of Reduced Form of Differential Equation ................................................... 34
2-2-3 Solution by Finite Difference Equations ..................................................................... 40
Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ......................................................... 49
3-1 Introduction ........................................................................................................................ 49
3-2 Experimental Measurements of p-y Curves........................................................................ 51
3-2-1 Direct Measurement of Soil Response ........................................................................ 51
3-2-2 Derivation of Soil Response from Moment Curves Obtained by Experiment ............ 51
3-2-3 Nondimensional Methods for Obtaining Soil Response ............................................. 53
3-3 p-y Curves for Cohesive Soils ............................................................................................ 54
3-3-1 Initial Slope of Curves................................................................................................. 54
3-3-2 Analytical Solutions for Ultimate Lateral Resistance ................................................. 57
3-3-3 Influence of Diameter on p-y Curves .......................................................................... 63
3-3-4 Influence of Cyclic Loading ........................................................................................ 64
iii
3-3-5 Introduction to Procedures for p-y Curves in Clays .................................................... 66
3-3-6 Procedures for Computing p-y Curves in Clay ........................................................... 69
3-3-7 Response of Soft Clay in the Presence of Free Water................................................. 69
3-3-8 Response of Stiff Clay in the Presence of Free Water ................................................ 75
3-3-9 Response of Stiff Clay with No Free Water ................................................................ 84
3-3-10 Modified p-y Formulation for Stiff Clay with No Free Water .................................. 88
3-3-11 Other Recommendations for p-y Curves in Clays ..................................................... 89
3-4 p-y Curves for Cohesionless Soils ...................................................................................... 89
3-4-1 Description of p-y Curves in Sands ............................................................................. 89
3-4-2 Reese, et al. (1974) Procedure for Computing p-y Curves in Sand ............................ 94
3-4-3 API RP 2A Procedure for Computing p-y Curves in Sand ....................................... 100
3-4-4 Other Recommendations for p-y Curves in Sand ...................................................... 106
3-5 p-y Curves for Liquefied Soils.......................................................................................... 107
3-5-1 Response of Piles in Liquefied Sand ......................................................................... 107
3-5-2 Method of Rollins et al. (2005a) ............................................................................... 108
3-5-3 Simplified Hybrid p-y Model .................................................................................... 110
3-5-4 Modeling of Lateral Spread....................................................................................... 116
3-6 p-y Curves for Loess Soils ................................................................................................ 116
3-6-1 Background ............................................................................................................... 116
3-6-2 Procedure for Computing p-y Curves in Loess ......................................................... 118
3-7 p-y Curves for Cemented Soils with Both Cohesion and Friction ................................... 125
3-7-1 Background ............................................................................................................... 125
3-7-2 Recommendations for Computing p-y Curves .......................................................... 126
3-7-3 Procedure for Computing p-y Curves in Soils with Both Cohesion and Internal
Friction ................................................................................................................................ 128
3-7-4 Discussion ................................................................................................................. 131
3-8 p-y Curves for Rock .......................................................................................................... 132
3-8-1 Introduction ............................................................................................................... 132
3-8-2 Descriptions of Two Field Experiments.................................................................... 133
3-8-3 Procedure for Computing p-y Curves in Vuggy Limestone ...................................... 138
3-8-4 Procedure for Computing p-y Curves in Weak Rock ................................................ 138
3-8-5 Case Histories for Drilled Shafts in Weak Rock ....................................................... 141
3-9 p-y Curves for Massive Rock ........................................................................................... 146
3-9-1 Introduction ............................................................................................................... 146
3-9-2 Shearing Properties of Massive Rock ....................................................................... 146
3-9-3 Determination of Rock Mass Modulus ..................................................................... 148
3-9-4 Determination of pus Near the Ground Surface ......................................................... 150
3-9-5 Determination of pud at Great Depth ......................................................................... 152
3-9-6 Determination of Initial Tangent Stiffness of p-y Curve Ki ...................................... 153
3-9-7 Procedure for Computing p-y Curves in Massive Rock ............................................ 153
3-10 p-y Curves in Piedmont Residual Soils .......................................................................... 154
3-11 Response of Layered Soils ............................................................................................. 155
3-11-1 Layering Correction Method of Georgiadis ............................................................ 156
3-11-2 Example p-y Curves in Layered Soils ..................................................................... 158
3-11-3 Modified Equations Using Equivalent Depth ......................................................... 163
3-12 Modifications to p-y Curves for Pile Batter and Ground Slope ..................................... 166
iv
3-12-1 Piles in Sloping Ground .......................................................................................... 166
3-12-2 Effect of Batter on p-y Curves in Clay and Sand .................................................... 169
3-12-3 Modeling of Piles in Short Slopes ........................................................................... 170
3-13 Shearing Force Acting at Pile Tip .................................................................................. 171
Chapter 4 Special Analyses ........................................................................................................ 172
4-1 Introduction ...................................................................................................................... 172
4-2 Computation of Top Deflection versus Pile Length ......................................................... 172
4-3 Analysis of Piles Loaded by Soil Movements .................................................................. 175
4-4 Analysis of Pile Buckling ................................................................................................. 176
4-4-1 Procedure for Analysis of Pile Buckling ................................................................... 176
4-4-2 Example of An Incorrect Pile Buckling Analysis ..................................................... 178
4-4-3 Evaluation of Pile Buckling Capacity ....................................................................... 179
4-5 Pushover Analysis of Piles ............................................................................................... 181
4-5-1 Procedure for Pushover Analysis .............................................................................. 181
4-5-2 Example of Pushover Analysis ................................................................................. 182
4-5-3 Evaluation of Pushover Analysis .............................................................................. 183
4-6 Computation of Foundation Stiffness Matrix ................................................................... 183
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity....................... 188
5-1 Introduction ...................................................................................................................... 188
5-1-1 Application ................................................................................................................ 188
5-1-2 Assumptions .............................................................................................................. 188
5-1-3 Stress-Strain Curves for Concrete and Steel ............................................................. 189
5-1-4 Cross Sectional Shape Types .................................................................................... 191
5-2 Beam Theory .................................................................................................................... 191
5-2-1 Flexural Behavior ...................................................................................................... 191
5-2-2 Axial Structural Capacity .......................................................................................... 194
5-3 Validation of Method........................................................................................................ 195
5-3-1 Analysis of Concrete Sections................................................................................... 195
5-3-2 Analysis of Steel Pipe Piles....................................................................................... 206
5-3-3 Analysis of Prestressed-Concrete Piles ..................................................................... 208
5-4 Discussion ......................................................................................................................... 211
5-5 Reference Information ...................................................................................................... 212
5-5-1 Standard Concrete Reinforcing Steel Sizes ............................................................... 212
5-5-2 Prestressing Strand Types and Sizes ......................................................................... 213
5-5-3 Steel H-Piles .............................................................................................................. 214
Chapter 6 Use of Vertical Piles to Stabilize a Slope ................................................................... 216
6-1 Introduction ...................................................................................................................... 216
6-2 Proposed Methods ............................................................................................................ 216
6-3 Review of Some Previous Applications ........................................................................... 218
6-4 Analytical Procedure ........................................................................................................ 218
6-5 Alternative Method of Analysis ....................................................................................... 221
6-6 Case Studies and Example Computation .......................................................................... 222
6-6-1 Case Studies .............................................................................................................. 222
6-6-2 Example Computation ............................................................................................... 222
v
6-6-3 Conclusions ............................................................................................................... 225
References ................................................................................................................................... 226
Name Index ................................................................................................................................. 234
vi
List of Figures
Figure 1-1 Example of Modeling a Bridge Foundation ............................................................... 18
Figure 1-2 Three-dimensional Soil-Pile Interaction .................................................................... 20
Figure 1-3 Coefficients of Pile-head Stiffness Matrix ................................................................. 20
Figure 2-1 Models of Piles Under Lateral Loading, (a) 3-Dimensional Finite Element
Mesh, and (b) Cross-section of 3-D Finite Element Mesh, (c) Brom’s Model,
(d) MFAD Model ..................................................................................................... 27
Figure 2-2 Model of a Pile Under Lateral Loading and p-y Curves ............................................ 29
Figure 2-3 Distribution of Stresses Acting on a Pile, (a) Before Lateral Deflection and (b)
After Lateral Deflection y ......................................................................................... 30
Figure 2-4 Variation of Shear Stresses in Pile and Soil for Displaced Pile ................................. 31
Figure 2-5 Illustration of General Procedure for Selecting a Pile to Sustain a Given Set of
Loads ........................................................................................................................ 33
Figure 2-6 Analysis of Pile Buckling........................................................................................... 34
Figure 2-7 Solving for Critical Pile Length ................................................................................. 35
Figure 2-8 Simplified Method of Analyzing a Pile for an Offshore Platform ............................. 36
Figure 2-9 Analysis of a Breasting Dolphin ................................................................................ 38
Figure 2-10 Loading On a Single Shaft Supporting a Bridge Deck ............................................ 39
Figure 2-11 Foundation Options for an Overhead Sign Structure ............................................... 40
Figure 2-12 Use of Piles to Stabilize a Slope Failure .................................................................. 41
Figure 2-13 Anchor Pile for a Flexible Bulkhead ........................................................................ 42
Figure 2-14 Element of Beam-Column (after Hetenyi, 1946) ..................................................... 43
Figure 2-15 Sign Conventions ..................................................................................................... 45
Figure 2-16 Form of Results Obtained for a Complete Solution ................................................. 46
Figure 2-17 Boundary Conditions at Top of Pile......................................................................... 47
Figure 2-18 Values of Coefficients A1, B1, C1, and D1 ................................................................ 50
Figure 2-19 Representation of deflected pile ............................................................................... 53
Figure 2-20 Case 1 of Boundary Conditions ............................................................................... 54
Figure 2-21 Case 2 of Boundary Conditions ............................................................................... 55
Figure 2-22 Case 3 of Boundary Conditions ............................................................................... 56
Figure 2-23 Case 4 of Boundary Conditions ............................................................................... 57
Figure 2-24 Case 5 of Boundary Conditions ............................................................................... 57
Figure 3-1 Conceptual p-y Curves ............................................................................................... 61
vii
Figure 3-2 p-y Curves from Static Load Test on 24-inch Diameter Pile (Reese, et al. 1975) ..... 64
Figure 3-3 p-y Curves from Cyclic Load Test on 24-inch Diameter Pile (Reese, et al.
1975) ......................................................................................................................... 65
Figure 3-4 Plot of Ratio of Initial Modulus to Undrained Shear Strength for Unconfinedcompression Tests on Clay ....................................................................................... 67
Figure 3-5 Variation of Initial Modulus with Depth .................................................................... 68
Figure 3-6 Assumed Passive Wedge Failure in Clay Soils, (a) Shape of Wedge, (b) Forces
Acting on Wedge ...................................................................................................... 69
Figure 3-7 Measured Profiles of Ground Surface Heave Near Piles Due to Static Loading,
(a) Ground Surface Heave at Maximum Load, (b) Residual Ground Surface
Heave ........................................................................................................................ 70
Figure 3-8 Ultimate Lateral Resistance for Clay Soils ................................................................ 72
Figure 3-9 Assumed Mode of Soil Failure Around Pile in Clay, (a) Section Through Pile,
(b) Mohr-Coulomb Diagram, (c) Forces Acting on Section of Pile ......................... 73
Figure 3-10 Values of Ac and As................................................................................................... 74
Figure 3-11 Development of Scour Around Pile in Clay During Cyclic Loading, (a)
Profile View, (b) Photograph of Turbulence Causing Erosion During Lateral
Load Test .................................................................................................................. 76
Figure 3-12 p-y Curves in Soft Clay,(a) Static Loading, (b) Cyclic Loading .............................. 82
Figure 3-13 Example p-y Curves in Soft Clay Showing Effect of J ............................................ 83
Figure 3-14 Shear Strength Profile Used for Example p-y Curves for Soft Clay ........................ 85
Figure 3-15 Example p-y Curves for Soft Clay with the Presence of Free Water ....................... 85
Figure 3-16 Annular Gapping Developed Around Pile After Cyclic Loading ............................ 87
Figure 3-17 Characteristic Shape of p-y Curves for Static Loading in Stiff Clay with Free
Water ........................................................................................................................ 88
Figure 3-18 Characteristic Shape of p-y Curves for Cyclic Loading of Stiff Clay with Free
Water ........................................................................................................................ 91
Figure 3-19 Example Shear Strength Profile for p-y Curves for Stiff Clay with No Free
Water ........................................................................................................................ 93
Figure 3-20 Example p-y Curves for Stiff Clay in Presence of Free Water for Cyclic
Loading ..................................................................................................................... 93
Figure 3-21 Characteristic Shape of p-y Curve for Static Loading in Stiff Clay without
Free Water ................................................................................................................ 95
Figure 3-22 Characteristic Shape of p-y Curves for Cyclic Loading in Stiff Clay with No
Free Water ................................................................................................................ 96
Figure 3-23 Ratio of Expansion versus Number of Cycles of Loading for Stiff Clay
without Free Water ................................................................................................... 97
viii
Figure 3-24 Example p-y Curves for Stiff Clay with No Free Water, Cyclic Loading .............. 98
Figure 3-25 Geometry Assumed for Passive Wedge Failure for Pile in Sand ........................... 101
Figure 3-26 Assumed Mode of Soil Failure by Lateral Flow Around Pile in Sand, (a)
Section Though Pile, (b) Mohr-Coulomb Diagram ................................................ 103
Figure 3-27 Characteristic Shape of p-y Curves for Static and Cyclic Loading in Sand ........... 105
Figure 3-28 Values of Coefficients Ac and As for Cohesionless Soils ................................... 106
Figure 3-29 Values of Coefficients Bc and Bs for Cohesionless Soils ....................................... 107
Figure 3-31 Example p-y Curves for Sand Below the Water Table, Static Loading ................. 110
Figure 3-32 Coefficients C1, C2, and C3 versus Angle of Internal Friction ............................... 112
Figure 3-33 Value of k for API Sand Procedure ........................................................................ 113
Figure 3-34 Value of k versus Friction Angle for Fine Sand Used in LPile .............................. 114
Figure 3-35 Example p-y Curves for API Sand Criteria ............................................................ 116
Figure 3-36 Example p-y Curve in Liquefied Sand ................................................................... 118
Figure 3-37 Recommended Method for Computing Residual Shear Strength of Liquefied
Soil for Use in Hybrid p-y Model ........................................................................... 122
Figure 3-38 Factor 50 as Function of SPT Blowcount .............................................................. 123
Figure 3-39 Possible Intersection Patterns of Residual and Dilative p-y Curves in Hybrid
p-y Model................................................................................................................ 124
Figure 3-40 Example of Non-intersecting Curves ..................................................................... 124
Figure 3-41 Example of Curves with One Intersection of Dilative and Residual Curves ......... 125
Figure 3-42 Example of Curve with One Intersection of Dilative Curve and Residual
Plateau .................................................................................................................... 125
Figure 3-43 Example of Curve with Two Intersection Points ................................................... 126
Figure 3-37 Idealized Tip Resistance Profile from CPT Testing Used for Analyses. ............... 128
Figure 3-38. Generic p-y curve for Drilled Shafts in Loess Soils .............................................. 129
Figure 3-39 Variation of Modulus Ratio with Normalized Lateral Displacement .................... 132
Figure 3-40 p-y Curves for the 30-inch Diameter Shafts ........................................................... 133
Figure 3-41 p-y Curves and Secant Modulus for the 42-inch Diameter Shafts. ........................ 133
Figure 3-42 Cyclic Degradation of p-y Curves for 30-inch Shafts ............................................ 134
Figure 3-43 Characteristic Shape of p-y Curves for c- Soil ..................................................... 136
Figure 3-44 Representative Values of k for c- Soil.................................................................. 139
Figure 3-46 p-y Curves for Cemented c- Soil .......................................................................... 142
Figure 3-47 Initial Moduli of Rock Measured by Pressuremeter for San Francisco Load
Test ......................................................................................................................... 145
ix
Figure 3-48 Modulus Reduction Ratio versus RQD (Bienawski, 1984) ................................... 146
Figure 3-49 Engineering Properties for Intact Rocks (after Deere, 1968; Peck, 1976; and
Horvath and Kenney, 1979) ................................................................................... 147
Figure 3-50 Characteristic Shape of p-y Curve in Strong Rock ................................................ 148
Figure 3-51 Sketch of p-y Curve for Weak Rock (after Reese, 1997) ....................................... 149
Figure 3-52 Comparison of Experimental and Computed Values of Pile-Head Deflection,
Islamorada Test (after Reese, 1997) ....................................................................... 152
Figure 3-53 Computed Curves of Lateral Deflection and Bending Moment versus Depth,
Islamorada Test, Lateral Load of 334 kN (after Reese, 1997) ............................... 153
Figure 3-54 Comparison of Experimental and Computed Values of Pile-Head Deflection
for Different Values of EI, San Francisco Test ...................................................... 154
Figure 3-55 Values of EI for three methods, San Francisco test ............................................... 155
Figure 3-56 Comparison of Experimental and Computed Values of Maximum Bending
Moments for Different Values of EI, San Francisco Test ...................................... 155
Figure 3-57 p-y Curve in Massive Rock .................................................................................... 156
Figure 3-58 Equation for Estimating Modulus Reduction Ratio from Geological Strength
Index ....................................................................................................................... 159
Figure 3-59 Poisson’s Ratio as Function of Stress Wave Velocity Ratio .................................. 160
Figure 3-60 Model of Passive Wedge for Drilled Shafts in Rock ............................................. 161
Figure 3-61 Degradation Plot for Es .......................................................................................... 165
Figure 3-62 p-y Curve for Piedmont Residual Soil.................................................................... 165
Figure 3-63 Illustration of Equivalent Depths in a Multi-layer Soil Profile .............................. 167
Figure 3-64 Soil Profile for Example of Layered Soils ............................................................. 169
Figure 3-65 Equivalent Depths of Soil Layers Used for Computing p-y Curves ...................... 169
Figure 3-66 Example p-y Curves for Layered Soil .................................................................... 170
Figure 3-67 Pile in Sloping Ground and Battered Pile .............................................................. 177
Figure 3-68 Soil Resistance Ratios for p-y Curves for Battered Piles from Experiment
from Kubo (1964) and Awoshika and Reese (1971) .............................................. 180
Figure 4-1 Pile and Soil Profile for Example Problem .............................................................. 183
Figure 4-2 Variation of Top Deflection versus Depth for Example Problem ............................ 183
Figure 4-3 Pile-head Load versus Deflection for Example........................................................ 184
Figure 4-4 Top Deflection versus Pile Length for Example ...................................................... 184
Figure 4-5 Evaluation of Soil Modulus from p-y Curve Displaced by Soil Movement ............ 186
Figure 4-6 Examples of Pile Buckling Curves for Different Shear Force Values ..................... 188
Figure 4-7 Examples of Correct and Incorrect Pile Buckling Analyses .................................... 189
x
Figure 4-8 Typical Results from Pile Buckling Analysis .......................................................... 190
Figure 4-9 Pile Buckling Results Showing a and b ................................................................... 190
Figure 4-10 LPile Dialog for Controls for Pushover Analysis .................................................. 191
Figure 4-11 Pile-head Shear Force versus Displacement from Pushover Analysis ................... 192
Figure 4-12 Maximum Moment Developed in Pile versus Displacement from Pushover
Analysis .................................................................................................................. 193
Figure 4-13 Example of Stiffness Matrix of Foundation ........................................................... 194
Figure 4-14 Coefficients of Pile-head Stiffness Matrix ............................................................. 195
Figure 5-1 Stress-Strain Relationship for Concrete Used by LPile ........................................... 199
Figure 5-2 Stress-Strain Relationship for Reinforcing Steel Used by LPile.............................. 200
Figure 5-3 Element of Beam Subjected to Pure Bending .......................................................... 202
Figure 5-4 Validation Problem for Mechanistic Analysis of Rectangular Section.................... 206
Figure 5-5 Free Body Diagram Used for Computing Nominal Moment Capacity of
Reinforced Concrete Section .................................................................................. 213
Figure 5-6 Bending Moment versus Curvature.......................................................................... 214
Figure 5-7 Bending Moment versus Bending Stiffness ............................................................. 215
Figure 5-8 Interaction Diagram for Nominal Moment Capacity ............................................... 215
Figure 5-9 Example Pipe Section for Computation of Plastic Moment Capacity ..................... 216
Figure 5-10 Moment versus Curvature of Example Pipe Section ............................................. 216
Figure 5-11 Elasto-plastic Stress Distribution Computed by LPile ........................................... 218
Figure 5-12 Stress-Strain Curves of Prestressing Strands Recommended by PCI Design
Handbook, 5th Edition............................................................................................. 219
Figure 5-13 Sections for Prestressed Concrete Piles Modeled in LPile .................................... 221
Figure 6-1 Scheme for Installing a Row of Piles in a Slope Subject to Sliding ........................ 227
Figure 6-2 Scheme for Stabilizing Piles with Grade Beam and Anchor Pile Group ................. 227
Figure 6-3 Forces from Soil Acting Against a Pile in a Sliding Slope, (a) Pile, Slope, and
Slip Surface Geometry, (b) Distribution of Mobilized Forces, (c) Free-body
Diagram of Pile Below the Slip Surface................................................................. 229
Figure 6-4 Influence of Stabilizing Pile on Factor of Safety Against Sliding ........................... 230
Figure 6-5 Matching of Computed and Assumed Values of hp ................................................. 231
Figure 6-6 Soil Conditions for Analysis of Slope for Low Water ............................................. 233
Figure 6-7 Preliminary Design of Stabilizing Piles ................................................................... 234
Figure 6-8 Load Distribution from Stabilizing Piles for Slope Stability Analysis .................... 235
xi
List of Tables
Table 3-1 Terzaghi’s Recommendations for Soil Modulus for Laterally Loaded Piles in
Stiff Clay (no longer recommended) ........................................................................ 79
Table 3-2 Representative Values of 50 for Soft to Stiff Clays .................................................... 81
Table 3-3 Representative Values of k for Stiff Clays................................................................... 89
Table 3-4 Representative Values of 50 for Stiff to Hard Clays ................................................... 89
Table 3-5 k Values Recommended by Terzaghi for Laterally Loaded Piles in Sand ............... 100
Table 3-6 Representative Values of k for Fine Sand Below the Water Table for Static and
Cyclic Loading ....................................................................................................... 108
Table 3-7 Representative Values of k for Fine Sand Above Water Table for Static and
Cyclic Loading ....................................................................................................... 108
Table 3-8 Results of Grout Plug Tests by Schmertmann (1977) ............................................... 144
Table 3-9 Values of Compressive Strength at San Francisco .................................................... 146
Table 3-10 Values of Material Index mi for Intact Rock, by Rock Group (from Hoek,
2001) ....................................................................................................................... 158
Table 3-11 Typical Properties for Rock Masses (from Hoek, 2001) ......................................... 160
Table 3-12 Tablulated Values of As as Function of z/b ............................................................ 171
Table 3-13 Computed Values of pu and F1 for the Sand in Figure 3-64 as Function of
Depth ...................................................................................................................... 172
Table 3-14 Equivalent Depths of Tops of Soil Layers Computed by LPile .............................. 172
Table 3-15 Equivalent Depths of Example p-y Curves Computed by Hand and by LPile ........ 173
Table 5-1 LPile Output for Rectangular Concrete Section ........................................................ 207
Table 5-2 Comparison of Results from Hand Computation versus Computer Solution............ 214
xii
Chapter 1
Introduction
1-1 Compatible Designs
The program LPile provides the capability to analyze individual piles for a variety of
applications in which lateral loading is applied. The analysis is based on solution of a differential
equation describing the behavior of a beam-column with nonlinear support. The solution
obtained ensures that the computed deformations and stresses in the foundation and supporting
soil are compatible and consistent. Analyses of this type have been in use in the practice of civil
engineering since the 1950’s and the analytical procedures that are used in LPile are widely
accepted.
The one goal of foundation engineering is to predict how a foundation will deform and
deflect in response to loading. In advanced analyses, the analysis of the foundation performance
can be combined with that those for the superstructure to provide a global solution in which both
equilibrium of forces and moments and compatibility of displacements and rotations is achieved.
Analyses of this type are possible because of the power of computer software for analysis
and computer graphics. Calibration and verification of the analyses is possible because of the
availability of sophisticated instrumentation and data acquisition systems for observing the
behavior of structural systems.
Some problems can be solved only by using the concepts of soil-structure interaction.
Presented herein are analyses for isolated piles that achieve the pile response while
simultaneously satisfying the appropriate nonlinear response of the soil. In these analyses, the
pile is treated as a beam-column and the soil is modelled with nonlinear Winkler-spring
mechanisms. These mechanisms can accurately predict the response of the soil and provide a
means of obtaining solutions to a number of practical problems.
1-2 Principles of Design
1-2-1 Introduction
The design of a pile foundation to sustain a combination of lateral and axial loading
requires the designing engineer to consider factors involving both performance of the foundation
to support loading and the costs and methods of construction for different types of foundations.
Presentation of complete designs as examples and a discussion many practical details related to
construction of piles is outside the scope for this manual.
The discussion of the analytical methods presented herein address two aspects of design
that are helpful to the user. These aspects of design are computation of the loading at which a
particular pile will fail as a structural member and identification of the level of loading that will
cause an unacceptable lateral deflection. The analysis made using LPile includes computation of
deflections, bending moments, and shear forces developed along the length of a pile under
1
Chapter 1 – Introduction
loading. Additional considerations that are useful are computation of the minimum required
length of a pile foundation, computation of pile-head stiffness relationships, and the evaluation
of the buckling capacity of a pile that extends above the ground line.
1-2-2 Modelling of Nonlinear Response of Soil
In one sense, the design of a pile under lateral loading is no different that the design of
any foundation. First, one needs to determine the loading of the foundation that will cause failure
and then apply a global factor of safety or alternatively to apply load and resistance factors to set
the allowable loading capacity of the foundation. What is different for analysis of lateral loading
is that the limit states for evaluating a design cannot be found by solving the equations of static
equilibrium alone. Instead, the lateral capacity of the foundation is found by first solving a
differential equation governing the pile’s behavior and then evaluating the results of the solution.
Furthermore, as noted below, use of a closed-form solution of the differential equation, as with
the use a constant modulus of subgrade reaction, is inappropriate in the vast majority of cases.
To illustrate the nonlinear response of soil to lateral loading of a pile, curves of response
of soil obtained from the results of a full-scale lateral load test of a steel-pipe pile are presented
in Chapter 2. This test pile was instrumented for measurement of bending moment and was
driven into overconsolidated clay with free water present above the ground surface. The results
of static load testing definitely show that the soil resistance is nonlinear with pile deflection and
increases with depth. With cyclic loading, frequently encountered in practice, the nonlinearity in
load-deflection response is greatly increased. Thus, if a linear analysis shows a tolerable level of
stress in a pile and of deflection, an increase in loading could cause a failure by collapse or by
excessive deflection. Therefore, a basic principle of compatible design is that nonlinear response
of the soil to lateral loading must be considered.
1-2-3 Limit States
In many instances, failure of a pile is initiated by a bending moment that would cause the
development of a plastic hinge. However, in other instances the failure could be due to excessive
deflection, or, in a small fraction of cases, by shear failure of the pile. Therefore, pile design is
based on a decision of what constitutes a limit state for structural failure or excessive deflection.
Then, computations are made to determine if the loading considered exceeds the limit states.
A global factor of safety is normally employed to find the allowable loading, the service
load level, or the working load level.
Alternatively, an approach using load and resistance factors may be employed. However,
analyses employed in applying load and resistance factors is implemented herein by using upperbound and lower-bound values of the important parameters.
1-2-4 Step-by-Step Procedure
1. Collect all relevant data, including the soil profile, soil properties, magnitude and type of
loading, and performance requirements for the structure.
2. Select a pile type and size for analysis.
3. Compute curves of nominal bending moment capacity as a function of axial thrust load and
curvature; compute the corresponding values of nonlinear bending stiffness.
2
Chapter1 – Introduction
4. Select p-y curve types for the analysis, along with average, upper bound, and lower bound
values of input variables.
5. Make a series of solutions, starting with a small load and increasing the load in increments,
with consideration of the manner the pile is fastened to the superstructure.
6. Obtain curves showing maximum moment in the pile and lateral pile-head deflection versus
lateral shear loading and curves of lateral deflection, bending moment, and shear force versus
depth along the pile.
7. Change the pile dimensions or pile type, if necessary and repeat the analyses until a range of
suitable pile types and sizes have been identified.
8. Identify the pile type and size for which the global factor of safety is adequate and the most
efficient cost of the pile and construction is estimate.
9. Compute behavior of pile under working loads.
Few of the examples presented in this manual need to follow all steps indicated above.
However, in most cases, the examples do show the curves that are indicated in Step 6.
1-2-5 Suggestions for the Designing Engineer
As will be explained in some detail, there are five sets of boundary conditions that can be
employed; examples will be shown for the use of these different boundary conditions. However,
the manner in which the top of the pile is fastened to the pile cap or to the superstructure has a
significant influence on deflections and bending moments that are computed. The engineer may
be required to perform an analysis of the superstructure, or request that one be made, in order to
ensure that the boundary conditions at the top of the pile are satisfied as well as possible.
With regard to boundary conditions at the pile head, it is important to note the versatility
of LPile. For example, piles that are driven with an accidental batter or an accidental eccentricity
can be easily analyzed. It is merely necessary to define the appropriate conditions for the
analysis.
As noted earlier, selection of upper and lower bound values of soil properties is a
practical procedure. Parametric solutions are easily done and relatively inexpensive and such
solutions are recommended. With the range of maximum values of bending moment that result
from the parametric studies, for example, the insight and judgment of the engineer can be
improved and a design can probably be selected that is both safe and economical. Alternatively,
one may perform a first-order, second moment reliability analysis to evaluate variance in
performance for selected random variables. For further guidance on this topic, the reader is
referred to the textbook by Baecher and Christian (2003).
If the axial load is small or negligible, it is recommended to make solutions with piles of
various lengths. In the case of short piles, the mobilization shear force at the bottom of the pile
can be defined along with the soil properties. In most cases, the installation of a few extra feet of
pile length will add little cost to the project and, if there is doubt, a pile with a few feet of
additional length could possibly prevent a failure due to excessive deflection. If the base of the
pile is founded in rock, available evidence shows that often only a short socket will be necessary
to anchor the bottom of the pile. In all cases, the designer must assure that the pile has adequate
bending stiffness over its full length.
3
Chapter 1 – Introduction
A useful activity for a designer is to use LPile to analyze piles for which experimental
results are available. It is, of course, necessary to know the appropriate details from the load
tests; pile geometry and bending stiffness, stratigraphy and soil properties, magnitude and point
of application of loading, and the type of loading (either static or cyclic). Many such experiments
have been run in the past. Comparison of the results from analysis and from experiment can yield
valuable information and insight to the designer. Some comparisons are provided in this
document, but those made by the user could be more site-specific and more valuable.
In some instances, the parametric studies may reveal that a field test is indicated. Such a
case occurs when a large project is planned and when the expected savings from an improved
design exceeds the cost of the testing. Savings in construction costs may be derived either by
proving a more economical foundation design is feasible, by permitting use of a lower factor of
safety or, in the case of a load and resistance factor design, use of an increased strength reduction
factor for the soil resistance.
There are two types of field tests. In one instance, the pile may be fully instrumented so
that experimental p-y curves are obtained. The second type of test requires no internal instrumentation in the pile but only the pile-head settlement, deflection, and rotation will be found as a
function of applied load. LPile can be used to analyze the experiment and the soil properties can
be adjusted until agreement is reached between the results from the computer and those from the
experiment. The adjusted soil properties can be used in the design of the production piles.
In performing the experiment, no attempt should be made to maintain the conditions at
the pile head identical to those in the design. Such a procedure could be virtually impossible.
Rather, the pile and the experiment should be designed so that the maximum amount of
deflection is achieved. Thus, the greatest amount of information can be obtained on soil
response.
The nature of the loading during testing; whether static, cyclic, or otherwise; should be
consistent for both the experimental pile and the production piles.
The two types of problems concerning the performance of pile groups of piles are
computation of the distribution of loading from the pile cap to a widely spaced group of piles and
the computation of the behavior of spaced-closely piles.
The first of these problems involves the solutions of the equations of structural mechanics
that govern the distribution of moments and forces to the piles in the pile group (Hrennikoff,
1950; Awoshika and Reese, 1971; Akinmusuru, 1980). For all but the most simple group
geometries, solution of this problem requires the use of a computer program developed for its
solution.
The second of the two problems is more difficult because less data from full-scale
experiments is available (and is often difficult to obtain). Some full-scale experiments have been
performed in recent years and have been reported (Brown, et al., 1987; Brown et al., 1988).
These and additional references are of assistance to the designer (Bogard and Matlock, 1983;
Focht and Koch, 1973; O’Neill, et al., 1977).
The technical literature includes significant findings from time to time on piles under
lateral loading. Ensoft will take advantage of the new information as it becomes available and
verified by loading testing and will issue new versions of LPile when appropriate. However, the
material that follows in the remaining sections of this document shows that there is an
4
Chapter1 – Introduction
opportunity for rewarding research on the topic of this document, and the user is urged to stay
current with the literature as much as possible.
1-3 Modeling a Pile Foundation
1-3-1 Introduction
As a problem in foundation engineering, the analysis of a pile under combined axial and
lateral loading is complicated by the fact that the mobilized soil reaction varies in proportion to
the pile movement, and the pile movement, on the other hand, is dependent on the soil response.
This is the basic problem of soil-structure interaction. The question about how to simulate the
behavior of the pile in the analysis arises when the foundation engineer attempts to use boundary
conditions for the connection between the structure and the foundation. Ideally, a program can be
developed by combining the structure, piles, and soils into a single model. However, special
purpose programs that permit development of a global model are currently unavailable. Instead,
the approach described below is commonly used for solving for the nonlinear response of the
pile foundation so that equilibrium and compatibility can be achieved with the superstructure.
The use of models for the analysis of the behavior of a bridge is shown in Figure 1-1(a).
A simple, two-span bridge is shown with spans in the order of 30 m and with piles supporting the
abutments and the central span. The girders and columns are modeled by lumped masses and the
foundations are modeled by nonlinear springs, as shown in Figure 1-1(b). If the loading is threedimensional, the pile head at the central span will undergo three translations and three rotations.
A simple matrix-formulation for the pile foundation is shown in Figure 1-1(c), assuming twodimensional loading, along with a set of mechanisms for the modeling of the foundation. Three
springs are shown as symbols of the response of the pile head to loading; one for axial load, one
for lateral load, and one for moment.
The assumption is made in analysis that the nonlinear curve for axial loading is not
greatly influenced by lateral loading (shear) and moment. This assumption is not strictly true
because lateral loading can cause gapping in overconsolidated clay at the top of the pile with a
consequent loss of load transfer in skin friction along the upper portion of the pile. However, in
such a case, the soil near the ground surface could be ignored above the first point of zero lateral
deflection. The practical result of such a practice in most cases is that the curve of axial load
versus settlement and the stiffness coefficient K11 are negligibly affected.
The curves representing the response to shear and moment at the top of the pile are
certainly multidimensional and unavoidably so. Figure 1-1(c) shows a curve and identifies one of
the stiffness terms K32. A single-valued curve is shown only because a given ratio of moment M1
and shear V1 was selected in computing the curve. Therefore, because such a ratio would be
unknown in the general case, iteration is required between the solutions for the superstructure
and the foundation.
The conventional procedure is to select values for shear and moment at the pile head and
to compute the initial stiffness terms so that the solution of the superstructure can proceed for the
most critical cases of loading. With revised values of shear and moment at the pile head, the
model for the pile can be resolved and revised terms for the stiffnesses can be used in a new
solution of the model for the superstructure. The procedure could be performed automatically if a
computer program capable of analyzing the global model were available but the use of
5
Chapter 1 – Introduction
independent models allows the designer to exercise engineering judgment in achieving
compatibility and equilibrium for the entire system for a given case of loading.
a. Elevation View
Lumped masses
Foundation springs
b. Analytical Model
K33
K22
Moment
M
K11
K33
Rotation
 K11
 0

 0

 K11
 x 
P 
0 0 0 0 Q
   x
 0

 y  
 V 
 K K 22K K23H







22
23
 0
K 32 K33     My
K 32
  
K 33  
M    
c. Stiffness Matrix
Figure 1-1 Example of Modeling a Bridge Foundation
6
Chapter1 – Introduction
The stiffness K11 is the stiffness of the axial load-settlement curve for the axial load P.
This stiffness is obtained either from load test results or from a numerical analysis using an axial
capacity analysis program like Shaft or APile from Ensoft, Inc.
1-3-2 Example Model of Individual Pile with Axial and Lateral Loading
An interesting presentation of the forces that resist the displacements of an individual pile
is shown in Figure 1-2 (Bryant, 1977). Figure 1-2(a) shows a single pile beneath a cap along with
the three-dimensional displacements and rotations. The assumption is made that the top of the
pile is fixed or partially fixed into the cap and that biaxial bending and torsion reactions will
develop because of the three-dimensional translation and rotation of the cap. The reactions of the
soil along the pile are shown in Figure 1-2(b), and the load-transfer curves are shown in Figure
1-2(c). The argument given earlier about the curve for axial displacement being single-value
pertains as well to the curve for axial torque. However, the curve for lateral deflection is
certainly a function of the shear forces and moments that cause such deflection. When computing
lateral deflection, a complication may arise because the loading and deflection may not be in a
two-dimensional plane. The recommendations that have been made for correlating the lateral
resistance with pile geometry and soil properties all depend on the results of loading in a twodimensional plane.
q
y
Axial
Py
x
u
Px
My
Mx
Axial Pile
Displacement, u
z
Mz
Pz
p
Axial Soil
Reaction, q
Lateral
y
Torsional Pile
Displacement, 
Lateral Soil
Reaction, p
t
Lateral Pile
Displacement, y
Torsional Soil
Reaction, t
(a) Three-dimensional
pile displacements
(b) Pile reactions
7
Torsional

(c) Nonlinear load-transfer
curves
Chapter 1 – Introduction
Figure 1-2 Three-dimensional Soil-Pile Interaction
1-3-3 Computation of Foundation Stiffness
Stiffness matrices are often used to model foundations in structural analyses and LPile
provides an option for evaluating the lateral stiffness of a deep foundation. This feature in LPile
allows the user to solve for coefficients, as illustrated by the sketches shown in Figure 1-3, of
pile-head movements and rotations as functions of incremental loadings. The program divides
the loads specified at the pile head into increments and then computes the pile head response for
each individual loading. The deflection of the pile head is computed for each lateral-load
increment with the rotation at the pile head being restrained to zero. Next, the rotation of the pile
head is computed for each bending-moment increment with the lateral deflection at the pile head
being restrained to zero. The user can thus define the stiffness matrix directly based on the
relationship between computed deformation and applied load. For instance, the stiffness
coefficient K33, shown in Figure 1-1(c), can be obtained by dividing the applied moment M by
the computed rotation θ at the pile top.
P
P
M
−M
−V
V
0
0
0
0
Stiffnesses K22 and K23 are computed using the
shear-rotation pile-head condition, for which the
user enters the lateral load V at the pile head.
LPile computes pile-head deflection  and
reaction moment −M at the pile head using zero
slope at the pile head (pile head rotation  = 0).
Stiffnesses K32 and K33 are computed using the
displacement-moment pile-head condition, for
which the user enters the moment M at the pile
head. LPile computes the lateral reaction force,
−H, and pile-head rotation  using zero deflection
at the pile head ( = 0).
K22 =  V/  and K32 = –M/ .
K23 = –V/  and K33 =  M/ .
Figure 1-3 Coefficients of Pile-head Stiffness Matrix
8
Chapter1 – Introduction
Most analytical methods in structural mechanics can employ either the stiffness matrix or
the flexibility matrix to define the support condition at the pile head. If the user prefers to use the
stiffness matrix in the structural model, Figure 1-3 illustrates basic procedures used to compute a
stiffness matrix. The initial coefficients for the stiffness matrix may be defined based on the
magnitude of the service load. The user may need to make several iterations before achieving
acceptable agreement.
1-3-4 Concluding Comments
The correct modeling of the problem of the single pile to respond to axial and lateral
loading is challenging and complex, and the modeling of a group of piles is even more complex.
However, in spite of the fact that research is continuing, the following chapters will demonstrate
that usable solutions are at hand.
New developments in computer technology allow a complete solution to be readily
developed, including automatic generation of the nonlinear responses of the soil around a pile
and iteration to achieve force equilibrium and compatibility.
1-4 Organization of Technical Manual
Chapters 2, 3, and 4 provide the user with background information on soil-pile interaction
for lateral loading and present the equations that are solved when obtaining a solution for the
beam-column problem when including the effects of the nonlinear response of the soil. In
addition, information on the verification of the validity of a particular set of output is given. The
user is urged to read carefully these latter two sections. Output from the computer should be
viewed with caution unless verified, and the user’s selection of the appropriate soil response (p-y
curves) is the most critical aspect of most computations.
Not all engineers will have a computer program available that can be used to predict the
level of bending moment in a reinforced-concrete section at which a plastic hinge will develop,
while taking into account the influence of axial thrust loading. Chapter 5 of this manual describes
features of LPile that are provided for this purpose. LPile can compute the flexural rigidity of the
pile sections as a function of the developed bending moments.
Finally, Chapter 6 includes the development of a solution that is designed to give the user
some guidance in the use of piles to stabilize a slope. While no special coding is necessary for
the purpose indicated, the number of steps in the solution is such that a separate section is
desirable rather than including this example with those in the User’s Manual for LPile.
9
Chapter 1 – Introduction
(This page was deliberately left blank)
10
Chapter 2
Solution for Pile Response to Lateral Loading
2-1 Introduction
Many pile-supported structures will be subjected to horizontal loads during their
functional lifetime. If the loads are relatively small, a design can be made by building code
provisions that list allowable loads for vertical piles as a function of pile diameter and properties
of the soil. However, if the load per pile is large, the piles are frequently installed at a batter. The
analyst may assume that the horizontal load on the structure is resisted by components of the
axial loads on the battered piles. The implicit assumption in the procedure is that the piles do not
deflect laterally which, of course, is not true. Rational methods for the analysis of single piles
under lateral load, where the piles are vertical or battered, will be discussed herein, and methods
are given for investigating a wide variety of parameters. The problem of the analysis of a group
of piles is discussed in another publication.
As a foundation problem, the analysis of a pile under lateral loading is complicated
because the soil reaction (resistance) at any point along a pile is a function of pile deflection. The
pile deflection, on the other hand, is dependent on the soil resistance; therefore, solving for the
response of a pile under lateral loading is one of a class of soil-structure-interaction problems.
The conditions of compatibility and equilibrium must be satisfied between the pile and soil and
between the pile and the superstructure. Thus, the deformation and movement of the
superstructure, ranging from a concrete mat to an offshore platform, and the manner in which the
pile is attached to the superstructure, must be known or computed in order to obtain a correct
solution to most problems.
2-1-1 Influence of Pile Installation and Loading on Soil Characteristics
2-1-1-1 General Review
The most critical factor in solving for the response of a pile under lateral loading is the
prediction of the soil resistance at any point along a pile as a function of the pile deflection. Any
serious attempt to develop predictions of soil resistance must address the stress-deformation
characteristics of the soil. The properties to be considered, however, are those that exist after the
pile has been installed. Furthermore, the influence of lateral loading on soil behavior must be
taken into account.
The deformations of the soil from the driving of a pile into clay cause important and
significant changes in soil characteristics. Different but important effects are caused by driving
of piles into granular soils. Changes in soil properties are also associated with the installation of
bored piles. While definitive research is yet to be done, evidence clearly shows that the soil
immediately adjacent to a pile wall is most affected. Investigators (Malek, et al., 1989) have
suggested that the direct-simple-shear test can be used to predict the behavior of an axially
loaded pile, which suggests that the soil just next to the pile wall will control axial behavior.
However, the lateral deflection of a pile will cause strains and stresses to develop from the pile
11
Chapter 2 – Solution for Pile Response to Lateral Loading
wall to several diameters away. Therefore, the changes in soil characteristics due to pile
installation are less important for laterally loaded piles than for axially loaded piles.
The influence of the loading of the pile on soil response is another matter. Four classes of
lateral loading can be identified: short-term, repeated, sustained, and dynamic. The first three
classes are discussed herein, but the response of piles to dynamic loading is beyond the scope of
this document. The use of a pseudo-horizontal load as an approximation in making earthquakeresistant designs should be noted, however.
The influence of sustained or cyclic loading on the response of the soil will be discussed
in some detail in Chapter 3; however, some discussion is appropriate here to provide a basis for
evaluating the models that are presented in this chapter. If a pile is in granular soil or
overconsolidated clay, sustained loading, as from earth pressure, will likely cause only a
negligible amount of long-term lateral deflection. A pile in normally consolidated clay, on the
other hand, will experience long-term deflection, but, at present, the magnitude of such
deflection can only be approximated. A rigorous solution requires solution of the threedimensional consolidation equation stepwise with time. At some time, the pile-head will
experience an additional deflection that will cause a change in the horizontal stresses in the
continuum.
Methods have been developed, as reviewed later, for getting answers to the problem of
short-term loading by use of correlations between soil response and the in situ undrained strength
of clay and the in-situ angle of internal friction for granular soil. Such “backbone” solutions are
important because they can be used for sustained loading in some cases and because an initial
condition is provided for taking the influence of repeated loading into account. Experience has
shown that the loss of lateral resistance due to repeated loading is significant, especially if the
piles are installed in clay below free water. The clay can be pushed away from the pile wall and
the soil response can be significantly decreased. Predictions for the effect of cyclic loading are
given in Chapter 3.
Four general types of loading are recognized above and each of these types is further
discussed in the following sections. The importance of consideration and evaluation of loading
when analyzing a pile subjected to lateral loading cannot be overemphasized.
Many of the load tests described later in this chapter were performed by applying a lateral
load in increments, holding that load for a few minutes, and reading all the instruments that gave
the response of the pile. The data that were taken allowed p-y curves to be computed; analytical
expressions are developed from the experimental results and these expressions yield p-y curves
that are termed “static” curves. Repeated loadings were applied as well, as will be discussed in a
following section.
2-1-1-2 Static Loading
The static p-y curves can be thought of as backbone curves that can be correlated to some
extent with soil properties. Thus, the curves are useful for providing some theoretical basis to the
p-y method.
From the standpoint of design, the static p-y curves have application in the following
cases: where loadings are short-term and not repeated (probably not encountered); and for
sustained loadings, as in earth-pressure loadings, where the soil around the pile is not susceptible
to consolidation and creep (overconsolidated clays, clean sands, and rock).
12
Chapter 2 – Solution for Pile Response to Lateral Loading
As will be noted later in this chapter, the use of the p-y curves for repeated loading, a type
of loading that is frequently encountered in practice, will often yield significant increases in pile
deflection and bending moment. The engineer may wish to make computations with both the
static curves and with the repeated (cyclic) curves so that the influence of the loading on pile
response can be seen clearly.
2-1-1-3 Repeated Cyclic Loading
The full-scale field tests that were performed included repeated or cyclic loading as well as the
static loading described above. An increment of load was applied, the instruments were read, and
the load was repeated a number of times. In some instances, the load was forward and backward,
and in other cases only forward. The instruments were read after a given number of cycles and
the cycling was continued until there was no obvious increase in ground line deflection or in
bending moments. Another increment was applied and the procedure was repeated. The final
load that was applied brought the maximum bending moment close to the moment that would
cause the steel to yield plastically.
Four specific sets of recommendations for p-y curves for cyclic loading are described in
Chapter 3. For three of the sets, the recommendations that are given are for the “lower-bound”
case. That is, the data that were used to develop the p-y curves were from cases where the
ground-line deflection had substantially ceased with repetitions in loading. In the other case, for
stiff clay where there was no free water at the ground surface, the recommendations for p-y
curves are based on the number of cycles of load application, as well as other factors.
The presence of free water at or near the ground surface for clay soils can be significant
in regard to the loss of soil resistance due to cyclic loading (Long, 1984). After a deflection is
exceeded that is based on the “elastic” response of the soil, a space opens between the pile and
the soil when the load is released. Free water moves into this space and on the next load
application, the water is ejected and soil may be eroded. This erosion causes a loss of soil
resistance in addition to the losses due to remolding of the soil resulting from the cyclic strains.
At this point, the use of judgment in the design of the piles under lateral load should be
emphasized. If, for example, the clay is below a layer of sand, or if a provision could be made to
supply sand around the pile, the sand will settle into the opening around the pile and partially
restore the soil resistance that was lost due to the cyclic loading.
Pile-supported structures are subjected to cyclic loading in many cases. Some common
cases are wind loading on overhead signs and high-rise buildings, traffic loads on bridges, wave
loadings on offshore structures, impact loads against docks and dolphin structures, and ice loads
against locks and dams. The nature of the loading must be considered carefully. Factors to be
considered are frequency, magnitude, duration, and direction. The engineer will be required to
use a considerable amount of judgment in the selection of the soil parameters and response
curves.
2-1-1-4 Sustained Loading
If the soil resisting the lateral deflection of a pile is overconsolidated clay, the influence
of sustained loading would probably be small. The maximum lateral stress from the pile against
the clay would probably be less than the previous lateral stress; thus, the additional deflection
due to consolidation and creep in the clay should be small or negligible.
13
Chapter 2 – Solution for Pile Response to Lateral Loading
If the soil that is effective in resisting lateral deflection of a pile is a granular material that
is freely-draining, the creep would be expected to be small in most cases. However, if the pile is
subjected to vibrations, there could be densification of the sand and a considerable amount of
additional deflection. Thus, the judgment of the engineer in making the design should be brought
into play.
If the soil resisting lateral deflection of a pile is soft, saturated clay, the stress applied by
the pile to the soil could cause a considerable amount of additional deflection due to
consolidation (if positive pore water pressures were generated) and creep. An initial solution
could be made, the properties of the clay could be employed, and an estimate could be made of
the additional deflection. The p-y curves could be modified to reflect the additional deflection
and a second solution obtained with the computer. In this manner, convergence could be
achieved. The writers know of no rational way to solve the three-dimensional, time-dependent
problem of the additional deflection that would occur so, again, the judgment and integrity of the
engineer will play an important role in obtaining an acceptable solution.
2-1-1-5 Dynamic Loading
Two types of problems involving dynamic loading are frequently encountered in design:
machine foundations and earthquakes. The deflection from the vibratory loading from machine
foundations is usually quite small and the problem would be solved using the dynamic properties
of the soil. Equations yielding the response of the structure under dynamic loading would be
employed and the p-y method described herein would not be employed.
With regard to earthquakes, a rational solution should proceed from the definition of the
free-field motion of the near-surface soil due to the earthquake. Thus, the p-y method described
herein could not be used directly. In some cases, an approximate solution to the earthquake
problem has been made by applying a horizontal load to the superstructure that is assumed to
reflect the effect of the earthquake. In such a case, the p-y method can be used but such solutions
would plainly be approximate.
2-1-2 Models Used in Analyses of Laterally Loaded Single Piles
A number of models have been developed for the pile and soil system. The following are
brief descriptions for a few of them.
2-1-2-1 Elastic Pile and Soil
The model shown in Figure 2-1(a) depicts a pile in an elastic soil. A model of this sort
has been widely used in analysis. Terzaghi (1955) gave values of subgrade modulus that can be
used to solve for deflection and bending moment, but he went on to qualify his
recommendations. The standard equation for a beam-column was employed in a manner that had
been suggested by earlier writers such as Hetenyi (1946). Terzaghi stated that the tabulated
values of subgrade modulus should not be used for cases where the computed soil resistance was
more than one-half of the bearing capacity of the soil. However, he provided no
recommendations for the computation of bearing capacity under lateral load, nor did he provide
any comparisons between the results of computations and experiments.
The values of subgrade moduli published by Terzaghi proved to be useful and provide
evidence that Terzaghi had excellent insight into the problem. However, in a private
conversation with Professor Lymon Reese, Terzaghi said that he had not been enthusiastic about
14
Chapter 2 – Solution for Pile Response to Lateral Loading
writing the paper and only did so in response to numerous requests. The method illustrated by
Figure 2-1(a) serves well in obtaining the response of a pile under small loads, in illustrating the
various interrelationships in the response, and in giving an overall insight into the nature of the
problem. The method cannot be employed without modification in solving for the loading at
which a plastic hinge will develop in the pile.
P
M
V
(a)
(b)
P
M
M
V
V
kh Lateral
Translational
Spring
k Vertical Side Shear
Moment Spring
Center of Rotation
kb Base Moment Spring
kb Base Shear Translational Spring
(c)
(d)
Figure 2-1 Models of Piles Under Lateral Loading, (a) 3-Dimensional Finite Element Mesh, and
(b) Cross-section of 3-D Finite Element Mesh, (c) Brom’s Model, (d) MFAD Model
15
Chapter 2 – Solution for Pile Response to Lateral Loading
2-1-2-2 Analysis Using the Finite Element Method
The case shown in Figure 2-1(b) is the same as the previous case except that the soil has
been modeled by finite elements. No attempt is made in the sketch to indicate an appropriate size
of the element mesh, boundary constraints, special interface elements, most favorable shape of
elements, or other details. The finite elements may be axially symmetric with non-symmetric
loading or full three-dimensional models. The elements of various types may be used.
In view of the computational power that is now available, the model shown in Figure 21(b) appears to be practical to solve the pile problem. The elements can be three-dimensional and
material models may be nonlinear. However, the selection of an appropriate material model for
the soil involves not only the parameters that define the model, but methods for dealing with
other factors such as volume change and unloading. These factors also include development of
tensile stresses in the soil, modeling of layered soils, development of separation and closure of
gapping between pile and soil during repeated loading, and the changes in soil characteristics
that are associated with the various types of loading and construction.
Yegian and Wright (1973) and Thompson (1977) used a plane-stress finite element model
and obtained soil-response curves that agree well with results at or near the ground surface from
full-scale experiments. The writers are aware of research that is underway with threedimensional, nonlinear, finite and boundary elements, and are of the opinion that in time such a
model will lead to results that can be used in practice.
2-1-2-3 Rigid Pile and Plastic Soil
Broms (1964a, 1964b, 1965) employed the model shown in Figure 2-1(c) to derive
equations for the loading that causes a failure, either because of excessive stresses in the soil or
because of a plastic hinge, or hinges, in the pile. The rigid pile is assumed and a solution is found
using the equations of statics for the distribution of ultimate resistance of the soil that puts the
pile in equilibrium. The soil resistance shown hatched in the Figure 2-1(c) is for cohesive soil,
and a solution was developed for cohesionless soil as well. After the ultimate loading is
computed for a pile of particular dimensions, Broms suggests that the deflection at the working
load may be computed by the use of the model shown in Figure 2-1(c).
Broms’ method makes use of several simplifying assumptions but is useful for the initial
selection of a pile for a given soil and for a given set of loads.
2-1-2-4 Rigid Pile and Four-Spring Model for Soil
The model shown in Figure 2-1 (d) was developed for the design of short, stiff piles that
support transmission towers (DiGioia, et al., 1989). The loading applied to the pile head includes
shear force, overturning moment and axial load. The four reaction spring types are:




a rotational spring at the pile tip that responds to the rotation of the tip,
a linear spring at the pile tip that responds to the axial movement of the tip,
a set of linear springs parallel to the pile wall that respond to vertical movement of the
pile, and
a set of linear springs normal to the sides of the pile that respond to lateral deflection.
The model was developed using analytical techniques and tested against a series of
experiments performed on short piles. However, the experimental procedures did not allow the
independent determination of the curves that give the forces as a function of the four different
16
Chapter 2 – Solution for Pile Response to Lateral Loading
types of movement. Therefore, the relative importance of the four types of soil resistance has not
been found by experiment, and the use of the model in practice has been limited to the design of
short, stiff foundations with length to diameter ratios typically less than five.
2-1-2-5 Nonlinear Pile and p-y Model for Soil
The model shown in Figure 2-2 represents the one utilized by LPile. The loading on the
pile is two-dimensional consisting of shear, overturning moment, and axial thrust. No torsion or
out-of-plane bending is included in the model. The horizontal lines across the pile are meant to
show that it is made up of different sections; for example, a steel pipe could be used with
changes in wall thickness or step-tapered as shown here. The difference-equation method is
employed for the solution of the beam-column equation to allow the different values of bending
stiffness to be addressed. In addition, it is possible to vary the bending stiffness with respect to
the bending curvature that is computed during the iterative solution.
P
M
y
V
p
y
p
y
p
y
p
y
p
y
x
Figure 2-2 Model of a Pile Under Lateral Loading and p-y Curves
An axial thrust load is included and is considered in the solution with respect to its effect
on bending, but not in respect to the development of axial settlement. However, as shown later in
this manual, the computational procedure allows for the determination of the magnitude of the
axial thrust load at which a pile will buckle.
The soil around the pile is replaced by a set of nonlinear springs that indicate that the soil
resistance p is a nonlinear function of pile deflection y. The nonlinear springs and the
corresponding curves that model their behavior are widely spaced in the figure, but are actually
spaced at every nodal point on the pile. As may be seen, the p-y curves are nonlinear with respect
to depth x along the pile and lateral deflection y. The top p-y curve is drawn to indicate that the
pile may deflect a finite distance with no soil resistance. The second curve from the top is drawn
to show that the soil resistance is deflection softening. There is no reasonable limit to the
variations in the resistance of the soil to the lateral deflection of a pile.
17
Chapter 2 – Solution for Pile Response to Lateral Loading
As will be shown later, the p-y method is versatile and provides a practical means for
design. The method was first suggested by McClelland and Focht (1956). Two technological
developments during the 1950’s made implementation of the method possible: the development
of digital computer programs for solving a nonlinear, fourth-order differential equation; and the
development of electrical resistance strain gauges for use in obtaining soil-response (p-y) curves
from full-scale lateral load tests of piles.
The p-y method was developed originally from proprietary research sponsored by the
petroleum industry in the 1950’s and 1960’s. At the time, large piles were being designed for to
support offshore oil production platforms that were to be subjected to exceptionally large
horizontal forces from storm waves and wind. Rules and recommendations for the use of the p-y
method for design of such piles are presented by the American Petroleum Institute (2010) and
Det Norske Veritas (1977).
The use of the method has been extended to the design of onshore foundations. For
example, the Federal Highway Administration (USA) has sponsored a reference publication
dealing with the design of piles for transportation facilities (Reese, 1984). The method is being
cited broadly by Jamiolkowski (1977), Baguelin, et al. (1978), George and Wood (1976), and
Poulos and Davis (1980). The method has been used with apparent success for the design of
piles; however, research is continuing up to the present.
2-1-2-6 Definition of p and y
The definitions of the quantities p and y as used in this document are necessary because
other definitions have been used. The sketch in Figure 2-3(a) shows a uniform distribution of
radial stresses, normal to the wall of a non-displaced cylindrical pile. This distribution of stresses
is correct for a pile that has been installed without bending. If the pile is displaced a distance y
(the amount of the displacement is exaggerated in the sketch for clarity), the distribution of
stresses becomes non-uniform and will be similar to that shown in Figure 2-3 (b). The stresses
will have decreased on the backside of the pile and increased on the front side. Some of the unit
stresses have both normal and shearing components.
p
y
(a)
(b)
Figure 2-3 Distribution of Stresses Acting on a Pile, (a) Before Lateral Deflection and (b) After
Lateral Deflection y
18
Chapter 2 – Solution for Pile Response to Lateral Loading
Integration of the unit stresses around the perimeter of the pile results in the lateral load
intensity p, which acts opposite to the direction of pile displacement y. The dimensions of p are
force per unit length of the pile. These definitions of p and y are convenient in the solution of the
differential equation and are consistent with those used in the solution of the elastic beam
equation.
Direction of Pile Displacement
The distribution of shear stresses in the soil around the pile is known to be more complex
than the simplified version shown in Figure 2-3. The results of a nonlinear finite element stress
analysis to determine the distribution of shear stresses around a laterally load pile is shown in
Figure 2-4.
Figure 2-4 Variation of Shear Stresses in Pile and Soil for Displaced Pile
The reader should note the fineness of the finite element mesh utilized in the analysis
presented in Figure 2-4. Experience has found that use of fine meshes is necessary to obtain
stress distributions in the pile that are accurate enough to permit evaluation of the bending
moment developed in the pile.
2-1-2-7 Procedure for Developing Experimental p-y Curves
Most models for p-y curves have been derived from analyses of full-size load tests. When
performing a load test on a pile subjected to lateral loading, strain gages may be installed along
the length of the pile. This permits direct measurement of strain and evaluation of the curvature
developed at the locations of the strain gages. Values of bending moment at the locations of the
strain gages can be computed from the values of curvature. Other direct measurements are the
applied lateral load and displacement of the pile head in terms of lateral displacement and pilehead rotation. All other structural responses in the pile and the inferred lateral load transfer from
the pile to the soil must be inferred from the available measurement of pile response under load.
19
Chapter 2 – Solution for Pile Response to Lateral Loading
The quality of the interpreted results from the load test depends on the quantity, accuracy, and
consistency of the direct measurements on pile response.
In contrast to the direct measurements on the pile behavior, it is neither simple nor
practical to make direct measurements in the soil and rock surrounding the pile. Usually, the soil
properties at the site are measured prior to the construction of the test pile, but no measurements
of soil behavior are measured during the performance of the load test.
The method used to develop the experimentally measured p-y curve is the following. At
each level of pile-head loading, first fit analytical curves to the measured values of pile curvature
and bending moment developed along the length of the pile. Compute values of lateral load
intensity, p, along the length of the pile by computing second derivative of bending moment
versus depth. Next, compute the lateral displacement profile along the length of the pile by
double integrating the curve of curvature along the length of the pile. Lastly, tabulate the
corresponding values of p and y at the depths of the measurements. After this has been done for
all levels of loading, it is possible to plot the p-y curves at each depth of measurement.
2-1-2-8 Comments on the p-y method
The most common criticism of the p-y method is that the soil is not treated as a
continuum, but instead as a series of discrete springs (i.e. the Winkler model). Several comments
can be given in response to this valid criticism.
The recommendations for the computation of p-y curves for use in the analysis of piles,
given in Chapter 3, are based for the most part on the results of full-scale experiments, where the
“continuum effect” was explicitly satisfied. Further, Matlock (1970) performed some tests of a
pile in soft clay where the pattern of pile deflection was varied along its length. The p-y curves
that were derived from each of the loading conditions were essentially the same. Thus, Matlock
found that experimental p-y curves from fully instrumented piles could predict, within reasonable
limits, the response of a pile whose head is free to rotate or is fixed against rotation.
The methods for computing p-y curves derived from correlations to the results of fullscale experiments have been used to make computations for the response of piles where only the
pile-head movements were recorded. These computations, some of which are shown in Chapter 6
of the User’s Manual for LPile, show reasonable to excellent agreement between computed
predictions and experimental measurements.
Finally, technology may advance so that the soil resistance for a given deflection at a
particular point along a pile can be modified quantitatively to reflect the influence of the
deflection of the pile above and below the point in question. In such a case, multi-valued p-y
curves can be developed at every point along the pile. The analytical solution that is presented
herein could be readily modified to deal with the multi-valued p-y curves.
In short, the p-y method has some limitations; however, there is much evidence to show
that the method yields information of considerable value to an analyst and designer.
2-1-3 Computational Approach for Single Piles
The general procedure to be used in computing the behavior of many piles under lateral
loading is illustrated in Figure 2-5. Figure 2-5 (a) shows a pile with a given size embedded in a
soil with known properties. A lateral load V, axial load P, and moment M are acting at the pile
head. The design loading presumably would have been found by considering the unfactored
20
Chapter 2 – Solution for Pile Response to Lateral Loading
loads acting on the superstructure. Each of the loads is decreased or increased by an appropriate
load factor and, for each combination of loads, a solution of the problem is found. A curve can
be plotted, such as shown by the solid line in Figure 2-5 (b), which will show the maximum
bending moment developed in the pile as a function of the level of loading. With the value of the
nominal bending moment capacity Mnom for the section that takes into account the axial loading,
the “failure loading” can be found. An assumption is made that development of a plastic hinge at
any point in the pile would not be acceptable. The failure loading is then divided by a global
factor of safety to find the allowable loading. The allowable loading is then compared to the
loading from the superstructure to determine if the pile that was selected was satisfactory.
An alternate approach makes use of the concept of partial safety factors. The parameters
that influence the resistance of the pile to lateral loading are factored and the curve shown by the
dashed line is computed. As shown in Figure 2-5, smaller values of the failure loading would be
found. The values of allowable loading would probably be about the same as before with the
loading being reduced by a smaller value of partial safety factor.
In the case of a very short pile, the performance failure might be due to excessive
deflection as the pile “plows” through the soil. The design engineer can then employ a global
factor of safety or partial factors of safety to set the allowable load capacity.
As shown in Figure 2-5(b), the bending moment is a nonlinear function of load; therefore,
the use of allowable bending stresses, for example, is inappropriate and perhaps unsafe. A series
of solutions is necessary in order to obtain the allowable loading on a pile; therefore, the use of a
computer is required.
P
M
Loading
V
Loading at Failure
Mult
Allowable
Loading
Maximum Bending Moment
(a)
(b)
Figure 2-5 Illustration of General Procedure for Selecting a Pile to Sustain a Given Set of Loads
The next step in the computational process is to solve for the deflection of the pile under
the allowable loading. The tolerable deflection is frequently limited by special project
requirements and probably should not be dictated by building codes or standards. Among factors
21
Chapter 2 – Solution for Pile Response to Lateral Loading
to be considered are machinery that is sensitive to differential deflection and the comfort of
humans on structures that move a sensible amount under loading.
The computation of the load at failure requires values of the nominal bending moment
capacity and flexural rigidity of the section. Because the analyses require the structural section to
be stressed beyond the linear-elastic range, a computer program is required to compute the
nonlinear properties of the section. These capabilities are included in the LPile program.
General guidelines about making computations for the behavior of a pile under lateral
loading are presented in this manual. In addition, several examples are presented in detail in the
User’s Manual for LPile. However, it should be emphasized that a full design involves
consideration of many other factors that are not addressed here.
2-1-4 Pile Buckling Analysis
A common design problem is the analysis of the pile buckling capacity. In this problem,
shown in Figure 2-6(a), a pile that extends above the ground line is subjected to a lateral load V
and an axial load P. As part of the design process, the engineer desires to evaluate the axial load
that will cause the pile to buckle. The lateral load is held constant at the maximum value and the
axial load is increased in increments. The deflection yt at the top of the pile is plotted as a
function of axial load, as shown in Figure 2-6(b). A value of axial load will be approached at
which the pile-head deflection will increase without limit. This load is selected for the buckling
load. It is important that the buckling load be found by starting the computer runs with smaller
values of axial load because the computer program fails to obtain a solution at axial loads above
the buckling load. An example analysis of pile buckling is presented in Section 4-4 of this
manual and an example problem is presented in the User’s Manual for LPile.
P
yt
P
V
Buckling Load
yt
(b)
(a)
Figure 2-6 Analysis of Pile Buckling
22
Chapter 2 – Solution for Pile Response to Lateral Loading
2-1-5 Analysis of Critical Pile Length
Another common design problem is illustrated in Figure 2-7. A pile is subjected to a
combination of loads, as shown in Figure 2-7(a), but the axial load is relatively small so that the
length of the pile is controlled by the magnitude of the lateral load. Factored values of the loads
are applied to the top of a pile that is relatively long and a computer run is made to solve for the
lateral deflection yt and a point may be plotted in Figure 2-7(b). A series of runs are made with
the length of the pile reduced in increments. Connecting the points for the deflection at the top of
the pile yields the curve in Figure 2-7 (b). These computations and generation of the critical
length curve can be automatically performed by LPile for individual load cases that include pilehead shear force(either shear and moment, shear and fixed rotation, or shear and rotational
stiffness pile-head loading conditions).
The curve in Figure 2-7 (b) shows that the value of yt is unchanged for pile lengths longer
than a length that is termed Lcrit and that values of lateral deflection are larger for smaller values
of pile length. The designer will normally select a pile for a particular application whose length
is somewhat greater than Lcrit.
Another use of the critical length is to determine the length of pile required not to
accumulate a permanent inclination of the pile after lateral loading. The shorter a pile is relative
to the critical length, the more likely it is to develop a permanent inclination after loading. Thus,
if it is required that a structure remain upright within a specified tolerance, the foundation piles
should be longer than the critical length.
P
M
yt
V
Lcrit
L
Lcrit
Pile Length
Figure 2-7 Solving for Critical Pile Length
2-1-6 Occurrences of Lateral Loads on Piles
Piles that sustain lateral loads of significant magnitude occur in offshore structures,
waterfront structures, bridges, buildings, industrial plants, navigation locks, dams, and retaining
walls. Piles can also be used to stabilize slopes against sliding that either have failed or have a
low factor of safety. The lateral loads may be derived from earth pressures, wind, waves and
23
Chapter 2 – Solution for Pile Response to Lateral Loading
currents, earthquakes, impact, moving vehicles, and the eccentric application of axial loads. In
numerous cases, the loading of the piles cannot be obtained without consideration of the stresses
and deformation in the particular superstructure.
Structures where piles are subjected to lateral loading are discussed briefly in the
following paragraphs. Some general comments are presented about analytical techniques. The
cases that are selected are not comprehensive but are meant to provide examples of the kinds of
problems that can be attacked with the methods presented herein. In each of the cases, the
assumption is made that the piles are widely spaced and the distribution of loading to each of the
piles in a group is neglected.
2-1-6-1 Offshore Platform
An offshore platform is illustrated in Figure 2-8(a). A three-dimensional analysis of such
a structure is sometimes necessary, but often a two-dimensional analysis is adequate. The
preferred method of analysis of the piles is to consider the full interaction between the
superstructure and the supporting piles. However, in many analyses, the piles are replaced by
nonlinear load-transfer reactions: axial load versus axial movement, lateral load versus lateral
deflection, and moment versus lateral deflection. A simplified method of analyzing a single pile
is illustrated in the figure.
h = 6.1 m
St 
h
Mt
3.5EI c
M
d = 838 mm
Ic = 5.876 x 10-3 m4
4m
V
V
M
d = 762 mm
Ip = 3.07 x 10-3 m4
E = 2 x 108 kPa
(a)
(b)
(c)
Figure 2-8 Simplified Method of Analyzing a Pile for an Offshore Platform
The second pile is shown in Figure 2-8(b). In typical conditions, the annular void
between the jacket leg and the head of the pile was sealed with a flexible gasket and the annular
space is filled with grout. Consequently, it is usually assumed that the bending and lateral
deflection in the pile and jacket leg will be continuous and have the same curvature.
The sketch in Figure 2-8(c) shows that the stiffness of the braces was neglected and that
the rotational restraint at the upper panel point was intermediate between being fully fixed and
24
Chapter 2 – Solution for Pile Response to Lateral Loading
fully free. The assumption is then made that the resultant force on the bent can be equally
divided among the four piles, giving a known value of Pt. The second boundary condition at the
top of the pile is the value of the rotational restraint, Mt/St, which is taken as 3.5EIc/h, where EIc
is the combined bending stiffness of the pile and the jacket leg. The p-y curves for the supporting
soil can be generated, and the deflection and bending moment along the length of the pile can be
computed.
The above method is approximate. However, a pile with the approximate geometry can
be rapidly modeled by the p-y method. In addition, there may be structures for which the pile
head is neither completely fixed nor free and the use of rotational restraint for the pile-head fixity
condition is required.
The implementation of the method outlined above is shown by Example 3 in the User’s
Manual for LPile provided with LPile. In addition to investigating the exact value of pile-head
rotational stiffness, the designer should consider the rotation of the superstructure due principally
to the movement of the piles in the axial direction. This rotation will affect the boundary
conditions at the top of the piles.
2-1-6-2 Breasting Dolphin
One application of a pile under lateral load is a large pile used as a foundation for a
breasting dolphin. Figure 2-9(a) depicts a vessel with mass m approaching a freestanding pile.
The velocity of the vessel is v and its kinetic energy on impact with the dolphin would be ½mv2.
The deflection of the dolphin could be computed by finding the area under the load-deflection
curve that would equate to the energy of the vessel.
The design engineer should be concerned with a number of parameters in the problem.
The level of water could vary due to tide levels, requiring a number of solutions. The pile could
be tapered to give it the proper strength to sustain the computed bending moment while at the
same time making it as flexible as possible.
With the first impact of a vessel, the soil will behave as if it were under static loading
(assuming no inertia effects in the soil) and would be relatively stiff. With repeated loading on
the pile from berthing, the soil will behave as if under cyclic loading. The appropriate p-y curves
would need to be used, depending on the number of applications of load.
A single pile, or a group of piles, could support a primary fender, but the exact types and
sizes of cushions or fenders to be used between the vessel and the pile need to be selected on the
basis of the vessel size and berthing velocity. It should be noted that fenders must be mounted
properly above the waterline to prevent damage to the berthing vessels and that the lateral
spacing of breasting dolphins will depend on the overall length of the vessel and the vessel’s
curvature near the bow and stern.
25
Chapter 2 – Solution for Pile Response to Lateral Loading
Load
m, v
Breasting
Dolphin
Deflection
Figure 2-9 Analysis of a Breasting Dolphin
2-1-6-3 Single-Pile Support for a Bridge
A common design used for the support of a bridge is shown in Figure 2-10. The design
provides more space under the bridge in an urban area and may be aesthetically more pleasing
than multiple columns.
As may be seen in the sketch, the primary loads that must be sustained by the pile lie in a
plane perpendicular to the axis of the bridge.
The loads may be resolved into an axial load, a lateral load, and a moment at the ground
surface or, alternately, at the top of the column.
The braking forces are shown properly in a plane parallel to the axis of the bridge and can
be large, if heavily loaded trucks are suddenly brought to a stop on a downward-sloping span.
The deflection that may be possible in the direction of the axis of the bridge is probably limited
to that allowed by the joints in the bridge deck. Thus, one of the boundary conditions for the
piles for such loading could be a limiting deflection.
If it is decided that significant loads can be acting simultaneously in perpendicular planes,
two independent solutions can be made, and the resulting bending moments can be added
algebraically. Such a procedure would not be perfectly rigorous but should yield results that will
be instructive to the designer.
26
Chapter 2 – Solution for Pile Response to Lateral Loading
Loads From Traffic
Loads From Braking
and Wind Forces
From Dead Loads
From Wind and
Other Forces
Figure 2-10 Loading On a Single Shaft Supporting a Bridge Deck
2-1-6-4 Pile-Supported Overhead Sign
The sketches in Figure 2-11 show two schemes for piles to support an overhead sign.
Many such structures are used in highways and in other transportation facilities. Similar schemes
could be used for the foundation of a tower that supports power lines.
The loadings on the foundation from the wind will be a lateral load and a relatively large
moment; a small axial load will result from the dead weight of the superstructure. The lateral
load and moment will be variable because the wind will blow intermittently and will gust during
a storm. The predominant direction of the wind will vary; these factors should be taken into
account in the analysis.
The sketch in Figure 2-11(a) shows a two-pile foundation. The lateral load and axial load
will be divided between the two piles, and the moment will be carried principally by tension in
one pile and compression in the other. The lateral load will cause each of the piles to deflect, and
there will be a bending moment along each pile. In performing the analysis for lateral loading, py curves must be derived for the supporting soil with repeated loading being assumed. A factored
load must be used, and the degree of fixity of the pile heads must be assessed. The connection
between the piles and the cap may be such that the pile heads are essentially free to rotate.
Alternatively, the design analysis may be made assuming that the pile heads are fixed against
rotation.
27
Chapter 2 – Solution for Pile Response to Lateral Loading
Wind
Load
Wind
Load
Column
Dead Load
Pile
Cap
Column
Dead Load
Two-Shaft
Foundation
(a)
Single-Shaft
Foundation
(b)
Figure 2-11 Foundation Options for an Overhead Sign Structure
The pile heads, under almost any designs, will likely be partially restrained, or at some
point between fixed and free. An interesting exercise is to take a free body of the pile from the
bottom of the cap and to analyze its behavior when a shear and a moment are applied at the end
of this “stub pile. “ The concrete in this instance will serve a similar function as the soil along the
lower portion of the pile. The rotational restraint provided by the concrete can be computed by
use of an appropriate model, perhaps by using finite elements. At present, an appropriate
analytical technique, when a pile head extends into a concrete cap or mat, is to assume various
degrees of pile-head fixity, ranging from completely fixed to completely free, and to design for
the worst conditions that results from the computer runs.
The sketch in Figure 2-11(b) shows a structure supported by a single pile. Shown in the
figure is a pattern of soil resistance that must result to put the pile into equilibrium. In performing
the analyses, the p-y curves must be derived as before, but, in this instance, the conditions at the
pile head are fully known. The loading will consist of a shear and a relatively large moment, and
the pile head will be free to rotate. Because the axial load will be relatively small, studies will
probably be necessary to determine the required penetration of the pile so that the tip deflection
will be small and the pile will not behave as a “fence post. “
Of the two schemes, selection of the most efficient scheme will depend on a number of
conditions. Two considerations are the deflection under the maximum load at the top of the
structure and the availability of equipment that can construct the large pile.
2-1-6-5 Use of Piles to Stabilize Slopes
An application for piles that is continuing interest is the stabilizing of slopes that have
failed or are judged to be near failure. The sketch in Figure 2-12 illustrates the application. A
bored pile is often employed because it can be installed with a minimum of disturbance of the
soil near the actual or potential sliding surface.
28
Chapter 2 – Solution for Pile Response to Lateral Loading
Figure 2-12 Use of Piles to Stabilize a Slope Failure
The procedures for the design of such a pile are described in some more detail later in
this manual. The special treatment accorded to this particular problem is due to its importance
and because the technical literature fails to provide much guidance to the designer.
2-1-6-6 Anchor Pile for a Tieback
The use of a pile as the anchor for a tieback anchor is illustrated in Figure 2-13. A
vertical pile is shown in the sketch with the tie rod attached below the top of the pile. The force
in the rod can be separated into components; one component indicates the lateral load on the pile
and the other the axial load.
The p-y curves are derived with proper attention to soil characteristics with respect to
depth below the ground surface. The loading will be sustained and a proper adjustment must be
made, if time-related deflection is expected.
The analysis will proceed by considering the loading to be applied at the top of the pile
or, preferably, as a distributed load along the upper portion of the pile. In the case of the anchor
that is shown in Figure 2-13, the load is applied at some distance below the pile head. The anchor
pile can be modelled using the methods presented in Section 3-8-1-6 in the User’s Manual for
LPile.
2-1-6-7 Other Uses of Laterally Loaded Piles
Piles under lateral loading occur in many structures or applications other than the ones
that were mentioned earlier. Some of these are high-rise buildings that are subjected to forces
from wind; structures subject to unbalanced earth pressures; pile-supported retaining walls; locks
and dams; waterfront structures such as piers and quay walls; supports for overhead pipes and for
other facilities found in industrial plants; and bridge abutments.
The method has the potential of analyzing the flexible bulkhead that is shown in Figure
2-13. The sheet piles (or tangent piles if drilled shafts or bored piles are used) can be analyzed as
a pile, if the p-y curves are modified to reflect the soil resistance versus deflection for a wall,
rather than of an individual pile. Research on the topic has been performed (Wang, 1986) and has
been implemented in the computer program PYWall from Ensoft, Inc.
29
Chapter 2 – Solution for Pile Response to Lateral Loading
Tie-back
Anchor Pile
(Dead Man)
Sheet Pile Wall
Figure 2-13 Anchor Pile for a Flexible Bulkhead
2-2 Derivation of Differential Equation for the Beam-Column and
Methods of Solution
The equation for the beam-column must be solved for implementation of the p-y method,
and a brief derivation is shown in the following section. An abbreviated version of the equation
can be solved by a closed-form method for some purposes, but a general solution can be made
only by a numerical procedure. Both of these kinds of solution are presented in this chapter.
2-2-1 Derivation of the Differential Equation
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 desirable to
include the axial loading in the analytical process. The derivation of the differential equation for
a beam-column foundation was presented by Hetenyi (1946). The derivation is shown in the
following paragraphs, though the notation differs from that used by Hetenyi.
The assumption is made that a bar on an elastic foundation is subjected not only to the
vertical loading, but also to the pair of compressive forces Px acting at the centroid 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 Figure 2-14), the equilibrium of moments (ignoring second-order terms)
leads to the equation

M  dM   M  Px dy  Vv dx  0 ..........................................(2-1)
or
30
Chapter 2 – Solution for Pile Response to Lateral Loading
y
x
y
Px
S
M
Vv
Vn
Vv
dx
Vv+dVv
y+dy
M+dM
Px
x
Figure 2-14 Element of Beam-Column (after Hetenyi, 1946)
dM
dy
 Px
 Vv  0 . ....................................................(2-2)
dx
dx
Differentiating Equation 2-2 with respect to x, the following equation is obtained
d 2M
d 2 y dVv
 Px 2 
 0 ................................................(2-3)
dx
dx 2
dx
The following definitions are noted:
d 2M
d4y
 EI 4
dx 2
dx
dVv
 p
dx
p  Es y
where Es is equal to the secant modulus of the soil-response curve.
And making the indicated substitutions, Equation 2-3 becomes
EI
d4y
d2y

P
 E s y  0 ...............................................(2-4)
x
dx 4
dx 2
The direction of the shearing force Vv is shown in Figure 2-14. The shearing force in the
plane normal to the deflection line can be obtained as
31
Chapter 2 – Solution for Pile Response to Lateral Loading
Vn = Vv cos S – Px sin S ..................................................(2-5)
Because S is usually small, we may assume the small angle relationships cos S = 1 and sin S =
tan S = dy/dx. Thus, Equation 2-6 is obtained.
Vn  Vv  Px
dy
.........................................................(2-6)
dx
Vn will mostly be used in computations, but Vv can be computed from Equation 2-6 where
dy/dx is equal to the rotation S.
The ability to allow a distributed force W per unit of length along the upper portion of a
pile is convenient in the solution of a number of practical problems. The differential equation
then becomes as shown below.
d4y
d2y
EI 4  Px 2  p  W  0 .............................................(2-7)
dx
dx
where:
Px = axial thrust load in the pile,
y
= lateral deflection of the pile at a point x along the length of the pile,
p = soil reaction per unit length,
EI = flexural rigidity, and
W = distributed load along the length of the pile.
Other beam formulas that are needed in analyzing piles under lateral loads are:
Vv  EI
d3y
dy
 Px
.....................................................(2-8)
3
dx
dx
d2y
M  EI 2 ...........................................................(2-9)
dx
and,
S
dy
.............................................................(2-10)
dx
where
Vv = horizontal shear in the pile,
M = bending moment in the pile, and
S = slope of the elastic curve relative to the x-axis of the pile.
32
Chapter 2 – Solution for Pile Response to Lateral Loading
Except for the axial load Px, the sign conventions that are used in the differential equation
and in subsequent development are the same as those commonly employed in the mechanics for
beams, with the axes for the pile rotated 90 degrees clockwise from the axes for the beam. The
axial load Px does not normally appear in the equations for beams. The sign conventions are
presented graphically in Figure 2-15. A solution of the differential equation yields a set of curves
such as shown in Figure 2-16, with a compressive axial load being defined as positive in sign.
The mathematical relationships for the various curves that give the response of the pile are
shown in the figure for the case where no axial load is applied.
Slope (L/L)
Deflection (L)
y
y(+)
S (+)
x
Moment (F*L)
y
y
M (+)
x
x
P (+)
Axial Force (F)
Soil Resistance (F/L)
Shear (F)
y
y
y
V (+)
p (+)
x
x
x
Figure 2-15 Sign Conventions
The assumptions that are made in deriving the differential equation are:
1. The pile is initially 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 in tension and compression,
6. Transverse deflections of the pile are small,
7. The pile is not subjected to dynamic loading, and
8. Deflections due to shearing stresses are small.
33
Chapter 2 – Solution for Pile Response to Lateral Loading
Assumption 8 can be addressed by including more terms in the differential equation, but
errors associated with omission of these terms are usually small. The numerical method
presented later can deal with the behavior of a pile made of materials with nonlinear stress-strain
properties.
y
S
M
V
p
Figure 2-16 Form of Results Obtained for a Complete Solution
2-2-2 Solution of Reduced Form of Differential Equation
A simpler form of the differential equation results from Equation 2-4, if the assumptions
are made that no axial load is applied, that the bending stiffness EI is constant with depth, and
that the soil modulus Es is constant with depth and equal to . The first two assumptions can be
satisfied in many practical cases; however, the last of the three assumptions seldom occurs or is
ever satisfied in practice.
The solution shown in this section is presented for two important reasons: (1) the
resulting equations demonstrate several factors that are common to any solution; thus, the nature
of the problem is revealed; and (2) the closed-form solution allows for a check of the accuracy of
the numerical solutions that are given later in this chapter.
If the assumptions shown above are employed and if the identity shown in Equation 2-11
is used, the reduced form of the differential equation is shown as Equation 2-12.
4 

4 EI

Es
.....................................................(2-11)
4 EI
d4y
 4 4 y  0 ......................................................(2-12)
4
dx
The solution to Equation 2-12 may be directly written as:
34
Chapter 2 – Solution for Pile Response to Lateral Loading
y  e x (C1 cos x  C 2 sin x)
 e  x (C 3 cos x  C 4 sin x)
..........................................(2-13)
The coefficients C1, C2, C3, and C4 must be evaluated for the various boundary conditions that are
desired. A pile of any length is considered later but, if one considers a long pile, a simple set of
equations can be derived. An examination of Equation 2-13 shows that C1 and C2 must approach
zero because the term ex will increase without limit.
The boundary conditions for the top of the pile that are employed for the solution of the
reduced form of the differential equation are shown by the simple sketches in Figure 2-17. A
more complete discussion of boundary conditions for a pile is presented in the next section.
Spring (takes no shear, but
restrains pile head rotation)
Mt
y
Vt
Vt
Free-head
y
Fixed-Head
(a)
Vt
y
Partially Restrained
(b)
(c)
Figure 2-17 Boundary Conditions at Top of Pile
2-2-2-1 Solution for Free-head Pile
The boundary conditions at the top of the pile selected for the first case are illustrated in
Figure 2-17(a) and in equation form are:
at x = 0,
d 2 y Mt

..........................................................(2-14)
dx 2
EI
d 3 y Vt

...........................................................(2-15)
dx 3 EI
The differentiations of Equation 2-13 are made and the substitutions indicated by Equation 2-14
yield the following.
35
Chapter 2 – Solution for Pile Response to Lateral Loading
C4 
 Mt
.........................................................(2-16)
2 EI 2
The substitutions indicated by Equation 2-15 yield the following.
C3  C 4 
Vt
.....................................................(2-17)
2 EI 3
Equations 2-16 and 2-17 are used and expressions for deflection y, slope S, bending moment M,
shear V, and soil resistance p can be written as shown in Equations 2-18 through 2-22.
y
2b 2 e -bx Vt

cos x  M t (cos x  sin x) ...............................(2-18)

 b

 2V  2

M
S  e   x  t
(sin x  cos x)  t cos x  ............................(2-19)
EI
 

V

M  e  x  t sin x  M t (sin x  cos x) .................................(2-20)


V  e  x Vt (cos x  sin x)  2M t  sin x ................................(2-21)
V

p  2 2 e  x  t cos x  M t (cos x  sin x) .............................(2-22)


It is convenient to define some functions that make it easier to write the above equations.
These are:
A1  e  x cos x  sin x  ..............................................(2-23)
B1  e  x cos x  sin x  ..............................................(2-24)
C1  e  x cos x ......................................................(2-25)
D1  e  x sin x ......................................................(2-26)
Using these functions, Equations 2-18 through 2-22 become:
y
2Vt 

C1 
36
Mt
B1 ...............................................(2-27)
2 EI  2
Chapter 2 – Solution for Pile Response to Lateral Loading
S
 2Vt  2

M
Vt

A1 
Mt
C1 ...............................................(2-28)
EI 
D1  M t A1 ....................................................(2-29)
V  Vt B1  2M t  D1 ..................................................(2-30)
p  2Vt  C1  2M t  2 B1 .............................................(2-31)
Values for A1, B1, C1, and D1, are shown in Figure 2-18 as a function of the nondimensional
distance x along the pile.
2-2-2-2 Solution for Fixed-head Pile
For a pile whose head is fixed against rotation, as shown in Figure 2-17(b), the solution
may be obtained by employing the boundary conditions as given in Equations 2-32 and 2-33.
dy
 0 .............................................................(2-32)
dx
At x = 0,
EI
d3y
 Vt .........................................................(2-33)
dx 3
Using the procedures as for the case where the boundary conditions were as shown in
Figure 2-5(a), the results are as follows.
C3  C 4 
Vt
....................................................(2-34)
4 EI  3
The solution for long piles is given in Equations 2-35 through 2-39.
y
S 
Vt 

A1 ..........................................................(2-35)
Vt
D1 ......................................................(2-36)
2 EI  2
37
Chapter 2 – Solution for Pile Response to Lateral Loading
A1, B1, C1, D1
-0.25
0.0
0.00
0.25
0.50
0.75
1.00
0.5
1.0
1.5
2.0
2.5
x
A11
B11
C11
D1
1
3.0
3.5
4.0
4.5
5.0
5.5
A1  e  x cos x  sin x 
B1  e  x cos x  sin x 
C1  e  x cos x
D1  e  x sin x
6.0
Figure 2-18 Values of Coefficients A1, B1, C1, and D1
M 
Vt
B1 .........................................................(2-37)
2
V  Vt C1 ............................................................(2-38)
p  Vt  A1 ........................................................(2-39)
2-2-2-3 Solution for Pile with Rotational Restraint
It is sometimes convenient to have a solution for a third set of boundary conditions
describing the rotational restraint of the pile head, as shown in Figure 2-17(c). For this boundary
38
Chapter 2 – Solution for Pile Response to Lateral Loading
condition, the rotational spring does not take any shear, but does restrain the rotation of the pile
head. These boundary conditions are given in Equations 2-40 and 2-41. At the pile head, where x
= 0, the rotational restrain is controlled by
EI
d2y
dx 2  M t ........................................................(2-40)
dy
St
dx
and the pile-head shear force is controlled by
d 3 y Vt

...........................................................(2-41)
dx 3 EI
Employing these boundary conditions, the coefficients C3 and C4 can be evaluated, and the
results are shown in Equations 2-42 and 2-43. For convenience in writing, the rotational restraint
Mt /St is given the symbol k.
C3 
Vt (2 EI   k )
..................................................(2-42)
EI (  4 3 k )
C4 
k Vt
..................................................(2-43)
EI (  4 3 k )
These expressions can be substituted into Equation 2-13, differentiation performed as
appropriate, and substitution of Equations 2-23 through 2-26 will yield a set of expressions for
the long pile similar to those in Equations 2-27 through 2-31 and 2-35 through 2-39.
Timoshenko (1941) stated that the solution for the “long” pile is satisfactory where L is
greater than 4; however, there are occasions when the solution of the reduced differential
equation is desired for piles that have a nondimensional length less than 4. The solution can be
obtained by using the following boundary conditions at the tip of the pile. At x = L,
d2y
 0 (bending moment, M, is zero at pile tip)............................(2-44)
dx 2
and
d3y
 0 (shear force, V, is zero at pile tip) .................................(2-45)
dx3
When the above boundary conditions are used, along with a set for the top of the pile, the
four coefficients C1, C2, C3, and C4 can be evaluated. The solutions are not shown here, but new
values of the parameters A1, B1, C1, and D1 can be computed as a function of L. Such
computations, if carried out, will show readily the influence of the length of the pile.
39
Chapter 2 – Solution for Pile Response to Lateral Loading
The reduced form of the differential equation will not normally be used for the solution
of problems encountered in design; however, the influence of pile length and other parameters
can be illustrated with clarity. Furthermore, the closed-form solution can be used to check the
accuracy of the numerical solution shown in the next section.
2-2-3 Solution by Finite Difference Equations
The solution of Equation 2-7 is necessary for dealing with numerous problems that are
encountered in practice. The formulation of the differential equation in finite difference form and
a solution by iteration mandates a computer program. In addition, the following improvements in
the solutions shown in the previous section are then possible.

The effect of the axial load on deflection and bending moment can be considered and
problems of pile buckling can be solved.

The bending stiffness EI of the pile can be varied along the length of the pile.

Perhaps of more importance, the soil modulus Es can vary with pile deflection and with the
depth of the soil profile.

Soil displacements around the pile due to slope movements, seepage forces, or other causes
can be taken into account.
In the finite difference formulations, the derivative terms are replaced by algebraic
expressions. The following central difference expressions have errors proportional to the square
of the increment length h.
dy ym 1  ym 1

dx
2h
d 2 y ym 1  2 ym  ym 1

dx 2
h2
d 3 y  ym  2  2 ym 1  2 ym 1  ym  2

dx3
2h 3
d 4 y ym  2  4 ym 1  6 ym  4 ym 1  ym  2

dx 4
h4
If the pile is subdivided in increments of length h, as shown in Figure 2-19, the governing
differential equation, Equation 2-7, in difference form with collected terms for y is as follows:
y m  2 Rm 1 
y m 1 (2 Rm 1  2 Rm  Px Qh 2 ) 
y m ( Rm 1  4 Rm  Rm 1  2 Px h 2  k m hH 4 )  .................................(2-46)
y m 1 (2 Rm  2 Rm 1  Px h 2 ) 
y m  2 Rm 1Wh 4  0
40
Chapter 2 – Solution for Pile Response to Lateral Loading
y
ym+2
h
h
ym+1
ym
h
ym-1
h
ym-2
x
Figure 2-19 Representation of deflected pile
where
Rm = EmIm (flexural rigidity of pile at point m) and
km = Esm.
The assumption is implicit in Equation 2-46 that the magnitude of Px is constant with
depth. Of course, that assumption is not strictly true. However, experience has shown that the
maximum bending moment usually occurs a relatively short distance below the ground line at a
point where the value of Px is undiminished. This fact plus the fact that Px, except in cases of
buckling, has little influence on the magnitudes of deflection and bending moment, leads to the
conclusion that the assumption of a constant Px is generally valid. For the reasons given, it is
thought to be unnecessary to vary Px in Equation 2-46; thus, a table of values of Px as a function
of x is not required.
If the pile is divided into n increments, n+1 equations of the sort as Equation 2-46 can be
written. There will be n+5 unknowns because two imaginary points will be introduced above the
top of the pile and two will be introduced below the bottom of the pile. If two equations giving
boundary conditions are written at the bottom and two at the top, there will be n+5 equations to
solve simultaneously for the n+5 unknowns. The set of algebraic equations can be solved by
matrix methods in any convenient way.
The two boundary conditions that are employed at the bottom of the pile involve the
moment and the shear. If the possible existence of an eccentric axial load that could produce a
moment at the bottom of the pile is discounted, the moment at the bottom of the pile is zero. The
assumption of a zero moment is believed to produce no error in all cases except for short rigid
piles that carry their loads in end bearing, and when the end bearing is applied eccentrically. (The
case where the moment at the bottom of a pile is not equal to zero is unusual and is not treated by
the procedure presented herein.) Thus, the boundary equation for zero moment at the bottom of
the pile requires
41
Chapter 2 – Solution for Pile Response to Lateral Loading
y1  2 y0  y1  0 .....................................................(2-47)
where y0 denotes the lateral deflection at the bottom of the pile. Equation 2-47 is expressing the
condition that EI(d2y/dx2) = 0 at x = L (The numbering of the increments along the pile starts
with zero at the bottom for convenience).
The second boundary condition involves the shear force at the bottom of the pile. The
assumption is made that soil resistance due to shearing stress can develop at the bottom of a short
pile as deflection occurs. It is further assumed that information can be developed that will allow
V0, the shear at the bottom of the pile, to be known as a function of y0 Thus, the second equation
for the zero-shear boundary condition at the bottom of the pile is
R0
 y 2  2 y 1  2 y1  y 2   Px  y 1  y1   V0 ..............................(2-48)
3
2h
2h
Equation 2-48 is expressing the condition that there is some shear at the bottom of the pile or that
EI(d3y/dx3) + Px (dy/dx) = V0 at x = L. The assumption is made in the equations that the pile
carries its axial load in end bearing only, an assumption that is probably satisfactory for short
piles for which V0 would be important. The value of V0 should be set equal to zero for long piles
(2 or more points of zero deflection along the length of the pile).
As noted earlier, two boundary equations are needed at the top of the pile. Four sets of
boundary conditions, each with two equations, have been programmed. The engineer can select
the set that fits the physical problem.
Case 1 of the boundary conditions at the top of the pile is illustrated graphically in Fig 220. (The axial load Px is not shown in the sketches, but Px is assumed to act at the top of the pile
for each of the four cases of boundary conditions.). For the condition where the shear at the top
of the pile is equal to Pt, the following difference equation is employed.
Vt
+M
+V
yt+2
yt+1
yt
yt-1
yt-2
h
Figure 2-20 Case 1 of Boundary Conditions
Vt 
Rt
 yt 2  2 yt 1  2 yt 1  yt 2   Px  yt 1  yt 1  .........................(2-49)
3
2h
2h
42
Chapter 2 – Solution for Pile Response to Lateral Loading
For the condition where the moment at the top of the pile is equal to Mt, the following difference
equation is employed.
Mt 
Rt
h2
 y t 1  2 y t  y t 1  .............................................(2-50)
Case 2 of the boundary conditions at the top of the pile is illustrated graphically in Figure
2-21. Here the pile is embedded into a concrete pile cap for which the rotation is known. Often
the rotation can be assumed as zero, at least for the initial trial analyses. Equation 2-49 is the first
of the two needed equations. The second needed equation is for the condition where the slope St
at the top of the pile is known.
yt+2
yt+1
yt
+Vt
St
yt-1
yt-2
1
Figure 2-21 Case 2 of Boundary Conditions
St 
yt 1  yt 1
.......................................................(2-51)
2h
Case 3 of the boundary conditions at the top of the pile is illustrated in Figure 2-22. It is
assumed that the pile continues into a superstructure and is a member in a frame. The solution for
this problem can proceed by cutting a free body at the bottom joint of the frame. A moment is
applied to the frame at that joint, and the rotation of the frame is computed (or estimated for the
initial trial analysis). The moment divided by the rotation, Mt /St, is the rotational stiffness
provided by the superstructure and becomes one of the boundary conditions.
To implement the boundary conditions in Case 3, it may be necessary to perform an
initial solution for the pile, with an estimate of Mt /St, to obtain a preliminary value of the
moment at the bottom joint of the superstructure. The superstructure can then be analyzed for a
more accurate value of Mt /St, and then the pile can be re-analyzed. One or two iterations of this
sort should be sufficient in most instances.
43
Chapter 2 – Solution for Pile Response to Lateral Loading
Pile extends above ground surface
and in effect becomes a column in
the superstructure
yt+2
yt+1
yt
+Vt
yt-1
h
yt-2
Figure 2-22 Case 3 of Boundary Conditions
Equation 2-49 is the first of the two equations that are needed for Case 3. The second
equation expresses the condition that the rotational restraint Mt /St is known.
Rt
 y  2 yt  yt 1 
M t h 2 t 1

..............................................(2-52)
yt 1  yt 1
St
2h
Case 4 of the boundary conditions at the top of the pile is illustrated in Figure 2-23. It is
assumed, for example, that a pile is embedded in a bridge abutment that moves laterally a given
amount; thus, the deflection yt at the top of the pile is known. It is further assumed that the
bending moment is known. If the embedment amount is small, the bending moment is frequently
assumed as zero. The first of the two equations expresses the condition that the moment Mt at the
pile head is known, and Equation 2-50 can be employed. The second equation merely expresses
the fact that the pile-head deflection is known.
yt  Yt .............................................................(2-53)
Case 5 of the boundary conditions at the top of the pile is illustrated in Figure 2-24. Both
the deflection yt the rotation St at the top of the pile are assumed to be known. This case is related
to the analysis of a superstructure because advanced models for structural analyses have been
recently developed to achieve compatibility between the superstructure and the foundation. The
boundary conditions in Case 5 can be conveniently used for computing the forces at the pile head
in the model for the superstructure. Equation 2-53 can be used with a known value of yt and
Equation 2-51 can be used with a known value of St.
44
Chapter 2 – Solution for Pile Response to Lateral Loading
Foundation
moves laterally
yt+2
Mt
yt+1
yt
yt-1
Pile-head moment is
known, may be zero
h
yt-2
Figure 2-23 Case 4 of Boundary Conditions
St
yt
yt+2
yt+1
yt
1
yt-1
yt-2
St
Figure 2-24 Case 5 of Boundary Conditions
The five boundary conditions at the top of a pile should be adequate for virtually any
situation but other cases can arise. However, the boundary conditions that are available in LPile,
with a small amount of effort, can produce the required solutions. For example, it can be
assumed that Vt and yt are known at the top of a pile and constitute the required boundary conditions (not one of the four cases). The Case 4 equations can be employed with a few values of Mt
being selected, along with the given value of yt. The computer output will yield values of Vt. A
simple plot will yield the required value of Mt that will produce the given boundary condition, Vt.
LPile solves the difference equations for the response of a pile to lateral loading.
Solutions of some example problems are presented in the User’s Manual for LPile. In addition,
case studies are included in which the results from computer solutions are compared with
experimental results. Because of the obvious approximations that are inherent in the difference45
Chapter 2 – Solution for Pile Response to Lateral Loading
equation method, a discussion is provided of techniques for the verification of the accuracy of a
solution that is essential to the proper use of the numerical method. The discussion will deal with
the number of significant digits to be used in the internal computations and with the selection of
the increment length h. However, at this point some brief discussion is in order about another
approximation in Equation 2-46.
The bending stiffness EI, denoted by R in the difference equations, is correctly
represented as a constant in the second-order differential equation, Equation 2.-9.
EI
d2y
 M ...........................................................(2-9)
dx 2
In finite difference form, Equation 2.9 becomes
Rm
ym 1  2 ym  ym 1
 M m ............................................. (2.54)
h2
In building up the higher ordered terms by differentiation, the value of R is made to
correspond to the central term for y in the second-order expression. The errors that are involved
in using the above approximation where there is a change in the bending stiffness along the
length of a pile are thought to be small, but may be investigated as necessary.
46
Chapter 2 – Solution for Pile Response to Lateral Loading
(This page was deliberately left blank)
47
Chapter 3
Lateral Load-Transfer Curves for Soil and Rock
3-1 Introduction
This chapter presents the formulation of expressions for p-y curves for soil and rock
under both static and cyclic loading. As part of this presentation, a number of fundamental
concepts are presented that are relevant to any method of analyzing deep foundations under
lateral loading. Chapter 1 presented the concept of the p-y method, and this chapter will present
details for the computation of load-transfer behavior for a pile under a variety of conditions.
A typical p-y curve is shown in Figure 3-1a. The p-y curve is just one of a family of p-y
curves that describe the lateral-load transfer along the pile as a function of depth and of lateral
deflection. It would be desirable if soil reaction could be found analytically at any depth below
the ground surface and for any value of pile deflection. Factors that might be considered are pile
geometry, soil properties, and whether the type of loading, static is cyclic, sustained, or dynamic.
Unfortunately, common methods of analysis are currently inadequate for solving all possible
problems. However, principles of geotechnical engineering can be helpful in gaining insight into
the evaluation of two characteristic portions of a p-y curve.
Soil Resistance, p
b
b
p
c
a
a
d
Pile Deflection, y
y
(b)
(a)
b
p
a
e
y
(c)
Figure 3-1 Conceptual p-y Curves
49
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The p-y curve in Figure 3-1(a) is meant to represent the case where a short-term
monotonic loading was applied to a pile. This case will be called “static” loading for
convenience and will seldom, if ever, be encountered in practice. However, the static loading
curve is useful because analytical procedures can be used to develop expressions to correlate
with some portions of the curve, and the static curve serves as a baseline for demonstrating the
effects of other types of loading.
The three curves in Figure 3-1 show a straight-line relationship between p and y from the
origin to point a. If it can be reasonably assumed that for small strains in soil there is a linear
relationship between p and y for small values of y. Analytical methods for computing the slopes
of the initial portion of the p-y curves, Esi, are discussed later.
Recommendations will be given in this chapter for the selection of the slope of the initial
portion of p-y curves for the various cases of soils and loadings that are addressed. The point
should be made, however, that the recommendations for the slope of the initial portion are meant
to be somewhat conservative because the deflection and bending moment of a pile under light
loads will probably be somewhat less than computed by use of the recommendations. There are
some cases in the design of piles under lateral loading when it will be unconservative to compute
more deflection than will actually occur; in such cases, a field load test must be made.
The portion of the curve in Figure 3-1(a) from points a to b shows that the value of p is
strain softening with respect to y. This behavior is reflecting the nonlinear portion of the stressstrain curve for natural soil. Currently, there are no accepted analytical procedures that can be
used to compute the a-b portion of a p-y curve. Rather, that portion of the curves is empirical and
based on results of full-scale tests of piles in a variety of soils with both monotonic and cyclic
loading.
The horizontal, straight-line portion of the p-y curve in Figure 3-1(a) implies that the soil
is behaving plastically with no loss of shear strength with increasing strain. Using this
assumption, some analytical models can be used to compute the ultimate resistance pu as a
function of pile dimensions, soil properties, and depth below the ground surface. One part of a
model is for soil resistance near the ground surface and assumes that at failure the soil mass
moves vertically and horizontally. The other part of the model is for the soil resistance deep
below the ground surface and assumes only horizontal movement of the soil mass around the
pile.
Figure 3-1(b) shows a shaded portion of the curve in Figure 3-1(a). The decreasing values
of p from point c to point d reflect the effects of cyclic loading. The curves in Figures 3-1(a) and
3-1(b) are identical up to point c, which implies that the soil behaves identically for both type of
loading at small deflections. The loss of resistance shown by the shaded area depends on the
number of cycles of loading.
A possible effect of sustained, long-term loading is shown in Figure 3-1(c). This figure
shows that there is a time-dependent increase in deflection with sustained loading. The
decreasing value of p implies that the resistance is shifted to other elements of soil along the pile
as the deflection occurs at some particular point. The effect of sustained loading should be
negligible for heavily overconsolidated clays and for granular soils. The effect for soft clays
must be approximated at present.
50
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-2 Experimental Measurements of p-y Curves
Methods of getting p-y curves from field experiments with full-sized piles will be
presented prior to discussing the use of analysis in getting soil response. The strategy that has
been employed for obtaining design criteria is to make use of theoretical 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.
3-2-1 Direct Measurement of Soil Response
A number of attempts have been made to make direct measurements in the field of p and
y. Measurement of lateral deflection involves the conceptually simple process using a slope
inclinometer system to measure lateral deflection along the length of the pile. The method is
cumbersome in practice and has not been very successful in the majority of tests in which it was
attempted.
Measurement of soil resistance directly involves the design of an instrument that will
integrate the soil stress around the circumference at a point along the pile. The design of such an
instrument has been proposed, but none has yet been built. Some attempts have been made to
measure total soil stress and pore water pressure at a few points around the exterior of a pile with
the view that the soil pressures at other points on the circumference can be estimated by
interpolation. The method has met with little success for a variety of reasons, including changes
in calibration when axial loads are applied to the pile and failure to survive pile installation.
The experimental method that has met with the greatest success is to instrument the pile
to measure bending strains along the length of the pile, typically using spacing of 6 to 12 inches
(150 to 300 mm) between levels of gages. The data reduction consists of converting the strain
measurements to bending curvature and bending moment, the obtaining lateral load-transfer than
double differentiation of the bending moment curve versus depth, and obtaining lateral deflection
by double integration of the bending curvature curve versus depth.
3-2-2 Derivation of Soil Response from Moment Curves Obtained by Experiment
Almost all successful load test experiments that have yielded p-y curves have measured
bending moment using electrical-resistance strain gages. In this method, curvature of the pile is
measured directly using strain gages. Bending moment in the pile is computed from the product
of curvature and the bending stiffness. Pile deflection can be obtained with considerable
accuracy by twice integrating curvature versus depth. The deflection and the slope of the pile at
the ground line are measured accurately. It is best if the pile is long enough so that there are at
least two points of zero deflection along the lower portion of the pile so that it can be reasonably
assumed that both moment and shear equal zero at the pile tip.
Evaluation of soil resistance mobilized along the length of the pile requires two
differentiations of a bending moment curve versus depth. Matlock (1970) made extremely
accurate measurements of bending moment and was able to do the differentiations numerically
(Matlock and Ripperger, 1958). This was possible by using a large number of gages and by
calibrating the instrumented pile in the laboratory prior to installation in the field. However, most
investigators fit analytical curves of various types through the points of experimental bending
moment and mathematically differentiate the fitted curves.
51
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The experimental p-y curves can be plotted once multiple of curves showing the
distribution of deflection and soil resistance for multiple levels of loading have been developed.
A check can be made of the accuracy of the analyses by using the experimental p-y curves to
compute bending-moment curves versus depth. The computed bending moments should agree
closely with those measured in the load test. In addition, computed values of pile-head slope and
deflection can be compared to the values measured during the load test. Usually, it is more
difficult to obtain agreement between computations and measurement of pile-head deflection and
slope over the full range of loading than for bending moment.
Examples of p-y curves that were obtained from a full-scale experiment with pipe piles
with a diameter of 641 mm (24 in.) and a penetration of 15.2 m (50 ft) are shown in Figures 3-2
and 3-3 (Reese et al., 1975) . The piles were instrumented for measurement of bending moment
at close spacing along the length and were tested in overconsolidated clay.
3,000
x = 12"
x = 24"
x = 36"
2,500
x = 48"
x = 60"
Soil Resistance, p, lb/in.
x = 72"
x = 96"
2,000
x = 120"
1,500
1,000
500
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Deflection, y, inches
Figure 3-2 p-y Curves from Static Load Test on 24-inch Diameter Pile (Reese, et al. 1975)
52
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3,000
x = 12"
x = 24"
x = 36"
x = 48"
x = 60"
x = 72"
Soil Resistance, p, lb/in.
2,500
x = 84"
x = 96"
x = 108"
x = 120"
2,000
1,500
1,000
500
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Deflection, y, inches
Figure 3-3 p-y Curves from Cyclic Load Test on 24-inch Diameter Pile (Reese, et al. 1975)
3-2-3 Nondimensional Methods for Obtaining Soil Response
Reese and Cox (1968) described a method for obtaining p-y curves for cases where only
pile-head measurements are made during lateral loading. They noted that nondimensional curves
could be obtained for many variations of soil modulus with depth. Equations for the soil modulus
involving two parameters were employed, such as shown in Equations 3-1 and 3-2.
Es  k1  k 2 x ..........................................................(3-1)
or
Es  kxn .............................................................(3-2)
Measurements of shear force, bending moment, pile-head deflection, and rotation at the
ground line are necessary. Then, either of the equations is selected to develop expressions for
53
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
pile-head deflection and rotation in terms of Es and the two parameters (either k1 and k2 or k and
n) are solved for a given applied load and moment. With an expression for soil modulus for a
particular load, the soil resistance and deflection along the length of the pile can be then
computed. The reader is referred to Reese and Cox (1968) for additional details.
The procedure is repeated for each of the applied loadings. 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 sizable number of such experiments can add significantly 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 methods of analyses, as indicated above, have assisted in the
development of p-y curves in some instances.
In the remaining portion of this chapter, procedures are presented for developing p-y
curves for clays and for sands. In addition, some discussion is presented for producing p-y curves
for other types of soil.
3-3 p-y Curves for Cohesive Soils
3-3-1 Initial Slope of Curves
The conceptual p-y curves in Figure 3-1 are characterized by an initial straight line from
the origin to point a. A mass of soil with an assumed linear relationship between compressive
stress and strain, Ei, for small strains can be considered. If a pile is caused to deflect a small
distance in such a soil, one can reasonably assume that principles of mechanics can be used to
find the initial slope Esi of the p-y curve. However, some difficulties are encountered in making
the computations.
For instance, the value of Ei for soil is not easily determined. Stress-strain curves from
unconfined compression tests were studied (see Figure 3-4) and it was found that the initial
modulus Ei ranged from about 40 to about 200 times the undrained shear strength c (Matlock, et
al., 1956; Reese, et al., 1968). There is a considerable amount of scatter in the ratio values. This
is probably due to the variability of the soils at the two sites that were studied. The ratios of Ei to
c would probably have been higher had an attempt been made to get precise values for the early
part of the curve. Stokoe (1989) reported that values of Ei in the order of 2,000 times c are found
routinely in resonant column tests when soil specimens are subjected to shearing strains below
0.01 percent. Johnson (1982) performed some tests with the self-boring pressuremeter and
computations with his results gave ratios of Ei to c that ranged from 1,440 to 2,840, with the
average of 1,990. These studies of the initial modulus from compressive-stress-strain curves of
clay seem to indicate that such curves are linear only over a limited range of strains.
If the assumption is made that a program of subsurface investigation and laboratory
testing can be used to obtain values of EI, the following equation for a beam of infinite length
(Vesić, 1961) can be used to gain some information on the subgrade modulus (initial slope of the
p-y curve):
54
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
4
 0.65   Ei b
E si  

 b   EpI p




1 / 12
 Ei 

 ...........................................(3-3)
2
 1  
Ei /c
0
100
200
300
0
Manor Road
Lake Austin
Depth, m
3
6
9
12
Figure 3-4 Plot of Ratio of Initial Modulus to Undrained Shear Strength
for Unconfined-compression Tests on Clay
Where:
b = pile diameter,
Ei = initial slope of stress-strain curve of soil,
Ep = modulus of elasticity of the pile, and
Ip = moment of inertia of pile, respectively, and
 = Poisson’s ratio.
While Equation 3-3 may appear to provide some useful information on the initial slope of the p-y
curves (the initial modulus of the soil in the p-y relationship), an examination of the initial slopes
of the p-y curves in Figures 3-2 and 3-3 clearly show that the initial slopes are strongly
influenced by depth below the ground surface. The initial slopes of those curves are plotted in
Figure 3-5 and the influence of depth below the ground surface is apparent.
Yegian and Wright (1973) and Thompson (1977) conducted some numerical studies
using two-dimensional finite element analyses. The plane-stress case was employed in these
studies to reflect the influence of the ground surface. Kooijman (1989) and Brown, et al. (1989)
55
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
used three-dimensional finite element analyses as a means of developing p-y curves. In addition
to developing the soil response for small deflections of a pile, all of the above investigators used
nonlinear elements in an attempt to gain information on the full range of soil response.
Initial Soil Modulus, Esi, MPa
0
200
400
600
800
0
Pile 1 Static
Depth, meters
0.6
1.2
1.8
2.4
Pile 2 (Cyclic)
3.0
Figure 3-5 Variation of Initial Modulus with Depth
Studies using finite element modeling have found the finite element method to be a
powerful tool that can supplement field-load tests as a means of producing p-y curves for
different pile dimensions, or perhaps can be used in lieu of load tests on instrumented piles if the
nonlinear behavior of the soil is well defined. However, some other problems may arise that are
unique to finite element analysis: selecting special interface elements, modeling the gapping
when the pile moves away from a clay soil (or the collapse of sand against the back of a pile),
modeling finite deformations when soil moves up at the ground surface, and modeling tensile
stresses during the iterations. Further development of general-purpose finite element software
and continuing improvements in computing hardware are likely to increase the use of the finite
element method in the future.
3-3-2 Analytical
Solutions for Ultimate Lateral Resistance
Two analyses are used to gain some insight into the ultimate lateral resistance pu that
develop near the ground surface in one case and at depth in the other case. The first analysis is
56
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
for values of ultimate lateral resistance near the ground surface and considers the resistance a
passive wedge of soil displaced by the pile. The second analysis is for values of lateral resistance
well beneath the ground surface and models the plane-strain (flow-around) behavior of the soil.
The first analytical model for clay near the ground surface is shown in Figure 3-6. Some
justification can be presented for making use of a model that assumes that the ground surface
will move upward. Contours of the measured rise of the ground surface during a lateral load test
are shown in Figure 3-7. The p-y curves for the overconsolidated clay in which the pile was
tested are shown in Figures 3-3 and 3-4. As shown in Figure 3-7(a) for a load of 596 kN (134
kips), the ground-surface moved upward out to a distance of about 4 meters (13 ft) from the axis
of the pile. After the load was removed from the pile, the ground surface subsided to the profile
as shown in Figure 3-7 (b).
y
Ft
Ft
W
Ff
x
H
Ft
Fn
Fp
Fs
Ff
W
Fp


Fn
Fs
b
(a)
(b)
Figure 3-6 Assumed Passive Wedge Failure in Clay Soils, (a) Shape of Wedge,
(b) Forces Acting on Wedge
The use of plane sliding surfaces, shown in Figure 3-6, will obviously not model the
movement that is indicated by the contours in Figure 3-7; however, a solution with the simplified
model should give some insight into the variation of the ultimate lateral resistance pu with depth.
Summing the forces in the vertical direction yields
Fn sin   W  Fs cos   2Ft cos   F f .....................................(3-4)
where
 = angle of the inclined plane with the vertical, and
W = the weight of the wedge.
57
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The expression for W is
bH 2
W 
tan  ........................................................(3-5)
2
25 mm
19 mm
3 mm
6 mm
596 kN
13 mm
(a) Heave at maximum load
3 mm
0 kN
13 mm
6 mm
(b) Residual heave
4
3
2
1
0
Scale, meters
Figure 3-7 Measured Profiles of Ground Surface Heave Near Piles Due to Static Loading,
(a) Ground Surface Heave at Maximum Load, (b) Residual Ground Surface Heave
where:
 = unit weight of soil,
b = width (diameter) of pile, and
H = depth of wedge.
The resultant shear force on the inclined plane Fs is
Fs  ca bH sec  ........................................................(3-6)
58
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
where ca is the average undrained shear strength of the clay over depth H.
The resultant shear force on a side plane is
Ft 
ca H 2
tan  ........................................................(3-7)
2
The frictional force between the wedge and the pile is
Ft   ca bH ...........................................................(3-8)
where  is a reduction factor.
The above equations are solved for Fp, and Fp is differentiated with respect to H to solve
for the soil resistance pc1 per unit length of the pile.
pc1  ca btan   (1   ) cot     bH  2ca H (tan  sin   cos  ) .................(3-9)
The value of  can be set to zero with some logic for the case of cyclic loading because
one can reason that the relative movement between pile and soil would be small under repeated
loads. The value of  can be taken as 45 degrees, if the soil is assumed to behave in an undrained
mode. With these assumptions, Equation 3-9 becomes
pc1  2ca b  bH  2.83ca H ............................................(3-10)
However, Thompson (1977) differentiated Equation 3-9 with respect to H and evaluated
the integrals numerically. His results are shown in Figure 3-8 with the assumption that the value
of the term /ca is negligible. The cases where  is assumed as zero and where  is assumed 1.0
are shown in the figure. Also shown in Figure 3-8 is a plot of Equation 3-10 with the same
assumption with respect to /ca. As shown, the differences in the plots are not great. The curve in
Figure 3-8 from Hansen (1961a, 1961b), indicated by the blue line, is discussed on page 72.
The equations developed above do not address the case of tension in the pile. If piles are
designed for a permanent uplift force, the equation for ultimate soil resistance should be
modified to reflect the effect of an uplift force at the face of the pile (Darr, et al., 1990).
The second of the two models for computing the ultimate resistance pu is shown in the
plan view in Figure 3-9(a). At some point below the ground surface, the maximum value of soil
resistance will occur with the soil moving horizontally. Movement in only one side of the pile is
indicated; but movement, of course, will be around both sides of the pile. Again, planes are
assumed for the sliding surfaces with the acceptance of some approximation in the results.
A cylindrical pile is indicated in the figure, but for ease in computation, a prismatic block
of soil is subjected to horizontal movement. Block 5 is moved laterally as shown and stress of
sufficient magnitude is generated in that block to cause failure. Stress is transmitted to Block 4
and on around the pile to Block 1, with the assumed movements indicated by the dotted lines.
Block 3 is assumed not to distort, but failure stresses develop on the sides of the block as it
slides.
59
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
pu
cb
0
5
10
15
20
25
30
0
0
2
 = 1.0
(Thompson)
pc1  2ca b  bH  2.83ca H
4
H
b
=0
(Thompson)
6
8
pu

cb
2.567  5.307
1  0.652
H
b
H
b
10
Figure 3-8 Ultimate Lateral Resistance for Clay Soils
The Mohr-Coulomb diagram for undrained, saturated clay is shown in Figure 3-9(b) and
a free body of the pile is shown in Figure 3-9(c). The ultimate soil resistance pc2 is independent
of the value of 1 because the difference in the stress on the front 6 and back 1 of the pile is
equal to 10c. The shape of the cross section of a pile will have some influence on the magnitude
of pc2; for the circular cross section, it is assumed that the resistance that is developed on each
side of the pile is equal to c (b/2), and
pc 2   6   1  c b  11c b ............................................(3-11)
Equation 3-11 is also shown plotted in Figure 3-8.
Thompson (1977) noted that Hansen (1961a, 1961b) formulated equations for computing
the ultimate resistance against a pile at the ground surface, at a moderate depth, and at a great
depth. Hansen considered the roughness of the wall of the pile, the angle of internal friction, and
unit weight of the soil. He suggested that the influence of the unit weight be neglected and
proposed the following equation for the  equals zero case for all depths.
60
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
5
2
c
4
5
6
3
1 1
1
c
5 
6
5
2
4 3 3
4
Pile Movement
2
(a)

c







2c
(b)
cb/2
6b
1b
pu
cb/2
(c)
Figure 3-9 Assumed Mode of Soil Failure Around Pile in Clay, (a) Section Through Pile,
(b) Mohr-Coulomb Diagram, (c) Forces Acting on Section of Pile
pu

cb
2.567  5.307
1  0.652
H
b
H
b .................................................(3-12)
Equation 3-12 is also shown plotted in Figure 3-8. The agreement with the “block” solutions is
satisfactory near the ground surface, but the difference becomes significant with depth.
61
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Equations 3-10 and 3-11 are similar to Equations 3-20 and 3-21, presented later, that are
used in the recommendations for two of the models of p-y curves. However, emphasis was
placed on experimental results. The values of pu obtained in the full-scale experiments were
compared to the analytical values, and empirical factors were found by which Equations 3-10
and 3-11 could be modified. The adjustment factors that were determined are shown in Figure 310 (see Section 3-3-8 for more discussion), and it can be seen that the experimental values of
ultimate resistance for overconsolidated clay below the water table were far smaller than the
computed values. The recommended method of computing the p-y curves for such clays is
presented later.
Ac and As
0
0.2
0.4
0.6
0.8
1.0
0
2
Ac
x
b
As
4
6
8
Figure 3-10 Values of Ac and As
3-3-3 Influence of Diameter on p-y Curves
The analytical developments presented to this point indicate that the term for the pile
diameter appears to the first power in the expressions for p-y curves. Reese, et al. (1975)
described tests of piles with diameters of 152 mm (6 in.) and 641 mm (24 in.) at the Manor site.
The p-y formulations developed from the results from the larger piles were used to analyze the
behavior of the smaller piles. The computation of bending moment led to good agreement
between analysis and experiment, but the computation of ground line deflection showed
considerable disagreement, with the computed deflections being smaller than the measured ones.
No explanation could be made to explain the disagreement.
O’Neill and Dunnavant (1984) and Dunnavant and O’Neill (1985) reported on tests
performed at a site where the clay was overconsolidated and where lateral-loading tests were
62
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
performed on piles with diameters of 273 mm (10.75 in.), 1,220 mm (48 in.), and 1,830 mm (72
in.). They found that the site-specific response of the soil could best be characterized by a
nonlinear function of the diameter.
There is good reason to believe that the diameter of the pile should not appear as a linear
function when piles in clays below the water table are subjected to cyclic loading. However, data
from experiments are insufficient at present to allow general recommendations to be made. The
influence of cyclic loading on p-y curves is discussed in the next section.
3-3-4 Influence of Cyclic Loading
Cyclic loading is specified in a number of the examples presented in Chapter 1; a notable
example is an offshore platform. Therefore, a number of the field tests employing fully
instrumented piles have employed cyclic loading in the experimental procedures. Cyclic loading
has invariably resulted in increased deflection and bending moment above the respective values
obtained in short-term loading. A dramatic example of the loss of soil resistance due to cyclic
loading may be seen by comparing the two sets of p-y curves in Figures 3-2 and 3-3.
Wang (1982) and Long (1984) did extensive studies of the influence of cyclic loading
on p-y curves for clays. Some of the results of those studies were reported by Reese, et al.
(1989). The following two reasons can be suggested for the reduction in soil resistance from
cyclic loading: the subjection of the clay to repeated strains of large magnitude, and scour from
the enforced flow of water near the pile. Long (1984) studied the first of these factors by
performing some triaxial tests with repeated loading using specimens from sites where piles had
been tested. The second of the effects is present when water is above the ground surface, and its
influence can be severe.
Welch and Reese (1972) report some experiments with a bored pile under repeated lateral
loading in overconsolidated clay with no free water present. During the cyclic loading, the
deflection of the pile at the ground line was in the order of 25 mm (1 in.). After a load was
released, a gap was revealed at the face of the pile where the soil had been pushed back. In
addition, cracks a few millimeters in width radiated away from the front of the pile. Had water
covered the ground surface, it is evident that water would have penetrated the gap and the cracks.
With the application of a load, the gap would have closed and the water carrying soil particles
would have been forced to the ground surface. This process was dramatically revealed during the
soil testing in overconsolidated clay at Manor (Reese, et al., 1975) and at Houston (O’Neill and
Dunnavant, 1984) .
The phenomenon of scour is illustrated in Figure 3-11. A gap has opened in the
overconsolidated clay in front of the pile and it has filled with water as load is released. With the
next cycle of loading on the pile, the water is forced upward from the space. The water exits
from the gap with turbulence and the clay is eroded from around the pile.
Wang (1982) constructed a laboratory device to investigate the scouring process. A
specimen of undisturbed soil from the site of a pile test was brought to the laboratory, placed in a
mold, and a vertical hole about 25 mm (1 in.) in diameter was cut in the specimen. A rod was
carefully fitted into the hole and hinged at its base. Water a few millimeters deep was kept over
the surface of the specimen and the rod was pushed and pulled by a machine at a given period
and a given deflection for a measured period. The soil that was scoured to the surface of the
specimen was carefully collected, dried, and weighed. The deflection was increased, and the
63
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
process was repeated. A curve was plotted showing the weight of soil that was removed as a
function of the imposed deflection. The characteristics of the curve were used to define the scour
potential of that particular clay.
Boiling and turbulence
as space closes
(a)
(b)
Figure 3-11 Development of Scour Around Pile in Clay During Cyclic Loading, (a) Profile
View, (b) Photograph of Turbulence Causing Erosion During Lateral Load Test
The device developed by Wang was far more discriminating about scour potential of a
clay than was the pinhole test (Sherard, et al., 1976), but the results of the test could not explain
fully the differences in the loss of resistance experienced at different sites where lateral-load tests
were performed in clay with water above the ground surface. At one site where the loss of
resistance due to cyclic loading was relatively small, it was observed that the clay included some
seams of sand. It was reasoned that the sand would not have been scoured readily and that
particles of sand could have partially filled the space that was developed around the pile. In this
respect, one experiment showed that pea gravel placed around a pile during cyclic loading was
effective in restoring most of the loss of resistance. However, O’Neill and Dunnavant (1984)
report that “placing concrete sand in the pile-soil gap formed during previous cyclic loading did
not produce a significant regain in lateral pile-head stiffness. “
While both Long (1984) and Wang (1982) developed considerable information about
the factors that influence the loss of resistance in clays under free water due to cyclic loading,
their work did not produce a definitive method for predicting the loss of resistance. Thus, the
analyst should be cautious when making use of the numerical results presented here with regard
to the behavior of piles in clay under cyclic loading. Full-scale experiments with instrumented
piles at a particular site are recommended for those cases where behavior under cyclic loading is
a critical design requirement.
64
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-3-5 Introduction to Procedures for p-y Curves in Clays
3-3-5-1 Early Recommendations for p-y Curves in Clay
Designers used all available information for selecting the sizes of piles to sustain lateral
loading in the period prior to the advent of instrumentation that allowed the development of p-y
curves from experiments with instrumented piles. The methods yielded values of soil modulus
that were employed principally with closed-form solutions of the differential equation. The work
of Skempton (1951) and the method proposed by Terzaghi (1955) were useful to the early
designers.
The method proposed by McClelland and Focht (1956), discussed later, appeared at the
beginning of the period when large research projects were conducted. This model is significant
because those authors were the first to present the concept of using p-y curves to model the
resistance of soil against lateral pile movement. Their paper is based on a full-scale experiment at
an offshore site where a moderate amount of instrumentation was employed.
3-3-5-2 Skempton (1951)
Skempton (1951) stated that “simple theoretical considerations” were employed to
develop a prediction model for load-settlement curves. The theory can be also used to obtain p-y
curves if it is assumed that the ground surface does not affect the results, that the state of stress is
the same in the horizontal and vertical directions, and that the stress-strain behavior of the soil is
isotopic.
The mean settlement, , of a foundation of width b on the surface of a semi-infinite
elastic solid is given by Equation 3-13.
  qbI 
I  2
......................................................(3-13)
E
where:
q = foundation pressure,
I = influence coefficient,
 = Poisson’s ratio of the solid, and
E = Young’s modulus of the solid.
In Equation 3-13, the value of Poisson’s ratio can be assumed to be 0.5 for saturated clays
if there is no change in water content, and I can be taken as /4 for a rigid circular footing on
the surface. Furthermore, for a rigid circular footing, the failure stress qf may be taken as equal
6.8 c, where c is the undrained shear strength. Making the substitutions indicated and setting  =
1 for the particular case
1
b

4c q
..........................................................(3-14)
E qf
65
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Skempton noted that the influence value I decreases with depth below the ground surface and
the bearing capacity factor increases; therefore, as a first approximation Equation 3-14 is valid at
any depth.
In an undrained compression test, the axial strain is given by

 1   3    
E
E
....................................................(3-15)
Where E is Young’s modulus at the principal stress difference of  1   3  .
For saturated clays with no change in water content, Equation 3-15 may be rewritten as

2c  1   3 
.................................................... (3-16)
E  1   3  f
Where  1   3  f is the principal stress difference at failure.
Equations 3-14 and 3-16 show 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.
1
b
 2
The equation above can be rearranged as
1  2  b .......................................................... (3-17)
Skempton’s reasoning was based on the theory of elasticity and on the actual behavior of
full-scale foundations, led to the following conclusion:
“Thus, to a degree of approximation (20 percent) comparable with the accuracy of the
assumptions, it may be taken that Equation 3-17 applies to a circular or any rectangular
footing.”
Skempton stated that the failure stress for a footing reaches a maximum value of 9c. If one
assumes the same value for 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
Equation 3-17 to obtain deflection and 4.5  b to obtain soil resistance. The procedure would
presumably be valid at depths beyond where the presence of the ground surface would not
reduce the soil resistance.
Skempton presented information about laboratory stress-strain curves to indicate that 50,
the strain corresponding to a stress of 50 percent of the ultimate stress, ranges from about 0.005
to 0.02. That information, and information about the general shape of a stress-strain curve,
allows an approximate curve to be developed if only the strength of the soil is available.
66
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-3-5-3 Terzaghi (1955)
In a widely referenced paper, Terzaghi discussed several important aspects of subgrade
reaction, including the resistance of soil to lateral loading of a pile. Unfortunately, while his
numerical recommendations reveal that his knowledge of the problem of the pile was extensive,
Terzaghi did not present experimental data or analytical procedures to validate his
recommendations.
Terzaghi’s recommendations for the coefficient of subgrade reaction for piles in stiff clay
were based on a concept that the deformational characteristics of stiff clay are “more or less
independent of depth.” Consequently, he proposed that p-y curves should be constant with depth
and that the ratio between p and y should be defined by a constant T. Therefore, his family of py curves (though not defined in such these terms) consisted of a series of straight lines, all with
the same slope, and passing through the origin of the p-y coordinate system.
Terzaghi recognized, of course, that the pile could not be deflected to an unlimited extent
with a linear increase in soil resistance and that a lateral bearing capacity exists for laterally
loaded piles. Terzaghi stated that the linear relationship between p and y was limited to values of
p that were smaller than about one-half of the maximum lateral load-transfer capacity.
Table 3-1 presents Terzaghi’s recommendations for stiff clay. The units have been
changed to reflect current practice. These values of T are independent of pile diameter, which is
consistent with theory for small deflections.
Table 3-1 Terzaghi’s Recommendations for Soil Modulus for Laterally Loaded Piles in Stiff
Clay (no longer recommended)
Consistency of Clay
Stiff
Very Stiff
Hard
qu, kPa
100-200
200-400
> 400
qu, tsf
1-2
2-4
>4
Soil Modulus, T, MPa
3.2-6.4
6.4-12.8
> 12.8
Soil Modulus, T, psi
460-925
925-1,850
> 1,850
3-3-5-4 McClelland and Focht (1956)
McClelland and Focht (1956) wrote the first paper that described the concept of nonlinear
lateral load-transfer curves, now referred to as p-y curves. In this paper, they presented the first
nonlinear p-y curves derived from a full-scale, instrumented, pile-load test. Significantly, this
paper shows conclusively that lateral load transfer is a function of lateral pile deflection and
depth below the ground surface, as well as of soil properties.
McClelland and Focht recommended testing of soil using consolidated-undrained triaxial
tests with the confining pressure set equal to the overburden pressure. The results of the shear
test should be plotted as the compressive stress difference,  , versus the axial compressive
strain, . The p-values of the p-y curve are then scaled from the stress-strain curve using
p  5.5 b   ......................................................... (3-18)
67
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
and the values of pile deflection y are scaled using
y  0.5  b .......................................................... (3-19)
These equations are similar in form to those developed by Skempton, but the factors used for
lateral defection are different (0.5 used by McClelland and Focht and 2 used by Skempton).
3-3-6 Procedures for Computing p-y Curves in Clay
Five procedures are provided for computing p-y curves for clay. Each procedure is based
on the analysis of the results of experiments using full-scale instrumented piles. In every case, a
comprehensive soil investigation was performed at each load test site and the best estimate of the
undrained shear strength of the clay was found. In addition, the physical dimensions and bending
stiffness of the piles were accurately evaluated. Experimental p-y curves were obtained by one or
more of the techniques described earlier. Euler-Bernoulli beam theory was used and
mathematical expressions were developed for p-y curves for use in a computer analysis to obtain
values of lateral pile deflection and bending moment versus depth that agreed well with the
experimental values.
Loadings in all load tests were both short-term (static) and cyclic. The p-y curves that
resulted from the two tests performed with water above the ground surface have been used
extensively in the design of offshore structures around the world.
3-3-7 Response of Soft Clay in the Presence of Free Water
3-3-7-1 Background
Matlock (1970) performed lateral-load tests with an instrumented steel-pipe pile that was
324 mm (12.75 in.) in diameter and 12.8 meters (42 ft) long. The test pile was driven into clay
near Lake Austin, Texas that had an average shear strength of about 38 kPa (800 psf). The test
pile was exhumed after the first test and taken to Sabine Pass, Texas, and driven into soft clay
with a shear strength that averaged about 14.4 kPa (300 psf) in the significant upper zone.
The initial loading was short-term. The load was applied to the pile long enough for
readings of strain gages to be taken by an extremely precise device. A rough balance of the
external Wheatstone bridge was obtained by use of a precision decade box and the final balance
was taken by rotating a 150-mm-diameter drum on which a copper wire had been wound. A
contact on the copper wire was read on the calibrated drum when a final balance was achieved.
The accuracy of the strain readings was less than one microstrain, but some time was required to
obtain readings sequentially from the top of the pile to the bottom and back up to the top again.
The pressure in the hydraulic ram that controlled the load was adjusted as necessary to maintain
a constant load because of the creep of the soil under the imposed loading. The two sets of
readings at each point along the pile were interpolated with time to find the value at a selected
time, assuming that the change in moment due to creep had a constant rate.
The accurate readings of bending moment allowed the soil resistance to be found by
numerical differentiation, which was a distinct advantage. The disadvantage was the somewhat
indeterminate influence of the creep of the soft clay.
The test pile was extracted, re-driven, and tested a second type with cyclic loading.
Readings of the strain gages were taken under constant load after specified numbers of cycles of
68
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
loading. The load was applied in two directions, with the load in the forward direction being
more than twice as large as the load in the backward direction. After a significant number of
cycles, the deflection at the top of the pile was either stable or creeping slowly, so an equilibrium
condition was assumed. The p-y curves for cyclic loading are intended to represent a lowerbound condition. Thus, a designer might possibly be computing an overly conservative response
of a pile, if the cyclic p-y curves are used and if there are only a small number of applications of
the design load (the factored load).
3-3-7-2 Procedure for Computing p-y Curves in Soft Clay for Static Loading
The following procedure is for short-term static loading and is illustrated by Figure 312(a). As noted earlier, the curves for static loading constitute the basis for indicating the
influence of cyclic loading and would be rarely used in design if cyclic loading is of concern.
1.
Obtain the best possible estimates of the variation of undrained shear strength c and
effective unit weight with depth. Also, obtain the value of 50, the strain corresponding to
one-half the maximum principal stress difference. If no stress-strain curves are available,
typical values of 50 are given in Table 3-2.
Table 3-2 Representative Values of 50 for Soft to Stiff Clays
2.
Consistency of Clay
50
Soft
0.020
Medium
0.010
Stiff
0.005
Compute the ultimate soil resistance per unit length of pile, using the smaller of the values
given by the equations below.

  avg
J 
pu  3 
x  x  cb .............................................. (3-20)
c
b 

pu  9 c b .......................................................... (3-21)
where
 = average effective unit weight from ground surface to p-y curve,1
 avg
x = depth from the ground surface to p-y curve,
c = shear strength at depth x, and
b = width of pile.
1
Matlock did not specify in his original paper whether the unit weight was total unit weight or
effective unit weight. However, API RP2A specifies that effective unit weight be used. Most
users have adopted the recommendation by API and this is the implementation chosen for LPile.
69
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
1
1
p
0.5
pu
0
 p 
 y 3
   0.5

 pu 
 y50 
0 1
y
y50
8.0
(a)
p
pu
1
For x  xn (Depth where
Flow AroundFailureGoverns
0.72
0.5
0.72
0
0
3
1
y
y50
x
xr
15
(b)
Figure 3-12 p-y Curves in Soft Clay,(a) Static Loading, (b) Cyclic Loading
Matlock (1970) stated that the value of J was determined experimentally to be 0.5 for soft
clay and about 0.25 for a medium clay. A value of 0.5 is frequently used for J for offshore
soils in the Gulf of Mexico. The value of pu is computed at each depth where a p-y curve is
desired, based on shear strength at that depth.
Equations 3-20 and 3-21 are solved simultaneously to find the transition depth, xr, where
the transition in definition of pu by Equation 3-20 to 3-21 occurs. In general, the minimum
value of xr should be 2.5 pile diameters (see API RP2A, 2010, Section 6.8.2). If the unit
weight and shear strength are constant in the soil layer, then xr is computed using
70
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
xr 
6cb
 2.5 ................................................... (3-22)
  b  Jc
LPile has two versions of the soft clay criteria. One version uses a value of J equal to 0.5
by default. This is the version used by most users. The second version is identical in
computations as the first, but the user may enter the value of J at the top and bottom of the
soil layer. LPile does not perform error checking on the input value of J. If the p-y curve
with variable J (API soft clay with user-defined J) is selected, the user should consider the
advice by Matlock for selecting the J value discussed on page 82.
The net effect of using a J value less than 0.5 is to reduce the strength of the p-y curve. An
example of the effect of J on a p-y curve at a depth of 5 feet for a 36-inch diameter pile in
soft clay with c = 1,000 psf and  = 55 pcf is shown in Figure 3-13.
1,200
1,000
p, lbs/inch
800
600
400
J = 0.5
J = 0.25
200
0
0
1
2
3
4
5
6
7
8
y, inches
Figure 3-13 Example p-y Curves in Soft Clay Showing Effect of J
3.
Compute deflection at one-half the ultimate soil resistance, y50, from the following
equation:
y50 = 2.5 50b ....................................................... (3-23)
4.
Compute points describing the p-y curve from the origin up to 8 y50 using
1
p
p u
2
 y 

 ...................................................... (3-24)
y
 50 
71
3
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The value of p remains constant for y values beyond 8 y50.
3-3-7-3 Procedure for Computing p-y Curves in Soft Clay for Cyclic Loading
The following procedure is for cyclic loading and is illustrated in Figure 3-12(b). As
noted earlier in this chapter, the presence of free water at the ground surface has a significant
influence on the behavior of a pile in clay under cyclic loading. If the clay is soft, the assumption
can be made that there is free water, otherwise the clay would have dried and become stiffer. A
question arises whether or not to use these recommendations if a thin stratum of stiff clay is
present above the soft clay and the water table is at the interface of the soft and the stiff clay. In
such a case, free water is unlikely to be ejected to the ground surface and erosion around the pile
due to scour would not occur. However, the free water in the excavation, under repeated
excursions of the pile, could cause softening of the clay. Therefore, the following
recommendations for p-y curves for cyclic loading can be used with the recognition that there
may be some conservatism in the results.
1.
Construct the p-y curve in the same manner as for short-term static loading for values of p
less than 0.72pu. For lateral displacements in this range, there is not significant degradation
of the p-y curve during cyclic loading.
2.
If the depth to the p-y curve is greater than or equal to xr (Equation 3-22), select p as 0.72pu
for y equal to 3y50 (Note that the number 0.72 is computed using Equation 3-24 as 1/2 *
31/3 = 0.721124785 ~ 0.72).
3.
If the depth of the p-y curve is less than xr, note that the value of p decreases from 0.72pu at
y = 3y50 down to the value given by Equation 3-25 at y = 15y50.
 x
p  0.72 pu 
 xr

 ..................................................... (3-25)

The value of p remains constant beyond y = 15y50.
3-3-7-4 Recommended Soil Tests for Soft Clays
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.
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 one-half the total maximum principalstress difference,
3. Miniature vane tests of samples in tubes, and
4. Unconfined compression tests. Tests must also be performed on the soil samples to
determine the total unit weight of the soil, water content, and effective unit weight.
3-3-7-5 Examples
An example set of p-y curves was computed for soft clay for a pile with a diameter of 610
mm (24 in.). The soil profile that was used is shown in Figure 3-14. The submerged unit weight
was 6.3 kN/m3 (40 pcf). In the absence of a stress-strain curve for the soil, 50 was taken as 0.02
72
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
for the full depth of the soil profile. The loading was assumed to be static. The p-y curves were
computed for the following depths below the ground surface: 1.5 m (5 ft), 3 m (10 ft), 6 m (20
ft), and 12 m (40 ft). The plotted curves are shown in Figure 3-15.
0
2
Depth, meters
4
6
8
10
12
14
16
0
10
20
30
40
50
Shear Strength, kPa
Figure 3-14 Shear Strength Profile Used for Example p-y Curves for Soft Clay
250
Load Intensity p, kN/m
200
150
Depth = 2.00 m
Depth = 3.00 m
Depth = 6.00 m
Depth = 12.00 m
100
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Lateral Deflection y, meters
Figure 3-15 Example p-y Curves for Soft Clay with the Presence of Free Water
73
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-3-8 Response of Stiff Clay in the Presence of Free Water
3-3-8-1 Background
Reese, Cox, and Koop (1975) performed lateral-load tests with instrumented steel pipe
piles that were 641 mm (24 in.) in diameter and 15.2 m (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 96 kPa (1 tsf) at the ground surface to about 290 kPa (3 tsf) at a depth of 3.7 m (12 ft).
The loading of the pile was applied in a similar manner to that described for the tests
performed by Matlock (1970). A significant difference was that a data-acquisition system was
employed that allowed a full set of readings of the strain gages to be taken in about a minute.
Thus, the creep of the piles under sustained loading was small or negligible. The disadvantage of
the system was that the accuracy of the curves of bending moment was such that curve fitting
was necessary in doing the differentiations.
As in the case of the Matlock recommendations for cyclic loading, the lower-bound pile
response is presented. Cyclic loading was continued until the lateral pile deflection and bending
moments appeared to stabilize. The number of cycles of loading was in the order of 100; and 500
cycles were applied later in a reloading test. O’Neill and Dunnavant (1984) reported that an
equilibrium condition was not reached during cyclic loading of piles at the Houston site. It is
likely that the same result would have been found at the Manor test site. However, the l00 cycles
of loading that were applied at Manor at a load at which the pile was near its ultimate bending
moment and the loading was more than would be expected during an offshore storm or under
other types of repeated loading.
The diameter appears to the first power in the equations for p-y curves for cyclic loading.
However, based on lateral tests performed later on piles of larger diameter, there is reason to
believe that a nonlinear relationship for diameter may be required for piles of greater diameter.
During the load test with cyclic loading, an annular gap developed between the soil and
the pile after deflection at the ground surface of perhaps 10 mm (0.4 in.) and scour of the soil at
the face of the pile due to ejected water, as shown in Figure 3-11, began at that time. The open
gap remained at the conclusion of the load test. A photograph showing the annular gap is shown
in Figure 3-16.
There is reason to believe that scour would be initiated in overconsolidated clays after a
given deflection at the mudline rather than at a given fraction of the pile diameter, as indicated
by the following recommendations. However, insufficient data are available at present to justify
a change in the recommended procedures. However, engineers could recommend a field test at a
particular site in recognition of some uncertainty regarding the influence of scour on p-y curves
for overconsolidated clays.
3-3-8-2 Recommendations for Use
A frequent question posed by engineers when selecting p-y criteria for stiff cohesive soils
is “Under what conditions should the criteria for stiff clay in the presence of free water be used?”
There is no definitive answer to this question, but general recommendations can be made to
guide the user.
74
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Erosion
Figure 3-16 Annular Gapping Developed Around Pile After Cyclic Loading
The user of LPile should consider several factors, including the position and depth of the
layer in the soil profile, access of free water to the stiff clay from the surface or adjacent or
interbedded water-bearing sand layers, and the presence of fissuring in the clay.
The position of the stiff clay in the soil profile is important. If the depth range of the stiff
soil is in the upper portion is below a depth equal to 10 to 12 piles diameters below the ground
surface, the lateral pile deflection is highly likely to be too small for an opening to develop
around the pile. In this case, the p-y model for stiff clay without free water should be chosen.
If the stiff clay layer is within the depth range of 10 to 12 pile diameters of the surface
and inflow of free water is possible from surface water, a high water table, or water-bearing sand
layers adjacent to the stiff clay. In such conditions, the development of an annulus around the
pile due to erosion of soil from around the pile during cyclic loading is more likely to occur. In
such conditions, the p-y model for stiff clay with free water should be chosen.
If the soil is highly fissured and has access to free water, the presence of fissuring will
contribute to the degradation of the lateral load-transfer from the pile to the soil. As such, the
presence of fissuring should encourage the selection of the p-y model for stiff clay with free
water.
If the soil is not fissured and is largely intact and dense, the development of erosion from
around the pile is much less likely.
Another important factor to consider is the possible presence of a clean sand layer (i.e.
sand without fines) above the stiff clay layer. If clean sand is present, it may be possible for some
sand from the layers above to fill any gap that develops around the pile in the stiff clay layer,
75
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
thereby counter-acting some of the negative effects of erosion. If this condition is present, a
reasonable choice would be to select the p-y model for stiff clay without free water.
3-3-8-3 Procedure for Computing p-y Curves for Static Loading
The following procedure is for computing p-y curves in stiff clay with free water for
short-term static loading and is illustrated by Figure 3-17. As before, these curves form the basis
for evaluating the effect of cyclic loading, and they may be used for sustained loading in some
circumstances.
1.
Obtain values of undrained shear strength c, effective unit weight , and pile diameter b at
depth x.
2.
Compute the average undrained shear strength ca over the depth x.
3.
Compute the soil resistance per unit length of pile, pc, using the smaller value of pct or pcd
computed using Equations 3-26 and 3-27.
pct  2ca b    bx  2.83ca x ............................................ (3-26)
pcd  11cb ........................................................ (3-27)
4.
Choose the appropriate value of As from Figure 3-10 on page 74 for modifying pct and pcd
and for shaping the p-y curves or compute As as a function of x/b using
As  0.2  0.4 tanh0.62 x / b ............................................ (3-28)
p
p
pc
2
y
y50
1.25
 y  As y50 

poffset  0.055 pc 
A
y
s
50


0.5pc
Ess  
y50   50b
0.0625 pc
y50
Esi  k s x
0
y50
As y50
6y50
18y50
y
Figure 3-17 Characteristic Shape of p-y Curves for Static Loading in Stiff Clay with Free Water
76
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
5.
Establish the initial linear portion of the p-y curve, using the appropriate value of ks for
static loading or kc for cyclic loading from Table 3-3 for k.
p  (kx) y ......................................................... (3-29)
Table 3-3 Representative Values of k for Stiff Clays
Average Undrained
Shear Strength*
ks (static loading)
kc (cyclic loading)
50-100 kPa
(1,000-2,000 psf)
135 MN/m3
(500 pci)
55 MN/m3
(200 pci)
100-200 kPa
(2,000-4,000 psf)
270 MN/m3
(1,000 pci)
110 MN/m3
(400 pci)
200-400 kPa
(4,000-6,000 psf)
540 MN/m3
(2,000 pci)
220 MN/m3
(800 pci)
*The average shear strength should be computed as the average of shear strength of the soil from the ground surface to a
depth of 5 pile diameters. It should be defined as one-half the maximum principal stress difference in an unconsolidatedundrained triaxial test. Note: Conversions of stress ranges are approximate in this table.
6. Compute y50 as
y50   50b .......................................................... (3-30)
using an appropriate value of 50 from results of laboratory tests or, in the absence of
laboratory tests, from Table 3-4. Note that the strain values of 50 are dimensionless.
Table 3-4 Representative Values of 50 for Stiff to Hard Clays
7.
Average Undrained
Shear Strength
50
50-100 kPa
(1,000-2,000 psf)
0.007
100-200 kPa
(2,000-4,000 psf)
0.005
200-400 kPa
(4,000-6,000 psf)
0.004
Compute the first parabolic portion of the p-y curve using the following equation. The
value of pc is computed using the smaller of the two values computed using Equations 326 for shallow wedge failure conditions or Equation 3-27 for deep flow-around failure
conditions.
77
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
 y 

p  0.5 pc 
y
 50 
0.5
.................................................... (3-31)
Equation 3-31 should define the portion of the p-y curve from the point of the intersection
with Equation 3-29 to a point where y is equal to Asy50 (see note in Step 10).
8.
Establish the second parabolic portion of the p-y curve,
 y 

p  0.5 pc 
 y50 
0.5
 y  As y50 

 0.055 pc 
 As y50 
1.25
............................... (3-32)
Equation 3-32 should define the portion of the p-y curve from the point where y is equal to
Asy50 to a point where y is equal to 6Asy50 (see note in Step 10).
9.
Establish the next straight-line portion of the p-y curve,
p  0.5 pc 6 As  0.411 pc 
0.0625
pc  y  6 As y 50  ........................ (3-33)
y 50
Equation 3-33 should define the portion of the p-y curve 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  0.5 pc 6 As  0.411 pc  0.75 pc As ................................... (3-34)
or


p  pc 1.225 As  0.75 As  0.411 ...................................... (3-35)
Equation 3-34 should define the portion of the p-y curve from the point where y is equal to
18Asy50 and for all larger values of y, see the following note.
Note: The p-y curve shown in Figure 3-17 is drawn, as if there is an intersection
between Equation 3-29 and 3-31. However, for small values of k there may be no
intersection of Equation 3-29 with any of the other equations defining the p-y curve.
Equation 3-29 defines the p-y curve until it intersects with one of the other equations
or, if no intersection occurs, Equation 3-29 defines the full p-y curve.
3-3-8-4 Procedure for Computing p-y Curves for Cyclic Loading
A second pile, identical in dimensions to the pile used for the static loading, was tested
under cyclic loading conditions. The following p-y computation procedure is for cyclic loading
conditions and its form is illustrated in Figure 3-18. As may be seen from an examination of the
p-y curves that are recommended, the results of load tests performed at the Manor site showed a
very large loss of soil resistance. The data from the tests have been studied carefully and the
recommended p-y curves for cyclic loading accurately reflect the behavior of the soil present at
the site. Nevertheless, the loss of resistance due to cyclic loading for the soils at Manor is much
78
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
more than has been observed elsewhere. Therefore, the use of the recommendations in this
section for cyclic loading will yield conservative results for many clays. Long (1984) was
unable to show precisely why the loss of resistance occurred during cyclic loading. One
observation was that the clay from Manor was found to lose volume by slaking when a specimen
was placed in fresh water. Thus, the clay at the site of the load test was quite susceptible to
erosion from the hydraulic action of the free water flushing from the annular gap around the pile
as the pile was pushed back and forth during cyclic loading.
p

y  0.45 y p

p  Ac pc 1 
0.45 y p


2.5 


 Esi  kc x
A c pc
Esc  
0.085 pc
y50
y p  4.1As y50
Esi  kc x
0
y50   50b
0.45yp 0.6yp
1.8yp
y
Figure 3-18 Characteristic Shape of p-y Curves for Cyclic Loading of Stiff Clay with Free Water
1. Obtain values of undrained shear strength c, effective unit weight , and pile diameter b.
2. Compute the average undrained shear strength ca over the depth x.
3. Compute the soil resistance per unit length of pile, pc, using the smaller of the pct or pcd from
Equations 3-26 and 3-27.
4. Choose the appropriate value of Ac from Figure 3-10 on page 74 or compute Ac as a function
of x/b using
Ac  0.2  0.1tanh1.5x / b ............................................. (3-36)
5. Compute yp using
y p  4.1Ac y50 ....................................................... (3-37)
6. Establish the initial linear portion of the p-y curve, using the appropriate value of ks for static
loading or kc for cyclic loading from Table 3-3 for k. and compute p using Equation 3-29.
79
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
7. Compute y50 using Equation 3-30.
8. Establish the parabolic portion of the p-y curve,

y  0.45 y p
p  Ac pc 1 
0.45 y p


2.5

 ........................................... (3-38)


Equation 3-38 should define the portion of the p-y curve from the point of the intersection
with Equation 3-29 to where y is equal to 0.6yp (see note in step 9).
8. Establish the next straight-line portion of the p-y curve,
p  0.936 Ac pc 
0.085
pc ( y  0.6 y p ) ................................... (3-39)
y 50
Equation 3-39 should define the portion of the p-y curve from the point where y is equal to
0.6yp to the point where y is equal to 1.8yp (see note on Step 9).
9. Establish the final straight-line portion of the p-y curve,
p  0.936 Ac pc 
0.102
pc y p ........................................... (3-40)
y 50
Equation 3-40 defines the p-y curve from the point where y equals 1.8yp and all larger values
of y (see following note).
Note: Figure 3-18 is drawn, as if there is an intersection between Equation 3-29 and
Equation 3-38. There may be no intersection of Equation 3-29 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.
3-3-8-5 Recommended Soil Tests for Stiff Clays in the Presence of Free Water
Triaxial compression tests of the unconsolidated-undrained type with confining pressures
equal to in-situ total stresses 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 onehalf the maximum total-principal-stress difference. The shear strength, c, should be interpreted as
one-half of the maximum total-principal-stress difference. Values obtained from 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.
3-3-8-6 Examples
Example p-y curves were computed for stiff clay for a pile with a diameter of 610 mm
(24 in.). The soil profile that was used is shown in Figure 3-19. The submerged unit weight of
the soil was 7.9 kN/m3 (50 pcf) over the full depth.
In the absence of a stress-strain curve, 50 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 135 MN/m3 (500 pci). The loading was assumed as cyclic. The p-y curves were computed for
80
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
the following depths below the ground surface: 0.6 m (0.2 ft), 1.5 m (5 ft), 3 m (10 ft), and 12 m
(40 ft). The plotted curves are shown in Figure 3-20.
0
2
Depth, meters
4
6
8
10
12
14
16
0
50
100
150
200
Shear Strength, kPa
Figure 3-19 Example Shear Strength Profile for p-y Curves for Stiff Clay with No Free Water
250
Depth = 1.00 m
Depth = 2.00 m
Depth = 3.00 m
Depth = 12.00 m
Load Intensity p, kN/m
200
150
100
50
0
0.0
0.005
0.01
0.015
0.02
0.025
Lateral Deflection y, meters
0.03
0.035
Figure 3-20 Example p-y Curves for Stiff Clay in Presence of Free Water for Cyclic Loading
81
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-3-9 Response of Stiff Clay with No Free Water
3-3-9-1 Background
A lateral-load test was performed at a site in Houston, Texas on a drilled shaft (bored
pile), with a diameter of 915 mm (36 in.). A 254-mm (10 in)-diameter steel pipe instrumented
with strain gages was positioned at the central axis of the pile before concrete was placed. The
embedded length of the pile was 12.8 m (42 ft). The average undrained shear strength of the clay
in the upper 6 m (20 ft) was approximately 105 kPa (2,200 psf). The experiments and their
interpretation were reported in the papers by Welch and Reese (1972) and by Reese and Welch
(1975).
The same experimental setup was used to develop both the static and the cyclic p-y
curves, contrary to the procedures employed for the two other experiments with piles in clays.
The load was applied in only one direction rather than in two directions, also in variance with the
other experiments.
A load was applied and maintained until the strain gages were read with a high-speed
data-acquisition system. The same load was then cycled for a number of times and held constant
while the strain gages were read at specific numbers of cycles of loading. The load was then
increased and the procedure was repeated. The difference in the magnitude of successive loads
was relatively large and the assumption was made that cycling at the previous load did not
influence the readings for the first cycle at the new higher load.
The p-y curves obtained for these load tests were relatively consistent in shape and
showed the increase in lateral deflection during cyclic loading. This permitted the expressions of
lateral deflection to be formulated in terms of the stress level and the number of cycles of
loading. Thus, the engineer can specify a number of cycles of loading (up to a maximum of
5,000 cycles of loading) in doing the computations for a particular design.
3-3-9-2 Procedure for Computing p-y Curves for Stiff Clay without Free Water for Static
Loading
The following procedure is for short-term static loading and the p-y curve for stiff clay
without free water is illustrated in Figure 3-21.
1.
Obtain values for undrained shear strength c, effective unit weight , and pile diameter b.
Also, obtain the values of 50 from stress-strain curves. If no stress-strain curves are
available, use a value of 50 of 0.010 or 0.005 as given in Table 3-2, the larger value being
more conservative.
2.
Compute the ultimate soil resistance, pu, per unit length of pile using the smaller of the
values given by Equation 3-20 or Equation 3-21. Note that in the use of Equation 3-20, the
shear strength is taken as the average between the ground surface and the depth being
considered, J is taken as 0.5, and the average effective unit weight of the soil should reflect
the position of the water table.
82
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
p
p  pu
 y 
p

 0.5

pu
 y50 
1
4
y
16y50
Figure 3-21 Characteristic Shape of p-y Curve for Static Loading in Stiff Clay without Free
Water


  avg
  avg
J 
0.5 
pu  3 
x  x  cb  3 
x
x  cb .............................(3-20)
b 
b 
 cavg
 cavg
pu  9 c b ...........................................................(3-21)
3.
Compute the deflection, y50, at one-half the ultimate soil resistance using Equation 3-23.
y50  2.5 50b ........................................................ (3-23)
4.
Compute points describing the p-y curve from the relationships below.
p  y 

p  u 
2  y50 
0.25
..................................................... (3-41)
4
  p 
y  y50 2  ....................................................(3-42)
  pu  
5.
Beyond y = 16y50, p is equal to pu for all values of y.
3-3-9-3 Procedure for Computing p-y Curves for Stiff Clay without Free Water for Cyclic
Loading
The following procedure is for cyclic loading and the p-y curve for stiff clay without free
water is illustrated in Figure 3-22.
83
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
pu
N1
N3
N2
yc = ys + y50 C log N3
yc = ys + y50 C log N2
yc = ys + y50 C log N1
16y50+9.6(y50)logN1
y
16y50+9.6(y50)logN3
16y50+9.6(y50)logN2
Figure 3-22 Characteristic Shape of p-y Curves for Cyclic Loading in Stiff Clay with No Free
Water
1.
Determine the p-y curve for short-term static loading by the procedure previously given.
2.
Determine the number of times the lateral load will be applied to the pile.
3.
Obtain the value of C for several values of p/pu, where C is the parameter describing the
effect of repeated loading on deformation. The value of C is found from a relationship
developed by laboratory tests, (Welch and Reese, 1972), or in the absence of tests, from
4
 p 
 ....................................................... (3-43)
C  9.6
 pu 
4.
At the value of p corresponding to the values of p/pu selected in Step 3, compute new
values of y for cyclic loading from
yc  y s  y50C log N ................................................. (3-44)
Cyclic deflection is computed from the ratio p/pu using
4
  p 4 
  p 
yc  y50 2   y50 9.6    log N ................................(3-45)
  pu  
  pu 
where:
yc = deflection under N-cycles of loading,
ys = deflection under short-term static loading,
84
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
y50 = deflection under short-term static loading at one-half the ultimate resistance
computed using Equation 3-23, and
N = number of cycles of loading.
The net effect of cyclic loading is to expand the p-y curve in the y-direction. The ratio of
expansion is defined as the ratio of cyclic deflection over static deflection for an equal ratio of
p/pu. The expansion ratio as a function of number of cycles of loading is shown in the figure
below. As an example, the width of a p-y curve for 2,000 cycles of loading is approximately
three times “wider” than the static p-y curve.
3.5
Ratio of Expansion
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1
10
100
1000
10000
Cycles of Loading
Figure 3-23 Ratio of Expansion versus Number of Cycles of Loading for Stiff Clay without Free
Water
3-3-9-4 Recommended Soil Tests for Stiff Clays
Triaxial compression tests of the unconsolidated-undrained type with confining stresses
equal to the overburden pressures at the elevations from which the samples were taken are
recommended to determine the shear strength. The value of 50 should be taken as the strain
during the test corresponding to the stress equal to one-half the maximum total-principal-stress
difference. The undrained shear strength, c, should be defined as one-half the maximum totalprincipal-stress difference. The unit weight of the soil must also be determined.
3-3-9-5 Examples
An example set of p-y curves was computed for stiff clay above the water table for a pile
with a diameter of 610 millimeters (24 in.). The soil profile that was used is shown in Figure 385
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
19. A unit weight for soil of 19.0 kN/m3 (125 pcf) was assumed for the entire soil profile. In the
absence of a stress-strain curve, 50 was taken as 0.005. Equation 3-43 was used to compute
values for the parameter C and it was assumed that there were to be 100 cycles of loading.
The p-y curves were computed for the following depths below the ground line: 0.6 m (2
ft), 1.5 m (5 ft), 3 m (10 ft), and 12 meters (40 feet). The plotted curves are shown in Figure 324.
400
Load Intensity p, kN/m
300
200
Depth = 0.60 m
Depth = 1.50 m
Depth = 3.00 m
Depth = 12.00 m
100
0
0.0
0.05
0.1
0.15
0.2
Lateral Deflection y, meters
0.25
0.3
Figure 3-24 Example p-y Curves for Stiff Clay with No Free Water,
Cyclic Loading
3-3-10 Modified p-y Formulation for Stiff Clay with No Free Water
The p-y formulation for stiff clay with no free water was described in Section 3-3-9. The
p-y curve for stiff clay with no free water is based on Equation 3-41, which does not contain an
initial stiffness parameter k. Although the formulation for stiff clay without free water has been
used successfully for many years, there have been cases reported from the Southeastern United
States where load tests on full-size piles have found that the initial slope of the load-deformation
response modeled using the original formulation is too stiff.
The ultimate load-transfer resistance pu used in the p-y formulation is consistent with the
theory of plasticity and has also correlated well with the results of load tests. However, the soil
resistance at small deflections is influenced by factors such as soil moisture content, clay
mineralogy, clay structure, possible desiccation, and pile diameter. Brown (2002) has
recommended using a field-calibrated k value to modify the initial portion of the p-y curves if
one has the results of lateral load test for calibration of the initial stiffness k. Judicious use of this
modified p-y formulation enables one to obtain improved predictions with experimental readings
that may be used later for design computations.
86
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The user may select an initial stiffness k based on Table 3-3 on page 89 or from a sitespecific lateral load test. LPile will use the lower of the p values computed using Equation 3-29
or Equation 3-41 for pile response as a function of lateral pile displacement.
One drawback of the modified p-y formulation for stiff clay with no free water is that pvalues for a p-y curve computed at the ground surface will always be zero. This is not the case
for the unmodified formulation.
3-3-11 Other Recommendations for p-y Curves in Clays
As noted earlier in this chapter, the selection of the set of p-y curves for a particular field
application is a critical feature of the method of analysis. The presentation of three particular
methods for clays does not mean the other recommendations are not worthy of consideration.
Some of these methods are mentioned here for consideration and their existence is an indication
of the level of activity with regard to the response of soil to lateral deflection.
Sullivan, et al. (1980) studied data from tests of piles in clay when water was above the
ground surface and proposed a procedure that unified the results from those tests. While the
proposed method was able to predict the behavior of the experimental piles with excellent
accuracy, two parameters were included in the method that could not be found by any rational
procedures. Further work could develop means of determining those two parameters.
Stevens and Audibert (1979) reexamined the available experimental data and suggested
specific procedures for formulating p-y curves. Bhushan, et al. (1979) described field tests on
drilled shafts under lateral load and recommended procedures for formulating p-y curves for stiff
clays. Briaud, et al. (1982) suggested a procedure for use of the pressuremeter in developing p-y
curves. A number of other authors have also presented proposals for the use of results of
pressuremeter tests for obtaining p-y curves.
O’Neill and Gazioglu (1984) reviewed all of the data that were available on p-y curves
for clay and presented a summary report to the American Petroleum Institute. The research
conducted by O’Neill and his co-workers (O’Neill and Dunnavant, 1984; Dunnavant and
O’Neill, 1985) at the test site on the campus of the University of Houston developed a large
volume of data on p-y curves. This work will most likely result in specific recommendations in
due course.
3-4 p-y Curves for Cohesionless Soils
3-4-1 Description of p-y Curves in Sands
3-4-1-1 Initial Portion of Curves
The initial stiffness of stress-strain curves for sand is a function of the confining pressure
and magnitude of shearing strain; therefore, the use of mechanics for obtaining the initial Es for
pile design sands is complicated due to the complex strain fields around a pile. The p-y curve at
the ground surface will be characterized by zero values of p for all values of y, because the
vertical effective stress at the surface is zero. The initial slope of the p-y curves and the ultimate
resistance on the p-y curve will increase approximately linearly with depth.
A presentation of the recommendations of Terzaghi (1955) is of interest here, but it is
now recognized that his coefficients probably are meant to reflect the slope of secants to p-y
87
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
curves rather than the initial moduli. As noted earlier, Terzaghi recommended the use of his
coefficients up to the point where the computed soil resistance was equal to about one-half of the
ultimate bearing stress.
In terms of p-y curves, Terzaghi recommended a series of straight lines with slopes that
increase linearly with depth, as indicated in Equation 3-46.
Es  k x .......................................................... (3-46)
where k is a constant giving the variation of soil modulus with depth, and x is the depth below
ground surface. Terzaghi’s recommended values for k values in both US customary units and SI
units are presented in Table 3-5.
The k values recommended by Terzaghi in Table 3-5 are now known to be too
conservative. Users of LPile are advised to use the values recommended by Reese and Matlock
presented later in this manual because those values are based on load tests of fully instrumented
piles and are supported by high quality soil investigations. Terzaghi’s recommended values were
based on a literature review conducted in the early 1950’s, not a direct evaluation of pile load
testing results, and should be recognized as being preliminary recommendations. Terzaghi later
acknowledged around 1958 that he had some doubts about the source data and he ceased
recommending use of the values shown in Table 3-5.
3-4-1-2 Analytical Solutions for Ultimate Resistance
Two models are used for computing the ultimate resistance for piles in sand. These
models follow a procedure similar to that used for clay. The first of the models for the soil
resistance near the ground surface is shown in Figure 3-25. The total lateral force Fpt (Figure 325(c)) may be computed by subtracting the active force Fa, computed by use of Rankine theory,
from the passive force Fp, computed for the model by assuming that the Mohr-Coulomb failure
condition is satisfied on vertical wedge side planes defined by ADE and BCF, and on the sloping
wedge surface defined by AEFB in Figure 3-25(a). The directions of the resultant forces are
shown in Figure 3-25(b). Solutions other than the ones shown here have been developed by
assuming a friction force on the pile-soil interface surface defined by DEFC (assumed to be zero
in the analysis shown here) and by assuming the water table to be within the wedge (the unit
weight is assumed to be constant in the analysis shown here).
Table 3-5 k Values Recommended by Terzaghi for Laterally Loaded Piles in Sand
Relative Density
Loose
Medium
Dense
k, MN/m3 (pci)
Dry or Moist Sand
Submerged Sand
0.95 - 2.8
0.53 - 1.7
(3.5 - 10.4)
(2.1 - 6.4)
3.5 - 10.9
2.2 - 7.3
(13.0 - 40.0)
(8.0 - 27.0)
13.8 - 27.7
8.3 - 17.9
(51.0 - 102.0)
(32.0 - 64.0)
88
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
B

Fs
A
y
Ff
F
Fs
C
Fn
x
D
W
H
Fp
Fn
Ft

Fp
(b)
Fp
(a)
E
b
Pile of
Diameter b
Fs
Fn
F

Ff
W
Fpt
Fa
(c)
Figure 3-25 Geometry Assumed for Passive Wedge Failure for Pile in Sand
The total lateral force Fpt may be computed by following a procedure similar to that used
to solve the equation in the clay model (Figure 3-6). The resulting equation is
 K H tan  tan 
tan    H

Fpt   H 2  0

  tan  tan  

 3 tan(   ) cos  tan(   )  2 3
............... (3-47)
K Ab 
2  K 0 H tan 
tan sin   tan  
 H 
3
2 

where:
 = the angle of the wedge in the horizontal direction
 = is the angle of the wedge with the ground surface,
b = is the pile diameter,
H = the height of the wedge,
K0 = coefficient of earth pressure at rest, and
KA = coefficient of active earth pressure.
The ultimate soil resistance near the ground surface per unit length of the pile is obtained by
differentiating Equation 3-47 with respect to depth.
89
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
 K H tan  sin 

tan 
b  H tan  tan  
( pu ) sa   H  0

 ............... (3-48)
 tan(   ) cos  s tan(   )
  H K0 H tan  tan  sin   tan    K Ab
Bowman (1958) performed some laboratory experiments with careful measurements and
suggested values of  from /3 to /2 for loose sand and up to  for dense sand. The value of  is
approximated by the following equation.

  45  ......................................................... (3-49)
2
The model for computing the ultimate soil resistance at some distance below the ground
surface is shown in Figure 3-26(a). The stress 1 at the back of the pile must be equal or larger
than the minimum active earth pressure; if not, the soil could fail by slumping. The assumption is
based on two-dimensional behavior; thus, it is subject to some uncertainty. If the states of stress
shown in Figure 3-26(b) are assumed, the ultimate soil resistance for horizontal movement of the
soil is
 pu sb  K AbH tan8   1 K0bH tan tan4 
............................ (3-50)
The equations for (pu)sa and (pu)sb are approximate because of the elementary nature of
the models that were used in the computations. However, the equations serve a useful purpose in
indicating the general form, if not the magnitude, of the ultimate soil resistance.
3-4-1-3 Influence of Diameter on p-y Curves
No studies have been reported on the influence of pile diameter on p-y curves in sand.
The reported case studies of piles in sand, some of which are of large diameter, do not reveal any
particular influence of the pile diameter. However, virtually all of the reported lateral-load tests,
except the ones described later, have used only static loading.
90
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
5
4
2
4
4
3
3
2
3
1
1
5
6
5
6
5
1
Pile Movement
2
(a)


   tan 
 
 



(b)
Figure 3-26 Assumed Mode of Soil Failure by Lateral Flow Around Pile in Sand,
(a) Section Though Pile, (b) Mohr-Coulomb Diagram
3-4-1-4 Influence of Cyclic Loading
As noted above, there are very few reports of testing of piles subjected to cyclic lateral
loading. In these reports, two types of behavior have been observed and noted.
There is clear evidence that the repeated loading on a pile in predominantly one direction
will result in a permanent deflection of the pile in the direction of loading. It has been observed
during load testing that when a relatively large cyclic load is applied in one direction, the upper
section of the pile will deflect enough to allow grains of cohesionless soil to fall into the open
91
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
gap at the back of the pile. Thus in such a case, the pile cannot return to its initial position after
cyclic loading ceases causing the development of the permanent deflection.
Observations of the shearing behavior of sand near the ground surface during cyclic
loading support the idea that the void ratio of sand is approaching a critical value. This means
that dense sand will loosen and loose sand will densify under cyclic loading.
A careful study of the two phenomena mentioned above should provide information of
use to engineers. Full-scale experiments with detailed studies of the nature of the sand around the
top of a pile, both before and after loading, would be a welcome contribution.
3-4-1-5 Early Recommendations
The values of subgrade moduli recommended by Terzaghi (1955) provided some basis
for computation of lateral pile response, but Terzaghi’s values could not be implemented into
practice until the digital computer and the required programs became widely available. There
was a period of a few years in the 1950’s when engineers were solving the difference equations
using mechanical calculators. The piles supporting some early offshore platforms constructed
during this era were designed using this method.
Parker and Reese (1971) performed some small-scale experiments, examined
unpublished data, and recommended procedures for predicting p-y curves for sand. The method
of Parker and Reese was little used in practice because the method of Cox, et al. (1974),
described later, was based on a comprehensive load testing program on full-sized piles and
became available shortly afterward.
3-4-1-6 Field Experiments
An extensive series of field tests were performed at a site on Mustang Island, near Corpus
Christi, Texas (Cox, et al., 1974). Two steel-pipe piles, 610 mm (24 in.) in diameter, were driven
into sand in a manner to simulate the driving of an open-ended pipe and were subjected to lateral
loading. The embedded length of the piles was 21 meters (69 feet). One of the piles was
subjected to short-term loading and the other to cyclic loading.
The soil at the test site was classified as SP using the Unified Soil Classification System,.
The sand was poorly graded, fine sand with an angle of internal friction of 39 degrees. The
effective unit weight was 10.4 kN/m3 (66 pcf). The water surface was maintained at 150 mm (6
in.) above the ground surface throughout the test program.
3-4-1-7 Response of Sand Above and Below the Water Table
The procedure for developing p-y curves for piles in sand is shown in detail in the next
section. The piles that were used in the experiments, described briefly below, were the ones used
at Manor, except that the piles at Manor had an extra wrap of steel plate.
3-4-2 Reese, et al. (1974) Procedure for Computing p-y Curves in Sand
The Reese, et al. (1974) following procedure is for both short-term static loading and for
cyclic loading for a flat ground surface and a vertical pile. The shape of the p-y curves computed
using this procedure is illustrated in Figure 3-27.
92
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
p
u
pu
m
m
pm
yu
ym
k
pk
yk
kx
y
3b/80
b/60
Figure 3-27 Characteristic Shape of p-y Curves for Static and Cyclic Loading in Sand
1. Obtain values for the depth of the p-y curve x, angle of internal friction , effective unit
weight of soil , and pile diameter b (Note: use effective unit weight for sand below the
water table and total unit weight for sand above the water table).
2. Compute the following parameters:


2
,   45 


, K0  0.4 , and K A  tan 2  45   ..................... (3-51)
2
2


3. Compute the ultimate soil resistance per unit length of pile, ps, using the smaller of pst or
psd
ps = min[pst, psd] .................................................... (3-52)
where:
 K x tan  sin 
tan 
pst   x  0

(b  x tan  tan  )
.................... (3-53)
 tan(   ) cos  tan(   )
 K 0 x tan  (tan sin   tan  )  K Ab

psd  K A b  x(tan8   1)  K0 b  x tan tan4  ............................. (3-54)
4. Compute the y value defining point u using
93
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
yu 
3b
........................................................... (3-55)
80
Compute pu defining point u for static loading conditions using
pu  As ps ......................................................... (3-56)
or for cyclic loading conditions using
pu  Ac ps .......................................................... (3-57)
Obtain the appropriate value of As or Ac from Figure 3-28 as a function of the
nondimensional depth and type of loading (either static or cyclic). Compute ps using the
appropriate equation, either Equation 3-53 or Equation 3-54.
As or Ac
2
1
0
3
0
1
2
x
b
Ac
As
3
4
5
x
 5.0, A  0.88
b
6
Figure 3-28 Values of Coefficients Ac and As for Cohesionless Soils
5. Compute the y-value at point m using
ym 
b
.......................................................... (3-58)
60
Compute pm at point m for static loading conditions using
94
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
pm  Bs ps .......................................................... (3-59)
or for cyclic loading conditions using
pm  Bc ps .......................................................... (3-60)
Obtain the appropriate value of Bs or Bc from Figure 3-29 as a function of the
nondimensional depth and the type of loading (either the static or cyclic). Use the
appropriate equation for ps. The two straight-line portions of the p-y curve, beyond the
point where y is equal to b/60, can now be determined.
Bs or Bc
2
1
0
3
0
1
Bs (static)
Bc (cyclic)
2
x
b
3
4
5
x
 5.0, Bc  0.55, Bs  0.50
b
6
Figure 3-29 Values of Coefficients Bc and Bs for Cohesionless Soils
6. Establish the initial straight-line portion of the p-y curve,
p  k x y ......................................................... (3-61)
Use the appropriate value of k from Table 3-6 or 3-7.
If the input value of k is left equal to zero, LPile will compute a default value using the
curves shown in Figure 3-34 on page 114. Whether the sand is above or below the water
table will be determined from the input value of effective unit weight. If the effective unit
weight is less than 77.76 pcf (12.225 kN/m3) the sand is considered to be below the water
table. If the input value of  is greater than 45 degrees, a k value corresponding to 45
degrees is used by LPile.
95
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Table 3-6 Representative Values of k for Fine Sand Below the Water Table for Static and Cyclic
Loading
Relative Density
Loose
Medium
Dense
5.4
16.3
34
(20.0)
(60.0)
(125.0)
Recommended k
MN/m3
(pci)
Table 3-7 Representative Values of k for Fine Sand Above Water Table for Static and Cyclic
Loading
Recommended k
Loose
6.8
(25.0)
MN/m3
(pci)
Relative Density
Medium
Dense
24.4
61.0
(90.0)
(225.0)
If the sand profile is coarse or well-graded sand, the user may consider using a higher
value of k that those suggested in the tables above. While experimental data for k in wellgraded sands is poorly documented, use of values 10 to 50 percent higher may be
appropriate in dense and very dense well-graded sands that do not contain any
compressible minerals such as mica.
7. Fit the parabola between point k and point m as follows:
a. Compute the slope of the p-y curve between point m and point u using
m
pu  p m
........................................................ (3-62)
yu  y m
b. Compute the power of the parabolic section using
n
pm
........................................................... (3-63)
m ym
C
pm
........................................................... (3-64)
y1m/ n
c. Compute the coefficient C using
8. Compute the y value defining point k using
n
 C  n 1
y k    ........................................................ (3-65)
 kx 
Compute the p value defining point k using
96
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
pk  k x yk
9. Compute p-values along the parabolic section of the p-y curve between points k and m
using
p  C y1/ n .......................................................... (3-66)
Note: The curve in Figure 3-27 is drawn as if there is an intersection between the initial
straight-line portion of the p-y curve and the parabolic portion of the curve at point k.
However, in some instances there may be no intersection with the parabola. Equation 361 defines the p-y curve until there is an intersection with another portion of the p-y curve
or if no intersection occurs, Equation 3-61 defines the complete p-y curve. If yk is in
between points ym and yu, the curve is tri-linear and if yk is greater than yu, the curve is bilinear as shown in Figure 3-30.
p
Lower k x
kx
Higher k x
kx
y
Figure 3-30 Illustration of Effect of k on p-y Curve in Sand
3-4-2-1 Recommended Soil Tests
Fully drained triaxial compression tests are recommended for obtaining the angle of
internal friction of the sand. Confining pressures should be used which are close to or equal to
the effective overburden stresses at the depths being considered in the analysis. Tests must be
performed to determine the unit weight of the sand. However, it may be impossible to obtain
undisturbed samples and frequently the angle of internal friction is estimated from results of
some type of in-situ test.
The procedure above 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 smallscale experiments were performed by Parker and Reese (1971) and recommendations for the p-y
curves for dry sand were developed from those experiments. The results from the Parker and
97
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Reese experiments should be useful in checking solutions from results of experiments with fullscale piles.
3-4-2-2 Example Curves
An example set of p-y curves was computed for sand below the water table for a pile with
a diameter of 610 mm (24 in.). The sand properties are assumed to be an angle of internal friction
of 35 degrees and a submerged unit weight of 9.81 kN/m3 (62.4 pcf). The loading was assumed
as static.
The p-y curves were computed for the following depths below the mudline: 1.5 m (5 ft), 3
m (10 ft), 6 m (20 ft), and 12 meters (40 feet). The plotted curves are shown in Figure 3-31.
4,000
Load Intensity, p, kN/m
3,500
3,000
2,500
2,000
1,500
1,000
500
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Lateral Deflection, y, meters
1.5 m
3.0 m
6.0 m
12 m
Figure 3-31 Example p-y Curves for Sand Below the Water Table, Static Loading
3-4-3 API RP 2A Procedure for Computing p-y Curves in Sand
3-4-3-1 Background of API Method for Sand
This procedure is recommended by the American Petroleum Institute in its manual for
recommended practice for designing fixed offshore platforms (API RP 2A). Thus, the method
has official recognition. The API procedure for p-y curves in sand was based on a number of
field experiments. There is no difference for ultimate resistance (pu) between the Reese et al.
criteria and the API procedure. The API method uses a hyperbolic tangent function for
computation of the curve. The main difference between those two criteria will be the initial
modulus of subgrade reaction and the shapes of the curves.
98
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-4-3-2 Procedure for Computing p-y Curves Using the API Sand Method
The following procedure is for both short-term static loading and for cyclic loading as
described in API RP2A (2010).
1.
Obtain values for the angle of internal friction , the effective unit weight of soil, , and
the pile diameter b.
2.
Compute the ultimate soil resistance at a selected depth x. The ultimate lateral bearing
capacity (ultimate lateral resistance pu) for sand has been found to vary from a value at
shallow depths determined by Equation 3-67 to a value at deep depths determined by
Equation 3-68. At a given depth, the equation giving the smallest value of pu should be
used as the ultimate bearing capacity. The value of pu is the lesser of pu at shallow depths,
pus, or pu at great depth, pud , where:
pus  (C1 x  C2b)  x ................................................. (3-67)
pud  C3b  x ........................................................ (3-68)
where:
pu = ultimate resistance (force/unit length), lb./in. (kN/m),
 = effective unit weight, pci (kN/m3),
x = depth, in. (m),
 = angle of internal friction of sand, degrees,
C1, C2, C3 = coefficients determined from Figure 3-32 as a function of , or from:



 1

C1  tan  K P tan   K 0 tan  sin  
 1  tan   
 cos  



C2  K P  K A
C3  K P2 K P  K 0 tan   K A
where


K p  tan 2  45  
2

and
K 0  0.4
99
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
5
100
90
4
80
3
60
C2
50
2
Values of C3
Values of C1 and C2
70
40
C1
30
C3
1
20
10
0
0
20
25
30
35
40
Angle of Friction, degrees
Figure 3-32 Coefficients C1, C2, and C3 versus Angle of Internal Friction
b = average pile diameter from surface to depth, in. (m).
3.
Compute the load-deflection curve based on the ultimate soil resistance pu which is the
minimum value of pu calculated in Step 2. The lateral soil resistance-deflection (p-y)
relationships for sand are nonlinear and, in the absence of more definitive information,
may be approximated at any specific depth x by the following expression:
 kx
p  A pu tanh
 A pu

y  ................................................ (3-69)

where
A = factor to account for cyclic or static loading. Evaluated by:
A = 0.9 for cyclic loading.
100
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
x

A   3.0  0.8   0.9 for static loading,
b

pu = smaller of values computed from Equation 3-67 or 3-68, lb./in. (kN/m),
k = initial modulus of subgrade reaction, pci (kN/m3). Determine k from Figure 3-33 as a
function of angle of internal friction, ,
y = lateral deflection, in. (m), and
x = depth, inches (m).
f, Friction Angle, degrees
28
29
Very
Loose
36
30
Medium
Loose
41
Dense
45
Very
Dense
300
80
70
Fine Sand Above
the Water Table
250
60
200
150
40
k, MN/m3
k, lb/in3
50
30
100
Fine Sand Below
the Water Table
20
50
10
0
0
20
40
60
80
0
100
Relative Density, %
Figure 3-33 Value of k for API Sand Procedure2
2
It should be noted that Figure 3-32 has been corrected and differs from a similar figure presented in API RP-2A.
The positions of the labels for relative density on the bottom axis have moved to their correct positions, the label for
friction angle at the division line between dense and very dense sand has be corrected to the correct value of 41
degrees, and the scale in SI units has been added.
101
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
LPile will assign a default value for k if the user enters a value of zero. The value of k is
determined from the angle of friction and it is assumed that the sand is fine. The equations used
by LPile to determine k as a function of friction angle for fine sand are shown in Figure 3-34.
Whether the sand is above or below the water table will be determined from the input value of
effective unit weight. If the effective unit weight is less than 77.76 pcf (12.225 kN/m 3) the sand
is considered to be below the water table. If the input value of  is greater than 45 degrees, a k
value corresponding to 45 degrees is used by LPile. The two correlation lines intersect at a
friction angle value of 27.6423 degrees and a k value of 10.2068 pci. If the input value of  is
less than 27.6423 degrees, the value of k linearly varies from a value of zero at zero degrees to a
value of 10.2068 pci at 27.6423 degrees
If the sand profile is coarse or well-graded sand, the user may consider using a higher
value of k that those suggested in the Figure 3-34. While experimental data for k in well-graded
sands is sparse, use of k values 10 to 50 percent higher may be appropriate in dense and very
dense well-graded sands that do not contain any compressible minerals such as mica.
400
350
Fine Sand Above the Water Table
k = 0.4168 2 - 8.1254 - 83.664
300
k, psi
250
200
150
100
50
Fine Sand Below the Water Table
k = 0.0166 3 - 1.5526 2 + 58.43 -769.18
0
25
30
35
40
, Angle of Friction, Degrees
45
Figure 3-34 Value of k versus Friction Angle for Fine Sand Used in LPile
3-4-3-3 Example Curves
An example set of p-y curves was computed for sand above the water table, using the API
criteria. The soil properties are unit weight  = 0.07 pci, and internal-friction angle  = 35
degrees. The sand layer exists from the ground surface to a depth of 40 feet. The pile is of
102
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
reinforced concrete; the geometry and properties are: pile length = 25 feet, diameter = 36 in.,
moment in inertia = 82,450 in.4 and the modulus of elasticity = 3.6  106 psi. The loading is
assumed as static. The p-y curves are computed for the following depths: 20 in., 40 in., and 100
inches.
A hand calculation for p-y curves at a depth of 20 in. was made to check the computer
solution, as shown in the following.
1.
List the soil and pile parameters
 = 0.070 pci
 = 35 degrees
b = 36 inches
2.
Obtain coefficients C1, C2, C3 from Figure 3-32.
C1 = 2.97
C2 = 3.42
C3 = 53.8
3.
Compute the ultimate soil resistance pu.
pus = (C1 x + C2 b)  x = [(2.97)(20 in.) + (3.42)(36 in.)](0.07 pci)(20 in) = 255 lb./in.
pud = C3 b  x = (53.8)(36 in. )(0.07 pci) (20 in.) = 2,711 lb./in.
pu = pus = 255 lb./in. (smaller value)
4.
Compute coefficient A
A = 3.0 – (0.8) (x)/(b) = 3.0 – (0.8)(20 in.)/(36 in.) = 2.56
5.
Compute p for different y values.
If y = 0.1 inch, k (above water table) = 140 pci (from Figure 3-33)
 kx
p  A pu tanh
 A pu

y 

 (140 lb./in. 3 )(20 in.)

p  (2.55)(255 lb./in. ) tanh
(0.1 in.) 
 (2.55)(255 lb./in. )

p  264 lb./in. (computer output = 264.012 lb./in.)
If y = 1.35 in.
 kx
p  A pu tanh
 A pu

y 

 (140)(20 in.)

p  (2.55)(255 lb./in.) tanh
(1.35 in.) 
3
 (2.55)(255 lb/in. )

p  653 lb./in. (computer output = 652.93 lb./in.)
103
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The check by hand computations yielded exact values for the two values of deflection that
were considered.
The computed curves are presented in Figure 3-35.
3,000
2,500
2,000
1,500
1,000
500
0
0.0
0.25
0.5
0.75
1.0
1.25
1.5
1.75
2.0
Lateral Deflection y, in.
Depth = 20.00 in.
Depth = 40.00 in.
Depth = 100.00 in.
Figure 3-35 Example p-y Curves for API Sand Criteria
3-4-4 Other Recommendations for p-y Curves in Sand
A survey of the available information of p-y curves for sand was made by O’Neill and
Murchison (1983), and some changes were suggested in the procedure given above. Their
suggestions were submitted to the American Petroleum Institute and modifications were adopted
by the API review committee.
Bhushan, et al. (1981) reported on lateral load tests of drilled piers in sand. A procedure
for predicting p-y curves was suggested.
A number of authors have discussed the use of the pressuremeter in obtaining p-y curves.
The method that is proposed is described in some detail by Baguelin, et al. (1978) .
3-5 p-y Curves for Liquefied Soils
3-5-1 Response of Piles in Liquefied Sand
The lateral resistance of deep foundations in liquefied sand is often critical to the design.
Although reasonable methods have been developed to define p-y curves for non-liquefied and,
104
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
considerable uncertainty remains regarding how much lateral load-transfer resistance can be
provided by liquefied sand. In some cases, liquefied sand is assumed to have no lateral
resistance. This assumption can be implemented in LPile either by using appropriate p-multiplier
values or by entering a very low friction angle for sand.
When sand is liquefied under undrained conditions, some suggest that it behaves in a
manner similar to the behavior of soft clay. Wang and Reese (1998) have studied the behavior of
piles in liquefied soil by modeling the liquefied sand as soft clay. The p-y curves were generated
using the model for soft clay by equating the cohesive strength equal to the residual strength of
liquefied sand. The strain factor 50 was set equal to 0.05 in their study.
Laboratory procedures cannot measure the residual shear strength of liquefied sand with
reasonable accuracy due to the unstable nature of the soil. Some case histories must be evaluated
to gather information on the behavior of liquefied deposit. Recognizing the need to use case
studies, Seed and Harder (1990) examined cases reported where major lateral spreading has
occurred due to liquefaction and where some conclusions can be drawn concerning the strength
and deformation of liquefied soil.
Unfortunately, cases are rare where data are available on strength and deformation of
liquefied soils. However, a limited number of such cases do exist, for which the residual
strengths of liquefied sand and silty sand can be determined with a reasonable accuracy. Seed
and Harder found that a residual strength of about 10 percent of the effective overburden stress
can be used for liquefied sand.
Although simplified methods based on engineering judgment have been used for design,
full-scale field tests are needed to develop a full range of p-y curves for liquefied sand. Rollins et
al. (2005b) have performed full-scale load tests on a pile group in liquefied sand with an initial
relative density between 45 and 55 percent. The p-y curves that were developed from these
studies have a concave upward shape, as illustrated in Figure 3-36. This characteristic shape
appears to result primarily from dilative behavior during shearing, although gapping effects may
also contribute to the observed load-transfer response. Rollins and his co-workers also found that
p-y curves for liquefied sand stiffen with depth (or initial confining stress). With increasing
depth, small displacement is required to develop significant resistance and the rate at which
resistance develops as a function of lateral pile displacement also increases.
105
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
p
y
150 mm
Figure 3-36 Example p-y Curve in Liquefied Sand
Following liquefaction, p-y curves in sand become progressively stiffer with the passage
of time as excess pore water pressures dissipate and return to hydrostatic levels. The shape of a
p-y curve appears to transition from concave up to concave down as pore water pressures
decrease. A model based on load tests has been developed by Rollins et al. (2003) to describe the
observed load-displacement response of piles in liquefied sand as a function depth.
3-5-2 Method of Rollins et al. (2005a)
The expression developed by Rollins et al. (2005a) for p-y curves in liquefied sands at
different depths is shown below is based on their fully-instrumented load tests. Coefficients for
these equations were fit to the test data using a trial and error process in which the errors between
the target p-y curves and those predicted by the equations were minimized. The resulting
equations were then compared, and the equation that produced the most consistent fit was
selected.
p0.3 m  A By  ≤ 15 kN/m .............................................(3-70)
C
A  3  107 z  1
6.05
...................................................(3-71)
B  2.80( z  1) 0.11 .....................................................(3-72)
C  2.85z  1
0.41
.....................................................(3-73)
Where:
p0.3 m = soil resistance in kN/meter for a reference pile with a diameter of 0.3 meters,
y = lateral deflection of the pile in millimeters,
z = depth in meters (see note in last paragraph of this section, and
106
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
A, B, C are functions of the depth in meters.
Note that the engineering units of pile diameter is in meters, pile displacement is in
millimeters, depth is in meters, and computed values lateral load transfer are in kilonewtons per
meter.
The end of the upward curve is at a displacement of 150 mm only if p0.3 m is less than 15
kN/m at a y value of 150 mm. If p0.3 m reaches 15 kN/m at a value of y smaller than 150 mm, the
upward curve is ended at that value of y.
If the pile diameter differs from 0.3 m, the value of p is scaled by a diameter modification
factor. The diameter modification factor is discussed below.
Rollins et al. (2005a) studied the diameter effects for different sizes of piles and
recommended using a modification factor for adjusting Equation 3-70. The modification factor
for pile diameters between 0.3 and 2.6 meters is
Pd  3.81 ln b  5.6 ...................................................(3-74)
where b is the diameter of the pile or drilled shaft in meters. The p value for the reference
diameter of 0.3 meters can be multiplied by Pd to obtain values for p values for piles of varying
diameters using the equation below.
p( y)  Pd p0.3 m .......................................................(3-75)
Note that the diameter modification factor has been experimentally validated for pile
diameters ranging from 0.3 to 2.6 meters.
For pile diameters smaller than 0.3 meters, the procedure is to compute p0.3 m for a
diameter of 0.3 meters then multiply by the ratio of the pile diameter over 0.3 meters. Thus, for
pile diameters less than 0.3 meters, the diameter modification factor is computed from
 b 
 b 
Pd  
 3.81ln 0.3 m  5.6  1.0129 
 .........................(3-76)
 0.3 m 
 0.3 m 
Where the pile diameter b is in meters.
For pile diameters greater than 2.6 meters, the value of Pd is constant and equal to 9.24.
Application of Equation 3-70 should generally be limited to conditions comparable to
those from which it was derived. These conditions are:

Relative density between 45 and 55 percent,

Lateral soil resistance less than 15 kN/meter for the reference diameter of 0.3 meters,

Lateral pile deflection less than 150 mm (0.15 m),

Depths of 6 meters or less, and

Position of the water table is near to or at the ground surface.
107
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
In some cases, the liquefying layer may not be at the surface. In such cases, the depth
variable (z) may be modified to a depth equal to the initial vertical effective stress divided by an
effective unit weight of 10 kN/m3, which is generally representative of the unit weight of the
sand at the site.
3-5-3 Simplified Hybrid p-y Model
Franke and Rollins (2013)
developed the simplified hybrid p-y spring model that
combines the features of the Rollins et al. (2005a) model and the residual strength model
suggested by Wang and Reese (1998) . In this model, two p-y curves are computed. One curve is
the curve computed using the Rollins et al. (2005a) method. The second curve is based on the
Matlock method for p-y curves in soft clay under static loading conditions in which the cohesive
strength of the soil is based on the Seed and Harder (1990) curves of residual strength of
liquefied soils.
Franke and Rollins (2013) developed a model that combines the curves for dilative
liquefied sand developed by Rollins, et al. (2005a) with the Reese and Wang (1998) concept of a
residual strength curve computed using the residual strength curve developed by Seed and
Harder (1990). This model is referred to as the hybrid model for liquefied sands.
The concept used to formulate the hybrid model is to compute the p-y curves for the
dilative behavior based on the equations developed by Rollins, et al. (2005a) and for the residual
strength behavior based on the soft clay equations for static loading conditions developed by
Matlock (1970). The hybrid model uses the lowest p-value computed using either model for a
given y-value.
All input values need to be converted to SI units before computation of the hybrid p-y
curve. The sole exception to this is that the correlation from SPT blowcount to residual strength
outputs residual strength in units of psf.
This model combines the Matlock soft clay model for static conditions using residual
strengths correlated to SPT blowcount and the Ashford and Rollins dilative liquefied sand model.
The model uses the lowest p-value computed using either model for a given y-value.
The equations for the dilative model are the same as those presented by Rollins, et a.
(2005a) before.
3-5-3-1 Equations for Simplified Hybrid p-y Curve
Define the engineering units to be depth z in meters, pile diameter b in meters, and y in
millimeters. The equations are:
108
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
p  ABy 
C
A  3  10 7  z  1
6.05
B  2.80 z  1
0.11
C  2.85  z  1
 0.41

p 0.3m  min A(150 B) C , 15 kN/m

Pd  3.81ln b  5.6 for diameters between 0.3 m and 2.6 m
p d  b / 0.3m for diameters less than 0.3 m
pu  Pd p 0.3m


p  y   Pd ABy   pu for y values up to 150 mm
C
Where:
A, B, and C are dimensionless functions of the depth of the curve z in meters,
p0.3m is the maximum dilative resistance for a 0.3 m diameter pile,
Pd is a dimensionless factor to adjust for pile diameter,
pu is the peak lateral load intensity, and
50 is the 50 value required to compute the Matlock soft clay curve.
The parameter p0.3m is the lesser of the computed dilative resistance at a displacement of 150 mm
or 15 kN/m.
The equations for the p-y curves using residual strengths are those developed by Matlock
(1970) for soft clays under static loading conditions. The equations and p-y curve for soft clay
are presented in Figure 3-12(a). The peak load-transfer capacity is computed using



 avg
J 
pu  min  3 
z  z  Surb, 9cb 
Sur
b 


given that J = 0.5 and Sur substituted for c.
The pile diameter is accounted for by the factor y50. This factor is computed using
y50  2.5 50b
The curved portion of the p-y curve is computed using
p
p u
2
 y 
 
 y50 
1
3
for y ≤ 8y50
The lateral load-transfer is constant and equal to pu for lateral pile deflections greater
than 8y50.
109
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The engineering units used in the three equations above are consistent units of force and
length. The units may be lbs and ft, lbs and inches, or kN and meters. See the hybrid model
equations for the correlation of SPT blowcounts to 50.
3-5-3-2 Correlation for 33% residual strength as function of SPT (N1)60-cs -value:
2,000
Earthquake-induced liquefaction and sliding case histories where
SPT data and residual strength parameters have been measured.
1,600
40
Earthquake-induced liquefaction and sliding case histories where
SPT data and residual strength parameters have been estimated.
Construction-induced liquefaction and sliding case histories.
1,200
30
`
20
800
Recommended
residual strengths
for use in the
hybrid p-y model
400
0
0
4
8
12
16
20
10
0
24
Residual Undrained Shear Strength, Sr, kPa
Residual Undrained Shear Strength, Sr, psf
The residual shear strength is taken as the 33rd percentile of the residual strength
correlation developed by Seed and Harder (1990) . This correlation is illustrated in Figure 3-37.
In the original recommendations for the hybrid p-y curve, the residual undrained shear strength
for soils less than 5 blows per foot was taken to be zero.
Equivalent Clean Sand SPT Blowcount, (N1)60-cs
Figure 3-37 Recommended Method for Computing Residual Shear Strength of Liquefied Soil
for Use in Hybrid p-y Model
The 33rd percentile for residual strength as a function of SPT blowcount is shown as the
dashed gray line the figure above. The 33rd percentile of residual strength is determined by:
Sur = 0 for (N1)60-cs –values less than 5 blows per foot
Sur (psf) = 0.083467(N1)60-cs3 + 2.000777(N1)60-cs 2 - 12.642774(N1)60-cs + 90.689977
Sur is constant for (N1)60-cs greater than 16 bpf = 742.4853 psf
The strain factor 50 is computed as a function of SPT blowcount using
110
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
0.0005   50  0.1537 e
0.229 N1 60cs 
 0.05 ...................................(3-77)
Note that the lower and upper limits on 50 are 0.0005 and 0.05. The correlation of 50 with SPT
blowcount is illustrated in Figure 3-38
0.06
0.05
0.04
50
0.03
0.02
0.01
0
0
5
10
15
20
25
30
(N1)60-cs
Figure 3-38 Factor 50 as Function of SPT Blowcount
There are four possible patterns for the over lapping of the dilative and residual p-y
curves. These patterns are illustrated in Figure 3-39, with the residual curves shown in red and
the dilative curve shown in blue. There are two patterns where there is one intersection point
between the two curves. These two patterns are indicated by points 1 and 2 in the graph. There is
one pattern in which the curves intersect at two points, indicated by points 3 and 4, and one
pattern in which the dilative curve underlies the entirety of the residual curve. LPile computes
the coordinates of the intersection points and includes them in the output report for the generated
p-y curves.
111
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
p
No intersections
Intersections at 2 and 3
4
3
Intersection at 2
2
Intersection at 1
1
y
Figure 3-39 Possible Intersection Patterns of Residual and Dilative p-y Curves in Hybrid p-y
Model
The four following figures illustrate these four curve patterns.
14
12
Pile Diam., b = 0.3 m
Depth, z = 4.0 m
Eff. Unit Wt., ’ = 10 kN/m3
(N1)60-cs = 7 bpf
Sur = 128.9 psf
p, kN/m
10
8
6
Residual p
4
Dilative p
Hybrid p
2
0
0
50
100
150
200
250
300
y, mm
Figure 3-40 Example of Non-intersecting Curves
112
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
16
14
Pile Diam., b = 0.3 m
Depth, z = 4.0 m
Eff. Unit Wt., ’ = 10 kN/m3
(N1)60-cs = 6 bpf
Sur = 104.9 psf
12
p, kN/m
10
8
y1i
6
Residual p
4
Dilative p
Hybrid p
2
0
0
100
200
300
400
y, mm
Figure 3-41 Example of Curves with One Intersection of Dilative and Residual Curves
10
9
8
Pile Diam., b = 0.3 m
Depth, z = 0 m
Eff. Unit Wt.,  = 10 kN/m3
(N1)60-cs = bpf
Sur = 99.8 psf
7
p, kN/m
6
5
4
Residual p
Dilative p
3
Hybrid p
2
1
0
0
50
100
150
200
250
y, mm
Figure 3-42 Example of Curve with One Intersection of Dilative Curve and Residual Plateau
113
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
18
16
Pile Diam., b = 0.3 m
Depth, z = 4.0 m
Eff. Unit Wt., ’ = 10 kN/m3
(N1)60-cs = 7 bpf
Sur = 128.9 psf
14
p, kN/m
12
10
8
6
Residual p
Dilative p
4
Hybrid p
2
0
0
50
100
150
200
250
300
y, mm
Figure 3-43 Example of Curve with Two Intersection Points
3-5-4 Modeling of Lateral Spread
When liquefaction occurs in sloping soil layers, it is possible for the ground to develop
large permanent deformations. This phenomenon is called lateral spreading. Lateral spreading
may develop even though the ground surface may be nearly flat. If the free-field soil movements
are greater than the pile displacements, the displaced soils will apply an additional lateral load to
the piles. The magnitude and direction of the forces acting on the pile by soil movement is
dependent on the relative displacement between the pile and soil. If the liquefaction causes the
upper layer to become unstable and move laterally, a model recommended by Isenhower (1992)
may be used to solve for the behavior of the pile. This method is described in Section 4-3. Other
references on the topic of foundation subjected to lateral spreading are Ashford, et al. (2011) and
California Department of Transportation (2011).
3-6 p-y Curves for Loess Soils
3-6-1 Background
A procedure was formulated by Johnson, et al. (2006) for loess soil that includes
degradation of the p-y curves by load cycling.
The soil strength parameter used in the model is the cone tip resistance (qc) from cone
penetration (CPT) testing. The p-y curve for lateral resistance with displacement is modeled as a
hyperbolic relationship. Recommendations are presented for selection of the needed model
114
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
parameters, as well as a discussion of their effect. The p-y curves were obtained from backfitting of lateral analyses using the computer program LPile to the results of the load tests.
3-6-1-1 Description of Load Test Program
Shafts were tested in pairs to provide reaction for each other. Both shafts used in the load
test were fully instrumented. Load tests were performed on one pair of 30-inch diameter loaded
statically, one pair of 42-inch diameter test shafts loaded statically, and one pair of 30-inch
diameter test shafts loaded cyclically. Lateral loads were maintained at constant levels for load
increments without inclinometer readings, and the hydraulic pressure supply to the hydraulic
rams was locked off during load increments with inclinometer readings to eliminate creep of the
deflected pile shape with depth while inclinometer readings were made.
13 and 15 load increments were used to load the 30-inch and 42 inch diameters pairs of
static test piles, respectively, while both sets of static test piles were unloaded in four
decrements. Six sets of inclinometer readings were performed for each static test pile, four of
which occurred at load increments. Load increments and decrements for the static test shafts
were sustained for approximately 5 minutes, with the exception of the load increments with
inclinometer readings where the duration was approximately 20 minutes (this allowed for
approximately 10 minutes for inclinometer measurements for each of the two test shafts in the
pair). Lateral loads were applied to the 30-inch and 42-inch diameter static test shafts in
approximately 10-kip and 15-kip increments, respectively.
There were four load increments (noted as “A” through “D”) on the 30-inch diameter
cyclic test shafts, with ten load cycles (N = 1 through 10) performed per load increment. The
lateral load for each load cycle were sustained for only a few seconds with the exception of load
cycles 1 and 10 which were sustained for approximately 15 to 20 minutes to allow time for the
inclinometer readings to be performed. For load cycles 2 through 9, the duration for each load
cycle was approximately 1 minute, 2 minutes, 3.5 minutes, and 6.5 minutes for load increments
A though D, respectively, as a greater time was required to reach the larger loads. The load was
reversed after each load cycle to return the top of pile to approximately the same location.
3-6-1-2 Soil Profile from Cone Penetration Testing
A back-fit model of the pile behavior using the available soil strength data obtained (from
both in-situ and laboratory tests) to the measured pile performance led to the conclusion that the
CPT testing provided the best correlation. Furthermore, CPT testing can be easily performed in
the loess soils being modeled and has become readily widely available.
Three cone penetration tests were performed by the Kansas Department of Transportation
at the test site location. A preliminary cone penetration test was performed in the general vicinity
of the test shafts (designated as CPT-1). Two additional cone penetration tests were performed
subsequent to the lateral load testing. A cone penetration test was performed between the 42-inch
diameter static test shafts (Shafts 1 and 2) shortly after on the same day the lateral load test was
performed on these shafts. A cone penetration test was performed between the 30-inch diameter
static test shafts (Shafts 3 and 4) two days after the completion of the load test performed on
these shafts. The locations of the cone penetration tests were a few feet from the test shafts.
Given the nature of the soil conditions and the absence of a ground water table, it is reasonable to
assume that the cone penetration tests were unaffected by any pore water pressure effects that
may have been induced by the load testing.
115
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
An idealized profile of cone tip resistance with depth interpreted as an average from the
cone penetration tests performed between the static test shafts is shown in Figure 3-44. This
profile is considered representative of the subsurface conditions for all the test shaft locations.
Note that it is most useful to break the idealized soil profile into layers wherein the cone tip
resistance is either constant with depth or linearly varies with depth as these two conditions are
easily accommodated by most lateral pile analyses software.
The cone tip resistance is reduced by 50% at the soil surface, and allowed to increase
linearly with depth to the full value at a depth of two pile diameters, as shown in Figure 3-44.
This is done to account for the passive wedge failure mechanism exhibited at the ground surface
that reduces the lateral resistance of the soil between the ground surface and a lower depth
(assumed at two shaft diameters). Below a depth of two shaft diameters, the lateral resistance is
considered as a flow around bearing failure mechanism.
The idealized cone tip resistance values were correlated with depth with the ultimate
lateral soil resistance (pu0) at corresponding depths.
Reduced by 50% at surface
0
2-D = 5 ft for 30-inch Diam. Shafts
5
2-D = 7 ft for 42-inch Diam. Shafts
Depth Below Grade (ft)
10
15
20
Used For Model
25
Between 30"
A.L.T. (6/9/2005)
30
Between 42"
A.L.T. (6/8/2005)
CPT-1
(8/12/2004)
35
40
0
20
40
60
80
100
120
140
160
180
200
qc (ksf)
qc, ksf
Figure 3-44 Idealized Tip Resistance Profile from CPT Testing Used for Analyses.
3-6-2 Procedure for Computing p-y Curves in Loess
3-6-2-1 General Description of p-y Curves in Loess
Procedures are provided to produce a p-y curve for loess, shown generically in Figure 345. The ultimate soil resistance (pu0) that can be provided by the soil is correlated to the cone tip
resistance at any given elevation. Note that to account for the passive wedge failure mechanism
exhibited at the ground surface, the cone tip resistance is reduced by 50% at the soil surface and
allowed to return to the full value at a depth equal to two pile diameters. The initial modulus of
116
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
the p-y curve, Ei, is determined from the ultimate lateral soil reaction expressed on a per unit
length of pile basis, pu, for the specified pile diameter, and specified reference displacement, yref.
A hyperbolic relationship is used to compute the secant modulus of the p-y curve, Es, at any
given pile displacement, y. The lateral soil reaction per unit pile length, p, for any given pile
displacement is determined by the secant modulus at that displacement. Provisions for the
degradation of the p-y curve as a function of the number of cycles loading, N, are incorporated
into the relationship for ultimate soil reaction.
The model is of a p-y curve that is smooth and continuous. This model is similar to the
lateral behavior of pile in loess soil measured in load tests.
3-6-2-2 Equations of p-y Model for Loess
The ultimate unit lateral soil resistance, pu0, is computed from the cone tip resistance
multiplied by the cone bearing capacity factor, NCPT using
puo  NCPT qc .........................................................(3-78)
p
pu
Ei
Es
y
yref
Figure 3-45. Generic p-y curve for Drilled Shafts in Loess Soils
where NCPT is dimensionless, and pu0 and qc are in consistent units of (force/length2)
The value of NCPT was determined from a best fit to the load test data. It is believed that
NCPT is relatively insensitive to soil type as this is a geotechnical property determined by in-situ
testing. The value of NCPT derived from the load test data is
N CPT  0.409 ........................................................(3-79)
117
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The ultimate lateral soil reaction, pu, is computed by multiplying the ultimate unit lateral
soil resistance by the pile diameter, b, and dividing by an adjustment term to account for cyclic
loading. The adjustment term for cyclic loading takes into account the number of cycles of
loading, N, and a dimensionless constant, CN.
pu 
puo b
....................................................(3-80)
1 CN log N
where:
b is the pile diameter in any consistent unit of length,
CN is a dimensionless constant,
N is the number of cycles of loading (1 to 10), and
pu is in units of (force/length).
CN was determined from a best fit of cyclic degradation for two 30-inch diameter test
shafts subjected to cyclic loading. CN is
C N  0.24 ...........................................................(3-81)
The cyclic degradation term (the denominator of Equation 3-80) equals 1 for N = 1
(initial cycle, or static load) and equals 1.24 for N = 10. The value of CN has a direct effect on the
amount of cyclic degradation to the p-y curve (i.e., a greater value of CN will allow greater
degradation of the p-y curve, resulting in a smaller pu). Note that the degradation of the ultimate
soil resistance per unit length of shaft parameter will also have the desired degradation effect
built into the computation of the p-y modulus values.
A parameter is needed to define the rate at which the strength develops towards its
ultimate value (pu0). The reference displacement, yref, is defined as the displacement at which the
tangent to the p-y curve at zero displacement intersects the ultimate soil resistance asymptote
(pu), as shown in Figure 3-45. The best fit to the load test data was obtained with the following
value for reference displacement.
yref = 0.117 inches = 0.0029718 meters .................................. (3-82)
Note that the suggested value for the reference displacement provided the best fit to the
piles tested at a single test site in Kansas for a particular loess formation. Unlike the ultimate unit
lateral resistance (pu0), it is believed that the rate at which the strength is mobilized may be
sensitive to soil type. Thus, re-evaluation of the reference displacement parameter is
recommended when performing lateral analyses for piles in different soil conditions because this
parameter is likely to have a substantial effect on the resulting pile deflections. The effect of the
reference displacement is proportional to pile performance that is a larger value of yref will allow
for larger pile head displacements at a given lateral load.
The initial modulus, Ei, is defined as the ratio of the ultimate lateral resistance expressed
on a per unit length of pile basis over the reference displacement.
118
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Ei 
pu
........................................................... (3-83)
yref
A secant modulus, Es, is determined for any given displacement, y, by the following
hyperbolic relationship of the initial modulus expressed on a per unit length of pile basis and a
hyperbolic term ( yh ) which is in turn a function of the given displacement (y), the reference
displacement (yref), and a dimensionless correlation constant (a).
Es 
 y
yh  
y
 ref
Ei
......................................................... (3-84)
1  yh



 1  a e  y ref


 y 




 .............................................. (3-85)


a  0.10 ............................................................(3-86)
where Es and Ei are in units of force/length2, and a and yh are dimensionless.
The constant a was found from a best fit to the load test data. Note that the constant a
primarily affects the secant modulus at small displacements (say within approximately 1 inch or
25 mm), and is inversely proportional to the stiffness response of the p-y curve (i.e., a larger
value of a will reduce the mobilization of soil resistance with displacement). Combining the two
equations above, one obtains
 y
yh  
y
 ref



 1  0.1e  y ref


 y 




 .............................................(3-87)


The modulus ratio (secant modulus over initial modulus, Es/Ei) versus displacement used
for p-y curves in loess is shown in Figure 3-46. Note that the modulus ratio is only a function of
the hyperbolic parameters of the constant (a) and the reference displacement (yref), thus the curve
presented is valid for all pile diameters and cone tip bearing values tested.
119
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
1.0
0.9
a = 0.1
0.8
0.7
0.6
Es
Ei
0.5
0.4
0.3
0.2
0.1
0
0.001
0.01
0.1
1.0
10
100
y
yref
Figure 3-46 Variation of Modulus Ratio with Normalized Lateral Displacement
Both the initial modulus and the secant modulus are proportional related to the pile
diameter because the ultimate soil resistance is proportional to a given pile size, as was shown in
Equation 3-80. It follows that the lateral response will increase in proportion to the pile diameter.
For a given pile displacement, the lateral soil resistance per unit length of pile is a
product of the pile displacement and the corresponding secant modulus at that displacement.
p  ES y ........................................................... (3-88)
where:
Es is the secant modulus in units of force/length2, and
y is the lateral pile displacement.
Several p-y curves obtained from the model described above is presented in Figure 3-47
for the 30-inch diameter shafts, and Figure 3-48 for the 42-inch diameters shafts. Note that there
are three sets of curves presented for each shaft diameter which correspond to the cone tip
resistance values of 11 ksf, 22 ksf, and 100 ksf (as was shown in Figure 3-44). These p-y curves
were used in the LPile analyses presented later.
120
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
9,000
8,000
7,000
p, lb/in. .
6,000
11 ksf
5,000
22 ksf
100 ksf
4,000
3,000
2,000
1,000
0
0
1
2
3
4
5
6
7
y , inches
Figure 3-47 p-y Curves for the 30-inch Diameter Shafts
14,000
12,000
p, lb/in. .
10,000
11 ksf
8,000
22 ksf
100 ksf
6,000
4,000
2,000
0
0
1
2
3
4
5
6
7
y , inches
Figure 3-48 p-y Curves and Secant Modulus for the 42-inch Diameter Shafts.
The static p-y curves shown in Figure 3-47 and 3-48 were degraded with load cycle
number (N) for use in the cyclic load analyses. Figure 3-49 presents the cyclic p-y curve
generated for the analyses of the 30-inch diameter shafts at the cone tip resistance value of 22
ksf.
121
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
2,000
1,800
1,600
p, lb/in. .
1,400
N= 1
1,200
N= 5
1,000
N = 10
800
600
400
200
0
0
1
2
3
4
5
6
7
y , inches
Figure 3-49 Cyclic Degradation of p-y Curves for 30-inch Shafts
3-6-2-3 Step-by-Step Procedure for Computing p-y Curves
A step-by-step procedure to generate p-y curves in using the model follows.
1. Develop an idealized profile of cone tip resistance with depth that is representative of the
local soil conditions. It is most useful to subdivide the soil profile into layers where the cone
tip resistance is either constant with depth or varies linearly with depth.
2. Reduce the cone tip resistance by 50% at the soil surface, and allowed the value to return to
the full measured value at a depth equal to two pile diameters. Linear interpolation may be
used between the surface and the depth of two pile diameters.
3. For each soil layer, compute the ultimate soil resistance from the cone tip resistance in
accordance with Equation 3-78 for both the top and the bottom of each layer.
4. Multiply the ultimate soil resistance by the pile diameter to obtain the ultimate soil reaction
per unit length of shaft (pu). For cyclic analyses, pu may be degraded for a given cycle of
loading (N) in accordance with Equation 3-80.
5. Select a reference displacement (yref) that will represent the rate at which the resistance will
develop.
6. Determine the initial modulus (Ei) in accordance with Equation 3-83.
7. Select a number of lateral pile displacements (y) for which a representative p-y curve is to be
generated.
8. Determine the secant modulus (Es) for each of the displacements selected in Step 7 in
accordance with Equations 3-84 and 3-85.
9. Determine the soil resistance per unit length of pile (p) for each of the displacements selected
in Step 7 in accordance with Equation 3-88.
122
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-6-2-4 Limitations on Conditions for Validity of Model
The p-y curve for static loading was based on best fits of data from full-scale load tests on
30-inch and 42-inch diameter shafts installed in a loess with average cone tip resistance values
ranging from 20 to 105 ksf (960 to 5,000 kPa).
Caution is advised when extrapolating the static model formulation for shaft diameters or
soil types and/or strengths outside these limits. In addition, the formulation for the cyclic
degradation model parameters are based on load tests with only ten cycles of loading (N = 1 to
10) obtained at four different load increments on an additional two 30-inch diameter shafts.
Caution is thus also warranted when extrapolating the cyclic model to predict results beyond 10
cycles of load (N > 10), particularly as the magnitude of loading increases.
3-7 p-y Curves for Cemented Soils with Both Cohesion and Friction
3-7-1 Background
The methods for p-y curves that were presented previously were for soils that can be
characterized as either as purely cohesive or purely cohesionless. There are currently no
generally accepted recommendations for developing p-y curves for soils that have both cohesion
due to cementation and frictional characteristics.
Among the reasons for the limitation on soil characteristics are the following. Firstly, in
foundation design where the p-y analysis has been used, the characterization of the soil by either
a value of cohesion or friction has been used, but not both. Secondly, the major experiments on
which the p-y predictions have been based have been performed in soils that can be described by
either cohesion (c) or friction (. However, there are numerous occasions when it is desirable,
and perhaps necessary, to describe the characteristics of the soil with both cohesion due to
cementation and friction.
One example of the need to have predictions for p-y curves for cemented c- soils is
when piles are used to stabilize a slope. A detailed explanation of the analysis procedure is
presented in Chapter 6. It is well known that most of the currently accepted methods of analysis
of slope stability characterize the soils in terms of c and  for long-term, drained conditions.
Therefore, it is inconsistent, and potentially unsafe or unconservative, to assume that the soil is
characterized by either c or  alone.
There are other instances in the design of piles under lateral loading where it is desirable
to have methods of prediction for p-y curves for c- soils. The shear strength of unsaturated,
cohesive soils generally is represented by strength components of both c and . In many practical
cases, however, there is the likelihood that the soil deposit might become saturated because of
seasonal rainfall and subsequent rise of the ground water table. However, there could well be
times when the ability to design for dry seasons is needed.
Cemented soils are frequently found in subsurface investigations. Some comments for the
response of laterally loaded piles in calcareous soils were presented by Reese (1988). It is
apparent that cohesion from the cementation will increase the shearing resistance significantly,
particularly for soils near the ground surface.
123
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
It should be noted that the procedure presented here has been revised from versions of
LPile earlier than data format 8 (LPile 2015).
3-7-2 Recommendations for Computing p-y Curves
The following procedure for computing p-y curves cemented c- soils is for short-term
static loading and for cyclic loading. The shape of the resulting p-y curve is illustrated in Figure
3-50. The p-y curve is composed of four segments; an initial slope between the origin and point k
defined by the slope k x, a parabolic section between points k and m, a straight line section
between points m and u and a flat section defined past point u. As will be noted, the suggested
procedure follows closely the procedure that which was recommended earlier for p-y curves in
sand.
p
m
pm
k
pk
yk
ym
u
pu
kx
yu
y
b/60
3b/80
Figure 3-50 Characteristic Shape of p-y Curves for c- Soil
Conceptually, the ultimate soil resistance, pu, is taken as the passive soil resistance acting
on the face of the pile in the direction of the horizontal movement, plus any sliding resistance on
the sides of the piles, less any active earth pressure force on the rear face of the pile. The force
from active earth pressure and the sliding resistance will generally be small compared to the
passive resistance, and will tend to cancel each other out. Evans and Duncan (1982)
recommended an approximate equation for the peak resistance of c- soils as:
p   p b  C p hb .................................................... (3-89)
where
p = passive pressure including the three-dimensional effect of the passive wedge (F/L2)
b = pile width (L),
The Rankine passive pressure for a wall of infinite length (F/L2),
124
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock




 h    x tan 2  45    2 c tan 45   ................................ (3-90)

2

2
 = effective unit weight of soil (F/L3),
x = depth at which the passive resistance is considered (L),
 = angle of internal friction (degrees),
c = cohesion (F/L2), and
Cp = dimensionless modifying factor to account for the three-dimensional effect of the
passive wedge at the ground surface.
The modifying factor Cp can be divided into two terms: Cp to represent the frictional
contribution to Equation 3-89 and Cpc to represent the cohesive contribution to Equation 3-89.
Equation 3-89 can then be written as Equation 3-91 as


 


p  C p   x tan 2  45    C pc c tan 45   b ........................... (3-91)
2
2 



Equation 3-91 will be rewritten as
p  A p  pc ........................................................ (3-92)
where A can be found for static and cyclic loading from Figure 3-28 on page 106.
The frictional component, p, is the smaller of ps or pd.
pu = min[ps, pd] .................................................... (3-93)
The terms ps and pd are defined by the two equations below:
 K x tan  sin 
tan 
ps   x  0

(b  x tan  tan  )
.................... (3-94)
 tan(   ) cos  tan(   )
 K 0 x tan  (tan  sin   tan  )  K Ab
pd  K A b  x (tan8   1)  K 0 b  x tan  tan 4  ............................ (3-95)
The cohesive component, pc, is the smaller of pcs or pcd.
pc  min  pcs , pcd  ................................................... (3-96)
where:

J 

pcs   3  x  x  c b .............................................. (3-97)
c
b 

125
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
where J is a dimensionless constant equal to 0.5 and
pcd  9 c b .......................................................... (3-98)
Furthermore, it is assumed that the contribution of cohesion due to cementation is lost as
the load-transfer curve transitions from the peak value to the residual value, so the strength of the
residual curve is due to the frictional component only and is
p  A pu ........................................................... (3-99)
3-7-3 Procedure for Computing p-y Curves in Soils with Both Cohesion and
Internal Friction
To develop the p-y curves, the procedures described earlier for sand by Reese et al (1974)
will be used because the stress-strain behavior of c- soils are believed to be closer to the stressstrain curve of cohesionless soil than for cohesive soil. The following procedures are used to
develop the p-y curves for soils with both cohesion and internal friction.
1.
Compute yu by the following equation:
yu 
2.
3b
...........................................................(3-100)
80
Compute pu for static loading using
pu  As pu ...................................................... (3-101)
or for cyclic loading using
pu  Ac pu ...................................................... (3-102)
Use the appropriate value of As or Ac from Figure 3-28 on page 106 for the particular
non-dimensional depth (x/b) and type of loading.
3.
Compute ym as
ym 
4.
b
......................................................... (3-103)
60
Compute pm for static by the following equation
pm  Bs pu  puc .................................................. (3-104)
or for cyclic loading by
pm  Bc pu  puc ................................................. (3-105)
126
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Use the appropriate value of Bs or Bc from Figure 3-29 on page 107 for the particular nondimensional depth and for either the static or cyclic case. Use the appropriate equations for
pu and puc. The two straight-line portions of the p-y curve, beyond the point where y is
equal to b/60, can now be established.
5.
Establish the initial straight-line portion of the p-y curve,
p  k x  y ....................................................... (3-106)
The value of k for Equation 3-106 may be found from the following equation and by
reference to Figure 3-51.
k  k c  k ....................................................... (3-107)
For example, if c is equal to 0.2 tsf and  is equal to 35 degrees for a layer of c- soil
above the water table, the recommended kc is 350 pci and k is 80 pci, yielding a value of k
of 430 pci.
2,000
1,500
kc (static)
400,000
kc (cyclic)
1,000
300,000
200,000
k (submerged)
500
100,000
k (above water table)
0
Initial Modulus k, kN/m3
Initial Modulus k, pci
500,000
0
c tsf
0
1
2
3
4
 deg.
0
28
32
36
40
c kPa
0
96
192
287
383
Figure 3-51 Representative Values of k for c- Soil
6.
The parabolic section of the p-y curve will be computed from
p  S y1 / n ........................................................ (3-108)
127
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
To fit the parabola between points k and m, compute the parameters m, n, S, and yk using
the following expressions:
a. Compute the slope of the line between point m and point u by,
m
pu  pm
...................................................... (3-109)
yu  y m
b. Compute the power of the parabolic section using
n
pm
........................................................ (3-110)
m ym
c. Compute the coefficient S using
S
pm
 ym 1/ n
....................................................... (3-111)
d. Compute displacement, yk, at the intersection of the initial slope defined by kx and the
parabolic section using
n
 S  n1
 ...................................................... (3-112)
yk  
k x
e. Compute points along the parabolic section by using
p  S y1 / n ........................................................ (3-108)
Note: The step-by-step procedure is outlined above as if there is an intersection between
the initial straight-line portion of the p-y curve and the parabolic portion of the curve at point k.
However, in some instances there may be no intersection with the parabola if the initial slope is
sufficiently small, as drawn below.
128
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
p
m
ym, pm
u
yu, pu
y
Figure 3-52 Possible Intersection Points of Initial Tangent Line Along p-y Curve
3-7-4 Discussion
An example of p-y curves was computed for cemented c- soils for a pile with a diameter
of 12 inches (0.3 meters). The c value is 400 psf (20 kPa) and a  value is 35 degrees. The unit
weight of soil is 115 pcf (18 kN/m3). The p-y curves were computed for depths of 39.4 in. (1 m),
78.7 in. (2 m), and 118.1 inches (3 meters). The p-y curves computed by using the simplified
procedure are shown in Figure 3-53. As can be seen, the ultimate resistance of the soil, based in
the model procedure, is higher than from the simplified procedure. Both of the p-y curves show a
peak strength, then drop to a residual strength at large deflections, as is expected. Because of a
lack of experimental data to calibrate the soil resistance, based on the model procedure, it is
recommended that the simplified procedure be used at present.
The point was made clearly at the beginning of this section that data are unavailable from
a specific set of experiments that was aimed at the response of c- soils. Such experiments would
have made use of instrumented piles. Further, little information is available in the literature on
the response of piles under lateral loading in such soils where response is given principally by
deflection of the pile at the point of loading.
Data from one such experiment, however, was available and the writers have elected to
use that data in an example to demonstrate the use of this criterion. A comparison was made
there between results from experiment and results from computations.
The reader will note that the procedure presented above does not reflect a severe loss of
soil resistance under cyclic loading that is a characteristic for clays below a free-water surface.
Rather, the procedures described above are for a material that is primarily granular in nature,
which does not reflect such loss of resistance. Therefore, if a c- soil has a very low value of 
and a relatively large value of c, the user is advised to ignore the  and to use the
recommendations for p-y curves for clay. Further, a relatively large factor of safety is
recommended in any case, and a field program of testing of prototype piles is certainly in order
for jobs that involve any large number of piles.
129
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
1,000
1m
2m
800
p, kN/meter
3m
600
400
200
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
y, meters
Figure 3-53 p-y Curves for Cemented c- Soil
3-8 p-y Curves for Rock
3-8-1 Introduction
The use of deep foundations in rock is frequently required for support of bridges,
transmission towers, or other structures that sustain lateral loads of significant magnitude.
Because the rock must be drilled in order to make the installation, drilled shafts are commonly
used. However, a steel pile could be grouted into the drilled hole. In any case, the designer must
use appropriate mechanics to compute the bending moment capacity and the variable bending
stiffness EI. Experimental results show conclusively that the EI must be reduced, as the bending
moment increases, in order to achieve a correct result (Reese, 1997).
In some applications, the axial load is negligible so the penetration is controlled by lateral
load. The designer will wish to initiate computations with a relatively large penetration of the
pile into the rock. After finding a suitable geometric section, the factored loads are employed and
computer runs are made with penetration being gradually reduced. The ground-line deflection is
plotted as a function of penetration and a penetration is selected that provides adequate security
against a sizable deflection of the bottom of the pile.
Concepts are presented in the following section that from the basis of computing the
response of piles in rock. The background for designing piles in rock is given and then two sets
of criteria are presented in this section, one for vuggy limestone and the other for weak rock.
130
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Much of the presentation follows the paper by Reese (1997) and more detail will be found in
that paper.
The secondary structure of rock is an overriding feature is respect to its response to
lateral loading. Thus, an excellent subsurface investigation is assumed prior to making any
design. The appropriate tools for investigating the rock are employed and the Rock Quality
Designation (RQD) should be taken, along with the compressive strength of intact specimens. If
possible, sufficient data should be taken to allow the computation of the Rock Mass Rating
(RMR). Sometimes, the RQD is so low that no specimens can be obtained for compressive tests.
The performance of pressuremeter tests in such instances is indicated.
If investigation shows that there are soil-filled joints or cracks in the rock, the procedures
suggested herein should not be used but full-scale testing at the site is recommended.
Furthermore, full-scale testing may be economical if a large number of piles are to be installed at
a particular site. Such field testing will add to the data bank and lead to improvements in the
recommendations shown below, which are to be considered as preliminary because of the
meager amount of experimental data that is available.
In most cases of design, the deflection of the drilled shaft (or other kind of pile) will be so
small that the ultimate strength pur of the rock is not developed. However, the ultimate resistance
of the rock should be predicted in order to allow the computation of the lateral loading that
causes the failure of the pile. Contrary to the predictions of p-y curves for soil, where the unit
weight is a significant parameter, the unit weight of rock is neglected in developing the
prediction equations that follow. While a pile may move laterally only a small amount under the
working loads, the prediction of the early portion of the p-y curve is important because the small
deflections may be critical in some designs.
Most intact rocks are brittle and develop shear planes at low shear strains. This fact leads
to an important concept about intact rock. The rock is assumed to fracture and lose strength
under small values of deflection of a pile. If the RQD of a stratum of rock is zero, or has a low
value, the rock is assumed to have already fractured and, thus, will deflect without significant
loss of strength. The above concept leads to the recommendation of two sets of criteria for rock,
one for strong rock and the other for weak rock. For the purposes of the presentations herein,
strong rock is assumed to have a compressive strength of 6.9 MPa (1,000 psi) or above.
The methods of predicting the response of rock is based strongly on a limited number of
experiments and on correlations that have been presented in technical literature. Some of the
correlations are inexact; for example, if the engineer enters the figure for correlation between
stiffness and strength with a value of stiffness from the pressuremeter, the resulting strength can
vary by an order of magnitude, depending on the curve that is selected. The inexactness of the
necessary correlations, plus the limited amount of data from controlled experiments, mean that
the methods for the analysis of piles in rock must be used with a good deal of both judgment and
caution. For major projects, full-scale load testing is recommended to verify foundation
performance and to evaluate the efficiency of proposed construction methods.
131
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-8-2 Descriptions of Two Field Experiments
3-8-2-1 Islamorada, Florida
An instrumented drilled shaft (bored pile) was installed in vuggy limestone in the Florida
Keys (Reese and Nyman, 1978) and was tested under lateral loads. The test was performed for
gaining information for the design of foundations for highway bridges.
Considerable difficulty was encountered in obtaining properties of the intact rock. Cores
broke during excavation and penetrometer tests were misleading because of the presence of vugs
or could not be performed. It was possible to test two cores from the site. The small
discontinuities in the outside surface of the specimens were covered 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 149 mm (5.88 in.) in diameter and
with heights of 302 mm (11.88 in.) for Specimen 1 and 265 mm (10.44 in.) for Specimen 2. The
undrained shear strength values of the specimens were taken as one-half the unconfined
compressive strength and were 1.67 MPa (17.4 tsf) and 1.30 MPa (13.6 tsf) for Specimens 1 and
2, respectively.
The rock at the site was also investigated by in-situ-grout-plug tests (Schmertmann,
1977). In these tests, a 140-mm (5.5 in.) 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 the interface of the grout and limestone. Tests were performed at three borings, and the results
shown in Table 3-8 were obtained. The average of the eight tests was 1.56 MPa (226 psi or 16.3
tsf). However, the rock was stronger in the zone where the deflections of the drilled shaft were
greatest and a shear strength of 1.72 MPa (250 psi or 18.0 tsf) was selected for correlation.
Table 3-8 Results of Grout Plug Tests by Schmertmann (1977)
Depth Range
meters
0.76-1.52
2.44-3.05
feet
2.5-5.0
8.0-10.0
5.49-6.10 18.0-20.0
Ultimate Resistance
MPa
psf
tsf
2.27
331
23.8
1.31
190
13.7
1.15
167
12.0
1.74
253
18.2
2.08
301
21.7
2.54
368
26.5
1.31
190
13.7
1.02
149
10.7
The bored pile was 1,220 mm (48 in.) in diameter and penetrated 13.3 m (43.7 ft) into the
limestone. The overburden of fill was 4.3 m (14 ft) thick and was cased. The load was applied at
3.51 m (11.5 ft) above the limestone. A maximum horizontal load of 667 kN (75 tons) was
132
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
applied to the pile. The maximum deflection at the point of load application was 18.0 mm (0.71
in.) and at the top of the rock (bottom of casing) it was 0.54 mm (0.0213 in.). While the curve of
load versus deflection was nonlinear, there was no indication of failure of the rock. Other details
about the experiment are shown in the Case Studies that follow.
3-8-2-2 San Francisco, California
The California Department of Transportation (Caltrans) performed lateral-load tests of
two drilled shafts near San Francisco (Speer, 1992). The results of these unpublished tests have
been provided by courtesy of Caltrans.
Two exploratory borings were made into the rock and sampling was done with a NWD4
core barrel in a cased hole with a diameter of 102 mm (4 in.). A 98-mm (3.88-in.) tri-cone roller
bit was used in drilling. The sandstone was medium to fine grained with grain sizes from 0.1 to
0.5 mm (0.004 to 0.02 in.), well sorted, and thinly bedded with thickness of 25 to 75 mm (1 to 3
in.). Core recovery was generally 100%. The reported values of RQD ranged from zero to 80,
with an average of 45. The sandstone was described by Speer (1992) as moderately to very
intensely fractured with bedding joints, joints, and fracture zones.
Pressuremeter tests were performed and the results were scattered. The results for moduli
values of the rock are plotted in Figure 3-54. The dashed lines in the figure show the average
values that were used for analysis. Correlations of RQD to modulus reduction ratio shown in
Figure 3-55 and the correlation of rock strength and modulus shown in Figure 3-56 were
employed in developing the correlation between the initial stiffness from Figure 3-54 and the
compressive strength, and the values were obtained as shown in Table 3-9.
Initial Modulus, Eir, MPa
0
800
400
1,200
1,600
2,000
0
2
186 MPa
Depth , meters
4
3.9 m
645 MPa
6
8
8.8 m
10
1,600 MPa
12
Figure 3-54 Initial Moduli of Rock Measured by Pressuremeter for San Francisco Load Test
133
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
1.2
Modulus Reduction Ratio
Emass/Ecore
1.0
0.8
0.6
0.4
0.2
?
?
?
0.0
0%
25%
50%
75%
100%
Rock Quality Designation (RQD), %
Figure 3-55 Modulus Reduction Ratio versus RQD (Bienawski, 1984)
Two drilled shafts, each with diameters of 2.25 m (7.38 ft), and with penetrations of 12.5
m (41 ft) and 13.8 m (45 ft), were tested simultaneously by pulling the shafts together. Lateral
loading was applied using hydraulic rams acting on high-strength steel bars that were passed
through tubes, transverse and perpendicular to the axes of the shafts. Lateral load was measured
using electronic load cells. Lateral deflections of the shaft heads were measured using
displacement transducers. The slope and deflection of the shaft heads were obtained by readings
from slope indicators.
The load was applied in increments at 1.41 m (4.6 ft) above the ground line for Pile A
and 1.24 m (4.1 ft) for Pile B. The pile-head deflection was measured at slightly different points
above the rock line, but the results were adjusted slightly to yield equivalent values for each of
the piles. Other details about the loading-test program are shown in the case studies that follow.
Table 3-9 Values of Compressive Strength at San Francisco
Depth Interval
Compressive Strength
m
ft
MPa
psi
0.0 to 3.9
0.0 to 12.8
1.86
270
3.9 to 8.8
12.8 to 28.9
6.45
936
below 8.8
below 28.9
16.0
2,320
The rock below 8.8 m (28.9 ft) is in the range of strong rock, but the rock
above that depth will control the lateral behavior of the drilled shaft.
134
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
(MPa)
10
1,000
Ra
us
50
0
100
ul
00
0
tio
Very Low
Low
Medium
High
Very High
1,
Rock Strength
Classification
(Deere)
100
M
od
1
0
20
0
10
100,000
10
Upper and
Middle Chalk
(Hobbs)
Concrete
(MPa)
Steel
Young’s Modulus – psi  106
Gneiss
1.0
Grades
of Chalk
(Ward et al.)
I
II
III
0.1
Limestone,
Dolomite
Basalt and other
Flow Rocks
Lower
Chalk
(Hobbs)
Deere
10,000
Sandstone
1,000
Trias (Hobbs)
IV
V
Keuper
100
Black Shale
0.01
Grey Shale
Hendron, et al.
10
Medium
0.001
Stiff
Very Stiff
Hard
0.01
0.1
Clay
1
1.0
10
100
Uniaxial Compressive Strength – psi 
103
Figure 3-56 Engineering Properties for Intact Rocks
(after Deere, 1968; Peck, 1976; and Horvath and Kenney, 1979)
135
1,000
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-8-3 Procedure for Computing p-y Curves in Vuggy Limestone
The p-y curve recommended for strong rock (vuggy limestone), with compressive
strength of intact specimens larger than 6.9 MPa (1,000 psi), shown in Figure 3-57. If the rock
increases in strength with depth, the strength at the top of the stratum will normally control.
Cyclic loading is assumed to cause no loss of resistance.
As shown in the Figure 3-57, load tests are recommended if deflection of the rock (and
pile) is greater than 0.0004b and brittle fracture is assumed if the lateral stress (force per unit
length) against the rock becomes greater than half the diameter times the compressive strength of
the rock.
The p-y curve shown in Figure 3-57 should be employed with 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 be controlled by joints, cracks, and secondary structure and not by the
strength of intact specimens.
Perform proof test if
deflection is in this range
p
pu = b su
Assume brittle fracture if
deflection is in this range
Es = 100su
Es = 2000su
NOT TO SCALE
y
0.0004b
0.0024b
Figure 3-57 Characteristic Shape of p-y Curve in Strong Rock
3-8-4 Procedure for Computing p-y Curves in Weak Rock
The p-y curve that is recommended for weak rock is shown in Figure 3-58. The
expression for the ultimate resistance pur for rock is derived from the mechanics for the ultimate
resistance of a wedge of rock at the top of the rock.
136
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
p
Mir
pur
y
yA
Figure 3-58 Sketch of p-y Curve for Weak Rock (after Reese, 1997)
x 

pur   r qurb1  1.4 r  for xr  3b .................................... (3-113)
b

pur  5.2 r qurb for xr  3b .......................................... (3-114)
where:
qur = compressive strength of the rock, usually lower-bound as a function of depth,
r =
strength reduction factor,
b =
diameter of the pile, and
xr =
depth below the rock surface.
The assumption is made that fracturing will occur at the surface of the rock under small
deflections, therefore, the compressive strength of intact specimens is reduced by multiplication
by r to account for the fracturing. The value of r is assumed to be 1.0 at RQD of zero and to
decrease linearly to a value of one-third for an RQD value of 100%. If RQD is zero, the
compressive strength may be obtained directly from a pressuremeter curve, or approximately
from Figure 3-56, by entering with the value of the pressuremeter modulus.


 r  1 
2 RQD% 
 ................................................ (3-115)
3 100% 
137
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
If one were to consider a strip from a beam resting on an elastic, homogeneous, and
isotropic solid, the initial modulus Mir (pi divided by yi) in Figure 3-58 may be shown to have the
following value (using the symbols for rock). 3
Mir  kir Eir ..................................................... (3-116)
where
Eir = the initial modulus of the rock, and
kir = dimensionless constant defined by Equation 3-117.
Equations 3-116 and 3-117 for the dimensionless constant kir are derived from data available
from experiment and reflect the assumption that the presence of the rock surface will have a
similar effect on kir as was shown for pur for ultimate resistance.
400 xr 

kir  100 
 for 0  xr  3b ..................................... (3-117)
3b 

kir = 500 for xr > 3b ................................................ (3-118)
With guidelines for computing pur and Mir, the equations for the three branches of the
family of p-y curves for rock in Figure 3-57 can be presented. The equation for the straight-line,
initial portion of the curves is given by Equation 3-119 and for the other branches by Equations
3-121 through 3-120.
p  M ir y for y  y A ...............................................(3-119)
yrm = rm b .........................................................(3-120)
p
p  ur
2
 y 


 y rm 
0.25
for y A  y , y < 16yrm, and p  pur ......................(3-121)
p  pur for y > 16yrm ................................................(3-122)
where
rm = a constant, typically ranging from 0.0005 to 0.00005 that serves to establish the upper
limit of the elastic range of the curves using Equation 3-120. The constant rm is
analogous to 50 used for p-y curves in clays. The stress-strain curve for the uniaxial
compressive test may be used to determine rm in a similar manner to that used to
determined 50.
The notation used here for Mir and rm differs from that used in Reese (1997). The notation was
changed to improve the clarity of the presentation.
3
138
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The value of yA is found by solving for the intersection of Equations 3-119 and 3-121, and the
solution is presented in Equation 3-123.

pur
y A  
0.25
 2 y rm  M ir
1.333




.............................................(3-123)
As shown in the case studies that follow, the equations from weak rock predict with
reasonable accuracy the behavior of single piles under lateral loading for the two cases that are
available. An adequate factor of safety should be employed in all cases.
The equations are based on the assumption that p is a function only of y. This assumption
appears to be valid if loading is static and resistance is only due to lateral stresses. However,
O’Neill (1996) noted that in large diameter drilled shafts, rotational moment is resisted in the
vertical shear couple produced by the vertical shear stresses caused by the rotation of the pile. In
rock, this effect could be significant, especially for small deflections, if the diameter of the pile is
large.
3-8-5 Case Histories for Drilled Shafts in Weak Rock
3-8-5-1 Islamorada
The drilled shaft was 1.22 m (48 in.) diameter and penetrated 13.3 m (43.7 ft) into
limestone. A layer of sand over the limestone was retained by a steel casing, and the lateral load
was applied at 3.51 m (11.5 ft) above the surface of the rock. A maximum lateral load of 667 kN
(150 kips) was applied and the measured curve of load versus deflection was nonlinear.
Values of the strengths of the concrete and steel were unavailable and the bending
stiffness of the gross section was used for the initial solutions. The following values were used to
compute the p-y curves:
qur = 3.45 MPa (500 psi),
r = 1.0, (RQD = 0%)
Eri = 7,240 MPa (1.05  106 psi),
rm = 0.0005,
b = 1.22 m (48 in.),
L = 15.2 m (50 ft), and
EI = 3.73  106 kN-m2 (1.3  109 ksi).
A comparison of pile-head deflection curves from experiment and from analysis is shown
in Figure 3-59. Excellent agreement between the elastic EI and experiment and is found for
loading levels up to about 350 kN (78.7 kips), where sharp change in the load-deflection curve
occurs. Above that level of loading, nonlinear EI is required to match the experimental values
reasonably well.
Curves giving deflection and bending moment as a function of depth were computed for a
lateral load of 334 kN (75 kips), one-half of the ultimate lateral load, and are shown in Figure 360. The plotting is shown for limited depths because the values to the full length are too small to
139
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
plot. The stiffness of the rock, compared to the stiffness of the pile, is reflected by a total of 13
points of zero deflection over the length of the pile of 15.2 meters (50 ft). However, for the data
employed here, the pile will behave as a long pile through the full range of loading.
800
EI = 37.3105 kN-m2
EI = 5.36105 kN-m2
EI = 6.23105 kN-m2
Lateral Load, kN
600
EI = 7.46105 kN-m2
EI = 9.33105 kN-m2
400
EI = 12.4105 kN-m2
200
Analysis with Elastic EI
Analysis with Reduced EI
Measured in Load Test
0
0
10
5
15
20
Groundline Deflection, mm
Figure 3-59 Comparison of Experimental and Computed Values of Pile-Head Deflection,
Islamorada Test (after Reese, 1997)
Values of EI were reduced gradually where bending moments were large to obtain
deflections that would agree fairly well with values from experiment. Values of lateral deflection
and bending moment versus depth are shown in Figure 3-60. The largest moment occurs close to
the top of rock, in the zone of about 2.5 m (8.2 ft) to 4.5 meters (14.8 ft). The following values of
load and bending stiffness were used in the analyses: 350 kN and below 3.73106 kN-m2; 400
kN, 1.24106 kN-m2; 467 kN, 9.33105 kN-m2; 534 kN, 7.46105 kN-m2; 601 kN, 6.23105 kNm2; and 667 kN, 5.36105 kN-m2. The computed bending moment curves were studied and
reductions were only made where the bending stiffness was expected to be in the nonlinear
range.
The lowest value of EI that was used is believed to be roughly equal to that for the fully
cracked section. The decrease in slope of the curve of yt versus Pt at Islamorada can reasonably
be explained by reduction in values of EI. The analysis of the tests at Islamorada gives little
guidance to the designer of piles in rock except for early loads. A study of the testing at San
Francisco that follows is more instructive.
3-8-5-2 San Francisco
The value of krm used in the analyses was 0.00005. For the beginning loads the value used
for EI was 35.15106 kN-m2 (12.25109 ksi, E=28.05106 kPa (4.07106 psi); I = 1.253 m4
(3.01105 in4)). The nominal bending moment capacity Mnom was computed to be 17,740 m-kN
140
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
(1.57105 in-kips) and values of EI were computed as a function of bending moment. Data from
Speer (1992) gave the following properties of the cross section: compressive strength of the
concrete was 34.5 MPa (5,000 psi), tensile strength of the rebars was 496 MPa (72,000 psi),
there were 40 bars with a diameter of 43 mm (1.69 in.), and cover thickness was 0.18 m (7.09
in.).
Bending Moment, M, kN-m
400
0
0
400
800
1,200
M
Depth, meters
2
y
Rock Surface
4
6
8
1
0
1
2
3
Lateral Deflection, y, mm
Figure 3-60 Computed Curves of Lateral Deflection and Bending Moment versus Depth,
Islamorada Test, Lateral Load of 334 kN (after Reese, 1997)
The data on deflection as a function of loads showed that the two piles behaved about the
same for the beginning loads but the curve for Pile B exhibited a large increase in pile-head
deflection at the largest load. The experimental curve for Pile B shown by the heavy solid line in
Figure 3-61 suggests that a plastic hinge developed at the ultimate bending moment of 17,740 mkN (157,012 in-kips).
Consideration was given to the probable reduction in the values of EI with increasing
load and three methods were used to predict the reduced values. The three methods were: the
analytical method as presented in Chapter 4, the approximate method of the American Concrete
Institute (ACI 318) which does not account for axial load and may be used here; and the
experimental method in which EI is found by trial-and-error computations that match computed
and observed deflections. The plots of the three methods are shown in Figure 3-62 and all three
curves show a sharp decrease in EI with increase in bending moment. For convenience in the
computations, the value of EI was changed for the entire length of the pile but errors in using
constant values of EI in the regions of low values of M are thought to be small.
141
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The computed and measured lateral load versus pile-head deflection curves are shown in
Figure 3-61. The computed load-deflection curve computed using EI values derived from the
load test agrees well with the load test curve, but the computed load-deflection curves using
other modeling methods are less (i.e. “stiffer”) than the load test values. However, if load factors
of 2.0 and higher are selected, the computed deflections would be about 2 or 3 mm (0.078 to
0.118 in.) with the experiment showing about 4 mm (0.157 in.). Thus, the differences are
probably not very important in the range of the service loading.
10,000
Lateral Load, kN
8,000
Pile B
6,000
4,000
Unmodified EI
Analytical
ACI
Experimental
2,000
0
0
10
20
30
40
50
Groundline Deflection, mm
Figure 3-61 Comparison of Experimental and Computed Values of Pile-Head Deflection for
Different Values of EI, San Francisco Test
Also shown in Figure 3-61 is a curve showing deflection as a function of lateral load with
no reduction in the values of EI. The need to reduce EI as a function of bending moment is
apparent.
Values of bending stiffness in Figure 3-62 along with EI of the gross section were used to
compute the maximum bending moment mobilized in the shaft as a function of the applied load
are shown in Figure 3-63. The close agreement between computations from all the methods is
striking. The curve based on the gross value of EI is reasonably close to the curves based on
adjusted values of EI, indicating that the computation of bending moment for this particular
example is not very sensitive to the selected values of bending stiffness.
142
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Bending Stiffness, kN-m2  106
40
Analytical
Experimental
ACI
30
20
10
0
5,000
0
10,000
15,000
20,000
Bending Moment, kN-m
Figure 3-62 Values of EI for three methods, San Francisco test
10,000
Lateral Load, kN
7,500
5,000
Unmodified EI
Analytical
ACI
Experimental
2,500
0
0
5,000
10,000
15,000
20,000
Bending Moment, kN-m
Figure 3-63 Comparison of Experimental and Computed Values of Maximum Bending
Moments for Different Values of EI, San Francisco Test
143
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-9 p-y Curves for Massive Rock
3-9-1 Introduction
Liang, Yang, and Nusairat (2009) developed a criterion for computing p-y curves for
drilled shafts in massive rock. This criterion is based on both full-scale load tests and threedimensional finite element modeling.
A hyperbolic equation is used as the basis for the p-y relationship
p
y
1
y

K i pu
.......................................................(3-124)
where pu is the ultimate lateral resistance of the rock mass and Ki is the initial slope of the p-y
curve. A drawing of the p-y curve for massive rock is presented in Figure 3-64. Both of these
parameters, Ki and pu, are computed using the properties of the rock mass. The ultimate lateral
resistance pu is computed for two conditions; near the ground surface and at great depth. The
lower of the two values of pu is used in computing the p-y curve.
p
pu
p
y
1
y

K i pu
Ki
y
Figure 3-64 p-y Curve in Massive Rock
3-9-2 Shearing Properties of Massive Rock
The shearing properties, c and , used in computing the p-y curve for massive rock are
defined using the Hoek-Brown (1980) strength criterion for rock. In the Hoek-Brown strength
criterion, the major and minor principal stresses are related by
144
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
a
 

 1   3   ci  mb 3  s  ............................................(3-125)
  ci

where 1 and 3 are the major and minor principal stresses at failure, ci is the uniaxial
compressive strength of intact rock, and the parameters mb, s, and a are material constants that
depend on the characteristics of the rock mass; s = 1 for intact rock, and a = 0.5 for most rock
types. The parameters mb and s can be determined for many types of rock using the
recommendations of Marinos and Hoek (2000).4 Parameter mb can be computed using the HoekBrown material index mi and the Geologic Strength Index, GSI, and blast damage factor Dr using
mb  mi e
 GSI 100 


 2814Dr 
...................................................(3-126)
Representative values for the Hoek-Brown material index are presented in Table 3-10. For deep
excavations like drilled shaft or bored piles, the blast damage factor Dr is assumed equal to zero.
Hoek (1990) provided a method for estimating the Mohr-Coulomb failure parameters c
and  of the rock mass from the principal stresses at failure. These parameters are:

2 
 .............................................(3-127)
  1   3 
   90  arcsin 
c     n tan   ....................................................(3-128)
1 can be found from Equation 3-125, and n and  are found from
 n   3 
1   3 2
2 1   3   0.5mb ci
   1   3  1 
4
........................................(3-129)
mb ci
...........................................(3-130)
2 1   3 
This reference may be obtained from the Internet at https://www.rocscience.com/education/hoeks_corner#.
145
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Table 3-10 Values of Material Index mi for Intact Rock, by Rock Group (from Hoek, 2001)
Rock
Type
Class
Group
Clastic
Carbonates
Sedimentary
Non-clastic
Evaporites
Organic
Non Foliated
Metamorphic
Slightly Foliated
Foliated**
Light
Plutonic
Dark
Igneous
Hypabyssal
Lava
Volcanic
Pyroclastic
Texture (values in parenthesis are estimates)
Coarse
Medium
Fine
Very Fine
Conglomerates*
Siltstones
Claystones
(21±3)
Sandstones
7±2
4±2
Breccias
14±2
Graywackes
Shales
(19±5)
(18±3)
(6±2)
Chrystalline
Sparitic
Micritic
Dolomites
limestones
Limestones
Limestones
(9±3)
(12±3)
(10±2)
(9±2)
Gypsum
Anhydrite
8±2
12±2
Chalk
7±2
Hornfels
Marble
(19±4)
Quartzites
9±3
Metasandstone
20±3
(19±3)
Migmatite
Amphibolites
(29±3)
26±6
Gneiss
Schists
Phyllites
Slates
28±5
12±3
(7±3)
7±4
Granite
Diorite
32±3
25±5
Granodiorite
(29±3)
Gabbro
27±3
Dolerite
Norite
(16±5)
20±5
Porphyries
Diabase
Peridotite
(20±5)
(15±5)
(25±5)
Rhyolite
Dacite
(25±5)
(25±3)
Obsidian
Andesite
Basalt
(19±3)
25±5
(25±5)
Agglomerate
Breccia
Tuff
(19±3)
(19±5)
(13±5)
* Conglomerates and breccias may present a wide range of mi values depending on the nature of the cementing material and degree of
cementation, so they may range from values similar to sandstone to values used for fine-grained sediments.
** These values for intact rock specimens tested normal to bedding or foliation. The values of mi will be significantly different if failure occurs
along a weakness plane.
3-9-3 Determination of Rock Mass Modulus
Two methods for evaluating the modulus of the rock mass are recommended by Liang et
al. One method is to compute rock mass modulus by multiplying the intact rock modulus
measured in the laboratory by the modulus reduction ratio Em / Ei , computed from the geological
strength index, GSI, using Equation 3-131.
E 
eGSI / 21.7
............................................(3-131)
Em  Ei  m   Ei
100
 Ei 
146
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
The experimental data and correlation for the modulus reduction ratio are shown as a
function of GSI in Figure 3-65.
100
Bieniawski (1978)
Serafin and Pereira (1983)
Ironton-Russell
Regression Line
GSI / 21.7
E
e
%
E
100
m
Em/Ei, (%)
Modulus Reduction Ratio
80
i
60
40
20
0
0
20
40
60
80
100
Geologic Strength Index
Figure 3-65 Equation for Estimating Modulus Reduction Ratio from Geological Strength Index
The second method recommended by Liang et al. for determining the modulus of the
rock mass is to perform an in-situ rock pressuremeter test. The difficulty in using this approach is
that many pressuremeter testing devices are not capable of reaching sufficiently large pressures
to deform the rock. If this is the case, interpretation of test results may be restricted because of
the limited range of expansion pressures achievable.
Values for Poisson’s ratio are also required to compute p-y curves in massive rock.
Values of Poisson’s ratio vary with the quality of the rock mass. Typical values for Poisson’s
ratio and other properties for rock masses reported by Hoek (2001) are shown in Table 3-11.
Values of Poisson ratio for the rock mass can be estimated by interpretation
measurements of in-situ stress wave velocities. Poisson’s ratio can be computed from the
compression and shear wave velocities using Equation 3-132 (Zhang, 2004). This relationship is
drawn in Figure 3-66.

(V p / Vs ) 2  2
2 [(V p / Vs ) 2  1]
147
................................................ (3-132)
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Table 3-11 Typical Properties for Rock Masses (from Hoek, 2001)
Property
Symbol
Intact Rock
Strength
Hoek-Brown
Constant
Geological Strength
Index
Friction
Angle
Cohesive
Strength
Poisson’s
Ratio
ci
Good Quality
Hard Rock Mass
150 MPa
21,750 psi
80 MPa
11,600 psi
Very Poor Quality
Rock Mass
20 MPa
2,900 psi
mi
25
12
8
GSI
75
50
30

46 deg.
33 deg.
24 deg.
c
13 MPa
1,885 psi
3.5 MPa
500 psi
0.55 MPa
80 psi

0.2
0.25
0.3
Average Rock Mass
0.5
0.4

0.3
0.2
0.1
0
0
1
2
3
Vp/Vs
4
5
6
Figure 3-66 Poisson’s Ratio as Function of Stress Wave Velocity Ratio
3-9-4 Determination of pus Near the Ground Surface
For a passive wedge type failure near the ground surface, as shown in Figure 3-67, the
ultimate lateral resistance per unit length, pus of the drilled shaft at depth H near the ground
surface is
pus  2C1 cos  sin   C2 sin   2C4 sin   C5 ...........................(3-133)
where   45 

2
and  

2
.
148
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock

Fs
Fs
H
Fnet
W
Fa
Fn

D
Figure 3-67 Model of Passive Wedge for Drilled Shafts in Rock
Liang, et al. note that the value of 3 can be taken as the effective overburden pressure at
a depth of H/3 for estimating  and c using Equations 3-127 and 3-128.
The following equations are used to compute parameters C1 through C5 with c =
effective cohesion,  = effective friction angle, and,  = effective unit weight respectively of the
rock mass.
 

K a  tan 2  45  
2

K 0  1  sin  
z0 

2c
 v0
  Ka  
H


C1  H tan  sec  c  K 0 v0 tan    K 0   tan   
2


C2  C3 tan    cD sec   2H tan  sec  tan 
D tan   v0  H   H tan 2  tan  2 v0  H   cD  2 H tan  tan    2C1 cos  cos 
C3 
sin   tan   cos 
149
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
 H 

C4  K 0 H tan  sec   v0 
 , and
2 

C5    K a H  z0  D , with the condition that C5  0
Equation 3-133 is valid for homogeneous rock mass. For layered rock mass,
representative properties can be computed by a weighted method based on the volume of the
failure wedge. Methods for obtaining the rock properties c and  are given in Section 3-9-2 on
page 156.
3-9-5 Determination of pud at Great Depth
The passive wedge failure mechanism is not likely to form if the overburden pressure is
sufficiently large. Studies of rock sockets using three-dimensional stress analysis using the finite
element method have concluded that at depth the rock failure first in tension, followed by failure
in friction between the shaft and rock, followed finally by failure of the rock in compression.
Therefore, the expression for ultimate resistance at depth is a function of the limiting pressure,
pL, and the peak frictional resistance max. The ultimate resistance at depth can be computed
using
2


pud   pL   max  pa  D ..........................................(3-134)
3
4

where pa is the active horizontal active earth pressure given by
pa  Ka v  2c Ka with the condition that pa  0 ........................(3-135)
The effective overburden pressure, v, at the depth under consideration includes the pressure
from overburden soils.
The limiting normal pressure of the rock mass, pL, is taken as the compressive strength of
the rock mass, 1, computed using Equation 3-125 and equating 3 equal to v,.
a



pL   v   ci  mb v  s  ............................................(3-136)
  ci

The limiting shear stress, max, is the maximum axial side resistance of the rock-shaft interface,
proposed by Rowe and Armitage (1987).
 max  0.45  ci .....................................................(3-137)
where both max and ci are in units of megapascals. Values of max in units of kPa and psi are
computed by LPile using the following equations:
For max and  ci in units of kPa:  max  14.23025  ci ......................(3-138)
150
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
For max and  ci in units of psi:  max  5.4194  ci .......................(3-139)
3-9-6 Determination of Initial Tangent Stiffness of p-y Curve Ki
The initial tangent stiffness of the p-y curve, Ki, is computed rock mass modulus,
Poisson’s ratio, pile diameter, and mobilized bending stiffness of the pile using
 D
Ki  Em 
D
 ref
  2  E p I p 
e 
 E D 4 

 m 

0.284
........................................(3-140)
where Em is the modulus of the rock mass,  is Poisson’s ratio of the rock mass, D is the diameter
of the drilled shaft, Dref is a reference shaft diameter equal to 0.3048 m or 12 inches, and E p I p is
the bending stiffness of the drilled shaft.
3-9-7 Procedure for Computing p-y Curves in Massive Rock
1. Obtain the values of the intact rock strength ci and the intact rock modulus Ei.
2. Obtain values for the rock mass modulus, Em, by either use of Equation 3-131 if
pressuremeter data are unavailable or from interpretation of pressuremeter testing results. If
Equation 3-131 is used, obtain values of GSI and mi according to the recommendations of
Marinos and Hoek (2000).
3. Obtain the value of Poisson’s ratio of the rock mass from in-situ measurements or estimated
from Table 3-11.
4. Select a shaft diameter, compressive strength of concrete, and reinforcing details.
5. Compute the bending stiffness and nominal moment capacity of the drilled shaft. Set the
value of bending stiffness equal to the cracked section bending stiffness at a level of loading
where the reinforcement is in the elastic range.
6. Compute Ki using Equation 3-140.
7. Compute pus at shallow depth using Equation 3-133 with 3 equal to the vertical effective
stress at H/3 when computing the values of  and c using Equations 3-127 and 3-128.
8. Compute pud at great depth using Equation 3-134 with pL computed using Equation 3-136
and equating 3 equal to v.
9. Compute pu as the smaller of pus computed using Equation 3-133 or pud computed using
Equation 3-134.
10. The values of the p can then be computed as a function of y, Ki, and pu using Equation 3-124
as a function of pile displacement y.
151
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-10 p-y Curves in Piedmont Residual Soils
The Piedmont residual soils are found east of the Appalachian ridge in a region extending
from southeastern Pennsylvania south through Maryland, central Virginia, eastern North
Carolina, eastern South Carolina, northern Georgia, into Alabama. It is a weathered in-place
rock, underlain by metamorphic rock. In general, the engineering behavior of Piedmont residual
soil is poorly understood, due to difficulties in obtaining undisturbed samples for laboratory
testing and relatively wide variability.
The degree of weathering varies with local conditions. Weathering is greatest at the
ground surface and decreases with depth until the unweathered, parent rock is found. The
residual soil profile is often divided into three zones: an upper zone of red, sandy clays, an
intermediate zone of micaceous silts, and a weathered zone of gravelly sands mixed with rock.
Often the boundaries of the zones are indistinct or inclined. Weathering is greatest near seepage
zones.
The method for computing p-y curves in Piedmont residual soils was developed by
Simpson and Brown (2006). This method was developed to use correlations for estimates of soil
modulus measured using four field testing methods: dilatometer, Menard pressuremeter,
Standard Penetration Test, and cone penetration tests. The basic method is described in the
following paragraphs.
Given a shaft diameter b, and soil modulus Es, the relationship between p and y is
p  Es b y  ......................................................(3-141)
This relationship is considered to be linear up to y/b = 0.001 (0.1 percent).
For y/b values greater than 0.001,

y/b 
 for 0.001  y/b  0.0375 ..........................(3-142)
Es  Esi 1   ln
0.001 

pu  b y 1 3.624  .................................................(3-143)
where  = –0.23, which gives
pu  1.834Esib y
Esi   Etest
152
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Es/Esi
1

ln y/b
Figure 3-68 Degradation Plot for Es
pu
y
0.001b
0.0375b
Figure 3-69 p-y Curve for Piedmont Residual Soil
3-11 Response of Layered Soils
There are conditions where the soil profile near the ground surface is composed of soil
layers of different types. If the layers are in the shallow range of depths where the soil would
move up and out as a wedge, some modifications are needed to the methods to compute the
ultimate soil resistance pu in the formulations for p-y curves for different soil layers.
The effect of a layered soil profile on lateral load-transfer behavior was investigated by
Allen (1985). However, the methods developed by Allen require the use of several specialpurpose computer programs. Full integration of the methods developed by Allen with the
153
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
methods shown herein must be delayed until a later date when this research can be implemented
in a readily usable form.
Earlier, a method for making layering corrections was proposed by Georgiadis (1983).
This method, while not addressing all possible combinations of commonly encountered
conditions, has been implemented as a feature in LPile. The following section describes this
method.
3-11-1 Layering Correction Method of Georgiadis
The layering correction method proposed by Georgiadis (1983) is based on the
determination of “equivalent” depths of all the layers below the upper layer. To do this, the
integral of ultimate resistances are computed over the depth of the upper layer using the methods
for homogeneous soils. The equivalent depth h2 to the top of the underlying layer is computed by
equating the integral of the ultimate resistances over the depth of the overlying layer, F0, to the
integral of ultimate resistance of the underlying layer, F1, assuming that the overlying layer is
composed of the same material properties as the underlying layer. Thus, the following two
integrals are equal and the unknown quantity is the upper limit for the integral, h2 for layer 2.
F0 

h1
pu1dz ..................................................... (3-144)
0
and
F1 

h2
pu 2 dz ......................................................(3-145)
0
The equivalent thickness h2 of the upper layer along with the soil properties of the second layer,
are then used to compute the p-y curves and integral of pu over the depth of the second layer.
The procedure to compute h2 is the following. First, the F0 integral is computed by
dividing the upper layer into 100 evenly thick slices and computing the F0 integral using
Simpson’s rule. Next, the upper limit of the F1 integral is computed using the trapezoidal
integration rule using layers of 0.01 m thickness and the test function equal to F1 minus F0 is
computed. The upper limit h2 is determined when the test function equals zero. This final value
of h2 is determined by determining the two trial values for h2 when the test function transitions
from being negative to positive in sign, then interpolating to determine the value of h2 for which
the test function is equal to zero.
If there are more layers requiring layering correction, the integrals F0 and F1 are added
together to get the new F0 and the equivalent depth of the next underlying layer is computed as
described above.
Note that the discussion above has assumed that the pile head is located at the ground
surface. If the pile head is above the ground surface, the equivalent depths of multiple soil layers
will be computed as the same values as if the pile head is located at the ground surface. If the
pile head is below the ground surface, the computed values for the equivalent depths will be
154
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
different due to the different profile of vertical effective stress versus depth and due to the F0
integral being computed over a fraction of the depth of the top soil layer between the pile head
and the bottom of the layer. The magnitude of the differences between the equivalent depths of
the different pile head elevations depend on the data defining the pile and soil layer properties.
The concepts presented here can be used to get the equivalent thicknesses of multiple,
dissimilar layers of soil overlying the layer for which the equivalent depth is desired. The
equivalent depths may be either smaller or greater than the actual depths of the soil layers and
will depend on the relative strengths of the layers in the soil profiles. This is illustrated in Figure
3-70.
F1 = Total force acting on pile above point i at
the point of soil failure
hi = Equivalent depth of top of layer i
Groundline
h3
h1
Soft Soil (Layer 1)
h2
1
F1
Stronger Soil Below Weaker Soil
(behaves as if shallower)
2
F2
Weaker Soil Below Stronger Soil
(behaves as if deeper)
F3
Figure 3-70 Illustration of Equivalent Depths in a Multi-layer Soil Profile
It should be noted that it is possible that conditions could be present that were not directly
addressed by the method of Georgiadis, but must be addressed in the layering correction
computations. A few of these conditions are:




sloping ground surface, which causes the p-y curves to be stronger for pile displacements
in the uphill direction and weaker in the downhill direction
battered piles (similar effect to sloping ground surface)
tapered piles for which pile diameter varies with depth
when the pile head is below the ground surface (how are the limits of the F0 integral
determined for top layer?)
155
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock



how values of effective stress are computed in the underlying layers (are the values of
effective stress equal to the true effective stress or an “equivalent” effective stress based
on the equivalent depth?)
soil properties that vary with depth (how should any increase in strength with depth in the
lower layer be extrapolated upward when matching the F1 integral for the upper layers?)
what to do when peak resistance is higher than the residual resistance, as is the case for
stiff clay with free water.
The basic approach is to follow the general procedure outlined by Georgiadis, with the
additional assumptions needed to handle the conditions listed above.
The ground surface is always considered as flat and the pile is considered as vertical so
the equivalent depths are equal for loading in both the uphill and downhill directions and the outbatter and in-batter directions. To do otherwise would cause abrupt changes in the equivalent
layer effects for small changes in ground slope when the ground surface is nearly flat or when
the pile is only slightly battered.
In the case of tapered piles and variable soil properties, the F1 integral is computed
assuming that the pile diameter is equal to the pile diameter at the top of the underlying layer and
the shear strength values at the top of the underlying layer are used.
Effective stresses in the expressions for pu are computed using the actual depths in all
computations.
In cases where the residual resistance is lower than the peak resistance, as is the case for
stiff clay with free water, the residual resistance is used.
The layering correction computations may yield results that are predictable in conditions
where pile diameter is constant and soil properties do not vary with depth in a layer and not
always predictable in other conditions where pile diameter and soil properties vary significantly
with depth.
3-11-2 Example p-y Curves in Layered Soils
In this section, an example problem will be presented to illustrate how the layering
correction computations are performed. In the first part of this section, the example will present a
hand solution and in the second part the results of the computer solution will be presented.
The example problem to demonstrate the manner in which layered soils are modeled is
shown in Figure 3-71. As seen in the sketch, a pile with a diameter of 610 mm (24 in.) is
embedded in soil consisting of an upper layer of soft clay, overlying a layer of loose sand, which
in turn overlays a layer of stiff clay. The water table is at the ground surface, and the loading is
static. This example was first presented in Single Piles and Pile Groups Under Lateral Loading
by Reese and Van Impe, 2011.
Four p-y curves are to be computed at depths A, B, C and D, as shown in Figure 3-72.
These four depths are at 1, 3, 5, and 9 m below the pile head. The four p-y curves computed for
the case of layered soils are shown in Figure 3-73. The curve at a depth of 1 m is in the upper
layer of soft clay; the curve at the depth of 3 m is in the sand layer below the soft clay; and the
curves for the depth of 5 m and 9 m are in the lower layer of stiff clay without free water. Note
156
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
that the legend shows the depths of the p-y curves below the pile head, not the equivalent depths.
The equivalent depths are listed in the output report for the LPile analysis.
0m
2.00 m
Soft Clay
2.00 m
Loose
Sand
c = 25 kPa
50 = 0.02
 = 8.0 kN/m3
2.00 m
 = 30 deg.
 = 8.0 kN/m3
4.00 m
c = 100 kPa
50 = 0.005
 = 10.0 kN/m3
Stiff Clay
without
Free Water
6.00 m
Static Loading
10.00 m
610 mm
Figure 3-71 Soil Profile for Example of Layered Soils
0m
Soft
Clay
A
Loose
Sand
B
xEQ = 2.388 m
2.0 m
xEQ = 2.036 m
4.0 m
C
Stiff
Clay
D
Point
Actual
Depth, m
Equivalent
Depth, m
A
1.0
1.000
B
3.0
3.388
C
5.0
3.036
D
9.0
9.000
10.0 m
610 mm
Figure 3-72 Equivalent Depths of Soil Layers Used for Computing p-y Curves
157
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
600
550
Load Intensity p, kN/m
500
450
400
350
300
250
200
150
100
50
0
0.00
0.05
0.10
0.15
0.20
0.25
Lateral Deflection y, m
Depth = 1.00 m
Depth = 3.00 m
Depth = 5.00 m
Depth = 9.00 m
Figure 3-73 Example p-y Curves for Layered Soil
3-11-2-1 Hand Computation Example
Following the method suggested by Georgiadis, the p-y curve for soft clay can be
computed as if the profile consists only of that soil. The value of pu was computed to be 63.1
kN/m and the p-y curve was computed using Equations 3-20 and 3-21, shown below.

  avg
J 
pu  3 
x  x  cb .............................................. (3-20)
c
b 

pu  9 c b .......................................................... (3-21)
When dealing with the layer of loose sand, an equivalent depth is found such that the
integrals of the ultimate soil resistance of an equivalent sand layer and for the soft clay are equal
at the interface. The first step is to employ Equations 3-20 and 3-21 of Section 3-3-7 for pu for
the soft clay and to compute the sum of pu values at the depth of 2 m. Preliminary computations
showed that the transition depth xr where Equations 3-20 and 3-21 are equal occurred at a depth
of 5.56 m, therefore Equation 3-20 would be used over the full depth of 2 m. The value of pu
158
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
varies linearly with depth and values of 45.7 kN/m and 80.5 kN/m were computed for depths of
0 and 2 m, respectively. The sum of values of pu was computed to be 126.2 kN at a depth of 2 m.
The next step is compute the depth of the sand with the properties shown in Figure 3-71
such that the integral of the computed values of pu for the sand will equal 126.2 kN. The
equations for pu in sand are nonlinear and the integration must be performed numerically.
Equation 3-53 and Equation 3-54 are employed, along with values of As , to compute values of
pu values as a function of depth. Figure 3-28 was employed and values of As are tabulated for
ease of computation in Table 3-12 below. Preliminary computations find that the intersection of
Equations 3-53 and 3-54 occurs at 8.3 m below the ground surface so Equation 3-53 is used for
all computations of pu. Tabulated values of pu and the F1 integral in sand are shown in Table 313 for each 0.5 m of depth. Interpolating between the values in Table 3-13 found that the F1
integral equaled the value of F0 at a depth of 2.35 m. Thus, the equivalent thickness of loose sand
to replace the 2.0 m of soft clay was found to be 2.35 meters. Thus, the equivalent depth to point
B in loose sand, 1m below the top of layer 2, is 3.35 meters.
 K x tan  sin 
tan 
pst   x  0

(b  x tan  tan  )
.................... (3-53)
 tan(   ) cos  tan(   )
 K 0 x tan  (tan sin   tan  )  K Ab

psd  K A b  x(tan8   1)  K0 b  x tan tan4  ............................. (3-54)
Table 3-12 Tablulated Values of As as Function of z/b
z/b
As
0
2.88
0.5
2.497
1.0
2.113
1.5
1.73
2.0
1.47
2.5
1.24
3.0
1.05
3.5
0.95
4.0
0.90
4.5
0.88
below 4.5
0.88
159
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Table 3-13 Computed Values of pu and F1 for the Sand in Figure 3-71 as Function of Depth
Depth, m
pu, kN/m
F1, kN
0
0
0
0.5
23.26
5.812
1.0
47.33
23.46
1.5
67.95
52.28
2.0
87.20
91.07
2.5
115.31
141.70
3.0
155.48
209.40
3.5
204.95
299.50
4.0
261.14
416.03
4.5
324.07
562.33
An equivalent depth of stiff clay was found such that the sum of the ultimate soil
resistance for the top of the stiff clay layer is equal to the sum of the ultimate soil resistance of
the loose sand and soft clay. In making the computations, the equivalent and actual thicknesses
of the loose sand, 2.35 m and 2.00 m, were replaced by 2.10 m of stiff clay. Thus, the actual
thicknesses of the soft clay and loose sand of 4.00 m were reduced by 1.90 m, leading to
equivalent depths in the stiff clay of point C of 3.10 m. Point D fell at a depth for which deep
conditions control, so the equivalent and actual depths were equivalent and equal to 9.00 m.
3-11-2-2 Computer Solution Example
The same problem was analyzed by LPile and the results computed for the equivalent
depths of the tops of layers are presented in Tables 3-14. The equivalent depths for the p-y curves
computed by hand and by LPile are presented in Table 3-15. The results are approximately
equivalent and the numerical differences are due to the thinner layers used in the computer
solution. The computed p-y curves are illustrated in Figure 3-72.
Table 3-14 Equivalent Depths of Tops of Soil Layers Computed by LPile
Layer No.
Top of Layer
Below Pile
Head,
meters
1
2
3
0.00
2.0000
4.0000
Equivalent
Top Depth
Same Layer
Layer is
Below Pile
Type As
Rock or is
Head,
Layer Above
Below Rock
meters
0.00
N.A.
No
2.3883
No
No
2.0053
No
No
*F0 for layer n+1 = F0 + F1 for layer n.
160
F0 Integral
for Layer,
kN*
F1 Integral
for Layer,
kN
0.00
126.2600
477.3266
126.2600
351.0666
N.A.
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
3-11-2-3 Comment on Choice of Layer Type for Stiff Clay
Another point of interest is that the recommendations for p-y curves for stiff clay in the
presence of no free water were used for the stiff clay. This decision was based on the assumption
that the sand above the stiff clay can move downward and fill any gap that might develop
between the clay and the pile. Furthermore, in the pile load test in stiff clay with free water, the
free water was flushed out of the annual space between the soil and pile with each cycle of
loading, thereby eroding soil from around the upper portion of the pile. The presence of soft clay
and sand to a depth of 4.00 m above the stiff clay is believed to adequate to suppress the erosion
of soil by the free water even if the sand does not fill in any potential gap around the pile.
Table 3-15 Equivalent Depths of Example p-y Curves Computed by Hand and by LPile
Equivalent
Depth of p-y
Curve by Hand,
m
Equivalent Depth
of p-y Curve by
LPile, m
p-y Curve
Layer
Depth of p-y
Curve, m
A
soft clay
1
1.00
1.0000
B
sand
3
3.35
3.3883
C
stiff clay
5
3.10
3.0053
D
stiff clay
9
9.00
9.0000
3-11-3 Modified Equations Using Equivalent Depth
The equations used to compute lateral load transfer at failure are the ultimate values for
flat ground surfaces and vertical piles.
Layering corrections are applied only to the p-y curve models that have different
expressions for ultimate lateral load transfer for shallow and deep conditions. The six p-y curve
models that have different expressions for shallow and deep conditions are:






soft clay by Matlock,
API soft clay,
Stiff clay with free water,
Stiff clay without free water,
Stiff clay without free water with user-defined k value, and
Cemented c- soil (cemented silt)
The following notation is used in the modified equations. Note that “elevation” is the
depth below the pile head. The equivalent depth is defined as a distance below the pile head and
that the equivalent depth may be either shallower or deeper than the actual depth below the pile
head.
xeq = equivalent depth
161
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
xeq = equivalent depth of top of soil layer + elevation of nodal point – Elevation of Layer Top
xact = Actual depth below ground surface
3-11-3-1 Soft Clays Under Static Loading Conditions
  x
J 
pu  3  avg act  xeq  cb .......................................... (3-146)
c
b 

pu  9 c b ........................................................ (3-147)
3-11-3-2 Soft Clay Under Cyclic Loading Conditions
xr 
6cb
..................................................... (3-148)
  b  Jc
x 
p  0.72 pu  eq  .................................................. (3-149)
 xr 
3-11-3-3 Stiff Clay with Free Water Under Static Loading conditions
pct  2cavgb   bxact  2.83cavg xeq ..................................... (3-150)
pcd  11c b ...................................................... (3-151)
pc  min  pct , pcd  ..................................................(3-152)


p  pc 1.225 As  0.75 As  0.411 .................................... (3-153)
Where As is determined from the ratio xeq/b
3-11-3-4 Stiff Clay with Free Water Under Cyclic Loading Conditions
p  0.936 Ac pc 
0.102
pc y p ........................................ (3-154)
y 50
Where Ac is determined from the ratio xeq/b
Stiff clay without free water for both static and cyclic loading:
162
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
 
J 
pus  3  avg xact  xeq  cb ......................................... (3-155)
c
b 

pud  9 c b ........................................................ (3-156)
pu  min  pus , pud  ................................................. (3-157)
3-11-3-5 Sand
 K x tan( ) sin(  )
tan( )
pst   xact  0

b  xeq tan( ) tan( )
............ (3-158)
 tan(   ) cos( ) tan(   )
 K 0 xeq tan( ) tan( ) sin(  )  tan( )  K Ab





psd  K A b  xact tan8 ( )  1  K0 b  xact tan( ) tan4 ( ) ..................... (3-159)
ps  min  pst , psd  ................................................. (3-160)
pu  As ps or pu  Ac ps ............................................ (3-161)
3-11-3-6 API Sand
pus  (C1 xeq  C2b)  xact ............................................. (3-162)
pud  C3b  xact .................................................... (3-163)
pu  min  pus , pud 
................................................. (3-164)
where:


 
 1

C1  tan( ) K p tan( )  K 0 tan( ) sin(  ) 
 1  tan( )  .......... (3-165)

 cos( ) 

 
C2  K p  K a
.................................................... (3-166)
C3  K p2 K p  K0 tan( )  Ka ........................................ (3-167)
3-11-3-7 Cemented c- Soils
 K x tan( ) sin(  )
tan( )
ps    xact  0 eq

b  xeq tan( ) tan( )
....................... (3-168)
 tan(   ) cos( ) tan(   )
 K 0 xeq tan( )tan( ) sin(  )  tan( )  K Ab

163

Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock


pd  K A b   xact tan8 ( )  1  K0 b  x tan tan4 ( ) ................................................ (3-169)
pu  min ps , pd  ................................................. (3-170)
Compute pu for static loading using
pu  As pu ...................................................... (3-171)
or for cyclic loading using
pu  Ac pu ...................................................... (3-172)
Use the appropriate value of As or Ac from Figure 3-28 for the particular non-dimensional
depth (xeq/b) and type of loading.
p  A pu ......................................................... (3-173)
3-12 Modifications to p-y Curves for Pile Batter and Ground Slope
3-12-1 Piles in Sloping Ground
The formulations for p-y curves presented to this manual were developed for a horizontal
ground surface. In order to allow designs to be made if a pile is installed on a slope,
modifications must be made to the p-y curves. The modifications involve revisions in the manner
in which the ultimate soil resistance is computed. In this regard, the assumption is made that the
flow-around failure that occurs at depth will not be influenced by sloping ground; therefore, only
the equations for the wedge-type failures near the ground surface need modification.
The modifications to p-y curves presented here are based on earth pressure theory and
should be considered as preliminary. Future changes may be needed once laboratory and field
study are completed.
3-12-1-1 Equations for Ultimate Resistance in Clay in Sloping Ground
The ultimate soil resistance at the ground surface for a pile in in saturated clay with a
horizontal ground surface was developed by Reese (1958) and is
( pu ) ca  2 ca b   b H  2.83 ca H ..................................... (3-174)
If the ground surface has a slope angle  as shown in Figure 3-74, the soil resistance in the
downhill direction of movement, following Reese’s approach is:
( pu ) ca  2 ca b   b H  2.83 ca H 
The soil resistance in the uphill direction of movement is:
164
1
............................. (3-175)
1  tan
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
( pu ) ca  2 ca b   b H  2.83 ca H 
cos 
....................... (3-176)
2 cos(45   )
+
+
Figure 3-74 Pile in Sloping Ground and Battered Pile
where:
(pu)ca = ultimate soil resistance near ground surface,
ca =
average undrained shear strength,
b =
pile diameter,
 =
average unit weight of soil,
H =
depth from ground surface to point along pile where soil resistance is computed, and
 =
angle of slope as measured in degrees from the horizontal.
A comparison of Equations 3-175 and 3-176 shows that the equations are identical except for the
terms at the right side of the parenthesis. If  is equal to zero, the equations become equal to the
original equation.
3-12-1-2 Equations for Ultimate Resistance in Sand
The ultimate soil resistance at the ground surface for a pile in in sand with a horizontal
ground surface is
165
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
 K H tan  sin 
tan 
( pu ) sa   H  0

(b  H tan  tan  )
............... (3-177)
 tan(   ) cos  tan(   )
 K 0 H tan  (tan  sin   tan  )  K Ab
If the ground surface has a slope angle , the ultimate soil resistance in the downhill direction is:
 K H tan  sin 
( pu ) sa   H  0
( 4 D13  3D12  1)
 tan(   ) cos 
tan 


bD2  H tan  tan D22 
tan(   )
............... (3-178)
 K 0 H tan  (tan  sin   tan  )( 4 D13  3D12  1)  K Ab
where:
D1 
tan  tan
................................................. (3-179)
tan  tan  1
D2  1  D1 , and ................................................... (3-180)
K A  cos 
cos   cos 2   cos 2 
cos   cos 2   cos 2 
.................................... (3-181)
where  is defined in Figure 3-74.
Note that the denominator of Equation 3-179 for D1 will equal zero when the sum of the
slope and friction angles is 90 degrees. This occurs when the inclination of the failure wedge is
parallel to the ground surface. In computations, the lower value of (pu)sa or to pu from Equation
3-54 is used, so no computational problem arises.
The ultimate soil resistance in the uphill direction is:
 K H tan  sin 
( pu ) sa   H  0
( 4 D33  3D32  1)
tan(



)
cos


tan 


bD4  H tan  tan D42 
tan(   )
............... (3-182)
 K 0 H tan  (tan  sin   tan  )( 4 D33  3D32  1)  K Ab
where
D3 
tan  tan 
................................................ (3-183)
1  tan  tan 
166
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
and
D4  D3  1 ....................................................... (3-184)
This completes the necessary derivations for modifying the equations for clay and sand to
analyze a pile under lateral load in sloping ground.
3-12-1-3 Effect of Direction of Loading on Output p-y Curves
The equations for computing maximum soil resistance for p-y curves in sand depend on
whether the pile is being pushed up or down the slope. LPile determines which case to compute
by using the values of lateral pile deflection and slope angle. Whenever, p-y curves are generated
for output, the curve that is output by the program is based on the lateral deflection computed for
loading case 1. If the user desires output of both sides of an unsymmetrical p-y curve it is
necessary to run an analysis twice, with the pile-head loadings for shear, moment, rotation, or
displacement reversed for the two analyses, while keeping the axial thrust force unchanged. The
user may then combine the two output curves together.
3-12-2 Effect of Batter on p-y Curves in Clay and Sand
Piles are sometimes constructed with an intentional inclination. This inclination or angle
is called batter and piles that are not vertical are called battered piles. Vertical piles are
sometimes referred to as “plumb” piles.
The effect of batter on the behavior of laterally loaded piles has been investigated in
several model test studies. The lateral, soil-resistance curves for a vertical pile in a horizontal
ground surface were modified by a modifying constant to account for the effect of the inclination
of the pile. The values of the modifying constant as a function of the batter angle were deduced
from the results of the model tests (Awoshika and Reese, 1971) and from results of full-scale
tests reported by Kubo (1964). The modifying constant to be used is shown by the solid line in
Figure 3-75.
167
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
Pile Batter Angle in LPile, degrees
Ratio of Soil Resistance
2.0
30
20
0
10
 10
 20
−
 30

Load
1.0
Kubo’s tests
Awoshika’s tests
0
30
20
10
0
10
20
30
Ground Slope Angle in LPile, degrees
Figure 3-75 Soil Resistance Ratios for p-y Curves for Battered Piles from Experiment
from Kubo (1964) and Awoshika and Reese (1971)
This modifying constant is used to increase or decrease the value of pult, which will in
turn cause the p-values to be modified proportionally. While it is likely that the values of pult for
the deeper soils are not affected by pile batter, the behavior of a pile is only slightly affected by
the resistance of the deeper soils; therefore, the use of the modifying constant for all depths of a
battered pile is believed to be satisfactory.
As shown in Figure 3-75, the agreement between the empirical curve and the experiments
for the outward batter piles ( is positive) agrees somewhat better that for the inward batter piles.
The data indicate that the use of the modifying constant for inward batter piles will yield results
that are somewhat doubtful; therefore, on important projects, full-scale load testing is desirable.
3-12-3 Modeling of Piles in Short Slopes
Whenever piles are installed in slopes, the user has two methods available in LPile to
model the pile and slope. One way is the specify the slope angle of the ground surface and the
other way is to use Figure 3-75 to determine what value of p-multiplier to use. The best choice of
which method to use depends on the elevation of the pile tip.
If the pile tip is above the toe of the slope, the user should just specify the ground slope
angle and pile batter angle. LPile will then compute the effective slope angle, e, as the
difference between the pile batter angle  and the ground slope angle i. LPile then uses e in
place of 
168
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
If the pile tip is below the toe of the slope, the user should specify a p-multiplier over the
depth range that is above the toe of the slope. For example, if the slope ratio is 3 to 1 and the pile
is loaded in the downslope direction, the p-multiplier is approximately 0.5.
3-13 Shearing Force Acting at Pile Tip
It is possible to include a shearing force at the bottom of the pile in the development of
the finite difference equations. Inclusion of the modelling of a shearing force at the pile tip
would be become important only to those cases where the pile is short; that is, where there is
only one point of zero deflection in the pile.
Currently, formulations to compute a curve of shearing force as a function of deflection
are unavailable. It is believed that construction techniques will have a major effect on the
development of shearing forces at the pile tip. It is not possible for design engineers to know
what these effects are since design computations are usually performed far in advance of
construction of the foundations.
At present, all that the geotechnical engineer can do is to make an estimate of the
necessary force-deflection curve by considering pile geometry and soil properties or to infer a
relationship from the results of load tests on piles of similar size at the site of the project.
169
Chapter 3 – Lateral Load-Transfer Curves for Soil and Rock
(This page was deliberately left blank)
170
Chapter 4
Special Analyses
4-1 Introduction
LPile has several options for making special analyses. This chapter provides explanations
about the various options and guidance for using the optional features for making special
analyses.
4-2 Computation of Top Deflection versus Pile Length
This option is available only in the conventional analysis mode and is not available in the
LRFD analysis mode.
The activation of this option is made by selecting the option when entering the load
definitions. Note that this option is not available if one of the pile head loading conditions is
displacement.
In the following example, shown in Figure 4-1, a pile with elastic bending properties is
loaded with five levels of pile-head shear at 0%, 50%, 100%, 150%, and 200% of the service
load. The following figures illustrate the problem conditions, lateral pile deflection versus depth,
pile-top deflection versus displacement, and curves of pile-top deflection versus pile length.
When the problem computes the curves of pile-top deflection versus pile length, the
program first computes pile-top deflection for the full length. The full pile length is 12 meters in
this example. Then LPile reduces the pile length in increments of 5 percent of the full length (0.6
meters in this example). Thus, the pile length values for which pile-top deflection is computed
for are 12 meters, 11.4 meters, 10.8 meters, and so on, until the computed pile-top deflection
becomes excessive.
A typical plot top deflection versus pile length for a pile in soil profile composed of
layers of clay and sand is shown in Figure 4-4. Usually, when LPile generates this graph, it uses
all of the computed values. However, in cases where there is a change in sign of lateral
deflection when the pile is shortened, LPile will omit all data points with an opposite sign from
the top deflection for the full length.
When examining the results in a graph of top deflection versus pile length, the design
engineer may find that the top deflection at full length is too large and that some change in the
dimensions of the pile are required. The manner in which this decision is made depends on the
shape of the curves in the graph.
If the right-hand portions of the curves are flat or nearly flat, it is not possible to reduce
pile-top deflection by lengthening the pile. The only available option is to increase the diameter
of the pile or to increase the number of piles, so that the average load per pile is reduced.
171
Chapter 4 – Special Analyses
250 kN DL + 100 kN LL = 350 kN
Service Loads Shown
80 kN DL + 20 kN LL = 100 kN
Soft Clay, 6 m
Sand, 9 m
M=0
c = 12 to 24 kPa
 = 8.95 kN/m3
 = 38 to 40
 = 9.50 kN/m3
Elastic Circular Pile with L = 12 m, D = 1 m, E = 27,500,000 kPa
Figure 4-1 Pile and Soil Profile for Example Problem
Figure 4-2 Variation of Top Deflection versus Depth for Example Problem
172
Chapter 4 – Special Analyses
200
Shear Force, kN
150
100
50
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
Top Deflection, m
Figure 4-3 Pile-head Load versus Deflection for Example
Figure 4-4 Top Deflection versus Pile Length for Example
If the right-hand portions of the curves are inclined, it is possible to reduce the pile-top
deflection by lengthening the pile. However, there are situations where other factors may need to
be considered. One common situation is when the pile-top deflection is acceptable as long as the
173
Chapter 4 – Special Analyses
pile tip is sufficiently embedded in a strong layer of soil or rock. In this case, the designer must
decide how reliably the depth of the strong layer can be predicted. In such a case, the designer
may wish to specify the depth of a drilled foundation long enough to penetrate into the strong
layer and add a requirement for the construction inspector to notify the design engineer if the
strong layer is not reached after drilling to the planned depth. In the case of a driven pile
foundation, the design engineer can set the pile length to be long enough to reach a specified
driving resistance that is based a pile driving analysis that is based on the presence of the strong
layer.
4-3 Analysis of Piles Loaded by Soil Movements
In general, a pile subjected to lateral loading is supported by the soil. However, there are
cases in which the soil is displaced and the load imparted by the displaced soil must be taken into
account.
Lateral soil movements can result from several causes. A few of the causes are slope
movements (probably the most common cause), nearby fill placement or excavation, and lateral
soil movements due to seepage forces resulting from water flowing through the soil in which the
pile is founded.
A number of cases involved with pile loaded by soil movements have been reported in
the literature. In many cases, the piles have supported bridge abutments for which the bridge
approach embankments were unstable.
Earthquakes are another source of lateral soil movements. Free-field displacements are
motions of the soil that may be induced by the earthquake, or by unstable slope movements or
lateral spreading triggered by the earthquake. This important problem can be extremely complex
to analyze. In such a case, the first step in the solution is to predict the soil movements with
depth below the soil surface using special analyses that may consider a synthetic acceleration
time history of the design earthquake.
Isenhower (1992) developed a method of analysis based on computing soil reaction as a
function of the relative displacement between the pile and soil. If the pile at a particular depth
undergoes greater displacement than the soil movement at that depth then the soil will provide
resistance to the pile. If the opposite occurs, the soil will then apply an extra lateral loading to the
pile.
If a pile is in a soil layer undergoing lateral movement, the soil reaction depends on the
relative movement of the pile and soil. The p-y modulus is evaluated for a pile displacement
relative to the soil displacement. This technique is illustrated in Figure 4-5.
The solution is implemented in LPile by modifying the governing differential equation to
EI
d4y
d2y

Q
 E py ( y  ys )  W  0 .......................................(4-1)
dx 4
dx 2
It should be noted that it is often difficult to determine the soil displacement profile for
use in the LPile analysis. Occasionally, it is possible to install slope inclinometer casings at a
project site to measure soil displacements as they develop. In other cases, the soil displacement
profile may be developed using the finite element method.
174
Chapter 4 – Special Analyses
p
ps
y
yys
ys
Epy
y
Figure 4-5 Evaluation of Soil Modulus from p-y Curve Displaced by Soil Movement
Analyses that include loading by soil movements is controlled by two options in the
Program Options and Settings dialog. These options are to input a single soil movement profile
that is applied to all cases of loading or to input multiple profiles of soil movement versus depth
that are applied to specified load cases.
The user should note that loading by soil movement profiles is available only for
conventional analysis and is not available for LRFD analyses.
4-4 Analysis of Pile Buckling
It is possible to use LPile to analyze pile buckling using an iterative procedure, combined
with evaluation of the computed results by the user. The following describes a typical procedure
and a potential difficulty caused by inappropriate input.
4-4-1 Procedure for Analysis of Pile Buckling
The procedure for analysis of pile buckling is the following.
1. In the Program Options and Settings, increase the maximum number iterations to 975 to
avoid early termination of an analysis
2. Make an initial conventional analysis in which the maximum loads expected for the
foundation are analyzed. Note the sign of the pile-head deflection, which will depend on
the sign of the applied loads. If the pile section is nonlinear (not elastic, elastic-plastic, or
user-input nonlinear bending), examine the output report to find the maximum axial
structural capacity for the pile. Use this axial structural capacity to estimate the maximum
axial thrust load to be applied in the buckling analysis.
175
Chapter 4 – Special Analyses
3. In the Program Options and Settings dialog, select a pile buckling analysis by checking
the Computational Options group.
4. Open the Controls for Pile Buckling Analysis dialog
5. Select the appropriate pile-head fixity condition for the pile buckling analysis.
6. Enter the maximum pile-head loading for the pile-head fixity condition.
7. Increase the magnitude of axial thrust force in even increments for the subsequent load
cases. An initial increment size may be 5 percent of the axial structural capacity. Up to
100 load steps may be specified.
8. Perform the analysis with the option for pile buckling analysis.
9. Examine the output report and pile buckling graph.
An example of a buckling study was performed.
The pile head is at the ground surface.
The soil profile is composed of three layers and is sand from 0 to 2 meters (API sand,  =
18 kN/m3,  = 30 degrees, and k = 13,550 kN/m3), soft clay from 2 to 8.5 meters ( = 7.19
kN/m3, c = 1 kPa, 50 = 0.06), and sand below 8.5 meters (API sand,  = 10 kN/m3,  = 40
degrees, k = 60,000 kN/m3).
The pile has a diameter of 0.15 meters, a length of 18 meters, a cross-sectional area of
0.0177 m2, a moment of inertia of 1.678  10-7, and a Young’s modulus of 200 GPa.
Two pile buckling curves are plotted in Figure 3-6. For one curve, the specified lateral
shear force is 0.1 kN and buckling failure occurs for thrust values above 218 kN. For the second
curve, the specified lateral shear force is 1.0 kN and buckling failure occurs for thrust values
above 121 kN. This graph illustrates that the buckling capacity is a function of the pile head
loading conditions, with the lower pile buckling capacity associated with the higher loading
condition.
These curves illustrate that the axial buckling capacity is a function of the specified
lateral shear force used in the analysis and that the buckling capacity is reduced as the lateral
shear force is increased. Thus, it is important to use the maximum expected load condition, if it is
known, since a range of computed buckling capacities is possible.
176
Chapter 4 – Special Analyses
250
V = 0.1 kN
Axial Thrust Force, kN
200
V = 1.0 kN
150
100
50
0
0
0.002
0.004
0.006
0.008
0.01
Pilehead Deflection, meters
Figure 4-6 Examples of Pile Buckling Curves for Different Shear Force Values
4-4-2 Example of An Incorrect Pile Buckling Analysis
The following is an example of an incorrect buckling analysis. In this analysis, the soil
and pile properties are the same as used in the example above. The shear force is specified as 5.0
kN (larger than the 0.1 and 1.0 kN thrust values used in the prior example).
If the section is either a drilled shaft (bored pile) or prestressed concrete pile with low
levels of reinforcement, it may be possible to obtain buckling results for axial thrust values
higher than the axial buckling capacity, but the sign will be reversed. The reason for this is a
large axial thrust value will create compression over the full section. This causes the moment
capacity to be controlled by crushing of the concrete and not by yielding of the reinforcement.
In the incorrect analysis shown in Figure 3-7, the incorrect analysis used a range of axial
thrust forces that was too large and the computed lateral deflections were on both positive and
negative as shown in Figure 3-7. In a correct buckling analysis, the computed lateral deflections
should always have the same sign. In the correct analysis, also shown in Figure 3-7, the axial
thrust values were increased in smaller increments and non-convergence due to excessive lateral
deflections occurred at a thrust levels higher than 39 kN.
177
Chapter 4 – Special Analyses
450
Correct
400
Incorrect
Axial Thrust Force, kN
350
300
250
200
150
100
50
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Pile-head Deflection, meters
Figure 4-7 Examples of Correct and Incorrect Pile Buckling Analyses
4-4-3 Evaluation of Pile Buckling Capacity
The analysis of buckling cannot calculate the buckling capacity theoretically. It can only
evaluated the buckling capacity approximately by simulating the pre-buckling behavior. The
results of an analysis can be interpreted using a technique based on the fitting of a hyperbolic
curve to the computed results for pre-buckling behavior.
A typical buckling-deformation curve for a given set of pile-head loading is shown in
Figure 4-8. The lateral deflection of the pile head is denoted by y0.
The equation for a hyperbolic curve that originates at y0 is
P
y  y0
.......................................................(4-2)
b  a y  y 0 
Where y is deflection, P is the axial thrust force and a and b are curve-fitting parameters. This
expression may be re-written as
y  y0
 b  a y  y 0  ...................................................(4-3)
P
178
Chapter 4 – Special Analyses
The pile deflections may be re-plotted in which values of y  y 0 are plotted along the xaxis and values of  y  y 0  / P are plotted along the y-axis. In many cases, this will result in a
straight line with a slope of a and a y-intercept of b as shown in Figure 4-9.
P
y0
Pile-head Deflection, y
Figure 4-8 Typical Results from Pile Buckling Analysis
y  y0
P
a
1
b
y – y0
Figure 4-9 Pile Buckling Results Showing a and b
The pile buckling capacity, Pcrit, is calculated from
179
Chapter 4 – Special Analyses
Pcrit 
1
...............................................................(4-4)
a
The estimate pile buckling capacity is computed from the shape of the pile-head response
curve and is not based on the magnitude of maximum moment compared to the plastic moment
capacity of the pile. For piles with nonlinear bending behavior, the estimated buckling capacity
may over-estimate the actual buckling capacity if the buckling capacity is controlled by the pile’s
plastic moment capacity. Thus, for analyses of nonlinear piles, the user should compare the
maximum moment developed in the pile to the plastic moment capacity. If the two values are
close, the buckling capacity should be reported as the last axial thrust value for which a solution
was reported.
4-5 Pushover Analysis of Piles
The program feature for pushover analysis has options for different pile-head fixity
options and the setting of the range and distribution of pushover deflection. The output of the
pushover analysis is displayed in graphs of pile-head shear force versus deflection and maximum
moment developed in the pile versus deflection.
The dialog for input of controls for performing a pushover analysis are shown in Figure
4-10. The control parameters allow the user to specify the pile-head fixity condition and how the
pushover displacement points are generated. Optionally, the user may specify the pushover
displacements to be used.
Figure 4-10 LPile Dialog for Controls for Pushover Analysis
180
Chapter 4 – Special Analyses
4-5-1 Procedure for Pushover Analysis
The pushover analysis is performed by running a series of analyses for displacement-zero
moment pile-head conditions for pinned-head piles and analyses for displacement-zero slope
pile-head conditions for fixed head piles. The displacements used are controlled by the maximum
and minimum displacement values specified and the displacement distribution method. The
displacement distribution method may be either logarithmic (which requires a non-zero, positive
minimum and maximum displacement values), arithmetic, or a set of user-specified pile-head
displacement values. The number of loading steps sets the number of pile-head displacement
values generated for the pushover analysis.
The axial thrust force used in the pushover analysis must be entered in the dialog. If the
pile being analyzed is not an elastic pile, the user should make sure that the axial thrust force
entered matches one the values for axial thrust entered in the conventional pile-head loadings
table to make sure that the correct nonlinear bending properties are used in the pushover analysis.
If the values do not match, the nonlinear bending properties for the next closest axial thrust will
be used by LPile for the pushover analysis.
4-5-2 Example of Pushover Analysis
Some typical results from a pushover analysis are presented in the following two figures.
Figure 4-11 presents the pile-head shear force versus displacement for pinned and fixed-head
conditions and indicates the maximum level of shear force that can be developed for the two
conditions. Similarly, Figure 4-12 presents the maximum moment developed in the pile (a
prestressed concrete pile in this example) versus displacement and shows that a plastic hinge
develops in the fixed-head pile at a lower displacement than for the pinned-head pile.
Formation of
plastic hinge
Figure 4-11 Pile-head Shear Force versus Displacement from Pushover Analysis
181
Chapter 4 – Special Analyses
Formation of
plastic hinge
Figure 4-12 Maximum Moment Developed in Pile versus Displacement from Pushover Analysis
In general, it is not possible to develop more than one plastic hinge in a pile if the pilehead condition is pinned. It is sometimes possible to develop two plastic hinges in the pile if the
pile-head condition is fixed against rotation and the axial load is zero.
4-5-3 Evaluation of Pushover Analysis
Evaluation of a pushover analysis requires examination of both graphs generated by the
analysis. It is important to identify the load levels at which plastic hinges form and the location
of the plastic hinges.
In many practical situations, the pile-head fixity conditions are neither fixed or free, but
may be close to one of these conditions. If actual conditions are close to being fixed-head
conditions, the amount of pile-head deflection required to develop a plastic hinge will be
somewhat greater than the value shown in the pushover analysis for fixed-head conditions.
Similarly, if actual conditions are close to being free-head, the amount of pile-head deflection
required to develop a plastic hinge will be somewhat less than the value shown in the pushover
analysis for free-head conditions.
4-6 Computation of Foundation Stiffness Matrix
Stiffness matrices are often used to model foundations in structural analyses and LPile
provides an option for evaluating the lateral stiffness of a deep foundation. This feature of LPile
allows the user to solve for stiffness coefficients, as illustrated by the sketches shown in Figure
4-13, of pile-head movements and rotations as functions of incremental loadings. The stiffness
coefficients are computed by dividing the applied pile-head loadings into increments and then
computing the pile head reactions for each level of loading. First, the deflection of the pile head
is computed for each lateral-load increment with the rotation at the pile head being restrained to
zero. Next, the rotation of the pile head is computed for each bending-moment increment with
the lateral deflection at the pile head being restrained to zero. The user can thus define the
182
Chapter 4 – Special Analyses
stiffness matrix directly based on the relationship between computed deformation and applied
load. For instance, the stiffness coefficient K33, shown in Figure 4-13, can be obtained by
dividing the applied moment M by the computed rotation θ at the pile head.
K33
K22
Moment
M
K11
K33
Rotation
 K11
 0

 0

 K11
 x 
P 
0 0 0 0 Q
   x
 0 K

K23  y  
 V 
22
K
K
H


y
 0 22 K 32 23K33   
M 
K 32
  
K 33  
M    
Figure 4-13 Example of Stiffness Matrix of Foundation
The definitions of the pile-head stiffness values and their engineering units computed by
LPile are the following:
pile - head shear force reaction
lbs
kN

or
pile - head deflection
inch
meter
pile - head moment reaction
in - lbs kN - m
K 32 

or
pile - head deflection
inch
meter
pile - head shear force reaction
lbs
kN
K 23 

or
pile - head rotation
radian
radian
pile - head moment reaction
in - lbs kN - m
K 33 

or
pile - head rotation
radian
radian
K 22 
The nature of the cross-coupled stiffness coefficients is illustrated in Figure 4-14.
183
Chapter 4 – Special Analyses
M
y=0
0
V
K 22 
V
y
K 23 
V
K 32 
M
y
K 33 
M
M


y0
V
Coupled shear
restores zero
deflection
=0
Coupled moment
restores zero
rotation
Figure 4-14 Coefficients of Pile-head Stiffness Matrix
Most analytical methods in structural mechanics can employ either the stiffness matrix or
the flexibility matrix to define the support condition at the pile head. If the user prefers to use the
stiffness matrix in the structural model, Figure 4-14 illustrates the procedures used to compute a
stiffness matrix. The initial coefficients for the stiffness matrix may be defined based on the
magnitude of the service load. The user may need to make several iterations before achieving
acceptable agreement.
The dialog for Controls for Computation of Stiffness Matrix is shown in Figure 4-15. The
feature for computation of pile-head stiffness matrix values has three options to control how the
values are computed. In the first method, which is identical to the method used in versions of
LPile prior to LPile 2013, the loads used for computation of pile-head stiffness are those
specified in Load Case 1 for conventional loading. This method did not allow the user to control
the lateral displacement and pile-head rotation, so the second and third options were added to
provide this capability. In the second method, the maximum displacement and rotation are set by
the values computed for Load Case 1 for conventional loading. In the third method, the user may
specify the maximum values of pile-head displacement and rotation.
184
Chapter 4 – Special Analyses
Figure 4-15 Dialog for Controls for Computation of Stiffness Matrix
In all three methods, the user may control the number of steps of loading and the method
used to compute the steps of loading. Up to 100 step of loading can be specified, with the default
value equal to 10 steps. The computation method used to compute the magnitudes of the steps of
loading may either be scaled in proportion of the logarithm of the specified loading or by evenly
spaced values (arithmetically distributed values).
185
Chapter 4 – Special Analyses
(This page was deliberately left blank)
186
Chapter 5
Computation of Nonlinear Bending Stiffness
and Moment Capacity
5-1 Introduction
5-1-1 Application
The designer of deep foundations under lateral loading must make computations to
ascertain that three factors of performance are within tolerable limits: combined axial and
bending stress, shear stress, and pile-head deflection. The flexural rigidity, EI, of the deep
foundation (bending stiffness) is an important parameter that influences the computations (Reese
and Wang, 1988; Isenhower, 1994).
In general, flexural rigidity of reinforced concrete varies nonlinearly with the level of
applied bending moment, and to employ a constant value of EI in the p-y analysis for a concrete
pile will result in some degree of inaccuracy in the computations.
The response of a pile is nonlinear with respect to load because the soil has nonlinear
stress-strain characteristics. Consequently, the load and resistance factor design (LRFD) method
is recommended when evaluating piles as structural members. This requires evaluation of the
nominal (i.e. unfactored) bending moment of the deep foundation.
Special features in LPile have been developed to compute the nominal-moment capacity
of a reinforced-concrete drilled shaft, prestressed concrete pile, or steel-pipe pile and to compute
the bending stiffness of such piles as a function of applied moment or bending curvature. The
designer can utilize this information to make a correct judgment in the selection of a
representative EI value in accordance with the loading range and can compute the ultimate lateral
load for a given cross-section.
5-1-2 Assumptions
The program computes the behavior of a beam or beam-column. It is of interest to note
that the EI of the concrete member will undergo a significant change in EI when tensile cracking
occurs. In the coding used herein, the assumption is made that the tensile strength of concrete is
minimal and that cracking will be closely spaced when it appears. Actually, such cracks will
initially be spaced at some distance apart and the change in the EI will not be so drastic. In
respect to the cracking of concrete, therefore, the EI for a beam will change more gradually than
is given by the coding.
The nominal bending moment of a reinforced-concrete section in compression is
computed at a compression-control strain limit in concrete of 0.003 and is not affected by the
crack spacing. The ultimate bending moment for steel, because of the large amount of
deformation of steel when stressed about the proportional limit, is taken at a maximum strain of
0.015, which is five times the crushing strain of concrete.
187
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
For reinforced-concrete sections in tension, the nominal moment capacity of a section is
computed at a compression-control strain limit of 0.003 or a maximum tension in the steel
reinforcement of 0.005.
5-1-3 Stress-Strain Curves for Concrete and Steel
Any number of models can be used for the stress-strain curves for concrete and steel. For
the purposes of the computations presented herein, some relatively simple curves are used. The
stress-strain curve for concrete is shown in Figure 5-1.
f c
0.15 fc
Ec
0
0.0038
fr
Figure 5-1 Stress-Strain Relationship for Concrete Used by LPile
The following equations are used to compute concrete stress. The value of concrete
compressive strength, fc, in these equations is specified by the engineer.
    2 
f c  f c 2     for 0     0 .......................................(5-1)
  0   0  
   0 
 for  0    0.0038 ............................(5-2)
f c  f c  0.15 f c 
 0.0038   0 
The modulus of rupture, fr, is the tensile strength of concrete in bending. The modulus of
rupture for drilled shafts and bored piles is computed using
f r  7.5 f c psi in USCS units
f r  19.7 f c kPa in SI units
.............................................(5-3)
The modulus of rupture for prestressed concrete piles is computed using
188
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
f r  4.0 f c psi in USCS units
f r  10.5 f c kPa in SI units
.............................................(5-4)
The modulus of elasticity of concrete, Ec, is computed using
Ec  57,000 f c psi in USCS units
Ec  151,000 f c kPa in SI units
..........................................(5-5)
The compressive strain at peak compressive stress, 0, is computed using
 0  1.7
f c
............................................................(5-6)
Ec
The tensile strain at fracture for concrete, t, is computed using

 t   0 1  1 

fr
f c

 ...................................................(5-7)


The stress-strain (-) curve for steel is shown in Figure 5-2. There is no practical limit to
plastic deformation in tension or compression. The stress-strain curves for tension and
compression are assumed identical in shape.

fy
y

Figure 5-2 Stress-Strain Relationship for Reinforcing Steel Used by LPile
189
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
The yield strength of the steel, fy, is selected according to the material being used, and the
following equations apply.
y 
fy
Es
...............................................................(5-8)
where Es = 200,000 MPa (29,000,000 psi).
The models and the equations shown here are employed in the derivations that are shown
subsequently.
5-1-4 Cross Sectional Shape Types
The following types of cross sections can be analyzed:
1. Square or rectangular, reinforced concrete,
2. Circular, reinforced concrete,
3. Circular, reinforced concrete, with permanent steel casing,
4. Circular, reinforced concrete, with permanent steel casing and tubular core,
5. Circular, steel pipe,
6. Round prestressed concrete
7. Round prestressed concrete with hollow circular core,
8. Square prestressed concrete,
9. Square prestressed concrete with hollow circular core,
10. Octagonal prestressed concrete,
11. Octagonal prestressed concrete with hollow circular core,
12. Elastic shapes with rectangular, round, tubular, strong H-sections, or weak H-sections,
and
13. Elastic-plastic shapes with rectangular, round, tubular, strong H-sections, or weak Hsections.
The computed output consists of a set of values for bending moment M versus bending
stiffness EI for different axial loads ranging from zero to the axial-load capacity for the column.
5-2 Beam Theory
5-2-1 Flexural Behavior
The flexural behavior of a structural element such as a beam, column, or a pile subjected
to bending is dependent upon its flexural rigidity, EI, where E is the modulus of elasticity of the
material of which it is made and I is the moment of inertia of the cross section about the axis of
bending. In some instances, the values of E and I remain constant for all ranges of stresses to
which the member is subjected, but there are situations where both E and I vary with changes in
stress conditions because the materials are nonlinear or crack.
190
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
The variation in bending stiffness is significant in reinforced concrete members because
concrete is weak in tension and cracks and because of the nonlinearity in stress-strain
relationships. As a result, the value of E varies; and because the concrete in the tensile zone
below the neutral axis becomes ineffective due to cracking, the value of I is also reduced. When
a member is made up of a composite cross section, there is no way to calculate directly the value
of E for the member as a whole.
The following is a description of the theory used to evaluate the nonlinear momentcurvature relationships in LPile.
Consider an element from a beam with an initial unloaded shape of abcd as shown by the
dashed lines in Figure 5-3. This beam is subjected to pure bending and the element changes in
shape as shown by the solid lines. The relative rotation of the sides of the element is given by the
small angle d and the radius of curvature of the elastic element is signified by the length 
measured from the center of curvature to the neutral axis of the beam. The bending strain x in
the beam is given by
x 

dx
...............................................................(5-9)
where:
 = deformation at any distance from the neutral axis, and
dx = length of the element along the neutral axis.

d
a
M
d
b
dx

M
c

Figure 5-3 Element of Beam Subjected to Pure Bending
The following equality is derived from the geometry of similar triangles
191
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity

dx


.............................................................(5-10)

where:
 = distance from the neutral axis, and
 = radius of curvature.
Equation 5-11 is obtained from Equations 5-9 and 5-10, as follows:
x 

dx

 dx 1 
 .................................................(5-11)
 dx 
From Hooke’s Law
 x  E x ...........................................................(5-12)
where:
x = unit stress along the length of the beam, and
E = Young’s modulus.
Substituting Equation 5-11 into Equation 5-12, we obtain
x 
E

............................................................(5-13)
From beam theory
x 
M
...........................................................(5-14)
I
where:
M = applied moment, and
I = moment of inertia of the section.
Equating the right sides of Equations 5-13 and 5-14, we obtain
M E

..........................................................(5-15)
I

Cancelling  and rearranging Equation 5-15
M 1
 .............................................................(5-16)
EI 
192
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Continuing with the derivation, it can be seen that dx =  d and
1


d
  .........................................................(5-17)
dx
For convenience here, the symbol  is substituted for the curvature 1/. The following equation
is developed from this substitution and Equations 5-16 and 5-17
EI 
M

............................................................(5-18)
and because  =  d and x = /dx, we may express the bending strain as
x =   .............................................................(5-19)
The computation for a reinforced-concrete section, or a section consisting partly or
entirely of a pile, proceeds by selecting a value of  and estimating the position of the neutral
axis. The strain at points along the depth of the beam can be computed by use of Equation 5-19,
which in turn will lead to the forces in the concrete and steel. In this step, the assumption is made
that the stress-strain curves for concrete and steel are those shown in Section 5-1-3.
With the magnitude of the forces, both tension and compression, the equilibrium of the
section can be checked, taking into account the external compressive loading. If the section is not
in equilibrium, a revised position of the neutral axis is selected and iterations proceed until the
neutral axis is found.
Bending moment in the section is computed by integrating the moments of forces in the
slices times the distances of the slices from the centroid. The value of EI is computed using
Equation 5-18. The maximum compressive strain in the section is computed and saved. The
computations are repeated by incrementing the value of curvature until a failure strain in the
concrete or steel pipe, is developed. The nominal (unfactored) moment capacity of the section is
found by interpolation using the values of maximum compressive strain.
5-2-2 Axial Structural Capacity
The axial structural capacity, or squash load capacity, is the load at which a short column
would fail. Usually, this capacity is so large that it exceeds the axial bearing capacity of the soil,
except in the case of rock that is stronger than concrete. Several design equations are used to
compute the axial structural capacity, depending on the type of section being analyzed. For
reinforced concrete sections (not including prestressed concrete piles) the nominal (unfactored)
axial structural capacity, Pn, is
Pn  0.85 f c( Ag  As )  As f y ............................................(5-20)
where Ag is the gross cross-sectional area of the section, As is the cross-sectional area of the
longitudinal steel, fc is the specified compressive strength of concrete and fy is the specified yield
strength of the longitudinal reinforcing steel.
193
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Common design practice in North America and Europe is to restrict the steel
reinforcement to be between 1 and 8 percent of the gross cross-sectional area for drilled shafts
without permanent casing. Usually, reinforcement percentages higher than 3.5 to 4 percent are
attainable only by a combination of bundling of bars and by reducing the maximum aggregate
size to be small enough to pass through the reinforcement cage. LPile has features that help the
user to identify the combinations of reinforcement details that satisfy requirement for
constructability.
For prestressed concrete piles, the equations for the nominal axial structural capacity
differ depending on the cross-sectional shape and the level of prestressing. As for uncased
reinforced concrete sections, the concrete stress at failure is assumed to be 0.85 fc. With axial
loading, the effective prestress in the section is lowered. At a compressive strain of 0.003, only
about 60 percent of the prestressing remains in the member. Thus, the nominal strength can be
computed as
Pn  0.85 f c  0.60 f ps Ag ...............................................(5-21)
where fpc is the effective prestress.
The service load capacity for short column piles established by the Portland Cement
Association is based on a factor of safety between 2 and 3 is
N  0.33 f c  0.27 f pc Ag ...............................................(5-22)
Conventional construction practice in North American is to use effective prestressing of
600 to 1,200 psi (4.15 to 8.3 MPa) for driven piling. The level of prestressed used varies with the
overall length of the pile and local practice. Usually, the designing engineer obtains the value of
prestress and fraction of losses from the pile supplier.
5-3 Validation of Method
5-3-1 Analysis of Concrete Sections
An example concrete section is shown in Figure 5-4. This rectangular beam-column has a
cross section of 510 mm in width and 760 mm in depth and is subjected to both bending moment
and axial compression. The compressive axial load is 900 kN. For this example, the compressive
strength of the concrete fc is 27,600 kPa, E of the steel is 200 MPa, and the ultimate strength fy
of the steel is 413,000 kPa. The section has ten No. 25M bars, each with a cross-sectional area of
0.0005 m2, and the row positions are shown in the Figure 5-4. The following pages show how the
values of M and EI as a function of curvature are computed.
The results from the solution of the problem by LPile are shown in Table 5-1. The first
block of lines include an echo-print of the input, plus several quantities computed from the input
data, including the computed squash load capacity (9,093.096 kN), which is the load at which a
short column would fail. The next portion of the output presents results of computations for
various values of curvature, starting with a value of 0.0000492 rad/m and increasing  by even
194
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
increments.5
The fifth column of the output shows the value of the position of the neutral axis, as
measured from the compression side of the member. Other columns in the output, for each value
of , give the bending moment, the EI, and the maximum compressive strain in the concrete. For
the validation that follows, only one line of output was selected.
0.510 m
0.076 m
0.203 m
0.203 m
0.760 m
0.203 m
0.076 m
No. 25M bars
Figure 5-4 Validation Problem for Mechanistic Analysis of Rectangular Section
5-3-1-1 Computations Using Equations of Section 5-2
An examination of the output in Table 5-1 finds that the maximum compressive strain
was 0.0030056 for a value of  of 0.0176673 rad/m. This maximum strain is close to 0.003, the
value selected for computation of the nominal bending moment capacity, and that line of output
was selected for the basis of the following hand computations.
5-3-1-2 Check of Position of the Neutral Axis
In Table 5-1, the neutral axis is 0.1701205 m from the top of the section. The computer
found this value by iteration by balancing the computed axial thrust force against the specified
axial thrust. For the hand computations, the computed axial thrust for this neutral axis position
will be checked against the specified axial thrust. In the hand computations, the value of the
depth to the neutral axis was rounded to 0.1701 m for convenience.
5
LPile uses an algorithm to compute the initial increment of curvature that is based on the depth of the pile section. This algorithm is designed to
obtain initial values of curvature small enough to capture the uncracked behavior for all pile sizes.
195
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Table 5-1 LPile Output for Rectangular Concrete Section
-------------------------------------------------------------------------------Computations of Nominal Moment Capacity and Nonlinear Bending Stiffness
-------------------------------------------------------------------------------Axial thrust values were determined from pile-head loading conditions
Number of Sections = 1
Section No. 1:
Dimensions and Properties of Rectangular Concrete Pile:
Length of Section
Depth of Section
Width of Section
Number of Reinforcing Bars
Yield Stress of Reinforcing Bars
Modulus of Elasticity of Reinforcing Bars
Compressive Strength of Concrete
Modulus of Rupture of Concrete
Gross Area of Pile
Total Area of Reinforcing Steel
Area Ratio of Steel Reinforcement
Nom. Axial Structural Capacity = 0.85 Fc Ac + Fs As
=
=
=
=
=
=
=
=
=
=
=
=
15.24000000
0.76000000
0.51000000
10
413686.
199948000.
27600.
-39.40177573
0.38760000
0.00500000
1.28998971
9093.096
m
m
m
bars
kPa
kPa
kPa
kPa
sq. m
sq. m
percent
kN
Reinforcing Bar Details:
Bar
Number
---------1
2
3
4
5
6
7
8
9
10
Bar
Index
-----------16
16
16
16
16
16
16
16
16
16
Bar Diam.
m
-----------0.025200
0.025200
0.025200
0.025200
0.025200
0.025200
0.025200
0.025200
0.025200
0.025200
Bar Area
sq. m
-----------0.000500
0.000500
0.000500
0.000500
0.000500
0.000500
0.000500
0.000500
0.000500
0.000500
Bar X
m
------------0.167500
0.000000
0.167500
-0.167500
0.167500
-0.167500
0.167500
-0.167500
0.000000
0.167500
Bar Y
m
-----------0.304800
0.304800
0.304800
0.101600
0.101600
-0.101600
-0.101600
-0.304800
-0.304800
-0.304800
Concrete Properties:
Compressive Strength of Concrete
Modulus of Elasticity of Concrete
Modulus of Rupture of Concrete
Compression Strain at Peak Stress
Tensile Strain at Fracture
Maximum Coarse Aggregate Size
=
=
=
=
=
=
27600.
24865024.
-3271.7136591
0.0018870
-0.0001154
0.0190500
kPa
kPa
kPa
m
Number of Axial Thrust Force Values Determined from Pile-head Loadings = 1
Number
-----1
Axial Thrust Force
kN
-----------------900.000
Definitions of Run Messages and Notes:
C = concrete has cracked in tension
Y = stress in reinforcement has reached yield stress
T = tensile strain in reinforcement exceeds 0.005 when compressive strain
in concrete is less than 0.003.
Bending stiffness = bending moment / curvature
Position of neutral axis is measured from compression side of pile
Compressive stresses are positive in sign. Tensile stresses are negative in sign.
Axial Thrust Force =
900.000 kN
Bending
Bending
Bending
Depth to
Max Comp
Max Tens
Max Concrete
Max Steel
Curvature
Moment
Stiffness
N Axis
Strain
Strain
Stress
Stress
rad/m
kN-m
kN-m2
m
m/m
m/m
kPa
kPa
------------- ------------- ------------- ------------- ------------- ------------- ------------- ------------0.0000492
28.3173948
575409.
1.9085538
0.0000939
0.0000565 2674.0029283
18743.
0.0000984
56.6333321
575395.
1.1451716
0.0001127
0.0000379 3188.4483827
22462.
.
. (deleted lines)
196
Run
Msg
---
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
.
0.0004429
0.0004921
0.0005413
0.0005906
253.1619332
280.6180646
280.6180646
280.6180646
.
. (deleted lines)
.
0.0038878
651.6508321
0.0039862
663.0531399
0.0040846
674.4235902
0.0041831
685.7618089
.
. (deleted lines)
.
0.0176673
907.1915259
.
. (deleted lines)
.
0.0239665
913.9027316
571583.
570216.
518378.
475180.
0.5542915
0.5375669
0.4727569
0.4548249
0.0002455
0.0002646
0.0002559
0.0002686
-0.0000911
-0.0001095
-0.0001555
-0.0001802
6671.6631466
7149.3433542
6926.7437852
7241.7196541
48751.
52522.
50760.
53257.
167614.
166336.
165112.
163937.
0.2450564
0.2440064
0.2430210
0.2420960
0.0009527
0.0009727
0.0009927
0.0010127
-0.0020020
-0.0020569
-0.0021117
-0.0021664
20619.
20904.
21183.
21458.
-397341.
-408237.
-413686.
-413686.
C
C
CY
CY
51349.
0.1701205
0.0030056
-0.0104216
27596.
413686.
CY
38132.
0.1658249
0.0039742
-0.0142403
27600.
413686.
CY
C
C
-------------------------------------------------------------------------------Summary of Results for Nominal (Unfactored) Moment Capacity for Section 1
-------------------------------------------------------------------------------Moment values interpolated at maximum compressive strain = 0.003
or maximum developed moment if pile fails at smaller strains.
Load
No.
---1
Axial Thrust
kN
---------------900.000
Nominal Mom. Cap.
kN-m
-----------------907.021
Max. Comp.
Strain
-----------0.00300000
Note that the values of moment capacity in the table above are not
factored by a strength reduction factor (phi-factor).
In ACI 318-08, the value of the strength reduction factor depends on whether the
transverse reinforcing steel bars are spirals or tied hoops.
The above values should be multiplied by the appropriate strength reduction
factor to compute ultimate moment capacity according to ACI 318-08, Section 9.3.2.2
or the value required by the design standard being followed.
5-3-1-3 Forces in Reinforcing Steel
The rows of steel in Figure 5-4 are numbered from the top downward. Therefore, Row 1
will be in compression and the other rows will be in tension. The strain in each row of bars is
computed using Equation 5-19, as follows (with the positive sign indicating compression).
1 =   = (0.0176673 rad/m) (0.1701 m  0.0755 m) = +0.001672
Similarly,
2 = 0.001915
3 = 0.005501
4 = 0.009088
In order to obtain the forces in the steel at each level, it is necessary to know if the steel is
in the elastic or plastic range. Thus, it is required to compute the value of yield strain y using
Equation 5-8.
y 
fy
Es

413,000
 0.002065 ..........................................(5-23)
2 108
197
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
This computation shows that the bars in rows 1 and 2 are in the elastic range and the bars in the
other two rows are in the plastic range. Thus, the forces in each row of bars are:
F1 = (3 bars) (5  104 m2/bar) (0. 001447) (2  108 kPa) =
501.51 kN
F2 = (2 bars) (5  104 m2/bar) (0. 002779) (2108 kPa) =
382.95 kN
F3 = (2 bars) (5  104 m2/bar) (0.007005) (413,000 kPa) =
413.00 kN
4
F4 = (3 bars) (5  10 m /bar) (0.007005) (413,000 kPa) =
619.50 kN
Total of forces in the reinforcing bars =
913.95 kN.
2
5-3-1-4 Forces in Concrete
In computing the internal force in the concrete, 10 slices that are 17.01 mm (0.670 in.) in
thickness are selected for computation of the 0.1701 m of the section in compression. The slices
are numbered from the top downward for convenience. The strain is computed at the mid-height
of each slice by making use of Equation 5-19.
1 =   = (0.0176673 rad/m) (0.1701 m  0.01701 m/2) = 0.00285529
The second value in the parentheses is the distance from the neutral axis to the mid-height of the
first slice. Similarly, the strains at the centers of the other slices are:
2 = 0.002554
3 = 0.002254
4 = 0.001954
5 = 0.001653
6 = 0.001353
7 = 0.001052
8 = 0.000751
9 = 0.000451
10 = 0.000150
The forces in the concrete are computed by employing Figure 5-4 and Equations 5-1
through 5-8. The first step is to compute the value of 0 from Equation 5-6 and to see the strains
are lower or greater than the strain for the peak stress.


27,600
  0.001870

151
,
000
27
,
600


 0  1.7
The strain in the top two slices show that stress can be found by use of the second branch
of the compressive portion of the curve in Figure 5-1 and the stress in the other slices can be
computed using Equation 5-1. From Figure 5-4, the following quantity is computed
0.15 f c  4,140 kPa
198
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Then, the following equation can be used to compute the stress along the descending section of
the stress-strain curve corresponding to 1 and 2.
   0.001870 
f c  27,600  4,140

 0.0038  0.001870 
From the above equation:
fc1 = 25,487 kPa
fc2 = 26,132 kPa
fc3 = 26,777 kPa
fc4 = 27,421 kPa
The strains in the other slices are less than 0 so the stresses in the concrete are on the
ascending section of the stress-strain curve. The stresses in these slices can be computed by
Equation 5-1.
2
 


 
 
f c 3  27,6002

 
 
  0.001870   0.001870  
The other values of fc are computed as follows:
fc5 = 27,227 kPa
fc6 = 25,484 kPa
fc7 = 22,315 kPa
fc8 = 17,721 kPa
fc9 = 11,702 kPa
fc10 = 4,257 kPa
The forces in each slice of the concrete due to the compressive stresses are computed by
multiplying the area of the slice by the computed stress. All of the slices have the area of 0.00740
m2 (0.0145 m  0.51 m). Thus, the computed forces in the slices are:
Fc1 = 221.13 kN
Fc2 = 226.72 kN
Fc3 = 232.32 kN
Fc4 = 237.91 kN
Fc5 = 236.23 kN
Fc6 = 221.10 kN
Fc7 = 193.61 kN
Fc8 = 153.75 kN
199
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Fc9 = 101.53 kN
Fc10 =
36.93 kN
There is a small section of concrete in tension. The depth of the tensile section is
determined by the strains up to the strain developed at the modulus of rupture (Equation 5-3).
f r  19.7 27,600  3,273 kPa
In this zone, it is assumed that the stress-stain curve in tension is defined by the average concrete
modulus (Equation 5-5).
The modulus of elasticity of concrete, Ec, is computed using
Ec  151,000 27,600  25,086,000 kPa
The strain at rupture is then
r 
 3,273
 0.0001305
25,086,000
The thickness of the tension zone is computed using Equation 5-19 as
h
 r  0.0001305

 0.07384 m

0.0176673
The force in tension is the product of average tensile stress is and the area in tension and is
 E 
Ft   r c  0.073840.510  6.16 kN
 2 
A reduction in the computed concrete force is needed because the top row of steel bars is
in compression zone. The compressive force computed in concrete for the area occupied by the
steel bars must be subtracted from the computed value. The compressive strain at the location of
the top row of bars is 0.001447, the area of the bars is 0.0015 m2, the concrete stress is 27,289
kPa, and the force is 40.93 kN.
Thus, the total force carried in the concrete is sum of the computed compressive forces
plus the tensile concrete force minus the correction for the area of concrete occupied by the top
row of reinforce is 1814.10 kN.
5-3-1-5 Computation of Balance of Axial Thrust Forces
The summation of the internal forces yields the following expression for the sum of axial
thrust forces:
200
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
F = 1814.10 kN  913.95 kN = 900.15 kN.
Taking into account the applied axial load in compression of 900 kN, the section is out of
balance by only 0.15 kN (33.7 lbs).
This hand computation confirms the validity of the computations made by LPile. The
selection of a thickness of the increments of concrete of 0.01701 m is thicker than that used in
LPile. LPile uses 100 slices of the full section depth in its computations, so the slice thickness
used by LPile is 0.0076 m for this example problem. In addition, some error was introduced by
the reduced precision used in the hand computations, whereas LPile uses 64-bit precision in all
computations.
5-3-1-6 Computation of Bending Moment and EI
Bending moment is computed by summing the products of the slice forces about the
centroid of the section. The axial thrust load does not cause a moment because it is applied with
no eccentricity. The moments in the steel bars and concrete can be added together because the
bending strains are compatible in the two materials.
The moments due to forces in the steel bars are computed by multiplying the forces in the
steel bars times the distances from the centroid of the section. The values of moment in the steel
bars are:
Moment due to bar row 1: (479.1 kN) (0.3045) =
152.71 kN-m
Moment due to bar row 2: (411.9 kN) (0.1015) =
38.87 kN-m
Moment due to bar row 3: (415.0 kN) ( 0.1015) =
41.92 kN-m
Moment due to bar row 4: (622.5 kN) ( 0.3045) =
188.64 kN-m
Total moment due to stresses in steel bars =
344.40 kN-m
The moments due to forces in the concrete are computed by multiplying the forces in the
concrete times the distances from the centroid of the section. The values of moments in the
concrete slices are:
Moment in slice 1: (241.37 kN) (0.3728 m) =
82.15 kN-m
Moment in slice 2: (248.29 kN) (0.3583 m) =
80.37 kN-m
Moment in slice 3: (255.21 kN) (0.3438 m) =
78.40 kN-m
Moment in slice 4: (257.61 kN) (0.3293 m) =
76.24 kN-m
Moment in slice 5: (247.22 kN) (0.3148 m) =
71.68 kN-m
Moment in slice 6: (226.19 kN) (0.3003 m) =
63.33 kN-m
Moment in slice 7: (194.53 kN) (0.2858 m) =
52.16 kN-m
Moment in slice 8: ( 152.24 kN) (0.2713 m) =
38.81 kN-m
Moment in slice 9: ( 99.32 kN) (0.2568 m) =
23.90 kN-m
Moment in slice 10: ( 35.76 kN) (0.2423 m) =
8.07 kN-m
Moment correction for top row of bars = (40.93 kN) (0.3045 m) =
201
12.46 kN-m
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Total moment due to stresses in concrete =
561.32 kN-m
Sum of moments in steel bars and concrete =
905.71 kN-m
As mentioned above, the summation of the moments in the steel bars and concrete is
possible because the bending strains in the two materials are compatible, i.e. the bending strains
are consistent with the positions of the steel bars and concrete slices.
5-3-1-7 Computation of Bending Stiffness Using Approximate Method
The drawing in Figure 5-5 shows the information used in computing the nominal moment
capacity. The forces in the steel were computed by multiplying the stress developed in the steel
by the area, for either of two or three bars in a row at each row position.
0.1701 m
0.076 m
501.51 kN
0.203 m
382.95 kN
0.203 m
0.760 m
413 kN
0.203 m
619.5 kN
0.076 m
Figure 5-5 Free Body Diagram Used for Computing Nominal Moment Capacity of Reinforced
Concrete Section
The value of bending stiffness is computed using Equation 5-18.
EI 
M


905.71 kN - m
 51.265.02 kN - m 2
0.01701205 rad/m
A comparison of results from hand versus computer solutions is summarized in Table 52. The moment computed by LPile was 907.19 kN-m. Thus, the hand calculation is within 0.16%
of the computer solution. The value of the EI is computed by LPile is 51,348.62 kN-m2. The
hand solution is within 0.16% of the computer solution. The hand solution for axial thrust is
within 0.0167% of the computer solution
The agreement is close between the values computed by hand using only a small number
of slices and the values from the computer solution computed using 100 slices. This example
hand computation serves to confirm of the accuracy of the computer solution for the problem
that was examined.
202
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Table 5-2 Comparison of Results from Hand Computation versus Computer Solution
Parameter
By LPile
By Hand
Hand Error, %
Moment Capacity, kN-m
907.19
905.71
0.16%
Bending Stiffness, EI, kN-m2
51,348.62
51,265.02
0.16%
Axial Thrust, kN
900.00*
900.15
+0.0167%
* Input value
The rectangular section used for above example solution was chosen because the
geometric shapes of the slices are easy to visualize and their areas and centroid positions are easy
to compute. In reality, the algorithms used in LPile for the geometrical computation are much
more powerful because of the circular and non-circular shapes considered in the computations.
For example, when a large number of slices are used in computations, individual bars are divided
by the slice boundaries. So, in the computations made by LPile, the areas and positions of
centroids in each circular segment of the bars are computed. In addition, the areas of bars and
strands in a slice are subtracted from the area of concrete in a slice.
The two following graphs are examples of the output from LPile for curves of moment
versus curvature and ending stiffness versus bending moment. These graphs are examples of the
output from the presentation graphics utility that is part of LPile. Both of these graphs were
exported as enhanced Windows metafiles, which were then pasted into this document.
Moment vs. Curvature - All Sections
1,000
950
900
850
800
750
700
Moment, kN-m
650
600
550
500
450
400
350
300
250
200
150
100
50
0
0.0
0.005
0.01
0.015
Curvature, radians/m
Figure 5-6 Bending Moment versus Curvature
203
0.02
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
600,000
550,000
500,000
450,000
400,000
EI, kN-m²
350,000
300,000
250,000
200,000
150,000
100,000
50,000
0
0
100
200
300
400
500
600
700
800
Bending Moment, kN-m
Figure 5-7 Bending Moment versus Bending Stiffness
9,000
8,500
8,000
7,500
7,000
6,500
6,000
5,500
5,000
4,500
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
0
0
200
400
600
800
1,000
1,200
Unfactored Bending Moment, kN-m
Figure 5-8 Interaction Diagram for Nominal Moment Capacity
204
900
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
5-3-2 Analysis of Steel Pipe Piles
The method of Section 5-3-1 can be used to make a computation of the plastic moment
capacity Mp of steel pipe piles to compare with the value computed using LPile. The pipe section
that was selected is shown in Figure 5-9. The pipe section has an outer diameter of 838 mm and
an inner diameter of 781.7 mm. The value of the nominal moment was selected as 7,488 kN-m
from Figure 5-10 at a maximum curvature of 0.015 radians/meter.
414,000 kPa
0.838 m
0.7817 m
Figure 5-9 Example Pipe Section for Computation of Plastic Moment Capacity
8,000
7,500
7,000
6,500
6,000
5,500
Moment, kN-m
5,000
4,500
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
0
0.0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.012
0.013
0.014
0.015
Curvature, radians/m
Figure 5-10 Moment versus Curvature of Example Pipe Section
In the computations shown below, the assumption was made that the strain was sufficient
to develop the ultimate strength of the steel, 4.14  105 kPa, over the entire section. From the
205
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
practical point of view, it is unrealistic to assume that the bending strains developed in a section
can be large enough to yield the condition that is assumed; however, the computation should
result in a value that is larger than 7,488 kN-m (5,863 ft-kips) but in the appropriate range.
The expression for the plastic moment capacity Mp is the product of the yield stress fy and
plastic modulus Z.
M p  f y Z ..........................................................(5-24)
Referring to the dimensions shown in Figure 5-9, the plastic modulus Z of the pipe is
Z
d
3
o

 di3
 1.847  10 2 m3
6
The computed moment capacity is



M p  4.14  105 kPa 1.847 m3  7,647 kN - m
As expected, the value of Mp computed from the plastic modulus is slightly larger than
the 7,488 kN-m from the computed solution at a strain of 0.0149 rad/m. However, the close
agreement and the slight over-estimation provide confidence that the computer code computes
the plastic moment capacity accurately.
Another check on the accuracy of the computations is to examine the computed bending
stiffness in the elastic range. From elastic theory, the bending stiffness for the example problem
is
EI  E

 d o4  d i4 
64
 2  10 kPa
8
 
 0.838 m 4  0.7817 m 4

64
 1,175,726 kN - m
2
The value computed by LPile is 1,175,686 kN-m2. The error in bending stiffness for the
computed solution is 0.0035 percent, which is amazingly accurate for a numerical computation.
Please note that the fifth through seventh digits in the above values are shown to be able to
illustrate the comparison and are not indicative of the precision possible in normal computations.
Often, engineers use specified material strengths that are usually exceeded in reality.
The reason that the bending stiffness value computed by LPile is slightly smaller than the
full plastic yield value is that the stresses and strains near the neutral axis remain in the elastic
range. The stress distribution for a curvature of 0.015 rad/m is shown in Figure 5-11.
Approximately, the middle third of this section is in the elastic range.
206
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
414,000 kPa
0.838 m
0.138 m
0.7817 m
 = 0.015 rad/m
Figure 5-11 Elasto-plastic Stress Distribution Computed by LPile
5-3-3 Analysis of Prestressed-Concrete Piles
Prestressed-concrete piles are widely used in construction where conditions are suitable
for pile driving. A prestressed-concrete pile has a configuration similar to a conventional
reinforced-concrete pile except that the longitudinal reinforcing steel is replaced by prestressing
steel. The prestressing steel is usually in the form of strands of high-strength wire that are placed
inside of cage of spiral steel to provide lateral reinforcement. As the term implies, prestressing
creates an initial compressive stress in the pile so the piles have higher capacity in bending and
greater tolerance of tension stresses developed during pile driving. Prestressed piles can usually
be made lighter and longer than reinforced-concrete piles of the same size.
An advantage of prestressed-concrete piles, compared to conventional reinforcedconcrete piles, is durability. Because the concrete is under continuous compression, hairline
cracks are kept tightly closed, making prestressed piles more resistant to weathering and
corrosion than conventionally reinforced piles. This characteristic of prestressed concrete
removes the need for special steel coatings because corrosion is not as serious a problem as for
reinforced concrete.
Another advantage of prestressing is that application of a bending moment results in a
reduction of compressive stresses on the tension side of the pile rather than resulting in cracking
as with conventional reinforced concrete members. Thus, there can be an increase in bending
stiffness of the prestressed pile as compared to a conventionally reinforced pile of equal size. The
use of prestressing leads to a reduction in the ability of the pile to sustain pure compression, a
factor that is usually of minor importance in service but must be considered in pile driving
analyses. One significant importance is that a considerable bending moment may be applied to a
reinforced pile before first cracking. Consequently, the pile-head deflection of the prestressed
pile in the uncracked state is substantially reduced, and its performance under service loads is
improved.
When analyzing a foundation consisting of prestressed piles, the designer must input a
value of the level of stress due to prestressing, Fps, after losses due to creep and other factors.
The value usually ranges from 600 to 1,200 psi (4,140 to 8,280 kPa), but accurate values can
only be found from the manufacturer of the piles. The value of prestress will vary by
207
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
manufacturer from region to region and will also vary with the shape, size, and compressive of
the concrete. For most commercially obtained prestressed piles, Fps can be estimated by
assuming some level of initial prestressing in the concrete. Given a value of Fps the program
solves the statically indeterminate problem of balancing the prestressing forces in the concrete
and reinforcement using the nonlinear stress-strain relationships selected for both concrete and
reinforcing steel.
The stress-strain relationships used in prestressed concrete is defined using the stressstrain curves of concrete recommended by the Design Handbook of the Prestressed Concrete
Institute (PCI), as shown in Figure 5-12 and in equation form in Equations 5-25 to 5-28.
270
270 ksi
250
250 ksi
Minimum yield strength = 243 ksi at 1%
Elongation for 270 ksi (ASTM A 416)
Stress, ksi
230
Minimum yield strength = 225 ksi at 1%
Elongation for 250 ksi (ASTM A 416)
210
190
170
150
0
0.005
0.01
0.015
0.02
0.025
0.03
Strain, in/in
Figure 5-12 Stress-Strain Curves of Prestressing Strands Recommended by PCI Design
Handbook, 5th Edition.
For 250 ksi 7-wire low-relaxation strands:
 ps  0.0076 : f ps  28,500 ps (ksi) .......................................(5-25)
 ps  0.0076; f ps  250 
208
0.04
(ksi) ................................(5-26)
 ps  0.0064
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
For 270 ksi 7-wire low-relaxation strands:
 ps  0.0086 : f ps  28,500 ps (ksi) .......................................(5-27)
 ps  0.0086 : f ps  270 
0.04
(ksi) .................................(5-28)
 ps  0.007
PCI does not have any recommendations for grade 300 strands, which are not widely
available. The above equations were used as a model to develop a stress-strain relationship for
grade 300 strands. The equations are:
 ps  0.0088846 : f ps  28,500 ps (ksi) ...................................(5-29)
 ps  0.0088846 : f ps  300 
0.0835
(ksi) .............................(5-30)
 ps  0.0071
For prestressing bars, an elastic-plastic stress-strain curve is used.
As noted earlier, the value of the concrete stress due to prestressing is found prior to
performance of the moment-curvature analysis. When prestressed concrete piles are analyzed,
the initial strains in the concrete and steel due to prestressing must be computed prior to
computation of bending stiffness. The corresponding level of prestressing force applied to the
reinforcement, Fps is computed by balancing the force carried in the concrete with the force
carried in the reinforcement. Thus,
Fps   c Ac ...........................................................(5-31)
where c is the prestress in the concrete and Ac is the cross-sectional area of the concrete.
The user should check the output report from the program to see if the computed level of
prestressed force in the concrete at the initial stage is in the desired range. The computation
procedures for stresses of concrete for a specific curvature of the cross section are the same as
that for ordinary concrete, described in a previous section, except the current state of stresses of
concrete and strands should take into account the initial stress conditions. The stress levels for
both concrete and strands under loading conditions should be checked to ensure that the stresses
are in the desired range.
Elementary considerations show that a distance from the end of a pile is necessary for the
full transfer of stresses from reinforcing steel to concrete. The development length of the strand
is not computed in LPile. Usually the zone of development is about 50  the axial strand
diameter from the end of the pile.
Typical cross sections of prestressed piles are square solid, square hollow, octagonal
solid, octagonal hollow, round solid, or round hollow, are shown in Figure 5-13.
209
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Figure 5-13 Sections for Prestressed Concrete Piles Modeled in LPile
5-4 Discussion
Use of the mechanistic method of analysis of moment-curvature relations by hand is
relatively straightforward for cases of simple cross sections. Use of this method becomes
significantly more laborious when using geometrical values for complex cross sections and
nonlinear stress-strain relationships of concrete and steel or when including the effect of
prestressing in the case of prestressed concrete piles. Thus, use of a computer program is a
necessary feature of the method of analysis presented here.
A new user to the program may wish to practice using LPile by repeating the solutions
for the example problems. When LPile is employed for any problem being addressed by the user,
some procedure should be employed to obtain an approximate solution of the section properties
in order to verify the results and to detect gross input errors.
210
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
5-5 Reference Information
5-5-1 Standard Concrete Reinforcing Steel Sizes
Name
US Std. #3
US Std. #4
US Std. #5
US Std. #6
US Std. #7
US Std. #8
US Std. #9
US Std. #10
US Std. #11
US Std. #14
US Std. #18
ASTM 10M
ASTM 15M
ASTM 20M
ASTM 25M
ASTM 30M
ASTM 35M
ASTM 45M
ASTM 55M
CEB 6 mm
CEB 8 mm
CEB 10 mm
CEB 12 mm
CEB 14 mm
CEB 16 mm
CEB 20 mm
CEB 25 mm
CEB 32 mm
CEB 40 mm
JD6
JD8
JD10
JD13
JD16
JD19
JD22
JD25
JD29
JD32
JD35
JD38
JD41
LPile
Index No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
D, in
Area, in
0.375
0.500
0.625
0.750
0.875
1.000
1.128
1.270
1.410
1.693
2.257
0.445
0.630
0.768
0.992
1.177
1.406
1.720
2.220
0.236
0.315
0.394
0.472
0.551
0.630
0.787
0.984
1.260
1.575
0.250
0.315
0.375
0.500
0.626
0.752
0.874
1.000
1.126
1.252
1.374
1.504
1.626
0.11
0.20
0.31
0.44
0.60
0.79
1.00
1.27
1.56
2.25
4.00
0.155
0.310
0.466
0.777
1.088
1.554
2.332
3.886
0.043
0.078
0.122
0.175
0.239
0.312
0.487
0.761
1.246
1.947
0.049
0.078
0.111
0.196
0.308
0.444
0.600
0.785
0.996
1.231
1.483
1.767
2.077
2
211
lbs/ft
D, mm
Area, mm
0.376
0.668
1.043
1.502
2.044
2.670
3.400
4.303
5.313
7.650
13.600
0.526
1.052
1.578
2.629
3.681
5.259
7.880
13.150
0.147
0.263
0.415
0.594
0.810
1.057
1.651
2.581
4.227
6.604
0.167
0.263
0.375
0.666
1.044
1.506
2.035
2.664
3.377
4.176
5.029
5.994
7.045
9.5
12.7
15.9
19.1
22.2
25.4
28.7
32.3
35.8
43.0
57.3
11.3
16.0
19.5
25.2
29.9
35.7
43.7
56.4
6.0
8.0
10.0
12.0
14.0
16.0
20.0
25.0
32.0
40.0
6.35
8.0
9.53
12.7
15.9
19.1
22.2
25.4
28.6
31.8
34.9
38.2
41.3
71.3
126.7
198.6
286.5
387.1
506.7
646.9
819.4
1006
1452
2579
100
200
300
500
700
1000
1500
2500
28
50
79
113
154
201
314
491
804
1256
31.67
50
71.33
126.7
198.6
286.5
387.1
506.7
642.4
794.2
956.6
1140
1340
2
kg/m
0.559
0.993
1.557
2.246
3.035
3.973
5.072
6.424
7.887
11.384
20.219
0.784
1.568
2.352
3.920
5.488
7.840
11.76
19.60
0.220
0.392
0.619
0.886
1.207
1.576
2.462
3.849
6.303
9.847
0.248
0.392
0.559
0.993
1.557
2.246
3.035
3.973
5.036
6.227
7.500
8.938
10.506
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Name
AS12
AS16
AS20
AS24
AS28
AS32
AS36
NZ6
NZ10
NZ12
NZ16
NZ20
NZ25
NZ32
NZ40
LPile
Index No.
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
D, in
Area, in
0.472
0.630
0.787
0.945
1.102
1.260
1.417
0.236
0.394
0.472
0.630
0.787
0.984
1.260
1.575
0.175
0.312
0.487
0.701
0.954
1.247
1.578
0.044
0.122
0.175
0.312
0.487
0.761
1.247
1.948
2
lbs/ft
D, mm
Area, mm
0.596
1.060
1.656
2.384
3.245
4.238
5.364
0.149
0.414
0.596
1.060
1.656
2.587
4.238
6.622
12.0
16.0
20.0
24.0
28.0
32.0
36.0
6.0
10.0
12.0
16.0
20.0
25.0
32.0
40.0
113
201
314
452
616
804
1020
28.3
78.5
113
201
314
491
804
1260
2
kg/m
0.888
1.580
2.470
3.550
4.830
6.310
7.990
0.222
0.617
0.888
1.580
2.470
3.850
6.310
9.860
In addition to the bar sizes shown in the table above, LPile also has generic bar sizes in
millimeters ranging from 3 mm to 90 mm. Included in this range of bar diameters are the sizes
for high strength bars with diameters of 2.5, 3.0, and 3.5 inches.
5-5-2 Prestressing Strand Types and Sizes
Name
5/16" 3-wire
1/4 7-wire
5/16 7-wire
3/8 7-wire
7/16 7-wire
1/2" 7-wire
0.6" 7-wire
5/16" 3-wire
3/8 7-wire
7/16 7-wire
1/2" 7-wire
1/2" 7-w spec
9/16" 7-wire
0.6" 7-wire
0.7" 7-wire
3/8" 7-wire
7/16" 7-wire
1/2" 7-wire
1/2" Super
0.6" 7-wire
3/4" smooth
7/8" smooth
1" smooth
LPile
Index
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Grade,
ksi
D, in
Area, in
250
250
250
250
250
250
250
270
270
270
270
270
270
270
270
300
300
300
300
300
145
145
145
0.340
0.250
0.3125
0.375
0.4375
0.500
0.600
0.34
0.375
0.4375
0.500
0.500
0.5625
0.600
0.700
0.375
0.438
0.500
0.500
0.600
0.750
0.875
1.000
0.058
0.036
0.058
0.080
0.108
0.144
0.216
0.058
0.085
0.115
0.153
0.167
0.192
0.217
0.294
0.085
0.115
0.153
0.167
0.217
0.442
0.601
0.785
212
2
lbs/ft
D, mm
Area,
2
mm
kg/m
0.2
0.122
0.197
0.272
0.367
0.49
0.737
0.2
0.29
0.39
0.52
0.58
0.65
0.74
1.01
0.29
0.39
0.52
0.58
0.74
1.5
2.04
2.67
8.6
6.4
7.9
9.5
11.1
12.7
15.2
8.6
9.5
11.1
12.7
12.7
14.3
15.2
17.8
9.5
11.1
12.7
12.7
15.2
19.1
22.2
25.4
37.4
23.2
37.4
51.6
69.7
92.9
138.7
37.4
54.8
74.2
98.7
107.7
123.9
138.7
189.7
54.8
74.2
98.7
107.7
140.0
285.2
387.7
506.5
0.298
0.182
0.293
0.405
0.546
0.729
1.096
0.298
0.431
0.580
0.774
0.863
0.967
1.101
1.505
0.431
0.580
0.774
0.863
1.101
2.232
3.035
3.972
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
1 1/8" smooth
1 1/4" smooth
1 3/8" smooth
Name
3/4" smooth
7/8" smooth
1" smooth
1 1/8" smooth
1 1/4" smooth
1 3/8" smooth
5/8" def. bar
1" def. bar
1" def. bar
1 1/4" def. bar
1 1/4" def. bar
1 3/8" def. bar
24
25
26
LPile
Index
No.
27
28
29
30
31
32
33
34
35
36
37
38
145
145
145
1.125
1.250
1.375
0.994
1.227
1.485
Grade,
ksi
D, in
Area, in
160
160
160
160
160
160
157
150
160
150
160
160
0.75
0.875
1
1.125
1.25
1.375
0.625
1
1
1.25
1.25
1.375
0.442
0.601
0.785
0.994
1.227
1.485
0.28
0.85
0.85
1.25
1.25
1.58
2
3.38
4.17
5.05
28.6
31.8
34.9
641.3
791.6
958.1
5.029
6.204
7.513
lbs/ft
D, mm
Area,
2
mm
kg/m
1.5
2.04
2.67
3.38
4.17
5.05
0.98
3.01
3.01
4.39
4.39
5.56
19.1
22.2
25.4
28.6
31.8
34.9
15.9
25.4
25.4
31.8
31.8
34.9
285.2
387.7
506.5
641.3
791.6
958.1
180.6
548.4
548.4
806.5
806.5
1019.4
2.232
3.035
3.972
5.029
6.204
7.513
1.458
4.478
4.478
6.531
6.531
8.272
5-5-3 Steel H-Piles
Section
HP 14
HP 360
HP 13
HP 330
HP 12
HP 310
Weight
Area, A
Depth, d
lbs/ft
kg/m
in2
cm2
in
mm
117
34.4
175
222
102
30
Thickness
Flange
Width, b
Ixx
Iyy
in4
cm4
in4
cm4
Compact
Section
Criteria
F'y
ksi
MPa
in
mm
Flange, tf
in.
mm
Web, tw
in.
mm
14.21
14.885
0.805
0.805
1220
443
361
378
20.4
20.4
50800
18400
341
14.01
14.785
0.705
0.705
1050
380
38.4
49.4
153
194
356
376
17.9
17.9
43700
15800
265
89
26.1
13.83
14.695
0.615
0.615
904
326
29.6
133
168
351
373
15.6
15.6
37600
13600
204
20.3
73
21.4
13.61
14.585
0.505
0.505
729
261
109
138
346
370
12.8
12.8
30300
10900
140
100
29.4
13.15
13.205
0.765
0.765
886
294
56.7
150
190
334
335
19.4
19.4
36878
12237
391
87
25.5
12.95
13.105
0.665
0.665
755
250
43.5
130
165
329
333
16.9
16.9
31425
10406
300
31.9
73
21.6
12.75
13.005
0.565
0.565
630
207
109
139
324
330
14.4
14.4
26223
8616
220
60
17.5
12.54
12.9
0.46
0.46
503
165
21.5
90
113
319
328
11.7
11.7
20936
6868
148
52.5
84
24.6
12.28
12.295
0.685
0.685
650
213
126
159
312
312
17.4
17.4
27100
8870
362
74
21.8
12.13
12.215
0.61
0.61
569
186
42.1
111
141
308
310
15.5
15.5
23700
7740
290
63
18.4
11.94
12.125
0.515
0.515
472
153
30.5
94
119
303
308
13.1
13.1
19600
6370
210
213
Chapter 5 – Computation of Nonlinear Bending Stiffness and Moment Capacity
Section
HP10
HP 250
HP 8
HP 200
53
15.5
11.78
12.045
0.435
0.435
393
127
22
79
100
299
306
11
11
16400
5290
152
Weight
Area, A
Depth, d
lbs/ft
kg/m
in2
cm2
in
mm
57
16.8
85
108
42
63
Flange
Width, b
Ixx
Iyy
in4
cm4
in4
cm4
294
101
Compact
Section
Criteria
F'y
ksi
MPa
51.6
12200
4200
356
29.4
Thickness
in
mm
Flange, tf
in.
mm
9.99
10.225
0.565
Web, tw
in.
mm
0.565
254
260
14.4
14.4
12.4
9.7
10.075
0.42
0.42
210
71.7
80
246
256
10.7
10.7
8740
2980
203
36
10.6
8.02
8.155
0.445
0.445
119
40.3
50.3
54
68.4
204
207
11.3
11.3
4950
1680
347
214
Chapter 6
Use of Vertical Piles to Stabilize a Slope
6-1 Introduction
The computation of slope stability is a problem often faced by geotechnical engineers.
Numerous computer methods have been developed for making the slope stability computation.
One of the first of these available as a computer solution was the simplified method of slices
developed by Bishop (1955). Over the years, there have been additional developments for
analyzing slope stability. The method of Morgenstern and Price (1965) was the first method of
analysis that was capable of solving all equations of force and moment equilibrium for a limit
analysis of slope stability. The widely used computer programs UTexas4, Slope/W, and Slide
implement modern developments in computation of slope stability. In view of advances in
methods of analysis, the availability of computer programs, and numerous comparisons of results
of analysis and observed slope failures, many engineers will obtain approximately identical
factors of safety for a particular problem of slope stability. This chapter is written with the
assumption that the user is familiar with the theory of slope stability computations and has a
computer program available for use.
In spite of the ability to make reasonable computations, there are occasions when
engineering judgment may indicate the need to increase the factor of safety for a particular slope.
There are a large number of methods for accomplishing such a purpose. For example, the factor
of safety may be increased by flattening the slope, if possible, or by providing subsurface
drainage to lower the water table in the slope.
The method proposed in this chapter presents the engineer with additional option that
might prove useful in some cases. Piles have been used in the past to increase the stability of a
slope, but without an analysis to judge their effectiveness. Thus, a method of analysis to
investigate the benefits of using piles for this purpose is a useful tool for engineers.
6-2 Proposed Methods
Any number of situations could develop that might dictate the use of piles to increase the
stability of a slope. A common occurrence is the appearance of cracks parallel to the crest of the
slope. Cracks of this type often indicate the initial movement associate with slope failure and can
provide a means for surface water to enter the soil and saturate the slope. This could result in a
reduced factor of safety for slope stability in the future. Slope stability analysis may show that
the factor of safety for the slope is near unity and some strengthening of the slope is needed
before additional slope movement occurs.
One possible solution is shown in Figure 6-1. In this method, a row of drilled shafts or
piles is constructed in the slope near the position of the lowest extent of the sliding surface. The
use of a drilled foundation is a favorable procedure because the installation of the shaft will
result in minimal disturbance to the soils in the slope.
215
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
Even if no distress may appear in a slope, analysis of some slopes after construction may
show the stability of a slope is questionable. The original slope stability analysis may be
superseded by a more accurate one, additional soil borings or construction may reveal a weak
stratum that was not found earlier, or changes in environmental conditions could have caused a
weakening of the soils in the slope. The use of drilled shaft foundations to strengthen the slope
might then be considered.
Figure 6-1 Scheme for Installing a Row of Piles in a Slope Subject to Sliding
A second scheme for the positioning of piles is shown in Figure 6-2. In this scheme, the
tops of one or more rows of stabilizing piles are restrained by a structural grade beam connected
to an anchor pile group. In this scheme, it is possible for the stabilizing piles to carry more
loading because they are restrained at the top by the grade beam and anchor pile group and the
stable soils below the slip surface.
Structural
Grade Beam
Anchor Pile
Group
Stabilizing
Piles
Figure 6-2 Scheme for Stabilizing Piles with Grade Beam and Anchor Pile Group
216
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
Available right-of-way in urban areas may be limited or extremely expensive to purchase
with the result that the design of a slope with an adequate factor of safety against sliding is
impossible or very expensive. A cost study could reveal whether or not it would be preferable to
install a retaining wall or to strengthen the slope with drilled shafts.
6-3 Review of Some Previous Applications
Fukuoka (1977) described three applications where piles were used to stabilize slopes in
Japan. Heavily steel pipe piles were used at Kanogawa Dam to stabilize a landslide. A series of
steel pipe piles, 458 mm (18 inches) in diameter were driven in pairs, 5 m (16.4 ft) apart, through
pre-bored holes near the toe of the slide. A plan view of the supporting structure showed that it
extended about 1,100 meters (3,600 ft) in a generally circular pattern. The installation, along
with a drainage tunnel, apparently stabilized the slide.
A slide developed at the Hokuriku Expressway in Fukue Prefecture when a cut to a depth
of 30 m (98 ft) was made. The cut extended to about 170 meters from the centerline of the
highway and was about 100 meters (328 ft) in length. After movement of the slope was
observed, a row of H-piles was installed, but the piles were damaged by an increased by an
increase of the velocity of movement of the slide due to a torrential downpour. Subsequently,
drainage of the slope was improved and four rows of piles were installed parallel to the slope to
stabilize the slide. Analyses showed that the factor of safety against sliding was increased from
near unity to 1.3.
Fukuoka reported that there were numerous examples in Niigata Prefecture where piles
had been used to stabilize landslides. A detailed discussion was presented about the use of piles
at the Higashi-tono landslide. The length of the slide in the direction of the slope was about 130
meters (427 ft), its width was about 40 meters (131 ft), and the sliding surface was found to be
about 5 meters (16.4 ft) below the ground surface. A total of 100 steel pipe piles, 319 mm (12.6
in.) in diameter were installed in the slide over a period of three years. Computations indicated
that the presence of the piles increased the factor of safety against sliding by about 0.18, which
was sufficient to prevent further movement. Strain gages were installed on five of the piles and
these piles were recovered after some time. Two of the piles were fractured due to excessive
bending moment.
Hassiotis and Chameau (1984) and Oakland and Chameau (1986) present brief
descriptions of a large number of cases where piles have been used to stabilize slopes. The
authors present a detailed discussion of the use of piles and drilled piers in the stabilization of
slopes.
6-4 Analytical Procedure
A drawing of a pile embedded in a slope is shown in Figure 6-3(a) where the depth to the
sliding surface is denoted by the symbol hp. The distributed lateral forces from the sliding soil
are shown by the arrows, parallel to the slope in Figure 6-3(b). The resultant of the horizontal
components of the forces from the sliding soil is denoted by the symbol Fs.
The loading for the portion of the pile in stable soil are denoted in Figure 6-3(c) as a
shear P and moment M. The portion of the pile below the sliding surface is caused to deflect
laterally by P and M and the resisting forces from the soil are shown in the lower section of
217
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
Figure 6-3(b). The behavior of the pile can be found by the procedures shown earlier for piles
under lateral loading and the assumptions discussed in the following paragraph.
M
hp
P
(a)
(b)
(c)
Figure 6-3 Forces from Soil Acting Against a Pile in a Sliding Slope, (a) Pile, Slope, and Slip
Surface Geometry, (b) Distribution of Mobilized Forces, (c) Free-body Diagram of Pile Below
the Slip Surface
The principles of limit equilibrium are usually employed in slope stability analysis. The
influence of stabilizing piles on the factor of safety against sliding is illustrated in Figure 6-4.
The resultant of the resistance of the pile, T can be included in the analysis of slope stability.
Therefore, a consistent assumption is that the sliding soil has moved a sufficient amount that the
peak resistance from the soil has developed against the pile. If one considers the force acting on a
pile from a wedge of soil with a sloping surface, the force parallel to the soil surface is larger
than if the surface were horizontal. However, a reasonable assumption is that the peak resistance
acting perpendicular to the pile can be found from the p-y curve formations presented in Chapter
3.
218
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
R
z
T
Safety factor for moment equilibrium considering the same forces as above,
plus the effect of the stabilizing pile is expressed as:
F
 cLR  P  uLR tan    Tz
WX
......................................(6-1)
Where T is the average total force per unit length horizontally resisting soil
movement and z is the distance from the centroid of resisting pressure to
center of rotation.
Figure 6-4 Influence of Stabilizing Pile on Factor of Safety Against Sliding
The discussion above leads to the following step-by-step procedure:
1. Find the factor of safety against sliding for the slope using an appropriate computer
program.
2. At the proposed position for the stabilizing pile, tabulate the relevant soil properties with
depth.
3. Select a pile with a selected diameter and structural properties and compute the bending
stiffness and nominal moment capacity. Compute the ultimate moment capacity (i.e.
factored moment capacity) by multiplying by an appropriate strength reduction factor
(typically around 0.65)
4. Assume that the sliding surface is the same as found in Step 1, then use LPile to compute
the p-y curves at selected depths above the sliding surface. Employ the peak soil reaction
versus depth as a distributed lateral force for depths above the sliding surface as shown in
Figure 6-3(b) and analyze the pile again using LPile.
5. Compare the maximum bending moment found in Step 4 with the nominal moment
capacity from Step 3. At this point, an adjustment of the size or geometry of the pile may
or may not be made, depending on the results of the comparison. Note that in general, the
presence of the piles may change the position of the sliding surface, which will also
change the maximum bending moment developed in the pile. However, in some cases,
219
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
the position of the sliding surface will be known because of the location of a weak soil
layer, and, in any case, it is unlikely that the position of the sliding surface will be
changed significantly by the presence of the piles.
6. Employ the resisting shear and moment in the slope stability analysis used in Step 1 and
find the new position of the sliding surface. While only one pile is shown in Figure 6-4,
one or more rows of piles are most likely to be used. In such a case, the forces due to a
single pile should be divided by the center-to-center spacing along the row of piles prior
to input to the slope stability analysis program because the two-dimensional slope
stability analysis is written assuming that the thickness of the third dimension is one unit.
Some programs for slope stability analysis can use the profile of distributed loads in the
computation of the new sliding surface.
7. Change the depth of sliding, hp, to the depth of sliding employed in Step 4, obtain new
values of M and P, and repeat the analyses until agreement is found between that surface
and the resisting forces for the piles. Also, the geometry of the piles should be adjusted so
that the maximum bending moment found in the analyses is close to the ultimate moment
capacity of the piles.
8. Finally, compare the factor of safety against sliding of the slope with no piles to that with
piles in place and determine whether or not the improvement in factor of safety justifies
the use of the piles.
1
Computed hp
1
Assumed hp
Figure 6-5 Matching of Computed and Assumed Values of hp
6-5 Alternative Method of Analysis
In the method discussed above, the stabilizing force provided by the piles was based on
the peak lateral resistance from the formation of the p-y curves. In some cases, an alternative
approach might be used that is based on an analysis with LPile using the soil movement option.
In this method, the user can draw the geometry of the slope failure and estimate the magnitude of
220
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
soil movement along a vertical alignment at the centerline of the stabilizing pile. The evaluation
of stabilizing forces then proceeds in the manner discussed previously. If the soil movements are
small, the magnitude of stabilizing forces is likely to be smaller than those computed before.
The advantage of using this more conservative method is that the magnitude of the slope
movement needed to mobilize the stabilizing forces is smaller. Thus, if the factor of safety for
the slope is raised to an acceptable level, less distortion of the slope after installation of the
stabilizing piles will occur.
6-6 Case Studies and Example Computation
6-6-1 Case Studies
Fukuoka (1977) described a field experiment that was performed at the landslide at
Higashi-tono in the Niigata Prefecture. A pile, instrumented with strain gages, was installed in a
slide that continued to move at a slow rate. The moving soil was a mudstone and the N-value
from the Standard Penetration Test, NSPT, near the sliding surface was found to be 20 bpf. The
pile was 22 m in length, had an outer diameter of 406 mm, and had a wall thickness of 12.7
millimeters. The bending moment in the pile increased rapidly after installation and appeared to
have reached the maximum value after being in place about three months. The strain gages
showed the maximum bending moment to occur at a depth of about 10 m below the ground
surface and to be about 220 kN-meters. The maximum bending stress in the pile, thus, was about
1.5  105 kPa, a value that shows the loading on the pile from the sliding soil to be very low.
Therefore, it was concluded that the driving force from the moving soil was far from its
maximum value. The positive conclusion from this field test is that the bending-moment curve
given by Fukuoka had the general shape that would be expected.
At another site at the Higashi-tono landslide, Fukuoka described an experiment where a
number of steel-pipe piles were used in a sliding soil. Some of them were removed after a
considerable period of time and found to have failed in bending. One of them had a diameter of
318.5 mm and a wall thickness of 10.3 mm. The collapse moment for the pipe was computed to
be 241 kN-m. Assuming a triangular distribution of earth pressure on the pile from the sliding
mass of soil, which had a thickness of 5 m, the undrained shear strength that was required to
cause the pile to fail was 10.7 kPa. The author merely stated that the soil had a NSPT that was less
than 10 bpf. That value of NSPT probably reflects an undrained shear strength that encompasses
the computed strength to cause the pile to fail.
6-6-2 Example Computation
The example that was selected for analysis is shown in Figure 6-6. The slope exists along
the bank of a river where rapid drawdown is possible. Prior slope failures had been observed at
numerous places along the river and it was desired to stabilize the slope to allow a bridge to be
constructed.
221
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
Elevation, m
80
75
Fill
c = 47.9 kPa
 = 19.6 kN/m3
70
Silt
c = 23.9 kPa
cresidual =12.4 kPa
 = 17.3 kN/3m3
65
60
Clay
c = 36.3 kPa
 = 17.3 kN/m3
Sand
 = 19.6 kN/m3
 = 30 to 40 deg.
55
Figure 6-6 Soil Conditions for Analysis of Slope for Low Water
The undrained analysis for the sudden-drawdown case was made based on the Spencer's
method, and the factor of safety was found to be 1.06, a value that is in reasonable agreement
with observations. Plainly, some method of design and construction would be necessary in order
for bridge piers to be placed at the site. The method described herein was employed to select
sizes and spacing of drilled shafts that could be used to achieve stability.
A preliminary design is shown in Figure 6-7, but not shown in the figure is the distance
along the river for which the slope was to be stabilized. Drilled shafts were selected that were
915 mm (3 ft) in diameter and penetrated well below the sliding surface, as shown in the figure.
Furthermore, it was found that the tops of the shafts had to be restrained by a grade beam
connected to anchor piles outside of the slide zone. The use of the grade beam was required
because of the depth of the slide. The stabilizing piles were modeled with restrained pile heads to
model the effect of the pile-head connection to the grade beam. The results of the analysis, for
each of the pile groups perpendicular to the river, gave the following loads at the top of the
drilled shafts: Rows 1, 2, and 3, +1,090 kN/shaft; Row 4, –1,310 kN/shaft; and Row 5, –1,690
kN/shaft. The grade beam connecting the tops of the five rows of piles would be designed to
sustain the indicated loading. The maximum bending moment computed for shafts in Row 5 on
the extreme right was 6,250 kN-m. This level of moment required heavy reinforcement in the
shaft. The computed bending moments for the other drilled shafts were much smaller.
With the piles in place and with the restraining forces of the piles against the sliding soil,
shown Figure 6-8, a second analysis was performed to find the new factor of safety against
sliding. The new factor of safety that was obtained was 1.82. This result was sufficient to show
that the technique was feasible. However, in a practical design, a series of analyses would have
been performed to find the most economical geometry and spacing for the piles in the group.
222
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
Pile Row 1
2
3
4
5
5.5 m
Pile diameter 915 mm
Grade Beam
30 m
4.6 m 4.6 m
15.2 m
15.2 m
Figure 6-7 Preliminary Design of Stabilizing Piles
Elevation, m
80
48 kPa
48 kPa
75
70
108 kPa
108 kPa
65
71 kPa
71 kPa
60
55
Figure 6-8 Load Distribution from Stabilizing Piles for Slope Stability Analysis
223
Chapter 6 – Use of Vertical Piles to Stabilize a Slope
6-6-3 Conclusions
The results predicted by the proposed design method were compared with results from
available full-scale experiments. The case studies yielded information on the applicability of the
proposed method of analysis.
A complete analysis for the stability of slopes with drilled shafts in place was presented.
The proposed method of analysis is considered practical and can be implemented by engineers
by using readily available methods of analysis. The benefits of using the method is that
rationality and convenience are provided that were not previously available.
224
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232
Name Index
Akinmusuru, J. O. ....................................... 4
Duncan, J. M. .......................................... 124
Allen, J. ................................................... 153
Dunnavant, T. W. .............. 62, 63, 64, 74, 87
American Petroleum Institute . 18, 70, 98, 99
Dunnigan, L. P. ......................................... 64
Armitage, H. H. ....................................... 150
Evans, L. T. ............................................. 124
Ashford, S. A. ......................... 105, 107, 108
Fenske, C. W. ............................................ 87
Audibert, J. M. E. ...................................... 87
Fitzgibbon, D. P. ....................................... 54
Awoshika, K. .............................................. 4
Focht, J. A., Jr. .......................... 4, 18, 65, 67
Azzouz, A. S. ............................................ 11
Fong, P. T. ................................................. 87
Baecher, G. B. ............................................. 3
Franke, K. W. .......................................... 108
Baguelin, F. ....................................... 18, 104
Fukuoka, M. .................................... 217, 221
Baligh, M. M. ............................................ 11
Gazioglu, S. M. ......................................... 87
Bhushan, K. ....................................... 87, 104
George, P................................................... 18
Bishop, A. W........................................... 215
Georgiadis, M. ........................................ 154
Bogard, D. ................................................... 4
Gerber, T. M. .......................................... 105
Bowman, E. R. .......................................... 90
Germaine, J. T. .......................................... 11
Briaud, J. L, ............................................... 87
Grime, D. B. ............................................ 104
Broms, B. B............................................... 16
Hales, L. J. ...................................... 107, 108
Brown, D. A. ................... 4, 55, 86, 114, 152
Haley, S. C. ............................................... 87
Brown, E. T. ............................................ 144
Hansen, J. B. ............................................. 59
Bryant, L. M. ............................................... 7
Harder, L. F. .................................... 105, 110
Chameau, J. L. ........................................ 217
Hassiotis, S.............................................. 217
Christian, J. T. ............................................. 3
Hetenyi, A. .......................................... 14, 30
Cox, W. R. .... 52, 53, 54, 62, 63, 74, 92, 126
Hoek, E. .......................... 144, 145, 147, 151
Dapp, S. D. .............................................. 114
Horvath, R. G. ......................................... 135
Darr, K. ..................................................... 59
Hrennikoff, A. ............................................. 4
Davis, E. H. ............................................... 18
Isenhower, W. M..................... 114, 174, 187
Decker, R. S. ............................................. 64
Jamiolkowski, M. ...................................... 18
Det Norske Veritas .................................... 18
Jezequel, J. F. .................................... 18, 104
DiGiola, A. M. .......................................... 16
Johnson, G. W. .......................................... 54
233
Name Index
Johnson, R. M. ........................................ 114
Kenney, T. C. .......................................... 135
126, 130, 131, 132, 137, 140, 141, 156,
164, 187
Koch, K. J. .................................................. 4
Ripperger, E. A. .................................. 51, 54
Kooijman, A. P. ........................................ 55
Rojas-Gonzalez, L..................................... 16
Koop, F. D........... 52, 54, 62, 63, 74, 92, 126
Rollins, K. M........................... 105, 107, 108
Kubo, K. .................................................. 167
Rowe, R. K. ............................................. 150
Lane, J. D. ............................................... 105
Schmertmann, J. H. ................................. 132
Lee, L. J................................................... 104
Seed, R. B. ...................................... 105, 110
Liang, R................................................... 144
Sherard, J. L. ............................................. 64
Long, J. H................................ 13, 63, 64, 79
Shields, D. H. .................................... 18, 104
Malek, A. M. ............................................. 11
Simpson, M. ............................................ 152
Marinos, P. ...................................... 145, 151
Skempton, A. W........................................ 65
Matlock, H. ..................................... 108, 109
Smith, T. D................................................ 87
Matlock, H. . 4, 20, 51, 54, 68, 70, 72, 74, 88
Speer, D........................................... 133, 141
McClelland, B. .............................. 18, 65, 67
Stevens, J. B. ............................................. 87
Meyer, B. J. ............................................... 87
Stokoe, K. H. ............................................. 54
Morgenstern, N. R................................... 215
Sullivan, W. R. .......................................... 87
Morrison, C. M. ........................................ 55
Terzaghi, K. ...................... 14, 65, 67, 87, 92
Murchison, J. M. ..................................... 104
Thompson, G. R. ................................. 16, 55
Newman, F. B. .......................................... 16
Timoshenko, S. P. ..................................... 39
Nusairat, J. .............................................. 144
Van Impe, W. F. ...................................... 156
Nyman, K. J. ........................................... 132
Vesić, A. S. ............................................... 54
O’Neill, M. W.4, 62, 63, 64, 74, 87, 104,
139
Wang, S.-T. ..... 29, 59, 63, 64, 105, 108, 187
Oakland, M. W. ....................................... 217
Wood, D. ................................................... 18
Parker, F., Jr. ....................................... 92, 97
Wright, S. G. ....................................... 16, 55
Parsons, R. L. .......................................... 114
Yang, K. .................................................. 144
Poulos, H. G. ............................................. 18
Yegian, M. .......................................... 16, 55
Price, V. E. .............................................. 215
Zhang, L. ................................................. 147
Welch, R. C. .................................. 63, 82, 84
Reese, L. C.4, 18, 52, 53, 54, 55, 59, 62, 63,
74, 82, 84, 87, 88, 92, 97, 105, 108, 123,
234