Lpile Technical Manual
Short Description
Lpile Technical Manual...
Description
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. ShinTower 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 11 Compatible Designs.............................................................................................................. 1 12 Principles of Design.............................................................................................................. 1 121 Introduction ................................................................................................................... 1 122 Nonlinear Response of Soil........................................................................................... 2 123 Limit States ................................................................................................................... 2 124 StepbyStep Procedure................................................................................................. 2 125 Suggestions for the Designing Engineer ....................................................................... 3 13 Modeling a Pile Foundation ................................................................................................. 5 131 Introduction ................................................................................................................... 5 132 Example Model of Individual Pile Under ThreeDimensional Loadings ..................... 7 133 Computation of Foundation Stiffness ........................................................................... 8 134 Concluding Comments.................................................................................................. 9 14 Organization of Technical Manual ....................................................................................... 9 Chapter 2 Solution for Pile Response to Lateral Loading ............................................................ 11 21 Introduction ........................................................................................................................ 11 211 Influence of Pile Installation and Loading on Soil Characteristics............................. 11 2111 General Review.................................................................................................... 11 2112 Static Loading ...................................................................................................... 12 2113 Repeated Cyclic Loading..................................................................................... 13 2114 Sustained Loading................................................................................................ 13 2115 Dynamic Loading................................................................................................. 14 212 Models for Use in Analyses of Single Piles................................................................ 14 2121 Elastic Pile and Soil ............................................................................................. 14 2122 Elastic Pile and Finite Elements for Soil ............................................................. 16 2123 Rigid Pile and Plastic Soil.................................................................................... 16 2124 Rigid Pile and FourSpring Model for Soil.......................................................... 16 2125 Nonlinear Pile and py Model for Soil................................................................. 17 2126 Definition of p and y ............................................................................................ 18 2127 Comments on the py method .............................................................................. 19 213 Computational Approach for Single Piles................................................................... 19 2131 Study of Pile Buckling......................................................................................... 21 2132 Study of Critical Pile Length ............................................................................... 21 214 Occurrences of Lateral Loads on Piles........................................................................ 22 2141 Offshore Platform ................................................................................................ 22 2142 Breasting Dolphin ................................................................................................ 23 2143 SinglePile Support for a Bridge.......................................................................... 24 2144 PileSupported Overhead Sign............................................................................. 25 2145 Use of Piles to Stabilize Slopes ........................................................................... 27 2146 Anchor Pile in a Mooring System........................................................................ 27 2147 Other Uses of Laterally Loaded Piles .................................................................. 27 iii
22 Derivation of Differential Equation for the BeamColumn and Methods of Solution....... 28 221 Derivation of the Differential Equation ...................................................................... 28 222 Solution of Reduced Form of Differential Equation................................................... 32 223 Solution by Finite Difference Equations..................................................................... 37 Chapter 3 Lateral LoadTransfer Curves for Soil and Rock......................................................... 45 31 Introduction ........................................................................................................................ 45 32 Experimental Measurements of py Curves........................................................................ 47 321 Direct Measurement of Soil Response ........................................................................ 47 322 Derivation of Soil Response from Moment Curves Obtained by Experiment............ 47 323 Nondimensional Methods for Obtaining Soil Response ............................................. 49 33 py Curves for Cohesive Soils ............................................................................................ 50 331 Initial Slope of Curves................................................................................................. 50 332 Analytical Solutions for Ultimate Lateral Resistance ................................................. 52 333 Influence of Diameter on py Curves .......................................................................... 58 334 Influence of Cyclic Loading........................................................................................ 59 335 Introduction to Procedures for py Curves in Clays.................................................... 61 3351 Early Recommendations for py Curves in Clay ................................................. 61 3352 Skempton (1951).................................................................................................. 61 3353 Terzaghi (1955).................................................................................................... 63 3354 McClelland and Focht (1956) .............................................................................. 63 336 Procedures for Computing py Curves in Clay ........................................................... 64 337 Response of Soft Clay in the Presence of Free Water................................................. 64 3371 Description of Load Test Program....................................................................... 64 3372 Procedure for Computing py Curves in Soft Clay for Static Loading................ 65 3373 Procedure for Computing py Curves in Soft Clay for Cyclic Loading .............. 68 3374 Recommended Soil Tests for Soft Clays ............................................................. 68 3375 Examples.............................................................................................................. 68 338 Response of Stiff Clay in the Presence of Free Water ................................................ 70 3381 Procedure for Computing py Curves for Static Loading .................................... 70 3382 Procedure for Computing py Curves for Cyclic Loading................................... 73 3383 Recommended Soil Tests..................................................................................... 74 3384 Examples.............................................................................................................. 75 339 Response of Stiff Clay with No Free Water................................................................ 75 3391 Procedure for Computing py Curves for Stiff Clay without Free Water for Static Loading ............................................................................................................................. 76 3392 Procedure for Computing py Curves for Stiff Clay without Free Water for Cyclic Loading ............................................................................................................................. 78 3393 Recommended Soil Tests for Stiff Clays............................................................. 79 3394 Examples.............................................................................................................. 79 3310 Modified py Criteria for Stiff Clay with No Free Water ......................................... 80 3311 Other Recommendations for py Curves in Clays..................................................... 80 34 py Curves for Sands........................................................................................................... 81 341 Description of py Curves in Sands............................................................................. 81 3411 Initial Portion of Curves....................................................................................... 81 3412 Analytical Solutions for Ultimate Resistance ...................................................... 82 3413 Influence of Diameter on py Curves................................................................... 83 iv
3414 Influence of Cyclic Loading ................................................................................ 84 3415 Early Recommendations ...................................................................................... 85 3416 Field Experiments ................................................................................................ 85 3417 Response of Sand Above and Below the Water Table ........................................ 85 342 Response of Sand ........................................................................................................ 85 3421 Procedure for Computing py Curves in Sand ..................................................... 86 3422 Recommended Soil Tests..................................................................................... 90 3423 Example Curves ................................................................................................... 91 343 API RP 2A Recommendation for Response of Sand Above and Below the Water Table ..................................................................................................................................... 91 3431 Background of API Method for Sand .................................................................. 91 3432 Procedure for Computing py Curves Using the API Sand Method.................... 92 3433 Example Curves ................................................................................................... 94 344 Other Recommendations for py Curves in Sand........................................................ 96 35 py Curves in Liquefied Sands............................................................................................ 96 351 Response of Piles in Liquefied Sand........................................................................... 96 352 Procedure for Computing py Curves in Liquefied Sand............................................ 98 353 Modeling of Lateral Spreading ................................................................................... 99 36 py Curves in Loess ............................................................................................................ 99 361 Background ................................................................................................................. 99 3611 Description of Load Test Program....................................................................... 99 3612 Soil Profile from Cone Penetration Testing....................................................... 100 362 Procedure for Computing py Curves in Loess ......................................................... 101 3621 General Description of py Curves in Loess ...................................................... 101 3622 Equations of py Model for Loess...................................................................... 101 3623 StepbyStep Procedure for Generating py Curves........................................... 106 3624 Limitations on Conditions for Validity of Model .............................................. 107 37 py Curves in Soils with Both Cohesion and Internal Friction......................................... 107 371 Background ............................................................................................................... 107 372 Recommendations for Computing py Curves .......................................................... 108 373 Procedure for Computing py Curves in Soils with Both Cohesion and Internal Friction................................................................................................................................ 109 374 Discussion ................................................................................................................. 112 38 Response of Vuggy Limestone Rock ............................................................................... 113 381 Introduction ............................................................................................................... 113 382 Descriptions of Two Field Experiments.................................................................... 114 3821 Islamorada, Florida ............................................................................................ 114 3822 San Francisco, California................................................................................... 115 383 Procedure for Computing py Curves for Strong Rock (Vuggy Limestone) ............ 119 384 Procedure for Computing py Curves for Weak Rock .............................................. 119 385 Case Histories for Drilled Shafts in Weak Rock....................................................... 122 3851 Islamorada.......................................................................................................... 122 3852 San Francisco..................................................................................................... 123 39 py Curves in Massive Rock............................................................................................. 125 391 Determination of pu Near Ground Surface................................................................ 127 392 Rock Mass Failure at Great Depth ............................................................................ 129
v
393 Initial Tangent to py Curve Ki .................................................................................. 129 394 Rock Mass Properties................................................................................................ 129 395 Procedure for Computing py Curves in Massive Rock............................................ 131 310 py Curves in Piedmont Residual Soils .......................................................................... 132 311 Response of Layered Soils ............................................................................................. 133 3111 Layering Correction Method of Georgiadis ............................................................ 134 3112 Example py Curves in Layered Soils ..................................................................... 134 312 Modifications to py Curves for Pile Batter and Ground Slope ..................................... 139 3121 Piles in Sloping Ground .......................................................................................... 139 31211 Equations for Ultimate Resistance in Clay in Sloping Ground ....................... 139 31212 Equations for Ultimate Resistance in Sand...................................................... 140 31213 Effect of Direction of Loading on Output py Curves ..................................... 141 3122 Effect of Batter on py Curves in Clay and Sand .................................................... 142 3123 Modeling of Piles in Short Slopes........................................................................... 143 313 Shearing Force Acting at Pile Tip .................................................................................. 143 Chapter 4 Special Analyses ........................................................................................................ 144 41 Introduction ...................................................................................................................... 144 42 Computation of Top Deflection versus Pile Length......................................................... 144 43 Analysis of Piles Loaded by Soil Movements.................................................................. 147 44 Analysis of Pile Buckling ................................................................................................. 148 441 Procedure for Analysis of Pile Buckling................................................................... 148 442 Example of Incorrect Analysis.................................................................................. 150 443 Evaluation of Pile Buckling Capacity ....................................................................... 151 45 Pushover Analysis of Piles ............................................................................................... 152 451 Procedure for Pushover Analysis .............................................................................. 153 452 Example of Pushover Analysis ................................................................................. 153 453 Evaluation of Pushover Analysis .............................................................................. 155 Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity....................... 157 51 Introduction ...................................................................................................................... 157 511 Application ................................................................................................................ 157 512 Assumptions .............................................................................................................. 157 513 StressStrain Curves for Concrete and Steel ............................................................. 158 514 Cross Sectional Shape Types .................................................................................... 160 52 Beam Theory .................................................................................................................... 160 521 Flexural Behavior...................................................................................................... 160 522 Axial Structural Capacity .......................................................................................... 163 53 Validation of Method........................................................................................................ 164 531 Analysis of Concrete Sections................................................................................... 164 5311 Computations Using Equations of Section 52.................................................. 165 5312 Check of Position of the Neutral Axis ............................................................... 165 5313 Forces in Reinforcing Steel................................................................................ 167 5314 Forces in Concrete ............................................................................................. 168 5315 Computation of Balance of Axial Thrust Forces ............................................... 170 5316 Computation of Bending Moment and EI.......................................................... 171 5317 Computation of Bending Stiffness Using Approximate Method....................... 172 vi
532 Analysis of Steel Pipe Piles....................................................................................... 175 533 Analysis of PrestressedConcrete Piles ..................................................................... 177 54 Discussion......................................................................................................................... 180 55 Reference Information...................................................................................................... 181 551 Concrete Reinforcing Steel Sizes.............................................................................. 181 552 Prestressing Strand Types and Sizes ......................................................................... 182 553 Steel HPiles.............................................................................................................. 183 Chapter 6 Use of Vertical Piles in Stabilizing a Slope ............................................................... 184 61 Introduction ...................................................................................................................... 184 62 Applications of the Method .............................................................................................. 184 63 Review of Some Previous Applications ........................................................................... 185 64 Analytical Procedure ........................................................................................................ 186 65 Alternative Method of Analysis ....................................................................................... 189 66 Case Studies and Example Computation.......................................................................... 189 661 Case Studies .............................................................................................................. 189 662 Example Computation............................................................................................... 190 663 Conclusions ............................................................................................................... 192 References ...................................................................................................................................194 Name Index .................................................................................................................................202
vii
List of Figures
Figure 11 Example of Modeling a Bridge Foundation................................................................. 6 Figure 12 Threedimensional SoilPile Interaction ...................................................................... 7 Figure 13 Coefficients of Stiffness Matrix ................................................................................... 8 Figure 21 Models of Pile Under Lateral Loading, (a) 3Dimensional Finite Element Mesh, and (b) Crosssection of 3(d) MFAD Model....................................................................................................... 15 Figure 22 Model of Pile Under Lateral Loading and py Curves ............................................... 17 Figure 23 Distribution of Stresses Acting on a Pile, (a) Before Lateral Deflection and (b) After Lateral Deflection y .................................................................................... 18 Figure 24 Illustration of General Procedure for Selecting a Pile to Sustain a Given Set of Loads.......................................................................................................................... 20 Figure 25 Solution for the Axial Buckling Load ........................................................................ 21 Figure 26 Solving for Critical Pile Length ................................................................................. 22 Figure 27 Simplified Method of Analyzing a Pile for an Offshore Platform............................. 23 Figure 28 Analysis of a Breasting Dolphin ................................................................................ 24 Figure 29 Loading On a Single Shaft Supporting a Bridge Deck .............................................. 25 Figure 210 Foundation Options for an Overhead Sign Structure ............................................... 26 Figure 211 Use of Piles to Stabilize a Slope Failure .................................................................. 27 Figure 212 Anchor Pile for a Flexible Bulkhead........................................................................ 28 Figure 213 Element of BeamColumn (after Hetenyi, 1946) ..................................................... 29 Figure 214 Sign Conventions ..................................................................................................... 31 Figure 215 Form of Results Obtained for a Complete Solution................................................. 32 Figure 216 Boundary Conditions at Top of Pile......................................................................... 33 Figure 217 Values of Coefficients A1, B1, C1, and D1 ................................................................ 35 Figure 218 Representation of deflected pile............................................................................... 38 Figure 219 Case 1 of Boundary Conditions ............................................................................... 40 Figure 220 Case 2 of Boundary Conditions ............................................................................... 41 Figure 221 Case 3 of Boundary Conditions ............................................................................... 41 Figure 222 Case 4 of Boundary Conditions ............................................................................... 42 Figure 223 Case 5 of Boundary Conditions ............................................................................... 43 Figure 31 Conceptual py Curves ............................................................................................... 45 viii
Figure 32 py Curves from Static Load Test on 24inch Diameter Pile (Reese, et al. 1975) .......................................................................................................................... 48 Figure 33 py Curves from Cyclic Load Tests on 24inch Diameter Pile (Reese, et al. 1975) .......................................................................................................................... 49 Figure 34 Plot of Ratio of Initial Modulus to Undrained Shear Strength for Unconfinedcompression Tests on Clay ........................................................................................ 51 Figure 35 Variation of Initial Modulus with Depth.................................................................... 52 Figure 36 Assumed Passive Wedge Failure in Clay Soils, (a) Shape of Wedge, (b) Forces Acting on Wedge ........................................................................................... 53 Figure 37 Measured Profiles of Ground Heave Near Piles Due to Static Loading, (a) Heave at Maximum Load, (b) Residual Heave ......................................................... 54 Figure 38 Ultimate Lateral Resistance for Clay Soils ................................................................ 56 Figure 39 Assumed Mode of Soil Failure Around Pile in Clay, (a) Section Through Pile, (b) MohrCoulomb Diagram, (c) Forces Acting on Section of Pile................. 57 Figure 310 Values of Ac and As................................................................................................... 58 Figure 311 Scour Around Pile in Clay During Cyclic Loading, (a) Profile View, (b) Photograph of Turbulence Causing Erosion During Lateral Load Test .................... 60 Figure 312 py Curves in Soft Clay,(a) Static Loading, (b) Cyclic Loading.............................. 66 Figure 313 Example py Curves in Soft Clay Showing Effect of J............................................ 67 Figure 314 Shear Strength Profile Used for Example py Curves for Soft Clay........................ 69 Figure 315 Example py Curves for Soft Clay with the Presence of Free Water....................... 69 Figure 316 Characteristic Shape of py Curves for Static Loading in Stiff Clay with Free Water.......................................................................................................................... 71 Figure 317 Characteristic Shape of py Curves for Cyclic Loading of Stiff Clay with Free Water ................................................................................................................. 73 Figure 318 Example Shear Strength Profile for py Curves for Stiff Clay with No Free Water.......................................................................................................................... 75 Figure 319 Example py Curves for Stiff Clay in Presence of Free Water for Cyclic Loading ...................................................................................................................... 76 Figure 320 Characteristic Shape of py Curve for Static Loading in Stiff Clay without Free Water ................................................................................................................. 77 Figure 321 Characteristic Shape of py Curves for Cyclic Loading in Stiff Clay with No Free Water ................................................................................................................. 78 Figure 322 Example py Curves for Stiff Clay with No Free Water, Cyclic Loading .............. 79 Figure 323 Geometry Assumed for Passive Wedge Failure for Pile in Sand............................. 82 Figure 324 Assumed Mode of Soil Failure by Lateral Flow Around Pile in Sand, (a) Section Though Pile, (b) MohrCoulomb Diagram................................................... 84 ix
Figure 325 Characteristic Shape of a Set of py Curves for Static and Cyclic Loading in Sand ........................................................................................................................... 86 Figure 326 Values of Coefficients
and
........................................................................... 88
Figure 327 Values of Coefficients Bc and Bs .............................................................................. 88 Figure 329 Example py Curves for Sand Below the Water Table, Static Loading................... 91 Figure 330 Coefficients C1, C2, and C3 versus Angle of Internal Friction ................................. 93 Figure 331 Value of k, Used for API Sand Criteria.................................................................... 94 Figure 332 Example py Curves for API Sand Criteria.............................................................. 96 Figure 333 Example py Curve in Liquefied Sand ..................................................................... 97 Figure 334 Idealized Tip Resistance Profile from CPT Testing Used for Analyses. ............... 101 Figure 335. Generic py curve for Drilled Shafts in Loess Soils.............................................. 102 Figure 336 Variation of Modulus Ratio with Normalized Lateral Displacement .................... 104 Figure 337 py Curves for the 30inch Diameter Shafts........................................................... 105 Figure 338 py Curves and Secant Modulus for the 42inch Diameter Shafts. ........................ 105 Figure 339 Cyclic Degradation of py Curves for 30inch Shafts ............................................ 106 Figure 340 Characteristic Shape of py Curves for c Soil..................................................... 108 Figure 341 Representative Values of k for c Soil.................................................................. 111 Figure 342 py Curves for c Soils.......................................................................................... 112 Figure 343 Initial Moduli of Rock Measured by Pressuremeter for San Francisco Load Test .......................................................................................................................... 116 Figure 344 Modulus Reduction Ratio versus RQD (Bienawski, 1984) ................................... 117 Figure 345 Engineering Properties for Intact Rocks (after Deere, 1968; Peck, 1976; and Horvath and Kenney, 1979)..................................................................................... 118 Figure 346 Characteristic Shape of py Curve in Strong Rock ................................................ 119 Figure 347 Sketch of py Curve for Weak Rock (after Reese, 1997)....................................... 120 Figure 348 Comparison of Experimental and Computed Values of PileHead Deflection, Islamorada Test (after Reese, 1997) ........................................................................ 123 Figure 349 Computed Curves of Lateral Deflection and Bending Moment versus Depth, Islamorada Test, Lateral Load of 334 kN (after Reese, 1997) ................................ 124 Figure 350 Comparison of Experimental and Computed Values of PileHead Deflection for Different Values of EI, San Francisco Test ....................................................... 125 Figure 351 Values of EI for three methods, San Francisco test ............................................... 126 Figure 352 Comparison of Experimental and Computed Values of Maximum Bending Moments for Different Values of EI, San Francisco Test ....................................... 126
x
Figure 353 py Curve in Massive Rock .................................................................................... 127 Figure 354 Model of Passive Wedge for Drilled Shafts in Rock ............................................. 128 Figure 355 Equation for Estimating Modulus Reduction Ratio from Geological Strength Index ........................................................................................................................ 131 Figure 356 Degradation Plot for Es .......................................................................................... 133 Figure 357 py Curve for Piedmont Residual Soil.................................................................... 133 Figure 358 Illustration of Equivalent Depths in a Multilayer Soil Profile.............................. 135 Figure 359 Soil Profile for Example of Layered Soils ............................................................. 135 Figure 360 Example py Curves for Layered Soil .................................................................... 136 Figure 361 Equivalent Depths of Soil Layers Used for Computing py Curves ...................... 136 Figure 362 Pile in Sloping Ground and Battered Pile .............................................................. 139 Figure 363 Soil Resistance Ratios for py Curves for Battered Piles from Experiment from Kubo (1964) and Awoshika and Reese (1971) ............................................... 142 Figure 41 Pile and Soil Profile for Example Problem .............................................................. 145 Figure 42 Variation of Top Deflection versus Depth for Example Problem............................ 145 Figure 43 Pilehead Load versus Deflection for Example........................................................ 146 Figure 44 Top Deflection versus Pile Length for Example...................................................... 146 Figure 45 py Curve Displaced by Soil Movement .................................................................. 148 Figure 46 Examples of Pile Buckling Curves for Different Shear Force Values ..................... 149 Figure 47 Examples of Correct and Incorrect Pile Buckling Analyses .................................... 150 Figure 48 Typical Results from Pile Buckling Analysis .......................................................... 151 Figure 49 Pile Buckling Results Showing a and b ................................................................... 152 Figure 410 Dialog for Controls for Pushover Analysis ............................................................ 153 Figure 411 Pilehead Shear Force versus Displacement from Pushover Analysis................... 154 Figure 412 Maximum Moment Developed in Pile versus Displacement from Pushover Analysis ................................................................................................................... 154 Figure 51 StressStrain Relationship for Concrete Used by LPile ........................................... 158 Figure 52 StressStrain Relationship for Reinforcing Steel Used by LPile.............................. 159 Figure 53 Element of Beam Subjected to Pure Bending .......................................................... 161 Figure 54 Validation Problem for Mechanistic Analysis of Rectangular Section.................... 165 Figure 55 Free Body Diagram Used for Computing Nominal Moment Capacity of Reinforced Concrete Section ................................................................................... 172 Figure 56 Bending Moment versus Curvature.......................................................................... 173 Figure 57 Bending Moment versus Bending Stiffness ............................................................. 174 xi
Figure 58 Interaction Diagram for Nominal Moment Capacity ............................................... 174 Figure 59 Example Pipe Section for Computation of Plastic Moment Capacity ..................... 175 Figure 510 Moment versus Curvature of Example Pipe Section ............................................. 175 Figure 511 Elastoplastic Stress Distribution Computed by LPile........................................... 177 Figure 512 StressStrain Curves of Prestressing Strands Recommended by PCI Design Handbook, 5th Edition.............................................................................................. 178 Figure 513 Sections for Prestressed Concrete Piles Modeled in LPile .................................... 180 Figure 61 Scheme for Installing Pile in a Slope Subject to Sliding.......................................... 185 Figure 62 Forces from Soil Acting Against a Pile in a Sliding Slope, (a) Pile, Slope, and Slip Surface Geometry, (b) Distribution of Mobilized Forces, (c) Freebody Diagram of Pile Below the Slip Surface.................................................................. 186 Figure 63 Influence of Stabilizing Pile on Factor of Safety Against Sliding ........................... 187 Figure 64 Matching of Computed and Assumed Values of hp ................................................. 189 Figure 65 Soil Conditions for Analysis of Slope for Low Water ............................................. 190 Figure 66 Preliminary Design of Stabilizing Piles ................................................................... 191 Figure 67 Load Distribution from Stabilizing Piles for Slope Stability Analysis .................... 192
xii
List of Tables
Table 31. Stiff Clay (no longer recommended) ......................................................................... 63 Table 32. Representative Values of
50 .......................................................................................
65
Table 33. Representative Values of k for Stiff Clays.................................................................. 71 Table 34. Representative Values of Table 35
50
for Stiff Clays............................................................... 72
ues of k for Laterally Loaded Piles in Sand ........................................................................................................................... 81
Table 36. Representative Values of k for Submerged Sand for Static and Cyclic Loading ....... 89 Table 37. Representative Values of k for Sand Above Water Table for Static and Cyclic Loading ...................................................................................................................... 89 Table 38. Results of Grout Plug Tests by Schmertmann (1977) .............................................. 115 Table 39. Values of Compressive Strength at San Francisco ................................................... 117 Table 51. LPile Output for Rectangular Concrete Section ....................................................... 166 Table 52. Comparison of Results from Hand Computation versus Computer Solution........... 173
xiii
Chapter 1 Introduction
11 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 beamcolumn 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 soilstructure 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 Winklertype mechanisms. These mechanisms can accurately predict the response of the soil and provide a means of obtaining solutions to a number of practical problems.
12 Principles of Design 121 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
122 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 closedform 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 fullscale lateral load test of a steelpipe 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 loaddeflection 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. 123 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 lowerbound values of the important parameters. 124 StepbyStep 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 py 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 pilehead 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. 125 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 firstorder, 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 sitespecific 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 py curves are obtained. The second type of test requires no internal instrumentation in the pile but only the pilehead 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 spacedclosely 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 fullscale experiments is available (and is often difficult to obtain). Some fullscale 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
13 Modeling a Pile Foundation 131 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 soilstructure 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 11(a). A simple, twospan 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 11(b). If the loading is threedimensional, the pile head at the central span will undergo three translations and three rotations. A simple matrixformulation for the pile foundation is shown in Figure 11(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 11(c) shows a curve and identifies one of the stiffness terms K32. A singlevalued 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 11 Example of Modeling a Bridge Foundation The stiffness K11 is the stiffness of the axial loadsettlement 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
132 Example Model of Individual Pile Under ThreeDimensional Loadings An interesting presentation of the forces that resist the displacement of an individual pile is shown in Figure 12 (Bryant, 1977). Figure 12(a) shows a single pile beneath a cap along with the threedimensional 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 threedimensional rotations of the cap. The various reactions of the soil along the pile are shown in Figure 12(b), and the loadtransfer curves are shown in Figure 12(c). The argument given earlier about the curve for axial displacement being singlevalue 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 twodimensional plane. The recommendations that have been made for correlating the lateral resistance with pile geometry and soil properties all depend on the results of loading in a twodimensional plane. q y
Axial
Py x
u
Px
My
Mx Axial Pile Displacement, u
z
Mz 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) Threedimensional pile displacements
(b) Pile reactions
Torsional
(c) Nonlinear loadtransfer curves
Figure 12 Threedimensional SoilPile Interaction
7
Chapter 1 Introduction
133 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 13, of pilehead 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 lateralload increment with the rotation at the pile head being restrained to zero. Next, the rotation of the pile head is computed for each bendingmoment 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 11(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 shearrotation pilehead condition, for which the user enters the lateral load V at the pile head. LPile computes pilehead 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 displacementmoment pilehead condition, for which the user enters the moment M at the pile head. LPile computes the lateral reaction force, H, and pilehead rotation using zero deflection at the pile head ( = 0).
K22 = V/
K23 = V/
and K32 = M/ .
and K33 = M/ .
Figure 13 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 13 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. 134 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.
14 Organization of Technical Manual Chapters 2 to 4 provide the user with the background information on soilpile interaction for lateral loading and present the equations that are solved when obtaining a solution for the beamcolumn 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 (py 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 reinforcedconcrete 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
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10
Chapter 2 Solution for Pile Response to Lateral Loading
21 Introduction Many pilesupported 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 soilstructureinteraction 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. 211 Influence of Pile Installation and Loading on Soil Characteristics 2111 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 stressdeformation 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 directsimpleshear 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: shortterm, 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 pseudohorizontal 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 longterm lateral deflection. A pile in normally consolidated clay, on the other hand, will experience longterm 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 pilehead 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 shortterm 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 py curves to be computed; analytical expressions are developed from the experimental results and these expressions yield py curves following section. 2112 Static Loading The static py 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 py method. From the standpoint of design, the static py curves have application in the following cases: where loadings are shortterm and not repeated (probably not encountered); and for sustained loadings, as in earthpressure 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 py 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. 2113 Repeated Cyclic Loading The fullscale 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 py 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 py curves were from cases where the groundline 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 py 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. Pilesupported structures are subjected to cyclic loading in many instances. Some common cases are wind load against overhead signs and highrise 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. 2114 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 freelydraining, 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 py 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 threedimensional, timedependent 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. 2115 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 py method described herein would not be employed. With regard to earthquakes, a rational solution should proceed from the definition of the freefield motion of the nearsurface soil due to the earthquake. Thus, the py 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 py method can be used but such solutions would plainly be approximate. 212 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. 2121 Elastic Pile and Soil The model shown in Figure 21(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 onehalf 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 21 Models of Pile Under Lateral Loading, (a) 3Dimensional Finite Element Mesh, and (b) Crosssection of 3D Finite Element Mesh,
15
Chapter 2 Solution for Pile Response to Lateral Loading
2122 Elastic Pile and Finite Elements for Soil The case shown in Figure 21(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 nonsymmetric loading or full threedimensional 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 threedimensional 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 planestress model and obtained soilresponse curves that agree well with results at or near the ground surface from fullscale experiments. The writers are aware of research that is underway with threedimensional, nonlinear, finite and boundary elements, and are of the opinion that in time such a model will lead to results that can be used in practice. More discussion on the use of the finiteelement method is presented in a later chapter where py curves are described. 2123 Rigid Pile and Plastic Soil Broms (1964a, 1964b, 1965) employed the model shown in Figure 21(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 21(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 21(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. 2124 Rigid Pile and FourSpring Model for Soil The model shown in Figure 21 (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
2125 Nonlinear Pile and py Model for Soil The model shown in Figure 22 represents the one utilized by the LPile software. The loading on the pile is general for the twodimensional case (no torsion or outofplane 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 differenceequation method is employed for the solution of the beamcolumn 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 22 Model of Pile Under Lateral Loading and py 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 py curves are nonlinear with respect to depth x along the pile and pile deflection y. The top py 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 py 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, fourthorder differential equation; and the development of electrical resistance strain gauges for use in obtaining soilresponse (py) curves from fullscale lateral load tests of piles. The py 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 py 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. 2126 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 23(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 nonuniform and will be similar to that shown in Figure 23 (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 23 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. 2127 Comments on the py method The most common criticism of the py 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 py curves for use in the analysis of piles, given in a subsequent chapter, are based for the most part on the results of fullscale experiments, lock (1970) performed some tests of a pile in soft clay where the pattern of pile deflection was varied along its length. The py curves that were derived from each of the loading conditions were essentially the same. Thus, Matlock found that experimental py 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 py curves derived from correlations to the results of fullscale experiments have been used to make computations for the response of piles where only the pilehead 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, multivalued py 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 multivalued py curves. In short, the py method has some limitations; however, there is much evidence to show that the method yields information of considerable value to an analyst and designer. 213 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 24. Figure 24 (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 24 (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 24 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 24, 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 24(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 linearelastic 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. 2131 Study of Pile Buckling A second computational problem is shown in Figure 25. A pile that extends above the ground line is subjected to a lateral load Pt and an axial load Q, as shown in Figure 25(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 25(b). A value of axial load will be approached at which the pilehead 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 44.
Q yt
Q
Pt
Buckling Load
yt (b)
(a)
Figure 25 Solution for the Axial Buckling Load 2132 Study of Critical Pile Length Another computational technique is illustrated in Figure 26. A pile is subjected to a combination of loads, as shown in Figure 26(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 26(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 26 (b). These computations and curve can be automatically performed by LPile in userselected options. The curve in Figure 26 (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 26 Solving for Critical Pile Length 214 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. 2141 Offshore Platform A bent from an offshore platform is shown in Figure 27(a). A threedimensional analysis of such a structure is sometimes necessary, but a twodimensional 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 loadtransfer 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 103 m4
4m
V V
M
d = 762 mm Ip = 3.07 x 103 m4 E = 2 x 108 kPa
(a)
(b)
(c)
Figure 27 Simplified Method of Analyzing a Pile for an Offshore Platform The second pile is shown in Figure 27(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 27(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 py 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 py 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 pilehead 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 pilehead 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. 2142 Breasting Dolphin An interesting use of a pile under lateral load is the pile uses as a foundation for a breasting dolphin. Figure 28(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 loaddeflection 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 py 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 28 Analysis of a Breasting Dolphin 2143 SinglePile Support for a Bridge A common design used for the support of a bridge is shown in Figure 29. 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 downwardsloping 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 29 Loading On a Single Shaft Supporting a Bridge Deck 2144 PileSupported Overhead Sign The sketches in Figure 210 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 210(a) shows a twopile 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
TwoShaft Foundation
(a)
SingleShaft Foundation
(b)
Figure 210 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 pilehead 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 210(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 py 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
2145 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 211 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 211 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. 2146 Anchor Pile in a Mooring System The use of a pile as the anchor for a tieback anchor is illustrated in Figure 212. 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 py 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 timerelated 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. 2147 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 highrise buildings that are subjected to forces from wind or from unbalanced earth pressures; pilesupported 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 212. The sheet piles (or tangent piles if bored piles are used) can be analyzed as a pile, if the py 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.
Tieback
Anchor Pile (Dead Man)
Sheet Pile Wall
Figure 212 Anchor Pile for a Flexible Bulkhead
22 Derivation of Differential Equation for the BeamColumn and Methods of Solution The equation for the beamcolumn must be solved for implementation of the py method, and a brief derivation is shown in the following section. An abbreviated version of the equation can be solved by a closedform 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. 221 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 beamcolumn 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 crosssections 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 213), the equilibrium of moments (ignoring secondorder terms) leads to the equation ...........................................(21) or
y x
y
Px
S
M
Vn
Vv
Vv dx
Vv+dVv y+dy
M+dM Px
x Figure 213 Element of BeamColumn (after Hetenyi, 1946) . ....................................................(22) Differentiating Equation 22 with respect to x, the following equation is obtained .................................................(23) 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 soilresponse curve. And making the indicated substitutions, Equation 23 becomes ...............................................(24) The direction of the shearing force Vv is shown in Figure 213. The shearing force in the plane normal to the deflection line can be obtained as Vn = Vv cos S
Q sin S ..................................................(25)
Because S is usually small, we may assume the small angle relationships cos S = 1 and sin S = tan S = dy/dx. Thus, Equation 26 is obtained. ..........................................................(26) Vn will mostly be used in computations, but Vv can be computed from Equation 26 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. ..............................................(27) 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: .....................................................(28)
...........................................................(29) and,
30
Chapter 2  Solution for Pile Response to Lateral Loading
.............................................................(210) 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 214. A solution of the differential equation yields a set of curves such as shown in Figure 215. 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 214 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 stressstrain properties. y
S
V
M
p
Figure 215 Form of Results Obtained for a Complete Solution 222 Solution of Reduced Form of Differential Equation A simpler form of the differential equation results from Equation 24, 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 closedform 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 211 is used, the reduced form of the differential equation is shown as Equation 212. .....................................................(211) 32
Chapter 2  Solution for Pile Response to Lateral Loading
......................................................(212) The solution to Equation 212 may be directly written as: ..........................................(213)
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 213 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 216. 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
Freehead (a)
Pt
y
FixedHead
Pt
y
Partially Restrained
(b)
(c)
Figure 216 Boundary Conditions at Top of Pile The boundary conditions at the top of the pile selected for the first case are illustrated in Figure 216(a) and in equation form are: at x = 0,
..........................................................(214)
33
Chapter 2 Solution for Pile Response to Lateral Loading
...........................................................(215) The differentiations of Equation 213 are made and the substitutions indicated by Equation 214 yield the following. .........................................................(216) The substitutions indicated by Equation 215 yield the following. .....................................................(217) Equations 216 and 217 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 218 through 222. ................................(218)
............................(219)
.................................(220) .................................(221) .............................(222) 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) ..............................................(223)
B1 = e
x
( cos x
sin x) ..............................................(224)
C1 = e
x
cos x ......................................................(225)
D1 = e
x
sin x ......................................................(226)
Using these functions, Equations 218 through 222 become:
34
Chapter 2  Solution for Pile Response to Lateral Loading
................................................(227)
...............................................(228)
.....................................................(229) V = PtB1 p =
2Mt D1 ..................................................(230)
2Pt C1 2Mt 2B1 ...............................................(231)
Values for A1, B1, C1, and D1, are shown in Figure 217 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 217 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 216(b), the solution may be obtained by employing the boundary conditions as given in Equations 232 and 233. At x = 0,
.............................................................(232)
.........................................................(233) Using the procedures as for the case where the boundary conditions were as shown in Figure 24(a), the results are as follows. .....................................................(234) The solution for long piles is given in Equations 235 through 239. ..........................................................(235)
......................................................(236)
.........................................................(237) V = Pt C1 ...........................................................(238) p = Pt A1 ..........................................................(239) 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 216(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 240 and 241. At the pile head, where x = 0, the rotational restrain is controlled by
........................................................(240)
and the pilehead shear force is controlled by ...........................................................(241)
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 242 and 243. For convenience in writing, the rotational restraint Mt /St is given the symbol k . ...................................................(242)
...................................................(243) These expressions can be substituted into Equation 213, differentiation performed as appropriate, and substitution of Equations 223 through 226 will yield a set of expressions for the long pile similar to those in Equations 227 through 231 and 235 through 239. 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)...........................................(244) and (shear force, V, is zero at pile tip).................................(245) 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 closedform solution can be used to check the accuracy of the numerical solution shown in the next section. 223 Solution by Finite Difference Equations The solution of Equation 27 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 218, the governing differential equation, Equation 27, in difference form with collected terms for y is as follows: y
ym+2 h h
ym+1 ym
h
ym1
h
ym2
x Figure 218 Representation of deflected pile
38
Chapter 2  Solution for Pile Response to Lateral Loading
..................................(246)
where Rm = EmIm (flexural rigidity of pile at point m) and km = Esm. The assumption is implicit in Equation 246 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 246; 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 246 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 .....................................................(247) where y0 denotes the lateral deflection at the bottom of the pile. Equation 247 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 zeroshear boundary condition at the bottom of the pile is ...............................(248) Equation 248 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 endbearing 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 yt1 yt2
h
Figure 219 Case 1 of Boundary Conditions
.........................(249) For the condition where the moment at the top of the pile is equal to Mt, the following difference equation is employed. .............................................(250) Case 2 of the boundary conditions at the top of the pile is illustrated graphically in Figure 220. 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 249 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
yt1 yt2
1
Figure 220 Case 2 of Boundary Conditions .......................................................(251) Case 3 of the boundary conditions at the top of the pile is illustrated in Figure 221. 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
yt1 yt2
Figure 221 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 reanalyzed. One or two iterations of this sort should be sufficient in most instances. Equation 249 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.
..............................................(252)
Case 4 of the boundary conditions at the top of the pile is illustrated in Figure 222. 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 250 can be employed. The second equation merely expresses the fact that the pilehead deflection is known. yt = Yt..............................................................(253) Foundation moves laterally
yt+2 Mt
yt+1 yt yt1
Pilehead moment is known, may be zero
h
yt2
Figure 222 Case 4 of Boundary Conditions Case 5 of the boundary conditions at the top of the pile is illustrated in Figure 223. 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 253 can be used with a known value of yt and Equation 251 can be used with a known value of St. St yt yt+2 yt+1 yt 1
yt1 yt2 St
Figure 223 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 differenceequation 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 246. The bending stiffness EI, changed to R in the difference equations, is correctly represented as a constant in the secondorder differential equation, Equation 2.9. ...........................................................(29) 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 secondorder 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 LoadTransfer Curves for Soil and Rock
31 Introduction This chapter presents the formulation of expressions for py 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 py method, and this chapter will present details for the computation of loadtransfer behavior for a pile under a variety of conditions. A typical py curve is shown in Figure 31a. The py curve is just one of a family of py curves that describe the lateralload 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 py curve. b
b
p c
a
a
d
Pile Deflection, y
y (b)
(a)
b
p
a
e
y (c)
Figure 31 Conceptual py Curves 45
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
The py curve in Figure 31(a) is meant to represent the case where a shortterm 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 31 show a straightline 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 py curves, Esi, are discussed later. Recommendations will be given in this chapter for the selection of the slope of the initial portion of py 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 31(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 ab portion of a py curve. Rather, that portion of the curves is empirical and based on results of fullscale tests of piles in a variety of soils with both monotonic and cyclic loading. The horizontal, straightline portion of the py curve in Figure 31(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 31(b) shows a shaded portion of the curve in Figure 31(a). The decreasing values of p from point c to point d reflect the effects of cyclic loading. The curves in Figures 31(a) and 31(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, longterm loading is shown in Figure 31(c). This figure shows that there is a timedependent 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 LoadTransfer Curves for Soil and Rock
32 Experimental Measurements of py Curves Methods of getting py curves from field experiments with fullsized 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 py curves from fullscale 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 py curves. 321 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 loadtransfer than double differentiation of the bending moment curve versus depth, and obtaining lateral deflection by double integration of the bending curvature curve versus depth. 322 Derivation of Soil Response from Moment Curves Obtained by Experiment Almost all successful load test experiments that have yielded py curves have measured bending moment using electricalresistance 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 LoadTransfer Curves for Soil and Rock
The experimental py 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 py curves to compute bendingmoment curves versus depth. The computed bending moments should agree closely with those measured in the load test. In addition, computed values of pilehead 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 pilehead deflection and slope over the full range of loading than for bending moment. Examples of py curves that were obtained from a fullscale 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 32 and 33 (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 32 py Curves from Static Load Test on 24inch Diameter Pile (Reese, et al. 1975)
48
Chapter 3 Lateral LoadTransfer 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 33 py Curves from Cyclic Load Tests on 24inch Diameter Pile (Reese, et al. 1975) 323 Nondimensional Methods for Obtaining Soil Response Reese and Cox (1968) described a method for obtaining py curves for cases where only pilehead 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 31 and 32. Es = k1 + k2 x, .........................................................(31) or Es = k1 xn .............................................................(32) Measurements of pilehead 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 LoadTransfer Curves for Soil and Rock
The procedure is repeated for each of the applied loadings. While the method is approximate, the py 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 py curves have involved the acquisition of experimental moment curves. However, nondimensional methods of analyses, as indicated above, have assisted in the development of py curves in some instances. In the remaining portion of this chapter, details are presented for developing py curves for clays and for sands. In addition, some discussion is presented for producing py curves for other types of soil.
33 py Curves for Cohesive Soils 331 Initial Slope of Curves The conceptual py curves in Figure 31 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 py curve. However, some difficulties are encountered in making the computations. For one thing, the value of Ei for soil is not easily determined. Stressstrain curves from unconfined compression tests were studied (see Figure 34) 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 selfboring 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 compressivestressstrain 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 py curve): E si
0.65 b
Ei b 4 Ep I p
50
1 / 12
Ei 1
2
..........................................(33)
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Ei /c 0
100
200
300
0
Manor Road Lake Austin
3
6
9
12
Figure 34 Plot of Ratio of Initial Modulus to Undrained Shear Strength for Unconfinedcompression Tests on Clay Where: b = pile diameter, Ei = initial slope of stressstrain curve of soil, Ep = modulus of elasticity of the pile, and Ip = moment of inertia of pile, respectively, and While Equation 33 may appear to provide some useful information on the initial slope of the py curves (the initial modulus of the soil in the py relationship), an examination of the initial slopes of the py curves in Figures 32 and 33 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 35 and the influence of depth below the ground surface is apparent. Yegian and Wright (1973) and Thompson (1977) conducted some numerical studies using twodimensional finite element analyses. The planestress case was employed in these studies to reflect the influence of the ground surface. Kooijman (1989) and Brown, et al. (1989) used threedimensional finite element analyses as a means of developing py 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 LoadTransfer 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 35 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 fieldload tests as a means of producing py 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 generalpurpose finite element software and continuing improvements in computing hardware are likely to increase the use of the finite element method in the future. 332 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 planestrain (flowaround) behavior of the soil.
52
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
The first analytical model for clay near the ground surface is shown in Figure 36. 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 37. The py curves for the overconsolidated clay in which the pile was tested are shown in Figures 33 and 34. As shown in Figure 37(a) for a load of 596 kN (134 kips), the groundsurface 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 37 (b).
y
Ft Ft
W
Ff
x
H
Fn Fp
Fs
b
(a)
(b)
Figure 36 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 36, will obviously not model the movement that is indicated by the contours in Figure 37; 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
.................................(34)
where = angle of the inclined plane with the vertical, and W = the weight of the wedge. An expression for W is ........................................................(35) 53
Chapter 3 Lateral LoadTransfer 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 37 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 ........................................................(36) 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 LoadTransfer Curves for Soil and Rock
........................................................(37) The frictional force between the wedge and the pile is ...........................................................(38) 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. .................(39) 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 39 becomes ............................................(310) However, Thompson (1977) differentiated Equation 39 with respect to H and evaluated the integrals numerically. His results are shown in Figure 38 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 38 is a plot of Equation 310 with the same assumption with respect to /ca. As shown, the differences in the plots are not great. The curve in Figure 38 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 39(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 LoadTransfer 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. 310
1 2 3 4
H/b 5 6 7 8 9 10
Figure 38 Ultimate Lateral Resistance for Clay Soils The MohrCoulomb diagram for undrained, saturated clay is shown in Figure 39(b) and a free body of the pile is shown in Figure 39(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 ............................................(311) Equation 311 is also shown plotted in Figure 38. 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 LoadTransfer Curves for Soil and Rock
(a)
c
2c
(b) cb/2
6b
pu
1b
cb/2
(c) Figure 39 Assumed Mode of Soil Failure Around Pile in Clay, (a) Section Through Pile, (b) MohrCoulomb Diagram, (c) Forces Acting on Section of Pile
................................................(312)
Equation 312 is also shown plotted in Figure 38 olutions is satisfactory near the ground surface, but the difference becomes significant with depth. 57
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Equations 310 and 311 are similar to Equations 320 and 321, shown later, that are used in the recommendations for two of the sets of py curves. However, the emphasis was placed directly on experimental results. The values of pu obtained in the fullscale experiments were compared to the analytical values, and empirical factors were found by which Equations 310 and 311 could be modified. The adjustment factors that were found are shown in Figure 310 (see Section 337 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 py 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 310 Values of Ac and As 333 Influence of Diameter on py 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 py 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 py 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 lateralloading tests were 58
Chapter 3 Lateral LoadTransfer 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 sitespecific 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 py curves is discussed in the next section. 334 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 shortterm loading. A dramatic example of the loss of soil resistance due to cyclic loading may be seen by comparing the two sets of py curves in Figures 32 and 33. Wang (1982) and Long (1984) did extensive studies of the influence of cyclic loading on py 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 311. 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 LoadTransfer 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 311 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 lateralload 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 pilehead 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. Fullscale 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 LoadTransfer Curves for Soil and Rock
335 Introduction to Procedures for py Curves in Clays 3351 Early Recommendations for py 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 py curves from experiments with instrumented piles. The methods yielded values of soil modulus that were employed principally with closedform 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 py curves to model the resistance of soil against lateral pile movement. Their paper is based on a fullscale experiment at an offshore site where a moderate amount of instrumentation was employed. 3352 Skempton (1951) Skempton (1951) stated develop a prediction model for loadsettlement curves. The theory can be also used to obtain py 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 stressstrain behavior of the soil is isotopic. The mean settlement, , of a foundation of width b on the surface of a semiinfinite elastic solid is given by Equation 313. ......................................................(313) where: q = foundation pressure, I = influence coefficient, E In Equation 313, 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 ..........................................................(314) E qf
61
Chapter 3 Lateral LoadTransfer 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 314 is valid at any depth. In an undrained compression test, the axial strain is given by ....................................................(315) Where E is Yo
principal stress difference of (
1
3).
For saturated clays with no change in water content, Equation 315 may be rewritten as .................................................... (316)
Where
is the principal stress difference at failure.
Equations 314 and 316 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 .......................................................... (317)
fullscale foundations, led to the following conclusion: assumptions, it may be taken that Equation 317 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 py curve could be obtained, then, by taking points from a laboratory stressstrain curve and using Equation 317 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 stressstrain 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 stressstrain curve, allows an approximate curve to be developed if only the strength of the soil is available.
62
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
3353 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 py 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 py 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 onehalf of the maximum lateral loadtransfer capacity. Table 31 changed to reflect current practice. These values of consistent with theory for small deflections.
T
are independent of pile diameter, which is
Table 31. for Laterally Loaded Piles in Stiff Clay (no longer recommended) Consistency of Clay
Stiff
Very Stiff
Hard
qu, kPa
100200
200400
> 400
qu, tsf
12
24
>4
3.26.4
6.412.8
> 12.8
460925
9251,850
> 1,850
Soil Modulus, Soil Modulus,
T,
MPa
T,
psi
3354 McClelland and Focht (1956) McClelland and Focht (1956) wrote the first paper that described the concept of nonlinear lateral loadtransfer curves, now referred to as py curves. In this paper, they presented the first nonlinear py curves derived from a fullscale, instrumented, pileload 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 consolidatedundrained 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 pvalues of the py curve are then scaled from the stressstrain curve using ........................................................ (318)
63
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
and the values of pile deflection y are scaled using .......................................................... (319) 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). 336 Procedures for Computing py Curves in Clay Five procedures are provided for computing py curves for clay. Each procedure is based on the analysis of the results of experiments using fullscale 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 py curves were obtained by one or more of the techniques described earlier. EulerBernoulli beam theory was used and mathematical expressions were developed for py 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 shortterm (static) and cyclic. The py 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. 337 Response of Soft Clay in the Presence of Free Water 3371 Description of Load Test Program Matlock (1970) performed lateralload tests with an instrumented steelpipe 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 shortterm. 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 150mmdiameter 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, redriven, 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 LoadTransfer 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 py 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 py curves are used and if there are only a small number of applications of the design load (the factored load). 3372 Procedure for Computing py Curves in Soft Clay for Static Loading The following procedure is for shortterm 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 onehalf the maximum principal stress difference. If no stressstrain curves are available, typical values of 50 are given in Table 32. Table 32. 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. .............................................. (320)
.......................................................... (321) where = average effective unit weight from ground surface to py curve,1 x = depth from the ground surface to py 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 LoadTransfer 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 312 py 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 py curve is desired, based on shear strength at that depth. Equations 320 and 321 are solved simultaneously to find the transition depth, xr, where the transition in definition of pu by Equation 320 to 321 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 LoadTransfer Curves for Soil and Rock
.................................................. (322) 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 py curve with variable J (API soft clay with userdefined 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 py curve. An example of the effect of J on a py curve at a depth of 5 feet for a 36inch diameter pile in soft clay with c = 1,000 psf and = 55 pcf is shown in Figure 313. 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 313 Example py Curves in Soft Clay Showing Effect of J 3.
Compute deflection at onehalf the ultimate soil resistance, y50, from the following equation: y50 = 2.5
4.
50b
....................................................... (323)
Compute points describing the py curve from the origin up to 8 y50 using ...................................................... (324)
67
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
The value of p remains constant for y values beyond 8 y50. 3373 Procedure for Computing py Curves in Soft Clay for Cyclic Loading The following procedure is for cyclic loading and is illustrated in Figure 312(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 py curves for cyclic loading can be used with the recognition that there may be some conservatism in the results. 1.
Construct the py curve in the same manner as for shortterm static loading for values of p less than 0.72pu. For lateral displacements in this range, there is not significant degradation of the py curve during cyclic loading.
2.
If the depth to the py curve is greater than or equal to xr (Equation 322), select p as 0.72pu for y equal to 3y50 (Note that the number 0.72 is computed using Equation 324 as 1/2 * 31/3 = 0.721124785 ~ 0.72).
3.
If the depth of the py 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 325 at y = 15y50. ..................................................... (325) The value of p remains constant beyond y = 15y50.
3374 Recommended Soil Tests for Soft Clays For determining the various shear strengths of the soil required in the py construction, Matlock (1970) recommended the following tests in order of preference. 1. Insitu vaneshear tests with parallel sampling for soil identification, 2. Unconsolidatedundrained triaxial compression tests having a confining stress equal to the overburden pressure with c being defined as onehalf 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. 3375 Examples An example set of py 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 314. The submerged unit weight was 6.3 kN/m3 (40 pcf). In the absence of a stressstrain curve for the soil, 50 was taken as 0.02 68
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
for the full depth of the soil profile. The loading was assumed to be static. The py 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 315. 0 2 4 6 8 10 12 14 16 0
10
20
30
40
50
Shear Strength, kPa
Figure 314 Shear Strength Profile Used for Example py 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 315 Example py Curves for Soft Clay with the Presence of Free Water
69
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
338 Response of Stiff Clay in the Presence of Free Water Reese, Cox, and Koop (1975) performed lateralload tests with steelpipe 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 dataacquisition 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 py 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 py curves for overconsolidated clays. 3381 Procedure for Computing py Curves for Static Loading The following procedure is for computing py curves in stiff clay with free water for shortterm static loading and is illustrated by Figure 316. 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 326 and 327. ............................................. (326)
70
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
p
0.5pc
0
y50
6y50
As y50
y
18y50
Figure 316 Characteristic Shape of py Curves for Static Loading in Stiff Clay with Free Water ........................................................ (327) 4.
Choose the appropriate value of As from Figure 310 on page 58 for modifying pct and pcd and for shaping the py curves or compute As using ............................................ (328)
5.
Establish the initial linear portion of the py curve, using the appropriate value of ks for static loading or kc for cyclic loading from Table 33 for k. p = (kx) y.......................................................... (329) Table 33. Representative Values of k for Stiff Clays Average Undrained Shear Strength* 50100 kPa 1,0002,000 psf
100200 kPa 2,0004,000 psf
200400 kPa 4,0006,000 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 onehalf 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 LoadTransfer Curves for Soil and Rock
.......................................................... (330) Using an appropriate value of laboratory tests, from Table 34.
50
from results of laboratory tests or, in the absence of
Table 34. Representative Values of
50
for Stiff Clays
Average Undrained Shear Strength
7.
50
50100 kPa
1,0002,000 psf
0.007
100200 kPa
2,0004,000 psf
0.005
200400 kPa
4,0006,000 psf
0.004
Establish the first parabolic portion of the py curve, using the following equation and obtaining pc from Equations 326 or 327. .................................................... (331) Equation 331 should define the portion of the py curve from the point of the intersection with Equation 329 to a point where y is equal to Asy50 (see note in Step 10).
8.
Establish the second parabolic portion of the py curve, ............................... (332) Equation 332 should define the portion of the py 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 straightline portion of the py curve, p
0.5 pc 6 As
0.411 pc
0.0625 p c y 6 As y 50 ........................ (333) y 50
Equation 333 should define the portion of the py 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 straightline portion of the py curve, ................................... (334) or ...................................... (335) 72
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Equation 334 should define the portion of the py curve from the point where y is equal to 18Asy50 and for all larger values of y, see the following note. Note: The py curve shown in Figure 316 is drawn, as if there is an intersection between Equation 329 and 331. However, for small values of k there may be no intersection of Equation 329 with any of the other equations defining the py curve. Equation 329 defines the py curve until it intersects with one of the other equations or, if no intersection occurs, Equation 329 defines the full py curve. 3382 Procedure for Computing py 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 317. As may be seen from a study of the py 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 py 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 317 Characteristic Shape of py 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 LoadTransfer 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 326 and 327. 4. Choose the appropriate value of Ac from Figure 310 on page 58 or compute Ac using ............................................. (336) 5. Compute yp using ....................................................... (337) 6. Establish the initial linear portion of the py curve, using the appropriate value of ks for static loading or kc for cyclic loading from Table 33 for k. and compute p using Equation 329. 7. Compute y50 using Equation 330. 8. Establish the parabolic portion of the py curve,
........................................... (338)
Equation 338 should define the portion of the py curve from the point of the intersection with Equation 329 to where y is equal to 0.6yp (see note in step 9). 8. Establish the next straightline portion of the py curve, ................................... (339) Equation 339 should define the portion of the py 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 straightline portion of the py curve, ........................................... (340) Equation 340 defines the py curve from the point where y equals 1.8yp and all larger values of y (see following note). Note: Figure 317 is drawn, as if there is an intersection between Equation 329 and Equation 338. There may be no intersection of Equation 329 with any of the other equations defining the py curve. If there is no intersection, the equation should be employed that gives the smallest value of p for any value of y. 3383 Recommended Soil Tests Triaxial compression tests of the unconsolidatedundrained type with confining pressures conforming to in situ pressures are recommended for determining the shear strength of the soil. 74
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
The value of 50 should be taken as the strain during the test corresponding to the stress equal to onehalf the maximum totalprincipalstress difference. The shear strength, c, should be interpreted as onehalf of the maximum totalprincipalstress 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. 3384 Examples Example py 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 318. The submerged unit weight of the soil was 7.9 kN/m3 (50 pcf) over the full depth. In the absence of a stressstrain curve, 50 was taken as 0.005 for the full depth of the soil profile. The slope of the initial portion of the py curve was established by assuming a value of k of 135 MN/m3 (500 pci). The loading was assumed to be cyclic. The py 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 319. 339 Response of Stiff Clay with No Free Water A lateralload test was performed at a site in Houston, Texas on a drilled shaft (bored pile), with a diameter of 915 mm (36 in.). A 254mm (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 318 Example Shear Strength Profile for py Curves for Stiff Clay with No Free Water
75
Chapter 3 Lateral LoadTransfer 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 319 Example py 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 py 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 highspeed dataacquisition 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 py 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. 3391 Procedure for Computing py Curves for Stiff Clay without Free Water for Static Loading The following procedure is for shortterm static loading and the py curve for stiff clay without free water is illustrated in Figure 320.
76
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
p
y 16y50 Figure 320 Characteristic Shape of py 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 stressstrain curves. If no stressstrain curves are available, use a value of 50 of 0.010 or 0.005 as given in Table 32, 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 320 and 321. (In the use of Equation 320, 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.) ...............................................(320)
...........................................................(321) 3.
Compute the deflection, y50, at onehalf the ultimate soil resistance from Equation 323. y50 = 2.5
4.
(323)
Compute points describing the py curve from the relationship below. p
5.
50b ........................................................
pu y 2 y50
0.25
..................................................... (341)
Beyond y = 16y50, p is equal to pu for all values of y.
77
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
3392 Procedure for Computing py Curves for Stiff Clay without Free Water for Cyclic Loading The following procedure is for cyclic loading and the py curve for stiff clay without free water is illustrated in Figure 321.
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 321 Characteristic Shape of py Curves for Cyclic Loading in Stiff Clay with No Free Water 1.
Determine the py curve for shortterm 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 ....................................................... (342)
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 ................................................. (343) where yc = deflection under Ncycles of load, ys = deflection under shortterm static load, y50 = deflection under shortterm static load at onehalf the ultimate resistance, and 78
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
N = number of cycles of load application. 5.
The py curve defines the soil response after Ncycles of loading.
3393 Recommended Soil Tests for Stiff Clays Triaxial compression tests of the unconsolidatedundrained 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 onehalf the maximum totalprincipalstress difference. The undrained shear strength, c, should be defined as onehalf the maximum totalprincipalstress difference. The unit weight of the soil must also be determined. 3394 Examples An example set of py 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 stressstrain curve, 50 was taken as 0.005. Equation 342 was used to compute values for the parameter C and it was assumed that there were to be 100 cycles of loading. The py 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 322 Example py Curves for Stiff Clay with No Free Water, Cyclic Loading
79
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
3310 Modified py Criteria for Stiff Clay with No Free Water The py criteria for stiff clay with no free water were described in Section 339. The py curve for stiff clay with no free water is based on Equation 341, 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 loaddeformation response is modeled too stiffly. The ultimate loadtransfer resistance pu used in the py 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 py curves if one has the results of lateral load test for local calibration of the initial stiffness k. Judicious use of this modified py 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 33 or from a sitespecific lateral load test. LPile will use the lower of the values computed using Equation 329 or Equation 341 for pile response as a function of lateral pile displacement. 3311 Other Recommendations for py Curves in Clays As noted earlier in this chapter, the selection of the set of py 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 py curves. Bhushan, et al. (1979) described field tests on drilled shafts under lateral load and recommended procedures for formulating py curves for stiff clays. Briaud, et al. (1982) suggested a procedure for use of the pressuremeter in developing py curves. A number of other authors have also presented proposals for the use of results of pressuremeter tests for obtaining py curves. and Gazioglu (1984) reviewed all of the data that were available on py curves for clay and presented a summary report to the American Petroleum Institute. The research conducted by and his coworkers ( 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 py curves. This work will most likely result in specific recommendations in due course.
80
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
34 py Curves for Sands 341 Description of py Curves in Sands 3411 Initial Portion of Curves The initial stiffness of stressstrain 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 py 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 py 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 onehalf of the ultimate bearing stress. In terms of py curves, Terzaghi recommends a series of straight lines with slopes that increase linearly with depth, as indicated in Equation 344. Es = kx............................................................ (344) 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 35. 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 35. Table 35 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 LoadTransfer Curves for Soil and Rock
3412 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 323. The total lateral force Fpt (Figure 323(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 MohrCoulomb failure condition is satisfied on planes, ADE, BCF, and AEFB (Figure 323(a)). The directions of the forces are shown in Figure 323(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 323 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 36). The resulting equation is
............... (345)
where: = the angle of the wedge in the horizontal direction
82
Chapter 3 Lateral LoadTransfer 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 345 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
................ (346)
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. ........................................................ (347) The model for computing the ultimate soil resistance at some distance below the ground surface is shown in Figure 324(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 twodimensional behavior; thus, it is subject to some uncertainty. If the states of stress shown in Figure 324(b) are assumed, the ultimate soil resistance for horizontal movement of the soil is ............................ (348) 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. 3413 Influence of Diameter on py Curves No studies have been reported on the influence of pile diameter on py 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 lateralload tests, except the ones described herein, have used only static loading.
83
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
(a)
(b) Figure 324 Assumed Mode of Soil Failure by Lateral Flow Around Pile in Sand, (a) Section Though Pile, (b) MohrCoulomb Diagram 3414 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 LoadTransfer 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. Fullscale 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. 3415 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 smallscale experiments, examined unpublished data, and recommended procedures for predicting py 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 fullsized piles and became available shortly afterward. 3416 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 steelpipe piles, 610 mm (24 in.) in diameter, were driven into sand in a manner to simulate the driving of an openended 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 shortterm 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. 3417 Response of Sand Above and Below the Water Table The procedure for developing py 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. 342 Response of Sand The following procedure is for both shortterm static loading and for cyclic loading for a flat ground surface and a vertical pile. The procedure is illustrated in Figure 325 (Reese, et al., 1974).
85
Chapter 3 Lateral LoadTransfer 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 325 Characteristic Shape of a Set of py Curves for Static and Cyclic Loading in Sand 3421 Procedure for Computing py Curves in Sand 1.
Obtain values for the depth of the py 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
..................... (349)
Compute the ultimate soil resistance per unit length of pile using the smaller of the values given by , where
.................... (350)
............................. (351)
86
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
4.
Establish Compute pu using: or
............................................... (352)
Use the appropriate value of or from Figure 326 for the particular nondimensional depth, and for either the static or cyclic case. Use the appropriate equation for ps, Equation 350 or Equation 351 by referring to the computation in Step 4. 5. Compute ym using .......................................................... (353) Compute pm by the following equation: ............................................... (354) Use the appropriate value of Bs or Bc from Figure 327 as a function of the nondimensional depth, and for either the static or cyclic case. Use the appropriate equation for ps. The two straightline portions of the py curve, beyond the point where y is equal to b/60, can now be established. 6.
Establish the initial straightline portion of the py curve, p = (k x) y ......................................................... (355) Use the appropriate value of k from Table 36 or 37.
87
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
2
1
0
3
0
1
2
3
4
5
6
Figure 326 Values of Coefficients
and
B 2
1
0
3
0
1
Bs (static) Bc (cyclic)
2
3
4
5
6
Figure 327 Values of Coefficients Bc and Bs
88
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Table 36. 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 37. 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 331 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 py curve, .......................................................... (356) 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, ........................................................ (357) b. Obtain the power of the parabolic section by, ........................................................... (358) c. Obtain the coefficient
as follows: ........................................................... (359)
d. Determine point the pile deflection at point k as
89
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
........................................................ (360) e. Compute appropriate number of points on the parabola by using Equation 356. Note: The curve in Figure 325 is drawn as if there is an intersection between the initial straightline portion of the py curve and the parabolic portion of the curve at point k. However, in some instances there may be no intersection with the parabola. Equation 355 defines the py curve until there is an intersection with another portion of the py curve or if no intersection occurs, Equation 355 defines the complete py curve. If yk is in between points ym and yu, the curve is trilinear and if yk is greater than yu, the curve is bilinear as shown in Figure 328. 3422 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 insitu 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 328 Illustration of Effect of k on py Curve in Sand
90
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
3423 Example Curves An example set of py 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 py 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 329. 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 329 Example py Curves for Sand Below the Water Table, Static Loading 343 API RP 2A Recommendation for Response of Sand Above and Below the Water Table 3431 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 py 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 LoadTransfer Curves for Soil and Rock
3432 Procedure for Computing py Curves Using the API Sand Method The following procedure is for both shortterm 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 361 to a value at deep depths determined by Equation 362. 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: ................................................... (361) ........................................................ (362) 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 330 as a function of , or
where
and
92
Chapter 3 Lateral LoadTransfer 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 330 Coefficients C1, C2, and C3 versus Angle of Internal Friction b = average pile diameter from surface to depth, in. (m). 3.
Compute the loaddeflection curve based on the ultimate soil resistance pu which is the minimum value of pu calculated in Step 2. The lateral soil resistancedeflection (py) 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: ................................................ (363) 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 361 or 362, lb./in. (kN/m), 93
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
k = initial modulus of subgrade reaction, pci (kN/m3). Determine k from Figure 331 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 331 Value of k, Used for API Sand Criteria 3433 Example Curves An example set of py curves was computed for sand above the water table, using the API criteria. The soil properties are unit weight = 0.07 pci, and internalfriction 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 py curves are computed for the following depths: 20 in., 40 in., and 100 inches.
94
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
A hand calculation for py 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 330. 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 331) 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 332.
95
Chapter 3 Lateral LoadTransfer 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 332 Example py Curves for API Sand Criteria 344 Other Recommendations for py Curves in Sand A survey of the available information of py 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 py curves was suggested. A number of authors have discussed the use of the pressuremeter in obtaining py curves. The method that is proposed is described in some detail by Baguelin, et al. (1978) .
35 py Curves in Liquefied Sands 351 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 py curves for nonliquefied and, considerable uncertainty remains regarding how much lateral loadtransfer 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 pmultiplier values or by entering a very low friction angle for sand.
96
Chapter 3 Lateral LoadTransfer 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 py 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, fullscale field tests are needed to develop a full range of py 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 py curves developed on the basis of these studies have a concave upward shape, as shown in Figure 333. This characteristic shape appears to result primarily from dilative behavior during shearing, although gapping effects may also contribute to the observed loadtransfer response. Rollins and his coworkers also found that py 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 333 Example py Curve in Liquefied Sand
97
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Following liquefaction, py curves in sand become progressively stiffer with the passage of time as excess pore water pressures dissipate. The shape of a py 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. 352 Procedure for Computing py Curves in Liquefied Sand The expression developed by Rollins et al. (2005a) for py curves in liquefied sands at different depths is shown below is based on their fullyinstrumented 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 py 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. ........................................................(364) ...................................................(365) .....................................................(366) .....................................................(367) 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 364, as shown below. ...................................................(368) where b is the diameter or width of the pile or drilled shaft in meters. The py curves for liquefied sand can be multiplied by Pd to obtain values for py 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 364 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 LoadTransfer 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. 353 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 freefield 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 43.
36 py Curves in Loess 361 Background A procedure was formulated by Johnson, et al. (2006) for loess soil that includes degradation of the py curves by load cycling. The soil strength parameter used in the model is the cone tip resistance (qc) from cone penetration (CPT) testing. The py 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 py curves were obtained from backfitting of lateral analyses using the computer program LPile to the results of the load tests. 3611 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 30inch diameter loaded statically, one pair of 42inch diameter test shafts loaded statically, and one pair of 30inch 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 30inch 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 30inch and 42inch diameter static test shafts in approximately 10kip and 15kip increments, respectively.
99
Chapter 3 Lateral LoadTransfer 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. 3612 Soil Profile from Cone Penetration Testing A backfit model of the pile behavior using the available soil strength data obtained (from both insitu 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 CPT1). Two additional cone penetration tests were performed subsequent to the lateral load testing. A cone penetration test was performed between the 42inch 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 30inch 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 334. 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 334. 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 LoadTransfer Curves for Soil and Rock
Reduced by 50% at surface 0
2D = 5 ft for 30inch Diam. Shafts
5
2D = 7 ft for 42inch 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) CPT1 (8/12/2004)
35
40 0
20
40
60
80
100
120
140
160
180
200
qc, ksf Figure 334 Idealized Tip Resistance Profile from CPT Testing Used for Analyses. 362 Procedure for Computing py Curves in Loess 3621 General Description of py Curves in Loess Procedures are provided to produce a py 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 py 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 py 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 py 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 py curve that is smooth and continuous. This model is similar to the lateral behavior of pile in loess soil measured in load tests. 3622 Equations of py 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 ........................................................(369)
101
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
p pu Ei
Es
y
yref
Figure 335. Generic py 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 insitu testing. The value of NCPT derived from the load test data is ........................................................(370) 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. ....................................................(371) 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 LoadTransfer Curves for Soil and Rock
CN was determined from a best fit of cyclic degradation for two 30inch diameter test shafts subjected to cyclic loading. CN is ...........................................................(372) The cyclic degradation term (the denominator of Equation 371) 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 py curve (i.e., a greater value of CN will allow greater degradation of the py 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 py 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 py curve at zero displacement intersects the ultimate soil resistance asymptote (pu), as shown in Figure 335. The best fit to the load test data was obtained with the following value for reference displacement. yref = 0.117 inches = 0.0029718 meters .................................. (373) 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, reevaluation 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. ........................................................... (374) 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). ......................................................... (375)
.............................................. (376)
103
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
............................................................(377) 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 py curve (i.e., a larger value of a will reduce the mobilization of soil resistance with displacement). Combining the two equations above, one obtains
.............................................(378)
The modulus ratio (secant modulus over initial modulus, Es/Ei) versus displacement used for py curves in loess is shown in Figure 336. 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 336 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 371. 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 LoadTransfer Curves for Soil and Rock
........................................................... (379) where: Es is the secant modulus in units of force/length2, and y is the lateral pile displacement. Several py curves obtained from the model described above is presented in Figure 337 for the 30inch diameter shafts, and Figure 338 for the 42inch 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 334). These py 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 337 py Curves for the 30inch 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 338 py Curves and Secant Modulus for the 42inch Diameter Shafts. 105
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
The static py curves shown in Figure 337 and 338 were degraded with load cycle number (N) for use in the cyclic load analyses. Figure 339 presents the cyclic py curve generated for the analyses of the 30inch 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 339 Cyclic Degradation of py Curves for 30inch Shafts 3623 StepbyStep Procedure for Generating py Curves A stepbystep procedure to generate py 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 369 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 371. 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 374. 7. Select a number of lateral pile displacements (y) for which a representative py curve is to be generated. 8. Determine the secant modulus (Es) for each of the displacements selected in Step 7 in accordance with Equations 375 and 376. 106
Chapter 3 Lateral LoadTransfer 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 379. 3624 Limitations on Conditions for Validity of Model The py curve for static loading was based on best fits of data from full scale load tests on 30inch and 42inch 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 30inch 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.
37 py Curves in Soils with Both Cohesion and Internal Friction 371 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 py curves for c soils. Among the reasons for the limitation on soil characteristics are the following. Firstly, in foundation design, where the py 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 py 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 py 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 longterm 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 py 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 LoadTransfer Curves for Soil and Rock
The strength envelope for consolidateddrained clay is represented by components of both c and . Therefore, a py method for c soils is needed for drained analysis. A complication for such an analysis is that there will be some timedependent lateral deflection of the pile as drainage occurs. 372 Recommendations for Computing py Curves The following procedure is for shortterm static loading and for cyclic loading and is illustrated in Figure 340. 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 340 Characteristic Shape of py 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.................................................... (380)
where p
= passive pressure including the threedimensional effect of the passive wedge (F/L2)
b = pile width (L), The Rankine passive pressure for a wall of infinite length (F/L2), ................................ (381) = unit weight of soil (F/L3), 108
Chapter 3 Lateral LoadTransfer 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 threedimensional effect of the passive wedge. The modifying factor Cp can be divided into two terms: Cp to modify the frictional term of Equation 380 and Cpc to modify the cohesion term of Equation 380. Equation 382 can then be written as: ........................... (382) The derivation of equations for developing py curves for c soil is based on a concept proposed by Evans and Duncan (1982). Equation 382 will be rewritten as ..................................................... (383) where can be found from Figure 326. The friction component (pu ) will be the smaller of the values given by the two equations below.
.................... (384)
.............................. (385) The cohesion component (puc) will be the smaller of the two equations below. .............................................. (386) .......................................................... (387) 373 Procedure for Computing py Curves in Soils with Both Cohesion and Internal Friction To develop the py curves, the procedures described earlier for sand by Reese et al (1974) will be used because the stressstrain 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 py curves for soils with both cohesion and internal friction.
109
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
1.
Compute yu by the following equation: ............................................................(388)
2.
Compute pu for static or cyclic loading by the following equation: or
.................................... (389)
Use the appropriate value of or from Figure 326 on page 88 for the particular nondimensional depth and for static or cyclic loading. 2.
Compute ym as ........................................................... (390) Compute pm by the following equation: or
.............................................. (391)
Use the appropriate value of Bs or Bc from Figure 327 on page 88 for the particular nondimensional depth, and for either the static or cyclic case. Use the appropriate equation for ps. The two straightline portions of the py curve, beyond the point where y is equal to b/60, can now be established. 3.
Establish the initial straightline portion of the py curve, p = (k x) y .......................................................... (392) The value of k for Equation 392 may be found from the following equation and by reference to Figure 341. k = (kc + k ) ......................................................... (393) 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 py curve, .......................................................... (394) 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 LoadTransfer Curves for Soil and Rock
........................................................ (395)
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 341 Representative Values of k for c Soil b. Obtain the power of the parabolic section by, ........................................................... (396) c. Obtain the coefficient
as follows: ........................................................ (397)
d. Determine point k as,
....................................................... (398) e. Compute appropriate number of points on the parabola by using Equation 394.
111
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Note: The stepbystep procedure is outlined as if there is an intersection between the initial straightline portion of the py 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 py curve until there is an intersection with another branch of the py curve or if no intersection occurs, Equation 391 defines the complete py curve. This completes the development of the py curve for the desired depth. Any number of curves can be developed by repeating the above steps for each desired depth. 374 Discussion An example of py 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 py curves were computed for depths of 39 in. (1 m), 79 in. (2 m), and 118 inches (3 meters). The py curves computed by using the simplified procedure are shown in Figure 342. 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 py 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 342 py 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 LoadTransfer 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 freewater 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 py 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.
38 Response of Vuggy Limestone Rock 381 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 groundline 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 soilfilled joints or cracks in the rock, the procedures suggested herein should not be used but fullscale testing at the site is recommended. 113
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Furthermore, fullscale 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 py 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 py 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, fullscale load testing is recommended to verify foundation performance and to evaluate the efficiency of proposed construction methods. 382 Descriptions of Two Field Experiments 3821 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 onehalf 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 LoadTransfer Curves for Soil and Rock
The rock at the site was also investigated by insitugroutplug tests (Schmertmann, 1977). In these tests, a 140mm (5.5 in.) hole was drilled into the limestone, a highstrength 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 38 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 38. Results of Grout Plug Tests by Schmertmann (1977) Depth Range meters
0.761.52
2.443.05
feet
2.55.0
8.010.0
5.496.10 18.020.0
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. 3822 San Francisco, California The California Department of Transportation (Caltrans) performed lateralload 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 98mm (3.88in.) tricone 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 LoadTransfer 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 343. 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 344 and the correlation of rock strength and modulus shown in Figure 345 were employed in developing the correlation between the initial stiffness from Figure 343 and the compressive strength, and the values were obtained as shown in Table 39. 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 highstrength 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 pilehead 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 loadingtest 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 343 Initial Moduli of Rock Measured by Pressuremeter for San Francisco Load Test
116
Chapter 3 Lateral LoadTransfer 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 344 Modulus Reduction Ratio versus RQD (Bienawski, 1984)
Table 39. 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 LoadTransfer 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 345 Engineering Properties for Intact Rocks (after Deere, 1968; Peck, 1976; and Horvath and Kenney, 1979)
118
1,000
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
383 Procedure for Computing py Curves for Strong Rock (Vuggy Limestone) The py curve recommended for strong rock (vuggy limestone), with compressive strength of intact specimens larger than 6.9 MPa (1,000 psi), shown in Figure 346. 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 346, 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 py curve shown in Figure 346 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 346 Characteristic Shape of py Curve in Strong Rock 384 Procedure for Computing py Curves for Weak Rock The py curve that is recommended for weak rock is shown in Figure 347. 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 LoadTransfer Curves for Soil and Rock
p Mir
pur
y
yA Figure 347 Sketch of py Curve for Weak Rock (after Reese, 1997) for xr
3b ...................................... (399) ........................................... (3100)
where: qur = compressive strength of the rock, usually lowerbound 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 onethird 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 345, by entering with the value of the pressuremeter modulus. ................................................ (3101)
120
Chapter 3 Lateral LoadTransfer 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 347 may be shown to have the following value (using the symbols for rock). 2 Mir
kir Eir ..................................................... (3102)
where Eir = the initial modulus of the rock, and kir = dimensionless constant defined by Equation 3103. Equations 3102 and 3103 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. .................................... (3103) kir = 500 for xr > 3b................................................ (3104) With guidelines for computing pur and Mir, the equations for the three branches of the family of py curves for rock in Figure 346 can be presented. The equation for the straightline, initial portion of the curves is given by Equation 3105 and for the other branches by Equations 3106 through 3108. for for
...............................................(3105) ...............................(3106)
for y > 16yrm ................................................(3107) yrm =
rm
b.........................................................(3108)
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 3108. rm is analogous to 50 used for py curves in clays. The stressstrain 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 3105 and 3106, and the solution is presented in Equation 3109.
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 LoadTransfer Curves for Soil and Rock
.............................................(3109) 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 385 Case Histories for Drilled Shafts in Weak Rock 3851 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 py 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 kNm2 (1.3
109 ksi).
A comparison of pilehead deflection curves from experiment and from analysis is shown in Figure 348. 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 loaddeflection 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), onehalf 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 LoadTransfer Curves for Soil and Rock
Figure 348 Comparison of Experimental and Computed Values of PileHead 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 349. 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 kNm2; 400 kN, 1.24 106 kNm2; 467 kN, 9.33 105 kNm2; 534 kN, 7.46 105 kNm2; 601 kN, 6.23 105 kNm2; and 667 kN, 5.36 105 kNm2. 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. 3852 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 kNm2 (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 mkN (1.57 105 inkips) 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 LoadTransfer Curves for Soil and Rock
Bending Moment, M, kNm 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 349 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 pilehead deflection at the largest load. The experimental curve for Pile B shown by the heavy solid line in Figure 350 suggests that a plastic hinge developed at the ultimate bending moment of 17,740 mkN (157,012 inkips). 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 trialanderror computations that match computed and observed deflections. The plots of the three methods are shown in Figure 351 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 pilehead deflection curves are shown in Figure 350. The computed loaddeflection curve computed using EI values derived from the load test agrees well with the load test curve, but the computed loaddeflection 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 LoadTransfer 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 350 Comparison of Experimental and Computed Values of PileHead Deflection for Different Values of EI, San Francisco Test Also shown in Figure 350 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 351 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 352. 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.
39 py Curves in Massive Rock Liang, Yang, and Nasairat (2009) developed a criterion for computing py curves for drilled shafts in massive rock. This criterion is based on both fullscale load tests and threedimensional finite element modeling.
125
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
40 Analytical Experimental ACI
30
20
10
0 5,000
0
10,000
15,000
20,000
Bending Moment, kNm
Figure 351 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, kNm
Figure 352 Comparison of Experimental and Computed Values of Maximum Bending Moments for Different Values of EI, San Francisco Test
126
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
A hyperbolic equation is used as the basis for the py relationship. .......................................................(3110)
where pu is the ultimate lateral resistance of the rock mass and Ki is the initial slope of the py curve. A drawing of the py curve for massive rock is presented in Figure 353.
p pu
Ki
y Figure 353 py Curve in Massive Rock 391 Determination of pu Near Ground Surface For a passive wedge type failure near the ground surface, as shown in Figure 354, the ultimate lateral resistance per unit length, pu of the drilled shaft at depth H is .............................(3111) 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 LoadTransfer Curves for Soil and Rock
y
F
H
Fs Fp Fn
W
D
Figure 354 Model of Passive Wedge for Drilled Shafts in Rock
, and , with the condition that
128
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Equation 3111 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. 392 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 threedimensional 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 ...........................................(3112) where pa is the active horizontal active earth pressure given by with the condition that
........................(3113)
= 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 rockshaft interface, proposed by Kulhawy and Phoon (1993) V
.....................................................(3114) where both
max
and
ci
are in units of megapascals.
393 Initial Tangent to py Curve Ki ........................................(3115) 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 394 Rock Mass Properties The shearing properties of the rock mass, c and strength criterion for rock mass.
, are defined using the HoekBrown
............................................(3116)
129
Chapter 3 Lateral LoadTransfer 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 MohrCoulomb failure parameters c of the rock mass from the principal stresses at failure. These parameters are: .............................................(3117)
....................................................(3118) 1
can be found from Equation 3116, and
n
and are found from ........................................(3119)
...........................................(3120) 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 3121 ..................................................(3121) The modulus reduction ratio and is shown as a function of GSI in Figure 355. The second method recommended for determining rock mass modulus is to perform an insitu 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/PublishedPapers.htm.
130
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
100 Bieniawski (1978) Serafin and Pereira (1983) IrontonRussell Regression Line
80
60
40
20
0
0
20
40
60
80
100
Geologic Strength Index Figure 355 Equation for Estimating Modulus Reduction Ratio from Geological Strength Index 395 Procedure for Computing py 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 3121 if pressuremeter data are unavailable. If Equation 3121 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 3115. 6. Compute pu at shallow depth using Equation 3111 with 3 equal to the vertical effective stress at H/3 when computing the values of and c using Equations 3117 and 3118. 7. Compute pu at great depth using Equation 3112 with pL taken as Equation 3116 and equating 3 equal to v.
1
computed using
8. Compute pu as the smaller of the values computed by Equations 3111 and 3112. 9. The values of the py curve can then be computed using 3110 for selected values of pile movement y. 131
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
310 py 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 inplace 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 py 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 ......................................................(3122) 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..........................(3123)
.................................................(3124)
132
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Figure 356 Degradation Plot for Es
pu
y 0.001b
0.0375b
Figure 357 py Curve for Piedmont Residual Soil where
= 0.23, which gives
311 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 LoadTransfer 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 py 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. 3111 Layering Correction Method of Georgiadis The method of Georgiadis (1983) is ba of all the layers existing below the upper layer. The py curves of the upper layer are determined according to the methods for homogeneous soils. To compute the py 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. .................................................... (3125)
and ......................................................(3126)
The equivalent thickness H2 of the upper layer along with the soil properties of the second layer, are used to compute the py 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 py 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 358. 3112 Example py Curves in Layered Soils The example problem to demonstrate the manner in which layered soils are modeled is shown in Figure 359. 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 LoadTransfer 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 358 Illustration of Equivalent Depths in a Multilayer 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 359 Soil Profile for Example of Layered Soils
135
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Four py curves for the case of layered soil are shown in Figure 360. The curves are for points A, B, C and D as shown in the sketch in Figure 361, 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 360 Example py 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 361 Equivalent Depths of Soil Layers Used for Computing py Curves 136
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
Following the method suggested by Georgiadis, the py 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 361. 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 361). Another point of considerable interest is that the recommendations for py 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 stiffclay 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 .............................................. (320)
.......................................................... (321) Soft Clay cyclic loading ....................................................... (324)
..................................................... (325) Stiff Clay with Free Water Static pct = 2cab + bx + 2.83 cax ............................................ (326) pcd = 11cb ......................................................... (327)
137
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
...................................... (335) Stiff Clay with Free Water Cyclic ........................................... (340) Stiff Clay without Free Water static and cyclic loading ...............................................(320)
...........................................................(321) Sand
..................... (350)
............................... (351) or
............................................... (352)
API Sand ................................................... (361) ........................................................ (362)
138
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
312 Modifications to py Curves for Pile Batter and Ground Slope 3121 Piles in Sloping Ground The formulations for py 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 py 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 flowaround failure that occurs at depth will not be influenced by sloping ground; therefore, only the equations for the wedgetype failures near the ground surface need modification. The modifications to py 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. 31211 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 3127. ....................................... (3127) If the ground surface has a slope angle as shown in Figure 362, the soil resistance at the front of the pile, following the Reese approach is:
+
+
Figure 362 Pile in Sloping Ground and Battered Pile
139
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
....................................... (3128) The soil resistance at the back of the pile is: ......................... (3129) 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 3127 and 3128 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. 31212 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:
.............. (3130)
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
............... (3131)
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
................................................. (3132) D2
1 D1 , and ................................................... (3133) .................................... (3134)
where
is defined in Figure 362.
Note that the denominator of Equation 3132 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 351 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
............. (3135)
3D32
1)
K Ab
where ................................................ (3136) and D4 = 1 + D3....................................................... (3137) This completes the necessary derivations for modifying the equations for clay and sand to analyze a pile under lateral load in sloping ground. 31213 Effect of Direction of Loading on Output py Curves The equations for computing maximum soil resistance for py 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, py 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 py curve it is necessary to run an analysis twice, with the pilehead 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 LoadTransfer Curves for Soil and Rock
3122 Effect of Batter on py 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, soilresistance 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 fullscale tests reported by Kubo (1964). The modifying constant to be used is shown by the solid line in Figure 363.
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 363 Soil Resistance Ratios for py 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 pvalues 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 363, 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, fullscale fieldtesting is desirable. 142
Chapter 3 Lateral LoadTransfer Curves for Soil and Rock
3123 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 363 to determine what value of pmultiplier 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
313 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 forcedeflection 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
41 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.
42 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 41, a pile with elastic bending properties is loaded with five levels of pilehead shear at 0%, 50%, 100%, 150%, and 200% of the service load. The following figures illustrate the problem conditions, lateral pile deflection versus depth, piletop deflection versus displacement, and curves of piletop deflection versus pile length. When the problem computes the curves of piletop deflection versus pile length, the program first computes piletop 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 piletop deflection is computed for are 12 meters, 11.4 meters, 10.8 meters, and so on, until the computed piletop 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 44. 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 righthand portions of the curves are flat or nearly flat, it is not possible to reduce piletop 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.
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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 41 Pile and Soil Profile for Example Problem
Figure 42 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 43 Pilehead Load versus Deflection for Example
Figure 44 Top Deflection versus Pile Length for Example If the righthand portions of the curves are inclined, it is possible to reduce the piletop deflection by lengthening the pile. However, there are situations where other factors may need to be considered. One common situation is when the piletop 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.
43 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. Freefield 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 py modulus is evaluated for a pile displacement relative to the soil displacement. This is illustrated in Figure 45 . The solution is implemented in LPile by modifying the governing differential equation to .......................................(41) 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 45 py Curve Displaced by Soil Movement
44 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. 441 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 pilehead deflection, which will depend on the sign of the applied loads. If the pile section is nonlinear (not elastic, elasticplastic, or userinput 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 pilehead fixity condition for the pile buckling analysis. 6. Enter the maximum pilehead loading for the pilehead fixity condition.
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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 crosssectional area of 0.0177 m2, a moment of inertia of 1.678 107 s modulus of 200 GPa. Two curves are plotted in Figure 36. 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 46 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
442 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 37, 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 37. In a correct buckling analysis, the computed lateral deflections should always have the same sign. In the correct analysis, also shown in Figure 37, the axial thrust values were increased in smaller increments and nonconvergence 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
Pilehead Deflection, meters
Figure 47 Examples of Correct and Incorrect Pile Buckling Analyses
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Chapter 4 Special Analyses
443 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 prebuckling 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 prebuckling behavior. A typical bucklingdeformation curve for a given set of pilehead loading is shown in Figure 48. The lateral deflection of the pile head is denoted by y0. The equation for a hyperbolic curve that originates at y0 is .......................................................(42) Where y is deflection, P is the axial thrust force and a and b are curvefitting parameters. This expression may be rewritten as ...................................................(43) The pile deflections may be replotted in which values of are plotted along the xaxis and values of are plotted along the yaxis. In many cases, this will result in a straight line with a slope of a and a yintercept of b as shown in Figure 49.
P
y0
Pilehead Deflection, y
Figure 48 Typical Results from Pile Buckling Analysis
151
Chapter 4 Special Analyses
a 1 b y
y0
Figure 49 Pile Buckling Results Showing a and b The pile buckling capacity, Pcrit, is calculated from ...............................................................(44) The estimate pile buckling capacity is computed from the shape of the pilehead 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.
45 Pushover Analysis of Piles The program feature for pushover analysis has options for different pilehead fixity options and the setting of the range and distribution of pushover deflection. The output of the pushover analysis is displayed in graphs of pilehead 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 410. The control parameters allow the user to specify the pilehead 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 410 Dialog for Controls for Pushover Analysis 451 Procedure for Pushover Analysis The pushover analysis is performed by running a series of analyses for displacementzero moment pilehead conditions for pinned head piles and analyses for displacementzero slope pilehead 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 nonzero, positive minimum and maximum displacement values), arithmetic, or a set of userspecified pilehead displacement values. The number of loading steps sets the number of pilehead 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 pilehead 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. 452 Example of Pushover Analysis Some typical results from a pushover analysis are presented in the following two figures. Figure 411 presents the pilehead 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 412 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.
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Chapter 4 Special Analyses
Formation of plastic hinge
Figure 411 Pilehead Shear Force versus Displacement from Pushover Analysis
Formation of plastic hinge
Figure 412 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 pilehead condition is fixed head and the axial load is zero.
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Chapter 4 Special Analyses
453 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 pilehead fixity conditions are neither fixed or free, but may be close to one of these conditions. If actual conditions are close to being fixedhead conditions, the amount of pilehead deflection required to develop a plastic hinge will be somewhat greater than the value shown in the pushover analysis for fixedhead conditions. Similarly, if actual conditions are close to being freehead, the amount of pilehead deflection required to develop a plastic hinge will be somewhat less than the value shown in the pushover analysis for freehead conditions.
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Chapter 4 Special Analyses
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156
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity
51 Introduction 511 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 pilehead 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 py 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 stressstrain 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 nominalmoment capacity of a reinforcedconcrete drilled shaft, prestressed concrete pile, or steelpipe 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 crosssection. 512 Assumptions The program computes the behavior of a beam or beamcolumn. 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 reinforcedconcrete section in compression is computed at a compressioncontrol 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 reinforcedconcrete sections in tension, the nominal moment capacity of a section is computed at a compressioncontrol strain limit of 0.003 or a maximum tension in the steel reinforcement of 0.005. 513 StressStrain Curves for Concrete and Steel Any number of models can be used for the stressstrain curves for concrete and steel. For the purposes of the computations presented herein, some relatively simple curves are used. The stressstrain curve for concrete is shown in Figure 51.
fc 0.15 f c
Ec
0.0038
fr Figure 51 StressStrain 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
.......................................(51)
............................(52)
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
.............................................(53)
The modulus of rupture for prestressed concrete piles is computed using 158
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity
.............................................(54)
The modulus of elasticity of concrete, Ec, is computed using ..........................................(55)
The compressive strain at peak compressive stress,
0,
is computed using
............................................................(56) The tensile strain at fracture for concrete, t, is computed using ...................................................(57)
The stressstrain (  ) curve for steel is shown in Figure 52. There is no practical limit to plastic deformation in tension or compression. The stressstrain curves for tension and compression are assumed identical in shape.
fy
y
Figure 52 StressStrain 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. ..............................................................(58) 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. 514 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 Hsections, or weak Hsections, and 13. Elasticplastic shapes with rectangular, round, tubular, strong Hsections, 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 axialload capacity for the column.
52 Beam Theory 521 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 stressstrain 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 53. 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 ...............................................................(59) 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 53 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
.............................................................(510) where: = distance from the neutral axis, and = radius of curvature. Equation 511 is obtained from Equations 59 and 510, as follows: .................................................(511)
...........................................................(512) where: x
= unit stress along the length of the beam, and
E=
s modulus.
Substituting Equation 511 into Equation 512, we obtain ............................................................(513) From beam theory ...........................................................(514) where: M = applied moment, and I = moment of inertia of the section. Equating the right sides of Equations 513 and 514, we obtain ..........................................................(515) Cancelling
and rearranging Equation 515 ............................................................(516)
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Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity
Continuing with the derivation, it can be seen that dx =
d and .........................................................(517)
For convenience here, the symbol is substituted for the curvature 1/ . The following equation is developed from this substitution and Equations 516 and 517 ............................................................(518) and because
=
d and
x
= /dx, we may express the bending strain as x
=
.............................................................(519)
The computation for a reinforcedconcrete 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 519, which in turn will lead to the forces in the concrete and steel. In this step, the assumption is made that the stressstrain curves for concrete and steel are those shown in Section 513. 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 518. 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. 522 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 ............................................(520) where Ag is the gross crosssectional area of the section, As is the crosssectional 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 crosssectional 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 crosssectional 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 ...............................................(521) 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 ...............................................(522) 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.
53 Validation of Method 531 Analysis of Concrete Sections An example concrete section is shown in Figure 54. This rectangular beamcolumn 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 crosssectional area of 0.0005 m2, and the row positions are shown in the Figure 54. 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 51. The first block of lines include an echoprint 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 54 Validation Problem for Mechanistic Analysis of Rectangular Section 5311 Computations Using Equations of Section 52 An examination of the output in Table 51 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. 5312 Check of Position of the Neutral Axis In Table 51, 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 51. LPile Output for Rectangular Concrete Section Computations of Nominal Moment Capacity and Nonlinear Bending Stiffness Axial thrust values were determined from pilehead 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 Pilehead 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 kNm kNm2 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. kNm 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 (phifactor). In ACI 31808, 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 31808, Section 9.3.2.2 or the value required by the design standard being followed.
5313 Forces in Reinforcing Steel The rows of steel in Figure 54 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 519, 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 58. ..........................................(523)
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.
5314 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 midheight of each slice by making use of Equation 519. 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 midheight 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 54 and Equations 51 through 58. The first step is to compute the value of 0 from Equation 56 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 51 and the stress in the other slices can be computed using Equation 51. From Figure 54, 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 stressstrain 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 stressstrain curve. The stresses in these slices can be computed by Equation 51. 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 53).
In this zone, it is assumed that the stressstain curve in tension is defined by the average concrete modulus (Equation 55). 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 519 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. 5315 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 64bit precision in all computations. 5316 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 kNm
Moment due to bar row 2: ( 411.9 kN) (0.1015) =
38.87 kNm
Moment due to bar row 3: ( 415.0 kN) ( 0.1015) =
41.92 kNm
Moment due to bar row 4: ( 622.5 kN) ( 0.3045) =
188.64 kNm
Total moment due to stresses in steel bars =
344.40 kNm
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 kNm
Moment in slice 2: (248.29 kN) (0.3583 m) =
80.37 kNm
Moment in slice 3: (255.21 kN) (0.3438 m) =
78.40 kNm
Moment in slice 4: (257.61 kN) (0.3293 m) =
76.24 kNm
Moment in slice 5: (247.22 kN) (0.3148 m) =
71.68 kNm
Moment in slice 6: (226.19 kN) (0.3003 m) =
63.33 kNm
Moment in slice 7: (194.53 kN) (0.2858 m) =
52.16 kNm
Moment in slice 8: ( 152.24 kN) (0.2713 m) =
38.81 kNm
Moment in slice 9: ( 99.32 kN) (0.2568 m) =
23.90 kNm
Moment in slice 10: ( 35.76 kN) (0.2423 m) =
8.07 kNm
Moment correction for top row of steel bars = ( 40.93 kN) (0.3045 m) = 12.46 kNm 171
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity
Total moment due to stresses in concrete =
561.32 kNm.
Sum of moments in steel bars and concrete = 905.71 kNm. 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. 5317 Computation of Bending Stiffness Using Approximate Method The drawing in Figure 55 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 55 Free Body Diagram Used for Computing Nominal Moment Capacity of Reinforced Concrete Section The value of bending stiffness is computed using Equation 518.
A comparison of results from hand versus computer solutions is summarized in Table 52. The moment computed by LPile was 907.19 kNm. 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 kNm2. The hand solution is within 0.16% of the computer solution. The hand solution for axial thrust is within 0.02% 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 52. Comparison of Results from Hand Computation versus Computer Solution Parameter
By LPile
By Hand
Hand Error, %
Moment Capacity, kNm
907.19
905.71
0.16%
Bending Stiffness, EI, kNm
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 noncircular 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 56 Bending Moment versus Curvature
173
0.02
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity
Figure 57 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, kNm
Figure 58 Interaction Diagram for Nominal Moment Capacity
174
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity
532 Analysis of Steel Pipe Piles The method of Section 531 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 59. 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 kNm from Figure 510 at a maximum curvature of 0.015 radians/meter. 414,000 kPa
0.838 m
0.7817 m
Figure 59 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 510 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 kNm (5,863 ftkips) 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. ..........................................................(524) Referring to the dimensions shown in Figure 59, 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 kNm from the computed solution at a strain of 0.0149 rad/m. However, the close agreement and the slight overestimation 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 kNm2. 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 511. 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 511 Elastoplastic Stress Distribution Computed by LPile 533 Analysis of PrestressedConcrete Piles Prestressedconcrete piles are widely used in construction where conditions are suitable for pile driving. A prestressedconcrete pile has a configuration similar to a conventional reinforcedconcrete pile except that the longitudinal reinforcing steel is replaced by prestressing steel. The prestressing steel is usually in the form of strands of highstrength 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 reinforcedconcrete piles of the same size. An advantage of prestressedconcrete 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 pilehead 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 stressstrain relationships selected for both concrete and reinforcing steel. The stressstrain 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 512 and in equation form in Equations 525 to 528.
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 512 StressStrain Curves of Prestressing Strands Recommended by PCI Design Handbook, 5th Edition. For 250 ksi 7wire lowrelaxation strands: .......................................(525)
ps
0.0076; f ps
250 ps
178
0.04 (ksi) ................................(526) 0.0064
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity
For 270 ksi 7wire lowrelaxation strands: .......................................(527) .................................(528) 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 stressstrain relationship for grade 300 strands. The equations are: ...................................(529) .............................(530) For prestressing bars, an elasticplastic stressstrain curve is used. As noted earlier, the value of the concrete stress due to prestressing is found prior to performance of the momentcurvature 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, ...........................................................(531) where
c
is the prestress in the concrete and Ac is the crosssectional 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 513.
179
Chapter 5 Computation of Nonlinear Bending Stiffness and Moment Capacity
Figure 513 Sections for Prestressed Concrete Piles Modeled in LPile
54 Discussion Use of the mechanistic method of analysis of momentcurvature 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 stressstrain 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
55 Reference Information 551 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
552 Prestressing Strand Types and Sizes Name 5/16" 3wire 1/4 7wire 5/16 7wire 3/8 7wire 7/16 7wire 1/2" 7wire 0.6" 7wire 5/16" 3wire 3/8 7wire 7/16 7wire 1/2" 7wire 1/2" 7w spec 9/16" 7wire 0.6" 7wire 0.7" 7wire 3/8" 7wire 7/16" 7wire 1/2" 7wire 1/2" Super 0.6" 7wire 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
553 Steel HPiles
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 61 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.
62 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 61. 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 61 Scheme for Installing Pile in a Slope Subject to Sliding Available rightofwater 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.
63 Review of Some Previous Applications Fukuoka (1977) described three applications where piles were used to stabilize slopes in Japan. Heavilyreinforced, 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 prebored 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 Hpiles 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 Higashitono 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.
64 Analytical Procedure A drawing of a pile embedded in a slope is shown in Figure 62(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 62(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 62(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 62(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 62 Forces from Soil Acting Against a Pile in a Sliding Slope, (a) Pile, Slope, and Slip Surface Geometry, (b) Distribution of Mobilized Forces, (c) Freebody 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 63. 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 py 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: ......................................(61) 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 63 Influence of Stabilizing Pile on Factor of Safety Against Sliding The discussion above leads to the following stepbystep 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 py 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 62(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 63, 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 centertocenter spacing along the row of piles prior to input to the slope stability analysis program because the twodimensional 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 64 Matching of Computed and Assumed Values of hp
65 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 py 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.
66 Case Studies and Example Computation 661 Case Studies Fukuoka (1977) described a field experiment that was performed at the landslide at Higashitono 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 Nvalue 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 kNmeters. 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 bendingmoment curve given by Fukuoka had the general shape that would be expected. At another site at the Higashitono landslide, Fukuoka described an experiment where a number of steelpipe 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 kNm. 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. 662 Example Computation The example that was selected for analysis is shown in Figure 65. 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 65 Soil Conditions for Analysis of Slope for Low Water The undrained analysis for the suddendrawdown 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 66, 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 kNm, 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 67, 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 66 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 67 Load Distribution from Stabilizing Piles for Slope Stability Analysis
663 Conclusions The results predicted by the proposed design method are compared with results from available fullscale 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
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193
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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. 12611276. Brown, D. A.; Reese, L. C. , M. W., 1987. ading of a LargeScale Journal of the Geotechnical Engineering Division, ASCE, Vol. 113, No. 11, pp. 13261343. Brown, D. A.; Shie, C. F.; and Kumar, M., 1989. py Curves for Laterally Loaded Piles Derived from ThreeDimensional Finite Elemen Proceedings, 3rd Intl. Symposium, Numerical Models in Geomechanics, Niagara Falls, Canada, Elsevier Applied Science, pp. 683690. 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 SidebySide and InLine 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. 459472. 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.; RojasGonzalez, L.; and Newman, F. B., 1989. Proceedings, Foundation Engineering: Current Principals and Practices, ASCE, Vol. 2, pp. 13381352. Dunnavant, T. W., and , M. W., 1985. Lateral Load Test of a 72InchDiameter Bored Pile in Overof Civil Engineering, University of HoustonUniversity Park, Houston, Texas, Report No. UHCE 854, September, 57 pages. Evans, L. T., and Duncan, J. M., 1982. No. UCB/GT/8204, 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. 701708. Fukuoka, M., 1977. The Effects of Horizontal Loads on Piles Due to Landslides, Proceedings, 9th International Conference, ISSMFE, Tokyo, Japan.
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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
RojasGonzalez, 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
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