Technical Manual for LPile 2013 (Using Data Format Version 7) 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. October 2013 Copyright © 2013 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: October 24, 2013 Table of Contents 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 Three-Dimensional Loadings ..................... 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 2-1-1-1 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 ................................................................................................ 23 2-1-4-3 Single-Pile Support for a Bridge.......................................................................... 24 2-1-4-4 Pile-Supported Overhead Sign............................................................................. 25 2-1-4-5 Use of Piles to Stabilize Slopes ........................................................................... 27 2-1-4-6 Anchor Pile in a Mooring System........................................................................ 27 2-1-4-7 Other Uses of Laterally Loaded Piles .................................................................. 27 iii 2-2 Derivation of Differential Equation for the Beam-Column and Methods of Solution....... 28 2-2-1 Derivation of the Differential Equation ...................................................................... 28 2-2-2 Solution of Reduced Form of Differential Equation................................................... 32 2-2-3 Solution by Finite Difference Equations..................................................................... 37 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock......................................................... 45 3-1 Introduction ........................................................................................................................ 45 3-2 Experimental Measurements of p-y Curves........................................................................ 47 3-2-1 Direct Measurement of Soil Response ........................................................................ 47 3-2-2 Derivation of Soil Response from Moment Curves Obtained by Experiment............ 47 3-2-3 Nondimensional Methods for Obtaining Soil Response ............................................. 49 3-3 p-y Curves for Cohesive Soils ............................................................................................ 50 3-3-1 Initial Slope of Curves................................................................................................. 50 3-3-2 Analytical Solutions for Ultimate Lateral Resistance ................................................. 52 3-3-3 Influence of Diameter on p-y Curves .......................................................................... 58 3-3-4 Influence of Cyclic Loading........................................................................................ 59 3-3-5 Introduction to Procedures for p-y Curves in Clays.................................................... 61 3-3-5-1 Early Recommendations for p-y Curves in Clay ................................................. 61 3-3-5-2 Skempton (1951).................................................................................................. 61 3-3-5-3 Terzaghi (1955).................................................................................................... 63 3-3-5-4 McClelland and Focht (1956) .............................................................................. 63 3-3-6 Procedures for Computing p-y Curves in Clay ........................................................... 64 3-3-7 Response of Soft Clay in the Presence of Free Water................................................. 64 3-3-7-1 Description of Load Test Program....................................................................... 64 3-3-7-2 Procedure for Computing p-y Curves in Soft Clay for Static Loading................ 65 3-3-7-3 Procedure for Computing p-y Curves in Soft Clay for Cyclic Loading .............. 68 3-3-7-4 Recommended Soil Tests for Soft Clays ............................................................. 68 3-3-7-5 Examples.............................................................................................................. 68 3-3-8 Response of Stiff Clay in the Presence of Free Water ................................................ 70 3-3-8-1 Procedure for Computing p-y Curves for Static Loading .................................... 70 3-3-8-2 Procedure for Computing p-y Curves for Cyclic Loading................................... 73 3-3-8-3 Recommended Soil Tests..................................................................................... 74 3-3-8-4 Examples.............................................................................................................. 75 3-3-9 Response of Stiff Clay with No Free Water................................................................ 75 3-3-9-1 Procedure for Computing p-y Curves for Stiff Clay without Free Water for Static Loading ............................................................................................................................. 76 3-3-9-2 Procedure for Computing p-y Curves for Stiff Clay without Free Water for Cyclic Loading ............................................................................................................................. 78 3-3-9-3 Recommended Soil Tests for Stiff Clays............................................................. 79 3-3-9-4 Examples.............................................................................................................. 79 3-3-10 Modified p-y Criteria for Stiff Clay with No Free Water ......................................... 80 3-3-11 Other Recommendations for p-y Curves in Clays..................................................... 80 3-4 p-y Curves for Sands........................................................................................................... 81 3-4-1 Description of p-y Curves in Sands............................................................................. 81 3-4-1-1 Initial Portion of Curves....................................................................................... 81 3-4-1-2 Analytical Solutions for Ultimate Resistance ...................................................... 82 3-4-1-3 Influence of Diameter on p-y Curves................................................................... 83 iv 3-4-1-4 Influence of Cyclic Loading ................................................................................ 84 3-4-1-5 Early Recommendations ...................................................................................... 85 3-4-1-6 Field Experiments ................................................................................................ 85 3-4-1-7 Response of Sand Above and Below the Water Table ........................................ 85 3-4-2 Response of Sand ........................................................................................................ 85 3-4-2-1 Procedure for Computing p-y Curves in Sand ..................................................... 86 3-4-2-2 Recommended Soil Tests..................................................................................... 90 3-4-2-3 Example Curves ................................................................................................... 91 3-4-3 API RP 2A Recommendation for Response of Sand Above and Below the Water Table ..................................................................................................................................... 91 3-4-3-1 Background of API Method for Sand .................................................................. 91 3-4-3-2 Procedure for Computing p-y Curves Using the API Sand Method.................... 92 3-4-3-3 Example Curves ................................................................................................... 94 3-4-4 Other Recommendations for p-y Curves in Sand........................................................ 96 3-5 p-y Curves in Liquefied Sands............................................................................................ 96 3-5-1 Response of Piles in Liquefied Sand........................................................................... 96 3-5-2 Procedure for Computing p-y Curves in Liquefied Sand............................................ 98 3-5-3 Modeling of Lateral Spreading ................................................................................... 99 3-6 p-y Curves in Loess ............................................................................................................ 99 3-6-1 Background ................................................................................................................. 99 3-6-1-1 Description of Load Test Program....................................................................... 99 3-6-1-2 Soil Profile from Cone Penetration Testing....................................................... 100 3-6-2 Procedure for Computing p-y Curves in Loess ......................................................... 101 3-6-2-1 General Description of p-y Curves in Loess ...................................................... 101 3-6-2-2 Equations of p-y Model for Loess...................................................................... 101 3-6-2-3 Step-by-Step Procedure for Generating p-y Curves........................................... 106 3-6-2-4 Limitations on Conditions for Validity of Model .............................................. 107 3-7 p-y Curves in Soils with Both Cohesion and Internal Friction......................................... 107 3-7-1 Background ............................................................................................................... 107 3-7-2 Recommendations for Computing p-y Curves .......................................................... 108 3-7-3 Procedure for Computing p-y Curves in Soils with Both Cohesion and Internal Friction................................................................................................................................ 109 3-7-4 Discussion ................................................................................................................. 112 3-8 Response of Vuggy Limestone Rock ............................................................................... 113 3-8-1 Introduction ............................................................................................................... 113 3-8-2 Descriptions of Two Field Experiments.................................................................... 114 3-8-2-1 Islamorada, Florida ............................................................................................ 114 3-8-2-2 San Francisco, California................................................................................... 115 3-8-3 Procedure for Computing p-y Curves for Strong Rock (Vuggy Limestone) ............ 119 3-8-4 Procedure for Computing p-y Curves for Weak Rock .............................................. 119 3-8-5 Case Histories for Drilled Shafts in Weak Rock....................................................... 122 3-8-5-1 Islamorada.......................................................................................................... 122 3-8-5-2 San Francisco..................................................................................................... 123 3-9 p-y Curves in Massive Rock............................................................................................. 125 3-9-1 Determination of pu Near Ground Surface................................................................ 127 3-9-2 Rock Mass Failure at Great Depth ............................................................................ 129 v 3-9-3 Initial Tangent to p-y Curve Ki .................................................................................. 129 3-9-4 Rock Mass Properties................................................................................................ 129 3-9-5 Procedure for Computing p-y Curves in Massive Rock............................................ 131 3-10 p-y Curves in Piedmont Residual Soils .......................................................................... 132 3-11 Response of Layered Soils ............................................................................................. 133 3-11-1 Layering Correction Method of Georgiadis ............................................................ 134 3-11-2 Example p-y Curves in Layered Soils ..................................................................... 134 3-12 Modifications to p-y Curves for Pile Batter and Ground Slope ..................................... 139 3-12-1 Piles in Sloping Ground .......................................................................................... 139 3-12-1-1 Equations for Ultimate Resistance in Clay in Sloping Ground ....................... 139 3-12-1-2 Equations for Ultimate Resistance in Sand...................................................... 140 3-12-1-3 Effect of Direction of Loading on Output p-y Curves ..................................... 141 3-12-2 Effect of Batter on p-y Curves in Clay and Sand .................................................... 142 3-12-3 Modeling of Piles in Short Slopes........................................................................... 143 3-13 Shearing Force Acting at Pile Tip .................................................................................. 143 Chapter 4 Special Analyses ........................................................................................................ 144 4-1 Introduction ...................................................................................................................... 144 4-2 Computation of Top Deflection versus Pile Length......................................................... 144 4-3 Analysis of Piles Loaded by Soil Movements.................................................................. 147 4-4 Analysis of Pile Buckling ................................................................................................. 148 4-4-1 Procedure for Analysis of Pile Buckling................................................................... 148 4-4-2 Example of Incorrect Analysis.................................................................................. 150 4-4-3 Evaluation of Pile Buckling Capacity ....................................................................... 151 4-5 Pushover Analysis of Piles ............................................................................................... 152 4-5-1 Procedure for Pushover Analysis .............................................................................. 153 4-5-2 Example of Pushover Analysis ................................................................................. 153 4-5-3 Evaluation of Pushover Analysis .............................................................................. 155 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity....................... 157 5-1 Introduction ...................................................................................................................... 157 5-1-1 Application ................................................................................................................ 157 5-1-2 Assumptions .............................................................................................................. 157 5-1-3 Stress-Strain Curves for Concrete and Steel ............................................................. 158 5-1-4 Cross Sectional Shape Types .................................................................................... 160 5-2 Beam Theory .................................................................................................................... 160 5-2-1 Flexural Behavior...................................................................................................... 160 5-2-2 Axial Structural Capacity .......................................................................................... 163 5-3 Validation of Method........................................................................................................ 164 5-3-1 Analysis of Concrete Sections................................................................................... 164 5-3-1-1 Computations Using Equations of Section 5-2.................................................. 165 5-3-1-2 Check of Position of the Neutral Axis ............................................................... 165 5-3-1-3 Forces in Reinforcing Steel................................................................................ 167 5-3-1-4 Forces in Concrete ............................................................................................. 168 5-3-1-5 Computation of Balance of Axial Thrust Forces ............................................... 170 5-3-1-6 Computation of Bending Moment and EI.......................................................... 171 5-3-1-7 Computation of Bending Stiffness Using Approximate Method....................... 172 vi 5-3-2 Analysis of Steel Pipe Piles....................................................................................... 175 5-3-3 Analysis of Prestressed-Concrete Piles ..................................................................... 177 5-4 Discussion......................................................................................................................... 180 5-5 Reference Information...................................................................................................... 181 5-5-1 Concrete Reinforcing Steel Sizes.............................................................................. 181 5-5-2 Prestressing Strand Types and Sizes ......................................................................... 182 5-5-3 Steel H-Piles.............................................................................................................. 183 Chapter 6 Use of Vertical Piles in Stabilizing a Slope ............................................................... 184 6-1 Introduction ...................................................................................................................... 184 6-2 Applications of the Method .............................................................................................. 184 6-3 Review of Some Previous Applications ........................................................................... 185 6-4 Analytical Procedure ........................................................................................................ 186 6-5 Alternative Method of Analysis ....................................................................................... 189 6-6 Case Studies and Example Computation.......................................................................... 189 6-6-1 Case Studies .............................................................................................................. 189 6-6-2 Example Computation............................................................................................... 190 6-6-3 Conclusions ............................................................................................................... 192 References ...................................................................................................................................194 Name Index .................................................................................................................................202 vii List of Figures Figure 1-1 Example of Modeling a Bridge Foundation................................................................. 6 Figure 1-2 Three-dimensional Soil-Pile Interaction ...................................................................... 7 Figure 1-3 Coefficients of Stiffness Matrix ................................................................................... 8 Figure 2-1 Models of Pile Under Lateral Loading, (a) 3-Dimensional Finite Element Mesh, and (b) Cross-section of 3(d) MFAD Model....................................................................................................... 15 Figure 2-2 Model of Pile Under Lateral Loading and p-y Curves ............................................... 17 Figure 2-3 Distribution of Stresses Acting on a Pile, (a) Before Lateral Deflection and (b) After Lateral Deflection y .................................................................................... 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 ..................................................................................................... 31 Figure 2-15 Form of Results Obtained for a Complete Solution................................................. 32 Figure 2-16 Boundary Conditions at Top of Pile......................................................................... 33 Figure 2-17 Values of Coefficients A1, B1, C1, and D1 ................................................................ 35 Figure 2-18 Representation of deflected pile............................................................................... 38 Figure 2-19 Case 1 of Boundary Conditions ............................................................................... 40 Figure 2-20 Case 2 of Boundary Conditions ............................................................................... 41 Figure 2-21 Case 3 of Boundary Conditions ............................................................................... 41 Figure 2-22 Case 4 of Boundary Conditions ............................................................................... 42 Figure 2-23 Case 5 of Boundary Conditions ............................................................................... 43 Figure 3-1 Conceptual p-y Curves ............................................................................................... 45 viii Figure 3-2 p-y Curves from Static Load Test on 24-inch Diameter Pile (Reese, et al. 1975) .......................................................................................................................... 48 Figure 3-3 p-y Curves from Cyclic Load Tests on 24-inch Diameter Pile (Reese, et al. 1975) .......................................................................................................................... 49 Figure 3-4 Plot of Ratio of Initial Modulus to Undrained Shear Strength for Unconfinedcompression Tests on Clay ........................................................................................ 51 Figure 3-5 Variation of Initial Modulus with Depth.................................................................... 52 Figure 3-6 Assumed Passive Wedge Failure in Clay Soils, (a) Shape of Wedge, (b) Forces Acting on Wedge ........................................................................................... 53 Figure 3-7 Measured Profiles of Ground Heave Near Piles Due to Static Loading, (a) Heave at Maximum Load, (b) Residual Heave ......................................................... 54 Figure 3-8 Ultimate Lateral Resistance for Clay Soils ................................................................ 56 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................. 57 Figure 3-10 Values of Ac and As................................................................................................... 58 Figure 3-11 Scour Around Pile in Clay During Cyclic Loading, (a) Profile View, (b) Photograph of Turbulence Causing Erosion During Lateral Load Test .................... 60 Figure 3-12 p-y Curves in Soft Clay,(a) Static Loading, (b) Cyclic Loading.............................. 66 Figure 3-13 Example p-y Curves in Soft Clay Showing Effect of J............................................ 67 Figure 3-14 Shear Strength Profile Used for Example p-y Curves for Soft Clay........................ 69 Figure 3-15 Example p-y Curves for Soft Clay with the Presence of Free Water....................... 69 Figure 3-16 Characteristic Shape of p-y Curves for Static Loading in Stiff Clay with Free Water.......................................................................................................................... 71 Figure 3-17 Characteristic Shape of p-y Curves for Cyclic Loading of Stiff Clay with Free Water ................................................................................................................. 73 Figure 3-18 Example Shear Strength Profile for p-y Curves for Stiff Clay with No Free Water.......................................................................................................................... 75 Figure 3-19 Example p-y Curves for Stiff Clay in Presence of Free Water for Cyclic Loading ...................................................................................................................... 76 Figure 3-20 Characteristic Shape of p-y Curve for Static Loading in Stiff Clay without Free Water ................................................................................................................. 77 Figure 3-21 Characteristic Shape of p-y Curves for Cyclic Loading in Stiff Clay with No Free Water ................................................................................................................. 78 Figure 3-22 Example p-y Curves for Stiff Clay with No Free Water, Cyclic Loading .............. 79 Figure 3-23 Geometry Assumed for Passive Wedge Failure for Pile in Sand............................. 82 Figure 3-24 Assumed Mode of Soil Failure by Lateral Flow Around Pile in Sand, (a) Section Though Pile, (b) Mohr-Coulomb Diagram................................................... 84 ix Figure 3-25 Characteristic Shape of a Set of p-y Curves for Static and Cyclic Loading in Sand ........................................................................................................................... 86 Figure 3-26 Values of Coefficients and ........................................................................... 88 Figure 3-27 Values of Coefficients Bc and Bs .............................................................................. 88 Figure 3-29 Example p-y Curves for Sand Below the Water Table, Static Loading................... 91 Figure 3-30 Coefficients C1, C2, and C3 versus Angle of Internal Friction ................................. 93 Figure 3-31 Value of k, Used for API Sand Criteria.................................................................... 94 Figure 3-32 Example p-y Curves for API Sand Criteria.............................................................. 96 Figure 3-33 Example p-y Curve in Liquefied Sand ..................................................................... 97 Figure 3-34 Idealized Tip Resistance Profile from CPT Testing Used for Analyses. ............... 101 Figure 3-35. Generic p-y curve for Drilled Shafts in Loess Soils.............................................. 102 Figure 3-36 Variation of Modulus Ratio with Normalized Lateral Displacement .................... 104 Figure 3-37 p-y Curves for the 30-inch Diameter Shafts........................................................... 105 Figure 3-38 p-y Curves and Secant Modulus for the 42-inch Diameter Shafts. ........................ 105 Figure 3-39 Cyclic Degradation of p-y Curves for 30-inch Shafts ............................................ 106 Figure 3-40 Characteristic Shape of p-y Curves for c- Soil..................................................... 108 Figure 3-41 Representative Values of k for c- Soil.................................................................. 111 Figure 3-42 p-y Curves for c- Soils.......................................................................................... 112 Figure 3-43 Initial Moduli of Rock Measured by Pressuremeter for San Francisco Load Test .......................................................................................................................... 116 Figure 3-44 Modulus Reduction Ratio versus RQD (Bienawski, 1984) ................................... 117 Figure 3-45 Engineering Properties for Intact Rocks (after Deere, 1968; Peck, 1976; and Horvath and Kenney, 1979)..................................................................................... 118 Figure 3-46 Characteristic Shape of p-y Curve in Strong Rock ................................................ 119 Figure 3-47 Sketch of p-y Curve for Weak Rock (after Reese, 1997)....................................... 120 Figure 3-48 Comparison of Experimental and Computed Values of Pile-Head Deflection, Islamorada Test (after Reese, 1997) ........................................................................ 123 Figure 3-49 Computed Curves of Lateral Deflection and Bending Moment versus Depth, Islamorada Test, Lateral Load of 334 kN (after Reese, 1997) ................................ 124 Figure 3-50 Comparison of Experimental and Computed Values of Pile-Head Deflection for Different Values of EI, San Francisco Test ....................................................... 125 Figure 3-51 Values of EI for three methods, San Francisco test ............................................... 126 Figure 3-52 Comparison of Experimental and Computed Values of Maximum Bending Moments for Different Values of EI, San Francisco Test ....................................... 126 x Figure 3-53 p-y Curve in Massive Rock .................................................................................... 127 Figure 3-54 Model of Passive Wedge for Drilled Shafts in Rock ............................................. 128 Figure 3-55 Equation for Estimating Modulus Reduction Ratio from Geological Strength Index ........................................................................................................................ 131 Figure 3-56 Degradation Plot for Es .......................................................................................... 133 Figure 3-57 p-y Curve for Piedmont Residual Soil.................................................................... 133 Figure 3-58 Illustration of Equivalent Depths in a Multi-layer Soil Profile.............................. 135 Figure 3-59 Soil Profile for Example of Layered Soils ............................................................. 135 Figure 3-60 Example p-y Curves for Layered Soil .................................................................... 136 Figure 3-61 Equivalent Depths of Soil Layers Used for Computing p-y Curves ...................... 136 Figure 3-62 Pile in Sloping Ground and Battered Pile .............................................................. 139 Figure 3-63 Soil Resistance Ratios for p-y Curves for Battered Piles from Experiment from Kubo (1964) and Awoshika and Reese (1971) ............................................... 142 Figure 4-1 Pile and Soil Profile for Example Problem .............................................................. 145 Figure 4-2 Variation of Top Deflection versus Depth for Example Problem............................ 145 Figure 4-3 Pile-head Load versus Deflection for Example........................................................ 146 Figure 4-4 Top Deflection versus Pile Length for Example...................................................... 146 Figure 4-5 p-y Curve Displaced by Soil Movement .................................................................. 148 Figure 4-6 Examples of Pile Buckling Curves for Different Shear Force Values ..................... 149 Figure 4-7 Examples of Correct and Incorrect Pile Buckling Analyses .................................... 150 Figure 4-8 Typical Results from Pile Buckling Analysis .......................................................... 151 Figure 4-9 Pile Buckling Results Showing a and b ................................................................... 152 Figure 4-10 Dialog for Controls for Pushover Analysis ............................................................ 153 Figure 4-11 Pile-head Shear Force versus Displacement from Pushover Analysis................... 154 Figure 4-12 Maximum Moment Developed in Pile versus Displacement from Pushover Analysis ................................................................................................................... 154 Figure 5-1 Stress-Strain Relationship for Concrete Used by LPile ........................................... 158 Figure 5-2 Stress-Strain Relationship for Reinforcing Steel Used by LPile.............................. 159 Figure 5-3 Element of Beam Subjected to Pure Bending .......................................................... 161 Figure 5-4 Validation Problem for Mechanistic Analysis of Rectangular Section.................... 165 Figure 5-5 Free Body Diagram Used for Computing Nominal Moment Capacity of Reinforced Concrete Section ................................................................................... 172 Figure 5-6 Bending Moment versus Curvature.......................................................................... 173 Figure 5-7 Bending Moment versus Bending Stiffness ............................................................. 174 xi Figure 5-8 Interaction Diagram for Nominal Moment Capacity ............................................... 174 Figure 5-9 Example Pipe Section for Computation of Plastic Moment Capacity ..................... 175 Figure 5-10 Moment versus Curvature of Example Pipe Section ............................................. 175 Figure 5-11 Elasto-plastic Stress Distribution Computed by LPile........................................... 177 Figure 5-12 Stress-Strain Curves of Prestressing Strands Recommended by PCI Design Handbook, 5th Edition.............................................................................................. 178 Figure 5-13 Sections for Prestressed Concrete Piles Modeled in LPile .................................... 180 Figure 6-1 Scheme for Installing Pile in a Slope Subject to Sliding.......................................... 185 Figure 6-2 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.................................................................. 186 Figure 6-3 Influence of Stabilizing Pile on Factor of Safety Against Sliding ........................... 187 Figure 6-4 Matching of Computed and Assumed Values of hp ................................................. 189 Figure 6-5 Soil Conditions for Analysis of Slope for Low Water ............................................. 190 Figure 6-6 Preliminary Design of Stabilizing Piles ................................................................... 191 Figure 6-7 Load Distribution from Stabilizing Piles for Slope Stability Analysis .................... 192 xii List of Tables Table 3-1. Stiff Clay (no longer recommended) ......................................................................... 63 Table 3-2. Representative Values of 50 ....................................................................................... 65 Table 3-3. Representative Values of k for Stiff Clays.................................................................. 71 Table 3-4. Representative Values of Table 3-5 50 for Stiff Clays............................................................... 72 ues of k for Laterally Loaded Piles in Sand ........................................................................................................................... 81 Table 3-6. Representative Values of k for Submerged Sand for Static and Cyclic Loading ....... 89 Table 3-7. Representative Values of k for Sand Above Water Table for Static and Cyclic Loading ...................................................................................................................... 89 Table 3-8. Results of Grout Plug Tests by Schmertmann (1977) .............................................. 115 Table 3-9. Values of Compressive Strength at San Francisco ................................................... 117 Table 5-1. LPile Output for Rectangular Concrete Section ....................................................... 166 Table 5-2. Comparison of Results from Hand Computation versus Computer Solution........... 173 xiii 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 beamcolumn 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 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 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 Chapter 1 Introduction 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 upperbound and lower-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. 2 Chapter 1 Introduction 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 3 Chapter 1 Introduction 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; , 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 current with the literature as much as possible. 4 Chapter 1 Introduction 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 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 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 independent models allows the designer to exercise engineering judgment in achieving compatibility and equilibrium for the entire system for a given case of loading. 5 Chapter 1 Introduction a. Elevation View Lumped masses Foundation springs b. Analytical Model K33 K22 M K33 K11 Rotation c. Stiffness Matrix Figure 1-1 Example of Modeling a Bridge Foundation 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. 6 Chapter 1 Introduction 1-3-2 Example Model of Individual Pile Under Three-Dimensional 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 into the cap and that bending moments and a torsion 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 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 two-dimensional plane. q y Axial Py x u Px My Mx Axial Pile Displacement, u z Mz P z 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 Torsional (c) Nonlinear load-transfer curves Figure 1-2 Three-dimensional Soil-Pile Interaction 7 Chapter 1 Introduction 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 at the pile top. P P M M V V 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/ K23 = V/ and K32 = M/ . and K33 = M/ . Figure 1-3 Coefficients of Stiffness Matrix 8 Chapter 1 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 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 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. 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 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 inimportant 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 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 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 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. 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 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, recommendations were 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 conversation with the senior writer, Terzaghi said that he had not been enthusiastic about writing 14 Chapter 2 - Solution for Pile Response to Lateral Loading the paper and only did so in response to numerous requests. The method illustrated by Figure 21(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 Mt Pt Pt (c) (d) Figure 2-1 Models of Pile Under Lateral Loading, (a) 3-Dimensional Finite Element Mesh, and (b) Cross-section of 3-D Finite Element Mesh, 15 Chapter 2 Solution for Pile Response to Lateral Loading 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 21(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 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 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 been found by experiment, and the use of the model in practice has not been extensive. 16 Chapter 2 - Solution for Pile Response to Lateral Loading 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. In addition, it is possible to vary the bending stiffness with respect to the bending moment that is computed during iteration. Q P M y p y p y p y p y p y x Figure 2-2 Model of Pile Under Lateral Loading and p-y Curves 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 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 close together in the analysis. As may be seen, the p-y curves are nonlinear with respect to depth x along the pile and pile 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. 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 developments development of digital computer programs for 17 Chapter 2 Solution for Pile Response to Lateral Loading 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 from proprietary research sponsored by the petroleum industry in th 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 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 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 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 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, lock (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 , 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 2-4. Figure 2-4 (a) shows a pile with a given geometry embedded in a soil with known characteristics. A lateral load Pt, axial load Q, and 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 ssumption 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. 19 Chapter 2 Solution for Pile Response to Lateral Loading Q M Loading Pt Loading at Failure Mult Allowable Loading Maximum Bending Moment (a) (b) Figure 2-4 Illustration of General Procedure for Selecting a Pile to Sustain a Given Set of Loads 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 2-4, 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 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 2-4(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. 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 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. 20 Chapter 2 - Solution for Pile Response to Lateral Loading However, it should be emphasized that the material presented herein is only a valuable tool for the designer and that a complete design involves many other factors that are not addressed here. 2-1-3-1 Study of Pile Buckling A second computational problem is shown in Figure 2-5. A pile that extends above the ground line is subjected to a lateral load Pt and an axial load Q, as shown in Figure 2-5(a). The engineer desires to solve for the axial load that will cause the pile to buckle. The lateral load is held constant 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 2-5(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. Q yt Q Pt Buckling Load yt (b) (a) Figure 2-5 Solution for the Axial Buckling Load 2-1-3-2 Study of Critical Pile Length Another computational technique is illustrated in Figure 2-6. A pile is subjected to a combination of loads, as shown in Figure 2-6(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 2-6(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 2-6 (b). These computations and curve can be automatically performed by LPile in user-selected options. The curve in Figure 2-6 (b) shows that the value of yt is unchanged above a pile length that is termed Lcrit, but that the deflection increases for smaller values of pile length. The 21 Chapter 2 Solution for Pile Response to Lateral Loading designer will normally select a pile for a particular application whose length is somewhat greater than Lcrit. Q M yt Pt Lcrit L Lcrit Pile Length Figure 2-6 Solving for Critical Pile Length 2-1-4 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 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. 2-1-4-1 Offshore Platform A bent from an offshore platform is shown in Figure 2-7(a). A three-dimensional analysis of such a structure is sometimes necessary, but a two-dimensional analysis indicated by the drawing is frequently adequate. The preferred method of analysis of the piles is to consider the full interaction between the superstructure and the supporting piles. The piles are assumed to be removed and 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 sketches. 22 Chapter 2 - Solution for Pile Response to Lateral Loading h = 6.1 m 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 2-7 Simplified Method of Analyzing a Pile for an Offshore Platform The second pile is shown in Figure 2-7(b). The assumption is made that the annular void between the jacket leg and the head of the pile was sealed with a flexible gasket, and that the annular space was filled with grout. Thus, in bending the pile and jacket leg will be continuous and have the same curvature. The sketch in Figure 2-7(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 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 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 Manual 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. 2-1-4-2 Breasting Dolphin An interesting use of a pile under lateral load is the pile uses as a foundation for a breasting dolphin. Figure 2-8(a) depicts a vessel with mass m approaching a freestanding pile. 23 Chapter 2 Solution for Pile Response to Lateral Loading The velocity of the vessel is v and its energy on contact would be ½mv2. The deflection of the pile could be computed by finding the area under the load-deflection curve that would equate to the energy of the vessel. The analyst would be concerned with a number of parameters in the problem. The level of water could vary, 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 the primary fenders, 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. m, v Breasting Dolphin Deflection Figure 2-8 Analysis of a Breasting Dolphin 2-1-4-3 Single-Pile Support for a Bridge A common design used for the support of a bridge is shown in Figure 2-9. 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. 24 Chapter 2 - Solution for Pile Response to Lateral Loading 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. Loads From Traffic Loads From Braking and Wind Forces From Dead Loads From Wind and Other Forces Figure 2-9 Loading On a Single Shaft Supporting a Bridge Deck 2-1-4-4 Pile-Supported Overhead Sign The sketches in Figure 2-10 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 2-10(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 25 Chapter 2 Solution for Pile Response to Lateral Loading 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. Wind Load Wind Load Column Dead Load Pile Cap Column Dead Load Two-Shaft Foundation (a) Single-Shaft Foundation (b) Figure 2-10 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 thi 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 2-10(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 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. 26 Chapter 2 - Solution for Pile Response to Lateral Loading 2-1-4-5 Use of Piles to Stabilize Slopes An application for piles that is continuing interest is the stabilizing of slopes that have moved or are judged to be near failure. The sketch in Figure 2-11 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. Figure 2-11 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. 2-1-4-6 Anchor Pile in a Mooring System The use of a pile as the anchor for a tieback anchor is illustrated in Figure 2-12. 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, the load is applied at some distance from the top of the pile. The analytical method can deal with the anchor pile by appropriate innovation. 2-1-4-7 Other Uses of Laterally Loaded Piles Piles under lateral loading occur in many structures or applications other than the ones that were earlier mentioned. Some of these are high-rise buildings that are subjected to forces from wind or from unbalanced earth pressures; pile-supported retaining walls; locks and dams; 27 Chapter 2 Solution for Pile Response to Lateral Loading waterfront structures such as piers and quay walls; support 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 2-12. The sheet piles (or tangent piles if 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 for a pile. Research on the topic has been undertaken (Wang, 1986) and has already been implemented in the computer program PYWall from Ensoft, Inc. Tie-back Anchor Pile (Dead Man) Sheet Pile Wall Figure 2-12 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) and is shown in the following paragraphs. 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 horizontal compressive forces Q acting in the center of gravity of the end cross-sections of the bar. 28 Chapter 2 - Solution for Pile Response to Lateral Loading If an infinitely small unloaded element, bounded by two verticals a distance dx apart, is cut out of this bar (see Figure 2-13), the equilibrium of moments (ignoring second-order terms) leads to the equation ...........................................(2-1) or y x y Px S M Vn Vv Vv dx Vv+dVv y+dy M+dM Px x Figure 2-13 Element of Beam-Column (after Hetenyi, 1946) . ....................................................(2-2) Differentiating Equation 2-2 with respect to x, the following equation is obtained .................................................(2-3) The following definitions are noted: 29 Chapter 2 Solution for Pile Response to Lateral Loading where Es is equal to the secant modulus of the soil-response curve. And making the indicated substitutions, Equation 2-3 becomes ...............................................(2-4) The direction of the shearing force Vv is shown in Figure 2-13. The shearing force in the plane normal to the deflection line can be obtained as Vn = Vv cos S Q 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. ..........................................................(2-6) 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. ..............................................(2-7) where: Q = 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: .....................................................(2-8) ...........................................................(2-9) and, 30 Chapter 2 - Solution for Pile Response to Lateral Loading .............................................................(2-10) where V = shear in the pile, M = bending moment in the pile, and S = slope of the elastic curve defined by the axis of the pile. Except for the axial load Q, the sign conventions that are used in the differential equation and in subsequent development are the same as those usually 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 Q does not normally appear in the equations for beams. The sign conventions are presented graphically in Figure 2-14. A solution of the differential equation yields a set of curves such as shown in Figure 2-15. 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 Q (+) Axial Force (F) Soil Resistance (F/L) Shear (F) y y y V (+) p (+) x x x Figure 2-14 Sign Conventions The assumptions that are made in deriving the differential equation are: 1. The pile is straight and has a uniform cross section, 2. The pile has a longitudinal plane of symmetry; loads and reactions lie in that plane, 3. The pile material is homogeneous, 4. The proportional limit of the pile material is not exceeded, 31 Chapter 2 Solution for Pile Response to Lateral Loading 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. 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 V M p Figure 2-15 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 is seldom or 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. .....................................................(2-11) 32 Chapter 2 - Solution for Pile Response to Lateral Loading ......................................................(2-12) The solution to Equation 2-12 may be directly written as: ..........................................(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 2-16. 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 Pt Free-head (a) Pt y Fixed-Head Pt y Partially Restrained (b) (c) Figure 2-16 Boundary Conditions at Top of Pile The boundary conditions at the top of the pile selected for the first case are illustrated in Figure 2-16(a) and in equation form are: at x = 0, ..........................................................(2-14) 33 Chapter 2 Solution for Pile Response to Lateral Loading ...........................................................(2-15) The differentiations of Equation 2-13 are made and the substitutions indicated by Equation 2-14 yield the following. .........................................................(2-16) The substitutions indicated by Equation 2-15 yield the following. .....................................................(2-17) 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. ................................(2-18) ............................(2-19) .................................(2-20) .................................(2-21) .............................(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: 34 Chapter 2 - Solution for Pile Response to Lateral Loading ................................................(2-27) ...............................................(2-28) .....................................................(2-29) V = PtB1 p = 2Mt D1 ..................................................(2-30) 2Pt C1 2Mt 2B1 ...............................................(2-31) Values for A1, B1, C1, and D1, are shown in Figure 2-17 as a function of the nondimensional distance x along the pile. A1, B1, C1, D1 -0.4 0.0 -0.2 0 0.2 0.4 0.6 0.8 1 0.5 1.0 1.5 2.0 2.5 A1 x B1 3.0 C1 D1 3.5 4.0 4.5 5.0 5.5 6.0 Figure 2-17 Values of Coefficients A1, B1, C1, and D1 35 Chapter 2 Solution for Pile Response to Lateral Loading For a pile whose head is fixed against rotation, as shown in Figure 2-16(b), the solution may be obtained by employing the boundary conditions as given in Equations 2-32 and 2-33. At x = 0, .............................................................(2-32) .........................................................(2-33) Using the procedures as for the case where the boundary conditions were as shown in Figure 2-4(a), the results are as follows. .....................................................(2-34) The solution for long piles is given in Equations 2-35 through 2-39. ..........................................................(2-35) ......................................................(2-36) .........................................................(2-37) V = Pt C1 ...........................................................(2-38) p = Pt A1 ..........................................................(2-39) 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 2-16(c). For this boundary 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 ........................................................(2-40) and the pile-head shear force is controlled by ...........................................................(2-41) 36 Chapter 2 - Solution for Pile Response to Lateral Loading 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 . ...................................................(2-42) ...................................................(2-43) 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) 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, (M is zero at pile tip)...........................................(2-44) and (shear force, V, is zero at pile tip).................................(2-45) 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. 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. 37 Chapter 2 Solution for Pile Response to Lateral Loading 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. d4y dx 4 ym 2 4 ym 1 6 ym 4 ym h4 1 ym 2 If the pile is subdivided in increments of length h, as shown in Figure 2-18, the governing differential equation, Equation 2-7, in difference form with collected terms for y is as follows: y ym+2 h h ym+1 ym h ym-1 h ym-2 x Figure 2-18 Representation of deflected pile 38 Chapter 2 - Solution for Pile Response to Lateral Loading ..................................(2-46) 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 Q 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 Q is undiminished. This fact plus the fact that Q, 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 Q is generally valid. For the reasons given, it is thought to be unnecessary to vary Q in Equation 2-46; thus, a table of values of Q 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 .....................................................(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 39 Chapter 2 Solution for Pile Response to Lateral Loading 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 ...............................(2-48) Equation 2-48 is expressing the condition that there is some shear at the bottom of the pile or that EI(d3y/dx3) + Q(dy/dx) = V0 at x = L. The assumption is made in these 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 219. (The axial load Q is not shown in the sketches, but Q is assumed to be acting 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. Pt +Mt +Pt yt+2 yt+1 yt yt-1 yt-2 h Figure 2-19 Case 1 of Boundary Conditions .........................(2-49) For the condition where the moment at the top of the pile is equal to Mt, the following difference equation is employed. .............................................(2-50) Case 2 of the boundary conditions at the top of the pile is illustrated graphically in Figure 2-20. The pile is assumed to be embedded in a concrete foundation for which the rotation is known. In many cases, the rotation can be assumed to be zero, at least for the initial solutions. 40 Chapter 2 - Solution for Pile Response to Lateral Loading Equation 2-49 is the first of the two equations that are needed. The second of the two needed equations reflects the condition that the slope St at the top of the pile is known. yt+2 yt+1 yt +Pt St yt-1 yt-2 1 Figure 2-20 Case 2 of Boundary Conditions .......................................................(2-51) Case 3 of the boundary conditions at the top of the pile is illustrated in Figure 2-21. It is assumed that the pile continues into the superstructure and becomes a member in a frame. The solution for the 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 solution). The moment divided by the rotation, Mt/St, is the rotational restraint provided by the superstructure and becomes one of the boundary conditions. The boundary condition has proved to be useful in some cases. Pile extends above ground surface and in effect becomes a column in the superstructure yt+2 yt+1 yt +Pt yt-1 yt-2 Figure 2-21 Case 3 of Boundary Conditions 41 h Chapter 2 Solution for Pile Response to Lateral Loading 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. 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. ..............................................(2-52) Case 4 of the boundary conditions at the top of the pile is illustrated in Figure 2-22. 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 to be 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) Foundation moves laterally yt+2 Mt yt+1 yt yt-1 Pile-head moment is known, may be zero h yt-2 Figure 2-22 Case 4 of Boundary Conditions Case 5 of the boundary conditions at the top of the pile is illustrated in Figure 2-23. 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 42 Chapter 2 - Solution for Pile Response to Lateral Loading 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. St yt yt+2 yt+1 yt 1 yt-1 yt-2 St Figure 2-23 Case 5 of Boundary Conditions The five sets of 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 Pt 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 Pt. A simple plot will yield the required value of Mt that will produce the given boundary condition, Pt. LPile solves the difference equations for the response of a pile to lateral loading. Solution included in which the results from computer solutions are compared with experimental results. Because of the obvious approximations that are inherent in the difference-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, changed to R in the difference equations, is correctly represented as a constant in the second-order differential equation, Equation 2.-9. ...........................................................(2-9) In finite difference form, Equation 2.9 becomes 43 Chapter 2 Solution for Pile Response to Lateral Loading .............................................(2.54) 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. 44 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. 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 45 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 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. 46 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. 47 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" 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) 48 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 3,000 x = 12" x = 24" x = 36" x = 48" 2,500 x = 60" x = 72" x = 84" x = 96" x = 108" 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 3-3 p-y Curves from Cyclic Load Tests 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 + k2 x, .........................................................(3-1) or Es = k1 xn .............................................................(3-2) Measurements of pile-head deflection and rotation at the ground line are necessary. Then, either of the equations is selected and the two parameters are computed for a given applied load and moment. With an expression for soil modulus for a particular load, the soil resistance and deflection along the pile are computed. 49 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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, details 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 one thing, 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 points, probably due to the heterogeneity of the soils at the two sites that were studied. The ratios of Ei/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 very small shearing strains below 0.01%. Johnson (1982) performed some tests with the self-boring pressuremeter and computations with his results gave values of Ei/c that ranged from 1,440 to 2,840, with the average of 1,990. The studies of the initial modulus from compressive-stress-strain curves of clay seem to indicate that such curves are linear only over a very small 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 ( , 1961) can be used to gain some information on the subgrade modulus (initial slope of the p-y curve): E si 0.65 b Ei b 4 Ep I p 50 1 / 12 Ei 1 2 ..........................................(3-3) Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Ei /c 0 100 200 300 0 Manor Road Lake Austin 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 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 the 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) 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. 51 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Initial Soil Modulus, Esi, MPa 0 200 400 600 800 0 Pile 1 Static 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 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. 52 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 Fn Fp 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 + 2 Ft cos + Ff .................................(3-4) where = angle of the inclined plane with the vertical, and W = the weight of the wedge. An expression for W is ........................................................(3-5) 53 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 25 mm 19 mm 3 mm 6 mm 596 kN 13 mm (a) Heave at maximum load 3 mm 6 mm 0 kN 13 mm (b) Residual heave 4 3 2 1 0 Scale, meters Figure 3-7 Measured Profiles of Ground Heave Near Piles Due to Static Loading, (a) Heave at Maximum Load, (b) Residual 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 ........................................................(3-6) where ca = average undrained shear strength of the clay over depth H. The resultant shear force on a side plane is 54 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ........................................................(3-7) The frictional force between the wedge and the pile is ...........................................................(3-8) where = 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. .................(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 ............................................(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 to be 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) is discussed on page 56. 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. 55 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock p u /cb 0 5 10 15 20 25 30 0 Hansen K = 0 Thompson K = 0.5 Thompson K = 1.0 Thompson Eq. 3-10 1 2 3 4 H/b 5 6 7 8 9 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 ............................................(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. 56 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock (a) c 2c (b) cb/2 6b pu 1b 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 ................................................(3-12) Equation 3-12 is also shown plotted in Figure 3-8 olutions is satisfactory near the ground surface, but the difference becomes significant with depth. 57 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, shown later, that are used in the recommendations for two of the sets of p-y curves. However, the emphasis was placed directly 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 310 and 3-11 could be modified. The adjustment factors that were found are shown in Figure 3-10 (see Section 3-3-7 on page 64 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 demonstrated later. Ac and As 0 0.2 0.4 0.6 0.8 1.0 0 2 Ac 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. ll and Dunnavant (1984) and Dunnavant and (1985) reported on tests performed at a site where the clay was overconsolidated and where lateral-loading tests were 58 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 ( 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 59 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 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, and Dunnavant (1984) -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. 60 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 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. ......................................................(3-13) where: q = foundation pressure, I = influence coefficient, E In Equation 3-13, the value of 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 61 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 ....................................................(3-15) Where E is Yo principal stress difference of ( 1 3). For saturated clays with no change in water content, Equation 3-15 may be rewritten as .................................................... (3-16) Where 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. Which can be rearranged as .......................................................... (3-17) full-scale foundations, led to the following conclusion: assumptions, it may be taken that Equation 3-17 applies to a circular or any rectangular footi 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. 62 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. were based on a concept that the defor 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 changed to reflect current practice. These values of consistent with theory for small deflections. T are independent of pile diameter, which is Table 3-1. 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 3.2-6.4 6.4-12.8 > 12.8 460-925 925-1,850 > 1,850 Soil Modulus, Soil Modulus, T, MPa T, psi 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 ........................................................ (3-18) 63 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock and the values of pile deflection y are scaled using .......................................................... (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 Description of Load Test Program 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 clays near Lake Austin, Texas that had an average shear strength of about 38 kPa (800 psf). The test pile was recovered after the first test and taken to Sabine Pass, Texas, and driven into 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 were less than one microstrain, but some time was required to obtain readings manually 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 interpreted to find the assumed reading at a particular 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 various numbers of cycles of 64 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 Consistency of Clay 2. 50 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. .............................................. (3-20) .......................................................... (3-21) where = average effective unit weight from ground surface to p-y curve,1 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. 65 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 1 0.5 0 0 1 8.0 (a) 1 0.72 0.5 0 0 3 15 1 (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 66 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock .................................................. (3-22) 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 66. 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 1,000 800 600 400 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 4. 50b ....................................................... (3-23) Compute points describing the p-y curve from the origin up to 8 y50 using ...................................................... (3-24) 67 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. ..................................................... (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 68 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 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 250 200 150 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 69 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 3-3-8 Response of Stiff Clay in the Presence of Free Water Reese, Cox, and Koop (1975) performed lateral-load tests with 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 carried out 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. In addition, as in the case of the Matlock recommendations for cyclic loading, the lowerbound case is presented. Cycling was continued until the deflection and bending moments appeared to stabilize. The number of cycles of loading was in the order of 100; and 500 cycles were applied in a reloading test. and Dunnavant (1984) report that an equilibrium condition could not be reached during cyclic loading of piles at the Houston site. It is likely that the same result would have been found at the Manor site; however, the l00 cycles or more that were applied at Manor, at a load at which the pile was near its ultimate bending moment, were 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, there is reason to believe that a nonlinear relationship for diameter is required. During the experiment with repeated loading, a 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 began at that time. 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, the data that are available at present do not allow such a change in the recommended procedures. However, analysts could well 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. 3-3-8-1 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 3-16. 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 of the pct or pcd from Equations 3-26 and 3-27. ............................................. (3-26) 70 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock p 0.5pc 0 y50 6y50 As y50 y 18y50 Figure 3-16 Characteristic Shape of p-y Curves for Static Loading in Stiff Clay with Free Water ........................................................ (3-27) 4. Choose the appropriate value of As from Figure 3-10 on page 58 for modifying pct and pcd and for shaping the p-y curves or compute As using ............................................ (3-28) 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* 50-100 kPa 1,000-2,000 psf 100-200 kPa 2,000-4,000 psf 200-400 kPa 4,000-6,000 psf ks (static) 135 MN/m3 (500 pci) 270 MN/m3 (1,000 pci) 540 MN/m3 (2,000 pci) kc (cyclic) 55 MN/m3 (200 pci) 110 MN/m3 (400 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 71 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock .......................................................... (3-30) Using an appropriate value of laboratory tests, from Table 3-4. 50 from results of laboratory tests or, in the absence of Table 3-4. Representative Values of 50 for Stiff Clays Average Undrained Shear Strength 7. 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 Establish the first parabolic portion of the p-y curve, using the following equation and obtaining pc from Equations 3-26 or 3-27. .................................................... (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, ............................... (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 p c 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, ................................... (3-34) or ...................................... (3-35) 72 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 3-16 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. 3-3-8-2 Procedure for Computing p-y Curves for Cyclic Loading A second pile, identical to the pile used for the static loading, was tested under cyclic loading. The following procedure is for cyclic loading and is illustrated in Figure 3-17. As may be seen from a study 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 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 clue was that the clay from Manor was found to lose volume by slaking when a specimen was placed in fresh water; thus, the clay 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 Ac pc 0 0.45yp 0.6yp 1.8yp y Figure 3-17 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. 73 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 58 or compute Ac using ............................................. (3-36) 5. Compute yp using ....................................................... (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. 7. Compute y50 using Equation 3-30. 8. Establish the parabolic portion of the p-y curve, ........................................... (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, ................................... (3-39) 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, ........................................... (3-40) 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 3-17 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. 3-3-8-3 Recommended Soil Tests Triaxial compression tests of the unconsolidated-undrained type with confining pressures conforming to in situ pressures are recommended for determining the shear strength of the soil. 74 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 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. 3-3-8-4 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 3-18. 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 to be cyclic. The p-y curves were computed for 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 3-19. 3-3-9 Response of Stiff Clay with No Free Water 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 Reese and Welch (1975). 0 2 4 6 8 10 12 14 16 0 50 100 150 200 Shear Strength, kPa Figure 3-18 Example Shear Strength Profile for p-y Curves for Stiff Clay with No Free Water 75 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 250 Depth = 1.00 m Depth = 2.00 m Depth = 3.00 m Depth = 12.00 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 3-19 Example p-y Curves for Stiff Clay in Presence of Free Water for Cyclic Loading 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. 3-3-9-1 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 3-20. 76 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock p y 16y50 Figure 3-20 Characteristic Shape of p-y Curve for Static Loading in Stiff Clay without Free Water 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 Equations 3-20 and 3-21. (In the use of Equation 3-20, the shear strength is taken as the average from the ground surface to the depth being considered and J is taken as 0.5. The unit weight of the soil should reflect the position of the water table.) ...............................................(3-20) ...........................................................(3-21) 3. Compute the deflection, y50, at one-half the ultimate soil resistance from Equation 3-23. y50 = 2.5 4. (3-23) Compute points describing the p-y curve from the relationship below. p 5. 50b ........................................................ pu y 2 y50 0.25 ..................................................... (3-41) Beyond y = 16y50, p is equal to pu for all values of y. 77 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 3-3-9-2 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 3-21. 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 3-21 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 ....................................................... (3-42) 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 ................................................. (3-43) where yc = deflection under N-cycles of load, ys = deflection under short-term static load, y50 = deflection under short-term static load at one-half the ultimate resistance, and 78 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock N = number of cycles of load application. 5. The p-y curve defines the soil response after N-cycles of loading. 3-3-9-3 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. 3-3-9-4 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 318. The unit weight of the soil was assumed to be 19.0 kN/m3 (125 pcf) for the entire depth. In the absence of a stress-strain curve, 50 was taken as 0.005. Equation 3-42 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 322. 400 300 200 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 3-22 Example p-y Curves for Stiff Clay with No Free Water, Cyclic Loading 79 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 3-3-10 Modified p-y Criteria for Stiff Clay with No Free Water The p-y criteria for stiff clay with no free water were 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 criteria for stiff clay without free water has been used successfully for many year, there have been some reported cases from the Southeastern United States where load tests have found that the initial load-deformation response is modeled too stiffly. The ultimate load-transfer resistance pu used in the p-y criteria 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 the use of a k value to modify the initial portion of the p-y curves if one has the results of lateral load test for local calibration of the initial stiffness k. Judicious use of this modified p-y criteria enables one to obtain improved predictions with experimental readings that may be used later for design computations. The user may select an initial stiffness k based on Table 3-3 or from a site-specific lateral load test. LPile will use the lower of the values computed using Equation 3-29 or Equation 3-41 for pile response as a function of lateral pile displacement. 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. 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 and his co-workers ( and Dunnavant, 1984; Dunnavant and , 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. 80 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 3-4 p-y Curves for Sands 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 Esi for sands is complicated. The p-y curve at the ground surface will be characterized by zero values of p for all values of y, and the initial slope of the curves and the ultimate resistance will increase approximately linearly with depth. The presentation of the recommendations of Terzaghi (1955) is of interest here, but it is recognized that his coefficients probably are meant to reflect the slope of secants to p-y 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 recommends a series of straight lines with slopes that increase linearly with depth, as indicated in Equation 3-44. Es = kx............................................................ (3-44) where k = constant giving variation of soil modulus with depth, and x = depth below ground surface. both US customary units and SI units are given in Table 3-5. k values 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 soil investigations of good stopped recommending use of the values shown in Table 3-5. Table 3-5 Type of Sand Dry or moist, k, MN/m3 (pci) Submerged, k, MN/m3 (pci) ations for Values of k for Laterally Loaded Piles in Sand Relative Density Loose Medium Dense 0.95 - 2.8 (3.5 - 10.4) 3.5 - 10.9 (13.0 - 40.0) 13.8 - 27.7 (51.0 - 102.0) 0.53 - 1.7 (2.1 - 6.4) 2.2 - 7.3 (8.0 - 27.0) 8.3 - 17.9 (32.0 - 64.0) 81 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 3-4-1-2 Analytical Solutions for Ultimate Resistance Two models are used for computing the ultimate resistance for piles in sand, following 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 3-23. The total lateral force Fpt (Figure 3-23(c)) may be computed by subtracting the active force Fa, computed by use of Rankine theory, from the passive force Fp, computed from the model by assuming that the Mohr-Coulomb failure condition is satisfied on planes, ADE, BCF, and AEFB (Figure 3-23(a)). The directions of the forces are shown in Figure 3-23(b). Solutions other than the ones shown here have been developed by assuming a friction force on the surface 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). B Fs A y Ff F Fs C Fn x D W H Fp Fn Ft W F Ff Fp E (b) b Pile of Diameter b Fs Fn Fp (a) Fpt Fa (c) Figure 3-23 Geometry Assumed for Passive Wedge Failure for Pile in Sand The 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 ............... (3-45) where: = the angle of the wedge in the horizontal direction 82 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock = 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 3-45 with respect to depth. ( pu ) sa H K 0 tan sin tan( ) cos H K 0 H tan s tan sin tan tan( tan ) b H tan tan ................ (3-46) 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. ........................................................ (3-47) The model for computing the ultimate soil resistance at some distance below the ground surface is shown in Figure 3-24(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 3-24(b) are assumed, the ultimate soil resistance for horizontal movement of the soil is ............................ (3-48) 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 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 herein, have used only static loading. 83 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock (a) (b) Figure 3-24 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, very few reports of tests of piles subjected to cyclic lateral loading have been reported. There is evidence that the repeated loading on a pile in predominantly one direction will result in a permanent deflection in the direction of loading. It has been observed that when a relatively large cyclic load is applied in one direction, the top of the pile will deflect a significant amount, allowing grains of cohesionless soil to fall into the open gap at the back of the pile. Thus in such a case, the pile cannot return to its initial position after cyclic loading ceases. 84 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Observations of the behavior of sand near the ground surface during cyclic loading support the idea that the void ratio of sand is approaching a critical value. That is, dense sand will loosens 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 o to practice until the digital computer and the required programs became widely available. There was a period of a few years when engineers were solving the difference equations using mechanical calculators. The piles for some early offshore platforms 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 received little use 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 Response of Sand The following procedure is for both short-term static loading and for cyclic loading for a flat ground surface and a vertical pile. The procedure is illustrated in Figure 3-25 (Reese, et al., 1974). 85 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock p x = x4 x = x3 x = x2 pu m k pk u x = x1 m pm ym yu b/60 3b/80 yk ksx y Figure 3-25 Characteristic Shape of a Set of p-y Curves for Static and Cyclic Loading in Sand 3-4-2-1 Procedure for Computing p-y Curves in Sand 1. Obtain values for the depth of the p-y curve x, the 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. Make the following preliminary computations. , 3. , , and ..................... (3-49) Compute the ultimate soil resistance per unit length of pile using the smaller of the values given by , where .................... (3-50) ............................. (3-51) 86 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 4. Establish Compute pu using: or ............................................... (3-52) Use the appropriate value of or from Figure 3-26 for the particular nondimensional depth, and for either the static or cyclic case. Use the appropriate equation for ps, Equation 3-50 or Equation 3-51 by referring to the computation in Step 4. 5. Compute ym using .......................................................... (3-53) Compute pm by the following equation: ............................................... (3-54) Use the appropriate value of Bs or Bc from Figure 3-27 as a function of the nondimensional depth, and for either the static or cyclic case. 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 established. 6. Establish the initial straight-line portion of the p-y curve, p = (k x) y ......................................................... (3-55) Use the appropriate value of k from Table 3-6 or 3-7. 87 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 2 1 0 3 0 1 2 3 4 5 6 Figure 3-26 Values of Coefficients and B 2 1 0 3 0 1 Bs (static) Bc (cyclic) 2 3 4 5 6 Figure 3-27 Values of Coefficients Bc and Bs 88 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Table 3-6. Representative Values of k for Submerged Sand for Static and Cyclic Loading Recommended k MN/m3 (pci) Relative Density Loose Medium Dense 5.4 16.3 34 (20.0) (60.0) (125.0) Table 3-7. Representative Values of k for Sand Above Water Table for Static and Cyclic Loading Recommended k MN/m3 (pci) Loose 6.8 (25.0) Relative Density Medium Dense 24.4 61.0 (90.0) (225.0) If the input value of k is left equal to zero, a default value will be computed by LPile using the curves shown in Figure 3-31 on page 94. 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 below the water table. If the input value of is greater than 40 degrees, a k value corresponding to 40 degrees is used by LPile. 7. Establish the parabolic section of the p-y curve, .......................................................... (3-56) Fit the parabola between point k and point m as follows: a. Compute the slope of the curve between point m and point u by, ........................................................ (3-57) b. Obtain the power of the parabolic section by, ........................................................... (3-58) c. Obtain the coefficient as follows: ........................................................... (3-59) d. Determine point the pile deflection at point k as 89 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ........................................................ (3-60) e. Compute appropriate number of points on the parabola by using Equation 3-56. Note: The curve in Figure 3-25 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 3-55 defines the p-y curve until there is an intersection with another portion of the p-y curve or if no intersection occurs, Equation 3-55 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 bi-linear as shown in Figure 3-28. 3-4-2-2 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 or equal to those 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 py curves for dry sand were developed from those experiments. The results from the Parker and Reese experiments should be useful in checking solutions from results of experiments with fullscale piles. p Lower k x kx Higher k x kx y Figure 3-28 Illustration of Effect of k on p-y Curve in Sand 90 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 3-4-2-3 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 is assumed to have 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 3-29. 5,000 4,000 3,000 2,000 1,000 0 0.0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Lateral Deflection y, m Depth = 1.50 m Depth = 3.00 m Depth = 6.00 m Depth = 12.00 m Figure 3-29 Example p-y Curves for Sand Below the Water Table, Static Loading 3-4-3 API RP 2A Recommendation for Response of Sand Above and Below the Water Table 3-4-3-1 Background of API Method for Sand This method 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 criteria. The API method uses a hyperbolic tangent function for computation. The main difference between those two criteria will be the initial modulus of subgrade reaction and the shape of the curves. 91 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 3-61 to a value at deep depths determined by Equation 3-62. 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: ................................................... (3-61) ........................................................ (3-62) 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 3-30 as a function of , or where and 92 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 100 100 5.0 5 90 80 80 4 4.0 70 3 3.0 60 60 C2 50 40 40 2 2.0 C1 30 C3 1 1.0 20 20 10 0 0.0 15 15 00 20 20 25 25 30 30 35 35 40 40 Angle of Internal Friction, , degrees Figure 3-30 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: ................................................ (3-63) where A = factor to account for cyclic or static loading. Evaluated by: A = 0.9 for cyclic loading. for static loading, pu = smaller of values computed from Equation 3-61 or 3-62, lb./in. (kN/m), 93 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock k = initial modulus of subgrade reaction, pci (kN/m3). Determine k from Figure 3-31 as function of angle of internal friction, , y = lateral deflection, in. (m), and x = depth, inches (m). , Friction Angle, degrees 28 29 Very Loose 300 36 30 Loose Medium Dense 40 Dense 45 Very Dense Sand above the water table 250 200 150 Sand below the water table 100 50 0 0 20 40 60 80 100 Relative Density, % Figure 3-31 Value of k, Used for API Sand Criteria 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 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. 94 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 3-30. C1 = 2.97 C2 = 3.42 C3 = 53.8 3. Compute the ultimate soil resistance pu. pus = (C1 x + C2 b) pud = C3 b x = [(2.97)(20 in.) + (3.42)(36 in.)](0.07 pci)(20 in) = 255 lb./in. 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 5. (0.8) (x)/(b) = 3.0 (0.8)(20 in.)/(36 in.) = 2.56 Compute p for different y values. If y = 0.1 inch, k (above water table) = 140 pci (from Figure 3-31) kx y A pu p A pu tanh p (2.55)(255 lb./in. ) tanh p 264 lb./in. (computer output = 264.012 lb./in.) (140 lb./in. 3 )(20 in.) (0.1 in.) (2.55)(255 lb./in. ) If y = 1.35 in. kx y A pu p A pu tanh p (2.55)(255 lb./in.) tanh p 653 lb./in. (computer output = 652.93 lb./in.) (140)(20 in.) (1.35 in.) (2.55)(255 lb/in. 3 ) The check by hand computations yielded exact values for the two values of deflection that were considered. The computed curves are presented in Figure 3-32. 95 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 3-32 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 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 in Liquefied Sands 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, 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 by either using appropriate p-multiplier values or by entering a very low friction angle for sand. 96 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 developed on the basis of these studies have a concave upward shape, as shown in Figure 3-33. 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. p y 150 mm Figure 3-33 Example p-y Curve in Liquefied Sand 97 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Following liquefaction, p-y curves in sand become progressively stiffer with the passage of time as excess pore water pressures dissipate. The shape of a p-y curve appears to transition from concave up to concave down as pore water pressure decreases. An equation based on the results of the load tests has been developed by Rollins et al. (2003) to describe the observed loaddisplacement response of liquefied sand as a function depth. 3-5-2 Procedure for Computing p-y Curves in Liquefied Sand 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. ........................................................(3-64) ...................................................(3-65) .....................................................(3-66) .....................................................(3-67) where p is the soil resistance in kN/meter, y is the lateral deflection of the pile in millimeters, z is the depth in meters (see note in last paragraph of this section), and Pd is the diameter correction discussed below. Rollins et al. (2005a) studied the diameter effects for different sizes of piles and recommended a modification factor for correcting Equation 3-64, as shown below. ...................................................(3-68) where b is the diameter or width of the pile or drilled shaft in meters. The p-y curves for liquefied sand can be multiplied by Pd to obtain values for p-y curves for deep foundations of varying diameters. Note that use of the diameter correction is limited to foundations between 0.3 and 2.6 meters in diameter. This limitation on diameter prevents implementation of the above relations to micropiles because their diameters are generally less than 0.3 meters. Application of Equation 3-64 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 Lateral pile deflection less than 150 mm (0.15 m), Depths of 6 meters or less, and 98 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Position of the water table near to or at the ground surface. In some cases, the liquefying layer may not be at the surface. In such cases, the depth variable (z) may be modified to equal the initial vertical effective stress divided by 10 kN/m3, which is generally representative of the unit weight of the sand at the site. 3-5-3 Modeling of Lateral Spreading 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 on the piles. The magnitude 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 moves 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. 3-6 p-y Curves in Loess 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 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. 99 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock There were -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. 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 3-34. 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 3-34. 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. 100 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 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 Figure 3-34 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 335. 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 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 ........................................................(3-69) 101 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock p pu Ei Es y yref Figure 3-35. 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 ........................................................(3-70) 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. ....................................................(3-71) 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). 102 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock CN was determined from a best fit of cyclic degradation for two 30-inch diameter test shafts subjected to cyclic loading. CN is ...........................................................(3-72) The cyclic degradation term (the denominator of Equation 3-71) 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 3-35. The best fit to the load test data was obtained with the following value for reference displacement. yref = 0.117 inches = 0.0029718 meters .................................. (3-73) 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. ........................................................... (3-74) 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 ( ) which is in turn a function of the given displacement (y), the reference displacement (yref), and a dimensionless correlation constant (a). ......................................................... (3-75) .............................................. (3-76) 103 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ............................................................(3-77) where Es and Ei are in units of force/length2, and a and 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 .............................................(3-78) The modulus ratio (secant modulus over initial modulus, Es/Ei) versus displacement used for p-y curves in loess is shown in Figure 3-36. 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. 1.0 0.9 a = 0.1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.001 0.01 0.1 1.0 10 100 Figure 3-36 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 3-71. 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. 104 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ........................................................... (3-79) 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 3-37 for the 30-inch diameter shafts, and Figure 3-38 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 3-34). These p-y curves were used in the LPile analyses presented later. 9,000 8,000 7,000 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 3-37 p-y Curves for the 30-inch Diameter Shafts 14,000 12,000 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 3-38 p-y Curves and Secant Modulus for the 42-inch Diameter Shafts. 105 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock The static p-y curves shown in Figure 3-37 and 3-38 were degraded with load cycle number (N) for use in the cyclic load analyses. Figure 3-39 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. 2,000 1,800 1,600 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 3-39 Cyclic Degradation of p-y Curves for 30-inch Shafts 3-6-2-3 Step-by-Step Procedure for Generating 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 3-69 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 3-71. 5. Select a reference displacement (yref) that will be representative of the rate at which the resistance will develop. 6. Determine the initial modulus (Ei) in accordance with Equation 3-74. 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 3-75 and 3-76. 106 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 9. Determine the soil resistance per unit length of pile (p) for each of the displacements selected in Step 7 in accordance with Equation 3-79. 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 soil formation 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 in Soils with Both Cohesion and Internal Friction 3-7-1 Background The previous methods that were presented were for soils that can be characterized as either cohesive or cohesionless (clay or sand, for example). There are currently no generally accepted recommendations for developing p-y curves for c- soils. Among the reasons for the limitation on soil characteristics are the following. Firstly, in foundation design, where the p-y analysis has been used mostly, the characterization of the soil by either a value of c or , but not both, has been used. Secondly, the major experiments on which the p-y predictions have been based have been performed in soils that can be described by either c or . However, there are now numerous occasions when it is desirable, and perhaps necessary, to describe the characteristics of the soil more carefully. An example of the need to have predictions for p-y curves for 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 or drained analysis. Therefore, it is inconsistent, and either unsafe or unconservative, to assume the pile to be in soil that is characterized either by 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 rainfall and rise of the ground water table. However, there could well be times when the ability to design for dry seasons is critical. 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 soil resistance significantly, especially for soils near the ground surface. 107 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock The strength envelope for consolidated-drained clay is represented by components of both c and . Therefore, a p-y method for c- soils is needed for drained analysis. A complication for such an analysis is that there will be some time-dependent lateral deflection of the pile as drainage occurs. 3-7-2 Recommendations for Computing p-y Curves The following procedure is for short-term static loading and for cyclic loading and is illustrated in Figure 3-40. As will be noted, the suggested procedure follows closely that which was recommended earlier for sand. p m pm k pk yk u ym pu yu ks y b/60 3b/80 Figure 3-40 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 ultimate resistance of c- soils as: p= p b = Cp h b.................................................... (3-80) 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), ................................ (3-81) = unit weight of soil (F/L3), 108 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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. The modifying factor Cp can be divided into two terms: Cp to modify the frictional term of Equation 3-80 and Cpc to modify the cohesion term of Equation 3-80. Equation 3-82 can then be written as: ........................... (3-82) The derivation of equations for developing p-y curves for c- soil is based on a concept proposed by Evans and Duncan (1982). Equation 3-82 will be rewritten as ..................................................... (3-83) where can be found from Figure 3-26. The friction component (pu ) will be the smaller of the values given by the two equations below. .................... (3-84) .............................. (3-85) The cohesion component (puc) will be the smaller of the two equations below. .............................................. (3-86) .......................................................... (3-87) 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. 109 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 1. Compute yu by the following equation: ............................................................(3-88) 2. Compute pu for static or cyclic loading by the following equation: or .................................... (3-89) Use the appropriate value of or from Figure 3-26 on page 88 for the particular nondimensional depth and for static or cyclic loading. 2. Compute ym as ........................................................... (3-90) Compute pm by the following equation: or .............................................. (3-91) Use the appropriate value of Bs or Bc from Figure 3-27 on page 88 for the particular nondimensional depth, and for either the static or cyclic case. 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 established. 3. Establish the initial straight-line portion of the p-y curve, p = (k x) y .......................................................... (3-92) The value of k for Equation 3-92 may be found from the following equation and by reference to Figure 3-41. k = (kc + k ) ......................................................... (3-93) 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. 4. Establish the parabolic section of the p-y curve, .......................................................... (3-94) Fit the parabola between point k and point m as follows: a. Get the slope of the line between point m and point u by, 110 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ........................................................ (3-95) 2,000 500,000 1,500 kc (static) 400,000 kc (cyclic) 1,000 300,000 200,000 k (submerged) 500 100,000 k (above water table) 0 0 0 1 2 3 4 deg. 0 28 32 36 40 c kPa 0 96 192 287 383 c tsf Figure 3-41 Representative Values of k for c- Soil b. Obtain the power of the parabolic section by, ........................................................... (3-96) c. Obtain the coefficient as follows: ........................................................ (3-97) d. Determine point k as, ....................................................... (3-98) e. Compute appropriate number of points on the parabola by using Equation 3-94. 111 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Note: The step-by-step procedure is outlined 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 391 defines the p-y curve until there is an intersection with another branch of the p-y curve or if no intersection occurs, Equation 3-91 defines the complete p-y curve. This completes the development of the p-y curve for the desired depth. Any number of curves can be developed by repeating the above steps for each desired depth. 3-7-4 Discussion An example of p-y curves was computed for 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 in. (1 m), 79 in. (2 m), and 118 inches (3 meters). The p-y curves computed by using the simplified procedure are shown in Figure 3-42. 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 an initial peak strength, then drop to a residual strength at a large deflection, 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. 1,400 Depth = 1.00 m Depth = 2.00 m Depth = 3.00 m 1,200 1,000 800 600 400 200 0 0.0 0.005 0.01 0.015 0.02 0.025 Lateral Deflection y, m Figure 3-42 p-y Curves for c- Soils. 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 112 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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. 3-8 Response of Vuggy Limestone 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, one for strong rock and the other for weak 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. 113 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 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 will 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. 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. 114 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 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. 115 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Pressuremeter tests were performed and the results were scattered. The results for moduli values of the rock are plotted in Figure 3-43. 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 3-44 and the correlation of rock strength and modulus shown in Figure 3-45 were employed in developing the correlation between the initial stiffness from Figure 3-43 and the compressive strength, and the values were obtained as shown in Table 3-9. 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. Initial Modulus, Eir, MPa 0 800 400 1,200 1,600 2,000 0 2 186 MPa 4 3.9 m 645 MPa 6 8 8.8 m 10 1,600 MPa 12 Figure 3-43 Initial Moduli of Rock Measured by Pressuremeter for San Francisco Load Test 116 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 1.2 1.0 0.8 0.6 0.4 0.2 ? ? ? 0.0 0% 25% 50% 75% 100% Rock Quality Designation (RQD), % Figure 3-44 Modulus Reduction Ratio versus RQD (Bienawski, 1984) 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. 117 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock (MPa) 1 10 Rock Strength Classification (Deere) 100 1,000 Very Low Low Medium High Very High 100 Upper and Middle Chalk (Hobbs) Concrete 10 (MPa) Steel 100,000 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 1.0 100 10 Uniaxial Compressive Strength psi 103 Figure 3-45 Engineering Properties for Intact Rocks (after Deere, 1968; Peck, 1976; and Horvath and Kenney, 1979) 118 1,000 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 3-8-3 Procedure for Computing p-y Curves for Strong Rock (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 3-46. 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 3-46, 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 3-46 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 3-46 Characteristic Shape of p-y Curve in Strong Rock 3-8-4 Procedure for Computing p-y Curves for Weak Rock The p-y curve that is recommended for weak rock is shown in Figure 3-47. 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. 119 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock p Mir pur y yA Figure 3-47 Sketch of p-y Curve for Weak Rock (after Reese, 1997) for xr 3b ...................................... (3-99) ........................................... (3-100) 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 3-45, by entering with the value of the pressuremeter modulus. ................................................ (3-101) 120 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 3-47 may be shown to have the following value (using the symbols for rock). 2 Mir kir Eir ..................................................... (3-102) where Eir = the initial modulus of the rock, and kir = dimensionless constant defined by Equation 3-103. Equations 3-102 and 3-103 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. .................................... (3-103) kir = 500 for xr > 3b................................................ (3-104) With guidelines for computing pur and Mir, the equations for the three branches of the family of p-y curves for rock in Figure 3-46 can be presented. The equation for the straight-line, initial portion of the curves is given by Equation 3-105 and for the other branches by Equations 3-106 through 3-108. for for ...............................................(3-105) ...............................(3-106) for y > 16yrm ................................................(3-107) yrm = rm b.........................................................(3-108) 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 3-108. 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 value of yA is found by solving for the intersection of Equations 3-105 and 3-106, and the solution is presented in Equation 3-109. 2 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. 121 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock .............................................(3-109) 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, (1996) noted -pull shear produced by the axial shears 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 rm 106 psi), = 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 3-48. 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 349. The plotting is shown for limited depths because the values to the full length are too small to 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. 122 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Figure 3-48 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 3-49. 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 (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.). 123 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Bending Moment, M, kN-m 400 0 400 800 1,200 0 M 2 y Rock Surface 4 6 8 1 0 1 2 3 Lateral Deflection, y, mm Figure 3-49 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 3-50 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 3-51 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. The computed and measured lateral load versus pile-head deflection curves are shown in Figure 3-50. 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 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. 124 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 10,000 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 3-50 Comparison of Experimental and Computed Values of Pile-Head Deflection for Different Values of EI, San Francisco Test Also shown in Figure 3-50 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 3-51 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 3-52. 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. 3-9 p-y Curves in Massive Rock Liang, Yang, and Nasairat (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. 125 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 40 Analytical Experimental ACI 30 20 10 0 5,000 0 10,000 15,000 20,000 Bending Moment, kN-m Figure 3-51 Values of EI for three methods, San Francisco test 10,000 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 3-52 Comparison of Experimental and Computed Values of Maximum Bending Moments for Different Values of EI, San Francisco Test 126 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock A hyperbolic equation is used as the basis for the p-y relationship. .......................................................(3-110) 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 3-53. p pu Ki y Figure 3-53 p-y Curve in Massive Rock 3-9-1 Determination of pu Near Ground Surface For a passive wedge type failure near the ground surface, as shown in Figure 3-54, the ultimate lateral resistance per unit length, pu of the drilled shaft at depth H is .............................(3-111) where , , c = effective cohesion, = effective friction angle, and, = effective unit weight respectively of the rock mass and the following equations are used to compute parameters C1 through C5: 127 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock y F H Fs Fp Fn W D Figure 3-54 Model of Passive Wedge for Drilled Shafts in Rock , and , with the condition that 128 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Equation 3-111 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 on page 129. 3-9-2 Rock Mass Failure 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 ...........................................(3-112) where pa is the active horizontal active earth pressure given by with the condition that ........................(3-113) = effective overburden pressure at the depth under consideration including the pressure from overburden soils, pL is the limiting normal pressure of the rock mass (discussed later), and max is the axial side resistance of the rock-shaft interface, proposed by Kulhawy and Phoon (1993) V .....................................................(3-114) where both max and ci are in units of megapascals. 3-9-3 Initial Tangent to p-y Curve Ki ........................................(3-115) where Em is the rock mass modulus, D is the diameter of the drilled shaft, Ep Ip is the bending stiffness of the drilled shaft, Dref is the reference shaft diameter equal to 0.305 m, and is 3-9-4 Rock Mass Properties The shearing properties of the rock mass, c and strength criterion for rock mass. , are defined using the Hoek-Brown ............................................(3-116) 129 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock where 1 and 3 are the major and minor principal stresses at failure, ci is the uniaxial compressive strength of intact rock, and 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. and Hoek (1990) provided a method for estimating the Mohr-Coulomb failure parameters c of the rock mass from the principal stresses at failure. These parameters are: .............................................(3-117) ....................................................(3-118) 1 can be found from Equation 3-116, and n and are found from ........................................(3-119) ...........................................(3-120) The parameters mb and s can be determined for many types of rock using the recommendations of Marinos and Hoek (2000).3 Two methods for evaluating rock mass modulus 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 using the geological strength index, GSI., using Equation 3-121 ..................................................(3-121) The modulus reduction ratio and is shown as a function of GSI in Figure 3-55. The second method recommended for determining rock mass modulus 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 large pressures, so difficulties might arise during their use. In addition, interpretation of test results may be difficult because of the limited range of expansion pressures possible. 3 This reference may be obtained from the Internet at http://www.rocscience.com/hoek/references/Published-Papers.htm. 130 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 100 Bieniawski (1978) Serafin and Pereira (1983) Ironton-Russell Regression Line 80 60 40 20 0 0 20 40 60 80 100 Geologic Strength Index Figure 3-55 Equation for Estimating Modulus Reduction Ratio from Geological Strength Index 3-9-5 Procedure for Computing p-y Curves in Massive Rock 1. Obtain the value of ci and the intact rock modulus, Ei. 2. Obtain values for the rock mass modulus, Em, by use of Equation 3-121 if pressuremeter data are unavailable. If Equation 3-121 is used, obtain values of GSI and mi according to the recommendations of Marinos and Hoek (2000) . 3. Select a shaft diameter and reinforcing detail. 4. 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. 5. Compute Ki using Equation 3-115. 6. Compute pu at shallow depth using Equation 3-111 with 3 equal to the vertical effective stress at H/3 when computing the values of and c using Equations 3-117 and 3-118. 7. Compute pu at great depth using Equation 3-112 with pL taken as Equation 3-116 and equating 3 equal to v. 1 computed using 8. Compute pu as the smaller of the values computed by Equations 3-111 and 3-112. 9. The values of the p-y curve can then be computed using 3-110 for selected values of pile movement y. 131 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 ......................................................(3-122) This relationship is considered to be linear up to y/b = 0.001 (0.1 percent). For y/b values greater than 0.001, for 0.001 y/b 0.0375..........................(3-123) .................................................(3-124) 132 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Figure 3-56 Degradation Plot for Es pu y 0.001b 0.0375b Figure 3-57 p-y Curve for Piedmont Residual Soil where = 0.23, which gives 3-11 Response of Layered Soils There are many cases where the soil near the ground profile is not homogeneous, but is layered. If the layers are in the zone where the soil would move up and out as a wedge, some 133 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock modifications would be needed in the method to compute the ultimate soil resistance pu, and consequently modifications would be needed in the p-y curves. The problem of the layered soil has been given intensive study by Allen (1985); however, the methods developed by Allen with the methods shown herein must be delayed until a later date when this research can be put in a readily usable form. 3-11-1 Layering Correction Method of Georgiadis The method of Georgiadis (1983) is ba of all the layers existing below the upper layer. The p-y curves of the upper layer are determined according to the methods for homogeneous soils. To compute the p-y curves of the second layer, the equivalent depth H2 to the top of the second layer has to be determined by summing the ultimate resistances of the upper layer and equating that value to the summation as if the upper layer had been composed of the same material as in the second layer. The values of pu are computed using the equations for homogeneous soils. Thus, the following two equations are solved simultaneously for H2. .................................................... (3-125) and ......................................................(3-126) The equivalent thickness H2 of the upper layer along with the soil properties of the second layer, are used to compute the p-y curves for the second layer. The concepts presented above can be used to get the equivalent thickness of two or more dissimilar layers of soil overlying the layer for whom the p-y curves are desired. One possible consequence is that the equivalent depths may be either smaller or greater than the actual depths of the soil layers, depending on the relative strengths of the layers of the soil profiles. This is illustrated in Figure 3-58. 3-11-2 Example p-y Curves in Layered Soils The example problem to demonstrate the manner in which layered soils are modeled is shown in Figure 3-59. 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. 134 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock F1 = Total force acting on pile above point i at the time of soil failure hi = Equivalent depth of top of layer i Groundline h3 h1 Soft Soil (Layer 1) h2 1 F1 Stiffer Soil Below Softer Soil (behaves as if shallower) 2 F2 Soft Soil Below Stiffer Soil (behaves as if deeper) Fi Figure 3-58 Illustration of Equivalent Depths in a Multi-layer Soil Profile 1.73 m Soft Clay 1.32 m Loose Sand 6.1 m Stiff Clay c = 23.9 kPa 50 = 0.02 = 7.9 kN/m3 = 30 deg. = 7.9 kN/m3 c = 95.8 kPa 50 = 0.005 = 9.4 kN/m3 k = 20,400 kPa Static Loading 0.61 m Figure 3-59 Soil Profile for Example of Layered Soils 135 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Four p-y curves for the case of layered soil are shown in Figure 3-60. The curves are for points A, B, C and D as shown in the sketch in Figure 3-61, at depths of 0.92 m (36 in.), 1.83 m (72 in.), 3.66 m (144 in.), and 7.32 m (288 in.), respectively. The curve at a depth of 0.92 m (36 in.) falls in the upper zone of soft clay; the curve for the depth of 1.83 m (72 in.) falls in the sand just below the soft clay; and the curves for depths of 3.66 m (144 in.) and 7.32 m (288 in.) fall in the lower zone of stiff clay. 400 350 300 Sof t Clay, x = 0.92 m Sand, x = 1.83 m 250 Stiff Clay, x = 3.66 m Stiff Clay, x = 7.32 m 200 150 100 50 0 0.0 0.01 0.02 0.03 Lateral Deflection y, meters 0.04 0.05 Figure 3-60 Example p-y Curves for Layered Soil Soft Clay A xEQ = 2.057 m B 1.73 m Loose Sand 3.05 m xEQ = 1.816 m C Stiff Clay D Actual Depth, m Equivalent Depth, m A 0.92 0.92 B 1.83 2.057 C 3.66 1.816 D 7.32 5.476 9.14 m 0.61 m Figure 3-61 Equivalent Depths of Soil Layers Used for Computing p-y Curves 136 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock Following the method suggested by Georgiadis, the p-y curve for soft clay can be computed as if the profile consists altogether of that soil. When dealing with the sand, an equivalent depth of sand 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 equivalent thickness of loose sand to replace the 1.73 m (68 in.) of soft clay was found to be 1.88 meters (74 in.). Thus, the equivalent depth to point B in loose sand is 1.98 meters (78 in.). A plot of the integrals of ultimate soil resistance and equivalent depths is presented in Figure 3-61. An equivalent depth of stiff clay was found such that the sum of the ultimate soil resistance for the stiff clay is equal to the sum of the ultimate soil resistance of the loose sand and soft clay. In making the computation, the equivalent and actual thicknesses of the loose sand, 1.88 m (74 in.) and 1.32 m (52 in.), respectively, were replaced by 1.14 m (45 in.) of stiff clay. Thus, the actual thicknesses of the soft clay and loose sand of 3.05 m (120 in.) were reduced by 1.91 m (75 in.), leading to equivalent depths in the stiff clay of points C and D of 1.75 m (69 in.) and 5.41 m (213 in.), respectively (Figure 3-61). Another point of considerable 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 is based on the assumption that the sand above the stiff clay can move downward and fill any gap that develops between the clay and the pile. Furthermore, in the stiff-clay experiment where free water was present, the free water moved upward along the face of the pile with each cycle of loading. The presence of soft clay and sand to a depth of 3.05 m (120 in.) above the stiff clay is believed to suppress the hydraulic action of free water even though the sand did not serve to close the potential gaps in the stiff clay. The equations used to compute lateral load transfer at failure are the ultimate values. Soft Clay static loading .............................................. (3-20) .......................................................... (3-21) Soft Clay cyclic loading ....................................................... (3-24) ..................................................... (3-25) Stiff Clay with Free Water Static pct = 2cab + bx + 2.83 cax ............................................ (3-26) pcd = 11cb ......................................................... (3-27) 137 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ...................................... (3-35) Stiff Clay with Free Water Cyclic ........................................... (3-40) Stiff Clay without Free Water static and cyclic loading ...............................................(3-20) ...........................................................(3-21) Sand ..................... (3-50) ............................... (3-51) or ............................................... (3-52) API Sand ................................................... (3-61) ........................................................ (3-62) 138 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 near the ground surface for saturated clay where the pile was installed in ground with a horizontal slope was derived by Reese (1958) and is shown in Equation 3-127. ....................................... (3-127) If the ground surface has a slope angle as shown in Figure 3-62, the soil resistance at the front of the pile, following the Reese approach is: + + Figure 3-62 Pile in Sloping Ground and Battered Pile 139 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ....................................... (3-128) The soil resistance at the back of the pile is: ......................... (3-129) where: (pu)ca = ultimate soil resistance near ground surface, ca = average undrained shear strength, b = pile diameter, = H = = average unit weight of soil, 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 3-127 and 3-128 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 near the ground surface for sand where the pile was installed in ground with a horizontal slope was derived earlier and is: .............. (3-130) If the ground surface has a slope angle , the ultimate soil resistance in the front of the pile is: ( pu ) sa H K 0 H tan sin (4 D13 3D12 1) tan( ) cos tan tan( ) bD2 H tan tan D22 tan )(4 D13 3D12 1) K Ab K 0 H tan (tan sin where 140 ............... (3-131) Chapter 3 Lateral Load-Transfer Curves for Soil and Rock ................................................. (3-132) D2 1 D1 , and ................................................... (3-133) .................................... (3-134) where is defined in Figure 3-62. Note that the denominator of Equation 3-132 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 3-51 is used, so no computational problem arises. The ultimate soil resistance in the back of the pile is: ( pu ) sa H K 0 H tan sin (4 D33 tan( ) cos tan tan( ) bD4 3D32 1) H tan tan D42 tan )(4 D33 K 0 H tan (tan sin ............. (3-135) 3D32 1) K Ab where ................................................ (3-136) and D4 = 1 + D3....................................................... (3-137) 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. 141 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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. Piles that are vertical are sometimes The effect of batter on the behavior of laterally loaded piles has been investigated in a model test studies performed. 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 (Awoskika 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 3-63. Pile Batter Angle in LPile, degrees 2.0 30 20 0 10 10 20 30 Load 1.0 0 30 20 0 10 10 20 30 Ground Slope Angle in LPile, degrees Figure 3-63 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 in turn will 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 depth is believed to be satisfactory. As shown in Figure 3-63, 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 field-testing is desirable. 142 Chapter 3 Lateral Load-Transfer Curves for Soil and Rock 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 3-63 to determine what value of p-multiplier to use. The 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 3-13 Shearing Force Acting at Pile Tip Data input can include a shearing force at the bottom of the pile in the development of the finite difference equations,. The shearing force would be applicable only to those cases where the pile is short; that is, where there is only one point of zero deflection. The formulations to compute the shearing force as a function of deflection are currently unavailable. It is believed that construction techniques have a major effect of the development of shearing forces at the pile tip. At present, 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 do it to make an estimate of the necessary force-deflection curve by considering pile geometry and soil properties or to derive a relationship from the results of pile load tests. A study is necessary in which experimental results from a number of tests of short piles are studied. It is hoped that methods can be developed to estimate the V0 versus y0 curves. 143 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. 144 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 145 Chapter 4 Special Analyses 200 150 100 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 146 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 length for a drilled foundation to be long enough to penetrate into the strong layer after considering the variability of the depth to the strong layer and add a requirement for the construction inspector to notify the design engineer if the strong layer is not encountered in the field after drilling to the full 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 moving 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 is illustrated in Figure 4-5 . The solution is implemented in LPile by modifying the governing differential equation to .......................................(4-1) 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. 147 Chapter 4 Special Analyses p ps y y ys ys Epy y Figure 4-5 p-y Curve Displaced by Soil Movement 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. 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. 148 Chapter 4 Special Analyses 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 buckling study was performed. The pile head is at the elevation of the ground surface. The soil profile 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 s modulus of 200 GPa. Two curves are plotted in Figure 3-6. For one curve, the specified shear force is 0.1 kN and buckling failure occurs for thrust values above 218 kN. For the second curve, the specified 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 a lower capacity associated with a greater loading condition. 250 V = 0.1 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 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. 149 Chapter 4 Special Analyses 4-4-2 Example of Incorrect 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. 450 Correct 400 Incorrect 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 150 Chapter 4 Special 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 .......................................................(4-2) 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 ...................................................(4-3) The pile deflections may be re-plotted in which values of are plotted along the xaxis and values of 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 151 Chapter 4 Special Analyses a 1 b y y0 Figure 4-9 Pile Buckling Results Showing a and b The pile buckling capacity, Pcrit, is calculated from ...............................................................(4-4) 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 overs 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. 152 Chapter 4 Special Analyses Figure 4-10 Dialog for Controls for Pushover Analysis 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. 153 Chapter 4 Special Analyses Formation of plastic hinge Figure 4-11 Pile-head Shear Force versus Displacement from Pushover Analysis 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 head and the axial load is zero. 154 Chapter 4 Special Analyses 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. 155 Chapter 4 Special Analyses (This page was deliberately left blank) 156 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 that of concrete. 157 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. fc 0.15 f c Ec 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. for for .......................................(5-1) ............................(5-2) 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 .............................................(5-3) The modulus of rupture for prestressed concrete piles is computed using 158 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity .............................................(5-4) The modulus of elasticity of concrete, Ec, is computed using ..........................................(5-5) The compressive strain at peak compressive stress, 0, is computed using ............................................................(5-6) The tensile strain at fracture for concrete, t, is computed using ...................................................(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 159 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. ..............................................................(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. 160 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 ...............................................................(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 161 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity .............................................................(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: .................................................(5-11) ...........................................................(5-12) where: x = unit stress along the length of the beam, and E= s modulus. Substituting Equation 5-11 into Equation 5-12, we obtain ............................................................(5-13) From beam theory ...........................................................(5-14) 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 ..........................................................(5-15) Cancelling and rearranging Equation 5-15 ............................................................(5-16) 162 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity Continuing with the derivation, it can be seen that dx = d and .........................................................(5-17) 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 ............................................................(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 ............................................(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. 163 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 ...............................................(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 ...............................................(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 164 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity increments.4 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. 4 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. 165 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) 166 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 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. ..........................................(5-23) 167 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 F4 = (3 bars) (5 4 2 10 m /bar) ( 0.007005) (413,000 kPa) = 619.50 kN Total of forces in the reinforcing bars = 913.95 kN. 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. 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 168 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. 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 then 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 c3 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 169 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). 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 The strain at rupture is then The thickness of the tension zone is computed using Equation 5-19 as The force in tension is the product of average tensile stress is and the area in tension and is 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: 170 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. Also, some error was introduced by the reduced precision 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 steel bars = ( 40.93 kN) (0.3045 m) = 12.46 kN-m 171 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 0.760 m 0.203 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. 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 0.0-2% 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. 172 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-m 51,348.62 51,265.02 0.16% Axial Thrust, kN 900.00* 900.15 +0.02% 2 * 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 1,000 950 900 850 800 750 700 650 600 550 500 450 400 350 300 250 200 150 100 50 0 0.0 0.005 0.01 0.015 Curvature, radians/m Figure 5-6 Bending Moment versus Curvature 173 0.02 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity 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 174 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 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 175 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. ..........................................................(5-24) Referring to the dimensions shown in Figure 5-9, the plastic modulus Z of the pipe is The computed moment capacity is 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 8 2 10 kPa 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. 176 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 177 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) 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: .......................................(5-25) ps 0.0076; f ps 250 ps 178 0.04 (ksi) ................................(5-26) 0.0064 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity For 270 ksi 7-wire low-relaxation strands: .......................................(5-27) .................................(5-28) 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: ...................................(5-29) .............................(5-30) 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, ...........................................................(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. 179 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. 180 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity 5-5 Reference Information 5-5-1 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, in2 Wt/ft D, mm Area, mm2 Kg/m 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 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 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 10.506 181 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity 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 1 1/8" smooth 1 1/4" smooth 1 3/8" smooth 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 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 Grade, ksi D, in Area, in Wt/ft D, mm Area, mm2 Kg/m 250 250 250 250 250 250 250 270 270 270 270 270 270 270 270 300 300 300 300 300 145 145 145 145 145 145 160 160 160 160 160 160 157 150 160 150 160 160 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 1.125 1.250 1.375 0.75 0.875 1 1.125 1.25 1.375 0.625 1 1 1.25 1.25 1.375 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 0.994 1.227 1.485 0.442 0.601 0.785 0.994 1.227 1.485 0.28 0.85 0.85 1.25 1.25 1.58 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 3.38 4.17 5.05 1.5 2.04 2.67 3.38 4.17 5.05 0.98 3.01 3.01 4.39 4.39 5.56 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 28.6 31.8 34.9 19.1 22.2 25.4 28.6 31.8 34.9 15.9 25.4 25.4 31.8 31.8 34.9 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 98.7 107.7 140.0 285.2 387.7 506.5 641.3 791.6 958.1 285.2 387.7 506.5 641.3 791.6 958.1 180.6 548.4 548.4 806.5 806.5 1019.4 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 5.029 6.204 7.513 2.232 3.035 3.972 5.029 6.204 7.513 1.458 4.478 4.478 6.531 6.531 8.272 2 182 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity 5-5-3 Steel H-Piles Section HP 14 HP 360 HP 13 HP 330 HP 12 HP 310 HP10 HP 250 HP 8 HP 200 Weight Area, A lb/ft kg/m in 2 cm in mm 117 34.4 175 222 102 30 2 Depth, d Thickness Flange Width, b Ixx 4 Iyy 4 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 in 4 cm in 4 cm 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 43.5 87 25.5 12.95 13.105 0.665 0.665 755 250 130 165 329 333 16.9 16.9 31425 10406 300 73 21.6 12.75 13.005 0.565 0.565 630 207 31.9 109 139 324 330 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 53 15.5 11.78 12.045 0.435 0.435 393 127 22 79 100 299 306 11 11 16400 5290 152 57 16.8 9.99 10.225 0.565 0.565 294 101 51.6 85 108 254 260 14.4 14.4 12200 4200 356 42 12.4 9.7 10.075 0.42 0.42 210 71.7 29.4 63 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 347 183 Chapter 6 Use of Vertical Piles in Stabilizing a Slope 6-1 Introduction The computation of slope stability is a problem often faced by geotechnical engineers. Numerous methods have been presented for making the necessary analyses; 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. For example, the method of Morgenstern and Price (1965) was the first method of analysis that was capable of solving all equations of 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 Applications of the Method 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 top 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 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. A drilled shaft or pile is placed in the slope near the position of the lowest extent of the sliding surface (if present or predicted by slope stability analysis). The use of a drilled foundation is a favorable procedure because the installation of the shaft will result in minimal disturbance to the soils present in the 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 184 Chapter 6 Use of Vertical Piles in Stabilizing a Slope 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 Pile in a Slope Subject to Sliding Available right-of-water in urban areas may be limited or extremely expensive with the result that the design of a slope with an adequate factor of safety against sliding is impossible. 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-reinforced, steel pipe piles were used at Kanogawa Dan 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 (12.6 185 Chapter 6 Use of Vertical Piles in Stabilizing a Slope 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. At least 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 6-2(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 6-2(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 6-2(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 Figure 6-2(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 6-2 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 6-3. 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 186 Chapter 6 Use of Vertical Piles in Stabilizing a Slope 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. R z T Safety factor for moment equilibrium considering the same forces as above, plus the effect of the stabilizing pile is expressed as: ......................................(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 6-3 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 187 Chapter 6 Use of Vertical Piles in Stabilizing a Slope versus depth as a distributed lateral force for depths above the sliding surface as shown in Figure 6-2(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, 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 6-3, 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 unity. 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. 188 Chapter 6 Use of Vertical Piles in Stabilizing a Slope Figure 6-4 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 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 189 Chapter 6 Use of Vertical Piles in Stabilizing a Slope 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 6-5. The slope exists along the bank of a river where sudden drawdown is possible. Slides had been observed along the river at numerous places and it was desirable to stabilize the slope to allow a bridge to be constructed. 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 6-5 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. 190 Chapter 6 Use of Vertical Piles in Stabilizing a Slope A preliminary design is shown in Figure 6-6, 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. Further, as shown in the figure, it was found that the tops of the shafts had to be restrained with grade beam anchored in stable soil. The use of the grade beam was required because of the depth of the slide. The results of the analysis, for each of the groups perpendicular to the river, gave the following loads at the top of the drilled shafts: Shafts 1, 2, and 3, +1,090 kN; Shaft 4, 1,310 kN; and Shaft 5, 1,690 kN. The member connecting the tops of the 5 piles would be designed to sustain the indicated loading. The maximum bending moment for Shaft 5 was about 6,250 kN-m, which would require heavy reinforcement. The computed bending moments for the other drilled shafts was much smaller. With the piles in place and with the restraining forces of the piles against the sliding soil, shown Figure 6-7, a second analysis was performed to find the new factor of safety against sliding. The value 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. 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 6-6 Preliminary Design of Stabilizing Piles 191 Chapter 6 Use of Vertical Piles in Stabilizing a Slope Elevation, m 80 48 kPa 48 kPa 75 70 108 kPa 108 kPa 65 71 kPa 71 kPa 60 55 Figure 6-7 Load Distribution from Stabilizing Piles for Slope Stability Analysis 6-6-3 Conclusions The results predicted by the proposed design method are compared with results from available full-scale experiments. The case studies yield information on the applicability of the proposed method of analysis. A complete analysis for the stability of slopes with drilled shafts in place is presented. The method of analysis is considered to be 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 indicated that have not been previously available. 192 Chapter 6 Use of Vertical Piles in Stabilizing a Slope (This page was intentionally left blank) 193 References Akinmusuru, J. O., n Piled Footings, Journal of the Geotechnical Engineering Division, ASCE, Vol. 106, No. GT11, November, pp. 1263-1268. Allen, J., 1985. p-y Austin, May. Ph.D. dissertation, The University of Texas at American Petroleum Institute, 2010. Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms - Working Stress Design, API RP 2A-WSD, 21st Edition, Errata and Supplement, 2010. Awoshika, K., and Reese, L. C., 1971. -Spaced Batter Research Report 11 7-3F, Center for Highway Research, The University of Texas at Austin, February. Awoshika, K., and Reese, L. C., 1971. Proceedings, Intl. Symposium on the Engineering Properties of Sea-Floor Soils and Their Geophysical Identification, University of Washington, Seattle. Baecher, G. B., and Christian, J. T., 2003. Reliability and Statistics in Geotechnical Engineering, Wiley, New York, 605 p. Baguelin, F.; Jezequel, J. F.; and Shields, D. H., 1978. The Pressuremeter and Foundation Engineering, Trans Tech Publications. Bhushan, K.; Haley, S. C.; and Fong, P. T., 1979. Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT8, pp. 969-985. Bhushan, K.; Lee, L. J.; and Grime, D. B., 1981. Preprint, ASCE Annual Meeting, St. Louis, Missouri. Bishop, A.W. The Use of the Slip Circle in the Stability Analysis of Slopes, Géotechnique, Vol. 5, No. 1, pp. 7-17 Bogard, D., and Matlock, H., 1983. Proceedings, Geotechnical Practice in Offshore Engineering, ASCE. Bowman, E. R., 1959. M.S. thesis, The University of Texas at Austin, January, 84 p. Briaud, J.-L.; Smith, T. D.; and Meyer, B. J., 1982. Loaded Piles Using Symposium, The Pressuremeter and Marine Applications, Paris. Broms, B. B., 1964a. Journal of the Soil Mechanics and Foundations Engineering Division, ASCE, Vol. 90, No. SM3, pp. 123-156. Broms, B. B., 1964b. Journal of the Soil Mechanics and Foundations Engineering Division, ASCE, Vol. 90, No. SM2, pp. 27-63. 194 References Broms, B. B., 1965. Journal of the Soil Mechanics and Foundations Engineering Division, ASCE, Vol. 91, No. SM3, pp. 79-99. Brown, D. A., 2002. k for Stiff Clay with No Brown, D. A.; Morrison, C. M.; and Reese, L. C., 1988. Journal of the Geotechnical Engineering Division, ASCE, Vol. 114, No. 11, pp. 1261-1276. Brown, D. A.; Reese, L. C. , M. W., 1987. ading of a Large-Scale Journal of the Geotechnical Engineering Division, ASCE, Vol. 113, No. 11, pp. 1326-1343. Brown, D. A.; Shie, C. F.; and Kumar, M., 1989. p-y Curves for Laterally Loaded Piles Derived from Three-Dimensional Finite Elemen Proceedings, 3rd Intl. Symposium, Numerical Models in Geomechanics, Niagara Falls, Canada, Elsevier Applied Science, pp. 683-690. Bryant, L. M., 1977. Ph.D. dissertation, The University of Texas at Austin, 95 p. Cox, W. R.; Dixon, D. A.; and Murphy, B. S., 1984. Lateral Load Tests of 25.4 mm Diameter Piles in Very Soft Clay in Side-by-Side and In-Line Groups, Laterally Loaded Deep Foundations: Analysis and Performance, ASTM, SPT835. Cox, W. R.; Reese, L. C.; and Grubbs, B. R., 1974. Proceedings, 6th Offshore Technology Conference, Vol. II, pp. 459-472. Darr, K.; Reese, L. C.; and Wang, S.-T., 1990. Effects of Uplift Loading and Lateral Proceedings, 22nd Offshore Technology Conference, pp. 443450. Det Norske Veritas, 1977. Rules for the Design, Construction, and Inspection of Offshore Structures, Veritasveien 1, 1322 Høvik, Norway. DiGiola, A. M.; Rojas-Gonzalez, L.; and Newman, F. B., 1989. Proceedings, Foundation Engineering: Current Principals and Practices, ASCE, Vol. 2, pp. 1338-1352. Dunnavant, T. W., and , M. W., 1985. Lateral Load Test of a 72-Inch-Diameter Bored Pile in Overof Civil Engineering, University of Houston-University Park, Houston, Texas, Report No. UHCE 85-4, September, 57 pages. Evans, L. T., and Duncan, J. M., 1982. No. UCB/GT/82-04, Geotechnical Engineering, Department of Civil Engineering, University of California, Berkeley. Focht, J. A., Jr. and Koch, K. J., 1973 Proceedings, 5th Offshore Technology Conference, Vol. II, pp. 701-708. Fukuoka, M., 1977. The Effects of Horizontal Loads on Piles Due to Landslides, Proceedings, 9th International Conference, ISSMFE, Tokyo, Japan. 195 References Gazioglu, S. M., and , M. W., 1984. Evaluation of p-y Relationships in Cohesive Soil, Symposium on Analysis and Design of Pile Foundations, ASCE, San Francisco, 192-213. George, P., and Wood, D., 1976. Offshore Soil Mechanics, Cambridge University Engineering Department. Georgiadis, M., 1983. p-y Geotechnical Practice in Offshore Engineering, ASCE, pp. 536-545. Hansen, J. B., 1961. Geotechnical Institute, Copenhagen, Denmark, pp. 3 8-46. Proceedings, Bulletin No. 11, The Danish Hansen, J. B., 1961. Bulletin No. 12, The Danish Geotechnical Institute, Copenhagen, Denmark, pp. 5-9. FHWA/IN/JHRP-84/8, Purdue University, 181 p. Hetenyi, M., 1946. Beams on Elastic Foundation, University of Michigan Press. -Coulomb Friction and Cohesion Values from the HoekInternational Journal of Rock Mechanics, Mining Sciences, and Geomechanics Abstracts, 27 (3), pp. 227-229. Horvath, R. G., and Kenney, T. C., 1979. Proceedings, Symposium on Deep Foundations, ASCE, pp. 182-184. Hrennikoff, A., 1950. Vol. 115, pp. 351-374. Transactions, ASCE, Isenhower, W. M., 1992. Report to Department of the Army, Waterways Experiment Station, Corps of Engineers, Contract No. DAAL03-91-C-0034, TCN 92-185, Scientific Services Program, 67 p. Isenhower, W. M., 1994. Proceedings, Intl. Conf. on Design and Construction of Bridge Foundations, Vol. 2, pp. 571-585. Jamiolkowski, M., 1977. Foundation Engineering, Tokyo. Intl. Conf. on Soil Mechanics and Japanese Society for Architectural Engineering, Committee for the Study of the Behavior of Piles During Earthquakes, 1965. Research Concerning Horizontal Resistance and Dynamic Response of Pile Foundations, (in Japanese). Johnson, G. W., 1982. -Boring Pressuremeter in Obtaining In-Situ Shear Moduli M.S. thesis, The University of Texas, Austin, Texas, August, 156 p. Johnson, R. M., Parsons, R. L., Dapp, S. D., and Brown, D. A., 2006. Soil Characterization and p-y Curve Development for Loess, Kansas Department of Transportation, Bureau of Materials and Research, July. Kooijman, A. P., 1989. -plastic Quasi Three-Dimensional Model for Proceedings, 3rd Intl. Symposium, Numerical Models in Geomechanics, Elsevier Applied Science, New York, pp. 675-682. 196 References Kubo, K., 1964. Transportation Technology Research Institute, Japan, Vol. 12, No. 2. Design & Performance of Deep Foundations: Piles & Piers in Soil & Soft Rock (GSP 38), ASCE, New York, Oct 1993, 172-183. p-y Geotechnical and Geoenvironmental Engineering, Vol. 135, No. 1, pp. 26-36. Journal of Lieng, J. T., 1988. Behavior of Laterally Loaded Piles in Sand - Large Scale Model Tests, Ph.D. Thesis, Department of Civil Engineering, Norwegian Institute of Technology, 206 p. Long, J. H., 1984. etitive Ph.D. dissertation, The University of Texas, Austin, Texas, 332 p. Malek, A. M.; Azzouz, A. S.; Baligh, M. M.; and Germaine, J. T., 1989. Journal of Geotechnical Engineering Division, ASCE, Vol. 115, No. 5, pp. 615-637. Marinos, P., and Hoek, E., A Geologically Friendly Tool for Rock Mass Strength Proceedings, GeoEngineering 2000 Conference, Melbourne, pp.1,422-1,442. Matlock, H., 1970. Proceedings, 2nd Offshore Technology Conference, Vol. I, pp. 577-594. Matlock, H., and Ripperger, E. A., 1958. Proceedings, ASTM, Vol. 58, pp. 1245-1259. Matlock, H.; Ripperger, E. A.; and Fitzgibbon, D. P. 1956. (unpublished), 167 p. -Loading of McClelland, B., and Focht, J. A., 1956. Journal of the Soil Mechanics and Foundations Division, ASCE, No. SM4, October, Paper 1081, 22 p., also published in Transactions, ASCE, 1958, Vol. 123, pp. 1049-1086. he Analysis of the Stability of General Slip Géotechnique, No. 1, Vol. 15, pp. 79-93. , M. W., 1996. Personal Communication. , M. W., and Dunnavant, T. W., 1984. Cyclic Degradability Department of Civil Engineering, University of Houston-University Park, Houston, Texas, Report No. UHCE 84-7, 368 p. , M. W., and Gazioglu, S. M., 1984. p-y Relationships Report to the American Petroleum Institute, PRAC 82-41-2, The University of HoustonUniversity Park, Houston. , M. W., and Murchison, J. M., 1983. p-y Report to the American Petroleum Institute, PRAC 82-41-1, The University of HoustonUniversity Park, Houston. 197 References Ghazzaly, O. I.; and Ha, H. B., 1977. Groups with Nonlinear Soil Response and Pile-SoilOffshore Technology Conference, Vol. II, pp. 245-256. -Dimensional Pile Proceedings, 9th No. FHWA/IN/JHRP-86/7, Purdue University, 305 p. Parker, F., Jr., and Reese, L. C., 1971. -Soil Interaction Cu Proceedings, Intl. Symposium on the Engineering Properties of Sea-Floor Soils and Their Geophysical Identification, The University of Washington, Seattle, pp. 212-223. Poulos, H. G., and Davis, E. H. 1980. Pile Foundation Analysis and Design, Wiley, New York. Reese, L. C., and J. A. Focht, Jr., Transactions, ASCE, Vol. 123, pp. 1071-1074. Reese, L. C., 1984. Handbook on Design of Piles and Drilled Shafts under Lateral Load, Report FHWA-IP84-11, FHWA, US Department of Transportation, Office of Research, Development and Technology, McLean, Virginia, , July, 1984, 360 p. Reese, L. C., 1985. Behavior of Piles and Pile Groups Under Lateral Load, Report No. FHWA/RD-85/106, FHWA, Office of Research, Development, and Technology, Washington, D. C. Proceedings, t, State of the Art Reports, Balkema, pp. 859-866. Reese, L. C., 1997. Journal of the Geotechnical and Geoenvironmental Engineering Division, ASCE, pp. 1010-1017. Reese, L. C., and Cox, W. R., 1968. lysis of Tests of Uninstrumented Piles Under Lateral Loading Proceedings, 71st Annual Meeting, ASTM, pp. 161-176. Reese, L. C., and Matlock, H., 1956. -Dimensional Solutions for Laterally Loaded Piles with Soil Modulus Assumed Proportional to Depth, Proceedings, 8th Texas Conf. on Soil Mechanics and Foundation Engineering, Special Publication No. 29, Bureau of Engineering Research, The University of Texas, 1956, 41 pages. Reese, L. C., and Nyman, K. J., 1978. Florida, (unpublished). Reese, L. C., and Wang, S.-T. 1988. havior Texas Civil Engineer, May, pp. 17-23. Reese, L. C., and Welch, R. C., 1975. Journal of the Geotechnical Engineering Division, ASCE, Vol. 101, No. GT7, pp. 633-649. Reese, L. C.; Cox, W. R.; and Koop, F. D., 1968. -Load Tests of Instrumented Piles in (unpublished), 303 p. Reese, L. C.; Cox, W. R.; and Koop, F. D., 1974. Proceedings, 6th Offshore Technology Conference, Vol. II, pp. 473-484. 198 References Reese, L. C.; Cox, W. R.; and Koop, F. D., 1975. Proceedings, 7th Offshore Technology Conference, pp. 671-690. Reese, L. C.; Wang, S.-T.; and Long, J. H. 1989. Proceedings, 21st Offshore Technology Conference, pp. 395-401. Rollins, K. M.; Gerber, T. M.; Lane, J. D.; and Ashford, S. A., 2005a. FullJournal of the Geotechnical Geoenvironmental Engineering Division, ASCE, Vol. 131, pp. 115-125. and p-y Curves for Large Diameter Shafts in Seismic Performance and Simulation of Pile Foundations in Liquefied and Laterally Spreading Ground, Geotechnical Special Publication No. 145, ASCE, p. 11-23. Schmertmann, J. H., 1977. opment of a Keys Limestone Shear Test for Drilled Florida, (unpublished). Seed, R. B. and Harder, L. F., 1990. and U Publishers Ltd., pp. 351-376 -Based Analysis of Cyclic Pore Pressure Generation H. Bolton Seed, Memorial Symposium, Vol. 2, BiTech Sherard, J. L.; Dunnigan, L. P.; and Decker, R. S., 1976. of the Geotechnical Engineering Division, ASCE, Vol. 102, No. GT1, pp. 69-86. Journal Simpson, M. and Brown, D. A., 2006. Personal Communication. Skempton, A. W., 1951. Proceedings, Building Research Congress, Division I, Building Research Congress, London. Speer, D., 1992. Department of Transportation. Stevens, J. B., and Audibert, J. M. E., 1979. -examination of p-y th Proceedings, 11 Offshore Technology Conference, Houston, Texas, Vol. I, pp. 397-403. Stokoe, K. H., 1989. Personal Communication, October. Sullivan, W. R.; Reese, L. C.; and Fenske, C. W., 1980. f Civil Engineers, London, pp. 135-146. Terzaghi, K., 1955. No. 4, pp. 297-326. Thompson, G. R., 1977. Curves for Satu Géotechnique, Vol. 5, p-y hesis, The University of Texas at Austin. Timoshenko, S. P., 1956. Strength of Materials, Part II, Advanced Theory and Problems, 3rd Edition, Van Nostrand, New York. Bending of Beams Resting on Isotropic Elastic Sol Engineering Mechanics Division, ASCE, Vol. 87, No. EN2, pp. 35-53. 199 Journal of the References Wang, S.-T., 1982. M.S. thesis, The University of Texas, Austin, Texas, 69 pages. Wang, S.-T., 1986. dissertation, The University of Texas at Austin, 1986. - Ph.D. Wang, S.-T., and Reese, L. C., 1998. Geotechnical Earthquake Engineering and Soil Dynamics III, Geotechnical Special Publication No. 75, ASCE, pp. 1331-1343. Weaver, T. J., 2001. Ph.D. Dissertation, The University of California at San Diego. -Scale Lateral Welch, R. C., and Reese, L. C., 1972. Report No. 3-5-65-89, Center for Highway Research, The University of Texas at Austin, May 1972. Yegian, M., and Wright, S. G., 1973. -Displacement Relationships for Proceedings, 5th Offshore Technology Conference, Vol. II, pp. 663-676. 200 References (This page was intentionally left blank) 201 Name Index Akinmusuru, J. O. ....................................... 4 Dunnigan, L. P. ......................................... 60 Allen, J. ................................................... 134 Evans, L. T...................................... 108, 109 American Petroleum Institute . 18, 66, 91, 92 Fenske, C. W............................................. 80 Ashford, S. A. ..................................... 97, 98 Fitzgibbon, D. P. ....................................... 50 Audibert, J. M. E....................................... 80 Focht, J. A., Jr. .......................... 4, 17, 61, 63 Awoshika, K. .............................................. 4 Fong, P. T.................................................. 80 Azzouz, A. S. ............................................ 11 Fukuoka, M. .................................... 185, 189 Baecher, G. B.............................................. 3 Gazioglu, S. M. ......................................... 80 Baguelin, F.......................................... 18, 96 George, P................................................... 18 Baligh, M. M............................................. 11 Georgiadis, M. ........................................ 134 Bhushan, K.......................................... 80, 96 Gerber, T. M. ............................................ 97 Bishop, A. W........................................... 184 Germaine, J. T........................................... 11 Bogard, D.................................................... 4 Grime, D. B............................................... 96 Bowman, E. R. .......................................... 83 Hales, L. J. ................................................ 98 Briaud, J. L,............................................... 80 Haley, S. C. ............................................... 80 Broms, B. B............................................... 16 Hansen, J. B. ............................................. 55 Brown, D. A...................... 4, 51, 80, 99, 132 Harder, L. F............................................... 97 Bryant, L. M................................................ 7 Hassiotis, S.............................................. 186 Chameau, J. L. ........................................ 186 Hetenyi, A. .......................................... 14, 28 Christian, J. T.............................................. 3 Hoek, E. .......................................... 130, 131 Cox, W. R. .... 48, 49, 50, 58, 59, 70, 85, 109 Horvath, R. G.......................................... 118 Dapp, S. D................................................. 99 Hrennikoff, A.............................................. 4 Darr, K. ..................................................... 55 Isenhower, W. M....................... 99, 147, 157 Davis, E. H................................................ 18 Jamiolkowski, M....................................... 18 Decker, R. S. ............................................. 60 Jezequel, J. F. ...................................... 18, 96 Det Norske Veritas.................................... 18 Johnson, G. W........................................... 50 DiGiola, A. M. .......................................... 16 Johnson, R. M. .......................................... 99 Duncan, J. M. .................................. 108, 109 Kenney, T. C. .......................................... 118 Dunnavant, T. W............... 58, 59, 60, 70, 80 Koch, K. J. .................................................. 4 202 References Kooijman, A. P. ........................................ 51 Reese, L. C.4, 18, 48, 49, 50, 51, 55, 58, 59, 70, 75, 78, 80, 81, 85, 90, 97, 109, 113, 114, 120, 123, 124, 139, 157 Koop, F. D........... 48, 50, 58, 59, 70, 85, 109 Kubo, K................................................... 142 Ripperger, E. A. .................................. 47, 50 Kulhawy, F. D......................................... 129 Rojas-Gonzalez, L..................................... 16 Lane, J. D. ................................................. 97 Rollins, K. M....................................... 97, 98 Lee, L. J..................................................... 96 Schmertmann, J. H.................................. 115 Liang, R................................................... 125 Seed, R. B. ................................................ 97 Long, J. H................................ 13, 59, 60, 73 Sherard, J. L. ............................................. 60 Malek, A. M.............................................. 11 Shields, D. H. ...................................... 18, 96 Marinos, P. ...................................... 130, 131 Simpson, M. ............................................ 132 Matlock, H. . 4, 19, 47, 50, 64, 66, 68, 70, 81 Skempton, A. W........................................ 61 McClelland, B. .............................. 17, 61, 63 Smith, T. D................................................ 80 Meyer, B. J................................................ 80 Speer, D........................................... 115, 123 Morgenstern, N. R................................... 184 Stevens, J. B.............................................. 80 Morrison, C. M. ........................................ 51 Stokoe, K. H.............................................. 50 Murchison, J. M. ....................................... 96 Sullivan, W. R........................................... 80 Newman, F. B. .......................................... 16 Terzaghi, K. ...................... 14, 61, 63, 81, 85 Nusairat, J. .............................................. 125 Thompson, G. R.................................. 16, 51 Nyman, K. J. ........................................... 114 Timoshenko, S. P. ..................................... 37 4, 58, 59, 60, 70, 80, 96, 122 ............................................... 50 Oakland, M. W........................................ 186 Wang, S.-T. ............... 28, 55, 59, 60, 97, 157 Parker, F., Jr. ....................................... 85, 90 Welch, R. C................................... 59, 75, 78 Parsons, R. L. ............................................ 99 Wood, D.................................................... 18 Phoon, K. K............................................. 129 Wright, S. G. ....................................... 16, 51 Poulos, H. G.............................................. 18 Yang, K. .................................................. 125 Price, V. E. .............................................. 184 Yegian, M. .......................................... 16, 51 203