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Pile Group Program for Full Material Modeling and Progressive Failure Final Report Report CA02 0076 December 2008 Division of Research & Innovation Pile Group Program for Full Material Modeling and Progressive Failure Final Report Report No. CA02 0076 December 2008 Prepared By: Department of Civil and Environmental Engineering University of Nevada, Reno Reno, NV 89557 Prepared For: California Department of Transportation Engineering Services Center 1801 30th Street Sacramento, CA 95816 California Department of Transportation Division of Research and Innovation, MS 83 1227 O Street Sacramento, CA 95814 DISCLAIMER STATEMENT This document is disseminated in the interest of information exchange. The contents of this report reflect the views of the authors who are responsible for the facts and accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California or the Federal Highway Administration. This publication does not constitute a standard, specification or regulation. This report does not constitute an endorsement by the Department of any product described herein. STATE OF CALIFORNIA DEPARTMENT OF TRANSPORTATION TECHNICAL REPORT DOCUMENTATION PAGE TR0003 ( REV. 10/ 98) 1. REPORT NUMBER CA02 0076 2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER 4. TITLE AND SUBTITLE Pile Group Program for Full Material Modeling and Progressive Failure 5. REPORT DATE December, 2008 6. PERFORMING ORGANIZATION CODE 7. AUTHOR( S) Mohamed Ashour, Gary Norris 8. PERFORMING ORGANIZATION REPORT NO. UNR / CCEER 01 02 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of Civil & Environmental Engineering University of Nevada Reno, NV 89557 0152 10. WORK UNIT NUMBER 11. CONTRACT OR GRANT NUMBER DRI Research Task No. 0076 Contract No. 59A0160 12. SPONSORING AGENCY AND ADDRESS California Department of Transportation Engineering Services Center 1801 30th Street Sacramento, CA 95816 California Department of Transportation Division of Research and Innovation, MS 83 1227 O Street Sacramento, CA 95814 13. TYPE OF REPORT AND PERIOD COVERED Final Report 14. SPONSORING AGENCY CODE 913 15. SUPPLEMENTAL NOTES 16. ABSTRACT Strain wedge ( SW) model formulation has been used, in previous work, to evaluate the response of a single pile or a group of piles ( including its pile cap) in layered soils to lateral loading. The SW model approach provides appropriate prediction for the behavior of an isolated pile and pile group under lateral static loading in layered soil ( sand and/ or clay). The SW model analysis covers the entire range of soil strain or pile deflection that may be encountered in practice. The method allows development of p y curves for the single pile based on soil pile interaction by considering the effect of both soil and pile properties ( i. e. pile size, shape, bending stiffness, and pile head fixity condition) on the nature of the p y curve. This study has extended the capability of the SW model in order to predict the response of a laterally loaded isolated pile and pile group considering the nonlinear behavior of pile material ( steel and/ or concrete) and its effect on the soil pile interaction. The incorporation of the nonlinear behavior of pile material has a significant influence on the lateral response of the pile/ shaft and its ultimate capacity. The reduction in pile lateral resistance due to degradation in the pile bending stiffness affects the nature of the accompanying p y curves, and the distribution of lateral deflections and bending moment along the pile. Contrary to the traditional Matlock Reese p y curve that does not account to the variations in the pile bending stiffness, the modulus of subgrade reaction ( i. e. the p y curve) assessed based on the SW model is a function of the pile bending stiffness. In addition, the ultimate value of soil pile reaction on the p y curve is governed by either the flow around failure of soil or the plastic hinge formation in the pile. The SW model analysis for a pile group has been modified in this study to assess the p y curves for an individual pile in a pile group. The technique presented is more realistic and evaluates the variations in the stress and strain ( i. e. Young’s modulus) in the soil around the pile in question due to the interference with the neighboring piles in a pile group in a mobilized fashion. The nonlinear behavior of pile material is also incorporated in the SW model analysis for a pile group. 17. KEY WORDS Laterally Loaded Deep Foundations, Pile Groups, Strain Wedge Model, Layered Soils, Nonlinear Behavior of Shaft Material 18. DISTRIBUTION STATEMENT No restrictions. This document is available to the public through the National Technical Information Service, Springfield, VA 22161 19. SECURITY CLASSIFICATION ( of this report) Unclassified 20. NUMBER OF PAGES 166 Pages 21. PRICE Reproduction of completed page authorized PILE GROUP PROGRAM FOR FULL MATERIAL MODELING AND PROGRESSIVE FAILURE CCEER 01 02 Prepared by: Mohamed Ashour Research Assistant Professor and Gary Norris Professor of Civil Engineering University of Nevada, Reno Department of Civil Engineering Prepared for: State of California Department of Transportation Contract No. 59A0160 July 2001 i ACKNOWLEDGMENTS The authors would like to thank Caltrans for its financial support of this project. The authors would also like to acknowledge Mr. Anoosh Shamsabadi, Dr. Saad El Azazy, Mr. Steve McBride, Mr. Bob Tanaka and Mr. Tom Schatz for their support and guidance as the Caltrans monitors for this project. ii DISCLAIMER The contents of this report reflect the views of the authors who are responsible for the facts and accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California or the Federal Highway Administration. This report does not constitute standard specifications, or regulations. iii ABSTRACT Strain wedge ( SW) model formulation has been used, in previous work, to evaluate the response of a single pile or a group of piles ( including its pile cap) in layered soils to lateral loading. The SW model approach provides appropriate prediction for the behavior of an isolated pile and pile group under lateral static loading in layered soil ( sand and/ or clay). The SW model analysis covers a wide range over the entire strain or deflection range that may be encountered in practice. The method allows development of p y curves for the single pile based on soil pile interaction by considering the effect of both soil and pile properties ( i. e. pile size, shape, bending stiffness, and pile head fixity condition) on the nature of the p y curve. This study has extended the capability of the SW model in order to predict the response of a laterally loaded isolated pile and pile group considering the nonlinear behavior of pile material ( steel and/ or concrete) and its effect on the soil pile interaction. The incorporation of the nonlinear behavior of pile material has a significant influence on the lateral response of the pile/ shaft and its ultimate capacity. The reduction in pile lateral resistance due to degradation in the pile bending stiffness affects the nature of the accompanying p y curves, and the distribution of lateral deflections and bending moment along the pile. Contrary to the traditional Matlock Reese p y curve that does not account to the variations in the pile bending stiffness, the modulus of subgrade reaction ( i. e. the p y curve) assessed based on the SW model is a function of the pile bending stiffness. In addition, the ultimate value of soil pile reaction on the p y curve is governed by either the flow around failure of soil or the plastic hinge formation in the pile. The SW model analysis for a pile group has been modified in this study to assess the p y curves for an individual pile in a pile group. The technique presented is more realistic and evaluates the variations in the stress and strain ( i. e. Young’s modulus) in the soil around the pile in question due to the interference with the neighboring piles in a pile group in a mobilized fashion. The nonlinear behavior of pile material is also incorporated in the SW model analysis for a pile group. iv TABLE OF CONTENTS CHAPTER 1 .............................................................................................................. 1 INTRODUCTION............................................................................................................... 1 CHAPTER 2 .............................................................................................................. 4 LATERAL LOADING OF A PILE IN LAYERED SOIL USING THE STRAIN WEDGE MODEL 2.1 INTRODUCTION.................................................................................................... 4 2.2 THE THEORETICAL BASIS OF STRAIN WEDGE MODEL CHARACTERIZATION ........................................................................................ 4 2.3 SOIL PASSIVE WEDGE CONFIGURATION IN UNIFORM SOIL ..................................................................................................... 5 2.4 STRAIN WEDGE MODEL IN LAYERED SOIL ............................................... 6 2.5 SOIL STRESS STRAIN RELATIONSHIP.......................................................... 8 2.5.1 Horizontal Stress Level ( SL) ....................................................................... 10 2.6 SHEAR STRESS ALONG THE PILE SIDES ( SLt) ............................................ 12 2.6.1 Pile Side Shear in Sand................................................................................ 12 2.6.2 Pile Side Shear Stress in Clay ..................................................................... 12 2.7 SOIL PROPERTY CHARACTERIZATION IN THE STRAIN WEDGE MODEL .......................................................................... 14 2.7.1 Properties Employed for Sand Soil ............................................................ 14 2.7.2 The Properties Employed for Normally Consolidated Clay ........................................................................................ 15 2.8 SOIL PILE INTERACTION IN THE STRAIN WEDGE MODEL.................................................................................................... 17 2.9 PILE HEAD DEFLECTION .................................................................................. 20 2.10 ULTIMATE RESISTANCE CRITERIA IN STRAIN v WEDGE MODEL.................................................................................................... 21 2.10.1 Ultimate Resistance Criterion in Sand Soil ............................................... 21 2.10.2 Ultimate Resistance Criterion in Clay Soil ................................................ 22 2.11 STABILITY ANALYSIS IN THE STRAIN WEDGE MODEL ........................ 22 2.11.1 Local Stability of a Soil Sublayer in the Strain Wedge Model .................................................................................... 23 2.11.2 Global Stability in the Strain Wedge Model ............................................. 23 2.12 APPROACH VERIFICATION.............................................................................. 24 2.12.1 Mustang Island Full Scale Load Test on a Pile in Submerged Dense Sand........................................................................... 25 2.12.2 Pyramid Building at Memphis, Tennessee, Full Scale Load Test on a Pile in Layered Clay Soil................................ 26 2.12.3 Sabine River Full Scale Load Tests on a Pile in Soft Clay....................... 28 2.13 SUMMARY.............................................................................................................. 31 FIGURES.................................................................................................................. 32 CHAPTER 3 ............................................................................................................... 46 PILE GROUPS IN LAYERED SOILS 3.1 INTRODUCTION.................................................................................................... 46 3.2 CHARACTRIZATION OF PILE GROUP INTERFERENCE ......................... 47 3.3 EVALUATION OF YOUN’S MODULUS, Eg ...................................................... 51 3.4 EVALUATION OF MODULUS OF SUBGRADE REACTION, EEg ................ 52 3.5 CASE STUDIES....................................................................................................... 54 3.5.1 Full Scale Load Test on a Pile Group in Layered Clay ............................ 54 3.5.2 Full Scale Load Test on a Pile Group in Sand .......................................... 54 3.5.3 Full Scale Load Test on a Pile Group in Layered Clay ............................ 55 3.5.4 Full Scale Load Test on a Pile Group with a Pile Cap in Layered Soil ............................................................................................. 56 3.5.5 Model Scale Load Test on a Pile Group in Loose and vi Medium Dense Sand .................................................................................... 57 3.6 SUMMARY.............................................................................................................. 57 FIGURES.................................................................................................................. 39 CHAPTER 4 NUMERICAL MATERIAL MODELING........................................................................ 70 4.1 INTRODUCTION.................................................................................................... 70 4.2 THE COMBINATION OF MATERIAL MODELING WITH THE SW MODEL .................................................................................................................... 72 4.2.1 Material Modeling for Concrete Strength and Failure Criteria ............. 73 4.2.2 Material Modeling of Steel Strength .......................................................... 76 4.3 MOMENT CURAVATURE RELATIONSHIP.................................................... 78 4.4 SOLUTION PROCEDURE.................................................................................... 79 4.4.1 Steel Pile........................................................................................................ 79 4.4.2 Reinforced Concrete Pile and Drilled Shaft .............................................. 84 4.4.3 Steel Pipe Pile Filled with Concrete ( Cast in Steel Shell, CISS) ........................................................................... 88 4.4.4 Steel Pipe Pile Filled with Reinforced Concrete ( Cast in Steel Shell, CISS) ........................................................................... 89 4.5 ILLUSTRATIVE EXAMPLES.............................................................................. 90 4.5.1 Example Problem, a Fixed Head Steel Pile Supporting a Bridge Abutment....................................................................................... 90 4.5.2 Example Problem, a Free Head Drilled Shaft Supporting a Bridge Abutment .................................................................. 91 4.5.3 Example Problem, a Fixed Head Drilled Shaft Supporting a Bridge Abutment .................................................................. 93 4.6 SUMMARY ............................................................................................................. 93 vii FIGURES.................................................................................................................. 95 CHAPTER 5 EFFECT OF NONLINEAR BEHAVIOR OF PILE MATERIAL ON PILE AND PILE GROUP LATERAL RESPONSE ............................................................................ 108 5.1 INTRODUCTION.................................................................................................... 108 5.2 EFFECT OF PILE MATERIAL NONLINEAR RESPONSE ON THE P Y CURVE ..................................................................................................................... 109 5.2.1 Steps of Constructing the p y Curve in the SW Model Analysis............. 111 5.2.2 Effect of Material Modeling on the p y Curve Ultimate .......................... 113 5.3 CASE STUDIES....................................................................................................... 115 5.3.1 Pyramid Building at Memphis, Tennessee, Dull Scale Load Test on a Pile in Layered Clay Soil ............................................................ 115 5.3.2 Houston Full Scale Load Test on a Reinforced Concrete Shaft in Stiff Clay......................................................................................... 118 5.3.3 Las Vegas Test on Drilled Shafts and Shaft Group in a Caliche Layer ........................................................................................ 121 5.3.4 Southern California Full Scale Load Test in Stiff Clay ........................... 122 5.3.5 Islamorada Full Scale load Test on a Pile Driven in Rock....................... 123 5.3.6 University of California at Los Angeles Full Scale Load Test on a Pile Driven in Stiff Clay .............................................................. 125 5.4 SUMMARY ............................................................................................................. 126 5.5 FIGURES.................................................................................................................. 127 1 CHAPTER 1 INTRODUCTION This report presents a summary of strain wedge ( SW) model assessment of the behavior of piles and pile groups subjected to lateral loading in layered soil considering the nonlinear behavior of pile material. A computer code attached to this report has been developed to assess the response of a single pile and pile group in layered soils ( sand, clay and/ or rock) and the associated p y curves for various soil and pile conditions. The main goal of this report is to address the influence of the nonlinear behavior of pile/ shaft material on the lateral response of isolated piles/ shafts and pile groups. The significance of accounting for the variations in strength of pile/ shaft is to identify the actual behavior and the ultimate capacity of such piles/ shafts. In addition, the associated p y curves will experience different effects due to the degradation in pile materials. The California Department of Transportation ( CALTRANS) sponsored a significant part of the SW model research through different phases of research project ( Ashour et al. 1996, Ashour and Norris 1998, and Ashour and Norris 2000). The SW model relates one dimensional beam on elastic foundation analysis to the three dimensional soil pile interaction response. It relates the deflection of a pile versus depth ( or its rotation) to the relative soil strain that exists in the growing passive wedge that develops in front of a pile under horizontal load. The SW model assumes that the deflection of a pile under increasing horizontal load is due solely to the deformation of the soil within the mobilized passive wedge, that plane stress change conditions exist within the wedge, and that soil strain is constant with depth in the current wedge. The passive wedge will exhibit a height that corresponds to the pivot point as determined by a linear approximation of the pile deflection. If the soil strain is known, an equivalent linear Young's modulus value, associated with the soil within the wedge at any depth, can be determined. Assuming plane stress change conditions exist, the increase in horizontal 2 stress can then be determined. In addition, the beam on elastic foundation line load reaction at any depth along the pile face is equivalent to the increase in horizontal stress times the wedge width at that depth plus the mobilized side shear resistance that develops at that depth along the pile faces parallel to the direction of movement. Since the geometry of the developing wedge is based on known soil properties and the current value of soil strain, the wedge width can be determined at any depth within the wedge. An equivalent face stress from beam on elastic foundation ( BEF) analysis can therefore be related to the horizontal stress change in the soil. The SW model relates one dimensional BEF analysis ( p y response) to a threedimensional soil pile interaction response. Because of this relation, the strain wedge model is also capable of determining the maximum moment and developing p y curves for a pile under consideration since the pile load and deflection at any depth along the pile can be determined. A detailed summary of the theory incorporated into the strain wedge model is presented in Chapter 2. The problem associated with analyzing a pile group is that loading one pile in the group can dramatically affect the response of other piles in the group. Since the SW model determines the geometry of the developing passive wedge, it allows any overlap between passive wedges within the group to be quantified. By knowing the amount of passive wedge overlap, the effective strain associated with the pile under consideration can be determined which ultimately reduces the lateral load capacity of the pile for a given level of deflection. Despite Ashour and Norris ( 2000) discussed the assessment of the lateral response of a pile group, a new treatment for the problem of a laterally loaded pile group is presented in this report to upgrade the capability of the SW model technique. This report illustrates the links between the single pile and the pile group analysis. This is different from the current procedure in common use that employs a p y multiplier technique. Such multiplier technique is based on reducing the stiffness of the traditional ( Matlock Reese) p y curve using a multiplier that reduces the stiffness of the p y curve of the single pile to yield a softer response for an individual pile in the group. A detailed 3 summary of the theory in which the SW model analyzes pile group behavior is presented in Chapter 3. A methodology to assess the response of an isolated pile and pile group in layered soil considering the nonlinear behavior of pile material and how the accompanying modulus of subgrade reaction is affected is presented in Chapters 4 and 5. The effect of pile properties, such as the pile bending stiffness, on the pile lateral response has been presented by Ashour et al. ( 1996). Such a study emphasized the need to study the influence of the variation in pile bending stiffness during the loading process on the soilpile interaction and therefore lateral response of the isolated pile and pile group. The effect of pile nonlinear behavior of pile material has been studied by other researches ( Reese 1994, and Reese and Wang 1991). However, the incorporation of the nonlinear behavior of pile material has not affected the shape of the p y curve or the soil pile interaction. In other words, the p y curve has not accounted for the variation in the pile bending stiffness. Several case studies are presented in this study to show the capability of the SW model and how the modeling of pile material ( steel and/ or concrete) is employed in the SW model analysis. A numerical model for confined concrete is employed with the SW model. Such a model accounts for the enhancement of the concrete strength due to the confinement of the transverse reinforcement. 4 CHAPTER 2 LATERAL LOADING OF A PILE IN LAYERED SOIL USING THE STRAIN WEDGE MODEL 2.1 INTRODUCTION The strain wedge ( SW) model is an approach that has been developed to predict the response of a flexible pile under lateral loading ( Norris 1986, Ashour et al. 1996 and Ashour et al. 1998). The main concept associated with the SW model is that traditional one dimensional Beam on Elastic Foundation ( BEF) pile response parameters can be characterized in terms of three dimensional soil pile interaction behavior. The strain wedge model was initially established to analyze a free head pile embedded in one type of uniform soil ( sand or clay). However, the SW model has been improved and modified through additional research to accommodate a laterally loaded pile embedded in multiple soil layers ( sand and clay). The strain wedge model has been further modified to include the effect of pile head conditions on soil pile behavior. The main objective behind the development of the SW model is to solve the BEF problem of a laterally loaded pile based on the envisioned soil pile interaction and its dependence on both soil and pile properties. The problem of a laterally loaded pile in layered soil has been solved by Reese ( 1977) as a BEF based on modeling the soil response by p y curves. However, as mentioned by Reese ( 1983), the p y curve employed does not account for soil continuity and pile properties such as pile stiffness, pile cross section shape and pile head conditions. 2.2 THE THEORETICAL BASIS OF STRAIN WEDGE MODEL CHARACTERIZATION The SW model parameters are related to an envisioned three dimensional passive wedge of soil developing in front of the pile. The basic purpose of the SW model is to relate stress strain strength behavior of the soil in the wedge to one dimensional BEF parameters. The SW model is, therefore, able to provide a 5 theoretical link between the more complex three dimensional soil pile interaction and the simpler onedimensional BEF characterization. The previously noted correlation between the SW model response and BEF characterization reflects the following interdependence: · the horizontal soil strain ( e ) in the developing passive wedge in front of the pile to the deflection pattern ( y versus depth, x) of the pile; · the horizontal soil stress change ( D s h) in the developing passive wedge to the soil pile reaction ( p) associated with BEF; and · the nonlinear variation in the Young's modulus ( E = D s h/ e ) of the soil to the nonlinear variation in the modulus of soil subgrade reaction ( Es = p/ y) associated with BEF characterization. The analytical relations presented above reflect soil pile interaction response characterized by the SW model that will be illustrated later. The reason for linking the SW model to BEF analysis is to allow the appropriate selection of BEF parameters to solve the following fourth order ordinary differential equation to proceed. The closed form solution of the above equation has been obtained by Matlock and Reese ( 1961) for the case of uniform soil. In order to appreciate the SW model’s enhancement of BEF analysis, one should first consider the governing analytical formulations related to the passive wedge in front of the pile, the soil’s stress strain relationship, and the related soil pile interaction. 2.3 SOIL PASSIVE WEDGE CONFIGURATION IN UNIFORM SOIL The SW model represents the mobilized passive wedge in front of the pile which is characterized by base angles, Q m and b m, the current passive wedge depth h, and the spread of the wedge fan angle, j m ( the mobilized friction angle). The horizontal stress change at the passive wedge face, D s h, and side shear, t , act as shown in Fig. 2.1. One of the main assumptions associated with the SW model is that the deflection = 0 d x + E ( x) y + P d y d x EI d y 2 2 4 s x 4 ÷ ø ö ç è æ ÷ ø ö ç è æ ( 2.1) 6 pattern of the pile is taken to be linear over the controlling depth of the soil near the pile top resulting in a linearized deflection angle, d , as seen in Fig. 2.2. The relationship between the actual ( closed form solution) and linearized deflection patterns has been established by Norris ( 1986). This assumption allows uniform horizontal and vertical soil strains to be assessed ( as seen later in a Fig. 2.6). Changes in the shape and depth of the passive wedge, along with changes in the state of loading and pile deflection, occur with change in the uniform strain in the developing passive wedge. The configuration of the wedge at any instant of load and, therefore, mobilized friction angle, j m, and wedge depth, h, is given by the following equation: or its complement The width, BC, of the wedge face at any depth is where x denotes the depth below the top of the studied passive wedge, and D symbolizes the width of the pile cross section ( see Fig. 2.1). It should be noted that the SW model is based upon an effective stress analysis of both sand and clay soils. As a result, the mobilized fanning angle, j m, is not zero in clay soil as assumed by Reese ( 1958, 1983). 2.4 STRAIN WEDGE MODEL IN LAYERED SOIL The SW model can handle the problem of multiple soil layers of different types. The approach employed, which is called the multi sublayer technique, is based upon dividing the soil profile and the loaded pile into sublayers and segments of constant thickness, respectively, as shown in Fig. 2.3. Each sublayer of soil is considered to behave as a uniform soil and have its own properties according to the sublayer location and soil type. In addition, the multi sublayer technique depends on the deflection pattern of the embedded pile 2 = 45  m m j Q ( 2.2) 2 = 45 + m m b j ( 2.3) b j m m BC = D + ( h  x) 2 tan tan ( 2.4) 7 being continuous regardless of the variation of soil types. However, the depth, h, of the deflected portion of the pile is controlled by the stability analysis of the pile under the conditions of soil pile interaction. The effects of the soil and pile properties are associated with the soil reaction along the pile by the Young's modulus of the soil, the stress level in the soil, the pile deflection, and the modulus of subgrade reaction between the pile segment and each soil sublayer. To account for the interaction between the soil and the pile, the deflected part of the pile is considered to respond as a continuous beam loaded with different short segments of uniform load and supported by nonlinear elastic supports along soil sublayers, as shown in Fig. 2.4. At the same time, the point of zero deflection ( Xo in Fig. 2.4a) for a pile in a particular layered soil varies according to the applied load and the soil strain level. The SW model in layered soil provides a means for distinguishing layers of different soil types as well as sublayers within each layer where conditions ( e 50, SL, j m) vary even though the soil and its properties ( ` g , e or Dr, j , etc.) remain the same. As shown in Fig. 2.5 , there may be different soil layers and a transition in wedge shape from one layer to the next, with all components of the compound wedge having in common the same depth h. In fact, there may be a continuous change over a given sublayer; but the values of stress level ( SL) and mobilized friction angle ( j m) at the middle of each sublayer of height, Hi, are treated as the values for the entire sublayer. As shown in Fig. 2.5, the geometry of the compound passive wedge depends on the properties and the number of soil types in the soil profile, and the global equilibrium between the soil layers and the loaded pile. An iterative process is performed to satisfy the equilibrium between the mobilized geometry of the passive wedge of the layered soil and the deflected pattern of the pile for any level of loading. While the shape of the wedge in any soil layer depends upon the properties of that layer and, therefore, satisfies the nature of a Winkler foundation of independent “ soil” springs in BEF analysis, realize that there is forced interdependence given that all components of the compound wedge have the same depth ( h) in common. Therefore, the mobilized depth ( h) of the compound wedge at any time is a function of the various soils ( and their stress levels), the bending stiffness ( EI), and head fixity conditions ( fixed, free, or other) of 8 the pile. In fact, the developing depth of the compound wedge can be thought of as a retaining wall of changing height, h. Therefore, the resultant “ soil” reaction, p, from any soil layer is really a “ soil pile” reaction that depends upon the neighboring soil layers and the pile properties as they, in turn, influence the current depth, h. In other words, the p y response of a given soil layer is not unique. The governing equations of the mobilized passive wedge shape are applied within each one or two foot sublayer i ( of a given soil layer I) and can be written as follows: where h symbolizes the entire depth of the compound passive wedge in front of the pile and xi represents the depth from the top of the pile or compound passive wedge to the middle of the sublayer under consideration. The equations above are applied at the middle of each sublayer. 2.5 SOIL STRESS STRAIN RELATIONSHIP The horizontal strain ( e ) in the soil in the passive wedge in front of the pile is the predominant parameter in the SW model; hence, the name “ strain wedge”. Consequently, the horizontal stress change ( D s h) is constant across the width of the rectangle BCLM ( of face width BC of the passive wedge ), as shown in Fig. 2.1. The stress strain relationship is defined based on the results of the isotropically consolidated drained ( sand) or undrained ( clay) triaxial test. These properties are summarized as follows: · The major principle stress change ( D s h) in the wedge is in the direction of pile movement, and it ( ) ( ) 2 = 45  m i m i j Q ( 2.5) ( ) ( ) 2 = 45 + m i m i j b ( 2.6) ( BC ) i = D + ( h  xi ) 2 ( m ) i ( m ) i tan b tan j ( 2.7) 9 is equivalent to the deviatoric stress in the triaxial test as shown in Fig. 2.2 ( assuming that the horizontal direction in the field is taken as the axial direction in the triaxial test). · The vertical stress change ( D s v) and the perpendicular horizontal stress change ( D s ph) equal zero, corresponding to the standard triaxial compression test where deviatoric stress is increased while confining pressure remains constant. · The initial horizontal effective stress is taken as where K= 1 due to pile installation effects. Therefore, the isotropic confining pressure in the triaxial test is taken as the vertical effective stress ( ` s vo) at the associated depth. · The horizontal stress change in the direction of pile movement is related to the current level of horizontal strain ( e ) and the associated Young's modulus in the soil as are the deviatoric stress and the axial strain to the secant Young’s modulus ( E = D s h/ e ) in the triaxial test. · Both the vertical strain ( e v ) and the horizontal strain perpendicular to pile movement ( e ph) are equal and are given as e v = e ph =  n e where n is the Poisson’s ratio of the soil. It can be demonstrated from a Mohr’s circle of soil strain, as shown in Fig. 2.6, that shear strain, g , is defined as The corresponding stress level ( SL) in sand ( see Fig. 2.7) is s ho s vo s vo = K = ( v ) Q m ( ) Q m 1 + 2 2 1  2 = 2 1 = 2 g e e sin e n sin ( 2.8) ( ) ( 45 + )  1 45 +  1 SL = = 2 m 2 hf h j j s s tan tan D D ( 2.9) 10 where the horizontal stress change at failure ( or the deviatoric stress at failure in the triaxial test) is In clay, where Su represents the undrained shear strength which may vary with depth. Determination of the values of SL and j m in clay requires the involvement of an effective stress analysis which is presented later in this chapter. The relationships above show clearly that the passive wedge response and configuration change with the change of the mobilized friction angle ( j m) or stress level ( SL) in the soil. Such behavior provides the flexibility and the accuracy for the strain wedge model to accommodate both small and large strain cases. A power function stress strain relationship is employed in SW model analysis for both sand and clay soils. It reflects the nonlinear variation in stress level ( SL) with axial strain ( e ) for the condition of constant confining pressure. To be applicable over the entire range of soil strain, it takes on a form that varies in stages as shown in Fig. 2.8. The advantage of this technique is that it allows the three stages of horizontal stress, described in the next section, to occur simultaneously in different sublayers within the passive wedge. 2.5.1 Horizontal Stress Level ( SL) Stage I ( e £ e 50% ) The relationship between stress level and strain at each sublayer ( i) in the first stage is assessed using the following equation, ú û ù ê ë é ÷ ø ö ç è D æ  1 2 = 2 45 + hf vo j s s tan ( 2.10) ; = 2 S SL = hf u hf h s s s D D D ( 2.11) 11 where 3.707 and l ( l = 3.19) represent the fitting parameters of the power function relationship, and e 50 symbolizes the soil strain at 50 percent stress level. Stage II ( e 50% £ e £ e 80 % ) In the second stage of the stress strain relationship, Eqn. 2.12 is still applicable. However, the value of the fitting parameter l is taken to vary in a linear manner from 3.19 at the 50 percent stress level to 2.14 at the 80 percent stress level as shown in Fig. 2.8b. Stage III ( e ³ e 80% ) This stage represents the final loading zone which extends from 80 percent to 100 percent stress level. The following Equation is used to assess the stress strain relationship in this range, where m= 59.0 and q= 95.4 e 50 are the required values of the fitting parameters. The three stages mentioned above are developed based on unpublished experimental results ( Norris 1977). In addition, the continuity of the stress strain relationship is maintained along the SL e curve at the merging points between the mentioned stages. As shown in Fig. 2.9, if e 50 of the soil is constant with depth ( x), then, for a given horizontal strain ( e ), SL from Eqns. 2.12 or 2.13 will be constant with x. On the other hand, since strength, D s hf, varies with depth ( e. g., see Eqns. 2.10 and 2.11), D s h (= SL D s hf ) will vary in a like fashion. However, e 50 is affected by confining pressure ( ` s vo) in sand and Su in clay. Therefore, SL for a given e will vary somewhat with depth. ( ) (  3.707 SL ) SL = i 50 i i i exp e l e ( 2.12) ( ) SL 0.80 m + q 100 SL = 0.2 + i i i i i ³ ú û ù ê ë é exp ln ; e e ( 2.13) 12 The Young’s modulus of the soil from both the shear loading phase of the triaxial test and the strain wedge model is It can be seen from the previous equations that stress level, strain and Young's modulus at each sublayer ( i) depend on each other, which results in the need for an iterative solution technique to satisfy the equilibrium between the three variables. 2.6 SHEAR STRESS ALONG THE PILE SIDES ( SLt) Shear stress ( t ) along the pile sides in the SW model ( see Fig. 2.1) is defined according to the soil type ( sand or clay). 2.6.1 Pile Side Shear in Sand In the case of sand, the shear stress along the pile sides depends on the effective stress ( s vo) at the depth in question and the mobilized angle of friction between the sand and the pile ( j s). The mobilized side shear depends on the stress level and is given by the following equation, In Eqn. 2.15, note that mobilized side shear angle, tan j s, is taken to develop at twice the rate of the mobilized friction angle ( tan j m) in the mobilized wedge. Of course, j s is limited to the fully developed friction angle ( j ) of the soil. 2.6.2 Pile Side Shear Stress in Clay ( ) ( ) e s e s = SL E = h i i hf i i D D ( 2.14) = ( ) ( ) ; where ( ) = 2 ( ) i vo i s i s i m i t s tan j tan j tan j 13 The shear stress along the pile sides in clay depends on the clay’s undrained shear strength. The stress level of shear along the pile sides ( SLt) differs from that in the wedge in front of the pile. The side shear stress level is function of the shear movement, equal to the pile deflection ( y) at depth x from the ground surface. This implies a connection between the stress level ( SL) in the wedge and the pile side shear stress level ( SLt) . Using the Coyle Reese ( 1966) “ t z” shear stress transfer curves ( Fig. 2.10), values for SLt can be determined. The shear stress transfer curves represent the relationship between the shear stress level experienced by a one foot diameter pile embedded in clay with a peak undrained strength, Su, and side resistance, t ult ( equal to z times the adhesional strength a Su), for shear movement, y. The shear stress load transfer curves of Coyle Reese can be normalized by dividing curve A ( 0 < x < 3 m) by z = 0.53, curve B ( 3 < x < 6 m) by z = 0.85, and curve C ( x > 6 m) by z = 1.0. These three values of normalization ( 0.53, 0.85, 1.0) represent the peaks of the curves A, B, and C, respectively, in Fig. 2.10a. Figure 2.10b shows the resultant normalized curves. Knowing pile deflection ( y), one can assess the value of the mobilized pile side shear stress ( t ) as where and a indicates the adhesion value after Tomlinson ( 1957). The normalized shear stress load transfer curves can be represented by the following equations. For the normalized curves A ( x < 3 m) and B ( 3 < x < 6 m), For the normalized curve C ( x > 6 m) where y is in cm and D in m. = ( SL ) ( ) t i t i t ult i ( 2.16) ( ) = S ) t ult i z ( a u i ( 2.17) SL = 12.9 y D  40.5 y D 2 2 t ( 2.18) SL = 32.3 y D  255 y D 2 2 t ( 2.19) 14 From the discussion above, it is obvious that SLt varies nonlinearly with the pile deflection, y, at a given soil depth, x. Also, SLt changes nonlinearly with soil depth for a given value of soil strain ( see Fig. 2.11). These concepts are employed in each sublayer of clay. 2.7 SOIL PROPERTY CHARACTERIZATION IN THE STRAIN WEDGE MODEL One of the main advantages of the SW model approach is the simplicity of the required soil properties necessary to analyze the problem of a laterally loaded pile. The properties required represent the basic and the most common properties of soil, such as the effective unit weight and the angle of internal friction or undrained strength. The soil profile is divided into one or two foot sublayers, and each sublayer is treated as an independent entity with its own properties. In this fashion, the variation in soil properties or response ( such as e 50 and j in the case of sand, or Su and ` j in the case of clay) at each sublayer of soil can be explored. It is obvious that soil properties should not be averaged at the midheight of the passive wedge in front of the pile for a uniform soil profile ( as in the earlier work of Norris 1986), or averaged for all sublayers of a single uniform soil layer of a multiple layer soil profile. 2.7.1 Properties Employed for Sand Soil · Effective unit weight ( total above water table, buoyant below), ` g · Void ratio, e, or relative density, Dr · Angle of internal friction, j · Soil strain at 50% stress level, e 50 While standard subsurface exploration techniques and available correlations may be used to evaluate or estimate ` g , e or Dr, and j , some guidance may be required to assess e 50. 15 The e 50 represents the axial strain ( e 1 ) at a stress level equal to 50 percent in the e 1 SL relationship that would result from a standard drained ( CD) triaxial test. The confining ( consolidation) pressure for such tests should reflect the effective overburden pressure ( ` s vo) at the depth ( x) of interest. The e 50 changes from one sand to another and also changes with density state. In order to obtain e 50 for a particular sand, one can use the group of curves shown in Fig. 2.12 ( Norris 1986) which show a variation based upon the uniformity coefficient, Cu, and void ratio, e. These curves have been assessed from sand samples tested with “ frictionless” ends in CD tests at a confining pressure equal to 42.5 kPa ( Norris 1977). Since the confining pressure changes with soil depth, e 50, as obtained from Fig. 2.12, should be modified to match the existing pressure as follows: where ` s vo should be in kPa. 2.7.2 The Properties Employed for Normally Consolidated Clay · Effective unit weight ` g · Plasticity index, PI · Effective angle of friction, ` j · Undrained shear strength, Su · Soil strain at 50% stress level, e 50 Plasticity index, PI, and undrained shear strength, Su, are considered the governing properties because the effective angle of internal friction, ` j , can be estimated from the PI based on Fig. 2.13. The e 50 from an undrained triaxial test ( UU at depth x or CU with s 3 = ` s vo) can be estimated based on Su as indicated in Fig. 2.14. ( ) ( ) ÷ ø ö ç è æ 42.5 = ( ) vo i 0.2 50 i 50 42.5 e e s ( 2.20) ( ) ( ) ú û ù ê ë é ÷ ø ö ç è D æ  1 2 = 2 45 + i hf i vo i j s s tan ( 2.21) 16 An effective stress ( ES) analysis is employed with clay soil as well as with sand soil. The reason behind using the ES analysis with clay, which includes the development of excess porewater pressure with undrained loading, is to define the three dimensional strain wedge geometry based upon the more appropriate effective stress friction angle, ` j . The relationship between the normally consolidated clay undrained shear strength, Su, and ` s vo is taken as assuming that Su is the equivalent standard triaxial test strength. The effective stress analysis relies upon the evaluation of the developing excess porewater pressure based upon Skempton's equation ( 1954), i. e. where B equals 1 for saturated soil. Accordingly, Note that D s 3 = 0 both in the shear phase of the triaxial test and in the strain wedge. Therefore, the mobilized excess porewater pressure is where D s 1 represents the deviatoric stress change in the triaxial test and D s h in the field, i. e. Therefore, using the previous relationships, the Skempton equation can be rewritten for any sublayer ( i) as follows: The initial value of parameter Au is 0.333 and occurs at very small strain for elastic soil response. In u s vo S = 0.33 ( 2.22) u = B [ + A (  ) ] D D s 3 u D s 1 D s 3 ( 2.23) u = + A (  ) D D s 3 u D s 1 D s 3 ( 2.24) u s 1 D u = A D ( 2.25) u s h D u = A D ( 2.26) ( u ) = ( A ) SL ( ) = ( A ) SL 2 ( S ) i u i i hf i u i i u i D D s ( 2.27) ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ 1  ( ) ( S ) 1 1 + 2 1 ( A ) = i vo i u i uf i j s sin ( 2.28) 17 addition, the value of parameter Auf that occurs at failure at any sublayer ( i) is given by the following relationship after Wu ( 1966) as indicated in Fig. 2.15. In Eqn. 2.28, ` j symbolizes the effective stress angle of internal friction; and, based on Eqn. 2.22, Su/ ` s vo equals 0.33. However, Au is taken to change with stress level in a linear fashion as By evaluating the value of Au, one can effectively calculate the excess porewater pressure, and then can determine the value of the effective horizontal stress, ( s – vo + D s h  D u), and the effective confining pressure, ( s – vo  D u) at each sublayer, as shown in Fig. 2.15. Note that the mobilized effective stress friction angle, j – m, can be obtained from the following relationship. The targeted values of ` j mi and SLi in a clay sublayer and at a particular level of strain ( e ) can be obtained by using an iterative solution that includes Eqns 2.11 through 2.13, and 2.27 through 2.30. 2.8 SOIL PILE INTERACTION IN THE STRAIN WEDGE MODEL The strain wedge model relies on calculating the modulus of subgrade reaction, Es, which reflects the soil pile interaction at any level of soil strain during pile loading. Es also represents the secant slope at any point on the p y curve, i. e. ( A ) = 0.333 + SL [ ( A )  0.333 ] u i i uf i ( 2.29) ( ) (  u ) +  u = 2 ( ) 45 + vo i 2 m i vo h i D D D ÷ ÷ ø ö ç ç è æ s j s s tan ( 2.30) 18 Note that p represents the force per unit length of the pile or the BEF soil pile reaction, and y symbolizes the pile deflection at that soil depth. In the SW model, Es is related to the soil’s Young's modulus, E, by two linking parameters, A and Y s. It should be mentioned here that the SW model establishes its own Es from the Young's modulus of the strained soil, and therefore, one can assess the p y curve using the strain wedge model analysis. Therefore, Es should first be calculated using the strain wedge model analysis to identify the p and y values. Corresponding to the horizontal slice ( a soil sublayer) of the passive wedge at depth x ( see Fig. 2.1), the horizontal equilibrium of horizontal and shear stresses is expressed as where S1 and S2 equal to 0.75 and 0.5, respectively, for a circular pile cross section, and equal to 1.0 for a square pile ( Briaud et al. 1984). Alternatively, one can write the above equation as follows: where A symbolizes the ratio between the equivalent pile face stress, p/ D, and the horizontal stress change, D s h, in the soil. ( In essence, it is the multiplier that, when taken times the horizontal stress change, gives the equivalent face stress.) From a different perspective, it represents a normalized width ( that includes side shear and shape effects) that, when multiplied by D s h yields p/ D. By combining the equations of the passive wedge geometry and the stress level with the above relationship, one finds that y p Es = ( 2.31) p = ( ) BC S + 2 D S i h i i 1 i 2 D s t ( 2.32) ( ) ( ) 2 S + D = BC S p D A = h i i 1 i 2 h i i i s t D s D / ( 2.33) ( ) ( ) ( ) ( ) ( ) in sand 2 S + D h  x 2 A = S 1 + h i i m m i 2 vo i s i i 1 s b j s f D ÷ ÷ ø ö ç ç è æ tan tan tan ( 2.34) 19 Here the parameter A is a function of pile and wedge dimensions, applied stresses, and soil properties. However, given that D s h = E e in Eqn. 2.33, The second linking parameter, Y s, relates the soil strain in the SW model to the linearized pile deflection angle, d . Referring to the normalized pile deflection shape shown in Figs. 2.2 and 2.6 and ( ) ( ) ( ) in clay SL + S SL D h  x 2 A = S 1 + i i m m i 2 t i 1 i ÷ ÷ ø ö ç ç è æ tan b tan j ( 2.35) p = A D ( s ) = A D E e i i h i i i D ( 2.36) 2 = d g ( 2.37) Q m 2 2 = 2 max sin g g ( 2.38) 20 where g denotes the shear strain in the developing passive wedge. Using Eqns. 2.38 and 2.39, Eqn. 2.37 can be rewritten as Based on Eqn. 2.40, the relationship between e and d can expressed as or The parameter Y varies with the Poisson's ratio of the soil and the soil's mobilized angle of internal friction ( j m) and the mobilized passive wedge angle ( Q m). Poisson's ratio for sand can vary from 0.1 at a very small strain to 0.5 or lager ( due to dilatancy) at failure, while the base angle, Q m, can vary between 45o ( for j m = 0 at e = 0) and 25o ( for, say, j m = 40o at failure), respectively. For this range in variation for n and j m, the parameter Y for sand varies between 1.81 and 1.74 with an average value of 1.77. In clay soil, Poisson's ratio is assumed to be 0.5 ( undrained behavior) and the value of the passive wedge base angle, Q m, can vary between 45o ( for j m = 0 at e = 0) and 32.5o ( for, say, ` j m = 25o at failure). Therefore, the value of the parameter Y will vary from 1.47 to 1.33, with an average value of 1.4. It is clear from the equations above that employing the multi sublayer technique greatly influences the values of soil pile interaction as characterized by the parameter, Ai, which is affected by the changing effective stress and soil strength from one sublayer to another. The final form of the modulus of subgrade reaction ( ) 2 1 + = 2  = 2 v g e e n e max ( 2.39) ( ) 2 1 + 2 d = e n sin Q m ( 2.40) d e Y = ( 2.41) ( ) Q Y m 1 + 2 2 = n sin ( 2.42) 21 can be expressed as It should be mentioned that the SW model develops its own set of non unique p y curves which are function of both soil and pile properties, and are affected by soil continuity ( layering) as presented by Ashour et al. ( 1996). 2.9 PILE HEAD DEFLECTION As mentioned previously, the deflection pattern of the pile in the SW model is continuous and linear. Based on this concept, pile deflection can be assessed using a simplified technique which provides an estimation for the linearized pile deflection, especially yo at the pile head. By using the multi sublayer technique, the deflection of the pile can be calculated starting with the base of the mobilized passive wadge and moving upward along the pile, accumulating the deflection values at each sublayer as shown in the following relationships and Fig. 2.16. where the Y s value changes according to the soil type ( sand or clay), and Hi indicates the thickness of sublayer i and n symbolizes the current number of sublayers in the mobilized passive wedge. The main point of interest is the pile head deflection which is a function of not only the soil strain but also of the depth of the compound passive wedge that varies with soil and pile properties and the level of soil strain. 2.10 ULTIMATE RESISTANCE CRITERIA IN STRAIN WEDGE MODEL The mobilized passive wedge in front of a laterally loaded pile is limited by certain constraint criteria in the ( ) ( ) ( ) D E h  x = A h  x = A D E y p E = i i i i i i i i s i Y d e ( 2.43) Y e yi = Hi d i = Hi ( 2.44) y = y i = 1 to n o i S ( 2.45) 22 SW model analysis. Those criteria differ from one soil to another and are applied to each sublayer. Ultimate resistance criteria govern the shape and the load capacity of the wedge in any sublayer in SW model analysis. The progressive development of the ultimate resistance with depth is difficult to implement without employing the multi sublayer technique. 2.10.1 Ultimate Resistance Criterion of Sand Soil The mobilization of the passive wedge in sand soil depends on the horizontal stress level, SL, and the pile side shear resistance, t . The side shear stress is a function of the mobilized side shear friction angle, j s, as mentioned previously, and reaches its ultimate value ( j s = j ) earlier than the mobilized friction angle, j m, in the wedge ( i. e. SLt ³ SL). This causes a decrease in the rate of growth of sand resistance and the fanning of the passive wedge as characterized by the second term in Eqns 2.32 and 2.34, respectively. Once the stress level in the soil of a sublayer of the wedge reaches unity ( SLi = 1), the stress change and wedge fan angle in that sublayer cease to grow. However, the width BC the face of the wedge can continue to increase as long as e ( and, therefore, h in Eqn. 2.7) increases. Consequently, soil pile resistance, p, will continue to grow more slowly until a condition of initial soil failure ( SLi = 1) develops in that sublayer. At this instance, p = pult where pult in sand, given as pult is a temporary ultimate condition, i. e. the fanning angle of the sublayer is fixed and equal to j i, but the depth of the passive wedge and, hence, BC continue to grow. The formulation above reflects that the near surface “ failure” wedge does not stop growing when all such sublayers reach their ultimate resistance at SL = 1 because the value of h at this time is not limited. Additional load applied at the pile head will merely cause the point at zero deflection and, therefore, h to move down the pile. More soil at full strength ( SL = 1) will be mobilized to the deepening wedge as BC, therefore, pult will increase until either flow around failure or a plastic hinge occurs. ( p ) = ( ) BC S + 2 ( ) D S ult i hf i i 1 f i 2 D s t ( 2.46) 23 Recognize that flow around failure occurs in any sublayer when it is easier for the sand at that depth to flow around the pile in a local bearing capacity failure than for additional sand to be brought to failure and added to the already developed wedge. However, the value at which flow failure occurs [ Ai = ( Ault) i , ( pult) i = ( D s hf) i ( Ault) i D] in sand is so large that it is not discussed here. Alternatively, a plastic hinge can develop in the pile when the pile material reaches its ultimate resistance at a time when SLi £ 1 and Ai < ( Ault) i. In this case, h becomes fixed, and BCi and pi will be limited when SLi becomes equal to 1. 2.10.2 Ultimate Resistance Criterion of Clay Soil The situation in clay soil differs from that in sand and is given by Gowda ( 1991) as a function of the undrained strength ( Su) i of the clay sublayer. Consequently, Ault indicates the limited development of the sublayer wedge geometry for eventual development of flow around failure ( SLi = 1) and, consequently, the maximum fanning angle in that sublayer becomes fixed, possibly at a value j m £ ` j . If a plastic hinge develops in the pile at SLi less than 1, then h will be limited, but BC, and pi will continue to grow until Ai is equal to Ault or pi is equal to ( pult) i. 2.11 STABILITY ANALYSIS IN THE STRAIN WEDGE MODEL The objective of the SW model is to establish the soil response as well as model the soil pile interaction through the modulus of subgrade reaction, Es. The shape and the dimensions of the passive wedge in front of the pile basically depend on two types of stability which are the local stability of the soil sublayer and the global stability of the pile and the passive wedge. However, the global stability of the passive wedge ( p ) = 10 ( S ) D S + 2 ( S ) D S ult i u i 1 u i 2 ( 2.47) ( ) ( ) ( ) ( ) ( ) = 5 S + S D 2 S p = D p A = 1 2 u i ult i hf i ult i ult i D s ( 2.48) 24 depends, in turn, on the local stability of the soil sublayers. 2.11.1 Local Stability of a Soil Sublayer in the Strain Wedge Model The local stability analysis in the strain wedge model satisfies equilibrium and compatibility among the pile segment deflection, soil strain, and soil resistance for the soil sublayer under consideration. Such analysis allows the correct development of the actual horizontal stress change, D s h , pile side shear stress, t , and soil pile reaction, p, associated with that soil sublayer ( see Fig. 2.1). It is obvious that the key parameters of local stability analysis are soil strain, soil properties, and pile properties. 2.11.2 Global Stability in the Strain Wedge Model The global stability, as analyzed by the strain wedge model, satisfies the general compatibility among soil reaction, pile deformations, and pile stiffness along the entire depth of the developing passive wedge in front of the pile. Therefore, the depth of the passive wedge depends on the global equilibrium between the loaded pile and the developed passive wedge. This requires a solution for Eqn. 2.1. The global stability is an iterative beam on elastic foundation ( BEF) problem that determines the correct dimensions of the passive wedge, the corresponding straining actions ( deflection, slope, moment, and shear) in the pile, and the external loads on the pile. Satisfying global stability conditions is the purpose of linking the three dimensional strain wedge model to the BEF approach. The major parameters in the pile global stability problem are pile stiffness, EI, and the modulus of subgrade reaction profile, Es, as determined from local stability in the strain wedge analysis. Since these parameters are determined for the applied soil strain, the stability problem is no longer a soil interaction problem but a one dimensional BEF problem. Any available numerical technique, such as the finite element or the finite difference method, can be employed to solve the global stability problem. The modeled problem, shown in Fig. 2.4c, is a BEF and can be solved to identify the depth, Xo, of zero pile deflection. 2.12 VERIFICATION OF APPROACH Based on the SW model concepts presented in this chapter and Ashour et al. ( 1996), a computer program 25 ( SWSG) has been developed to solve the problem of a laterally loaded isolated pile and a pile group in layered soil ( Ashour et al. 1996). Any verification of the methodology and algorithms employed should incorporate comparisons to field and laboratory tests for single piles and pile groups. The results presented below demonstrate the capability of the SW model approach and SWM program ( Ashour et al 1997 and 1998) in solving problems of laterally loaded piles relative to different soil and pile properties. It should be noted that pile and soil properties employed with the SW model analyses for the following field tests are the same properties mentioned in the references below. 2.12.1 Mustang Island Full Scale Load Test on a Pile in Submerged Dense Sand ( Reese et al. 1974 and Cox et al. 1974) As reported by Reese et al ( 1974), a series of full scale lateral load tests was performed on two single piles in sand at Mustang Island near Corpus Christi, Texas in 1966. The results obtained from those tests were used to develop criteria for the design of laterally loaded piles in sand and to establish a family of p y curves at different depths in the sand soil. In addition, the field results were used to characterize the pile head load deflection curve at the ground surface. Pile Configuration and Material Properties Tests were performed on two 0.61 m outside diameter ( O. D.) steel pipe piles ( A 53) with a wall thickness of 9.5 mm. The two piles were driven to a penetration of 21 m below the ground surface. The two closed end piles were instrumented along their lengths for the measurement of bending moment. Each pile tested consisted of a 11.6 m uninstrumented section, a 9.75 m instrumented section, and a 3 m uninstrumented section. The piles maintained an approximate stiffness, EI, of 167168 kN m2. Connecting flanges of 91.5 x 51 x 3.81 cm were welded to the instrumented section and to the 3 m 26 section. Small holes were cut in the pile wall just below the diaphragm to allow water and air to escape from the bottom of the 11.65 m section during driving. More details about the lateral load testing can be obtained by referring to Cox et al ( 1974). Table 1. Pile Properties Employed in the SWM Program Pile Type Shape Length Diam. Wall thick. Stiffness, EI Head Fixity Steel Pipe Round 21 m 0.61 m 9.5 mm 167168 KN m2 Free head Foundation Material Characterization Two soil borings were taken at the test site which were at the Shell Oil Company battery of tanks on the Mustang Island near Port Arkansas, Texas. As mentioned in Cox et al ( 1974), a comparison of the logs of borings 1 and 2 indicate that there was a slight variation in the soil profile between the two locations. In the top 12.2 m of boring 1, the sand strata was classified as a fine sand, while the soil in the top 12.2 m of boring 2 was classified as a silty fine sand. This difference in soil material was also reflected in the plot of the number of blows, N, of the standard penetration test ( Cox et al, 1974). The N values at boring 2 from 0 to 12.2 m are generally lower than those from boring 1. The sand from 0 to 6.1 m was classified as a medium dense sand, from 6.1 to 12.2 m as a dense sand, and from 15.2 m to 21.4 m as a dense sand. Laboratory tests were run on samples from boring 1 obtained using a piston sampler. More details of soil properties and the laboratory tests are documented in Cox et al ( 1974). The angle of internal friction was found to be 39 degree and the submerged unit weight of sand was 27 10.3 kN/ m3. The axial strain of the sand at 50 percent stress level, e 50, characterized based on Fig. 2.14 was 0.003 based on the assessed sand. Table 2. Soil Properties Employed in the SWM Program Soil type Thickness Effective Unit Weight Friction Angle, f e 5 0 Medium dense 21 m 10.3 kn/ m3 39 degree 0.003 Figure 2.17 presents a comparison of field results versus SW model results and results obtained using the computer program COM624 ( Reese 1977). Note that it is from this specific field test that the COM624 p y curves for sand were derived and, therefore, a good correspondence between COM624 and measured results is to be expected. The SW model results of pile head response shown in Fig. 2.17 are in excellent agreement at lower pile head deflections ( lower strain levels) and within 5 percent at higher levels of deflection ( higher strain levels). The SW model predicted maximum moment of Fig. 2.17 is in excellent agreement with measured results throughout. 2.12.2 Pyramid Building at Memphis, Tennessee, Full Scale Load Test on a Pile in Layered Clay Soil ( Reuss et al. 1992) A lateral load test was performed on a full scale pile in downtown Memphis. In order to improve the lateral capacity of the piles associated with this building, 1.8 meters of soft soil around the piles was removed and replaced with stiff compacted clay. Since the improved soil profile consisted of different types of soil, the corresponding test represents a layered field case study. Pile Configuration and Material Properties A 400 mm diameter reinforced concrete pile was installed to a total penetration of 22 meters. An inclinometer casing was installed in the pile to measure the lateral deflection. For a composite material such as reinforced concrete, the pile stiffness, EI, is a function of bending moment on the pile cross section. The experimental values of EI as a function of the bending moment are reported by Reuss et al. ( 1992). The 28 selected value of EI lies, in general, between the uncracked EI value and the cracked EI value. An average value for EI equal to 38,742 m2 kN was characterized for the pile. Additional concrete was cast around the pile to restrain it against excessive deflection when it was reloaded, and the pile head was free to rotate. Table 3. Pile Properties Employed in the SWM Program Pile Type Shape Length Diam. Average Stiffness, EI Head Fixity R/ C Rounded 22 m 0.4 m 38,742 KN m2 Free head Foundation Material Characterization The lateral load test conducted was performed at a location where the subsurface soil conditions could be approximated using information from a nearby soil boring. The soil profile, which consisted of different types of soils at this site, was the main advantage of this pile test. As documented by Reuss et al ( 1992), the top 1.8 meters of loose soil was replaced with a compacted gravely clay for the lateral load test. The fill soil consisted of cinders, bricks, concrete, gravel, and sand intermixed with varying percentages of clay to 1.8 meters below the ground surface. The first soil stratum ( fill soil) exhibited an undrained shear strength ( Su) of 47.9 kPa, a soil density ( g ) of 18.08 kN/ m3, and an e 50 of 0.005. The fill soil was underlain by soft to firm dark gray clay and silt clay with occasional silt and sand lenses. This soil layer ( the second stratum) extended from approximately 1.8 to 13.1 meters below the ground surface. The second stratum of soil exhibited an Su of 24 kPa, a g of 9.11 kN/ m3 , and an e 50 of 0.02. Standard penetration N values for stratum 2 varied from 3 through 10 with an average of about 5 blows per 0.3 meters. A third stratum between a depth of 13.1 meters and 19.9 meters below the ground surface had a reported Su value of 38.3 kPa, g of 9.11 kN/ m3 , and an e 50 value of 0.01 were reported. This stratum exhibited a greater frequency of silty and clayey sand lenses and increased strength as evidenced by penetration resistance N values ranging from 4 to 16 and averaging 10. The fourth stratum lay a depth of 19.9 meters and consisted of stiff silty clay and silty sand lenses. This stratum exhibited an Su value of 71.8 kPa, g of 9.11 kN/ m3 , and e 50 29 of 0.005. Table 4. Soil Properties Employed in the SWM Program Soil Layer # Soil type Thickness Effective Unit Weight e 5 0 Su 1 Sand mixed with clay, ciders and gravel 1.8 m 18.08 kN/ m3 0.005 47.9 kPa 2 Dark gray clay and silt clay 11.3 m 9.11 kN/ m3 0.02 24 kPa 3 Silty clay sand 6.8 m 9.11 kN/ m3 0.01 38.3 kPa 4 stiff silty clay Below 19.9 m depth 9.11 kN/ m3 0.005 71.8 kPa The soil properties of the fill soils and the second stratum ( the natural clay soil) were modified by Reuss et al. ( 1992) to force good agreement between the results assessed with COM624 ( Reese 1977) and the field results ( see Fig. 2.18a). The measured values of the undrained shear strength of the first and second strata were increased by 40 percent and 20 percent, respectively, to achieve such agreement. The measured soil properties were employed with the SW model to analyze the response of the pile in the improved soil profile. Figure 2.18a shows good agreement between the measured values and SW model predicted pilehead response in the improved soil profile. Figure 2.18b shows the pile head response predicted by COM624 and SW model analysis for the same pile in the original soil profile ( natural clay at its measured undrained strength with no fill layer). 2.12.3 Sabine River Full Scale Load Tests on a Pile in Soft Clay ( Matlock 1970) The benefit of the Sabine River tests derives from having load tests on piles of both free and fixed head conditions. Note that the results of the free head test were performed to establish the p y curve criteria for 30 piles in soft clay ( Matlock 1970). Pile Configuration and Material Properties The same pile was driven twice, and two complete series of static and cyclic loading tests were performed at the Lake Austin site and then at the Sabine River site. Only the static loading tests are considered in this study. The driven pile was a steel pipe pile of 0.32 m diameter and a 12.8 m embedded length. The pile maintained an approximate stiffness, EI, of 31,255 kN m2. The piles was tested under free head conditions at both sites ( Lake Austin and Sabine River) and fixed head conditions at Sabine River site. The Sabine River tests were used to develop the p y curves for short term static loading in soft clay. Table 5. Pile Properties Employed in the SWM Program Pile Type Shape Length Diam. Wall thick. Stiffness, EI Head Fixity Steel Pipe Rounded 12.8 m 0.32 m 12.75 mm 31,255 KN m2 Free head ( 1) Fixed head ( 2) Foundation Material Characterization As noted in Matlock et al ( 1970), extensive sampling and testing of the soils were undertaken at the Sabine River site. In situ vane shear tests as well as laboratory triaxial compression tests were performed to determine stress strain characteristics. Sabine clay is typical of a slightly overconsolidated marine deposit, and exhibited lower Vane shear strengths averaging about 14.33 kPa in the significant upper zone. According to Matlock et al ( 1970), the values of e 50 for most clays may be assumed to be between 0.005 and 0.02. An intermediate value of 0.01 is probably satisfactory for most purposes. Average values of 0.012 and 0.007 for e 50 were estimated from the soil stress strain curves at Sabine River. 31 Table 4. Soil Properties Employed in the SWM Program Soil Layer # Soil type Thickness Effective Unit Weight e 5 0 Su 1 Soft clay 12.8 m 7.8 kN/ m3 0.007 14.33 kPa As seen in Fig. 2.19a, the predicted free head SW model results are in good agreement with the observed results for the Sabine River site. At higher levels of deflection, the results calculated using the SW model fall approximately 5 to 10 percent below those measured in the field. By comparison, the SW model predicted and the observed fixed head pile response are in excellent agreement as shown in Fig. 2.19b. SW model results were established for two cases of the clay based on having a single average Su and, separately, for a varying Su. 2.13 SUMMARY The SW model approach presented here provides an effective method for solving the problem of a laterally loaded pile in layered soil. This approach assesses its own nonlinear variation in modulus of subgrade reaction or p y curves. The strain wedge model allows the assessment of the nonlinear p y curve response of a laterally loaded pile based on the envisioned relationship between the three dimensional response of a flexible pile in the soil to its one dimensional beam on elastic foundation parameters. In addition, the strain wedge model employs stress strain strength behavior of the soil as established from the triaxial test in an effective stress analysis to evaluate mobilized soil behavior. Compared to empirically based approaches which rely upon a limited number of field tests, the SW approach depends on well known or accepted principles of soil mechanics ( the stress strain strength relationship) in conjunction with effective stress analysis. Moreover, the required parameters to solve the problem of the laterally loaded pile are a function of basic soil properties that are typically available to the designer. 32 Fig. 2.1 The Basic Strain Wedge in Uniform Soil 33 Fig. 2.2 Deflection Pattern of a Laterally Loaded Long Pile and the Associated Strain Wedge Fig. 2.3 The Linearized Deflection Pattern of a Pile Embedded in Soil Using the Multi Sublayer Strain Wedge Model 34 Fig. 2.4 Soil Pile Interaction in the Multi Sublayer Technique 36 Fig. 2.5 The Proposed Geometry of the Compound Passive Wedge 37 Fig. 2.6 Distortion of the Wedge a), The Associated Mohr Circle ofStrain b), and the Relationship Between Pile Deflection and Wedge Distortion c) 38 Fig. 2.7 Relationship Between Horizontal Stress Change, Stress Level, and Mobilized Friction Angle 39 40 Fig. 2.8 The Developed Stress Strain Relationship in 41 Fig. 2.9 The Nonlinear Variation of Stress Level Along the Depth of Soil at Constant Strain _ Fig. 2.10 The Employed Side Shear Stress Displacement Curve in Clay 42 Fig. 2.11. The Nonlinear Variation of Shear Stress Level ( SLt) Along the Depth of Soil Fig. 2.12 Relationship Between e 50, Uniformity coefficient ( Cu) and Void Ratio ( e) ( Norris 1986) 0 0.25 0.5 0.75 1 1.25 1.5 Void Ratio, e 0 0.25 0.5 0.75 1 1.25 1.5 g 50 (%) Uniformity Coefficient, Cu 10 6 2 1.18 43 Fig. 2.13 Relationship Between Plasticity Index ( PI) and Effective j ( US Army Corps of Engineers 1996) Fig. 2.14 Relationship Between e 50 and Undrained Shear Strength , Su 50 u 0.00 0.01 0.02 g 100 1000 10000 200.00 300.00 500.00 2000.00 3000.00 5000.00 S ( psf) Range of Suggested Values ( After Reese 1980) 44 ( Evans and Duncan 1982) 45 Fig. 2.15 Relationship Between Effective Stress and Total Stress Conditions 46 Fig. 2.16 The Assembling of Pile Head Deflection Using the Multi Sublayer Technique 47 48 Strain Wedge Model Measured COM624 4 Fig. 2.17 The Measured and Predicted Response of a Laterally Loaded Pile in Sand at the Mustang Island Test. 0 2 4 6 Maximum Moment, kN cm * 10 0 100 200 300 400 Pile Head Load, Po, kN 0 1 2 3 4 5 Ground Deflection, Yo, cm 0 100 200 300 400 Pile Head Load, Po, kN A B 49 SW Model Measured COM624 ( Using Modified Soil Properties) R/ C Rounded Pile ( EI) = 1.35E7 kips in2 D = 15.75 in. L = 72 f t Improved Soil Strain Wedge Model COM624 ( Reuss et al., 1992) Original Soil Fig. 2.18 The Measured and the Predicted Response of the Loaded Pile in the Improved and the Original Soils at the Pyramid Building, Memphis, Test. 0 1 2 3 4 Ground Deflection, Yo, cm 0 50 100 150 200 Pile Head Load, Po, kN 0 4 8 12 Ground Deflection, Yo, cm 0 50 100 150 200 Pile Head Load, Po, kN A B 50 Fig. 2.19 The Measured and the SW Model Results of the Loaded Pile at the Sabine River Test. 0 2 4 6 8 Ground Deflection, Yo, cm 0 50 100 150 Pile Head Load, Po, kN SW with Su average SW with Su Profile Measured Free Head Pile Fixed Head Pile 0 2 4 6 8 10 Ground Deflection, Yo, cm 0 25 50 75 100 Pile Head Load, Po, kN A B 51 46 CHAPTER 3 PILE GROUP IN LAYERED SOILS 3.1 INTRODUCTION As presented in Chapter 2, the prediction of single pile response to lateral loading using the SW model correlates traditional one dimensional beam on an elastic foundation ( BEF) response to threedimensional soil pile interaction. In particular, the Young's modulus of the soil is related to the corresponding horizontal subgrade modulus; the deflection of the pile is related to the strain that exists in the developing passive wedge in front of the pile; and the BEF line load for a given deflection is related to the horizontal stress change acting along the face of the developing passive wedge. The three dimensional characterization of the laterally loaded pile in the SW model analysis provides an opportunity to study the interference among the piles in a pile group in a realistic fashion. The influence of the neighboring piles on an individual pile in the group will be a function of soil and pile properties, pile spacing, and the level of loading. These parameters are employed together in the SW model analysis to reflect the pile soil pile interaction on pile group behavior. The work presented illustrates the links between the single pile and the pile group analysis. The pile group procedure commonly used today employs the p y multiplier technique ( Brown et al. 1988). Such procedure is based on reducing the stiffness of the traditional ( Matlock Reese) p y curve by using a multiplier ( fm < 1), as seen in Fig. 3.1. The value of the p y curve multiplier should be assumed and is based on the data collected from full scale field tests on pile groups which are few ( Brown et al. 1988). Consequently, a full scale field test ( which is costly) is strongly recommended in order to determine the value of the multiplier ( fm) of the soil profile at the site under consideration. Moreover, the suggested value of the multiplier ( fm) is taken to be constant for each soil layer at all levels of loading. 47 In essence, this is quite similar to the traditional approach given in NAVFAC ( DM 7.2, 1982) in which the subgrade modulus, Es, is reduced by a factor ( Rm) taken as a function of pile spacing ( Rm = 1 at 8 diameter pile spacing varying linearly to 0.25 at 3 diameters). The difference is that fm has been found to vary with pile row ( leading, second, third and higher); and is taken to be constant with lateral pile displacement, y. By contrast, Davisson ( 1970) suggested that Rm should be taken constant with pile head load such that displacement y increases. In any case, neither fm, nor Rm, reflects any change with load or displacement level, soil layering, pile stiffness, pile position ( e. g. leading corner versus leading interior pile, etc.), differences in spacing both parallel and normal to the direction of load, and pile head fixity. As seen in Fig. 3.2, the interference among the piles in a group varies with depth, even in the same uniform soil, and increases with level of loading as the wedges grow deeper and fan out farther. Therefore, the use of a single multiplier that is both constant with depth and constant over the full range of load/ deflection would seem to involve significant compromise. The assessment of the response of a laterally loaded pile group based on soil pile interaction is presented herein. The strain wedge ( SW) model approach, developed to predict the response of a long flexible pile under lateral loading ( Ashour et al. 1998; and Ashour and Norris 2000), is extended in this paper to analyze the behavior of a pile group in uniform or layered soil. Several field and experimental tests reported in the literature are used to demonstrate the validity of the approach. 3.2 CHARACTERIZATION OF PILE GROUP INTERFERENCE The pile group is characterized in terms of the three dimensional pile soil pile interaction ( Pilling 1997) and then converted into its equivalent one dimensional BEF model with associated parameters ( i. e. an ever changing modulus of sugrade reaction profile). Therefore, the interference among the piles in a group is determined based on the geometry of the developing passive wedge of soil in front of the pile in addition to the pile spacing. A fundamental concept of the SW model is that the size and shape ( geometry) of the passive wedge of soil changes in a mobilized fashion as a function of 48 both soil and pile properties, at each level of loading, and is expressed as follows: As seen in Fig. 3.3, BC is the width of the wedge face at any depth, x. D is the width of the pile cross section, h is the current depth of the passive wedge which depends on the lateral deflection of the pile and, in turn, on the pile properties such as pile stiffness ( EI) and pile head fixity. ϕm is the mobilized fan angle of the wedge ( also the mobilized effective stress friction angle of the soil) and is a function of the current stress level ( SL) or strain ( ε) in the soil as presented by ( Ashour et al. 1998). The overlap of shear zones among the piles in a group varies along the length of the pile as shown in Figs. 3.3 and 3.4. Also, the interference among the piles grows with the increase in lateral load. The modulus of subgrade reaction, which is determined based on the SW model approach, will account for the additional strains ( i. e. stresses) in the adjacent soil due to pile interference within the group ( Figs. 3.4 and 3.5). Thus the modulus of subgrade reaction ( i. e. the secant slope of the p y curve) of an individual pile in a group will be reduced in a mobilized fashion according to pile and soil properties, pile spacing and position, the level of loading, and depth, x. No single reduction factor ( fm or Rm) for the p y curve ( commonly, assumed to be a constant value with depth and level of loading) is needed or advised. The SW model also allows direct evaluation of the nonlinear variation in pile group stiffness as required, for instance, for the seismic analysis of a pile supported highway bridge. The multi sublayer technique developed by Ashour et al. ( 1996 and 1998) and presented in Chapter 2 provides a means to determine the interference among the passive wedges of piles in a group and the additional stress/ strain induced in the soil in these wedges. As seen in Fig. 3.3, the soil around the piles in the group interferes horizontally with that of adjacent piles by an amount that varies with 2 = 45 + m m ϕ β ( 3.1) β ϕ m m BC = D + ( h  x) 2 tan tan ( 3.2) 49 depth. The multi sublayer technique allows the SW model to determine the overlap of the wedges of neighboring piles in different sublayers over the depth of the interference as shown in Figs. 3.4 and 3.5. This provides a great deal of flexibility in the calculation of the growth in stress ( and, therefore, strain) in the overlap zones which increases with the growth of the passive wedges. The main objective in the calculation of the area of overlap among the piles is to determine the increase in soil strain within the passive wedge of the pile in question. A value of horizontal soil strain ( ε) is assumed for the soil profile within the developing passive wedge. The response of a single pile ( similar to the piles in the group) in the same soil profile is determined at this value of soil strain. The shape and the dimensions of the mobilized passive wedge are assessed ( i. e. ϕm, βm, h and BC in Fig. 3.3) as presented in Chapter 2. This will include the values of stress level in each soil sublayer i ( SLi), Young’s modulus ( Ei), and the corresponding modulus of subgrade reaction ( Es) i. Wedges will overlap and interact with the neighboring ones, as seen in Figs. 3.3 and 3.5. At a given depth ( see Fig. 3.5), zones of overlap will exhibit larger values of soil strains and stresses. The increase in average soil strain attributable to the passive wedge of a given pile will depend upon the number and area of interfering wedges overlying the wedge of the pile in question ( Fig. 3.6). Such interference depends on the position of the pile in the group. The type of pile ( by position) is based on the location of the pile by row ( leading/ trailing row) and the location of the pile in its row ( side/ interior pile) as seen in Fig. 3.5. The average value of deviatoric stress accumulated at the face of the passive wedge at a particular soil sublayer i ( sand or clay) is 50 The average stress level in a soil layer ( SLg) due to passive wedge interference is evaluated based on the following empirical relationship, where j is the number of neighboring passive wedges in soil layer i that overlap the wedge of the pile in question. R is the ratio between the length of the overlapped portion of the face of the passive wedge and the total length of the face of the passive wedge ( BC). R ( which is less than 1) is determined from all the neighboring piles to both sides and in front of the pile in question ( Fig. 3.6). SLg and the associated soil strain ( εg) will be assessed for each soil sublayer in the passive wedge of each pile in the group. εg is ≥ ε of the isolated pile ( no wedge overlap) and is determined based on the stress strain relationship ( σ vs. ε) presented in Chapter 2. It should be noted that the angles and dimensions of the passive wedge ( ϕm, βm, and BC) obtained from Eqns. 3.1 through 3.4 will be modified for group effect according to the calculated value of SLg and εg ( Fig. 3.7). For instance, the relationship between the corresponding stress level ( SLg) and the associated mobilized effective stress friction angle ( ϕm) in a soil sublayer i is where ( Δσh) g is the current horizontal stress change ( due to pile head lateral load and pile group interference), and Δσhf is the unchanged value of the deviatoric stress at failure for the full friction angle ϕ. The mobilized friction angle ϕm calculated in Eqn. 3.5 reflects the stresses in the soil ( sand or clay) around the pile in question at depth x for the corresponding pile head ( group) deflection with ( Δσ h ) g = SLg Δσ hf ( 3.3) ( SL ) = SL ( 1 + R ) 0.5 1 g i i Σ j ≤ ( 3.4)  1 2 45 +  1 2 45 + ( ) = ( ) ( SL ) = 2 i 2 m i hf h g i g i Δ Δ ϕ ϕ σ σ tan tan ( 3.5) 51 consideration of the stresses from neighboring piles ( Figs. 3.5 and 3.6). Consequently, the geometry of the passive wedge is modified according to the current state of soil stress and strain ( Fig. 3.7). It should be noted that the behavior of clay is assessed based on the effective stress analysis in which the developing excess porewater pressure is evaluated in Chapter 2 and Ashour et al. ( 1996). 3.3 EVALUATION OF THE YOUNG’S MODULUS, Eg The change in the soil Young’s modulus and, therefore, the change in moulus of subgrade reaction in each sublayer due to group interference is assessed. Once the modified variation of the modulus of subgrade reaction along the individual pile is predicted, the pile is analyzed as an equivalent isolated pile ( considering all piles in the group have the same pile head deflection). Based on the modified value of soil strain assessed at depth x ( for the wedge of the pile of interest) at the current level of loading, the value of Young’s modulus, ( Eg) i, of the soil sublayer i is expressed, i. e. It should be noted the Young’s modulus ( Eg) calculated using Eqn. 3.6 results from the original strain in the passive wedge ( ε) as an isolated pile and the additional soil strain ( Δε) which develops due to overlap zones between the pile in question and its neighboring piles ( Fig. 3.8), i. e. According to the amount of interference among the piles in the group, the value of the Young’s modulus ( Eg) should be less or equal to the associated modulus ( E) for the isolated pile. 3.4 EVALUATION OF THE MODULUS OF SUBGRADE REACTION, Esg Based on the concepts of the SW model, the modulus of subgrade reaction for an individual pile in a group can be expressed as ( ) ( SL ) ( ) ( E ) = g i g i hf i g i ε Δσ ( 3.6) ( ε g ) i = ε i + Δε i ( 3.7) 52 where x is the depth of a soil sublayer i below the pile head. δ is the linearized deflection angle of the deflection pattern as presented by Ashour et al. ( 1996). Ag is a parameter that governs the growth of the passive wedge and flow around failure, and is a function of soil and pile properties ( Ashour and Norris 2000). S1 and S2 are shape factors equal to 0.75 and 0.5, respectively, for a circular pile cross section, and equal to 1.0 for a square pile ( Briaud et al. 1984). τ is the mobilized shear stress along the pile sides in the SW model ( see Fig. 3.7) and is defined according to the soil type ( sand or clay). ϕs is the mobilized side shear angle, SLt is is the stress level of shear along the pile sides, and τult is ( ) ( h  x ) ( A ) D ( ) ( E ) = y E = p i i g i g i g i i i s g i δ ε ( 3.8) [ ] [ ( ) ] + 2 S D = BC S ( ) p D ( A ) = h g i i 1 i 2 h g i i g i σ τ Δσ Δ / ( 3.9) = ( ) ( ); where ( ) = 2 ( ) sand τ i σ vo i tan ϕ s i tan ϕ s i tan ϕ m i tanϕ i ( 3.10) τ i = ( SLt ) i ( τ ult ) i clay ( 3.11) Therefore, ( )( ) sand ( ) + 2 S D h  x 2 ( A ) = S 1 + h g vo s i 2 i m m i g i 1 Δ σ tanβ tanϕ σ tanϕ ( 3.12) ( )( ) clay SL + S SL D h  x 2 ( A ) = S 1 + g t i 2 i m m i 1 i g tanβ tanϕ ( 3.13) 53 the ultimate shear resistance ( Coyle Reese 1966, and Ashour et al. 1998). Compared to the case of a single pile, the developing passive wedge of a pile in a group will be larger than or equal to that of the single pile ( depending on the amount of pile interference). However, the criteria presented in Chapter 2 and Ashour and Norris ( 2000) continue to govern the development of flow around failure; and variation of the BEF soil pile reaction ( p) and lateral deflection ( y) in the single pile analysis continue to be employed in the pile group analysis. It should be expected that the resulting modulus of subgrade reaction of a pile in a group, Esg, is equal to or softer than the Es of an isolated pile at the same depth ( Fig 3.9). The value of Es will vary with the level of loading and the growth of the soil stress in the developing passive wedge. Thus, there is no constant variation or specific pattern for changes in Es of the individual piles in the pile group. Based on the predicted values of Esg, the approach presented has the capability of assessing the p y curve for any pile in the group. The modulus of subgrade reaction of a pile in a group should reflect the mutual resistance between the soil and the pile. However, a portion of the pile deformation ( Δyi) results from the additional stresses in the soil ( and, therefore, strains, Δε) which result from the effect of the neighboring piles ( Figs. 3.5 and 3.6). Therefore, under a particular lateral load, the pile in the group will yield deflections more than those of the single pile. The additional deflection at any pile segment, ( Δyi), due to Δεi derives solely from the presence of neighboring piles, not the pile in question. The soilpile reaction ( p) is affected by the changes in stress and strain in the soil, and the varying geometry of the passive wedge. Having reduced values of Es along individual piles in the group, each pile is then analyzed as an equivalent isolated pile by BEF analysis. The piles in a group, at a particular step of loading, must experience equal deflections at the pile cap. For each pile in the group, the interference among the piles and the changes in the Es profile ( i. e geometry and dimensions of the passive wedge, and the internal stresses) will continue in an iterative process until the pile in question provides a pile head 54 deflection equal to that of the group. As a reference, the group deflection is linked to the pile head deflection ( Yo) of the isolated pile at the original soil strain ( ε). This technique provides great flexibility to analyze each pile in the group independently in order to develop equal pile head deflections ( group deflection) which are the shared factor among the piles in a group. 3.5 CASE STUDIES The original SW model program ( Ashour et al. 1997 and 1998) for analyzing lateral loaded piles has been modified to incorporate the technique presented above. The modified SWM program allows the assessment of the lateral response ( deflection, moment and shear force distribution) of an isolated pile and a pile group including the p y curve along the length of the isolated pile and the individual piles in the pile group. 3.5.1 Full Scale Load Test on a Pile Group in Layered Clay A static lateral load test was performed on a full scale 3 x 3 pile group having a three diameter center to center spacing ( Rollins et al. 1998). The driven pipe piles were 0.305 m I. D., 9.5 mm wall thickness, and 9.1 m in length. The Young’s modulus of the steel was 200 GPa, and the yield stress was 331 MPa. The soil profile along the length of the pile consisted of different types of clay and sand silt soils as described by Rollins et al. ( 1998). Figure 3.10 shows good agreement between the measured and SWM program predicted response for the single pile and average pile in the group ( pile group response is 9 times the average load at the same deflection). The P Multiplier was used by Rollins et al. 1998 to differentiate between the average response of different piles by row. Accordingly, fm values were varied arbitrarily to obtain the best match between the traditionally assessed p y curve and averaged observed behavior. The predicted response assessed using SWM, averaged by pile row, shows reasonable agreement with the reported behavior as seen in Fig. 3.11. The deviation between predicted and observed behavior in the 10 to 40 mm range for the isolated pile carries over to that of the average pile in the group over the same range. SWM response was obtained based on the given pile and soil properties, and pile group layout; no adjustment was made to obtain better fit. 55 3.5.2 Full Scale Load Test on a Pile Group in Sand A full scale lateral load test on a 3 x 3 pile group in sand overlying overconsolidated clay was conducted at the University of Houston, Texas ( Morrison and Reese, 1986). The results obtained from this load test were used to develop values of fm for use in the P multiplier approach for laterally loaded pile groups in sand ( Brown et al. 1988). This pile group of three diameter pile spacing was embedded in approximately 3 m of a dense to very dense uniform sand overlying an overconsolidated clay. The piles consisted of steel pipe with an outside diameter of 0.275 m, a wall thickness of 9.3 mm, a 13 m embedded length, and a bending stiffness ( EI) of 1.9 x 104 kN m2. The soil properties, including the buoyant unit weight and the angle of internal friction suggested by Morrison and Reese ( 1986), were used in the SW model analysis. Figures 3.12 and 3.13 show a comparison between the field data and the results obtained using the SWM program. As seen in Figs. 3.12 and 3.13, the observed and predicted responses of an average pile in the tested pile group are in good agreement. The good match of the predicted and observed behavior carries over to the average pile in the group. 3.5.3 Full Scale Load Test on a Pile Group in Layered Clay A full scale 3 x 3 pile group was driven in layered overconsolidated clay ( Brown and Reese 1985). The pile group tested had a three diameter pile spacing and was laterally loaded 0.3 m above ground surface. The nine pipe piles tested had the same properties as the piles used in the preceding case study. The soil properties ( ε50, the soil unit weight, and the undrained shear strength of clay) evaluated by Brown and Reese were employed in the SW model analysis. As shown in Figs. 3.14 and 3.15, the SW model provides good agreement with observed behavior for both the single and average pile in the group for pile head load versus deflection and pile head load versus maximum bending moment. It should be noted that this case represents a layered clay profile which exhibits different levels of wedge interference in each soil layer that then changes with the level of loading. 56 The procedure presented here has the capability to predict the pile head response, deflection, and bending moment for every individual pile in the group ( type 1 through type 4 based on pile location, as seen in Fig. 3.5) not just the average pile response. Previous comparisons in terms of average pile in the group or average by row reflect what is reported in the literature. Likewise, the SWM program can assess the additional contribution to pile group resistance due to the presence of an embedded pile cap ( not presented in this study) at any level of lateral loading. The effect of pile cap resistance on the lateral resistance of pile group can be judged from the following case study. 3.5.4 Full Scale Load Test on a Pile Group with a Pile Cap in Layered Soil A series of high amplitude load tests were performed on the Rose Creek bridge near Winnemucca, Nevada ( Douglas and Richardson 1984). The stiffnesses of four pile groups, with pile caps of the foundation system were backfigured from system identification analysis of the collected accelerometer data. The soil profile and results of the tests are discussed by Norris ( 1994). Although the contribution of the embedded pile cap to the lateral resistance of the pile group has not been discussed in this paper, its effect on the lateral stiffness of pile groups is undertaken in the results predicted using the SWM program. Piers 1 and 4 are each supported by a 3 x 5 pile group with 3 diameter pile spacings embedded in layered silt and clay soil, while piers 2 and 3 are each supported by a 4 x 5 pile group with 3 diameter pile spacings in the same layered silt and clay soil. Pile caps ( 4.57 x 2.75 m and 1.3 m thickness) associated with piers 1 and 4 are founded at 1.5 m depth below finished grade in a medium dense sand silt soil. Pile caps ( 4.57 x 3.65 m and 1.3 m thickness) associated with piers 2 and 3 are founded at 0.92 m below finished grade in a medium dense sand silt soil. The piers extend from the pile caps have a width of 1.22 m. The piles are steel pipe piles of 0.32 m outer diameter backfilled with concrete and a bending stiffness of 3.38 x 104 kN m2. The piles associated with piers 1 through 4 were driven to 8, 7.8, 7.3 and 7 m below the bottom of the pile cap. All pile groups were loaded laterally in the direction 57 normal to length ( long side) of the pile cap. Full details on soil and pile properties are presented by Norris ( 1994). The pile heads are embedded 0.3 m into the pile cap. The piles in the group are treated as fixed head piles in the SW Model analysis. Even if the depth of pile head embedment into the pile cap was not adequate to provide complete restriction on the pile head rotation, the pile head in the group would exhibit fixed head conditions at the very low values of lateral deflection observed during the bridge load tests. Figure 3.16 shows the agreement between the measured ( backcalculated) and predicted pile group stiffnesses for groups 1 through 4 using the SWM program. It should be noted that the pile cap contribution to the total resistance of the group is a function of the pile cap dimensions and its embedment depth, properties of surrounding soil, and the level of lateral loading. 3.5.5 Model Scale Load Test on a Pile Group in Loose and Medium Dense Sand A series of load tests were performed using centrifuge tests on a model isolated pile, and on a model 3 by 3 pile group with piles spaced at 3 and 5 pile diameters within the group, embedded in a poorly graded loose ( Dr = 33%) and medium dense sand ( Dr = 0.55) ( McVay et al 1995). The prototype model piles, simulated using the centrifuge and a 1/ 45 ( i. e. 45g) scale consisted of steel pipe piles with a diameter of 0.43 m and an overall length of 13.3 m. The pile had a bending stiffness, EI, of 72.1 MN m2. The point of lateral load application to the pile groups was approximately 1.68 m above finished grade, while the point of lateral load application to the isolated pile was approximately 2.2 m above finished grade. Although a pile cap was associated with the pile group tests, McVay et al ( 1995) reported that the group tests simulated free headed piles. Very good agreement, between measured and predicted results, is shown in Figs. 3.17 and 3.18. Slight differences are observed between the measured and predicted capacity of the pile rows ( leading, middle and trailing rows) in the group. It should be noted that the procedure presented herein has the capability of assessing the capacity of three different pile rows ( leading, middle and 58 trailing rows). Therefore, 6 types of piles by position ( instead of 4 types as seen in Fig. 3.5) should be analyzed. However, at low and medium level of pile head deflection, no significant differences are observed between the lateral resistance of the middle and trailing row. 3.6 SUMMARY Assessment of the response of a laterally loaded pile group based on soil pile interaction is presented. The behavior of a pile group in uniform and layered soil ( sand and/ or clay) is predicted based on the strain wedge ( SW) model approach that was developed to analyze the response of a flexible long pile under lateral loading. Accordingly, the pile’s response is characterized in terms of three dimensional soil pile interaction which is then transformed into its one dimensional beam on elastic foundation equivalent with associated parameters ( modulus of sugrade reaction). The interference among the piles in a group is determined based on the geometry and interaction of the mobilized passive wedges of soil in front of the piles in association with the pile spacing. The overlap of shear zones among the piles in the group varies along the length of the pile and changes from one soil layer to another in the soil profile. Also, the interference among the piles grows with the increase in lateral loading, and the increasing depth and fan angles of the developing wedges. The modulus of subgrade reaction determined will account for the additional strains ( i. e. stresses) in the adjacent soil due to pile interference within the group. Based on the approach presented, the p y curve for individual piles in the pile group can be determined. The reduction in the capacity of the individual piles in a group compared to the isolated pile is governed by soil and pile properties, level of loading, and pile spacing. 59 Fig. 3.1 P Multiplier ( fm) Concept for Pile Group ( Brown et al. 1988) 60 Fig. 3.2 Lateral Interference Between Two Neighboring Piles 61 Fig. 3.3 Mobilized Passive Wedges and Associated Pile Group Interference Fig. 3.4 Front Overlap Among Soil Sublayers in Two adjacent Passive Wedges ( Section J J in Fig. 3.3.) Developing Passive Soil Wedges Pile in Question Pile Type 1 Pile Type 2 Pile Type 3 Pile Type 4 By Position Loading Direction Leading Row Trailing Row Trailing Row 62 Fig. 3.5 The Initial Interference Among Piles in a Pile Group at a Given Depth Fig. 3.6 Example of Overlap Ratio Calculation Among piles in a Pile Group ** At the same level of pile head deflection, fm for an isolated pile is ≤ fm for an individual pile in a pile group Pile in Question Loading Direction C D L1 L2 R1 = L1 CD R2 = L2 CD 63 Fig. 3.7 Stress and Geometry Change in a Slice of an Individual Pile in a Pile Group Fig. 3.8 Changes in Soil Young’s Modulus Due to Pile Interference in a Pile Group at a Particular Level Of Loading 64 Fig. 3.9 Change in the Modulus of Subgrade Reaction ( i. e. the p y Curve) due to Pile Interference in the Pile Group at Different Levels of Loading according to the SW Model 65 Leading Row SW Model Measured Single Pile Group 0 20 40 60 80 Deflection at Load Point, Yo, mm. 0 40 80 120 160 200 Average Load per Pile, Po, kN 0 20 40 60 80 Deflection at Load Point, Yo, mm. 0 50 100 150 Average Load per Row, Po, kN SW Model Measured Trailing Row Fig. 3.10 Average lateral load versus deflection curves for isolated pile and average pile in a 3 x 3 group ( after Rollins et al. 1998) Fig. 3.11 Measured and predicted average lateral load per row versus deflection ( after Rollins et al. 1998) 66 Single Pile Single Pile Group 0 20 40 60 Deflection at Load Point, Y o , mm. 0 25 50 75 100 Average Load per Pile, P o , kN Compression Stroke Tension Stroke 0 50 100 150 200 Maximum Bending Moment, kN m 0 25 50 75 100 Average Load per Pile, P o , kN Group SW Model Measured ( Compression Stroke) ( Tension Stroke) SW Model Measured ( Compression Stroke) ( Tension Stroke) Fig. 3.12 Lateral pile head lateral load vs. deflection for an isolated pile and an average pile in a 3 x 3 group in sand ( after Morrison and Reese, 1986) Fig. 3.13 Lateral load vs. maximum bending moment for isolated pile and an average pile in a 3 x 3 group in sand ( after Morrison and Reese, 1986) 67 Single Pile SW Model Measured Single Pile Group 0 20 40 60 80 100 Deflection at Load Point, Yo, mm. 0 20 40 60 80 100 Average Load per Pile, Po, kN Group 0 40 80 120 160 200 Maximum Bending Moment, kN m 0 50 100 150 Average Load per Pile, Po, kN SW Model Measured Fig. 3.14 Lateral load vs. deflection for isolated pile and an average pile in a 3 x 3 group in clay.( after Brown and Reese, 1985) Fig. 3.15 Lateral load vs. maximum bending moment for isolated pile and an average pile in a 3 x 3 group in clay.( after Brown and Reese, 1985). 68 0.001 0.01 0.1 1 10 Pile Head deflection, Yo , mm 0 100 200 300 400 500 Pile Group Stiffness, Kg , kN/ mm Measured and Predicted Stiffness of Pile Group 1 0.001 0.01 0.1 1 10 Pile Head deflection, Yo , mm 0 400 800 1200 1600 Pile Group Stiffness, Kg , kN/ mm Measured and Predicted Stiffness of Pile Group 2 0.001 0.01 0.1 1 10 Pile Head deflection, Yo , mm 0 400 800 1200 1600 2000 Pile Group Stiffness, Kg , kN/ mm Measured and Predicted Stiffness of Pile Group 3 0.001 0.01 0.1 1 10 Pile Head deflection, Yo , mm 0 100 200 300 400 500 Pile Group Stiffness, Kg , kN/ mm Measured and Predicted Stiffness of Pile Group 4 Measured SW Model Measured SW Model Measured SW Model Measured SW Model Fig. 3.16 Measured and Predicted Group Stiffness for Rose Creek Bridge Foundation ( After Norris 1994) 69 Pile Spacing = 3D Dr = 33% Pile Spacing = 5D Dr = 33% 0 20 40 60 80 100 Lateral Deflection, Yo, mm 0.0 0.4 0.8 1.2 1.6 Group Lateral Load, MN Fig. 3.17 A Pile Group Model ( 3x3) in Medium Loose Sand ( after McVay et al. 1995) 0 20 40 60 80 0.0 0.4 0.8 1.2 Group Lateral Load, MN Capacity % Row Location Lead Middle Tail Predicted 36 32 32 Measured 37 33 30 33 33 33 35 33 31 70 o Pile Spacing = 3D Dr = 55% Pile Spacing = 5D Dr = 55% Fig. 3.18 A Pile Group Model ( 3x3) in Medium Dense Sand ( after McVay et al. 1995) Capacity % Row Location Lead Middle Tail Predicted 36 32 32 Measured 41 32 27 33 33 33 36 33 31 0 20 40 60 80 100 0.0 0.4 0.8 1.2 1.6 Lateral Load, MN 0 20 40 60 80 100 Lateral Deflection, Yo, mm 0.0 0.4 0.8 1.2 1.6 2.0 Group Lateral Load, MN 69 70 CHAPTER 4 NUMERICAL MATERIAL MODELING 4.1 INTRODUCTION Deformations in any structural element depend upon the characteristics of the load, the element shape and its material properties. With laterally loaded piles and shafts, the flexural deformations are based on the applied moment and the flexural stiffness of the pile at the cross section in question. In addition, the flexural stiffness ( EI) of the pile is a function of the Young’s modulus ( E), moment of inertia ( I) of the pile cross section and the properties of the surrounding soil. Given the type of material, concrete and/ or steel, the properties of pile material vary according to the level of the applied stresses. Behavior of piles under lateral loading is basically influenced by the properties of both the soil and pile ( pile material and shape). The nonlinear modeling of pile material, whether it is steel and/ or concrete, should be employed in order to predict the value of the lateral load and the realistic associated bending moment and pile deflection especially at large values of pile head deflection and the onset of pile material failure. It is known that the variation in the bending stiffness ( EI) of a laterally loaded pile is a function of the bending moment distribution along the pile ( moment curvature, M F , relationship) as seen in Fig. 4.1. Consequently, some of the pile cross sections which are subjected to high bending moment experience a reduction in bending stiffness and softer interaction with the surrounding soil. Such behavior is observed with drilled shafts and steel piles at advanced levels of loading and has an impact on the lateral response and capacity of the loaded pile. The pile bending stiffnesses along the deflected pile change with the level of loading, the M F relationship of the pile material, and the soil reaction which affects the pattern of pile deflection. Therefore, the equilibrium among the distributions of pile deflection, bending moment, bending stiffness, and soil reaction along the pile should be maintained. 71 In the case of a steel pile, the Young’s modulus remains constant ( elastic zone) until reaching the yield stress, fy ( indicating the initial yielding), at which time the steel starts to behave elastic plastically with different values of the secant Young’s modulus. Once a plastic hinge develops, the pile cross section responds in plastic fashion under a constant plastic moment. But, in the case of a concrete pile or shaft, the stress strain relationship varies in a nonlinear fashion producing a simultaneous reduction in Young’s modulus and, in turn, the stiffness of the pile cross section. Furthermore, once it reaches a critical value of strain, the concrete ruptures catastrophically. The technique suggested by Reese ( 1984), which employs the Matlock Reese p y curves, requires separate evaluation of the M F relationship of the pile cross section and then adoption of a reduced bending stiffness ( EIr) to replace the original pile bending stiffness ( EI). The suggested procedure utilizes this reduced bending stiffness ( EIr) over the full length of the pile at all levels of loading. Assuming a reasonable reduction in bending stiffness, particulary with drilled shafts, is a critical matter that requires guidance from the literature which has only limited experimental data. At the same time, the use of one constant reduced bending stiffness for the pile/ shaft does not reflect the real progressive deformations and forces associated with the steps of lateral loading. However, this technique may work quite well with the steel H pile which fails approximately once the pile flange reaches the yielding stage ( occurs rapidly). In general, the response of the pile/ drilled shaft ( pile head load vs. deflection, and pile head load vs. maximum moment) is assessed based on a constant bending stiffness ( EIr) and is truncated at the ultimate bending moment of the original pile/ drilled shaft cross section. The moment curvature relationship, and thus the maximum bending moment carried by the pile cross section should be evaluated first. Reese and Wang ( 1994) enhanced the technique presented above by computing the bending moment distribution along the pile and the associated value of EI at each increment of loading. Reese and Wang ( 1994) concluded that the bending moment along the pile does not depend strongly on structural characteristics and that the moment differences due to EI variations are small. It should be noted that the effect of the varying EI on the bending moment values along the drilled shaft was not obvious because the 72 EI of the drilled shaft had no effect on the p y curves ( i. e. modulus of subgrade reaction) employed in their procedure. Therefore, it was recommended that a single value of EI of the cracked section ( constant value) be used for the upper portion of the pile throughout the analysis. Contrary to Reese and Wang’s assumption, the variation in the value of EI has a significant effect on the nature of the p y curve and modulus of subgrade reaction [ Ashour and Norris ( 2000); Yoshida and Yoshinaka ( 1972); and Vesic ( 1961)] specially in the case of drilled shafts. The main purpose in this chapter is to assess the moment curvature relationship ( M f ) of the loaded pile or shaft in a convenient and simplified fashion considering the soil pile interaction. The prediction of the moment curvature curve allows one to realistically determine the variation of pile stiffness ( EI) as a function of bending moment. The SW model allows the designer to include the nonlinear behavior of the pile material and, as a result, to find out the effect of material types on the pile response and its ultimate capacity based on the concepts of soil pile interaction. 4.2 THE COMBINATION OF MATERIAL MODELING WITH THE STRAIN WEDGE MODEL The bending moment distribution along the deflected length of a laterally loaded pile varies as shown in Fig. 4.1. This profile of moment indicates the associated variation of pile stiffness with depth if the stress strain relationship of pile material is nonlinear. The strain wedge model is capable of handling the nonlinear behavior of pile material as well as the surrounding soil. The multi sublayer technique, presented in Chapter 2, allows one to provide an independent description for each soil sublayer and the associated pile segment. The effect of pile material is considered with the global stability of the loaded pile and the shape of the developing passive wedge of soil in front of the pile. During the iteration process using the SW model, the stiffness of each pile segment, which has a length equal to the depth of the soil sublayer, is a function of the 73 calculated bending moment at the associated pile segment, as seen in Fig. 4.1. Therefore, the pile is divided into a number of segments of different values of flexural stiffness under a particular lateral load. In order to incorporate the effect of material nonlinearity, numerical material models should be employed with the SW model. A unified stress strain approach for confined concrete has been employed with the reinforced concrete pile as well as the steel pipe pile filled with concrete. In addition, steel is modeled using an elastic perfectly plastic uniaxial stress strain relationship which is commonly used to describe steel behavior. The procedure presented provides the implementation of soil pile interaction in a fashion more sophisticated than that followed in the linear analysis with the SW model presented in Chapter 3. The approach developed will allow one to load the pile to its actual ultimate capacity for the desired lateral load and bending moment according to the variation of pile material properties along the pile length. 4.2.1 Material Modeling of Concrete Strength and Failure Criteria Based upon a unified stress strain approach for the confined concrete proposed by Mander et al. ( 1984 and 1988), a concrete model is employed with circular and rectangular concrete sections. The proposed model, which is shown in Fig. 4.2, has been employed for a slow strain rate and monotonic loading. The longitudinal compressive concrete stress fc is given by where fcc symbolizes the compressive strength of confined concrete. r  1 + x f x r f = r cc c ( 4.1) 74 where e c indicates the axial compressive strain of concrete. where e cc is the axial strain at the peak stress. fco and e co represent the unconfined ( uniaxial) concrete strength and the corresponding strain, respectively. Generally, e co can be assumed equal to 0.002, and where and e e cc x = c ( 4.2) ú ú û ù ê ê ë é ÷ ÷ ø ö ç ç è æ  1 f f = 1 + 5 co cc e cc e co ( 4.3) E  E r = E c c sec ( 4.4) E = 57,000 ( f ) ( p s i) 0.5 c co ( 4.5) 75 Ec denotes the initial modulus of elasticity of the concrete under slowly applied compression load. As mentioned by Paulay and Priestly ( 1992), the strain at peak stress given by Eqn. 5.3 does not represent the maximum useful strain for design purposes. The concrete strain limits occur when transverse confining steel fractures. A conservative estimate for ultimate compression strain ( e cu) is given by where e sm is the steel strain at maximum tensile stress ( ranges from 0.1 to 0.15), and r s is the volumetric ratio of confining steel. Typical values for e cu range from 0.012 to 0.05. fyh represents the yield stress of the transverse reinforcement. In order to determine the compressive strength of the confined concrete ( fcc), a constitutive model ( Mander et al. 1988) is directly related to the effective confining stress ( fl) that can be developed at the yield of the transverse reinforcement. e cc cc f Esec = ( 4.6) f 1.4 f = 0.004 + cc s yh sm cu r e e ( 4.7) ú ú û ù ê ê ë é ÷ ÷ ø ö ç ç è æ f 2 f  f 7.94 f f = f  1.254 + 2.254 1 + co l co l 0.5 cc co ( 4.8) 76 For circular and square section of concrete, fl is given by · Monotonic tensile loading Although concrete tension strength is ignored in flexural strength calculation, due to the effect of concrete confinement it would be more realistic if it were considered in the calculation. As suggested by Mander et al. ( 1988), a linear stress strain relationship is assumed in tension up to the tensile strength ( ftu). The tensile stress is given by and where If tensile strain e t is greater than the ultimate tensile strain ( e tu), ft is assumed to be equal to zero. f = 0.95 f l s yh r ( 4.9) f = E for f f t c c t tu e £ ( 4.10) E f = c tu e tu ( 4.11) f = 9 ( f ) ( p s i ) 0.5 tu co ( 4.12) 77 4.2.2 Material Modeling of Steel Strength There are different numerical models to represent the stress strain relationship of steel. The model employed for steel in this study is linearly elastic perfectly plastic, as shown in Fig. 4.3. The complexity of this numerical model is located in the plastic portion of the model which dose not include any strain hardening ( perfectly plastic). The elastic behavior of the steel is limited by the linearly elastic zone of this model at which the strain is less than the yield strain where fy is the yield stress of steel, and e y is the value of the steel strain at the end of the elastic zone where the stress is equal to fy. Eso is the elastic Young’s modulus of steel which is equal to 29,000 kips/ inch2. When the value of steel stress ( fs) at any point on the cross section reaches the yield stress, the Young’s modulus becomes less than Eso of the elastic zone. The initial yielding takes place when the stress at the farthest point from the neutral axis on the steel cross section ( point A) becomes equal to the yield stress ( fy), as shown in Fig. 4.4a. The initial yielding indicates the beginning of the elastic plastic response of the steel section. By increasing the load, other internal points on the cross section will satisfy the yield stress to respond plastically under a constant yield stress ( fy), as seen in Figure 4.4b. Once all points on the steel section satisfy a normal stress ( fs) equal to the yield stress ( fy) or a strain value larger than the yield strain ( e y), the steel section responds as a plastic hinge with an ultimate plastic moment ( Mp) indicating the complete yielding of the steel section, as presented in Fig. 4.4c. E f = so y e y ( 4.13) 78 During the elastic plastic stage ( after the initial yielding and before complete yielding) some points on the steel section respond elastically ( fs £ fy) and the others respond plastically ( fs = fy) with different values of Young’s modulus ( Es) , as presented in Fig. 4.3. The values of normal strain are assumed to vary linearly over the deformed cross section of steel. If the strain at any point on the steel cross section is larger than the yield strain ( e y), the plastic behavior will be governed by the flow of the steel under a constant stress ( fy) at the point in question. Regardless of whether the section is under elastic, elastic plastic or plastic states, the strain is linearly distributed over the whole steel section. In addition, the strain at any point is controlled by the values of strain at other locations in order to keep the strain distribution linear. Generally, the external and internal moments over the steel section should be in a state of equilibrium. 4.3 MOMENT CURVATURE ( M F ) RELATIONSHIP The aim of developing the moment curvature relationship of the pile material is to determine the variation of the flexural stiffness ( EI) at every level of loading. The normal stress ( s x) at any cross section along the pile length is linked to the bending moment ( M) and curvature ( f ) by the following equations: = M d x EI d y 2 2 ( 4.14) = M EI EI = r f ( 4.15) 79 where z = the distance from the neutral axis to the longitudinal fiber in question r o = the radius of curvature of the deflected axis of the pile e x = the normal strain at the fiber located z distance from the neutral axis. The above equations are based on the assumption of a linear variation of strain across the pile cross section. In addition, the pile cross section is assumed to remain perpendicular to the pile axis before and after deforming, as shown in Fig. 4.5. 4.4 SOLUTION PROCEDURE The solution procedure adopted consists of calculating the value of bending moment ( Mi) at each cross section associated with a profile of the soil modulus of subgrade reaction which is induced by the applied load at the pile top. Then, the associated curvature ( f ), stiffness ( EI), normal stress ( s x) and normal strain z = d x = d y x 2 2 f e ( 4.16) r e o x z =  ( 4.17) = E = E z x x s e f ( 4.18) 80 ( e x) can be obtained. This procedure depends on the pile material. The profile of moment distribution along the deflected portion of the pile is modified in an iterative fashion along with the values of the strain, stress, bending stiffness and curvature to satisfy the equilibrium among the applied load and the associated responses of the soil and pile. Based on the concepts of the SW model, the modulus of subgrade reaction ( i. e. p y curve) is influenced by the variations in the pile bending stiffness at every pile segment. This procedure guarantees the incorporation of soil pile interaction with the material modeling. The technique presented strives for a more realistic assessment of the pile deflection pattern under lateral loading and due to the nonlinear response of pile material and soil resistance. 4.4.1 Steel Pile Steel piles involved in this study have either circular ( pipe) or H shape cross sections, as seen in Fig. 4.6. The cross section of the steel pipe pile is divided into a number of horizontal strips ( equal to a total of 2m) parallel to the neutral axis. Each strip has a depth equal to the thickness of the pipe pile skin, as seen in Fig. 4.7. The cross section of the steel H pile is divided into horizontal strips of a width equal to one half the thickness of the H section flange, as seen in Fig. 4.7. The moment applied over the cross section of the pile segment ( i) is Mi, and the normal stress at a strip ( n) is ( fs) n ( 1 £ n £ m). Using Eqns. 4.17 and 4.18, the stress and strain distributions over the cross section of each pile segment can be determined as ( E I ) = M i i i f ( 4.19) ( ) = z 1 n m s n n i e f £ £ ( 4.20) 81 where Es £ Eso; f i is the curvature at pile segment ( i) which is constant over the steel cross section at the current level of loading; zn indicates the distance from the neutral axis to the midpoint of strip n; ( e s) n represents the strain at strip n; ( EI) i represents the initial stiffness of the pile segment ( i); I is the moment of inertia of the steel cross section of the pile segment ( i) which is always constant; and Eso symbolizes the elastic Young’s modulus of the steel. 1. Elastic Stage The Young’s modulus of any strip of the steel section ( i) is equal to the steel elastic modulus ( 29x106 psi) as
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Title  Pile group program for full material modeling and progressive failure 
Subject  Piling (Civil engineering)Testing.; Shear strength of soilsTesting. 
Description  Title from PDF title page (viewed on January 28, 2010).; "December 2008."; "July 2001"Second title page; Reprint. Originally published in 2001.; Includes bibliographical references (p. 141146).; Final report.; Text document (PDF).; Performed by University of Nevada, Reno, Dept of Civil Engineering for California Dept. of Transportation, Engineering Services Center and California Dept. of Transportation, Division of Research and Innovation under contract no. 
Creator  Ashour, Mohamed. 
Publisher  California Department of Transportation; Available through the National Technical Information Service 
Contributors  Norris, G. M. (Gary M.); California. Dept. of Transportation. Engineering Service Center.; California. Dept. of Transportation. Division of Research and Innovation.; University of Nevada, Reno. Civil and Environmental Engineering. 
Type  Text 
Identifier  http://www.dot.ca.gov/newtech/researchreports/reports/2008/020076.pdf 
Language  eng 
Relation  http://worldcat.org/oclc/502286157/viewonline 
DateIssued  2008] 
FormatExtent  vii, 146 p. : digital, PDF file (1 MB) with charts (some col.). 
RelationRequires  Mode of access: World Wide Web. 
Transcript  Pile Group Program for Full Material Modeling and Progressive Failure Final Report Report CA02 0076 December 2008 Division of Research & Innovation Pile Group Program for Full Material Modeling and Progressive Failure Final Report Report No. CA02 0076 December 2008 Prepared By: Department of Civil and Environmental Engineering University of Nevada, Reno Reno, NV 89557 Prepared For: California Department of Transportation Engineering Services Center 1801 30th Street Sacramento, CA 95816 California Department of Transportation Division of Research and Innovation, MS 83 1227 O Street Sacramento, CA 95814 DISCLAIMER STATEMENT This document is disseminated in the interest of information exchange. The contents of this report reflect the views of the authors who are responsible for the facts and accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California or the Federal Highway Administration. This publication does not constitute a standard, specification or regulation. This report does not constitute an endorsement by the Department of any product described herein. STATE OF CALIFORNIA DEPARTMENT OF TRANSPORTATION TECHNICAL REPORT DOCUMENTATION PAGE TR0003 ( REV. 10/ 98) 1. REPORT NUMBER CA02 0076 2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER 4. TITLE AND SUBTITLE Pile Group Program for Full Material Modeling and Progressive Failure 5. REPORT DATE December, 2008 6. PERFORMING ORGANIZATION CODE 7. AUTHOR( S) Mohamed Ashour, Gary Norris 8. PERFORMING ORGANIZATION REPORT NO. UNR / CCEER 01 02 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of Civil & Environmental Engineering University of Nevada Reno, NV 89557 0152 10. WORK UNIT NUMBER 11. CONTRACT OR GRANT NUMBER DRI Research Task No. 0076 Contract No. 59A0160 12. SPONSORING AGENCY AND ADDRESS California Department of Transportation Engineering Services Center 1801 30th Street Sacramento, CA 95816 California Department of Transportation Division of Research and Innovation, MS 83 1227 O Street Sacramento, CA 95814 13. TYPE OF REPORT AND PERIOD COVERED Final Report 14. SPONSORING AGENCY CODE 913 15. SUPPLEMENTAL NOTES 16. ABSTRACT Strain wedge ( SW) model formulation has been used, in previous work, to evaluate the response of a single pile or a group of piles ( including its pile cap) in layered soils to lateral loading. The SW model approach provides appropriate prediction for the behavior of an isolated pile and pile group under lateral static loading in layered soil ( sand and/ or clay). The SW model analysis covers the entire range of soil strain or pile deflection that may be encountered in practice. The method allows development of p y curves for the single pile based on soil pile interaction by considering the effect of both soil and pile properties ( i. e. pile size, shape, bending stiffness, and pile head fixity condition) on the nature of the p y curve. This study has extended the capability of the SW model in order to predict the response of a laterally loaded isolated pile and pile group considering the nonlinear behavior of pile material ( steel and/ or concrete) and its effect on the soil pile interaction. The incorporation of the nonlinear behavior of pile material has a significant influence on the lateral response of the pile/ shaft and its ultimate capacity. The reduction in pile lateral resistance due to degradation in the pile bending stiffness affects the nature of the accompanying p y curves, and the distribution of lateral deflections and bending moment along the pile. Contrary to the traditional Matlock Reese p y curve that does not account to the variations in the pile bending stiffness, the modulus of subgrade reaction ( i. e. the p y curve) assessed based on the SW model is a function of the pile bending stiffness. In addition, the ultimate value of soil pile reaction on the p y curve is governed by either the flow around failure of soil or the plastic hinge formation in the pile. The SW model analysis for a pile group has been modified in this study to assess the p y curves for an individual pile in a pile group. The technique presented is more realistic and evaluates the variations in the stress and strain ( i. e. Young’s modulus) in the soil around the pile in question due to the interference with the neighboring piles in a pile group in a mobilized fashion. The nonlinear behavior of pile material is also incorporated in the SW model analysis for a pile group. 17. KEY WORDS Laterally Loaded Deep Foundations, Pile Groups, Strain Wedge Model, Layered Soils, Nonlinear Behavior of Shaft Material 18. DISTRIBUTION STATEMENT No restrictions. This document is available to the public through the National Technical Information Service, Springfield, VA 22161 19. SECURITY CLASSIFICATION ( of this report) Unclassified 20. NUMBER OF PAGES 166 Pages 21. PRICE Reproduction of completed page authorized PILE GROUP PROGRAM FOR FULL MATERIAL MODELING AND PROGRESSIVE FAILURE CCEER 01 02 Prepared by: Mohamed Ashour Research Assistant Professor and Gary Norris Professor of Civil Engineering University of Nevada, Reno Department of Civil Engineering Prepared for: State of California Department of Transportation Contract No. 59A0160 July 2001 i ACKNOWLEDGMENTS The authors would like to thank Caltrans for its financial support of this project. The authors would also like to acknowledge Mr. Anoosh Shamsabadi, Dr. Saad El Azazy, Mr. Steve McBride, Mr. Bob Tanaka and Mr. Tom Schatz for their support and guidance as the Caltrans monitors for this project. ii DISCLAIMER The contents of this report reflect the views of the authors who are responsible for the facts and accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California or the Federal Highway Administration. This report does not constitute standard specifications, or regulations. iii ABSTRACT Strain wedge ( SW) model formulation has been used, in previous work, to evaluate the response of a single pile or a group of piles ( including its pile cap) in layered soils to lateral loading. The SW model approach provides appropriate prediction for the behavior of an isolated pile and pile group under lateral static loading in layered soil ( sand and/ or clay). The SW model analysis covers a wide range over the entire strain or deflection range that may be encountered in practice. The method allows development of p y curves for the single pile based on soil pile interaction by considering the effect of both soil and pile properties ( i. e. pile size, shape, bending stiffness, and pile head fixity condition) on the nature of the p y curve. This study has extended the capability of the SW model in order to predict the response of a laterally loaded isolated pile and pile group considering the nonlinear behavior of pile material ( steel and/ or concrete) and its effect on the soil pile interaction. The incorporation of the nonlinear behavior of pile material has a significant influence on the lateral response of the pile/ shaft and its ultimate capacity. The reduction in pile lateral resistance due to degradation in the pile bending stiffness affects the nature of the accompanying p y curves, and the distribution of lateral deflections and bending moment along the pile. Contrary to the traditional Matlock Reese p y curve that does not account to the variations in the pile bending stiffness, the modulus of subgrade reaction ( i. e. the p y curve) assessed based on the SW model is a function of the pile bending stiffness. In addition, the ultimate value of soil pile reaction on the p y curve is governed by either the flow around failure of soil or the plastic hinge formation in the pile. The SW model analysis for a pile group has been modified in this study to assess the p y curves for an individual pile in a pile group. The technique presented is more realistic and evaluates the variations in the stress and strain ( i. e. Young’s modulus) in the soil around the pile in question due to the interference with the neighboring piles in a pile group in a mobilized fashion. The nonlinear behavior of pile material is also incorporated in the SW model analysis for a pile group. iv TABLE OF CONTENTS CHAPTER 1 .............................................................................................................. 1 INTRODUCTION............................................................................................................... 1 CHAPTER 2 .............................................................................................................. 4 LATERAL LOADING OF A PILE IN LAYERED SOIL USING THE STRAIN WEDGE MODEL 2.1 INTRODUCTION.................................................................................................... 4 2.2 THE THEORETICAL BASIS OF STRAIN WEDGE MODEL CHARACTERIZATION ........................................................................................ 4 2.3 SOIL PASSIVE WEDGE CONFIGURATION IN UNIFORM SOIL ..................................................................................................... 5 2.4 STRAIN WEDGE MODEL IN LAYERED SOIL ............................................... 6 2.5 SOIL STRESS STRAIN RELATIONSHIP.......................................................... 8 2.5.1 Horizontal Stress Level ( SL) ....................................................................... 10 2.6 SHEAR STRESS ALONG THE PILE SIDES ( SLt) ............................................ 12 2.6.1 Pile Side Shear in Sand................................................................................ 12 2.6.2 Pile Side Shear Stress in Clay ..................................................................... 12 2.7 SOIL PROPERTY CHARACTERIZATION IN THE STRAIN WEDGE MODEL .......................................................................... 14 2.7.1 Properties Employed for Sand Soil ............................................................ 14 2.7.2 The Properties Employed for Normally Consolidated Clay ........................................................................................ 15 2.8 SOIL PILE INTERACTION IN THE STRAIN WEDGE MODEL.................................................................................................... 17 2.9 PILE HEAD DEFLECTION .................................................................................. 20 2.10 ULTIMATE RESISTANCE CRITERIA IN STRAIN v WEDGE MODEL.................................................................................................... 21 2.10.1 Ultimate Resistance Criterion in Sand Soil ............................................... 21 2.10.2 Ultimate Resistance Criterion in Clay Soil ................................................ 22 2.11 STABILITY ANALYSIS IN THE STRAIN WEDGE MODEL ........................ 22 2.11.1 Local Stability of a Soil Sublayer in the Strain Wedge Model .................................................................................... 23 2.11.2 Global Stability in the Strain Wedge Model ............................................. 23 2.12 APPROACH VERIFICATION.............................................................................. 24 2.12.1 Mustang Island Full Scale Load Test on a Pile in Submerged Dense Sand........................................................................... 25 2.12.2 Pyramid Building at Memphis, Tennessee, Full Scale Load Test on a Pile in Layered Clay Soil................................ 26 2.12.3 Sabine River Full Scale Load Tests on a Pile in Soft Clay....................... 28 2.13 SUMMARY.............................................................................................................. 31 FIGURES.................................................................................................................. 32 CHAPTER 3 ............................................................................................................... 46 PILE GROUPS IN LAYERED SOILS 3.1 INTRODUCTION.................................................................................................... 46 3.2 CHARACTRIZATION OF PILE GROUP INTERFERENCE ......................... 47 3.3 EVALUATION OF YOUN’S MODULUS, Eg ...................................................... 51 3.4 EVALUATION OF MODULUS OF SUBGRADE REACTION, EEg ................ 52 3.5 CASE STUDIES....................................................................................................... 54 3.5.1 Full Scale Load Test on a Pile Group in Layered Clay ............................ 54 3.5.2 Full Scale Load Test on a Pile Group in Sand .......................................... 54 3.5.3 Full Scale Load Test on a Pile Group in Layered Clay ............................ 55 3.5.4 Full Scale Load Test on a Pile Group with a Pile Cap in Layered Soil ............................................................................................. 56 3.5.5 Model Scale Load Test on a Pile Group in Loose and vi Medium Dense Sand .................................................................................... 57 3.6 SUMMARY.............................................................................................................. 57 FIGURES.................................................................................................................. 39 CHAPTER 4 NUMERICAL MATERIAL MODELING........................................................................ 70 4.1 INTRODUCTION.................................................................................................... 70 4.2 THE COMBINATION OF MATERIAL MODELING WITH THE SW MODEL .................................................................................................................... 72 4.2.1 Material Modeling for Concrete Strength and Failure Criteria ............. 73 4.2.2 Material Modeling of Steel Strength .......................................................... 76 4.3 MOMENT CURAVATURE RELATIONSHIP.................................................... 78 4.4 SOLUTION PROCEDURE.................................................................................... 79 4.4.1 Steel Pile........................................................................................................ 79 4.4.2 Reinforced Concrete Pile and Drilled Shaft .............................................. 84 4.4.3 Steel Pipe Pile Filled with Concrete ( Cast in Steel Shell, CISS) ........................................................................... 88 4.4.4 Steel Pipe Pile Filled with Reinforced Concrete ( Cast in Steel Shell, CISS) ........................................................................... 89 4.5 ILLUSTRATIVE EXAMPLES.............................................................................. 90 4.5.1 Example Problem, a Fixed Head Steel Pile Supporting a Bridge Abutment....................................................................................... 90 4.5.2 Example Problem, a Free Head Drilled Shaft Supporting a Bridge Abutment .................................................................. 91 4.5.3 Example Problem, a Fixed Head Drilled Shaft Supporting a Bridge Abutment .................................................................. 93 4.6 SUMMARY ............................................................................................................. 93 vii FIGURES.................................................................................................................. 95 CHAPTER 5 EFFECT OF NONLINEAR BEHAVIOR OF PILE MATERIAL ON PILE AND PILE GROUP LATERAL RESPONSE ............................................................................ 108 5.1 INTRODUCTION.................................................................................................... 108 5.2 EFFECT OF PILE MATERIAL NONLINEAR RESPONSE ON THE P Y CURVE ..................................................................................................................... 109 5.2.1 Steps of Constructing the p y Curve in the SW Model Analysis............. 111 5.2.2 Effect of Material Modeling on the p y Curve Ultimate .......................... 113 5.3 CASE STUDIES....................................................................................................... 115 5.3.1 Pyramid Building at Memphis, Tennessee, Dull Scale Load Test on a Pile in Layered Clay Soil ............................................................ 115 5.3.2 Houston Full Scale Load Test on a Reinforced Concrete Shaft in Stiff Clay......................................................................................... 118 5.3.3 Las Vegas Test on Drilled Shafts and Shaft Group in a Caliche Layer ........................................................................................ 121 5.3.4 Southern California Full Scale Load Test in Stiff Clay ........................... 122 5.3.5 Islamorada Full Scale load Test on a Pile Driven in Rock....................... 123 5.3.6 University of California at Los Angeles Full Scale Load Test on a Pile Driven in Stiff Clay .............................................................. 125 5.4 SUMMARY ............................................................................................................. 126 5.5 FIGURES.................................................................................................................. 127 1 CHAPTER 1 INTRODUCTION This report presents a summary of strain wedge ( SW) model assessment of the behavior of piles and pile groups subjected to lateral loading in layered soil considering the nonlinear behavior of pile material. A computer code attached to this report has been developed to assess the response of a single pile and pile group in layered soils ( sand, clay and/ or rock) and the associated p y curves for various soil and pile conditions. The main goal of this report is to address the influence of the nonlinear behavior of pile/ shaft material on the lateral response of isolated piles/ shafts and pile groups. The significance of accounting for the variations in strength of pile/ shaft is to identify the actual behavior and the ultimate capacity of such piles/ shafts. In addition, the associated p y curves will experience different effects due to the degradation in pile materials. The California Department of Transportation ( CALTRANS) sponsored a significant part of the SW model research through different phases of research project ( Ashour et al. 1996, Ashour and Norris 1998, and Ashour and Norris 2000). The SW model relates one dimensional beam on elastic foundation analysis to the three dimensional soil pile interaction response. It relates the deflection of a pile versus depth ( or its rotation) to the relative soil strain that exists in the growing passive wedge that develops in front of a pile under horizontal load. The SW model assumes that the deflection of a pile under increasing horizontal load is due solely to the deformation of the soil within the mobilized passive wedge, that plane stress change conditions exist within the wedge, and that soil strain is constant with depth in the current wedge. The passive wedge will exhibit a height that corresponds to the pivot point as determined by a linear approximation of the pile deflection. If the soil strain is known, an equivalent linear Young's modulus value, associated with the soil within the wedge at any depth, can be determined. Assuming plane stress change conditions exist, the increase in horizontal 2 stress can then be determined. In addition, the beam on elastic foundation line load reaction at any depth along the pile face is equivalent to the increase in horizontal stress times the wedge width at that depth plus the mobilized side shear resistance that develops at that depth along the pile faces parallel to the direction of movement. Since the geometry of the developing wedge is based on known soil properties and the current value of soil strain, the wedge width can be determined at any depth within the wedge. An equivalent face stress from beam on elastic foundation ( BEF) analysis can therefore be related to the horizontal stress change in the soil. The SW model relates one dimensional BEF analysis ( p y response) to a threedimensional soil pile interaction response. Because of this relation, the strain wedge model is also capable of determining the maximum moment and developing p y curves for a pile under consideration since the pile load and deflection at any depth along the pile can be determined. A detailed summary of the theory incorporated into the strain wedge model is presented in Chapter 2. The problem associated with analyzing a pile group is that loading one pile in the group can dramatically affect the response of other piles in the group. Since the SW model determines the geometry of the developing passive wedge, it allows any overlap between passive wedges within the group to be quantified. By knowing the amount of passive wedge overlap, the effective strain associated with the pile under consideration can be determined which ultimately reduces the lateral load capacity of the pile for a given level of deflection. Despite Ashour and Norris ( 2000) discussed the assessment of the lateral response of a pile group, a new treatment for the problem of a laterally loaded pile group is presented in this report to upgrade the capability of the SW model technique. This report illustrates the links between the single pile and the pile group analysis. This is different from the current procedure in common use that employs a p y multiplier technique. Such multiplier technique is based on reducing the stiffness of the traditional ( Matlock Reese) p y curve using a multiplier that reduces the stiffness of the p y curve of the single pile to yield a softer response for an individual pile in the group. A detailed 3 summary of the theory in which the SW model analyzes pile group behavior is presented in Chapter 3. A methodology to assess the response of an isolated pile and pile group in layered soil considering the nonlinear behavior of pile material and how the accompanying modulus of subgrade reaction is affected is presented in Chapters 4 and 5. The effect of pile properties, such as the pile bending stiffness, on the pile lateral response has been presented by Ashour et al. ( 1996). Such a study emphasized the need to study the influence of the variation in pile bending stiffness during the loading process on the soilpile interaction and therefore lateral response of the isolated pile and pile group. The effect of pile nonlinear behavior of pile material has been studied by other researches ( Reese 1994, and Reese and Wang 1991). However, the incorporation of the nonlinear behavior of pile material has not affected the shape of the p y curve or the soil pile interaction. In other words, the p y curve has not accounted for the variation in the pile bending stiffness. Several case studies are presented in this study to show the capability of the SW model and how the modeling of pile material ( steel and/ or concrete) is employed in the SW model analysis. A numerical model for confined concrete is employed with the SW model. Such a model accounts for the enhancement of the concrete strength due to the confinement of the transverse reinforcement. 4 CHAPTER 2 LATERAL LOADING OF A PILE IN LAYERED SOIL USING THE STRAIN WEDGE MODEL 2.1 INTRODUCTION The strain wedge ( SW) model is an approach that has been developed to predict the response of a flexible pile under lateral loading ( Norris 1986, Ashour et al. 1996 and Ashour et al. 1998). The main concept associated with the SW model is that traditional one dimensional Beam on Elastic Foundation ( BEF) pile response parameters can be characterized in terms of three dimensional soil pile interaction behavior. The strain wedge model was initially established to analyze a free head pile embedded in one type of uniform soil ( sand or clay). However, the SW model has been improved and modified through additional research to accommodate a laterally loaded pile embedded in multiple soil layers ( sand and clay). The strain wedge model has been further modified to include the effect of pile head conditions on soil pile behavior. The main objective behind the development of the SW model is to solve the BEF problem of a laterally loaded pile based on the envisioned soil pile interaction and its dependence on both soil and pile properties. The problem of a laterally loaded pile in layered soil has been solved by Reese ( 1977) as a BEF based on modeling the soil response by p y curves. However, as mentioned by Reese ( 1983), the p y curve employed does not account for soil continuity and pile properties such as pile stiffness, pile cross section shape and pile head conditions. 2.2 THE THEORETICAL BASIS OF STRAIN WEDGE MODEL CHARACTERIZATION The SW model parameters are related to an envisioned three dimensional passive wedge of soil developing in front of the pile. The basic purpose of the SW model is to relate stress strain strength behavior of the soil in the wedge to one dimensional BEF parameters. The SW model is, therefore, able to provide a 5 theoretical link between the more complex three dimensional soil pile interaction and the simpler onedimensional BEF characterization. The previously noted correlation between the SW model response and BEF characterization reflects the following interdependence: · the horizontal soil strain ( e ) in the developing passive wedge in front of the pile to the deflection pattern ( y versus depth, x) of the pile; · the horizontal soil stress change ( D s h) in the developing passive wedge to the soil pile reaction ( p) associated with BEF; and · the nonlinear variation in the Young's modulus ( E = D s h/ e ) of the soil to the nonlinear variation in the modulus of soil subgrade reaction ( Es = p/ y) associated with BEF characterization. The analytical relations presented above reflect soil pile interaction response characterized by the SW model that will be illustrated later. The reason for linking the SW model to BEF analysis is to allow the appropriate selection of BEF parameters to solve the following fourth order ordinary differential equation to proceed. The closed form solution of the above equation has been obtained by Matlock and Reese ( 1961) for the case of uniform soil. In order to appreciate the SW model’s enhancement of BEF analysis, one should first consider the governing analytical formulations related to the passive wedge in front of the pile, the soil’s stress strain relationship, and the related soil pile interaction. 2.3 SOIL PASSIVE WEDGE CONFIGURATION IN UNIFORM SOIL The SW model represents the mobilized passive wedge in front of the pile which is characterized by base angles, Q m and b m, the current passive wedge depth h, and the spread of the wedge fan angle, j m ( the mobilized friction angle). The horizontal stress change at the passive wedge face, D s h, and side shear, t , act as shown in Fig. 2.1. One of the main assumptions associated with the SW model is that the deflection = 0 d x + E ( x) y + P d y d x EI d y 2 2 4 s x 4 ÷ ø ö ç è æ ÷ ø ö ç è æ ( 2.1) 6 pattern of the pile is taken to be linear over the controlling depth of the soil near the pile top resulting in a linearized deflection angle, d , as seen in Fig. 2.2. The relationship between the actual ( closed form solution) and linearized deflection patterns has been established by Norris ( 1986). This assumption allows uniform horizontal and vertical soil strains to be assessed ( as seen later in a Fig. 2.6). Changes in the shape and depth of the passive wedge, along with changes in the state of loading and pile deflection, occur with change in the uniform strain in the developing passive wedge. The configuration of the wedge at any instant of load and, therefore, mobilized friction angle, j m, and wedge depth, h, is given by the following equation: or its complement The width, BC, of the wedge face at any depth is where x denotes the depth below the top of the studied passive wedge, and D symbolizes the width of the pile cross section ( see Fig. 2.1). It should be noted that the SW model is based upon an effective stress analysis of both sand and clay soils. As a result, the mobilized fanning angle, j m, is not zero in clay soil as assumed by Reese ( 1958, 1983). 2.4 STRAIN WEDGE MODEL IN LAYERED SOIL The SW model can handle the problem of multiple soil layers of different types. The approach employed, which is called the multi sublayer technique, is based upon dividing the soil profile and the loaded pile into sublayers and segments of constant thickness, respectively, as shown in Fig. 2.3. Each sublayer of soil is considered to behave as a uniform soil and have its own properties according to the sublayer location and soil type. In addition, the multi sublayer technique depends on the deflection pattern of the embedded pile 2 = 45  m m j Q ( 2.2) 2 = 45 + m m b j ( 2.3) b j m m BC = D + ( h  x) 2 tan tan ( 2.4) 7 being continuous regardless of the variation of soil types. However, the depth, h, of the deflected portion of the pile is controlled by the stability analysis of the pile under the conditions of soil pile interaction. The effects of the soil and pile properties are associated with the soil reaction along the pile by the Young's modulus of the soil, the stress level in the soil, the pile deflection, and the modulus of subgrade reaction between the pile segment and each soil sublayer. To account for the interaction between the soil and the pile, the deflected part of the pile is considered to respond as a continuous beam loaded with different short segments of uniform load and supported by nonlinear elastic supports along soil sublayers, as shown in Fig. 2.4. At the same time, the point of zero deflection ( Xo in Fig. 2.4a) for a pile in a particular layered soil varies according to the applied load and the soil strain level. The SW model in layered soil provides a means for distinguishing layers of different soil types as well as sublayers within each layer where conditions ( e 50, SL, j m) vary even though the soil and its properties ( ` g , e or Dr, j , etc.) remain the same. As shown in Fig. 2.5 , there may be different soil layers and a transition in wedge shape from one layer to the next, with all components of the compound wedge having in common the same depth h. In fact, there may be a continuous change over a given sublayer; but the values of stress level ( SL) and mobilized friction angle ( j m) at the middle of each sublayer of height, Hi, are treated as the values for the entire sublayer. As shown in Fig. 2.5, the geometry of the compound passive wedge depends on the properties and the number of soil types in the soil profile, and the global equilibrium between the soil layers and the loaded pile. An iterative process is performed to satisfy the equilibrium between the mobilized geometry of the passive wedge of the layered soil and the deflected pattern of the pile for any level of loading. While the shape of the wedge in any soil layer depends upon the properties of that layer and, therefore, satisfies the nature of a Winkler foundation of independent “ soil” springs in BEF analysis, realize that there is forced interdependence given that all components of the compound wedge have the same depth ( h) in common. Therefore, the mobilized depth ( h) of the compound wedge at any time is a function of the various soils ( and their stress levels), the bending stiffness ( EI), and head fixity conditions ( fixed, free, or other) of 8 the pile. In fact, the developing depth of the compound wedge can be thought of as a retaining wall of changing height, h. Therefore, the resultant “ soil” reaction, p, from any soil layer is really a “ soil pile” reaction that depends upon the neighboring soil layers and the pile properties as they, in turn, influence the current depth, h. In other words, the p y response of a given soil layer is not unique. The governing equations of the mobilized passive wedge shape are applied within each one or two foot sublayer i ( of a given soil layer I) and can be written as follows: where h symbolizes the entire depth of the compound passive wedge in front of the pile and xi represents the depth from the top of the pile or compound passive wedge to the middle of the sublayer under consideration. The equations above are applied at the middle of each sublayer. 2.5 SOIL STRESS STRAIN RELATIONSHIP The horizontal strain ( e ) in the soil in the passive wedge in front of the pile is the predominant parameter in the SW model; hence, the name “ strain wedge”. Consequently, the horizontal stress change ( D s h) is constant across the width of the rectangle BCLM ( of face width BC of the passive wedge ), as shown in Fig. 2.1. The stress strain relationship is defined based on the results of the isotropically consolidated drained ( sand) or undrained ( clay) triaxial test. These properties are summarized as follows: · The major principle stress change ( D s h) in the wedge is in the direction of pile movement, and it ( ) ( ) 2 = 45  m i m i j Q ( 2.5) ( ) ( ) 2 = 45 + m i m i j b ( 2.6) ( BC ) i = D + ( h  xi ) 2 ( m ) i ( m ) i tan b tan j ( 2.7) 9 is equivalent to the deviatoric stress in the triaxial test as shown in Fig. 2.2 ( assuming that the horizontal direction in the field is taken as the axial direction in the triaxial test). · The vertical stress change ( D s v) and the perpendicular horizontal stress change ( D s ph) equal zero, corresponding to the standard triaxial compression test where deviatoric stress is increased while confining pressure remains constant. · The initial horizontal effective stress is taken as where K= 1 due to pile installation effects. Therefore, the isotropic confining pressure in the triaxial test is taken as the vertical effective stress ( ` s vo) at the associated depth. · The horizontal stress change in the direction of pile movement is related to the current level of horizontal strain ( e ) and the associated Young's modulus in the soil as are the deviatoric stress and the axial strain to the secant Young’s modulus ( E = D s h/ e ) in the triaxial test. · Both the vertical strain ( e v ) and the horizontal strain perpendicular to pile movement ( e ph) are equal and are given as e v = e ph =  n e where n is the Poisson’s ratio of the soil. It can be demonstrated from a Mohr’s circle of soil strain, as shown in Fig. 2.6, that shear strain, g , is defined as The corresponding stress level ( SL) in sand ( see Fig. 2.7) is s ho s vo s vo = K = ( v ) Q m ( ) Q m 1 + 2 2 1  2 = 2 1 = 2 g e e sin e n sin ( 2.8) ( ) ( 45 + )  1 45 +  1 SL = = 2 m 2 hf h j j s s tan tan D D ( 2.9) 10 where the horizontal stress change at failure ( or the deviatoric stress at failure in the triaxial test) is In clay, where Su represents the undrained shear strength which may vary with depth. Determination of the values of SL and j m in clay requires the involvement of an effective stress analysis which is presented later in this chapter. The relationships above show clearly that the passive wedge response and configuration change with the change of the mobilized friction angle ( j m) or stress level ( SL) in the soil. Such behavior provides the flexibility and the accuracy for the strain wedge model to accommodate both small and large strain cases. A power function stress strain relationship is employed in SW model analysis for both sand and clay soils. It reflects the nonlinear variation in stress level ( SL) with axial strain ( e ) for the condition of constant confining pressure. To be applicable over the entire range of soil strain, it takes on a form that varies in stages as shown in Fig. 2.8. The advantage of this technique is that it allows the three stages of horizontal stress, described in the next section, to occur simultaneously in different sublayers within the passive wedge. 2.5.1 Horizontal Stress Level ( SL) Stage I ( e £ e 50% ) The relationship between stress level and strain at each sublayer ( i) in the first stage is assessed using the following equation, ú û ù ê ë é ÷ ø ö ç è D æ  1 2 = 2 45 + hf vo j s s tan ( 2.10) ; = 2 S SL = hf u hf h s s s D D D ( 2.11) 11 where 3.707 and l ( l = 3.19) represent the fitting parameters of the power function relationship, and e 50 symbolizes the soil strain at 50 percent stress level. Stage II ( e 50% £ e £ e 80 % ) In the second stage of the stress strain relationship, Eqn. 2.12 is still applicable. However, the value of the fitting parameter l is taken to vary in a linear manner from 3.19 at the 50 percent stress level to 2.14 at the 80 percent stress level as shown in Fig. 2.8b. Stage III ( e ³ e 80% ) This stage represents the final loading zone which extends from 80 percent to 100 percent stress level. The following Equation is used to assess the stress strain relationship in this range, where m= 59.0 and q= 95.4 e 50 are the required values of the fitting parameters. The three stages mentioned above are developed based on unpublished experimental results ( Norris 1977). In addition, the continuity of the stress strain relationship is maintained along the SL e curve at the merging points between the mentioned stages. As shown in Fig. 2.9, if e 50 of the soil is constant with depth ( x), then, for a given horizontal strain ( e ), SL from Eqns. 2.12 or 2.13 will be constant with x. On the other hand, since strength, D s hf, varies with depth ( e. g., see Eqns. 2.10 and 2.11), D s h (= SL D s hf ) will vary in a like fashion. However, e 50 is affected by confining pressure ( ` s vo) in sand and Su in clay. Therefore, SL for a given e will vary somewhat with depth. ( ) (  3.707 SL ) SL = i 50 i i i exp e l e ( 2.12) ( ) SL 0.80 m + q 100 SL = 0.2 + i i i i i ³ ú û ù ê ë é exp ln ; e e ( 2.13) 12 The Young’s modulus of the soil from both the shear loading phase of the triaxial test and the strain wedge model is It can be seen from the previous equations that stress level, strain and Young's modulus at each sublayer ( i) depend on each other, which results in the need for an iterative solution technique to satisfy the equilibrium between the three variables. 2.6 SHEAR STRESS ALONG THE PILE SIDES ( SLt) Shear stress ( t ) along the pile sides in the SW model ( see Fig. 2.1) is defined according to the soil type ( sand or clay). 2.6.1 Pile Side Shear in Sand In the case of sand, the shear stress along the pile sides depends on the effective stress ( s vo) at the depth in question and the mobilized angle of friction between the sand and the pile ( j s). The mobilized side shear depends on the stress level and is given by the following equation, In Eqn. 2.15, note that mobilized side shear angle, tan j s, is taken to develop at twice the rate of the mobilized friction angle ( tan j m) in the mobilized wedge. Of course, j s is limited to the fully developed friction angle ( j ) of the soil. 2.6.2 Pile Side Shear Stress in Clay ( ) ( ) e s e s = SL E = h i i hf i i D D ( 2.14) = ( ) ( ) ; where ( ) = 2 ( ) i vo i s i s i m i t s tan j tan j tan j 13 The shear stress along the pile sides in clay depends on the clay’s undrained shear strength. The stress level of shear along the pile sides ( SLt) differs from that in the wedge in front of the pile. The side shear stress level is function of the shear movement, equal to the pile deflection ( y) at depth x from the ground surface. This implies a connection between the stress level ( SL) in the wedge and the pile side shear stress level ( SLt) . Using the Coyle Reese ( 1966) “ t z” shear stress transfer curves ( Fig. 2.10), values for SLt can be determined. The shear stress transfer curves represent the relationship between the shear stress level experienced by a one foot diameter pile embedded in clay with a peak undrained strength, Su, and side resistance, t ult ( equal to z times the adhesional strength a Su), for shear movement, y. The shear stress load transfer curves of Coyle Reese can be normalized by dividing curve A ( 0 < x < 3 m) by z = 0.53, curve B ( 3 < x < 6 m) by z = 0.85, and curve C ( x > 6 m) by z = 1.0. These three values of normalization ( 0.53, 0.85, 1.0) represent the peaks of the curves A, B, and C, respectively, in Fig. 2.10a. Figure 2.10b shows the resultant normalized curves. Knowing pile deflection ( y), one can assess the value of the mobilized pile side shear stress ( t ) as where and a indicates the adhesion value after Tomlinson ( 1957). The normalized shear stress load transfer curves can be represented by the following equations. For the normalized curves A ( x < 3 m) and B ( 3 < x < 6 m), For the normalized curve C ( x > 6 m) where y is in cm and D in m. = ( SL ) ( ) t i t i t ult i ( 2.16) ( ) = S ) t ult i z ( a u i ( 2.17) SL = 12.9 y D  40.5 y D 2 2 t ( 2.18) SL = 32.3 y D  255 y D 2 2 t ( 2.19) 14 From the discussion above, it is obvious that SLt varies nonlinearly with the pile deflection, y, at a given soil depth, x. Also, SLt changes nonlinearly with soil depth for a given value of soil strain ( see Fig. 2.11). These concepts are employed in each sublayer of clay. 2.7 SOIL PROPERTY CHARACTERIZATION IN THE STRAIN WEDGE MODEL One of the main advantages of the SW model approach is the simplicity of the required soil properties necessary to analyze the problem of a laterally loaded pile. The properties required represent the basic and the most common properties of soil, such as the effective unit weight and the angle of internal friction or undrained strength. The soil profile is divided into one or two foot sublayers, and each sublayer is treated as an independent entity with its own properties. In this fashion, the variation in soil properties or response ( such as e 50 and j in the case of sand, or Su and ` j in the case of clay) at each sublayer of soil can be explored. It is obvious that soil properties should not be averaged at the midheight of the passive wedge in front of the pile for a uniform soil profile ( as in the earlier work of Norris 1986), or averaged for all sublayers of a single uniform soil layer of a multiple layer soil profile. 2.7.1 Properties Employed for Sand Soil · Effective unit weight ( total above water table, buoyant below), ` g · Void ratio, e, or relative density, Dr · Angle of internal friction, j · Soil strain at 50% stress level, e 50 While standard subsurface exploration techniques and available correlations may be used to evaluate or estimate ` g , e or Dr, and j , some guidance may be required to assess e 50. 15 The e 50 represents the axial strain ( e 1 ) at a stress level equal to 50 percent in the e 1 SL relationship that would result from a standard drained ( CD) triaxial test. The confining ( consolidation) pressure for such tests should reflect the effective overburden pressure ( ` s vo) at the depth ( x) of interest. The e 50 changes from one sand to another and also changes with density state. In order to obtain e 50 for a particular sand, one can use the group of curves shown in Fig. 2.12 ( Norris 1986) which show a variation based upon the uniformity coefficient, Cu, and void ratio, e. These curves have been assessed from sand samples tested with “ frictionless” ends in CD tests at a confining pressure equal to 42.5 kPa ( Norris 1977). Since the confining pressure changes with soil depth, e 50, as obtained from Fig. 2.12, should be modified to match the existing pressure as follows: where ` s vo should be in kPa. 2.7.2 The Properties Employed for Normally Consolidated Clay · Effective unit weight ` g · Plasticity index, PI · Effective angle of friction, ` j · Undrained shear strength, Su · Soil strain at 50% stress level, e 50 Plasticity index, PI, and undrained shear strength, Su, are considered the governing properties because the effective angle of internal friction, ` j , can be estimated from the PI based on Fig. 2.13. The e 50 from an undrained triaxial test ( UU at depth x or CU with s 3 = ` s vo) can be estimated based on Su as indicated in Fig. 2.14. ( ) ( ) ÷ ø ö ç è æ 42.5 = ( ) vo i 0.2 50 i 50 42.5 e e s ( 2.20) ( ) ( ) ú û ù ê ë é ÷ ø ö ç è D æ  1 2 = 2 45 + i hf i vo i j s s tan ( 2.21) 16 An effective stress ( ES) analysis is employed with clay soil as well as with sand soil. The reason behind using the ES analysis with clay, which includes the development of excess porewater pressure with undrained loading, is to define the three dimensional strain wedge geometry based upon the more appropriate effective stress friction angle, ` j . The relationship between the normally consolidated clay undrained shear strength, Su, and ` s vo is taken as assuming that Su is the equivalent standard triaxial test strength. The effective stress analysis relies upon the evaluation of the developing excess porewater pressure based upon Skempton's equation ( 1954), i. e. where B equals 1 for saturated soil. Accordingly, Note that D s 3 = 0 both in the shear phase of the triaxial test and in the strain wedge. Therefore, the mobilized excess porewater pressure is where D s 1 represents the deviatoric stress change in the triaxial test and D s h in the field, i. e. Therefore, using the previous relationships, the Skempton equation can be rewritten for any sublayer ( i) as follows: The initial value of parameter Au is 0.333 and occurs at very small strain for elastic soil response. In u s vo S = 0.33 ( 2.22) u = B [ + A (  ) ] D D s 3 u D s 1 D s 3 ( 2.23) u = + A (  ) D D s 3 u D s 1 D s 3 ( 2.24) u s 1 D u = A D ( 2.25) u s h D u = A D ( 2.26) ( u ) = ( A ) SL ( ) = ( A ) SL 2 ( S ) i u i i hf i u i i u i D D s ( 2.27) ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ 1  ( ) ( S ) 1 1 + 2 1 ( A ) = i vo i u i uf i j s sin ( 2.28) 17 addition, the value of parameter Auf that occurs at failure at any sublayer ( i) is given by the following relationship after Wu ( 1966) as indicated in Fig. 2.15. In Eqn. 2.28, ` j symbolizes the effective stress angle of internal friction; and, based on Eqn. 2.22, Su/ ` s vo equals 0.33. However, Au is taken to change with stress level in a linear fashion as By evaluating the value of Au, one can effectively calculate the excess porewater pressure, and then can determine the value of the effective horizontal stress, ( s – vo + D s h  D u), and the effective confining pressure, ( s – vo  D u) at each sublayer, as shown in Fig. 2.15. Note that the mobilized effective stress friction angle, j – m, can be obtained from the following relationship. The targeted values of ` j mi and SLi in a clay sublayer and at a particular level of strain ( e ) can be obtained by using an iterative solution that includes Eqns 2.11 through 2.13, and 2.27 through 2.30. 2.8 SOIL PILE INTERACTION IN THE STRAIN WEDGE MODEL The strain wedge model relies on calculating the modulus of subgrade reaction, Es, which reflects the soil pile interaction at any level of soil strain during pile loading. Es also represents the secant slope at any point on the p y curve, i. e. ( A ) = 0.333 + SL [ ( A )  0.333 ] u i i uf i ( 2.29) ( ) (  u ) +  u = 2 ( ) 45 + vo i 2 m i vo h i D D D ÷ ÷ ø ö ç ç è æ s j s s tan ( 2.30) 18 Note that p represents the force per unit length of the pile or the BEF soil pile reaction, and y symbolizes the pile deflection at that soil depth. In the SW model, Es is related to the soil’s Young's modulus, E, by two linking parameters, A and Y s. It should be mentioned here that the SW model establishes its own Es from the Young's modulus of the strained soil, and therefore, one can assess the p y curve using the strain wedge model analysis. Therefore, Es should first be calculated using the strain wedge model analysis to identify the p and y values. Corresponding to the horizontal slice ( a soil sublayer) of the passive wedge at depth x ( see Fig. 2.1), the horizontal equilibrium of horizontal and shear stresses is expressed as where S1 and S2 equal to 0.75 and 0.5, respectively, for a circular pile cross section, and equal to 1.0 for a square pile ( Briaud et al. 1984). Alternatively, one can write the above equation as follows: where A symbolizes the ratio between the equivalent pile face stress, p/ D, and the horizontal stress change, D s h, in the soil. ( In essence, it is the multiplier that, when taken times the horizontal stress change, gives the equivalent face stress.) From a different perspective, it represents a normalized width ( that includes side shear and shape effects) that, when multiplied by D s h yields p/ D. By combining the equations of the passive wedge geometry and the stress level with the above relationship, one finds that y p Es = ( 2.31) p = ( ) BC S + 2 D S i h i i 1 i 2 D s t ( 2.32) ( ) ( ) 2 S + D = BC S p D A = h i i 1 i 2 h i i i s t D s D / ( 2.33) ( ) ( ) ( ) ( ) ( ) in sand 2 S + D h  x 2 A = S 1 + h i i m m i 2 vo i s i i 1 s b j s f D ÷ ÷ ø ö ç ç è æ tan tan tan ( 2.34) 19 Here the parameter A is a function of pile and wedge dimensions, applied stresses, and soil properties. However, given that D s h = E e in Eqn. 2.33, The second linking parameter, Y s, relates the soil strain in the SW model to the linearized pile deflection angle, d . Referring to the normalized pile deflection shape shown in Figs. 2.2 and 2.6 and ( ) ( ) ( ) in clay SL + S SL D h  x 2 A = S 1 + i i m m i 2 t i 1 i ÷ ÷ ø ö ç ç è æ tan b tan j ( 2.35) p = A D ( s ) = A D E e i i h i i i D ( 2.36) 2 = d g ( 2.37) Q m 2 2 = 2 max sin g g ( 2.38) 20 where g denotes the shear strain in the developing passive wedge. Using Eqns. 2.38 and 2.39, Eqn. 2.37 can be rewritten as Based on Eqn. 2.40, the relationship between e and d can expressed as or The parameter Y varies with the Poisson's ratio of the soil and the soil's mobilized angle of internal friction ( j m) and the mobilized passive wedge angle ( Q m). Poisson's ratio for sand can vary from 0.1 at a very small strain to 0.5 or lager ( due to dilatancy) at failure, while the base angle, Q m, can vary between 45o ( for j m = 0 at e = 0) and 25o ( for, say, j m = 40o at failure), respectively. For this range in variation for n and j m, the parameter Y for sand varies between 1.81 and 1.74 with an average value of 1.77. In clay soil, Poisson's ratio is assumed to be 0.5 ( undrained behavior) and the value of the passive wedge base angle, Q m, can vary between 45o ( for j m = 0 at e = 0) and 32.5o ( for, say, ` j m = 25o at failure). Therefore, the value of the parameter Y will vary from 1.47 to 1.33, with an average value of 1.4. It is clear from the equations above that employing the multi sublayer technique greatly influences the values of soil pile interaction as characterized by the parameter, Ai, which is affected by the changing effective stress and soil strength from one sublayer to another. The final form of the modulus of subgrade reaction ( ) 2 1 + = 2  = 2 v g e e n e max ( 2.39) ( ) 2 1 + 2 d = e n sin Q m ( 2.40) d e Y = ( 2.41) ( ) Q Y m 1 + 2 2 = n sin ( 2.42) 21 can be expressed as It should be mentioned that the SW model develops its own set of non unique p y curves which are function of both soil and pile properties, and are affected by soil continuity ( layering) as presented by Ashour et al. ( 1996). 2.9 PILE HEAD DEFLECTION As mentioned previously, the deflection pattern of the pile in the SW model is continuous and linear. Based on this concept, pile deflection can be assessed using a simplified technique which provides an estimation for the linearized pile deflection, especially yo at the pile head. By using the multi sublayer technique, the deflection of the pile can be calculated starting with the base of the mobilized passive wadge and moving upward along the pile, accumulating the deflection values at each sublayer as shown in the following relationships and Fig. 2.16. where the Y s value changes according to the soil type ( sand or clay), and Hi indicates the thickness of sublayer i and n symbolizes the current number of sublayers in the mobilized passive wedge. The main point of interest is the pile head deflection which is a function of not only the soil strain but also of the depth of the compound passive wedge that varies with soil and pile properties and the level of soil strain. 2.10 ULTIMATE RESISTANCE CRITERIA IN STRAIN WEDGE MODEL The mobilized passive wedge in front of a laterally loaded pile is limited by certain constraint criteria in the ( ) ( ) ( ) D E h  x = A h  x = A D E y p E = i i i i i i i i s i Y d e ( 2.43) Y e yi = Hi d i = Hi ( 2.44) y = y i = 1 to n o i S ( 2.45) 22 SW model analysis. Those criteria differ from one soil to another and are applied to each sublayer. Ultimate resistance criteria govern the shape and the load capacity of the wedge in any sublayer in SW model analysis. The progressive development of the ultimate resistance with depth is difficult to implement without employing the multi sublayer technique. 2.10.1 Ultimate Resistance Criterion of Sand Soil The mobilization of the passive wedge in sand soil depends on the horizontal stress level, SL, and the pile side shear resistance, t . The side shear stress is a function of the mobilized side shear friction angle, j s, as mentioned previously, and reaches its ultimate value ( j s = j ) earlier than the mobilized friction angle, j m, in the wedge ( i. e. SLt ³ SL). This causes a decrease in the rate of growth of sand resistance and the fanning of the passive wedge as characterized by the second term in Eqns 2.32 and 2.34, respectively. Once the stress level in the soil of a sublayer of the wedge reaches unity ( SLi = 1), the stress change and wedge fan angle in that sublayer cease to grow. However, the width BC the face of the wedge can continue to increase as long as e ( and, therefore, h in Eqn. 2.7) increases. Consequently, soil pile resistance, p, will continue to grow more slowly until a condition of initial soil failure ( SLi = 1) develops in that sublayer. At this instance, p = pult where pult in sand, given as pult is a temporary ultimate condition, i. e. the fanning angle of the sublayer is fixed and equal to j i, but the depth of the passive wedge and, hence, BC continue to grow. The formulation above reflects that the near surface “ failure” wedge does not stop growing when all such sublayers reach their ultimate resistance at SL = 1 because the value of h at this time is not limited. Additional load applied at the pile head will merely cause the point at zero deflection and, therefore, h to move down the pile. More soil at full strength ( SL = 1) will be mobilized to the deepening wedge as BC, therefore, pult will increase until either flow around failure or a plastic hinge occurs. ( p ) = ( ) BC S + 2 ( ) D S ult i hf i i 1 f i 2 D s t ( 2.46) 23 Recognize that flow around failure occurs in any sublayer when it is easier for the sand at that depth to flow around the pile in a local bearing capacity failure than for additional sand to be brought to failure and added to the already developed wedge. However, the value at which flow failure occurs [ Ai = ( Ault) i , ( pult) i = ( D s hf) i ( Ault) i D] in sand is so large that it is not discussed here. Alternatively, a plastic hinge can develop in the pile when the pile material reaches its ultimate resistance at a time when SLi £ 1 and Ai < ( Ault) i. In this case, h becomes fixed, and BCi and pi will be limited when SLi becomes equal to 1. 2.10.2 Ultimate Resistance Criterion of Clay Soil The situation in clay soil differs from that in sand and is given by Gowda ( 1991) as a function of the undrained strength ( Su) i of the clay sublayer. Consequently, Ault indicates the limited development of the sublayer wedge geometry for eventual development of flow around failure ( SLi = 1) and, consequently, the maximum fanning angle in that sublayer becomes fixed, possibly at a value j m £ ` j . If a plastic hinge develops in the pile at SLi less than 1, then h will be limited, but BC, and pi will continue to grow until Ai is equal to Ault or pi is equal to ( pult) i. 2.11 STABILITY ANALYSIS IN THE STRAIN WEDGE MODEL The objective of the SW model is to establish the soil response as well as model the soil pile interaction through the modulus of subgrade reaction, Es. The shape and the dimensions of the passive wedge in front of the pile basically depend on two types of stability which are the local stability of the soil sublayer and the global stability of the pile and the passive wedge. However, the global stability of the passive wedge ( p ) = 10 ( S ) D S + 2 ( S ) D S ult i u i 1 u i 2 ( 2.47) ( ) ( ) ( ) ( ) ( ) = 5 S + S D 2 S p = D p A = 1 2 u i ult i hf i ult i ult i D s ( 2.48) 24 depends, in turn, on the local stability of the soil sublayers. 2.11.1 Local Stability of a Soil Sublayer in the Strain Wedge Model The local stability analysis in the strain wedge model satisfies equilibrium and compatibility among the pile segment deflection, soil strain, and soil resistance for the soil sublayer under consideration. Such analysis allows the correct development of the actual horizontal stress change, D s h , pile side shear stress, t , and soil pile reaction, p, associated with that soil sublayer ( see Fig. 2.1). It is obvious that the key parameters of local stability analysis are soil strain, soil properties, and pile properties. 2.11.2 Global Stability in the Strain Wedge Model The global stability, as analyzed by the strain wedge model, satisfies the general compatibility among soil reaction, pile deformations, and pile stiffness along the entire depth of the developing passive wedge in front of the pile. Therefore, the depth of the passive wedge depends on the global equilibrium between the loaded pile and the developed passive wedge. This requires a solution for Eqn. 2.1. The global stability is an iterative beam on elastic foundation ( BEF) problem that determines the correct dimensions of the passive wedge, the corresponding straining actions ( deflection, slope, moment, and shear) in the pile, and the external loads on the pile. Satisfying global stability conditions is the purpose of linking the three dimensional strain wedge model to the BEF approach. The major parameters in the pile global stability problem are pile stiffness, EI, and the modulus of subgrade reaction profile, Es, as determined from local stability in the strain wedge analysis. Since these parameters are determined for the applied soil strain, the stability problem is no longer a soil interaction problem but a one dimensional BEF problem. Any available numerical technique, such as the finite element or the finite difference method, can be employed to solve the global stability problem. The modeled problem, shown in Fig. 2.4c, is a BEF and can be solved to identify the depth, Xo, of zero pile deflection. 2.12 VERIFICATION OF APPROACH Based on the SW model concepts presented in this chapter and Ashour et al. ( 1996), a computer program 25 ( SWSG) has been developed to solve the problem of a laterally loaded isolated pile and a pile group in layered soil ( Ashour et al. 1996). Any verification of the methodology and algorithms employed should incorporate comparisons to field and laboratory tests for single piles and pile groups. The results presented below demonstrate the capability of the SW model approach and SWM program ( Ashour et al 1997 and 1998) in solving problems of laterally loaded piles relative to different soil and pile properties. It should be noted that pile and soil properties employed with the SW model analyses for the following field tests are the same properties mentioned in the references below. 2.12.1 Mustang Island Full Scale Load Test on a Pile in Submerged Dense Sand ( Reese et al. 1974 and Cox et al. 1974) As reported by Reese et al ( 1974), a series of full scale lateral load tests was performed on two single piles in sand at Mustang Island near Corpus Christi, Texas in 1966. The results obtained from those tests were used to develop criteria for the design of laterally loaded piles in sand and to establish a family of p y curves at different depths in the sand soil. In addition, the field results were used to characterize the pile head load deflection curve at the ground surface. Pile Configuration and Material Properties Tests were performed on two 0.61 m outside diameter ( O. D.) steel pipe piles ( A 53) with a wall thickness of 9.5 mm. The two piles were driven to a penetration of 21 m below the ground surface. The two closed end piles were instrumented along their lengths for the measurement of bending moment. Each pile tested consisted of a 11.6 m uninstrumented section, a 9.75 m instrumented section, and a 3 m uninstrumented section. The piles maintained an approximate stiffness, EI, of 167168 kN m2. Connecting flanges of 91.5 x 51 x 3.81 cm were welded to the instrumented section and to the 3 m 26 section. Small holes were cut in the pile wall just below the diaphragm to allow water and air to escape from the bottom of the 11.65 m section during driving. More details about the lateral load testing can be obtained by referring to Cox et al ( 1974). Table 1. Pile Properties Employed in the SWM Program Pile Type Shape Length Diam. Wall thick. Stiffness, EI Head Fixity Steel Pipe Round 21 m 0.61 m 9.5 mm 167168 KN m2 Free head Foundation Material Characterization Two soil borings were taken at the test site which were at the Shell Oil Company battery of tanks on the Mustang Island near Port Arkansas, Texas. As mentioned in Cox et al ( 1974), a comparison of the logs of borings 1 and 2 indicate that there was a slight variation in the soil profile between the two locations. In the top 12.2 m of boring 1, the sand strata was classified as a fine sand, while the soil in the top 12.2 m of boring 2 was classified as a silty fine sand. This difference in soil material was also reflected in the plot of the number of blows, N, of the standard penetration test ( Cox et al, 1974). The N values at boring 2 from 0 to 12.2 m are generally lower than those from boring 1. The sand from 0 to 6.1 m was classified as a medium dense sand, from 6.1 to 12.2 m as a dense sand, and from 15.2 m to 21.4 m as a dense sand. Laboratory tests were run on samples from boring 1 obtained using a piston sampler. More details of soil properties and the laboratory tests are documented in Cox et al ( 1974). The angle of internal friction was found to be 39 degree and the submerged unit weight of sand was 27 10.3 kN/ m3. The axial strain of the sand at 50 percent stress level, e 50, characterized based on Fig. 2.14 was 0.003 based on the assessed sand. Table 2. Soil Properties Employed in the SWM Program Soil type Thickness Effective Unit Weight Friction Angle, f e 5 0 Medium dense 21 m 10.3 kn/ m3 39 degree 0.003 Figure 2.17 presents a comparison of field results versus SW model results and results obtained using the computer program COM624 ( Reese 1977). Note that it is from this specific field test that the COM624 p y curves for sand were derived and, therefore, a good correspondence between COM624 and measured results is to be expected. The SW model results of pile head response shown in Fig. 2.17 are in excellent agreement at lower pile head deflections ( lower strain levels) and within 5 percent at higher levels of deflection ( higher strain levels). The SW model predicted maximum moment of Fig. 2.17 is in excellent agreement with measured results throughout. 2.12.2 Pyramid Building at Memphis, Tennessee, Full Scale Load Test on a Pile in Layered Clay Soil ( Reuss et al. 1992) A lateral load test was performed on a full scale pile in downtown Memphis. In order to improve the lateral capacity of the piles associated with this building, 1.8 meters of soft soil around the piles was removed and replaced with stiff compacted clay. Since the improved soil profile consisted of different types of soil, the corresponding test represents a layered field case study. Pile Configuration and Material Properties A 400 mm diameter reinforced concrete pile was installed to a total penetration of 22 meters. An inclinometer casing was installed in the pile to measure the lateral deflection. For a composite material such as reinforced concrete, the pile stiffness, EI, is a function of bending moment on the pile cross section. The experimental values of EI as a function of the bending moment are reported by Reuss et al. ( 1992). The 28 selected value of EI lies, in general, between the uncracked EI value and the cracked EI value. An average value for EI equal to 38,742 m2 kN was characterized for the pile. Additional concrete was cast around the pile to restrain it against excessive deflection when it was reloaded, and the pile head was free to rotate. Table 3. Pile Properties Employed in the SWM Program Pile Type Shape Length Diam. Average Stiffness, EI Head Fixity R/ C Rounded 22 m 0.4 m 38,742 KN m2 Free head Foundation Material Characterization The lateral load test conducted was performed at a location where the subsurface soil conditions could be approximated using information from a nearby soil boring. The soil profile, which consisted of different types of soils at this site, was the main advantage of this pile test. As documented by Reuss et al ( 1992), the top 1.8 meters of loose soil was replaced with a compacted gravely clay for the lateral load test. The fill soil consisted of cinders, bricks, concrete, gravel, and sand intermixed with varying percentages of clay to 1.8 meters below the ground surface. The first soil stratum ( fill soil) exhibited an undrained shear strength ( Su) of 47.9 kPa, a soil density ( g ) of 18.08 kN/ m3, and an e 50 of 0.005. The fill soil was underlain by soft to firm dark gray clay and silt clay with occasional silt and sand lenses. This soil layer ( the second stratum) extended from approximately 1.8 to 13.1 meters below the ground surface. The second stratum of soil exhibited an Su of 24 kPa, a g of 9.11 kN/ m3 , and an e 50 of 0.02. Standard penetration N values for stratum 2 varied from 3 through 10 with an average of about 5 blows per 0.3 meters. A third stratum between a depth of 13.1 meters and 19.9 meters below the ground surface had a reported Su value of 38.3 kPa, g of 9.11 kN/ m3 , and an e 50 value of 0.01 were reported. This stratum exhibited a greater frequency of silty and clayey sand lenses and increased strength as evidenced by penetration resistance N values ranging from 4 to 16 and averaging 10. The fourth stratum lay a depth of 19.9 meters and consisted of stiff silty clay and silty sand lenses. This stratum exhibited an Su value of 71.8 kPa, g of 9.11 kN/ m3 , and e 50 29 of 0.005. Table 4. Soil Properties Employed in the SWM Program Soil Layer # Soil type Thickness Effective Unit Weight e 5 0 Su 1 Sand mixed with clay, ciders and gravel 1.8 m 18.08 kN/ m3 0.005 47.9 kPa 2 Dark gray clay and silt clay 11.3 m 9.11 kN/ m3 0.02 24 kPa 3 Silty clay sand 6.8 m 9.11 kN/ m3 0.01 38.3 kPa 4 stiff silty clay Below 19.9 m depth 9.11 kN/ m3 0.005 71.8 kPa The soil properties of the fill soils and the second stratum ( the natural clay soil) were modified by Reuss et al. ( 1992) to force good agreement between the results assessed with COM624 ( Reese 1977) and the field results ( see Fig. 2.18a). The measured values of the undrained shear strength of the first and second strata were increased by 40 percent and 20 percent, respectively, to achieve such agreement. The measured soil properties were employed with the SW model to analyze the response of the pile in the improved soil profile. Figure 2.18a shows good agreement between the measured values and SW model predicted pilehead response in the improved soil profile. Figure 2.18b shows the pile head response predicted by COM624 and SW model analysis for the same pile in the original soil profile ( natural clay at its measured undrained strength with no fill layer). 2.12.3 Sabine River Full Scale Load Tests on a Pile in Soft Clay ( Matlock 1970) The benefit of the Sabine River tests derives from having load tests on piles of both free and fixed head conditions. Note that the results of the free head test were performed to establish the p y curve criteria for 30 piles in soft clay ( Matlock 1970). Pile Configuration and Material Properties The same pile was driven twice, and two complete series of static and cyclic loading tests were performed at the Lake Austin site and then at the Sabine River site. Only the static loading tests are considered in this study. The driven pile was a steel pipe pile of 0.32 m diameter and a 12.8 m embedded length. The pile maintained an approximate stiffness, EI, of 31,255 kN m2. The piles was tested under free head conditions at both sites ( Lake Austin and Sabine River) and fixed head conditions at Sabine River site. The Sabine River tests were used to develop the p y curves for short term static loading in soft clay. Table 5. Pile Properties Employed in the SWM Program Pile Type Shape Length Diam. Wall thick. Stiffness, EI Head Fixity Steel Pipe Rounded 12.8 m 0.32 m 12.75 mm 31,255 KN m2 Free head ( 1) Fixed head ( 2) Foundation Material Characterization As noted in Matlock et al ( 1970), extensive sampling and testing of the soils were undertaken at the Sabine River site. In situ vane shear tests as well as laboratory triaxial compression tests were performed to determine stress strain characteristics. Sabine clay is typical of a slightly overconsolidated marine deposit, and exhibited lower Vane shear strengths averaging about 14.33 kPa in the significant upper zone. According to Matlock et al ( 1970), the values of e 50 for most clays may be assumed to be between 0.005 and 0.02. An intermediate value of 0.01 is probably satisfactory for most purposes. Average values of 0.012 and 0.007 for e 50 were estimated from the soil stress strain curves at Sabine River. 31 Table 4. Soil Properties Employed in the SWM Program Soil Layer # Soil type Thickness Effective Unit Weight e 5 0 Su 1 Soft clay 12.8 m 7.8 kN/ m3 0.007 14.33 kPa As seen in Fig. 2.19a, the predicted free head SW model results are in good agreement with the observed results for the Sabine River site. At higher levels of deflection, the results calculated using the SW model fall approximately 5 to 10 percent below those measured in the field. By comparison, the SW model predicted and the observed fixed head pile response are in excellent agreement as shown in Fig. 2.19b. SW model results were established for two cases of the clay based on having a single average Su and, separately, for a varying Su. 2.13 SUMMARY The SW model approach presented here provides an effective method for solving the problem of a laterally loaded pile in layered soil. This approach assesses its own nonlinear variation in modulus of subgrade reaction or p y curves. The strain wedge model allows the assessment of the nonlinear p y curve response of a laterally loaded pile based on the envisioned relationship between the three dimensional response of a flexible pile in the soil to its one dimensional beam on elastic foundation parameters. In addition, the strain wedge model employs stress strain strength behavior of the soil as established from the triaxial test in an effective stress analysis to evaluate mobilized soil behavior. Compared to empirically based approaches which rely upon a limited number of field tests, the SW approach depends on well known or accepted principles of soil mechanics ( the stress strain strength relationship) in conjunction with effective stress analysis. Moreover, the required parameters to solve the problem of the laterally loaded pile are a function of basic soil properties that are typically available to the designer. 32 Fig. 2.1 The Basic Strain Wedge in Uniform Soil 33 Fig. 2.2 Deflection Pattern of a Laterally Loaded Long Pile and the Associated Strain Wedge Fig. 2.3 The Linearized Deflection Pattern of a Pile Embedded in Soil Using the Multi Sublayer Strain Wedge Model 34 Fig. 2.4 Soil Pile Interaction in the Multi Sublayer Technique 36 Fig. 2.5 The Proposed Geometry of the Compound Passive Wedge 37 Fig. 2.6 Distortion of the Wedge a), The Associated Mohr Circle ofStrain b), and the Relationship Between Pile Deflection and Wedge Distortion c) 38 Fig. 2.7 Relationship Between Horizontal Stress Change, Stress Level, and Mobilized Friction Angle 39 40 Fig. 2.8 The Developed Stress Strain Relationship in 41 Fig. 2.9 The Nonlinear Variation of Stress Level Along the Depth of Soil at Constant Strain _ Fig. 2.10 The Employed Side Shear Stress Displacement Curve in Clay 42 Fig. 2.11. The Nonlinear Variation of Shear Stress Level ( SLt) Along the Depth of Soil Fig. 2.12 Relationship Between e 50, Uniformity coefficient ( Cu) and Void Ratio ( e) ( Norris 1986) 0 0.25 0.5 0.75 1 1.25 1.5 Void Ratio, e 0 0.25 0.5 0.75 1 1.25 1.5 g 50 (%) Uniformity Coefficient, Cu 10 6 2 1.18 43 Fig. 2.13 Relationship Between Plasticity Index ( PI) and Effective j ( US Army Corps of Engineers 1996) Fig. 2.14 Relationship Between e 50 and Undrained Shear Strength , Su 50 u 0.00 0.01 0.02 g 100 1000 10000 200.00 300.00 500.00 2000.00 3000.00 5000.00 S ( psf) Range of Suggested Values ( After Reese 1980) 44 ( Evans and Duncan 1982) 45 Fig. 2.15 Relationship Between Effective Stress and Total Stress Conditions 46 Fig. 2.16 The Assembling of Pile Head Deflection Using the Multi Sublayer Technique 47 48 Strain Wedge Model Measured COM624 4 Fig. 2.17 The Measured and Predicted Response of a Laterally Loaded Pile in Sand at the Mustang Island Test. 0 2 4 6 Maximum Moment, kN cm * 10 0 100 200 300 400 Pile Head Load, Po, kN 0 1 2 3 4 5 Ground Deflection, Yo, cm 0 100 200 300 400 Pile Head Load, Po, kN A B 49 SW Model Measured COM624 ( Using Modified Soil Properties) R/ C Rounded Pile ( EI) = 1.35E7 kips in2 D = 15.75 in. L = 72 f t Improved Soil Strain Wedge Model COM624 ( Reuss et al., 1992) Original Soil Fig. 2.18 The Measured and the Predicted Response of the Loaded Pile in the Improved and the Original Soils at the Pyramid Building, Memphis, Test. 0 1 2 3 4 Ground Deflection, Yo, cm 0 50 100 150 200 Pile Head Load, Po, kN 0 4 8 12 Ground Deflection, Yo, cm 0 50 100 150 200 Pile Head Load, Po, kN A B 50 Fig. 2.19 The Measured and the SW Model Results of the Loaded Pile at the Sabine River Test. 0 2 4 6 8 Ground Deflection, Yo, cm 0 50 100 150 Pile Head Load, Po, kN SW with Su average SW with Su Profile Measured Free Head Pile Fixed Head Pile 0 2 4 6 8 10 Ground Deflection, Yo, cm 0 25 50 75 100 Pile Head Load, Po, kN A B 51 46 CHAPTER 3 PILE GROUP IN LAYERED SOILS 3.1 INTRODUCTION As presented in Chapter 2, the prediction of single pile response to lateral loading using the SW model correlates traditional one dimensional beam on an elastic foundation ( BEF) response to threedimensional soil pile interaction. In particular, the Young's modulus of the soil is related to the corresponding horizontal subgrade modulus; the deflection of the pile is related to the strain that exists in the developing passive wedge in front of the pile; and the BEF line load for a given deflection is related to the horizontal stress change acting along the face of the developing passive wedge. The three dimensional characterization of the laterally loaded pile in the SW model analysis provides an opportunity to study the interference among the piles in a pile group in a realistic fashion. The influence of the neighboring piles on an individual pile in the group will be a function of soil and pile properties, pile spacing, and the level of loading. These parameters are employed together in the SW model analysis to reflect the pile soil pile interaction on pile group behavior. The work presented illustrates the links between the single pile and the pile group analysis. The pile group procedure commonly used today employs the p y multiplier technique ( Brown et al. 1988). Such procedure is based on reducing the stiffness of the traditional ( Matlock Reese) p y curve by using a multiplier ( fm < 1), as seen in Fig. 3.1. The value of the p y curve multiplier should be assumed and is based on the data collected from full scale field tests on pile groups which are few ( Brown et al. 1988). Consequently, a full scale field test ( which is costly) is strongly recommended in order to determine the value of the multiplier ( fm) of the soil profile at the site under consideration. Moreover, the suggested value of the multiplier ( fm) is taken to be constant for each soil layer at all levels of loading. 47 In essence, this is quite similar to the traditional approach given in NAVFAC ( DM 7.2, 1982) in which the subgrade modulus, Es, is reduced by a factor ( Rm) taken as a function of pile spacing ( Rm = 1 at 8 diameter pile spacing varying linearly to 0.25 at 3 diameters). The difference is that fm has been found to vary with pile row ( leading, second, third and higher); and is taken to be constant with lateral pile displacement, y. By contrast, Davisson ( 1970) suggested that Rm should be taken constant with pile head load such that displacement y increases. In any case, neither fm, nor Rm, reflects any change with load or displacement level, soil layering, pile stiffness, pile position ( e. g. leading corner versus leading interior pile, etc.), differences in spacing both parallel and normal to the direction of load, and pile head fixity. As seen in Fig. 3.2, the interference among the piles in a group varies with depth, even in the same uniform soil, and increases with level of loading as the wedges grow deeper and fan out farther. Therefore, the use of a single multiplier that is both constant with depth and constant over the full range of load/ deflection would seem to involve significant compromise. The assessment of the response of a laterally loaded pile group based on soil pile interaction is presented herein. The strain wedge ( SW) model approach, developed to predict the response of a long flexible pile under lateral loading ( Ashour et al. 1998; and Ashour and Norris 2000), is extended in this paper to analyze the behavior of a pile group in uniform or layered soil. Several field and experimental tests reported in the literature are used to demonstrate the validity of the approach. 3.2 CHARACTERIZATION OF PILE GROUP INTERFERENCE The pile group is characterized in terms of the three dimensional pile soil pile interaction ( Pilling 1997) and then converted into its equivalent one dimensional BEF model with associated parameters ( i. e. an ever changing modulus of sugrade reaction profile). Therefore, the interference among the piles in a group is determined based on the geometry of the developing passive wedge of soil in front of the pile in addition to the pile spacing. A fundamental concept of the SW model is that the size and shape ( geometry) of the passive wedge of soil changes in a mobilized fashion as a function of 48 both soil and pile properties, at each level of loading, and is expressed as follows: As seen in Fig. 3.3, BC is the width of the wedge face at any depth, x. D is the width of the pile cross section, h is the current depth of the passive wedge which depends on the lateral deflection of the pile and, in turn, on the pile properties such as pile stiffness ( EI) and pile head fixity. ϕm is the mobilized fan angle of the wedge ( also the mobilized effective stress friction angle of the soil) and is a function of the current stress level ( SL) or strain ( ε) in the soil as presented by ( Ashour et al. 1998). The overlap of shear zones among the piles in a group varies along the length of the pile as shown in Figs. 3.3 and 3.4. Also, the interference among the piles grows with the increase in lateral load. The modulus of subgrade reaction, which is determined based on the SW model approach, will account for the additional strains ( i. e. stresses) in the adjacent soil due to pile interference within the group ( Figs. 3.4 and 3.5). Thus the modulus of subgrade reaction ( i. e. the secant slope of the p y curve) of an individual pile in a group will be reduced in a mobilized fashion according to pile and soil properties, pile spacing and position, the level of loading, and depth, x. No single reduction factor ( fm or Rm) for the p y curve ( commonly, assumed to be a constant value with depth and level of loading) is needed or advised. The SW model also allows direct evaluation of the nonlinear variation in pile group stiffness as required, for instance, for the seismic analysis of a pile supported highway bridge. The multi sublayer technique developed by Ashour et al. ( 1996 and 1998) and presented in Chapter 2 provides a means to determine the interference among the passive wedges of piles in a group and the additional stress/ strain induced in the soil in these wedges. As seen in Fig. 3.3, the soil around the piles in the group interferes horizontally with that of adjacent piles by an amount that varies with 2 = 45 + m m ϕ β ( 3.1) β ϕ m m BC = D + ( h  x) 2 tan tan ( 3.2) 49 depth. The multi sublayer technique allows the SW model to determine the overlap of the wedges of neighboring piles in different sublayers over the depth of the interference as shown in Figs. 3.4 and 3.5. This provides a great deal of flexibility in the calculation of the growth in stress ( and, therefore, strain) in the overlap zones which increases with the growth of the passive wedges. The main objective in the calculation of the area of overlap among the piles is to determine the increase in soil strain within the passive wedge of the pile in question. A value of horizontal soil strain ( ε) is assumed for the soil profile within the developing passive wedge. The response of a single pile ( similar to the piles in the group) in the same soil profile is determined at this value of soil strain. The shape and the dimensions of the mobilized passive wedge are assessed ( i. e. ϕm, βm, h and BC in Fig. 3.3) as presented in Chapter 2. This will include the values of stress level in each soil sublayer i ( SLi), Young’s modulus ( Ei), and the corresponding modulus of subgrade reaction ( Es) i. Wedges will overlap and interact with the neighboring ones, as seen in Figs. 3.3 and 3.5. At a given depth ( see Fig. 3.5), zones of overlap will exhibit larger values of soil strains and stresses. The increase in average soil strain attributable to the passive wedge of a given pile will depend upon the number and area of interfering wedges overlying the wedge of the pile in question ( Fig. 3.6). Such interference depends on the position of the pile in the group. The type of pile ( by position) is based on the location of the pile by row ( leading/ trailing row) and the location of the pile in its row ( side/ interior pile) as seen in Fig. 3.5. The average value of deviatoric stress accumulated at the face of the passive wedge at a particular soil sublayer i ( sand or clay) is 50 The average stress level in a soil layer ( SLg) due to passive wedge interference is evaluated based on the following empirical relationship, where j is the number of neighboring passive wedges in soil layer i that overlap the wedge of the pile in question. R is the ratio between the length of the overlapped portion of the face of the passive wedge and the total length of the face of the passive wedge ( BC). R ( which is less than 1) is determined from all the neighboring piles to both sides and in front of the pile in question ( Fig. 3.6). SLg and the associated soil strain ( εg) will be assessed for each soil sublayer in the passive wedge of each pile in the group. εg is ≥ ε of the isolated pile ( no wedge overlap) and is determined based on the stress strain relationship ( σ vs. ε) presented in Chapter 2. It should be noted that the angles and dimensions of the passive wedge ( ϕm, βm, and BC) obtained from Eqns. 3.1 through 3.4 will be modified for group effect according to the calculated value of SLg and εg ( Fig. 3.7). For instance, the relationship between the corresponding stress level ( SLg) and the associated mobilized effective stress friction angle ( ϕm) in a soil sublayer i is where ( Δσh) g is the current horizontal stress change ( due to pile head lateral load and pile group interference), and Δσhf is the unchanged value of the deviatoric stress at failure for the full friction angle ϕ. The mobilized friction angle ϕm calculated in Eqn. 3.5 reflects the stresses in the soil ( sand or clay) around the pile in question at depth x for the corresponding pile head ( group) deflection with ( Δσ h ) g = SLg Δσ hf ( 3.3) ( SL ) = SL ( 1 + R ) 0.5 1 g i i Σ j ≤ ( 3.4)  1 2 45 +  1 2 45 + ( ) = ( ) ( SL ) = 2 i 2 m i hf h g i g i Δ Δ ϕ ϕ σ σ tan tan ( 3.5) 51 consideration of the stresses from neighboring piles ( Figs. 3.5 and 3.6). Consequently, the geometry of the passive wedge is modified according to the current state of soil stress and strain ( Fig. 3.7). It should be noted that the behavior of clay is assessed based on the effective stress analysis in which the developing excess porewater pressure is evaluated in Chapter 2 and Ashour et al. ( 1996). 3.3 EVALUATION OF THE YOUNG’S MODULUS, Eg The change in the soil Young’s modulus and, therefore, the change in moulus of subgrade reaction in each sublayer due to group interference is assessed. Once the modified variation of the modulus of subgrade reaction along the individual pile is predicted, the pile is analyzed as an equivalent isolated pile ( considering all piles in the group have the same pile head deflection). Based on the modified value of soil strain assessed at depth x ( for the wedge of the pile of interest) at the current level of loading, the value of Young’s modulus, ( Eg) i, of the soil sublayer i is expressed, i. e. It should be noted the Young’s modulus ( Eg) calculated using Eqn. 3.6 results from the original strain in the passive wedge ( ε) as an isolated pile and the additional soil strain ( Δε) which develops due to overlap zones between the pile in question and its neighboring piles ( Fig. 3.8), i. e. According to the amount of interference among the piles in the group, the value of the Young’s modulus ( Eg) should be less or equal to the associated modulus ( E) for the isolated pile. 3.4 EVALUATION OF THE MODULUS OF SUBGRADE REACTION, Esg Based on the concepts of the SW model, the modulus of subgrade reaction for an individual pile in a group can be expressed as ( ) ( SL ) ( ) ( E ) = g i g i hf i g i ε Δσ ( 3.6) ( ε g ) i = ε i + Δε i ( 3.7) 52 where x is the depth of a soil sublayer i below the pile head. δ is the linearized deflection angle of the deflection pattern as presented by Ashour et al. ( 1996). Ag is a parameter that governs the growth of the passive wedge and flow around failure, and is a function of soil and pile properties ( Ashour and Norris 2000). S1 and S2 are shape factors equal to 0.75 and 0.5, respectively, for a circular pile cross section, and equal to 1.0 for a square pile ( Briaud et al. 1984). τ is the mobilized shear stress along the pile sides in the SW model ( see Fig. 3.7) and is defined according to the soil type ( sand or clay). ϕs is the mobilized side shear angle, SLt is is the stress level of shear along the pile sides, and τult is ( ) ( h  x ) ( A ) D ( ) ( E ) = y E = p i i g i g i g i i i s g i δ ε ( 3.8) [ ] [ ( ) ] + 2 S D = BC S ( ) p D ( A ) = h g i i 1 i 2 h g i i g i σ τ Δσ Δ / ( 3.9) = ( ) ( ); where ( ) = 2 ( ) sand τ i σ vo i tan ϕ s i tan ϕ s i tan ϕ m i tanϕ i ( 3.10) τ i = ( SLt ) i ( τ ult ) i clay ( 3.11) Therefore, ( )( ) sand ( ) + 2 S D h  x 2 ( A ) = S 1 + h g vo s i 2 i m m i g i 1 Δ σ tanβ tanϕ σ tanϕ ( 3.12) ( )( ) clay SL + S SL D h  x 2 ( A ) = S 1 + g t i 2 i m m i 1 i g tanβ tanϕ ( 3.13) 53 the ultimate shear resistance ( Coyle Reese 1966, and Ashour et al. 1998). Compared to the case of a single pile, the developing passive wedge of a pile in a group will be larger than or equal to that of the single pile ( depending on the amount of pile interference). However, the criteria presented in Chapter 2 and Ashour and Norris ( 2000) continue to govern the development of flow around failure; and variation of the BEF soil pile reaction ( p) and lateral deflection ( y) in the single pile analysis continue to be employed in the pile group analysis. It should be expected that the resulting modulus of subgrade reaction of a pile in a group, Esg, is equal to or softer than the Es of an isolated pile at the same depth ( Fig 3.9). The value of Es will vary with the level of loading and the growth of the soil stress in the developing passive wedge. Thus, there is no constant variation or specific pattern for changes in Es of the individual piles in the pile group. Based on the predicted values of Esg, the approach presented has the capability of assessing the p y curve for any pile in the group. The modulus of subgrade reaction of a pile in a group should reflect the mutual resistance between the soil and the pile. However, a portion of the pile deformation ( Δyi) results from the additional stresses in the soil ( and, therefore, strains, Δε) which result from the effect of the neighboring piles ( Figs. 3.5 and 3.6). Therefore, under a particular lateral load, the pile in the group will yield deflections more than those of the single pile. The additional deflection at any pile segment, ( Δyi), due to Δεi derives solely from the presence of neighboring piles, not the pile in question. The soilpile reaction ( p) is affected by the changes in stress and strain in the soil, and the varying geometry of the passive wedge. Having reduced values of Es along individual piles in the group, each pile is then analyzed as an equivalent isolated pile by BEF analysis. The piles in a group, at a particular step of loading, must experience equal deflections at the pile cap. For each pile in the group, the interference among the piles and the changes in the Es profile ( i. e geometry and dimensions of the passive wedge, and the internal stresses) will continue in an iterative process until the pile in question provides a pile head 54 deflection equal to that of the group. As a reference, the group deflection is linked to the pile head deflection ( Yo) of the isolated pile at the original soil strain ( ε). This technique provides great flexibility to analyze each pile in the group independently in order to develop equal pile head deflections ( group deflection) which are the shared factor among the piles in a group. 3.5 CASE STUDIES The original SW model program ( Ashour et al. 1997 and 1998) for analyzing lateral loaded piles has been modified to incorporate the technique presented above. The modified SWM program allows the assessment of the lateral response ( deflection, moment and shear force distribution) of an isolated pile and a pile group including the p y curve along the length of the isolated pile and the individual piles in the pile group. 3.5.1 Full Scale Load Test on a Pile Group in Layered Clay A static lateral load test was performed on a full scale 3 x 3 pile group having a three diameter center to center spacing ( Rollins et al. 1998). The driven pipe piles were 0.305 m I. D., 9.5 mm wall thickness, and 9.1 m in length. The Young’s modulus of the steel was 200 GPa, and the yield stress was 331 MPa. The soil profile along the length of the pile consisted of different types of clay and sand silt soils as described by Rollins et al. ( 1998). Figure 3.10 shows good agreement between the measured and SWM program predicted response for the single pile and average pile in the group ( pile group response is 9 times the average load at the same deflection). The P Multiplier was used by Rollins et al. 1998 to differentiate between the average response of different piles by row. Accordingly, fm values were varied arbitrarily to obtain the best match between the traditionally assessed p y curve and averaged observed behavior. The predicted response assessed using SWM, averaged by pile row, shows reasonable agreement with the reported behavior as seen in Fig. 3.11. The deviation between predicted and observed behavior in the 10 to 40 mm range for the isolated pile carries over to that of the average pile in the group over the same range. SWM response was obtained based on the given pile and soil properties, and pile group layout; no adjustment was made to obtain better fit. 55 3.5.2 Full Scale Load Test on a Pile Group in Sand A full scale lateral load test on a 3 x 3 pile group in sand overlying overconsolidated clay was conducted at the University of Houston, Texas ( Morrison and Reese, 1986). The results obtained from this load test were used to develop values of fm for use in the P multiplier approach for laterally loaded pile groups in sand ( Brown et al. 1988). This pile group of three diameter pile spacing was embedded in approximately 3 m of a dense to very dense uniform sand overlying an overconsolidated clay. The piles consisted of steel pipe with an outside diameter of 0.275 m, a wall thickness of 9.3 mm, a 13 m embedded length, and a bending stiffness ( EI) of 1.9 x 104 kN m2. The soil properties, including the buoyant unit weight and the angle of internal friction suggested by Morrison and Reese ( 1986), were used in the SW model analysis. Figures 3.12 and 3.13 show a comparison between the field data and the results obtained using the SWM program. As seen in Figs. 3.12 and 3.13, the observed and predicted responses of an average pile in the tested pile group are in good agreement. The good match of the predicted and observed behavior carries over to the average pile in the group. 3.5.3 Full Scale Load Test on a Pile Group in Layered Clay A full scale 3 x 3 pile group was driven in layered overconsolidated clay ( Brown and Reese 1985). The pile group tested had a three diameter pile spacing and was laterally loaded 0.3 m above ground surface. The nine pipe piles tested had the same properties as the piles used in the preceding case study. The soil properties ( ε50, the soil unit weight, and the undrained shear strength of clay) evaluated by Brown and Reese were employed in the SW model analysis. As shown in Figs. 3.14 and 3.15, the SW model provides good agreement with observed behavior for both the single and average pile in the group for pile head load versus deflection and pile head load versus maximum bending moment. It should be noted that this case represents a layered clay profile which exhibits different levels of wedge interference in each soil layer that then changes with the level of loading. 56 The procedure presented here has the capability to predict the pile head response, deflection, and bending moment for every individual pile in the group ( type 1 through type 4 based on pile location, as seen in Fig. 3.5) not just the average pile response. Previous comparisons in terms of average pile in the group or average by row reflect what is reported in the literature. Likewise, the SWM program can assess the additional contribution to pile group resistance due to the presence of an embedded pile cap ( not presented in this study) at any level of lateral loading. The effect of pile cap resistance on the lateral resistance of pile group can be judged from the following case study. 3.5.4 Full Scale Load Test on a Pile Group with a Pile Cap in Layered Soil A series of high amplitude load tests were performed on the Rose Creek bridge near Winnemucca, Nevada ( Douglas and Richardson 1984). The stiffnesses of four pile groups, with pile caps of the foundation system were backfigured from system identification analysis of the collected accelerometer data. The soil profile and results of the tests are discussed by Norris ( 1994). Although the contribution of the embedded pile cap to the lateral resistance of the pile group has not been discussed in this paper, its effect on the lateral stiffness of pile groups is undertaken in the results predicted using the SWM program. Piers 1 and 4 are each supported by a 3 x 5 pile group with 3 diameter pile spacings embedded in layered silt and clay soil, while piers 2 and 3 are each supported by a 4 x 5 pile group with 3 diameter pile spacings in the same layered silt and clay soil. Pile caps ( 4.57 x 2.75 m and 1.3 m thickness) associated with piers 1 and 4 are founded at 1.5 m depth below finished grade in a medium dense sand silt soil. Pile caps ( 4.57 x 3.65 m and 1.3 m thickness) associated with piers 2 and 3 are founded at 0.92 m below finished grade in a medium dense sand silt soil. The piers extend from the pile caps have a width of 1.22 m. The piles are steel pipe piles of 0.32 m outer diameter backfilled with concrete and a bending stiffness of 3.38 x 104 kN m2. The piles associated with piers 1 through 4 were driven to 8, 7.8, 7.3 and 7 m below the bottom of the pile cap. All pile groups were loaded laterally in the direction 57 normal to length ( long side) of the pile cap. Full details on soil and pile properties are presented by Norris ( 1994). The pile heads are embedded 0.3 m into the pile cap. The piles in the group are treated as fixed head piles in the SW Model analysis. Even if the depth of pile head embedment into the pile cap was not adequate to provide complete restriction on the pile head rotation, the pile head in the group would exhibit fixed head conditions at the very low values of lateral deflection observed during the bridge load tests. Figure 3.16 shows the agreement between the measured ( backcalculated) and predicted pile group stiffnesses for groups 1 through 4 using the SWM program. It should be noted that the pile cap contribution to the total resistance of the group is a function of the pile cap dimensions and its embedment depth, properties of surrounding soil, and the level of lateral loading. 3.5.5 Model Scale Load Test on a Pile Group in Loose and Medium Dense Sand A series of load tests were performed using centrifuge tests on a model isolated pile, and on a model 3 by 3 pile group with piles spaced at 3 and 5 pile diameters within the group, embedded in a poorly graded loose ( Dr = 33%) and medium dense sand ( Dr = 0.55) ( McVay et al 1995). The prototype model piles, simulated using the centrifuge and a 1/ 45 ( i. e. 45g) scale consisted of steel pipe piles with a diameter of 0.43 m and an overall length of 13.3 m. The pile had a bending stiffness, EI, of 72.1 MN m2. The point of lateral load application to the pile groups was approximately 1.68 m above finished grade, while the point of lateral load application to the isolated pile was approximately 2.2 m above finished grade. Although a pile cap was associated with the pile group tests, McVay et al ( 1995) reported that the group tests simulated free headed piles. Very good agreement, between measured and predicted results, is shown in Figs. 3.17 and 3.18. Slight differences are observed between the measured and predicted capacity of the pile rows ( leading, middle and trailing rows) in the group. It should be noted that the procedure presented herein has the capability of assessing the capacity of three different pile rows ( leading, middle and 58 trailing rows). Therefore, 6 types of piles by position ( instead of 4 types as seen in Fig. 3.5) should be analyzed. However, at low and medium level of pile head deflection, no significant differences are observed between the lateral resistance of the middle and trailing row. 3.6 SUMMARY Assessment of the response of a laterally loaded pile group based on soil pile interaction is presented. The behavior of a pile group in uniform and layered soil ( sand and/ or clay) is predicted based on the strain wedge ( SW) model approach that was developed to analyze the response of a flexible long pile under lateral loading. Accordingly, the pile’s response is characterized in terms of three dimensional soil pile interaction which is then transformed into its one dimensional beam on elastic foundation equivalent with associated parameters ( modulus of sugrade reaction). The interference among the piles in a group is determined based on the geometry and interaction of the mobilized passive wedges of soil in front of the piles in association with the pile spacing. The overlap of shear zones among the piles in the group varies along the length of the pile and changes from one soil layer to another in the soil profile. Also, the interference among the piles grows with the increase in lateral loading, and the increasing depth and fan angles of the developing wedges. The modulus of subgrade reaction determined will account for the additional strains ( i. e. stresses) in the adjacent soil due to pile interference within the group. Based on the approach presented, the p y curve for individual piles in the pile group can be determined. The reduction in the capacity of the individual piles in a group compared to the isolated pile is governed by soil and pile properties, level of loading, and pile spacing. 59 Fig. 3.1 P Multiplier ( fm) Concept for Pile Group ( Brown et al. 1988) 60 Fig. 3.2 Lateral Interference Between Two Neighboring Piles 61 Fig. 3.3 Mobilized Passive Wedges and Associated Pile Group Interference Fig. 3.4 Front Overlap Among Soil Sublayers in Two adjacent Passive Wedges ( Section J J in Fig. 3.3.) Developing Passive Soil Wedges Pile in Question Pile Type 1 Pile Type 2 Pile Type 3 Pile Type 4 By Position Loading Direction Leading Row Trailing Row Trailing Row 62 Fig. 3.5 The Initial Interference Among Piles in a Pile Group at a Given Depth Fig. 3.6 Example of Overlap Ratio Calculation Among piles in a Pile Group ** At the same level of pile head deflection, fm for an isolated pile is ≤ fm for an individual pile in a pile group Pile in Question Loading Direction C D L1 L2 R1 = L1 CD R2 = L2 CD 63 Fig. 3.7 Stress and Geometry Change in a Slice of an Individual Pile in a Pile Group Fig. 3.8 Changes in Soil Young’s Modulus Due to Pile Interference in a Pile Group at a Particular Level Of Loading 64 Fig. 3.9 Change in the Modulus of Subgrade Reaction ( i. e. the p y Curve) due to Pile Interference in the Pile Group at Different Levels of Loading according to the SW Model 65 Leading Row SW Model Measured Single Pile Group 0 20 40 60 80 Deflection at Load Point, Yo, mm. 0 40 80 120 160 200 Average Load per Pile, Po, kN 0 20 40 60 80 Deflection at Load Point, Yo, mm. 0 50 100 150 Average Load per Row, Po, kN SW Model Measured Trailing Row Fig. 3.10 Average lateral load versus deflection curves for isolated pile and average pile in a 3 x 3 group ( after Rollins et al. 1998) Fig. 3.11 Measured and predicted average lateral load per row versus deflection ( after Rollins et al. 1998) 66 Single Pile Single Pile Group 0 20 40 60 Deflection at Load Point, Y o , mm. 0 25 50 75 100 Average Load per Pile, P o , kN Compression Stroke Tension Stroke 0 50 100 150 200 Maximum Bending Moment, kN m 0 25 50 75 100 Average Load per Pile, P o , kN Group SW Model Measured ( Compression Stroke) ( Tension Stroke) SW Model Measured ( Compression Stroke) ( Tension Stroke) Fig. 3.12 Lateral pile head lateral load vs. deflection for an isolated pile and an average pile in a 3 x 3 group in sand ( after Morrison and Reese, 1986) Fig. 3.13 Lateral load vs. maximum bending moment for isolated pile and an average pile in a 3 x 3 group in sand ( after Morrison and Reese, 1986) 67 Single Pile SW Model Measured Single Pile Group 0 20 40 60 80 100 Deflection at Load Point, Yo, mm. 0 20 40 60 80 100 Average Load per Pile, Po, kN Group 0 40 80 120 160 200 Maximum Bending Moment, kN m 0 50 100 150 Average Load per Pile, Po, kN SW Model Measured Fig. 3.14 Lateral load vs. deflection for isolated pile and an average pile in a 3 x 3 group in clay.( after Brown and Reese, 1985) Fig. 3.15 Lateral load vs. maximum bending moment for isolated pile and an average pile in a 3 x 3 group in clay.( after Brown and Reese, 1985). 68 0.001 0.01 0.1 1 10 Pile Head deflection, Yo , mm 0 100 200 300 400 500 Pile Group Stiffness, Kg , kN/ mm Measured and Predicted Stiffness of Pile Group 1 0.001 0.01 0.1 1 10 Pile Head deflection, Yo , mm 0 400 800 1200 1600 Pile Group Stiffness, Kg , kN/ mm Measured and Predicted Stiffness of Pile Group 2 0.001 0.01 0.1 1 10 Pile Head deflection, Yo , mm 0 400 800 1200 1600 2000 Pile Group Stiffness, Kg , kN/ mm Measured and Predicted Stiffness of Pile Group 3 0.001 0.01 0.1 1 10 Pile Head deflection, Yo , mm 0 100 200 300 400 500 Pile Group Stiffness, Kg , kN/ mm Measured and Predicted Stiffness of Pile Group 4 Measured SW Model Measured SW Model Measured SW Model Measured SW Model Fig. 3.16 Measured and Predicted Group Stiffness for Rose Creek Bridge Foundation ( After Norris 1994) 69 Pile Spacing = 3D Dr = 33% Pile Spacing = 5D Dr = 33% 0 20 40 60 80 100 Lateral Deflection, Yo, mm 0.0 0.4 0.8 1.2 1.6 Group Lateral Load, MN Fig. 3.17 A Pile Group Model ( 3x3) in Medium Loose Sand ( after McVay et al. 1995) 0 20 40 60 80 0.0 0.4 0.8 1.2 Group Lateral Load, MN Capacity % Row Location Lead Middle Tail Predicted 36 32 32 Measured 37 33 30 33 33 33 35 33 31 70 o Pile Spacing = 3D Dr = 55% Pile Spacing = 5D Dr = 55% Fig. 3.18 A Pile Group Model ( 3x3) in Medium Dense Sand ( after McVay et al. 1995) Capacity % Row Location Lead Middle Tail Predicted 36 32 32 Measured 41 32 27 33 33 33 36 33 31 0 20 40 60 80 100 0.0 0.4 0.8 1.2 1.6 Lateral Load, MN 0 20 40 60 80 100 Lateral Deflection, Yo, mm 0.0 0.4 0.8 1.2 1.6 2.0 Group Lateral Load, MN 69 70 CHAPTER 4 NUMERICAL MATERIAL MODELING 4.1 INTRODUCTION Deformations in any structural element depend upon the characteristics of the load, the element shape and its material properties. With laterally loaded piles and shafts, the flexural deformations are based on the applied moment and the flexural stiffness of the pile at the cross section in question. In addition, the flexural stiffness ( EI) of the pile is a function of the Young’s modulus ( E), moment of inertia ( I) of the pile cross section and the properties of the surrounding soil. Given the type of material, concrete and/ or steel, the properties of pile material vary according to the level of the applied stresses. Behavior of piles under lateral loading is basically influenced by the properties of both the soil and pile ( pile material and shape). The nonlinear modeling of pile material, whether it is steel and/ or concrete, should be employed in order to predict the value of the lateral load and the realistic associated bending moment and pile deflection especially at large values of pile head deflection and the onset of pile material failure. It is known that the variation in the bending stiffness ( EI) of a laterally loaded pile is a function of the bending moment distribution along the pile ( moment curvature, M F , relationship) as seen in Fig. 4.1. Consequently, some of the pile cross sections which are subjected to high bending moment experience a reduction in bending stiffness and softer interaction with the surrounding soil. Such behavior is observed with drilled shafts and steel piles at advanced levels of loading and has an impact on the lateral response and capacity of the loaded pile. The pile bending stiffnesses along the deflected pile change with the level of loading, the M F relationship of the pile material, and the soil reaction which affects the pattern of pile deflection. Therefore, the equilibrium among the distributions of pile deflection, bending moment, bending stiffness, and soil reaction along the pile should be maintained. 71 In the case of a steel pile, the Young’s modulus remains constant ( elastic zone) until reaching the yield stress, fy ( indicating the initial yielding), at which time the steel starts to behave elastic plastically with different values of the secant Young’s modulus. Once a plastic hinge develops, the pile cross section responds in plastic fashion under a constant plastic moment. But, in the case of a concrete pile or shaft, the stress strain relationship varies in a nonlinear fashion producing a simultaneous reduction in Young’s modulus and, in turn, the stiffness of the pile cross section. Furthermore, once it reaches a critical value of strain, the concrete ruptures catastrophically. The technique suggested by Reese ( 1984), which employs the Matlock Reese p y curves, requires separate evaluation of the M F relationship of the pile cross section and then adoption of a reduced bending stiffness ( EIr) to replace the original pile bending stiffness ( EI). The suggested procedure utilizes this reduced bending stiffness ( EIr) over the full length of the pile at all levels of loading. Assuming a reasonable reduction in bending stiffness, particulary with drilled shafts, is a critical matter that requires guidance from the literature which has only limited experimental data. At the same time, the use of one constant reduced bending stiffness for the pile/ shaft does not reflect the real progressive deformations and forces associated with the steps of lateral loading. However, this technique may work quite well with the steel H pile which fails approximately once the pile flange reaches the yielding stage ( occurs rapidly). In general, the response of the pile/ drilled shaft ( pile head load vs. deflection, and pile head load vs. maximum moment) is assessed based on a constant bending stiffness ( EIr) and is truncated at the ultimate bending moment of the original pile/ drilled shaft cross section. The moment curvature relationship, and thus the maximum bending moment carried by the pile cross section should be evaluated first. Reese and Wang ( 1994) enhanced the technique presented above by computing the bending moment distribution along the pile and the associated value of EI at each increment of loading. Reese and Wang ( 1994) concluded that the bending moment along the pile does not depend strongly on structural characteristics and that the moment differences due to EI variations are small. It should be noted that the effect of the varying EI on the bending moment values along the drilled shaft was not obvious because the 72 EI of the drilled shaft had no effect on the p y curves ( i. e. modulus of subgrade reaction) employed in their procedure. Therefore, it was recommended that a single value of EI of the cracked section ( constant value) be used for the upper portion of the pile throughout the analysis. Contrary to Reese and Wang’s assumption, the variation in the value of EI has a significant effect on the nature of the p y curve and modulus of subgrade reaction [ Ashour and Norris ( 2000); Yoshida and Yoshinaka ( 1972); and Vesic ( 1961)] specially in the case of drilled shafts. The main purpose in this chapter is to assess the moment curvature relationship ( M f ) of the loaded pile or shaft in a convenient and simplified fashion considering the soil pile interaction. The prediction of the moment curvature curve allows one to realistically determine the variation of pile stiffness ( EI) as a function of bending moment. The SW model allows the designer to include the nonlinear behavior of the pile material and, as a result, to find out the effect of material types on the pile response and its ultimate capacity based on the concepts of soil pile interaction. 4.2 THE COMBINATION OF MATERIAL MODELING WITH THE STRAIN WEDGE MODEL The bending moment distribution along the deflected length of a laterally loaded pile varies as shown in Fig. 4.1. This profile of moment indicates the associated variation of pile stiffness with depth if the stress strain relationship of pile material is nonlinear. The strain wedge model is capable of handling the nonlinear behavior of pile material as well as the surrounding soil. The multi sublayer technique, presented in Chapter 2, allows one to provide an independent description for each soil sublayer and the associated pile segment. The effect of pile material is considered with the global stability of the loaded pile and the shape of the developing passive wedge of soil in front of the pile. During the iteration process using the SW model, the stiffness of each pile segment, which has a length equal to the depth of the soil sublayer, is a function of the 73 calculated bending moment at the associated pile segment, as seen in Fig. 4.1. Therefore, the pile is divided into a number of segments of different values of flexural stiffness under a particular lateral load. In order to incorporate the effect of material nonlinearity, numerical material models should be employed with the SW model. A unified stress strain approach for confined concrete has been employed with the reinforced concrete pile as well as the steel pipe pile filled with concrete. In addition, steel is modeled using an elastic perfectly plastic uniaxial stress strain relationship which is commonly used to describe steel behavior. The procedure presented provides the implementation of soil pile interaction in a fashion more sophisticated than that followed in the linear analysis with the SW model presented in Chapter 3. The approach developed will allow one to load the pile to its actual ultimate capacity for the desired lateral load and bending moment according to the variation of pile material properties along the pile length. 4.2.1 Material Modeling of Concrete Strength and Failure Criteria Based upon a unified stress strain approach for the confined concrete proposed by Mander et al. ( 1984 and 1988), a concrete model is employed with circular and rectangular concrete sections. The proposed model, which is shown in Fig. 4.2, has been employed for a slow strain rate and monotonic loading. The longitudinal compressive concrete stress fc is given by where fcc symbolizes the compressive strength of confined concrete. r  1 + x f x r f = r cc c ( 4.1) 74 where e c indicates the axial compressive strain of concrete. where e cc is the axial strain at the peak stress. fco and e co represent the unconfined ( uniaxial) concrete strength and the corresponding strain, respectively. Generally, e co can be assumed equal to 0.002, and where and e e cc x = c ( 4.2) ú ú û ù ê ê ë é ÷ ÷ ø ö ç ç è æ  1 f f = 1 + 5 co cc e cc e co ( 4.3) E  E r = E c c sec ( 4.4) E = 57,000 ( f ) ( p s i) 0.5 c co ( 4.5) 75 Ec denotes the initial modulus of elasticity of the concrete under slowly applied compression load. As mentioned by Paulay and Priestly ( 1992), the strain at peak stress given by Eqn. 5.3 does not represent the maximum useful strain for design purposes. The concrete strain limits occur when transverse confining steel fractures. A conservative estimate for ultimate compression strain ( e cu) is given by where e sm is the steel strain at maximum tensile stress ( ranges from 0.1 to 0.15), and r s is the volumetric ratio of confining steel. Typical values for e cu range from 0.012 to 0.05. fyh represents the yield stress of the transverse reinforcement. In order to determine the compressive strength of the confined concrete ( fcc), a constitutive model ( Mander et al. 1988) is directly related to the effective confining stress ( fl) that can be developed at the yield of the transverse reinforcement. e cc cc f Esec = ( 4.6) f 1.4 f = 0.004 + cc s yh sm cu r e e ( 4.7) ú ú û ù ê ê ë é ÷ ÷ ø ö ç ç è æ f 2 f  f 7.94 f f = f  1.254 + 2.254 1 + co l co l 0.5 cc co ( 4.8) 76 For circular and square section of concrete, fl is given by · Monotonic tensile loading Although concrete tension strength is ignored in flexural strength calculation, due to the effect of concrete confinement it would be more realistic if it were considered in the calculation. As suggested by Mander et al. ( 1988), a linear stress strain relationship is assumed in tension up to the tensile strength ( ftu). The tensile stress is given by and where If tensile strain e t is greater than the ultimate tensile strain ( e tu), ft is assumed to be equal to zero. f = 0.95 f l s yh r ( 4.9) f = E for f f t c c t tu e £ ( 4.10) E f = c tu e tu ( 4.11) f = 9 ( f ) ( p s i ) 0.5 tu co ( 4.12) 77 4.2.2 Material Modeling of Steel Strength There are different numerical models to represent the stress strain relationship of steel. The model employed for steel in this study is linearly elastic perfectly plastic, as shown in Fig. 4.3. The complexity of this numerical model is located in the plastic portion of the model which dose not include any strain hardening ( perfectly plastic). The elastic behavior of the steel is limited by the linearly elastic zone of this model at which the strain is less than the yield strain where fy is the yield stress of steel, and e y is the value of the steel strain at the end of the elastic zone where the stress is equal to fy. Eso is the elastic Young’s modulus of steel which is equal to 29,000 kips/ inch2. When the value of steel stress ( fs) at any point on the cross section reaches the yield stress, the Young’s modulus becomes less than Eso of the elastic zone. The initial yielding takes place when the stress at the farthest point from the neutral axis on the steel cross section ( point A) becomes equal to the yield stress ( fy), as shown in Fig. 4.4a. The initial yielding indicates the beginning of the elastic plastic response of the steel section. By increasing the load, other internal points on the cross section will satisfy the yield stress to respond plastically under a constant yield stress ( fy), as seen in Figure 4.4b. Once all points on the steel section satisfy a normal stress ( fs) equal to the yield stress ( fy) or a strain value larger than the yield strain ( e y), the steel section responds as a plastic hinge with an ultimate plastic moment ( Mp) indicating the complete yielding of the steel section, as presented in Fig. 4.4c. E f = so y e y ( 4.13) 78 During the elastic plastic stage ( after the initial yielding and before complete yielding) some points on the steel section respond elastically ( fs £ fy) and the others respond plastically ( fs = fy) with different values of Young’s modulus ( Es) , as presented in Fig. 4.3. The values of normal strain are assumed to vary linearly over the deformed cross section of steel. If the strain at any point on the steel cross section is larger than the yield strain ( e y), the plastic behavior will be governed by the flow of the steel under a constant stress ( fy) at the point in question. Regardless of whether the section is under elastic, elastic plastic or plastic states, the strain is linearly distributed over the whole steel section. In addition, the strain at any point is controlled by the values of strain at other locations in order to keep the strain distribution linear. Generally, the external and internal moments over the steel section should be in a state of equilibrium. 4.3 MOMENT CURVATURE ( M F ) RELATIONSHIP The aim of developing the moment curvature relationship of the pile material is to determine the variation of the flexural stiffness ( EI) at every level of loading. The normal stress ( s x) at any cross section along the pile length is linked to the bending moment ( M) and curvature ( f ) by the following equations: = M d x EI d y 2 2 ( 4.14) = M EI EI = r f ( 4.15) 79 where z = the distance from the neutral axis to the longitudinal fiber in question r o = the radius of curvature of the deflected axis of the pile e x = the normal strain at the fiber located z distance from the neutral axis. The above equations are based on the assumption of a linear variation of strain across the pile cross section. In addition, the pile cross section is assumed to remain perpendicular to the pile axis before and after deforming, as shown in Fig. 4.5. 4.4 SOLUTION PROCEDURE The solution procedure adopted consists of calculating the value of bending moment ( Mi) at each cross section associated with a profile of the soil modulus of subgrade reaction which is induced by the applied load at the pile top. Then, the associated curvature ( f ), stiffness ( EI), normal stress ( s x) and normal strain z = d x = d y x 2 2 f e ( 4.16) r e o x z =  ( 4.17) = E = E z x x s e f ( 4.18) 80 ( e x) can be obtained. This procedure depends on the pile material. The profile of moment distribution along the deflected portion of the pile is modified in an iterative fashion along with the values of the strain, stress, bending stiffness and curvature to satisfy the equilibrium among the applied load and the associated responses of the soil and pile. Based on the concepts of the SW model, the modulus of subgrade reaction ( i. e. p y curve) is influenced by the variations in the pile bending stiffness at every pile segment. This procedure guarantees the incorporation of soil pile interaction with the material modeling. The technique presented strives for a more realistic assessment of the pile deflection pattern under lateral loading and due to the nonlinear response of pile material and soil resistance. 4.4.1 Steel Pile Steel piles involved in this study have either circular ( pipe) or H shape cross sections, as seen in Fig. 4.6. The cross section of the steel pipe pile is divided into a number of horizontal strips ( equal to a total of 2m) parallel to the neutral axis. Each strip has a depth equal to the thickness of the pipe pile skin, as seen in Fig. 4.7. The cross section of the steel H pile is divided into horizontal strips of a width equal to one half the thickness of the H section flange, as seen in Fig. 4.7. The moment applied over the cross section of the pile segment ( i) is Mi, and the normal stress at a strip ( n) is ( fs) n ( 1 £ n £ m). Using Eqns. 4.17 and 4.18, the stress and strain distributions over the cross section of each pile segment can be determined as ( E I ) = M i i i f ( 4.19) ( ) = z 1 n m s n n i e f £ £ ( 4.20) 81 where Es £ Eso; f i is the curvature at pile segment ( i) which is constant over the steel cross section at the current level of loading; zn indicates the distance from the neutral axis to the midpoint of strip n; ( e s) n represents the strain at strip n; ( EI) i represents the initial stiffness of the pile segment ( i); I is the moment of inertia of the steel cross section of the pile segment ( i) which is always constant; and Eso symbolizes the elastic Young’s modulus of the steel. 1. Elastic Stage The Young’s modulus of any strip of the steel section ( i) is equal to the steel elastic modulus ( 29x106 psi) as 



B 

C 

I 

S 


