Deformation analysis methods for drilled shaft foundations subjected to lateral and overturning moment loads

Drilled shaft foundations are widely used in many civil engineering applications where deep foundations are required because they are relatively easy to construct and are suitable for resisting lateral, axial, and overturning moment loads. While the analysis of drilled shafts subjected to axial loads is fairly straightforward, it is much more difficult to analyze drilled shafts subjected to lateral and overturning moment loads due to the complex nature of the soil-structure interaction. It has been suggested that the p-y model, which is currently the most commonly used model for performing these types of analyses, considerably overestimates deformation of semi-rigid to rigid drilled shafts subjected to lateral and overturning moment loads. While the p-y model has been shown to reasonably predict deformation of flexible steel pipe piles and drilled shafts, it has not been verified for rigid to semi-rigid drilled shafts. The major objectives of this investigation were to identify other methods in current use that might be more appropriate for analyzing this type of drilled shaft and to assess the accuracy of each analysis method by comparing the results from each method to the results of large-scale load tests. The literature review revealed several analysis methods, which range from simple analytical methods to complex numerical methods. In addition to the p-y model, other commonly used analysis methods include the strain wedge model and the four-spring model. All of these models are semi-empirical and rely to some extent on experimentally-observed data and simplifying assumptions about the soil-structure interaction. The p-y model, the strain wedge model, and the four-spring model are implemented in the commercial software packages LPile, DFSAP, and MFAD, respectively. Several large-scale load tests for rigid to semi-rigid drilled shafts were also identified in the literature review. The information from these load tests was used to perform analyses using the LPile, DFSAP, and MFAD program, and the results were compared to the experimentally-observed results. The results suggest that MFAD is the most accurate model for granular soils and that DFSAP is the most accurate model for cohesive soils. For the foundations considered in this investigation, there was no apparent correlation between the accuracy of the DFSAP or MFAD results and the rigidity of the foundation; however, the accuracy of the LPile results tended to decrease as the foundation rigidity increased. A parametric study was conducted to investigate how the soil input properties affect the results, and how sensitive the models are to variations in these properties. The parametric study showed that LPile and DFSAP are most sensitive to input properties of the angle of internal friction (&phis;) and undrained shear strength (Su) of the soil, while DFSAP is most sensitive to the modulus of deformation (E p) of the soil. A statistical analysis of the combined data from each analysis method resulted in design equations for estimating semi-rigid to rigid drilled shaft deflection using LPile, DFSAP , or MFAD for a target level of reliability.

DEFORMATION ANALYSIS METHODS FOR DRILLED SHAFT FOUNDATIONS SUBJECTED TO LATERAL AND OVERTURNING MOMENT LOADS by Byron Holth Foster A thesis submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Master of Science Department of Civil and Environmental Engineering The University of Utah May 2016 Copyright © Byron Holth Foster 2016 All Rights Reserved The University of Utah Graduate School STATEMENT OF THESIS APPROVAL The thesis of Byron Holth Foster has been approved by the following supervisory committee members: Evert C. Lawton Chair 1/14/2016 Date Approved Steven F. Bartlett Member 1/14/2016 Date Approved Chris P. Pantelides Member 1/14/2016 Date Approved and by Michael E. Barber the Department of Chair of Civil and Environmental Engineering and by David B. Kieda, Dean of the Graduate School ABSTRACT Drilled shaft foundations are widely used in many civil engineering applications where deep foundations are required because they are relatively easy to construct and are suitable for resisting lateral, axial, and overturning moment loads. While the analysis of drilled shafts subjected to axial loads is fairly straightforward, it is much more difficult to analyze drilled shafts subjected to lateral and overturning moment loads due to the complex nature of the soil-structure interaction. It has been suggested that the p-y model, which is currently the most commonly used model for performing these types of analyses, considerably overestimates deformation of semi-rigid to rigid drilled shafts subjected to lateral and overturning moment loads. While the p-y model has been shown to reasonably predict deformation of flexible steel pipe piles and drilled shafts, it has not been verified for rigid to semi­rigid drilled shafts. The major objectives of this investigation were to identify other methods in current use that might be more appropriate for analyzing this type of drilled shaft and to assess the accuracy of each analysis method by comparing the results from each method to the results of large-scale load tests. The literature review revealed several analysis methods, which range from simple analytical methods to complex numerical methods. In addition to the p-y model, other commonly used analysis methods include the strain wedge model and the four-spring model. All of these models are semi-empirical and rely to some extent on experimentally-observed data and simplifying assumptions about the soil-structure interaction. The p-y model, the strain wedge model, and the four-spring model are implemented in the commercial software packages LPile, DFSAP, and MFAD, respectively. Several large-scale load tests for rigid to semi-rigid drilled shafts were also identified in the literature review. The information from these load tests was used to perform analyses using the LPile, DFSAP, and MFAD program, and the results were compared to the experimentally-observed results. The results suggest that MFAD is the most accurate model for granular soils and that DFSAP is the most accurate model for cohesive soils. For the foundations considered in this investigation, there was no apparent correlation between the accuracy of the DFSAP or MFAD results and the rigidity of the foundation; however, the accuracy of the LPile results tended to decrease as the foundation rigidity increased. A parametric study was conducted to investigate how the soil input properties affect the results, and how sensitive the models are to variations in these properties. The parametric study showed that LPile and DFSAP are most sensitive to input properties of the angle of internal friction (<f) and undrained shear strength (su) of the soil, while DFSAP is most sensitive to the modulus of deformation (Ep) of the soil. A statistical analysis of the combined data from each analysis method resulted in design equations for estimating semi-rigid to rigid drilled shaft deflection using LPile, DFSAP, or MFAD for a target level of reliability. iv To my family, for their unwavering love and support. TABLE OF CONTENTS ABSTRACT............................................................................................................................... iii LIST OF SYMBOLS AND NOTATION............................................................................. viii ACKNOWLEDGEMENTS.................................................................................................... xiv 1 INTRODUCTION................................................................................................................. 1 1.1! Background..................................................................................................................... 1 1.2 Objectives of the Research.......................................................................................... 5 1.3 Organization of the Thesis........................................................................................... 6 1.4 Limitations of the Investigation...................................................................................7 2 LITERATURE REVIEW..................................................................................................... 8 2.1 Introduction................................................................................................................... 8 2.2 Drilled Shaft Behavior................................................................................................12 2.3 Beam on Elastic Foundation (BEF) Analysis..........................................................14 2.4 LPile and thep-y Method.......................................................................................... 27 2.5 DFSAP and the Strain Wedge Model.......................................................................63 2.6 MFAD and the Four-Spring Model.......................................................................... 81 2.7 Large-Scale Load Tests in the Literature................................................................ 88 3 METHODOLOGY..............................................................................................................96 3.1 Experimental Criteria.................................................................................................96 3.2 General Modeling Approach for Each Analysis Method....................................101 3.3 Determination of Soil and Foundation Properties................................................106 3.4 Foundation Properties...............................................................................................122 3.5 Full-Scale Load Test Details...................................................................................124 3.6 Parametric Study...................................................................................................... 193 4 RESULTS.......................................................................................................................... 196 4.1 Load Test 11 Results................................................................................................197 4.2 Load Test 18 Results................................................................................................200 4.3 Load Test 19 Results................................................................................................203 4.4 Load Test 20 Results................................................................................................222 4.5 Load Test 22 Results................................................................................................235 4.6 Load Test 23 Results................................................................................................241 4.7 Load Test 75 Results................................................................................................247 4.8 Load Test 76 Results................................................................................................250 5 DISCUSSION.....................................................................................................................264 5.1 Discussion of LPile Results.....................................................................................265 5.2 Discussion of DFSAP Results.................................................................................270 5.3 Discussion of MFAD Results..................................................................................275 5.4 Comparison of Analysis Methods.......................................................................... 280 5.5 Regression Analysis and Reliability.......................................................................290 6 CONCLUSIONS AND RECOMMENDATIONS....................................................... 302 6.1 Drilled Shaft Analysis in Granular Soils............................................................... 302 6.2 Drilled Shaft Analysis in Cohesive Soils..............................................................303 6.3 Recommendations for Future Research................................................................. 305 REFERENCES........................................................................................................................307 vii LIST OF SYMBOLS AND NOTATION As,c ultimate soil resistance correction factor ■‘A1 v,min = minimum area of transverse reinforcement a = depth to center of rotation of pile B = diameter (or width) of pile or footing Bs,c = nondimensional coefficient for determining pm Wc = width of mobilized passive wedge BEF = beam-on-elastic-foundation bw = diameter of concrete column C = curve fitting coefficient Ci = curve fitting coefficient Cu = coefficient of uniformity CPT = cone penetration test D = depth of pile embedment; pile diameter D/B = ratio of embedded depth to diameter of pile DMT = dilatometer test D50 = mean particle size E I = flexural rigidity of pile or footing E = secant modulus of soil from laboratory stress-strain curve Eds = drained secant modulus Ep = modulus of deformation Es = modulus of subgrade reaction Es, = secant modulus of initial linear portion ofp-y curve EUS = undrained secant modulus EpMT = pressuremeter modulus e = void ratio of soil f c = compressive strength of concrete fyt = yield strength of transverse reinforcement fr = rupture strength of concrete H = height of passive wedge in front of pile h = depth of passive wedge i = sub-layer index J = nondimensional coefficient for clay p-y curves Ko = coefficient of lateral earth pressure for at-rest conditions Ka = coefficient of lateral earth pressure for active conditions k = coefficient of subgrade reaction kb = base shear spring kh = horizontal coefficient of subgrade reaction kht = current tangent value of the horizontal subgrade modulus ke = vertical side shear moment spring keb = base moment spring [kB] = beam stiffness matrix [kh] = lateral spring stiffness matrix [kE] = stiffness matrix for each pile element [ke] = vertical side shear moment stiffness matrix L = pile length LL = liquid limit M - internal bending moment of pile Mult = ultimate moment capacity of pile Np = lateral bearing factor Nk = CPT cone factor nh = constant of subgrade reaction P = soil reaction per unit area PI = plasticity index PMT = pressuremeter test p = soil reaction per unit length p (x) = soil reaction as a function of axial direction x pa = atmospheric pressure pc = theoretical ultimate soil resistance p cd = theoretical ultimate soil resistance from lateral flow of soil around pile p ct = theoretical ultimate soil resistance from a passive wedge in sand p m = soil resistance at point m in Figure 2.12 po = applied lateral load at pile head p u = ultimate soil resistance per unit length p ult = theoretical ultimate soil resistance per unit length x Quit = ultimate shear capacity of pile Qv = axial (vertical) force acting on pile or footing qc = CPT cone tip resistance qu = unconfined compressive strength of soil r = radius of pile R = reliability of pile design Si = pile shape adjustment factor S2 = pile shape adjustment factor SL = horizontal stress level within the passive wedge SLt = stress level of shear along pile sides s = center-to-center spacing of transverse reinforcement su = undrained shear strength of cohesive soils UC = unconfined compression test UU = unconsolidated undrained triaxial test Au = excess porewater pressure V = internal shear force in the pile V, = incremental lateral load applied to pile head W = distributed load along the length of the pile w(x) = distributed load along a beam as a function of axial direction x wn = natural water content of soil Xo = depth to zero-deflection point as measured from the top of the pile x = distance along the length of the pile or footing xr = transition depth from wedge-type failure to flow-around failure y yo y 50 y (x) a = P = Pm y ' s Sprobable Spredieted £ = £d £50 z e e(x) em V = Ps Pd lateral deflection of the pile at point x along the length of the pile pile deflection profile as a function of axial direction x pile head deflection deflection at 50% of the ultimate soil resistance, p u angle of passive wedge from the vertical pile face; angle of passive wedge from a line parallel to applied lateral load; pile adhesion factor angle of passive wedge from the vertical pile face mobilized angle of passive wedge from the vertical pile face submerged (effective) unit weight of soil linearized deflection angle of pile from vertical probabilistic pile deflection pile deflection predicted from a particular analysis method horizontal strain within the passive wedge of soil; axial strain axial strain corresponding to od strain at 50% of the maximum stress on a laboratory stress-strain curve normalization factor slope of the pile pile slope profile as a function of axial direction x mobilized angle of passive wedge from horizontal curve fitting parameter Poisson's ratio longitudinal steel reinforcement ratio dry density of soil xii X Aah Aahf aj as ad a vo T = Tult r - t - $ m ts = change in horizontal stress in passive wedge change in horizontal stress at failure in passive wedge major principal stress minor principal stress deviatoric stress from triaxial test vertical effective stress shear stress in soil; shear stress along pile sides ultimate shear stress along pile sides effective angle of internal friction of soil angle of internal friction of soil; curvature of pile mobilized effective friction angle; angle of passive wedge from a line parallel to applied lateral load mobilized friction angle between pile sides and sand xiii ACKNOWLEDGEMENTS I would like to give special thanks to the University of Utah and the department of Civil Engineering for providing me with an incredible opportunity to receive a world-class education. I would like to especially thank my advisor, Dr. Evert Lawton, for providing me with the opportunity and resources to work on this research, for his mentorship over the years, and for always making his classes exciting and challenging. I would also like to thank Dr. Steven Bartlett and Dr. Chris Pantelides for their guidance and support in my academic pursuits and their participation on my committee. This research would not have been possible without financial support from PacifiCorp. My sincerest thanks to Pete Singh and Ben Fowler of PacifiCorp for their support and assistance with this project. 1 INTRODUCTION 1.1 Background A drilled shaft is a type of deep foundation that is widely used in many civil engineering applications. They are constructed by drilling a hole to a specified depth and diameter, placing a rebar reinforcing cage inside the hole, filling the hole with concrete, and allowing sufficient time to elapse for the concrete to develop its prescribed design strength. Drilled shafts are commonly used to support buildings, bridges, transmission line structures, radio towers, oil platforms, windmills, and many other types of structures. They are used in lieu of other types of deep foundations, such as steel pipe piles, because they are fairly easy to construct, they are durable, the materials are readily available, and they do not require special transportation considerations for very large foundations. Drilled shafts can be used where relatively small deep foundations are required, but oftentimes they are used when foundations with very large lengths and diameters are required, such as for single-pole transmission line structures and windmills. These foundations become very expensive very quickly due to the specialized equipment and personnel required for their construction, the large quantities of concrete and steel reinforcement required, and the additional challenges associated with mobilizing large pieces of equipment to areas that are remote or otherwise not easily accessible. As such, it is desirable to design the drilled shaft to be large enough for the application in which it will be used, including the required factors of safety, but not larger. It was recently brought to the attention of the author by the director of engineering of a regional power company that several of the drilled shaft foundations that have been designed for their single-pole transmission line structures are considerably larger than one would reasonably expect for the anticipated subsurface and loading conditions. He explained that the allowable deflection and rotation of the top of the shaft typically governs the design, and that a computer program known as LPile was being used to conduct the lateral load analysis and foundation design. He also explained that the geotechnical investigation for most projects was probably not rigorous enough to give the design engineer the confidence to minimize the conservatism in the soil properties that are used for design. Naturally, the following questions arose from the observation that these foundations might be overdesigned: 1) Are the foundations actually being overdesigned? 2) Can the anticipated overdesign be attributed to the model, the input parameters (soil properties), or a combination of both? 3) What alternative methods are available for performing such analyses, and how do the results of each method compare with each other? 4) Which method actually provides the most accurate results? These questions were the impetus for this thesis. A literature review was conducted to gain an understanding of how the LPile model is used in practice and how it was formulated. It turns out that LPile is a semi-empirical model that employs load transfer functions known as p-y curves to represent the stress-strain-strength characteristics of the soil. These p-y curves were developed from the results of a small number of large-scale lateral load tests that were performed in different types of soil. The popularity of LPile can likely be attributed, at least in part, to the following: 2 1) It is easy to use; 2) it is fairly inexpensive; 3) it requires relatively few input parameters; 4) it predicts deflection and rotation at the top of the foundation, which typically controls the design; 5) it reasonably accounts for the nonlinearity of the shaft and the soil; 6) the algorithms employed to obtain solutions are, in part, formulated from the results of large-scale load tests, which gives engineers confidence in the model's ability to provide accurate results; 7) it predicts the shear force and bending moment along the length of the shaft, which allows the engineer to properly design a foundation that has sufficient shear and moment capacity; and 8) it is a very efficient program, which allows the engineer to perform parametric studies quickly and easily. As mentioned previously, LPile is based on the results of large-scale load tests. Therefore, the results of any analysis obtained using this model will reflect the conditions of the experiments from which the p-y curves were derived. If the subsurface characteristics, foundation characteristics, and loading characteristics of the foundation being analyzed are similar to the same characteristics of the large-scale load test from which these curves were derived, the results should be reasonably accurate; however, it is difficult to gauge the accuracy of such an analysis if this criterion is not satisfied. The large-scale load tests that were used to derive the p-y curves were performed on steel pipe piles and one drilled shaft with very large length-to-diameter ratios. It was discovered during the literature review that the soil response is considerably influenced by the rigidity of the foundation, which decreases as the length-to-diameter ratio increases. The literature review also revealed that limited research has been conducted to investigate whether it is appropriate to conduct lateral load analyses of semi-rigid drilled shafts using p-y curves that were established for flexible foundations. As such, one of the main objectives of this investigation is to gain insight into the accuracy of the LPile 3 model for these types of foundations. The literature review revealed that several models have been developed for analyzing laterally loaded drilled shafts and piles. Most of these models were developed over the last century, and range in complexity from simple equilibrium models to advanced numerical models. Many of these models were derived to predict the ultimate capacity of a laterally loaded drilled shaft or pile. In most cases, the tolerable deflection or rotation of the top of the foundation is reached before the ultimate capacity, so these models are of limited value in the final design of a laterally loaded drilled shaft. These models will not be discussed in detail in this thesis. The alternative design methods that are being considered for this thesis must provide, at a minimum, the deflection and rotation at the top of the shaft, as well as the soil resistance along the length of the shaft. The alternative design models that were identified in the literature review that satisfy these criteria include FB-Multipier (formerly FLPier), Moment Foundation Analysis and Design (MFAD), and Deep Foundation System Analysis Package (DFSAP). General Finite Element Method (FEM) and Finite Difference Method (FDM) codes such as ABAQUS and FLAC3D, respectively, have also been used to perform analyses of laterally loaded drilled shafts, but they are not being considered in this investigation. The FB-Multipier program uses the same p-y curves as LPile to model the soil-structure interaction (Hoit, Hays, & McVay, 1997), and as such, it will not be included in this investigation. Therefore, the models that will be included in this investigation are LPile, DFSAP, and MFAD. In order to assess the accuracy of these models for semi-rigid drilled shafts, a large-scale lateral load test must be performed at a site where a rigorous geotechnical investigation has been conducted. Because research funds were not available for 4 conducting such an experiment, it was decided that the literature review would also be conducted to discover whether similar experiments have been performed and published by other researchers. The information from these experiments can be used to perform analyses using the aforementioned methods, and the results of the analyses can be compared to the observed results. Several large-scale lateral load tests were identified in the published literature, and the information and results from these load tests were used as the basis of comparison between the models being considered in this investigation. 1.2 Objectives of the Research The purpose of this research is to identify and evaluate several methods that are used to analyze laterally loaded semi-rigid drilled shafts. This will be accomplished by achieving the following objectives: • Determine which analysis methods are currently being used - primarily by practicing engineers - by conducting a thorough review of the literature; • Compile a list of large-scale load tests reported in the literature and determine which load tests are suitable for performing a comparison between analytical and observed results; • Perform an analysis of each suitable large-scale load test using select analysis methods currently used in professional practice and the reported foundation and subsurface information; • Conduct a parametric study to gain insight into how the input parameters affect the results of the analysis methods considered in this thesis; • Compare the results of the large-scale load tests with the results of each 5 analysis method and make recommendations regarding the use of each method for analyzing semi-rigid drilled shafts; 1.3 Organization of the Thesis This thesis is organized into six chapters. An overview of each chapter is as follows: • 1 Introduction - Discusses the genesis of the project, the statement of the problem, why the results are important, and a brief review of the relevant literature; • 2 Literature Review - Presents the findings of the literature review including the relevant analysis methods and the database of large-scale lateral load tests; • 3 Methodology - Presents relevant information about the large-scale lateral load tests that were used for this investigation, how the models for each large-scale load test were developed, and how the input parameters for each model were estimated; • 4 Results - Presents the results of each analysis along with the results of the large-scale load tests, and the results of the parametric study; • 5 Discussion - Discusses the simulations of each large-scale load test and how the simulation results from each analysis method compare with the observed results; discusses the results of the parametric study and how the input parameters affect the results; discusses the results of the statistical analysis of the data and the resulting regression equations; • 6 Conclusions and Recommendations - Presents the conclusions reached from 6 the analysis, recommendations based on the conclusions, and recommendations for future research. 1.4 Limitations of the Investigation It is acknowledged that the comparison of each large-scale load test simulation with the observed data should be based on the moment, displacement, and soil resistance profiles in addition to the deflection and rotation of the top of the foundation; however, there are insufficient data available in the literature to make such a comparison. As such, the basis of comparison for the models used in this thesis is the deflection of the top of the foundation only. Although this is not ideal, deflection and rotation of the top of the foundation control most designs, so comparison of deflections at the top of the foundation is still very useful. There are numerous ways of conducting a geotechnical field investigation, and likewise, there are numerous ways of estimating soil parameters from the results obtained from the field investigation and concomitant laboratory testing. Rigorous estimation of soil properties from the information available is outside the scope of this research. Soil properties that are reported along with the results of large-scale load tests will generally be used without modification, and a reasonable effort will be made to estimate the required soil properties that are not directly reported using methods that are consistent with the current standard of geotechnical engineering practice. 7 2 LITERATURE REVIEW A literature review was conducted to 1) identify alternative analysis methods currently being used that might be more suitable for analyzing laterally loaded rigid to semi-rigid drilled shafts subjected to large lateral or overturning moment loads; 2) gain further insight and understanding of how the p-y method and each of the alternative analysis methods were formulated, their basic assumptions, and their limitations; and 3) identify the large-scale lateral load tests of rigid to semi-rigid drilled shafts that are reported in the published literature. The results of the literature review are presented in this section. 2.1 Introduction The prediction of the interaction between drilled shafts and the surrounding soil and the overall foundation response to lateral loading is among the most complex topics in geotechnical engineering (Janoyan & Whelan, 2004). In order to analyze a laterally loaded drilled shaft, the stress-strain-strength characteristics of the foundation and surrounding soil must be evaluated (Chen & Kulhawy, 1994). In most civil engineering applications, the forces acting upon or caused by the superstructure impart lateral loads, axial loads (uplift and compression), and overturning moments on the head of the foundation. The combination of these loads and the resulting soil resistance that develops is a highly complex three-dimensional problem as shown in Figure 2.1. This three­ 9 dimensional problem is difficult to solve, and it is difficult to realistically represent this three-dimensional soil-structure interaction problem as a two-dimensional problem that is easier to solve (Phoon & Kulhawy, 2005). Several methods have been developed for analyzing laterally loaded piles and drilled shafts. In general, the purpose of the analysis is to estimate the ultimate capacity of the foundation. The ultimate capacity can be defined in terms of the maximum allowable deflection or rotation of the top of the foundation or an ultimate limit state. The ultimate limit state is reached by failure of the soil, failure of the foundation, or excessive deformation that will cause loss of structural integrity of the superstructure (Salgado, 2008). No standard definition for ultimate capacity exists, but Hirany and Kulhawy (1988) have discussed several of the proposed methods shown in Figure 2.2. Shaft center line Figure 2.1. Load and soil resistance components of a drilled shaft (Chen & Kulhawy, 1994) Applied Ground-Line Moment (k ip - f t ) (rotation) 10 % B 2 * (rotation) 5% B Loterol or moment 4 0 0 0 ° B s Foundation diameter =4.5ft (I 4m) 3 0 0 0 - displacement equol to: 1000 - Moment ot butt 2 0 % B Displacement at Top of Shoft (in.) Figure 2.2. Comparison of lateral load interpretation criteria (Hirany & Kulhawy, 1988) (M N -m) Most of the models that have been developed over the past several decades [ (Broms, 1964a), (Broms, 1964b), (Poulos, 1971a), (Poulos, 1971b) ] are based on a two-dimensional representation of the forces that act on the foundation. The forces from the soil are estimated from the strength characteristics of the soil, which are typically defined by the undrained shear strength (su) for undrained analyses, and angle of internal friction (<f) for drained analyses, and an assumed stress distribution along the foundation. The ultimate capacity is calculated by satisfying limit equilibrium for the applied forces and the soil resistance forces. The limit equilibrium models are still used to some extent in current design practices, and research has shown that these models generally predict the loads required to reach the ultimate limit state of laterally loaded drilled shafts quite well when used properly (Davidson, Cass, Khilji, & McQuade, 1982). Unfortunately, most of these models have limited, if any, capacity to predict the deflection and rotation that occurs at any point along the shaft. The amount of tolerable deformation in most designs is typically on the order of 5 to 50 mm, which is less than the amount of deformation required to reach the ultimate limit state of most drilled shafts (Salgado, 2008). As such, these limit equilibrium methods are of limited value in most design problems. Furthermore, the basis of comparison between methods for this investigation is the lateral deflection at the top of the foundation, which precludes the use of these methods for this investigation. The methods that are considered in this investigation provide, at a minimum, the deflection and rotation of the foundation head and the internal bending moment and shearing stress at all points along the foundation. The deflection and rotation is used for the purposes of meeting the specified limit state as previously discussed, and the 11 maximum internal bending moment and shearing stress are required to perform the structural analysis of the foundation. The analysis methods that will be considered for this investigation must provide values for these parameters, at a minimum. 2.2 Drilled Shaft Behavior The deflection and rotation that occurs at the top of the drilled shaft is dependent, in part, upon the fixity conditions of the foundation head, the rigidity of the foundation, and the stiffness of the soil [ (Kasch, Coyle, Bartoskewitz, & Sarver, 1977), (Chen & Kulhawy, 1994) ]. Foundations with a fixed head, such as foundations that are integrated into a pile cap, are outside of the scope of this thesis and will not be discussed. The flexural rigidity (El) of the drilled shaft increases as the embedded-depth-to-diameter (D/B) ratio decreases, and conversely, the E l decreases as D/B increases. No standard method has been developed to describe the flexural rigidity of drilled shafts, but several criteria have been developed as described by Chen and Kulhawy (1994). In general, shafts with D/B ratios greater than 10 are flexible, while shafts with D/B ratios less than 10 are semi-rigid. Shafts with D/B ratios less than 4 are typically considered to be rigid, but for this thesis they are considered to be semi-rigid. The typical behavior of a rigid, semi-rigid, and flexible drilled shaft subjected to lateral loading is shown in Figure 2.3. For rigid drilled shafts, it is assumed that bending of the foundation does not occur. Instead, the foundation rotates about a fixed point during lateral or moment loading. The soil resistance is primarily comprised of passive earth pressure on the opposite side of the direction of loading above the point of rotation and on the same side as the direction of loading below the point of rotation. Additionally, shearing resistance develops along the sides of the foundation and along the tip of the 12 13 RIGID SEMI-RIGID FLEXIBLE H V H H ORIGINAL SHAPE DEFORMED SHAPE APPLIED HORIZONTAL LOAD H Figure 2.3. Behavior of rigid, semi-rigid, and flexible drilled shafts subjected to lateral loading under free head conditions [After Kulhawy and Chen (1995)] foundation as shown in Figure 2.1. Research conducted by Smith and Slyh (1986) and Janoyan and Whelan (2004), among others, have suggested that shearing resistance is responsible for most of the soil resistance at small deformation and that passive earth pressure resistance is responsible for most of the soil resistance at large deformations. Semi-rigid drilled shafts experience flexural bending about a single point. For drilled shafts, the bending is usually extensive enough to cause cracking of the concrete. The soil resistance develops in much the same way as rigid drilled shafts, but there is not as much movement at the base of the foundation. Flexible drilled shafts exhibit bending behavior that is demonstrably different from rigid and semi-rigid drilled shafts. Flexural bending occurs about multiple points of rotation, and considerably more bending occurs. As with semi-rigid drilled shafts, the amount of bending that occurs typically causes cracking of the concrete. One of the most important differences between semi-rigid and flexible drilled shafts is that the bases of flexible drilled shafts are assumed to be fixed. The soil resistance that develops within the upper portion of the drilled shaft is comprised of passive earth pressure resistance and shearing resistance. As previously discussed in Section 1.1, the focus of this investigation is primarily on semi-rigid drilled shafts, and the criterion that will be used to characterize a drilled shaft as semi-rigid is that it must have a D/B ratio less than 10. 2.3 Beam on Elastic Foundation (BEF) Analysis A common design problem that arises in geotechnical engineering is the analysis and design of beam-like structures supported by the subgrade, such as strip footings or grade beams. This analysis can be performed using the Euler-Bernoulli beam equation, which is a special case of the theory of elasticity (Salgado, 2008). The Euler-Bernoulli beam equation is a fourth-order differential equation that was derived to represent the relationship between the deflection and applied load of a simplified, one-dimensional beam (Beer, Johnston, & DeWolf, 2006). An example of this derivation is shown by Beer et al. (2006), and the resulting equation is d! y 14 El d.F = - " (X) ^ where: El = flexural rigidity or bending stiffness of the beam o)(x) = distributed load as a function of x The complete solution to Equation (2.1) yields the deflection (y), slope (Q), curvature ($), moment (M), and shear (V) at any point along the beam for a given distributed load rn(x). For the case of a beam subjected to arbitrary loading and supported by the subgrade, the distributed load in Equation (2.1) can be replaced by the soil reaction, p(x), which is a function of the contact pressure. The general distribution of contact pressure for a smooth, perfectly rigid footing with width B is shown for both clay and sand in Figure 2.4. 15 y P(x) a) stress distribution for clay b) stress distribution for sand Figure 2.4. Contact pressure beneath footing for a) clay and b) sand [After Terzaghi (1955)] Qv Qv A A 7\ A A 7f\ A /\ According to Terzaghi et al. (1996), the relation between the stress-deformation characteristics of the subgrade and the contact pressure on the base of a perfectly smooth and rigid footing is by no means simple. Furthermore, the relation becomes even more complicated if the footing is not rigid. Winkler (1867) hypothesized that the soil reaction at any point along the beam is proportional to the displacement of the beam at the same point as shown in Equation (2.2): P = k y (2.2) where: y = deflection at a point along the beam k = coefficient of subgrade reaction P = soil reaction per unit area In Equation (2.2), k can be thought of as the constant of proportionality between the soil reaction and beam deflection. If it is assumed that the soil reaction is uniform across the width of the beam, both sides of Equation (2.2) can be multiplied by B and Equation (2.2) can be written as p = kB y = Es y (2.3) where: p = soil reaction per unit length Es = kB = modulus of subgrade reaction If the soil reaction (p) from Equation (2.3) is substituted into Equation (2.1) for the distributed load of rn(x), the resulting equation is the so-called BEF equation, which is 16 17 (2.4) The soil reaction represented by Equation (2.3) is shown in Figure 2.5(a). Unlike the soil reaction shown in Figure 2.4, the soil reaction in Figure 2.5(a) is not dependent upon soil type or the location of the soil reaction with respect to the location along the beam. This illustrates one of the major assumptions in the Winkler (1867) hypothesis, which is that the soil reaction is dependent only upon the displacement of the soil and is independent of the soil conditions or displacement of adjacent points along the beam. For this reason, the soil reaction along a beam given by Equation (2.3) has often been represented as a series of uncoupled linear springs acting at discrete, evenly spaced points along the beam as shown in Figure 2.5(b). The resulting soil reaction is a series of uniformly distributed loads as shown in Figure 2.5(c). a Q\ p_ y (a) B Q> (b) (c) Figure 2.5. Soil reaction along a flexible footing [After Terzaghi et al. (1996) and Salgado (2008)] The theory for beams on elastic foundation can be extended to model vertical beam-columns embedded in the subgrade. In this scenario, a pile that is subjected to lateral and axial loads is at least partially embedded in the subgrade, and the soil reaction that develops is a function of pile deflection. An additional term can be added to the BEF equation shown in Equation (2.4) to account for an axially applied load (Qv). Furthermore, an additional term can be added to account for a distributed lateral load being applied to some portion of the pile. The modified form of the BEF equation that includes these additional terms is presented in Equation (2.5). 18 £ ' S + + W = 0 <2'5) Es - k h B - - ^ <2.6) where: El = flexural rigidity or bending stiffness of the pile x = distance along the length of the pile y = lateral deflection of the pile at a point x along the axis of the pile B = diameter or width of the pile kh = horizontal coefficient of subgrade reaction Es = modulus of subgrade reaction P = soil reaction per unit length Qv = axial (vertical) force acting at a point x along the axis of the pile W = distributed load acting along the length of the pile 19 Equation (2.5) is amenable to closed-form solution for simple loading conditions and constant or linearly varying values of EI and Es as shown by Hetenyi (1946). The complete solution to Equation (2.5) yields profiles for deflection (y), slope (6), curvature (<f), moment (M), shear (V), and soil resistance (p) as shown in Figure 2.6 (curvature not shown). Solutions to Equation (2.5) can be obtained when the boundary conditions for y, 6, M, and V are known at the ground surface (Janoyan, Stewart, & Wallace, 2001), or if they can be reasonably assumed at some other point along the pile, such as at the base of a very long pile. The equations for y, 6, M, and V are show in Equations (2.7) through Until about 1956, the BEF equation was typically solved using a constant value of Es and EI out of mathematical necessity, despite the well-known fact that Es is not, in fact, constant (McClelland & Focht, 1958). The assumption that E I and Es are constant yields an elastic solution, which is not realistic for the vast majority of real engineering (2.10). (2.7) dd d 2 y (2.8) M = d>EI = E I-^r d x z d 2 y (2.9) (2.10) 20 y 6 M V Figure 2.6 Profiles of complete solution of a laterally loaded drilled shaft (Isenhower & Wang, 2010) problems. A more realistic solution to the BEF equation can be obtained using relationships for E I and Es that vary as a function of both pile deflection and depth; however, this makes it more difficult, if not impossible, to obtain a closed form solution. For this reason, numerical methods are the preferred technique for obtaining more realistic solutions to the BEF equation. McClelland and Focht (1958) discussed early attempts at increasing the versatility of the BEF equation by using the finite difference technique to obtain a solution, but that the correctness of the solution ultimately depends on the stress-strain relationship assigned to the soil for the purpose of analysis. Matlock (1970) stated that, with the availability of numerical solutions to the BEF equation, the most important but difficult part of the problem is to express the soil resistance characteristics, which implies difficulty in expressing Es. 2.3.1 Modulus of Subgrade Reaction, Es One of the earliest and most comprehensive discussions on the topic of subgrade reaction was presented in the seminal paper by Terzaghi (1955). Terzaghi (1955) stated that there is an erroneous conception that is widespread among engineers that the numerical value of k, and by extension Es, depends exclusively on the nature of the subgrade. Terzaghi (1955) suggested that k is not actually a fundamental soil property, but that its value depends on the characteristics of the soil, the area acted upon by the subgrade reaction, and the load applied to the foundation. According to Terzaghi (1955), Es decreases as the loaded area increases. Furthermore, he suggested that Es is somewhat independent of depth for stiff clays and increases with depth for soft clays and sands. While Terzaghi (1955) presented an in-depth discussion of the factors that affect the subgrade modulus, he describes the subgrade reaction as a fictitious pressure that satisfies Equation (2.6) in contrast to the real soil pressure (Terzaghi, Peck, & Mesri, 1996). He showed that the subgrade reaction resulting from a constant subgrade modulus is not the same as the real contact pressure on the foundation. For example, the soil reaction of a smooth, perfectly rigid beam that satisfies Equation (2.6) would be a uniformly distributed load, which is in contrast to the actual contact pressure presented in Figure 2.4. This variance is even more pronounced if the load Qv shown in Figure 2.4 does not act at the centroid of the footing, and acts, for example, towards the left side of the beam. In this case, the real contact pressure increases towards the left side of the beam and decreases towards the right side to satisfy static equilibrium, but the soil reaction from Equation (2.6) is still a uniformly distributed load as the settlement of a rigid footing is uniform, and thus static equilibrium is not satisfied (Terzaghi, Peck, & Mesri, 1996). Nevertheless, Terzaghi (1955) acknowledged the usefulness of the 21 subgrade reaction approach within its limitations, particularly for flexible foundations, and presented methods for characterizing k from plate load tests. Terzaghi (1955) also presented a procedure for conducting a lateral load test of a steel pipe pile and characterizing the subgrade modulus from the results. One of the first attempts to characterize Es from the results of a large-scale lateral load test using the method presented by Terzaghi (1955) was performed by McClelland and Focht (1958). McClelland and Focht (1958) obtained experimental data from a large-scale lateral load test of a steel pipe pile embedded in soft marine clay at an offshore site in Texas. The test pile had a diameter and embedded length of 2 ft and 75 ft, respectively, and was instrumented with several strain gauges placed in diametric pairs along the length of the foundation. Curvature profiles were estimated from the strain gauge data for each load increment by taking the difference in strain between diametric pairs and dividing the result by the diameter of the pile. A procedure similar to the one described later in Section 2.4.1 was used to obtain p and y profiles from the curvature data, with the most notable difference being that integration and differentiation of the curvature profiles was performed using graphical methods. For the range of lateral loads applied to the top of the test pile, McClelland and Focht (1958) plotted p-y data pairs at several depths along the pile. The resulting soil reaction vs. deflection curves, referred to as p-y curves, were used to evaluate Es at each depth from the relationship shown in Equation (2.6). McClelland and Focht (1958) performed triaxial compression tests on several samples from a range of depths at the test site, and showed that a correlation existed between the p-y curves and the stress-strain curves from the triaxial tests. The p-y curves were converted into so-called field stress-strain curves by converting y into one-dimensional strain, e, by somewhat arbitrarily 22 dividingy by the pile radius, r, as shown in Equation (2.11). 23 (2.11) It was then shown that the field stress-strain curve at a given depth could be estimated by multiplying the deviatoric stress from a triaxial compression test performed at confining stress equal to the same depth by 5.5 for all values of strain. This relationship is shown in Equation (2.12). deviatoric stress at failure is equal to 2c for purely cohesive soils, and therefore, the ultimate soil resistance at any depth is equal to 11c when od = 2c in Equation (2.12). Equation (2.11) and (2.12) can be rearranged to determine Es from the secant modulus of the triaxial test as Because E from a triaxial compression test is nonlinear, the estimated value of Es from Equation (2.13) must also be nonlinear. Thus, using the numerical procedure outlined by McClelland and Focht (1958), nonlinear solutions could be obtained for the laterally loaded pile problem using estimations of Es from laboratory testing. Although the work performed by McClelland and Focht (1958) was largely - = 5.5a! B ! (2.12) where od = deviatoric stress from the triaxial compression test. It should be noted that the (2.13) praised as being a much-needed step forward in characterization of the subgrade modulus and performing nonlinear analysis of laterally loaded piles, it was met with considerable skepticism and criticism as seen in the discussion of the paper. Ripperger (1958) posited the question of whether such a soil modulus actually exists, and if so, if its value can be uniquely defined for a specific soil. He concluded that it is true that a secant modulus can be obtained mathematically in the sense that the soil reaction at any point along a pile can be divided by the corresponding deflection, but that Es cannot be unique for a soil because it depends not only on the elastic characteristics of the soil, but on many other factors that are not easily characterized. Ripperger (1958) further stated that, in light of all of the factors that contribute to the relationship between subgrade reaction and deflection, it does not seem likely that Es, even if uniquely defined, could be related to E by a simple numerical factor. Reese (1958) suggested that only the ultimate soil resistance could be approximated using a rational method, and that ultimate soil resistance could be modeled as either a flow-around failure or a mobilized passive wedge in front of the pile near the ground surface. Reese (1958) showed that the ultimate soil resistance could be has high as 12c for flow around failure and as low as 2c for passive wedge failure, and suggested that the ultimate soil resistance of 11c found by McClelland and Focht (1958) was therefore reasonable at depths where flow-around failure occurs, but that it is probably too high near the ground surface where passive wedge-type failure occurs. McClelland and Focht (1958) acknowledged in the discussion that this numerical factor of 5.5 in Equation (2.12), and thus 11 in Equation (2.13), is not unique to the soil type, and that it will most certainly change for a different soil type and pile configuration; however, they suggest that the only way to truly evaluate Es for any soil-pile interaction 24 is through a load test of that pile, and that using the proposed correlation for obtaining Es from laboratory testing is a reasonable way of approximating Es without having to conduct and actual full-scale load test. Furthermore, they state that the theoretical values of ultimate soil resistance of 12c and 11.42c estimated by Reese (1958) and Meyerhof (1951), respectively, at least somewhat substantiate their proposed value of 11c. It should also be noted that Skempton (1951) performed a similar comparison between laboratory test data and observed soil reaction-settlement data from field tests of various types of structures, and concluded that a reasonable value of the ultimate soil resistance of soft clay is 9c. This provides further evidence that the value of 11c proposed by McClelland and Focht (1958) is reasonable. Considerable research has been conducted since McClelland and Focht first attempted to correlate Es with the results of laboratory testing. Most of this research has been performed with the same objective, which is to correlate Es with in-situ or lab testing data, while attempting to account for the factors thought to influence Es. McClelland and Focht (1958) concluded that Es generally depends on depth, deflection, pile diameter, pile stiffness, pile length, soil type, and load magnitude. Welch and Reese (1972) suggest that Es is also a function of shear strength, moisture content, stress history, and the effective stress state of the surrounding soil. Ashour and Norris (2000) have shown theoretically how Es depends on cross-sectional shape, and fixity of the pile head. Clearly, there is strong theoretical and experimental evidence to suggest that Es is a not a fundamental soil property, and that its value is entirely dependent upon the specific characteristics of a given soil-structure interaction. Despite the obvious limitations of subgrade reaction theory and the challenges associated with characterizing Es for a particular soil-structure interaction problem, 25 subgrade modulus theory and the BEF equation provide a convenient mathematical framework for performing a nonlinear analysis of a laterally loaded pile (McClelland & Focht, 1958). The concept of solving the BEF equation using numerical methods and a nonlinear representation of Es and E I is still appealing to this day. Brown et al. (1994) state that this approach has been widely accepted because of its simplicity and ability to capture the essential aspects of pile behavior, including nonlinear soil resistance, gapping around the pile, and variable soil and pile properties. According to Norris (1986), the BEF solution technique for laterally loaded piles is often preferable to elastic continuum or finite element methods because the formulation is simple, it can readily handle both layered and nonlinear soil response, parameter input is well-documented in the literature, and the method has been found to predict response that compares favorably with field behavior over a large range of deflection. The literature review that was conducted for this investigation, in addition to the author's consultation with several practicing engineers in the field of deep foundation design, revealed that virtually all of the methods that are currently used to perform these types of analyses are still based on this approach. There are examples of other methods of analysis being used, such as the finite element method and finite difference method, but these methods appear to only be used in special circumstances, and are far from routine. The details of the three methods being considered in this analysis and how they approach solving the BEF equation with nonlinear representation of Es and EI will be discussed in Section 2.4 through 2.6. 26 2.3.2 Nonlinear Solution Techniques Closed-form solutions to the BEF equation for a pile at least partially embedded in the subgrade, as shown in Equation (2.4), have been presented by Hetenyi (1946); however, these closed-form solutions are tedious, and require unrealistic assumptions about Es and E I that preclude their use from routine analyses. More realistic solutions can be obtained using numerical methods, such as the finite difference method (1970). These details of these techniques are outside the scope of this investigation; however, the general solution scheme proposed by McClelland and Focht (1958) will be discussed within this section because the solution scheme used by LPile and DFSAP is very similar. McClelland and Focht (1958) describe a procedure for obtaining nonlinear solutions for the BEF equation using p-y curves as follows: A pile deflection curve is assumed for a given loading condition. The strain is calculated at several points along the deflection curve above the point of zero deflection using Equation (2.11), and corresponding soil reaction is computed from a triaxial compression test performed at the same depth and Equation (2.12). An Es profile is estimated by computing Es for each p and y data pair using Equation (2.13). A simplification of the Es vs. depth profile is made, and the BEF equation in difference form is solved using this simplified relationship. The Es profile is computed from thep and y profiles obtained from the BEF solution and compared to the simplified Es profile. This procedure is iterated until convergence is achieved. 2.4 LPile and the p-y Method The so-called p-y method is an extension of the work performed by McClelland and Focht (1958) described in Section 2.3, and is described as such for the p-y curves that 27 are used to obtain nonlinear solutions to the BEF equation. Recall from Section 2.3 that a p-y curve for a particular point along a laterally loaded pile represents the subgrade reaction per unit length, p , and the corresponding pile deflection, y , at that point for a range of applied loads. Because p-y curves represent a force-displacement relationship, they can be used to mathematically represent the behavior of a nonlinear spring. A series of these springs can be used to model the subgrade reaction as shown in Figure 2.7. The subgrade modulus, Es, is the secant modulus of a p-y curve, and its value can be estimated at any point along the p-y curve using Equation (2.6). Thus, if the p-y curves are known for several points along the foundation, Es can be characterized in terms of p and y and numerical methods can be used to obtain nonlinear solutions to the BEF equation for a pile at least partially embedded in the subgrade shown in Equation (2.5). Modern computers have made it fairly straightforward to solve the BEF equation using numerical methods (1970); however, there is still considerable difficulty in estimating the "correct" p-y curves for any particular pile embedded in the subgrade. Because p-y curves are simply an expression of subgrade reaction theory discussed in Section 2.3.1, they are subject to the same challenges associated with characterizing Es. As such, it is not currently possible to determine the "correct" p-y curves for a particular soil-pile interaction based on the fundamental physics of the problem. Furthermore, it is not possible to directly measure p-y curves in a large-scale load test due to the complexity of the soil response to loading, as shown in Figure 2.1, and the inadequacy of sensor technology to directly measure all of the components that contribute to the total soil reaction (Janoyan & Whelan, 2004). It follows from the discussion in Section 2.3 that p-y curves are unique to every 28 29 Q V Figure 2.7 soil-structure interaction, and therefore, the only way to estimate the "correct" p-y curves for a particular pile or drilled shaft is by back-calculating the p-y curves from a large-scale load test using techniques described in Section 2.4.1. The cost to conduct such a load test is prohibitively high for most projects (Kulhawy, et al., 1983). As such, it is desirable to be able to estimate the p-y curves using some other technique. McClelland and Focht (1958) showed that it is possible to back-calculatep-y curves at any particular depth from the results of a large-scale load test of a steel pipe pile, and subsequently proposed a method to estimate p-y curves from the results of laboratory triaxial tests. Despite early criticism of the viability of correlating the subgrade reaction with S.XV' y y y y Model of subgrade reaction for a laterally loaded pile using independent springs and associated nonlinear p-y curves (Isenhower & Wang, 2010) laboratory test results, as discussed in Section 2.3, considerable research has since been performed to establish methods for estimatingp-y curves from in-situ tests and laboratory tests to obviate the need for conducting full-scale load tests. Methods that are commonly used to back-calculate p-y curves from large-scale load tests are presented in Section 2.4.1, which is followed by a discussion of how back-calculated p-y curves from several large-scale load tests were used to formulate semi-empirical relationships between p-y curves and soil properties that are readily available from laboratory or in-situ testing in Section 2.4.2. In-situ tests, such as the pressuremeter test (PMT) and dilatometer test (DMT), can be used to establish soil properties for estimating p-y curves using semi-empirical methods, but they have also been used to estimate p-y curves directly [ (Anderson, Townsend, & Grajales, 2003), (Briaud, Smith, & Tucker, 1985), (Briaud, Smith, & Meyer, 1983), (Robertson, Davies, & Campanella, 1989) ]. These methods will be discussed in greater detail in Section 2.4.3. 2.4.1 Estimation of p-y Curves from Large-Scale Load Tests The p-y curve for a particular foundation at a given depth can be back calculated from the results of a well-instrumented large-scale lateral load test. The instrumentation of a large-scale lateral load test typically includes inclinometers, strain gauges, displacement transducers, and load cells (Reese, Cox, & Koop, 1975). The inclinometer measurements can be obtained by individual inclinometers placed along the pile within a guide casing, or by placing an inclinometer casing along the length of the pile and using a single instrument to take measurements along the casing for each loading increment [ (Brown, Hidden, & Zhang, 1994), (Geokon, Inc., 2012) ]. The strain gauges are placed in diametric pairs along the line of loading and at several depths along the length of the 30 pile. The displacement and applied load are measured at the pile head using displacement transducers and a load cell, respectively. The basic procedure for computing p-y curves from inclinometer data is outlined by Brown et al. (1994). The slope profile along the pile is calculated directly from the inclinometer data. This slope profile is integrated once using the known deflection of the pile head as a boundary condition to obtain the deflection profile along the pile as shown in Equation (2.14). The slope profile is differentiated three times and multiplied by E I to obtain the soil resistance profile along the pile as shown in Equation (2.15). 31 yO ) = J0(x) dx (2.14) P(x) = EI y(x) (2.15) where: x = distance along the length of the pile EI = flexural rigidity of the shaft y (x ) = deflection of the shaft as a function of x p (x ) = soil reaction per unit length as a function of x Q(x) = slope of the shaft as a function of x The basic procedure for computing p-y curves from strain gauge data is outlined by Yang and Liang (2006). The curvature (<f) is calculated at each pair of strain gauges as the difference between the measured strains divided by the distance between the strain gauges. The fifth-order polynomial shown in Equation (2.16) is then used to fit the 32 Shaft Deflection, y Figure 2.8 Typical p-y curve obtained from back analysis of inclinometer or strain gauge data for each incremental load discrete curvature data points to obtain the curvature profile of the pile, $(x). Deflection as a function of depth, y(x), is obtained by double integration of the curvature profile as shown in Equation (2.17) using the known slope and deflection at the pile head as boundary conditions. The moment profile is obtained by multiplying the discrete values of $ by E I as shown in Equation (2.18). A third-order piecewise polynomial is then fitted to each 5 successive values of $ using the least-squares method as shown in Equation (2.19). Each piecewise polynomial is twice differentiated with respect to x as shown in Equation (2.20) to obtain the soil resistance profile, p(x). As with the procedure for inclinometer data, the procedure is repeated for each loading increment, Vi, to obtain the p-y curve for a specified depth as shown in Figure 2.8. 0 (x ) = C1 + C2x + C3 x 2,5+C4x 3 + C5x 4 + C6x 5 (2.16) y ( *) = f ( f 0 ( *) dx (2.17) 33 M = QEI (2.18) M(x)i = C1 + C2x + C3 x 2+C4x ! (2.19) P t o i = M(x)i (2.20) where: x = distance along the length of the shaft M = internal bending moment of the shaft El = flexural rigidity of the shaft <p(x) = curvature of shaft as a function of x y (x ) = deflection of the shaft as a function of x M(x)i = Internal bending moment as a function of x for 5 successive data curvature data points C = unknown curve fitting coefficients p(x)i = soil resistance as a function of x for 5 successive curvature data points Brown et al. (1994) state that the use of inclinometers is advantageous because they are much less expensive and much less susceptible to damage than strain gauges; however, the soil resistance data are prone to reduction error because the slope distribution must be differentiated three times in order to obtain p . They further state that it is perfectly acceptable to compute the deflection profile from inclinometer data, but that the soil resistance profile will likely have too much error to be useful. Other researchers such as Lin and Liao (2006) have devised more sophisticated methods for obtainingp from inclinometer data that have been shown to have much less error. Strain gauges are generally considered to give the most accurate estimation of curvature; however, Matlock (1958) showed that even a 1% difference in moment values can result in estimated values of p that are nearly 200% greater due to the error associated with double-differentiation. As such, Matlock (1958) suggested that careful consideration be given to strain gauge spacing and that the pile itself be calibrated against known moment profiles before being installed and tested. A large-scale lateral load test was performed by Janoyan et al. (Janoyan, Stewart, & Wallace, 2001) on a drilled shaft that was instrumented with inclinometers, strain gauges, and other instruments in order to quantify the variability of p . Their results showed that there was considerable variability in p when estimated using data from different instrumentation, and that redundant sources of instrumentation should always be used when possible. 2.4.2 Estimation of p-y Curves Using Semi-Empirical Methods There are several methods that can be used to develop the p-y curves required to perform an analysis of a laterally loaded pile or drilled shaft. The semi-empirical methods were developed to allow the designer to establish p-y curves without having to back-calculate p-y curves from a full-scale lateral load test as described in in Section 2.4.1. Semi-empirical methods are most commonly used in current practice because they are easy to use, require only a few input properties from the soil in which the foundation will be constructed, and give designers an added degree of confidence in the results 34 35 because they were partially developed from the results of large-scale load tests (Brown, Morrison, & Reese, 1988). As discussed in Section 2.3, McClelland and Focht (1958) developed the first semi-empirical method for estimating p-y curves from the results of a large-scale lateral load test of a steel pipe pile. They back-calculated p-y curves from the range of applied loads using a procedure similar to the one described later in Section 2.4.1, with the most notable difference being that integration and differentiation of the curvature profiles was performed using graphical methods. These back-calculated p-y curves were correlated with stress-strain curves from triaxial testing, and the resulting equations for p, y, and Es from triaxial stress-strain curves are where: p = soil resistance per unit length B = diameter (width) of shaft y = deflection of the shaft corresponding to p &d = deviator stress from a triaxial compression test £d = axial strain corresponding to od p = 5.5 Bad (2.21) y = 0.5 Bed (2.22) (2.23) Es = modulus of subgrade reaction E = secant modulus of soil from triaxial compression test Several semi-empirical methods have been developed in the decades since McClelland and Focht (1958) first proposed their methodology for estimating p-y curves from laboratory test data. As discussed in Section 2.3, Es is not a fundamental soil property, but is a unique property of each individual soil-structure interaction. As such, methods for developing p-y curves must necessarily take into consideration the properties of the soil and the properties of the foundation. The semi-empirical methods that are currently available in programs such as LPile and FB-Multipier were generally developed within the theoretical framework of soil mechanics and experimental observation, and most were developed in conjunction with large-scale load tests of steel pipe piles or drilled shafts. The relationships for p and y were parameterized to account for variables thought to affect the p-y curves, and these relationships were modified until there was sufficient agreement between the observed and predicted results. The semi-empirical methods are generally described by the type of soil or rock for which they are intended to be used, as well as any distinguishable characteristics of the particular method. For example, the four methods that are available for clay are described as soft clay (Matlock, 1970), stiff clay with free water (Reese, Cox, & Koop, 1975), stiff clay without free water [ (Welch & Reese, 1972), (Reese & Welch, 1975) ], and modular stiff clay without free water (Isenhower & Wang, 2010). The semi-empirical p-y curves that are currently available in LPile 13 are as follows: • Soft Clay (Matlock, 1970) • Stiff Clay with Free Water (Reese, Cox, & Koop, 1975) 36 • Stiff Clay without Free Water [ (Welch & Reese, 1972), (Reese & Welch, 1975)] • Modular Stiff Clay without Free Water (Isenhower & Wang, 2010) • Sand (Reese, Cox, & Koop, 1974) • API Sand (O'Neill & Murchison, 1983) • Liquefied Sand (Rollins, Hales, & Ashford, 2005) • Weak Rock (Reese L. C., 1997) • Strong Rock (Vuggy Limestone) (Reese & Nyman, 1978) • Piedmont Residual • Silt (cemented c-phi) • Loess • Elastic Subgrade The data available in the literature for large-scale lateral load tests conducted on semi-rigid drilled shafts are limited. As such, the semi-empirical methods for soft clay, stiff clay without free water, and sand are the only methods that will be included in this investigation. The formulation of the API Sand p-y curve (1983) is similar to the formulation of the Reese et al. (1975) p-y curve and in the author's experience, the difference in results tends to be small. The Reese et al. (1975)p-y curve was chosen for use in this investigation. The development of these semi-empirical methods and the procedures used to estimate p-y curves using these methods are discussed within Section 2.4.2.1 through 2.4.2.3. 37 2.4.2.1 Soft Clay (Matlock, 1970) Matlock (1970) conducted a large-scale lateral load test of a steel pipe pile that was 12.75 inches in diameter and 42 feet long. The pile had a D/B ratio of 39.5, which suggests the pile behaved as a very flexible pile. The pile was instrumented with 35 pairs of electric strain gauges, with spacing ranging from 6 inches in the upper section to 4 feet in the lower section. The pile was calibrated before being installed to provide extremely accurate determinations of internal bending moment. The pile was driven into soft clay with average su of approximately 800 psf at Lake Austin, Texas. A total of two lateral load tests were conducted, with the first being a static load test and the second being a cyclic load test. The load was applied to the pile at the ground surface under free-head conditions with the use of a hydraulic ram. After the load tests were completed, the pile was recovered and re-driven near Sabine Pass, Texas. The soil at Sabine Pass consisted of soft clay with average su of approximately 300 psf. The soil was somewhat overconsolidated due to desiccation, and fissures and cracks were present in the soil structure. The testing procedures were similar to those at Lake Austin, but two static and two cyclic load tests were performed - one each for free-head conditions, and one each for fixed-head conditions. Free water was present above the ground surface at both testing locations. The p-y curves from all of the load tests were back calculated using a procedure similar the strain gauge procedure described previously in Section 2.4.1. Values for p and y were obtained for each loading increment from the strain gauge data using numerical differentiation and integration, respectively. Matlock (1970) stated that he was quite confident in the profiles of p and y because of the calibration of the pile and instrumentation that was performed prior to the pile being installed and tested. 38 Based on Matlock's observations during the load tests, he developed generalized shapes of p-y curves for both static and cyclic loading. The p-y curve for static loading is shown in Figure 2.9. Cyclic loading is outside the scope of this investigation, and as such, the p-y curves for cyclic loading are not discussed. One of the major objectives of Matlock's research was to develop parametric equations for these p-y curves that were functions of parameters that can be obtained from laboratory or field tests. The following paragraphs describe how the equations for these curves were formulated. Several researchers, including McClelland and Focht (1958), Reese (1958), Hansen (1961), and Broms (1964b), among others, have spent considerable effort 39 Pu Figure 2.9 General shape of p-y curve for static loading in soft clay (Isenhower & Wang, 2010) 40 studying the ultimate strength of cohesive soils. Based on the general consensus of these researchers, Matlock estimated the ultimate soil resistance per unit length (pu), which is the horizontal straight-line portion of the p-y curves shown in Figure 2.9, using Equation (2.24). p u = ultimate soil resistance per unit length Np = lateral bearing factor B = pile diameter su = undrained shear strength of the soil Several values have historically been used for the lateral bearing factor, Np, but Matlock opted to use a value of 9 as proposed by Broms (1964b) and Skempton (1951). Note that Equation (2.24) is equivalent to Equation (2.21) where od = 2su. Reese (1958) suggested that Equation (2.24) is valid for flow-around type of failure shown in Figure 2.10(a), but that a passive wedge-type failure shown in the free body diagram of Figure 2.10(b) occurs near the ground surface. If the shearing resistance between the pile and the soil is assumed to be zero, and the angle of the passive wedge is assumed to be 45 degrees for undrained conditions at failure, Reese showed that Pu Np Bsu (2.24) where: x + 2.83 - B (2.25) 41 o. o, Is V • c c . 0 \ N \ t1 / ' - 'K II kx I \ T^- n i o - 1 \ - T °s Pile Movement a i (a) (b) Figure 2.10. Ultimate soil resistance for (a) flow-around failure, and (b) passive wedge-type failure for cohesive soil [After (Reese L. C., 1958)] where: y'avg = average effective unit weight from ground surface to p-y curve su = undrained shear strength of the soil x = distance along the length of the shaft to the p-y curve B = pile diameter It should be noted that Equation (2.25) was derived from the passive wedge shown in Figure 2.10(b) by differentiating the sum of the horizontal forces. Matlock accepted that Np is affected by the presence of the ground surface, as proposed by Reese (1958), and that the close proximity to the ground surface tends to reduce the value of Np; however, he modified Equation (2.25) by increasing the first term from 2 to 3 and by replacing the third term of 2.83 with a nondimensional coefficient, J, as shown in Equation (2.26). 42 Np = (2.26) Matlock (1970) stated that the first term in Equation (2.26) ranges from 2 to 4 for a flat plate and a square pile, respectively, where the increased value of 4 accounts for the side shear that develops along the sides of the square pile. Although not stated explicitly, in can be inferred from Matlock's chosen value of 3 for this term that he thought that cylindrical pile behavior was somewhere between a flat plate and a square pile. Reese (1958) suggested that an appropriate value for J can be as high as 2.83, as shown in Equation (2.25). Matlock used a value of 0.5 for J to obtain good agreement between the predicted and observed results of the Sabine Pass tests; however, he had to use a value of 0.25 to obtain good agreement between the predicted and observed results in the stiffer clay at Lake Austin. The depth at which failure transitions from wedge-type failure to flow-around failure, xr, can be estimated from Equation (2.26) by selecting a value for J, setting Np equal to 9, and solving for x. The p-y curves above this depth are estimated from Equation (2.24), while the p-y curves below this depth are estimated from Equation As previously discussed, Terzaghi (1955) stated that the coefficient of subgrade (2.26). 43 reaction was only valid up to approximately half of the ultimate soil resistance. Based on the ultimate soil resistance (y50) could be related to the strain at one half of the maximum stress on a laboratory stress-strain curve (s50) using the equation shown in Equation where: yso = deflection at one half of the ultimate soil resistance, p u strain at one half of the maximum stress on a laboratory stress-strain £50 = J curve Finally, from Equations (2.24) and (2.27), the equation for p less than p u, as shown in Figure 2.9, is It can be seen in Figure 2.9 that the value of p is assumed to be the constant value of pu beyond y =8y5Q. The coefficient of 2.5 in Equation (2.27) was suggested by Skempton (1951) for uniformly loaded footings with a length-to-diameter ratio of 10. He suggests values between 1.7 and 2.5 for length-to-diameter ratios less than 10. McClelland and Focht (1956) have suggested values as small as 0.5 should be used. Bhushan et al. (1979) conducted several large-scale load tests on semi-rigid and rigid drilled shafts in stiff clay, and they found that p-y curves computed with a value of 2.0 instead of 0.5 for J, 2.0 the work of Skempton (1951), Matlock assumed the deflection of the pile at one half of (2.27). (2.27) (2.28) instead of 2.5 in Equation (2.27), and 1/2 instead of 1/3 in Equation (2.28) were in much better agreement with the experimental results. It should be emphasized that these tests were conducted in stiff clay, but the importance that these parameters can have substantial variability, which can significantly affect the results, cannot be understated. Furthermore, these parameters cannot be changed in LPile, and as such, it is expected that unrealistic results can occur when analyzing drilled shafts with D/B ratios less than 10. As previously discussed in this section, Matlock reduced J from the theoretical maximum value of 2.83 proposed by Reese (1958) to 0.5 and 0.25 to obtain good agreement between the observed and predicted results for the Sabine Pass and Lake Austin load tests, respectively. Although Matlock states that J should be thought of as a rational but essentially empirical constant, it is worth noting that the suggested value of 2.83 by Reese (1958) does have important physical significance. The horizontal component of the shearing resistance along plane ABEF of the passive wedge shown in in Figure 2.10(b) is 2cH sec(a) where J = 2 sec(a). When the passive wedge is fully mobilized, a = 45 degrees for undrained conditions and J = 2sec(45°) = 2.83. Because J is a function of a, a reduction of its value suggests that a would also have to be reduced; however, this would result in nonvanishing trigonometric terms in the first two terms of Equation (2.26), and would therefore result in a different formulation for Np. Furthermore, the state of stress on the passive wedge can only be defined at failure using Mohr-Coulomb theory, and as such, there is difficulty in theoretically justifying the use of a different value of a and a passive wedge that has not yet reached failure. The reduction of the theoretical value of J helped Matlock achieve better agreement between observed and predicted p-y curves; however, the implication of 44 modifying J without consideration of the rest of the terms in Equation (2.26) is that J truly is reduced to an empirical adjustment factor, as Matlock suggested. It should be understood that the values of 0.5 and 0.25 that were used in this method are a significant deviation from theory. By inspection of Equation (2.26), the reduction of J from 2.83 to some lesser value will result in a deeper transition point from wedge-type failure to flow-around failure. For this experiment, xr is increased by a factor of approximately 4 and 7 for J = 0.5 and 0.25, respectively. This results in lower ultimate soil resistance over a greater depth from the ground surface because Np is less than the limiting value of 9 within the passive wedge. It is not clear why Matlock chose to modify J instead of applying a correction factor to the entire equation for Np; however, it can be inferred that a reduction in the soil reaction over a greater depth than the depth of the passive wedge was required to obtain satisfactory agreement between observed and predicted results. Furthermore, it is not clear whether this deviation from theory highlights a limitation of the theory itself or whether it was unique to this load test. There are insufficient large-scale load test data available to determine whether J of 0.5 is an appropriate correction factor to correct for a potential limitation of the theory, or whether it is simply a curve-fitting parameter for the specific load tests conducted by Matlock. As previously discussed, Bhushan et al. (1979) conducted several large-scale load tests on semi-rigid and rigid drilled shafts in stiff clay, and found that p-y curves estimated with a value of 2.0 for J were in much better agreement with the back-calculated p-y curves. Therefore, there is some evidence to suggest that a single value of J is not appropriate for all soil-pile configurations. It can be seen in this section that Matlock formulated the relationships forp and y within the theoretical framework established by Reese (1958) and Skempton (1951), 45 respectively. These equations are similar in form to the equations proposed by McClelland and Focht (1958), with the major difference being in the estimation of the shape of the p-y curves from laboratory stress-strain curves and the ultimate soil resistance. McClelland and Focht (1958) estimated the shape and ultimate soil resistance of the p-y curves by scaling a stress-strain curve from a triaxial test performed at equivalent confining stress by a constant factor of 5.5. Matlock predicted the shape of the p-y curves from e50, which is a single value on the laboratory stress-strain curve, and observation of the shapes of back-calculated p-y curves from a large-scale load test. Matlock's estimation of the ultimate soil resistance was nearly identical to the ultimate soil resistance predicted by McClelland and Focht (1958); however, he reduced the soil resistance near the ground surface to account for the passive wedge formation proposed by Reese (1958). Based on the results of the large-scale load test, Matlock further reduced the soil resistance and increased the depth over which the soil resistance should be reduced by reducing J in Equation (2.26) to the point where good agreement between observed and predicted results was achieved. While the approach taken by Matlock is supposedly an improvement upon the work of McClelland and Focht (1958), the method is truly semi-empirical in nature, and there are tradeoffs that have been made that increase reliance on one specific large-scale load test, and therefore reduce the reliance on theory and laboratory test data. 2.4.2.2 Stiff Clay without Free Water (Welch & Reese, 1972) The empirical curves for stiff clays were derived for two conditions: with free water present and without free water present. In this case, the presence of free water is defined as water being present above the ground surface or a considerable chance that 46 water will fill the gap that develops behind the foundation during lateral loading (Isenhower & Wang, 2010). Different curves were developed for stiff clays with free water because soils in this condition experience a loss of shear strength during cyclic loading due to soil particles being expelled from around the shaft as water is squeezed out of the gap between the soil and the foundation. None of the large-scale load tests that were used in this investigation were conducted in stiff clay with free water, and as such, only stiff clay without free water will be discussed in this section. A large-scale lateral load test was conducted by Welch and Reese (1972) on a drilled shaft that had a diameter of 30 inches and an embedded depth of 42 feet. The D/B ratio for this shaft was approximately 16.8, which suggests the behavior of the shaft can be categorized as flexible. A reinforcement cage was constructed from twenty 14-S deformed bars and 1/2-inch diameter transverse spiral reinforcing. The diameter of the cage was 24 inches and the spacing of the reinforcing spiral was 6 inches. A steel pipe with an outside diameter of 10 3/4 inches and a thickness of 1/4 inch was placed in the center of the drilled shaft for attachment of the strain gauges. A total of 31 pairs of strain gauges were placed on the steel pipe with an arbitrary spacing of 15 inches for the upper section of the shaft and 30 inches for the lower section of the shaft. The drilled shaft was installed near Houston, Texas, in stiff, overconsolidated clay with a groundwater depth of 18 feet. The undrained shear strength of the clay was estimated by conducting several triaxial tests on samples trimmed from 4-inch diameter thin wall tube samples. The triaxial compression tests were conducted by consolidating samples to the effective in-situ overburden stress and then shearing them in undrained conditions. The samples were tested with orientation both perpendicular to the ground surface and parallel to the ground surface to simulate lateral compression of the soil that 47 occurs during lateral loading of a drilled shaft. No significant differences were observed in the triaxial tests results due to sample orientation, but there was a wide variation in su due to the slickensided structure of the clay. There was no discernable pattern of strength variation with depth, and the average value of su was reported as 2,200 psf. The average value of e50 was reported as 0.005. Both static and cyclic lateral loads were applied to the foundation, with the load being applied at approximately 2 feet above ground level. The p-y curves at various depths were back calculated using a similar procedure to the one described in Section 2.4.1. The moment profile was established from the strain gauge data using Equation (2.18) and values of EI that were determined by directly measuring the bending stiffness of the shaft. The bending stiffness of the shaft was estimated after the completion of all of the tests by excavating the soil around the shaft to a depth of 20 ft and applying a series of lateral loads to the top of the foundation. By calculating the moment from the applied lateral load and measuring the curvature in the shaft at the bottom of the excavation, the bending stiffness can be estimated directly from Equation (2.18). A seventh-order polynomial was fitted to the moment profile data, and differentiation and numerical integration yieldedp and y profiles, respectively, for each loading increment. Welch and Reese (1972) estimated p-y curves for the static loading case from the laboratory test data using the same equations used by Matlock (1970), which are shown in Equations (2.24) through (2.28). After comparing the predicted results with the results obtained from the large-scale load test, they modified these equations to obtain better agreement between the observed and predicted results as discussed in the following paragraphs. 48 The exponent 1/3 in Equation (2.28) was changed from 1/3 to 1/4 as shown in Equation (2.29). 49 (2.29) The value of 1/4 was selected because it resulted in a better fit of the data when p/pu was plotted against y/y50 in logarithmic space. It should be noted, however, that the ultimate strength of the soil was not reached during the test because the shaft was not strong enough to resist the bending induced stress. As such, the researchers assumed a value for p u by doubling the value of p at y50. The deflection required to reach the ultimate soil resistance for soft clays is y = 8y50 as shown in Figure 2.9, but this deflection was increased by a factor of two to y = 16y50 as shown in Figure 2.11. This implies that stiff soils require more displacement than soft soils to reach their ultimate strength. P i Figure 2.11 General shape of p-y curve for static loading in stiff clay without free water (Isenhower & Wang, 2010) It should also be noted that the default and unchangeable value for J is 0.5 in LPile, despite the fact that Matlock (1970) suggested that J was less than 0.5 for medium stiff soils, and Bhushan et al. (1979) showed that J could be as high as 2.0 for stiff soils. The p-y curves are estimated at a particular depth for static loading from Equation (2.29), wherep u and y 50 are calculated from Equations (2.24) and (2.27), respectively. According to Isenhower and Wang (2010), several tests were reported in the Southeastern United States that exhibited much less initial stiffness than was estimated using the p-y curves for stiff clays. It was suggested that k should be incorporated to explicitly account for the stiffness of the soil. As such, the p-y curves for stiff clay were modified to allow a value of k to be specified for the soil layer under consideration. The p-y curve is calculated as before using Equation (2.29), but the initial portion is also calculated using Equation (2.30): p = (k x)y (2.30) The lesser value of p computed from both equations is used in the final formulation of the p-y curves. As previously discussed in Section 2.4.2.1, the "constant" parameters used in the equations that were developed to predictp and y for stiff clays are not really constant, but they can assume a wide range of values. A potential limitation of LPile is that these parameters cannot be changed for conditions where different values might be more appropriate. 50 2.4.2.3 Sand (Cox, Reese, & Grubbs, 1974) A large-scale lateral load test was conducted by Cox et al. (1974) at Mustang Island near Corpus Christi, Texas. Two 24 in. diameter steel pipe piles were driven open-ended into the sand at the test site to a depth of approximately 70 ft and inundated to simulate offshore conditions. The D/B ratio for these piles was approximately 35, which suggests the behavior of the piles can be categorized as flexible. Each pile was instrumented with strain gauges to measure the curvature during application of the lateral load. Rotation and deflection were measured at the top of the foundation, in addition to the magnitude of the applied load. The soil at the site was described as varying from clean fine sand to silty fine sand with high relative density. Laboratory tests were performed on "undisturbed" samples, and the effective angle of internal friction (^ ') and submerged (effective) unit weight (y r) for the entire subsurface profile was reported as 39 degrees and 66 pcf, respectively. The piles were tested under both static and cyclic loads, which were applied in increasing increments. A set of p-y curves was computed by Reese et al. (1974) from the curvature profile of each loading increment for both static and cyclic loading using Equations (2.17), (2.18) and (2.20). The general shape of the set of p-y curves from this method are shown in Figure 2.12. Theoretical p-y curves for sand are represented by three linear segments and one parabolic segment, as shown in Figure 2.12. The initial segment for y < y k is approximately linear, and was established from observations made by Terzaghi (1955) that the soil modulus for granular soils increases in proportion to confinement, which increases with depth as shown in Equation (2.31). 51 52 Figure 2.12. General shape of a set of p-y curves for static and cyclic loading in sand (Isenhower & Wang, 2010) Esi ^h % (2.31) where: Esi = soil modulus of initial linear portion of p-y curve nh = constant of subgrade reaction x = depth below ground surface It should be noted that the constant of subgrade reaction, nh, is equal to ks in Figure 2.12. Terzaghi (1955) suggested numerical values for the constant of subgrade reaction based on the relative density of the soil and the effective unit weight. The results of the large-scale load tests suggested that these values are much too conservative. The values of the initial soil modulus of the p-y curves that were computed from the results of the large-scale load test were 2.5 and 3.9 times higher than the highest values suggested by Terzaghi (1955) for the static and cyclic load tests, respectively. As such, Reese et al. (1974) suggested using values that are approximately 2.5 times higher than those proposed by Terzaghi (1955). These higher values are the default values used by LPile in the absence of user input values and are shown in Table 2.1. The constant of subgrade reaction is related to the horizontal coefficient of subgrade reaction as ^h = nh (^ ) (2.32) where: kh = horizontal coefficient of subgrade reaction nh = constant of subgrade reaction x = depth below ground surface B = diameter or width of pile As discussed by Habibagahi and Langer (1984), several methods and correlations have been developed for predicting the horizontal coefficient of subgrade reaction, which generally confirm that the values proposed by Terzaghi (1955) are too conservative. It should be noted, however, that Terzaghi (1955) stated that the coefficient of subgrade reaction is a function of pile width or diameter, and that the values he proposed were for a 1 ft wide strip. It should also be noted that the values proposed by Reese et al. (1974) in Table 2.1 do not take into account the diameter of the foundation. Additionally, Alizadeh and Davisson (1970) have shown from the results of large-scale load tests that the constant of subgrade reaction is not actually constant, but a function of pile deflection for deflections less than approximately 0.5 in. 53 Table 2.1. Values of k suggested by Reese et al. (1974) 54 Relative Density Recommended k (pci) Submerged Above GWT Loose 20 25 Medium 60 90 Dense 125 225 The horizontal linear portion of the curve represents the ultimate soil resistance (p u). Reese et al. (1974) formulated the ultimate resistance of sand using a theory similar to that of clay, as previously discussed in Section 2.4.2.1, whereby the ultimate soil resistance near the ground surface is represented by a passive wedge, and transitions to a flow-around type of failure at some greater depth. A schematic of the flow-around type of failure, which is based on Mohr-Coulomb failure criterion and virtually identical to flow-around failure for clays shown in Figure 2.10(a), is shown in Figure 2.13(a). The corresponding Mohr's circles for the state of stress in each idealized soil element are shown in Figure 2.13(b). Reese et al. (1974) showed that the ultimate soil resistance that develops from lateral flow of soil around the pile (pcd) can be estimated as Pcd = By'H[Ka(tan8(£) - 1) + K0tan ($ ')tan 4(£)] (233) where: Pcd = theoretical ultimate soil resistance from lateral flow of soil around pile H = depth (height) of passive wedge B = width or diameter of pile 0 ' = effective angle of internal friction P = angle of passive wedge from the vertical face of the pile K = at-rest earth pressure coefficient K = active earth pressure coefficient 55 > r 5 °2 > f °4 -► °6 -► I " ' °4 °3 \ - " O r \ \ ' t - 0 ­ ✓ ' III s^ n - 0 ': ' l s T Pile Movement (a) T °s <7 (b) Figure 2.13. Assumed mode of soil failure by lateral flow around the pile (Reese, Cox, & Koop, 1974) 56 The passive wedge at failure for sand is shown in Figure 2.14(a). The most notable difference between the passive wedge for sand and the passive wedge for clay, shown in Figure 2.10(b), is the development of a fan in the horizontal direction at an angle of a. The horizontal soil reaction that develops within the passive wedge can be solved from the geometry of the wedge, assuming Mohr-Coulomb failure is valid for sand (Reese, Cox, & Koop, 1974). By summing the forces that act on the passive wedge and differentiating with respect to x, the soil resistance per unit length within the wedge (pct) is where: p ct = theoretical ultimate soil resistance from a passive wedge in sand H = depth (height) of passive wedge B = width or diameter of pile p ' = effective angle of internal friction /? = angle of passive wedge from the vertical face of the pile a = angle of passive wedge from a line parallel to the applied load K0 = at-rest earth pressure coefficient K = active earth pressure coefficient The ultimate soil resistance at any point along the pile is the lesser of the two values calculated by Equation (2.33) and (2.34). Pct = Y' (2.34) 57 B Figure 2.14. Assumed passive wedge type of soil failure (Reese, Cox, & Koop, 1974) Reese (1962) suggested that a has a maximum value equal to however, Bowman (1958) states that the results of plate load tests in sand show that a is a function of void ratio, and that a is likely to be 0'/3 to 0'/2 for loose sand, and for dense sand. The results of the laboratory experiments performed by Reese (1962) on small, laterally loaded tubes in loose sand showed better agreement with analytical results when a was assumed to be instead of Bowman's suggested value of $'/2. Reese et al. (1974) stated that contours of the wedge that formed in front of the test piles in this load test indicated that the value of a was equal to about $'/3 for static loading and about 3$'/4 for cyclic loading. As such, they state that a can be assumed to be $'/2, and this is the assumption that is made by LPile when estimating p-y curves for sand. The angle ( is computed using Rankine's familiar lateral earth pressure theory, which, for lateral earth pressure in the two-dimensional passive case is 58 (2.35) Reese et al. (1974) state that the criteria for using Rankine's theory are not strictly satisfied for the case of a laterally loaded pile; however, they state that some model experiments indicate that Equation (2.35) gives a fairly good approximation of the slope of the failure surface. It should be noted that Ka in Equation (2.34) is also computed using Rankine's theory as There is no recommendation for establishing Ko in the LPile manual; however, Reese et al. (1974) suggest using a value of 0.4 for loose sand and 0.5 for dense sand in the absence of precise methods for determining relative density in the field. A value of 0.4 was used for the large-scale load test, and it is assumed that a constant value of 0.4 is used by LPile. According to Reese et al. (1974), the geometric parameters of the passive wedge can assume a wide range of values. They stated that the estimated ultimate soil resistance was in poor agreement with the ultimate soil resistance observed in the load test, even when these parameters were varied over a reasonable range. As such, a simple correction factor (As,c) was introduced to adjust the ultimate soil resistance as (2.36) 59 Pu Asc pc (2.37) where: A s,c = ultimate soil resistance correction factor p u = ultimate soil resistance p c = theoretical ultimate soil resistance Values of As,c were computed at several depths from the observed and theoretical ultimate soil resistance, and incorporated into the design aid shown in Figure 2.15. The authors did not state how they determined that the soil resistance at a particular depth was the ultimate resistance; however, the computed values of As,c were only for the upper 10 ft of soil, or x/B < 5, which seems reasonable. For depths greater than 5B, the correction factor approaches a constant value of 0.88 for both static and cyclic loading. This might be satisfactory for flexible foundations where the soil reaches ultimate capacity near the surface, but is probably unsatisfactory for semi-rigid foundations where the displacement of the pile tip might be sufficient to develop ultimate soil capacity. A 0 1.0 2.0 Figure 2.15. Correction factor A for ultimate soil resistance vs. depth for cyclic and static loading (Reese, Cox, & Koop, 1974) Further inspection of Figure 2.15 shows that A = 1 at x ~ 3.2B for static loading. It follows from Equation (2.37) that the ultimate soil resistance for static loading is increased from theory by a factor of Bs,c for x < 3.2B and decreased from theory by a factor of Bs,c for x > 3.2B. This seems to be a more rational approach for adjusting thep-y curves to match experimental observations than the approach used for soft clays, which, as discussed in Section 2.4.2.1, was to modify the geometric term J in the passive wedge equation. It should be noted, however, that the values shown in Figure 2.15 are curve-fitting parameters for the load test from which they were derived, and it is not presently known how these factors might change for a different soil and pile configuration. It can be seen in Figure 2.12 that there is a parabolic section and a linear section of the p-y curve between y = y k and y = 3B/60. According to Reese et al. (1974), these two segments of the p-y curve were selected empirically to yield a shape consistent with the experimental p-y curves. The linear portion of the curve located between points m and u in Figure 2.12 was established by the points ym and yu. The points ym and y u were observed in the large-scale load tests at 0.4 and 0.9 in. of pile head deflection, respectively. These deflections were normalized to the pile diameter, which resulted in values ofym and y u of B/60 and 3B/80, respectively. The soil resistance at point m, p m, is estimated from the theoretical ultimate soil resistance, pc, and the nondimensional coefficient Bs,c as Pm Bsc pc (2.38) where: Bs,c = nondimensional coefficient for determining pm 60 p c = theoretical ultimate soil resistance Pm = soil resistance at point m in Figure 2.12 The coefficient Bs,c was computed from the observed data for the upper 10 ft of soil, or x/B < 5 as shown in Figure 2.16. For depths greater than 5B, the coefficient Bs,c assumes a constant value of 0.5 for static loading and 0.55 for cyclic loading. This might be satisfactory for flexible foundations where the soil reaches ultimate capacity near the surface, but is probably unsatisfactory for semi-rigid foundations where the displacement of the pile tip might be sufficient to develop ultimate soil capacity. Additionally, as for the correction factor As,c, these values are unique to the load test from which they were derived, and it is not presently known how these correction factors might change for other soil-pile configurations. 61 B 0 10 2.0 Figure 2.16. Nondimensional coefficient B for calculating soil resistance at point m in Figure 2.12 for cyclic and static loading (Reese, Cox, & Koop, 1974) The parabolic portion of the p-y curves shown in Figure 2.12 is determined by fitting a parabola from the origin through point m with a slope at m equal to the slope of the linear portion between points m and u. The intersection of the parabolic curve and the initial straight-line portion of the p-y curve establishes the point k in Figure 2.12. Reese et al. (1974) have acknowledged that the method presented for establishing p-y curves for sands is based heavily on empiricism, and that the method might not be applicable for all foundations or soil conditions. They state, for example, that the presence of clay in a sufficient amount to give some cohesion will cause the soil to behave in an entirely different manner. Furthermore, they specifically admonish the use of caution and judgment when using these curves, and that serious errors can be made by inexperienced analysts. 2.4.3 Estimation of p-y curves from In-situ Tests The in-situ tests that are most often used for computing p-y curves are the dilatometer test (DMT) and the pressuremeter test (PMT). The p-y curves that are estimated from PMT test and DMT tests have the advantage of being based on lateral deformation properties of the soil (Anderson & Townsend, 1999); however, these tests do not directly capture the effects of the foundation on the p-y curves. According to Robertson et al. (1983), the pressuremeter offers an almost ideal in-situ modeling tool for determining directly the p-y curves for a pile. The PMT test is an excellent test for computing p-y curves because the test is similar to a laterally loaded pile (Briaud, Smith, & Tucker, 1985). Additionally, the results of the PMT test can be used to derive curves in soils and rock for which no empirical p-y curves exist (Little & Briaud, 1988). The DMT test is also an excellent test for computing p-y curves because the test 62 conditions simulate a driven pile, it is inexpensive, and it can be used near the ground surface, where the soil has the most influence on the response of the pile (Robertson, Davies, & Campanella, 1989). Unfortunately, both PMT and DMT tests are small-strain tests, and only a small amount of the soil is tested. Also, these tests only measure the passive resistance of the soil directly, and do not measure the skin friction, which is a major contributor to the overall soil resistance [ (Smith & Slyh, 1986), (Janoyan & Whelan, 2004), (Briaud, Smith, & Meyer, 1983) ]. Regardless, the pressuremeter has been shown to provide good results in several case histories as shown by Robertson et al. (1986). The dilatometer has also been shown to provide good results in several case histories as shown by Robertson et al. (1989) and Gabr et al. (1994). Anderson et al. (2003) also compared the results from analyses of laterally loaded piles and drilled shafts using p-y curves derived from the DMT and PMT tests. The comparison showed that the results from the dilatometer were in good agreement with the results of the large-scale load tests at small displacements, but that the error increased as the displacement increased. They explained that this makes sense because the membrane on the dilatometer is only pushed 1 mm into the soil during testing. The comparison also showed that the results from the pressuremeter were in good agreement with the results of the large-scale load tests, except for the cases where excess pore water pressure was present. 2.5 DFSAP and the Strain Wedge Model The strain wedge (SW) model was developed to establish a method for evaluating the modulus of subgrade reaction (Es) in order to solve the BEF equation for a laterally loaded pile based on the envisioned soil-pile interaction and its dependence on both soil 63 and pile properties (Ashour, Norris, & Piling, 2002). The SW model was originally formulated by Norris (1986) for sands, and subsequently expanded by Norris and his colleagues to include cohesive soils, "c-fi" soils, and rock [ (Ashour, Norris, & Pilling, 1998), (Ashour et al., 2001) ]. Norris (1986) stated that there is some reluctance to extrapolating existing methods for determining BEF parameters to conditions for which there is little or no field evidence. Subsequent publications on the SW model specifically express skepticism for employing the semi-empirical p-y methods discussed in Section 2.4.2 for analyzing foundations outside of the field-calibrated range. Ashour and Norris (2000) state that these semi-empirical p-y methods do not account for a change in pile properties such as pile bending stiffness, pile cross-sectional shape, pile-head fixity, and pile-head embedment below the ground surface. Ashour and Norris (2000) reiterate the importance of Es being dependent upon the soil-pile interaction, but that it would be prohibitively expensive to systematically evaluate all of these effects through load tests. As such, they suggest that the characteristics of the soil-pile interaction that are known to affect Es can be evaluated through the semi-theoretical means of the SW model. A survey of the published literature for the SW model revealed that the model uses a very similar solution scheme to the one proposed by McClelland and Focht (1958), and draws heavily from the theoretical basis of the semi-empirical p-y curve methods discussed in Section 2.4.2. A schematic for the SW representation of the three­dimensional soil-pile interaction and the resulting soil reaction for a laterally loaded pile is shown in Figure 2.17. The equivalent one-dimensional soil-pile reaction is shown in Figure 2.18. 64 65 SLICE OF WEDGE AT DEPTH x a. Basic Strain Wedge in Uniform Soil b. Deflection Pattern o f a Laterally Loaded Pile and Associated Strain Wedge r« SUBLAYER i SUBLAYER 3 / 1----- ------------ • PASSIVE WEDGE LINEARIZED DEFLECTION PATTERN c. Strain Wedge Model Sublayers Figure 2.17. Schematic of SW model (Ashour & Norris, 2000) ---------------- ^■Cvwvw\| Esi ^(www| Es2 y<AAAAAAA^ Es3 <www| Es4 A^AWVVsI y w v w w ^ Esn ^(/wvww^ AaAA/NAA/^ Esm (a ) PILE DEFLECTION PATTERN (b ) SOIL REACTION DISTRIBUTION ALONG THE PILE (c ) SOIL- PILE MODELING Figure 2.18. One-dimensional distribution of soil-pile reaction along deflected pile (Ashour & Norris, 2000) It can be seen in Figure 2.17(a) that the soil reaction of a laterally loaded pile in the SW model is a function of the passive soil wedge that forms in front of the pile and the shearing resistance that develops along the length of the pile. It can also be seen in Figure 2.17(a) that the passive wedge is fully defined in terms of the mobilized effective friction angle (0'm) pile diameter (D) and depth of the passive wedge (h) (Ashour & Norris, 2000). Ashour and Norris (2000) state that h is a function of the pile bending stiffness, diameter, pile head fixity, and pile shape. The width of the mobilized passive wedge, BC, is calculated as BC = 2(h - x) tan (^m) tan(^tf'm) + D (2.39) where Pm = 45 + ^ (2.40) It should be noted that the geometry can be defined for each sub-layer, and that the passive wedge for multilayered soils with different strength properties is discontinuous as shown in Figure 2.19. Even though the compound passive wedge is discontinuous across sub-layer boundaries, the strain within the overall passive wedge still remains constant. Furthermore, Ashour et al. (1998) state that the interaction between the sub-layers is accounted for by the continuity of the deflected length of the pile. The passive wedge representation of the soil reaction is virtually identical to the one proposed by Reese (1962), which is shown previously in Figure 2.14. The biggest difference between the way the passive wedge is used in the SW model and the way the passive wedge is used in the semi-empirical p-y methods are: 1) the passive wedge shown 66 67 Th* Goomotry of Sublayer Figure 2.19. Configuration of compound passive wedge for multilayered soil (Ashour, Norris, & Pilling, 1998) in Figure 2.14 is for sand, whereas the SW model uses this same passive wedge for both sand and clay, which will be discussed later;2) the passive wedge is only used to estimate the ultimate soil resistance in the semi-empirical p-y method, and provides no information about the intermediate soil reaction, i.e., it is not directly related to the stress-strain behavior of the soil up to failure, whereas the passive wedge in the SW model is a function of the stress-strain relationship of the soil and directly provides the soil resistance for any increment of pile deflection up to failure; 3) the passive wedge is defined at failure in the semi-empirical p-y method, but it is shown to "grow" from zero pile deflection up to failure with increasing pile deflection and resulting soil strain in the SW model. The increasing soil strain in the passive wedge is responsible for the growth of the passive wedge up to failure, which is shown by the "mobilized" value of terms with subscript m in Figure 2.17(a). These terms are defined only at failure for the semi-empirical p-y methods, and are thus constant at failure. The method used by the SW model to estimate Es is very similar to the method proposed by McClelland and Focht (1958). According to the McClelland and Focht (1958) method discussed in Section 2.3, the first step in evaluating Es for a laterally loaded pile is to assume the deflected shape of the pile and to estimate the deflection at several discrete points along the pile within the deflected zone. The strain at each point is calculated from the assumed magnitude of displacement, and the laboratory stress for an equivalent strain is estimated from a laboratory stress-strain curve for the same confining stress. The soil reaction p is estimated at each discrete point by multiplying the laboratory stress by a constant value of 5.5, and Es is calculated at each point using Equation (2.13). This procedure is repeated for several load increments to establish the Es profile, i.e., the p-y curve, for each discrete point along the length of the pile. These Es profiles are then used to solve the BEF equation. The specific steps for evaluating Es using the SW model are somewhat different from the method just described, but the general approach is remarkably similar. Figure 2.17(b) and (c) show that the deflected shape of the upper portion of the pile is assumed to be linear in the SW model; however, it should be noted that this assumption is only made for the assessment of Es. The final deflected shape is not linear in the upper portion of the pile as shown. The passive wedge in the SW model is discretized into sub-layers, as shown in Figure 2.17(c), and the strain within each sub-layer is estimated from the shear strain. For the assumed linear deflection profile, the shear strain is approximately equal to the angle of pile deflection (S). Because the shear strain is constant over the 68 entire depth of the passive wedge, e is also constant over the entire depth of the passive wedge, hence the name "strain wedge" model (Ashour, Norris, & Pilling, 1998). Like the method proposed by McClelland and Focht (1958), the SW model also relates the change in horizontal stress (Aoh) within each sub-layer to e through the triaxial test. Because e is constant within the passive wedge, Aoh is also constant within the passive wedge. The stress-strain relationship between Aoh and e is related by the secant modulus of the soil (E) from laboratory testing, where 69 E = - (2.41) The stress-strain behavior of each sub-layer within the passive wedge is represented by an axial compression triaxial test. In a typical triaxial compression test, the confining stress (o3) is held constant, while the axial compressive stress (&i) is increased until failure or some other predefined level of strain is reached. In the SW model, Aon is equal to the deviatoric stress (od) where od = o1 - o3. The confining stress is equal to the overburden stress acting on the sub-layer, and it is assumed that this confining stress does not change as the horizontal stress changes (Ashour, Norris, & Pilling, 1998). The stress-strain relationship for the sub-layers of the passive wedge in the SW model are represented by a triaxial compression test behavior, but they are not directly related to the actual laboratory stress-strain curves from the pile installation site like they are in the method proposed by McClelland and Focht (1958). In other words, stress-strain curves obtained from triaxial tests performed at confining stress equal to the sub­layer depths are not inputs for the SW model; instead, the stress-strain curves for each sub-layer are represented by a hyperbolic stress-strain relationship for both sand and clay as shown in Figure 2.20(a). The curves shown in Figure 2.20(a) and (b) are based on the results of laboratory experiments (Ashour, Norris, & Pilling, 1998). The hyperbolic curve shown in Figure 2.20(a) also represents the so-called stress level (SL) within the passive wedge. The SL is defined as the change in horizontal stress (or deviatoric stress in the triaxial test) normalized by the change in horizontal stress at failure (Aohf) as shown in Equation (2.42). For sands and clays, Aohf is calculated using Equations (2.43) and (2.44), respectively. 70 STRESS LEVEL (SL) (a) PARAMETER X Figure 2.20. Representative hyperbolic stress-strain curve for soil (Ashour, Norris, & Pilling, 1998) 71 tan2 (45 + 0 'm) - 1 tan2( 45 + 0 ') - 1 (2.42) Aahf = a'vo ta n 2 I 45 + - I - 1 (sands) (2.43) A ahf = 2 su (clays) (2.44) where: SL = stress level of sub-layer Aah = horizontal stress change within sub-layer Aahf = horizontal stress change at failure within sub-layer pp'm = mobilized effective friction angle of the soil p ' = effective friction angle of granular soils a'Vo = vertical effective stress acting at the top of the sub-layer su = undrained shear strength of cohesive soils The curve for SL shown in in Figure 2.20(a) is entirely defined by e50, £, and the curve fitting parameter 1 shown in Figure 2.20(b). For each sub-layer i, SLi is calculated from Equation (2.42) or (2.43). SL, = exp (-3 .7 0 7SL,); SL, < 0.80 (£5o)i (2.45) 72 SLi = exp , , 100£ ln(022 + 59£ + 95.4 (£50) J ; SLi > 0.80 (2.46) where: SLi = stress level of sub-layer i Ai = curve-fitting parameter for sub-layer i £ = horizontal strain within the passive wedge (£50)1 = strain in the soil for sub-layer i at 50% of peak strength It should be noted that e50 is the only input property for SL in the SW model. The general shape and curve-fitting parameters of Equation (2.42) and (2.43) are based on experimental results and are said to describe a typical stress-strain curve for both sand and clay (Ashour, Norris, & Pilling, 1998). This assumption seems somewhat simplified, as the general shape of the stress-strain curve shown in Figure 2.20(a) is only representative of normally consolidated soils and soils with large confining stress. As such, this curve does not accurately represent both sands and clays in all conditions. It should be noted that the SW model is an effective stress model, and that the strain wedge geometry is defined in terms of for both granular and cohesive soils. The difficulty of an effective stress analysis for undrained loading conditions, as in the case of a laterally loaded pile in cohesive soil, arises from the difficulty of estimating the excess porewater pressure (An) that is generated during pile deflection. The excess porewater pressure is estimated in the SW model using the well-known equations developed by Skempton (1954) for estimating the porewater pressure that develops within a soil mass subjected to an applied load. The specific procedures used in the SW model are discussed in further detail in (Ashour, Norris, & Pilling, 1998). The SW model includes the shearing resistance that develops along the length of the pile in the formulation of the soil reaction. In the semi-empirical p-y method, some of the shearing resistance is included by means of the empirical corrections made to the soil resistance equations; however, shearing resistance along the pile is not explicitly included in the soil resistance equations. This is one of the major criticisms of the developers of the SW model because they suggest that it is a major component of the overall soil reaction - particularly for large-diameter piles (Ashour, Norris, & Pilling, 1998). Smith and Slyh (Smith & Slyh, 1986) and Janoyan and Whelan (2004), among others, have shown that most of the soil reaction at small strains results from shearing resistance. This makes sense from a soil mechanics perspective because shear stress mobilizes much more quickly than the normal stresses in the passive zone (Salgado, 2008). The shear stress component of the soil reaction in the SW model depends on whether the soil is classified as sand or clay. For sands, the shear stress along the pile sides is a function of effective stress and mobilized friction angle between the sand and pile (0s) (Ashour, Norris, & Pilling, 1998). The shear stress along the pile within sub­layer i (t ) is calculated as Tj (o' !0)jtQft(0 s)j (2.47) where: Ti = shear stress along pile sides within sub-layer i (cf'vo,) i = vertical effective stress within sub-layer I (4>'s)i = mobilized effective friction angle at the soil-pile interface within sub­layer i 73 According to Ashour et al. (1998), $'s develops at twice the rate of $'m, such that Equation (2.63) becomes t ! = 2(a'vo) ! ta n (0 'm) f (2.48) The value of $'m, is, of course, limited to the value of of the soil, and the ultimate value of shear stress along the pile sides (rult) for sands is fruit) i = fr'vo) i tan((p')! (2.49) For clays, the shear stress along the pile sides is a function of su, SL, and pile adhesion (a) (Ashour, Norris, & Pilling, 1998). The stress level of shear along the pile sides (SLt) is different from the SL in the passive wedge (Ashour, Norris, & Pilling, 1998). In the SW model, SLt is estimated using the Coyle and Reese (1966) "t-z" shear stress transfer curves shown in Figure 2.21(a). These curves represent the ratio of the side shear stress of a 1-foot diameter pile embedded in clay to the theoretical shear strength along the interface (asu) as a function of pile displacement. Curves A, B, and C in Figure 2.21(a) represent depth ranges of 0 - 3m, 3 - 6m, and > 6m, respectively. Ashour, Norris, and Piling (1998) normalized and simplified these curves by multiplying asu by a normalization factor (Z), which is equal to the peak values of 0.53, 0.85, and 1.0 for curves A, B, and C, respectively. The normalized and simplified curves are shown in Figure 2.21(b). These curves are used to estimate SLt, for a given magnitude of pile displacement, y. Once SLt is estimated from Figure 2.21(b) and the assumed pile deflection within sub-layer i, Ti for clays can be calculated as 74 75 (b) pile displacement, y (in.) Figure 2.21. Estimation of stress level (SL) along pile sides using (a) Coyle and Reese (1966) shear-transfer curves, and (b) Normalized shear-transfer curves where: fault) i i (2.51) 76 T = shear stress along pile sides within sub-layer i (Tult)i = ultimate shear strength along pile sides within sub-layer i D ) = stress level of shear along pile sides within sub-layer i (su)i = undrained shear strength of the soil within sub-layer i z = normalization factor a = pile adhesion factor The values of pile adhesion factor, a, used in the SW model were recommended by Tomlinson (1957). The total soil reaction within each sub-layer i is calculated for each sub-layer as Vi = O 'h ) i BCt + 2t; DS2 (2.52) Note that the first and second terms in Equation (2.53) represent the normal and shearing resistance, respectively. It should also be noted that the active earth pressure is not accounted for in Equation (2.52) (Ashour, Norris, & Piling, 2002). The SW model makes some effort in Equation (2.52) to account for the shape of the pile by including the shape adjustment factors S1 and S2 proposed by Briaud et al. (1984). S1 and S2 are equal to 0.75 and 0.5, respectively, for a circular pile cross section, and 1.0 for a square pile cross section. Alternately, Equation (2.52) can also be written as Pi = Ai D(Aa'h) ! = Ai DEt e (2.53) From Equation (2.52) and the soil strength and geometric properties defined in Equations (2.39) through (2.51), it can be shown that 77 A _ r („ (h - xi) 2 (tanfim ta n y 'm 2 S2(Aa'v0) f (tanq)'s ) ^ ^ ^ ■ M 1 + D J + ( ^ ' ! i (2.54) in sand, and „ „ ( . (h - 2 (tanPmtany'm)A S2(SLt) A‘ = M 1 + ---------------- D---------------- + ~ S l T (2-55) in clay. It can be seen that Equation (2.53) takes the same form as Equation (2.12) proposed by McClelland and Focht (1958), were At = 5.5. By definition, the ultimate soil resistance, p u, is reached when SL = 1. In the case where SL = 1 for sands, Equation (2.52) becomes ( p j i = ( ^ ! / ) i Si + 2(T«it) i DS2 (2.56) In the case where SL = 1 for clays, Equation (2.52) becomes ( P J i = 1OO J i DS! + 2 ( s J ! DS2 (2.57) It can be seen in Equation (2.56) that p u for sand is dependent upon BC, which will continue to grow as the depth of the passive wedge increases with increasing pile deflection. As such, p u will continue to increase even after SL = 1 (Ashour & Norris, 2000). This differs from p u for sands in the semi-empirical model discussed in Section 2.4.2.3, which reaches a constant value of the lesser of Equation (2.33) and (2.34). For circular piles embedded in clay, Equation (2.57) reduces to p u = 8.5Dsu. For rectangular piles embedded in clay, the coefficient of 8.5 is increased to 12. Recall from 78 Equation (2.24) and the discussion in Section 2.4.2.1 that Matlock (1970) called this coefficient Np. According to Matlock (1970), the value of Np is depth dependent within the passive wedge and ranges from 3 at the ground surface to 9 at the transition to flow-around failure for clays, where it remains constant at greater depths. From Equation (2.57), it appears the value of Np is constant within each sub-layer in the SW model, and thusp u is constant. The pile deflection in the SW model is estimated from the horizontal strain, s, in the passive wedge and the linearized pile deflection angle (^) shown in Figure 2.17(b). Using Mohr's circle for strain, Ashour et al. (1998) show that Recall that v is Poisson's ratio for the soil within each sub-layer. From geometry, the deflection of each sub-layer is calculated as 2 ^ s (1 + v) sin (2 em) (2.58) where (2.59) (2.60) where: Hi = vertical thickness of sub-layer i Si = linearized pile deflection angle of sub-layer I 79 The pile head deflection (yo) is calculated by summing the pile deflection at each sub-layer within the passive wedge as for n sub-layers within the passive wedge. In the SW model, Es is calculated for each sub-layer i using an incremental form of Equation (2.6) as The values of p t and (E ) from Equations (2.52) and (2.62), respectively, are dependent upon the geometry of the passive wedge that results from the assumed value of e; therefore, iteration must be used to obtain a convergent solution to the BEF equation shown in Equation (2.4). The general solution scheme for the SW model is described by Ashour et al. (2002), and is summarized according to the author's understanding in the following solution steps: 1) Estimate the soil properties of ^', su, and e50 for the subsurface profile. Select values of embedment depth