Technical Manual for
LPile, Version 6
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.
Copyright © 2012 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.
Chapter 1 Introduction ... 1
1-1 Compatible Designs... 1
1-2 Principles of Design... 1
1-2-1 Introduction ... 1
1-2-2 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 Under Complex Loadings... 7
1-3-3 Computation of Foundation Stiffness ... 7
1-3-4 Concluding Comments... 7
1-4 Organization of Technical Manual ... 8
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
2-1-1-1 General Review... 11
2-1-1-2 Static Loading ... 12
2-1-1-3 Repeated Cyclic Loading... 13
2-1-1-4 Sustained Loading... 13
2-1-1-5 Dynamic Loading... 14
2-1-2 Models for Use in Analyses of Single Piles... 14
2-1-2-1 Elastic Pile and Soil ... 14
2-1-2-2 Elastic Pile and Finite Elements for Soil ... 16
2-1-2-3 Rigid Pile and Plastic Soil... 16
2-1-2-4 Rigid Pile and Four-Spring Model for Soil... 16
2-1-2-5 Nonlinear Pile and p-y Model for Soil... 17
2-1-2-6 Definition of p and y ... 18
2-1-2-7 Comments on the p-y method ... 19
2-1-3 Computational Approach for Single Piles... 19
2-1-3-1 Study of Pile Buckling... 21
2-1-3-2 Study of Critical Pile Length ... 21
2-1-4 Occurrences of Lateral Loads on Piles... 22
2-1-4-1 Offshore Platform ... 22
2-1-4-2 Breasting Dolphin ... 24
2-1-4-3 Single-Pile Support for a Bridge... 25
2-1-4-4 Pile-Supported Overhead Sign... 25
2-1-4-5 Use of Piles to Stabilize Slopes ... 27
2-2-1 Derivation of the Differential Equation ... 29
2-2-2 Solution of Reduced Form of Differential Equation... 33
2-2-3 Solution by Finite Difference Equations... 40
Chapter 3 Lateral Load-Transfer Curves for Soil and Rock... 47
3-1 Introduction ... 47
3-2 Experimental Measurements of p-y Curves... 49
3-2-1 Direct Measurement of Soil Response... 49
3-2-2 Derivation of Soil Response from Moment Curves Obtained by Experiment... 49
3-2-3 Nondimensional Methods for Obtaining Soil Response... 50
3-3 p-y Curves for Cohesive Soils ... 51
3-3-1 Initial Portion of Curves... 51
3-3-2 Analytical Solutions for Ultimate Lateral Resistance ... 55
3-3-3 Influence of Diameter on p-y Curves ... 61
3-3-4 Influence of Cyclic Loading... 62
3-3-5 Introduction to Procedures for p-y Curves in Clays... 63
3-3-5-1 Early Recommendations for p-y Curves in Clay ... 63
3-3-5-2 Skempton (1951)... 64
3-3-5-3 Terzaghi (1955)... 66
3-3-5-4 McClelland and Focht (1958) ... 66
3-3-6 Step-by-Step Procedures for p-y Curves in Clay ... 67
3-3-7 Response of Soft Clay in the Presence of Free Water... 67
3-3-7-1 Detailed Procedure for Computing p-y Curves in Soft Clay for Static Loading . 68 3-3-7-2 Detailed Procedure for Computing p-y Curves in Soft Clay for Cyclic Loading 70 3-3-7-3 Recommended Soil Tests for Soft Clays ... 71
3-3-7-4 Examples... 71
3-3-8 Response of Stiff Clay in the Presence of Free Water ... 73
3-3-8-1 Detailed Procedure for Computing p-y Curves for Static Loading... 73
3-3-8-2 Detailed Procedure for Computing p-y Curves for Cyclic Loading ... 76
3-3-8-3 Recommended Soil Tests... 78
3-3-8-4 Examples... 79
3-3-9 Response of Stiff Clay with No Free Water... 80
3-3-9-1 Procedure for Computing p-y Curves for Stiff Clay without Free Water for Static Loading ... 81
3-3-9-2 Detailed Procedure for Computing p-y Curves for Stiff Clay without Free Water for Cyclic Loading ... 82
3-3-9-3 Recommended Soil Tests for Stiff Clays... 83
3-3-9-4 Examples... 83
3-3-10 Modified p-y Criteria for Stiff Clay with No Free Water ... 83
3-3-11 Other Recommendations for p-y Curves in Clays... 84
3-4 p-y Curves for Sands... 85
3-4-1 Description of p-y Curves in Sands... 85
3-4-1-1 Initial Portion of Curves... 85
3-4-1-4 Influence of Cyclic Loading ... 90
3-4-1-5 Early Recommendations ... 90
3-4-1-6 Field Experiments ... 90
3-4-1-7 Response of Sand Above and Below the Water Table ... 90
3-4-2 Response of Sand ... 91
3-4-2-1 Detailed Procedure for Computing p-y Curves in Sand... 91
3-4-2-2 Recommended Soil Tests... 95
3-4-2-3 Example Curves ... 95
3-4-3 API RP 2A Recommendation for Response of Sand Above and Below the Water Table ... 96
3-4-3-1 Background of API Method for Sand ... 96
3-4-3-2 Procedure for Computing p-y Curves Using the API Sand Method... 96
3-4-3-3 Example Curves ... 98
3-4-4 Other Recommendations for p-y Curves in Sand... 100
3-5 p-y Curves in Liquefied Sands... 101
3-5-1 Response of Piles in Liquefied Sand... 101
3-5-2 Procedure for p-y Curves in Liquefied Sand... 103
3-5-3 Modeling of Lateral Spreading ... 104
3-6 p-y Curves in Loess ... 105
3-6-1 Background ... 105
3-6-1-1 Description of Load Test Program... 105
3-6-1-2 Soil Profile from Cone Penetration Testing... 105
3-6-2 Detailed Procedure for p-y Curves in Loess... 107
3-6-2-1 General Description of p-y Curves in Loess ... 107
3-6-2-2 Equations of p-y Model for Loess... 108
3-6-2-3 Step-by-Step Procedure for Generating p-y Curves... 114
3-6-2-4 Limitations on Conditions for Validity of Model ... 114
3-7 p-y Curves in Soils with Both Cohesion and Internal Friction... 114
3-7-1 Background ... 114
3-7-2 Recommendations for Computing p-y Curves ... 115
3-7-3 Detailed Procedure for Computing p-y Curves in Soils with Both Cohesion and Internal Friction ... 117
3-7-4 Discussion ... 120
3-8 Response of Vuggy Limestone Rock ... 122
3-8-1 Introduction ... 122
3-8-2 Descriptions of Two Field Experiments... 123
3-8-2-1 Islamorada, Florida ... 123
3-8-2-2 San Francisco, California... 124
3-8-3 Recommendations for Computing p-y Curves for Strong Rock (Vuggy Limestone)127 3-8-4 Recommendations for Computing p-y Curves for Weak Rock... 128
3-8-5 Case Histories for Drilled Shafts in Weak Rock... 131
3-8-5-1 Islamorada... 131
3-8-5-2 San Francisco ... 133
3-9 p-y Curves in Massive Rock... 136
3-9-1 Determination of pu Near Ground Surface ... 136
3-9-5 Step-by-Step Procedure for Computing p-y Curves in Massive Rock... 141
3-10 p-y Curves in Piedmont Residual Soils ... 142
3-11 Response of Layered Soils ... 144
3-11-1 Layering Correction Method of Georgiadis ... 144
3-11-2 Example p-y Curves in Layered Soils ... 145
3-12 Modifications to p-y Curves for Pile Batter and Ground Slope ... 149
3-12-1 Piles in Sloping Ground ... 149
3-12-1-1 Equations for Ultimate Resistance in Clay in Sloping Ground ... 150
3-12-1-2 Equations for Ultimate Resistance in Sand... 151
3-12-1-3 Effect of Direction of Loading on Output p-y Curves ... 152
3-12-2 Effect of Batter on p-y Curves in Clay and Sand ... 153
3-12-3 Modeling of Piles in Short Slopes... 154
3-13 Shearing Force Acting at Pile Tip ... 154
Chapter 4 Special Analyses ... 155
4-1 Introduction ... 155
4-2 Computation of Top Deflection versus Pile Length... 155
4-3 Analysis of Piles Loaded by Soil Movements... 158
4-4 Analysis of Pile Buckling... 159
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity... 164
5-1 Introduction ... 164
5-1-1 Application ... 164
5-1-2 Assumptions ... 164
5-1-3 Stress-Strain Curves for Concrete and Steel ... 165
5-1-4 Cross Sectional Shapes That Can Be Analyzed ... 167
5-2 Beam Theory ... 168
5-2-1 Flexural Behavior... 168
5-2-2 Axial Structural Capacity... 171
5-3 Validation of Method... 172
5-3-1 Analysis of Concrete Sections... 172
5-3-1-1 Computations Using Equations of Section 5-2... 173
5-3-1-2 Check of Position of the Neutral Axis ... 173
5-3-1-3 Forces in Reinforcing Steel... 175
5-3-1-4 Forces in Concrete ... 176
5-3-1-5 Computation of Balance of Axial Thrust Forces ... 179
5-3-1-6 Computation of Bending Moment and EI... 180
5-3-1-7 Computation of Bending Stiffness Using Approximate Method... 181
5-3-2 Analysis of Steel Pipes... 184
5-3-3 Analysis of Prestressed-Concrete Piles ... 187
5-4 Discussion... 190
5-5 Reference Information... 191
5-5-1 Concrete Reinforcing Steel Sizes... 191
Chapter 6 Use of Vertical Piles in Stabilizing a Slope ... 195
6-1 Introduction ... 195
6-2 Applications of the Method ... 195
6-3 Review of Some Previous Applications ... 196
6-4 Analytical Procedure ... 197
6-5 Alternative Method of Analysis ... 200
6-6 Case Studies and Example Computation... 200
6-6-1 Case Studies ... 200
6-6-2 Example Computation... 201
6-6-3 Conclusions ... 203
References ...205
Figure 1-1 Example of Modeling a Bridge ... 6
Figure 1-2 Three-dimensional Soil-Pile Interaction ... 8
Figure 1-3 Coefficients of Stiffness Matrix ... 9
Figure 2-1 Finite Element Model of Pile Under Lateral Loading, (a) 3-Dimensional Mesh, and (b) Mesh Cross-section of 3-D Mesh, (c) Brom’s Model, (d) MFAD Model... 15
Figure 2-2 Model for Pile Under Lateral Loading with p-y Curves ... 17
Figure 2-3 Distribution of Normal Stresses Against a Pile, (a) Before Lateral Deflection and (b) After Lateral Deflection ... 18
Figure 2-4 Illustration of General Procedure for Selecting a Pile to Sustain a Given Set of Loads... 20
Figure 2-5 Solution for the Axial Buckling Load ... 21
Figure 2-6 Solving for Critical Pile Length ... 22
Figure 2-7 Simplified Method of Analyzing a Pile for an Offshore Platform ... 23
Figure 2-8 Analysis of a Breasting Dolphin ... 24
Figure 2-9 Loading On a Single Shaft Supporting a Bridge Deck ... 25
Figure 2-10 Foundation Options for an Overhead Sign Structure ... 26
Figure 2-11 Use of Piles to Stabilize a Slope Failure ... 27
Figure 2-12 Anchor Pile for a Flexible Bulkhead... 28
Figure 2-13 Element of Beam-Column (after Hetenyi, 1946) ... 29
Figure 2-14 Sign Conventions ... 32
Figure 2-15 Form of Results Obtained for a Complete Solution... 33
Figure 2-16 Boundary Conditions at Top of Pile... 35
Figure 2-17 Values of A1, B1, C1, D1... 37
Figure 2-18 Representation of deflected pile ... 41
Figure 2-19 Case 1 of Boundary Conditions ... 43
Figure 2-20 Case 2 of Boundary Conditions ... 43
Figure 2-21 Case 3 of Boundary Conditions ... 44
Figure 2-22 Case 4 of Boundary Conditions ... 45
Figure 3-2 p-y Curves Developed from Static Load Test on 24-inch Diameter Pile
(Reese, et al. 1975)... 51
Figure 3-3 p-y Curves developed from Cyclic Load Tests on 24-inch Diameter Pile (Reese, et al. 1975)... 52
Figure 3-4 Plot of Ratio of Initial Modulus to Undrained Shear Strength for Unconfined-compression Tests on Clay ... 53
Figure 3-5 Variation of Initial Modulus with Depth... 54
Figure 3-6 Assumed Passive Wedge Failure in Clay Soils, (a) Shape of Wedge, (b) Forces Acting on Wedge... 55
Figure 3-7 Measured Profiles of Ground Heave Near Piles Due to Static Loading, (a) Heave at Maximum Load, (b) Residual Heave... 56
Figure 3-8 Ultimate Lateral Resistance for Clay Soils ... 58
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 ... 59
Figure 3-10 Values of Ac and As... 61
Figure 3-11 Scour Around Pile in Clay During Cyclic Loading ... 63
Figure 3-12 p-y Curves in Soft Clay,(a) Static Loading, (b) Cyclic Loading... 69
Figure 3-13 Shear Strength Profile Used for Example p-y Curves for Soft Clay... 72
Figure 3-14 Example p-y Curves for Soft Clay with the Presence of Free Water ... 72
Figure 3-15 Characteristic Shape of p-y Curves for Static Loading in Stiff Clay with Free Water... 74
Figure 3-16 Characteristic Shape of Cyclic p-y Curves for Loading of Stiff Clay with Free Water... 77
Figure 3-17 Example Shear Strength Profile for p-y Curves for Stiff Clay with No Free Water... 79
Figure 3-18 Example p-y Curves for Stiff Clay in Presence of Free Water for Cyclic Loading ... 80
Figure 3-19 Characteristic Shape of p-y Curve for Static Loading in Stiff Clay with No Free Water... 81
Figure 3-20 Characteristic Shape of p-y Curves for Cyclic Loading in Stiff Clay with No Free Water... 82
Figure 3-21 Example p-y Curves for Stiff Clay with No Free Water, Cyclic Loading ... 84
Figure 3-22 Geometry Assumed for Passive Wedge Failure for Pile in Sand... 87
Figure 3-23 Assumed Mode of Soil Failure by Lateral Flow Around Pile in Sand, (a) Section Though Pile, (b) Mohr-Coulomb Diagram ... 89
Figure 3-24 Characteristic Shape of a Set of p-y Curves for Static and Cyclic Loading in Sand... 91
Figure 3-27 Example p-y Curves for Sand Below the Water Table, Static Loading ... 96
Figure 3-28 Coefficients C1, C2, and C3 versus Angle of Internal Friction ... 98
Figure 3-29 Initial Modulus of Subgrade Reaction, k, Used for API Sand Criteria ... 99
Figure 7-30 Example p-y Curves for API Sand Criteria ... 101
Figure 3-31 Example p-y Curve in Liquefied Sand ... 102
Figure 3-32 Characteristic Shape of p-y Curves for c-φ Soil... 116
Figure 3-33 Representative Values of k for c-φ Soil... 119
Figure 3-34 p-y Curves for c-φ Soils... 121
Figure 3-35 Initial Moduli of Rock Measured by Pressuremeter for San Francisco Load Test... 125
Figure 3-36 Modulus Reduction Ratio (Bienawski, 1984) ... 125
Figure 3-37 Engineering Properties for Intact Rocks (after Deere, 1968; Peck, 1976; and Horvath and Kenney, 1979)... 126
Figure 3-38 Characteristic Shape of p-y Curve in Strong Rock ... 128
Figure 3-39 Sketch of p-y Curve for Weak Rock (after Reese, 1997)... 128
Figure 3-40 Comparison of Experimental and Computed Values of Pile-Head Deflection, Islamorada Test (after Reese, 1997) ... 132
Figure 3-41 Computed Curves of Lateral Deflection and Bending Moment versus Depth, Islamorada Test, Lateral Load of 334 kN (after Reese, 1997)... 132
Figure 3-42 Comparison of Experimental and Computed Values of Pile-Head Deflection for Different Values of EI, San Francisco Test... 134
Figure 3-43 Values of EI for three methods, San Francisco test ... 135
Figure 3-44 Comparison of Experimental and Computed Values of Maximum Bending Moments for Different Values of EI, San Francisco Test ... 135
Figure 3-45 Illustration of Equivalent Depths in a Multi-layer Soil Profile ... 145
Figure 3-46 Soil Profile for Example of Layered Soils ... 146
Figure 3-47 Example p-y Curves for Layered Soil ... 147
Figure 3-48 Equivalent Depths of Soil Layers Used for Computing p-y Curves ... 147
Figure 3-49 Pile in Sloping Ground and Battered Pile ... 150
Figure 3-51 p-y Curve Displaced by Soil Movement ... 159
Figure 4-1 Stress-Strain Relationship for Concrete Used by LPile ... 165
Figure 4-2 Stress-Strain Relationship for Reinforcing Steel Used by LPile... 167
Figure 4-3 Element of Beam Subjected to Pure Bending ... 169
Figure 4-4 Validation Problem for Mechanistic Analysis of Rectangular Section... 173
Figure 4-6 Moment vs. Curvature... 182
Figure 4-7 Bending Moment vs. Bending Stiffness... 183
Figure 4-8 Interaction Diagram for Nominal Moment Capacity ... 183
Figure 4-9 Example Pipe Section for Computation of Plastic Moment Capacity ... 184
Figure 4-10 Moment vs. Curvature of Example Pipe Section ... 185
Figure 4-11 Elasto-plastic Stress Distribution Computed by LPile... 187
Figure 4-12 Stress-Strain Curves of Prestressing Strands Recommended by PCI Design Handbook, 5th Edition... 188
Figure 4-13 Sections for Prestressed Concrete Piles Modeled in LPile ... 190
Figure 5-1 Scheme for Installing Pile in a Slope Subject to Sliding... 196
Figure 5-2 Forces from Soil Against Pile in a Sliding Slope... 197
Figure 5-3 Influence of Stabilizing Pile on Factor of Safety Against Sliding ... 198
Figure 5-4 Matching of Computed and Assumed Values of hp... 200
Figure 5-5 Soil Conditions for Analysis of Slope for Low Water ... 201
Figure 5-6 Preliminary Design... 202
Table 3-1. Terzaghi’s Recommendations for Soil Modulus for Laterally Loaded Piles in
Stiff Clay... 66
Table 3-2. Representative Values of ε50... 68
Table 3-3. Representative Values of k for Stiff Clays ... 75
Table 3-4. Representative Values of ε50 for Stiff Clays... 75
Table 3-5 Terzaghi’s Recommendations for Values of k for Laterally Loaded Piles in Sand... 86
Table 3-6 Representative Values of k for Submerged Sand ... 94
Table 3-7 Representative Values of k for Sand Above Water Table for Static and Cyclic Loading ... 94
Table 3-8 Results of Grout Plug Tests by Schmertmann... 124
Table 3-9 Values of Compressive Strength at San Francisco... 127
Table 4-1 LPile Output for Rectangular Concrete Section... 174
Chapter 1
Introduction
1-1 Compatible Designs
The program LPile provides the capability to analyze piles for a variety of applications in which lateral loading is applied to a deep foundation. 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 agree. Analyses of this type have been in use in the practice of civil engineering for some time and the analytical procedures that are used 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 moment 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 instruments 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 satisfying simultaneously the appropriate nonlinear response of the soil. The pile is treated as a beam-column and the soil is replaced with nonlinear Winkler-type 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 IntroductionThe 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 of the consideration of many practical details related to construction 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 deflection, bending moment, and shear force along the length of a pile under loading. Additional considerations that are useful are selection of the minimum required length of a pile foundation and evaluation of the buckling capacity of a pile that extends above the ground line.
1-2-2 Nonlinear Response of Soil
In one sense, the design of a pile under lateral loading is no different that the design of any foundation. One needs to determine first the loading of the foundation that will cause failure and then to apply a global factor of safety or load and resistance factors to set the allowable loading for the foundation. What is different for analysis of lateral loading is that the failure cannot be found by solving the equations of static equilibrium. Instead, the lateral capacity of the foundation can only be found by solving a differential equation governing its behavior and then evaluating the results of the solution. Furthermore, as noted below, 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 installed into overconsolidated clay with free water present above the ground surface. The results for 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 most instances, failure of a pile is initiated by a bending moment that would cause the development of a plastic hinge. However, in some 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.
An approach using partial load and resistance factors may be employed. However, analyses employed in applying load and resistance factors is implemented herein by using upper-bound and lower-upper-bound values of the important parameters.
1-2-4 Step-by-Step Procedure
1. Assemble all relevant data, including soil properties, magnitude and nature of the 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.
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.
Virtually none of the examples in this manual 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.
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 instru-mentation 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 opportunity for rewarding research on the topic of this document, and the user is urged to stay
1-3 Modeling a Pile Foundation
1-3-1 IntroductionAs a foundation engineering problem, the analysis of a pile under axial and lateral loading is complicated by the fact that the mobilized soil reaction is 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 step-wise 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 three-dimensional, 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 two-dimensional 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 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 Lumped masses Foundation springs b. Analytical Model Rotation
θ
Mome nt M K33 K33 K11 K22 Rotationθ
Mome nt M K33 K33 K11 K22 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ M V P K K K K K y x θ δ δ 33 32 23 22 11 0 0 0 0 c. Stiffness MatrixFigure 1-1 Example of Modeling a Bridge
The stiffness K11 is the stiffness of the axial load-settlement curve for the axial load P.
1-3-2 Example Model of Individual Pile Under Complex Loadings
An interesting presentation of the forces that resist the displacement 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 to the cap so that bending moments and a torque will develop as a result of the three-dimensional rotations of the cap. The various reactions of the soil along the pile are shown in Figure 1-2(b), and the soil-resistance 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 two-dimensional plane. 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 stiffness of a pile foundation. This option 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 10 unequal 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. The rotation of the pile head is computed for each bending-moment increment with the 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 Mi by the computed
rotation θi at the pile cap.
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.
Torsional Pile Displacement, θ Lateral Pile Displacement, y (a) Three-dimensional pile displacements Axial Soil Reaction, q Torsional Soil Reaction, t Lateral Soil Reaction, p y x z Axial Pile Displacement, u
(b) Pile reactions (c) Nonlinear load-transfer curves Px Py Pz My Mx Mz q u p y t θ Axial Lateral Torsional Torsional Pile Displacement, θ Lateral Pile Displacement, y (a) Three-dimensional pile displacements Axial Soil Reaction, q Torsional Soil Reaction, t Lateral Soil Reaction, p y x z Axial Pile Displacement, u
(b) Pile reactions (c) Nonlinear load-transfer curves Px Py Pz My Mx Mz q u p y t θ Axial Lateral Torsional
Figure 1-2 Three-dimensional Soil-Pile Interaction
1-4 Organization of Technical Manual
Chapters 2 to 4 provide the user with the 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. Also, 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 4 of this manual describes a program feature that can be provided for this purpose. The program can compute the flexural rigidity of the section as a function of the bending moment.
| LPile computes K22, K23, K32, and K33given the lateral load, H, and the bending moment, M, at the pile head. Stiffnesses K22 and K23are 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 δ and reaction moment −M at the pile head using zero slope at the pile head (pile head rotation θ = 0). Then K22= | V/δ | and K32= |–M/δ |. −
M
P
V
δ
≠ 0
θ
= 0
Stiffnesses K32and K33are 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 θ using zero deflection at the pile head (δ = 0).
Then K23= |–V/θ | and K33= | M/θ |.
M
P
−V
δ
= 0
θ
≠ 0
LPile computes K22, K23, K32, and K33given the lateral load, H, and the bending moment, M, at the pile head. Stiffnesses K22 and K23are 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 δ and reaction moment −M at the pile head using zero slope at the pile head (pile head rotation θ = 0). Then K22= | V/δ | and K32= |–M/δ |. −
M
P
V
δ
≠ 0
θ
= 0
Stiffnesses K32and K33are 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 θ using zero deflection at the pile head (δ = 0).
Then K23= |–V/θ | and K33= | M/θ |.
M
P
−V
δ
= 0
θ
≠ 0
Figure 1-3 Coefficients of Stiffness Matrix
If one is performing an elastic analysis, it is suggested that reduced values of flexural rigidity be used in the region of maximum bending moment for each value of lateral load because the flexural rigidity varies as a function of the bending moment. However, experience has often found that the lateral response of a pile is not critically dependent on the value of flexural rigidity for smaller lateral loads. Recommendations are provided for the selection of flexural rigidity that will yield results that are considered to be acceptable. However, the user could use the results from Chapter 4 as input to the coding for Chapter 2 to investigate the importance of entering accurate values of flexural rigidity.
Finally, Chapter 5 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 LPile User’s Manual.
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
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 earthquake-resistant 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 three-dimensional 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
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 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 develops 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 bringing soil particles with it. This erosion causes a loss of soil resistance in addition to the losses due to remolding of the soil as a result of the cyclic strains. At this point the use of judgment in the design of the piles under lateral load should be emphasized. For example, if the clay is below a layer of sand, or if provision could be made to supply sand around the pile, the sand will settle around the pile, and probably restore the soil resistance that was lost due to the cyclic loading.
Pile-supported structures are subjected to cyclic loading in many instances. Some common cases are wind load against overhead signs and high-rise buildings, traffic loads on bridge structures, wave loads against 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.
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 quite approximate.
2-1-2 Models for Use in Analyses of Single Piles
A number of models have been proposed 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 was employed in a manner that had been suggested earlier by such writers as Hetenyi (1946). Terzaghi stated that the tabulated values of subgrade modulus could not be used for cases where the computed soil resistance was more than one-half of the bearing capacity of the soil. However, a recommendation was not included for the computation of the bearing capacity under lateral load. Nor were any comparisons given between the results of computations and experiments.
The values of subgrade moduli published by Terzaghi have proved to be useful and provide evidence that Terzaghi had excellent insight into the problem. However, in a private
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.
(a) (b) Mt Pt Mt Pt Mt Pt Mt Pt (c) (d)
Figure 2-1 Finite Element Model of Pile Under Lateral Loading, (a) 3-Dimensional Mesh, and (b) Mesh Cross-section of 3-D Mesh,
2-1-2-2 Elastic Pile and Finite Elements for Soil
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 map, boundary constraints, special interface elements, most favorable shape and size of elements, or other details. The finite elements may be axially symmetric with non-symmetric loading or full three-dimensional models. The elements may be selected as linear or nonlinear.
In view of the computational power that is now available, the model shown in Figure 2-1(b) appears to be practical to solve the pile problem. The elements can be three-dimensional and nonlinear. However, the selection of an appropriate constitutive model for the soil involves not only the parameters that define the model but methods of dealing with tensile stresses, modeling layered soils, separation between pile and soil during repeated loading, and the changes in soil characteristics that are associated with the various types of loading.
Yegian and Wright (1973) and Thompson (1977) used a plane-stress 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 three-dimensional, 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. More discussion on the use of the finite-element method is presented in a later chapter where p-y curves are described.
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 pile is assumed to be rigid, and a solution is found by use of 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(a).
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 piles that support transmission towers (DiGioia, et al., 1989). The loading shown at the top of the pile includes an axial load. As shown in the sketch, the four springs are: a spring at the pile tip that responds to the rotation of the tip, a spring at the pile tip that responds to the axial movement of the tip, a set of springs parallel to the wall that respond to vertical movement of the pile, and a set of springs normal to the wall that respond to lateral deflection.
The model was developed by 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 types of movement. Therefore, the relative importance of the four types of soil resistance has not
2-1-2-5 Nonlinear Pile and p-y Model for Soil
The model shown in Figure 2-2 represents the one utilized by the LPile software. The loading on the pile is general for the two-dimensional case (no torsion or out-of-plane bending). 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. 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. Also, it is possible to vary the bending stiffness with respect to the bending moment that is computed during iteration.
An axial load is indicated and is considered in the solution with respect to its effect on bending and not in respect to axial settlement. However, as shown later in this manual, the computational procedure is such that it allows for the determination of the axial load at which a pile will buckle.
The soil around the pile is replaced by a set of mechanisms that merely indicate that the soil resistance p is a nonlinear function of pile deflection y. The mechanisms, and the corresponding curves that represent their behavior, are widely spaced in the sketch but are considered to be close together in the analysis. As may be seen, the p-y curves are fully nonlinear with respect to distance x along the pile and pile deflection y. The curve for x = x1 is drawn to
indicate that the pile may deflect a finite distance with no soil resistance. The curve at x = x2 is
drawn to show that the soil is deflection-softening. There is no reasonable limit to the variations that can be employed in representing the response of the soil to the lateral deflection of a pile.
p y p y p y p y p y
P
M
Q
y
x
p y p y p y p y p yP
M
Q
y
x
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 (1958), B. “. Two developments during the 1950’s made the method possible: the digital computer for solving a nonlinear, fourth-order differential equation; and the remote-reading strain gauge for use in obtaining soil-response (p-y) curves from full-scale lateral load tests of piles.
The p-y method evolved first from research sponsored by the petroleum industry in the
1950’s and 1960’s. Piles were designed for the support of platforms that were to be subjected to exceptionally large horizontal forces from 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 (1987) 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 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. At the Foundation Engineering Congress, ASCE, Evanston, Illinois, 1989, one of the keynote papers and 14 percent of the 125 papers dealt with some aspect of piles subjected to lateral loading.
2-1-2-6 Definition of p and y
The definition of the quantities p and y as used in this document is 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 cylindrical pile. This distribution of stresses is correct for a pile that has been installed without bending. If the pile is deflected a distance y (exaggerated in the sketch for clarity), the distribution of unit 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 a normal and a shearing component.
y
p
(a)
(b)
y
p
(a)
(b)
Integration of the unit stresses results in the quantity p which acts opposite in direction to
y. The dimensions of p are load 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 ordinary beam equation.
2-1-2-7 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 as a series of discrete springs (the Winkler model). Several comments can be given in response to this valid criticism.
The recommendations for the prediction of p-y curves for use in the analysis of piles, given in a subsequent chapter, 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 will predict within reasonable limits the response of a pile whose head is free to rotate or is fixed against rotation.
The methods of predicting p-y curves that were derived from correlations with results of full-scale experiments have been used to make computations for the response of piles where only the pile-head movements were recorded. These comparisons, some of which are shown later in this document, show reasonable to excellent agreement between computed and experimental results.
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 2-4. Figure 2-4 (a) shows a pile with a given geometry embedded in a soil with known characteristics. A lateral load Pt, an axial load Q, and a moment M are
acting at the pile head. The loading presumably would have been found by considering the unfactored loading on the superstructure. Each of the loads is decreased or increased by the same multiplier 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 2-4 (b), which will show the maximum bending moment at some point along the pile as a function of the 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. The assumption is made that a plastic hinge at any point along the length of the pile would not be tolerable. 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.
