NUMERICAL INVESTIGATION OF LOAD TRANSFER MECHANISM IN SLOPES REINFORCED WITH PILES

Transcription

1 NUMERICAL INVESTIGATION OF LOAD TRANSFER MECHANISM IN SLOPES REINFORCED WITH PILES A Dissertation Presented to the Faculty of the Graduate School University of Missouri-Columbia In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy By ENG-CHEW ANG Dr. J. Erik Loehr, Dissertation Supervisor DECEMBER, 25

2 The undersigned, appointed by the Dean of Graduate School, have examined the dissertation entitled NUMERICAL INVESTIGATION OF LOAD TRANSFER MECHANISM IN SLOPES REINFORCED WITH PILES Presented by Eng-Chew Ang A candidate for the degree of Doctor of Philosophy in Civil Engineering And hereby certify that in their opinion it is worthy of acceptance. epartmeqt of Civil and Environmental Engineering Douglas E. smith Department of Mechanical and Aerospace Engineering Department of Civil and Environmental Engineering ~e~artment of civil and Environmental Engineering wil?ysn J. Likos Department of Civil and Environmental Engineering -%t of Civil and Environmental Engineering

3 ACKNOWLEDGEMENTS It has been a great pleasure for me to have the opportunity to pursue my graduate study in the Geotechnical program at the University of Missouri-Columbia. I have been given many great opportunities by Dr. Erik Loehr, my advisor, in doing exciting research projects as well as in teaching. His willingness to share his experience and thoughts on all projects that we had worked together makes my study even more meaningful. I would like to take this opportunity to express my sincere appreciation to my supervising professor, Dr. J. Erik Loehr, for his patient to share his valuable experience, thoughts, supports, and advice. It is my honor to study under all the professors, Dr. Erik Loehr, Dr. John Bowders, Dr. William Likos, Dr. Brent Rosenblad and Dr. Cynthia Finley, in the Geotechnical program. I would also like to express my gratitude to all the staffs working in the Civil Engineering department for their help and supports. I would like to thank Dr. Hani Salim, Dr. Douglas Smith, and other committee members for their valuable comments and suggestions for the dissertation. I would like to thank Dr. Bowders, who always has his door open whenever I need help, for his precious advice, teaching, and supports. I would also like to thank my friends, Cheng-Wei Chen, Awilda Blanco, Rafael Baltodano, Ramesh Bobba, Elisabeth Freeman, Alexandra Wayllace and other students in the geotechnical program for their precious friendship and support. Finally, I would like to thank my parents, my wife and her parents, and my sons for their invaluable support and caring for me during my study. Without their caring and support I will not be able to fulfill my study. ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS... ii ABSTRACT... iii LIST OF TABLES... iv LIST OF ILLUSTRATIONS...v Chapter Page 1. INTRODUCTION LITERATURE REVIEW Introduction Limit Equilibrium Analyses of Pile-Reinforced Slopes Uncoupled Analysis of Slope Reinforced with Piles Pressure-Based Method Displacement-Based Method Calculation of Factor of Safety of Piled-Slopes Methods for Predicting Limit Soil Pressure Ito and Matsui s Theoretical Method Broms Empirical Method Pan et al. 3-D Numerical Analysis Comparison of Limit Soil Pressure from Different Methods Summary and Conclusions TWO DIMENSIONAL, PLANE STRAIN FINITE ELEMENT MODEL Introduction...2 iii

5 3.2 Idealized Piled-Slope Model Mesh Convergence Distributions of Contact Stresses and Lateral Force on Piles Results from 2-D, Plane Strain Finite Element Analyses Effect of Initial States of Stress and Interface Roughness Effect of Sliding Depth Effect of s/d Ratio Effect of Dilation Angle Effect of Soil Constitutive Model Effect of Loading Simulation Loading and Failure Patterns Development of Yielding Patterns Contours of Vertical Stress (Out-of-Plane Stress) Summary and Conclusions THREE-DIMENSIONAL THIN HORIZONTAL SLICE FINITE ELEMENT MODEL Introduction D Model of a Horizontal Slice Distribution of Contact Normal and Shear Stresses Results from 3-D Model of a Horizontal Slice Effect of Initial States of Stress and Interface Roughness Effect of Sliding Depth Effect of s/d Ratio Effect of Dilation Angle...55 iv

6 4.4.5 Effect of Soil Constitutive Model Effect of Loading Simulation by Applying Displacements at Different Boundaries Effect of Loading Simulation by Horizontal Body Load Loading and Failure Patterns Development of Yielding Patterns Contours of Vertical Stress Deformed Shape Summary and Conclusions INTERPRETATION OF RESULTS FOR 2-D, PLANE-STRAIN AND 3-D THIN HORIZONTAL SLICE FINITE ELEMENT MODELS Introduction Interpretation of Results for 2-D, Plane-Strain and 3-D Thin Horizontal Slice Effect of Initial Stress Condition, Interface Roughness and s/d Ratio Effect of Dilation Angle Stress Paths Summary and Conclusions THREE-DIMENSIONAL FINITE ELEMENT MODEL INCLUDING THE WHOLE SLIDING ZONE Introduction D Model Including the Whole Sliding Zone Distribution of Contact Normal and Shear Stresses Results from 3-D Model Including the Whole Sliding Zone Effect of Boundary Conditions...88 v

7 6.4.2 Effect of Initial States of Stress and Interface Roughness Effect of Sliding Depth and s/d Ratio Effect of Pile Inclination Failure Patterns from 3-D Model Including the Whole Sliding Zone Development of Yielding in the Soil Mass Contours of Vertical Stress Summary and Conclusions COMPLETE THREE-DIMENSIONAL FINITE ELEMENT MODEL Introduction Complete 3-D Finite Element Model of Piled-Slope Distribution of Contact Normal and Shear Stresses Results from the Complete 3-D Finite Element Model Results for Loading by Displacing Both the Right and Left Boundaries Results for Loading by Applying Horizontal Body Load Failure Patterns from 3-D Model Including the Whole Sliding Zone Summary and Conclusions COMPARISON AND INTERPRETATION OF RESULTS FOR THE 3-D MODEL INCLUDING THE SLIDING ZONE ONLY AND THE COMPLETE 3-D MODEL Introduction Comparison and Interpretation of the Results Limit Soil Pressure in the Sliding Zone Lateral Load Response in the Sliding Zone vi

8 8.2.3 Axial Load Response in the Sliding Zone Summary and Conclusions SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK Introduction Conclusions Recommendations for Future Work REFERENCES VITA vii

9 LIST OF TABLES Table Page 3.1 Soil properties used for the Mohr-Coulomb failure criterion Variables selected for parametric studies for each s/d ratio Parameters used for the Extended Cam-Clay model Soil properties used for the Mohr-Coulomb failure criterion Parameters used for the Extended Cam-Clay model Summary of factors affecting limit lateral force on piles Soil properties used for the Mohr-Coulomb failure criterion Variables selected for parametric studies for vertical piles Variables selected for parametric studies for inclined piles Soil properties used for the Mohr-Coulomb failure criterion Variables selected for loading simulated by prescribed displacements Variables selected for loading simulated by applying horizontal body load viii

10 LIST OF ILLUSTRATIONS Figure Page 1.1 Maximum lateral force acting on pile as a function of s/d Effect of adding reinforcing members to increase stability of a slope Limit lateral resistance curves due to different failure modes Graphical illustration of method for computing limit soil resistance: (a) integral of limiting soil pressure, and (b) equivalent total resisting force Graphical illustration for computing limit anchorage resistance: (a) integral of limiting soil pressure, and (b) equivalent total resisting force Relative displacements between the pile and soil (top figures) and their corresponding distributions of bending moment, distributed load, and limiting pressure with depth generated (bottom figures) by due to different failure modes: a) Flow mode, b) Intermediate mode, c) Short Pile mode, and d) Long Pile mode. (from Hull et al. 1991) Shear force induced at sliding depth as a function of the normalized sliding depth by the pile length at the ultimate state. (from Hull et al. 1991) Ito and Matsui s state of plastic deformation in the ground around the piles lateral force acting on pile as a function of s/d (Loehr et al. 24) (a) Plan, and (b) profile views of a slope reinforced with one row of piles Two-dimensional, plane strain finite element model (a) Finer mesh, and (b) coarser mesh generated to study mesh convergence Force induced versus displacement at midpoint between piles Contact normal and shear stresses at the interface at ultimate state Lateral forces on pile versus displacement for s/d = 4 at z s = 3-m Maximum force on pile as a function of depth for s/d = Maximum force on pile as a function of s/d at z s = 3-m Influence of dilation angle on reaction force on pile for s/d = ix

11 3.1 Effects of soil constitutive model on maximum lateral force on pile Effects of other loading simulations on maximum lateral force on pile Development of yielding pattern for s/d=4, z s =3-m, K o =.5, and µ=.577 at end displacements of a) 1-mm, b) 3-mm, c) 8-mm, d) 11-mm, and e) 19.1-mm Development of yielding pattern for s/d=4, z s =3-m, K o =.5, and µ=. at end displacements of a) 1-mm, b) 4-mm, c) 8-mm, d) 11-mm, and e) 18-mm Contours of vertical stress for s/d=4, z s =3-m, K o =.5, and µ=.577 at end displacements of a) -mm, b) 1-mm, and c) 19.1-mm a) Plan and b) schematic view of 3-D horizontal slice model for s/d= Comparison of results with full and reduced integration formulations Contact normal and shear stresses at ultimate state along the a) top edge, b) middle, and c) bottom edge of the pile-soil interface Transverse distance in the y-direction from the center of the pile (plan view) Load-displacement response for s/d=4 at depth of 3-m below ground Load-displacement response for s/d=2 at depth of 3-m below ground Axial force-displacement response for s/d=4 at a depth of 3-m below ground Maximum lateral force on pile as a function of depth for s/d= Maximum lateral force as a function of s/d at 3-m depth below ground Influence of dilation angle on lateral force on pile for s/d= Load-displacement response for s/d = 4 and K o = 1. at sliding depths of a) 3-m, b) 6-m, and c) 9-m below ground Maximum lateral force as a function of s/d at 6-m below ground with extended Cam-clay model when moving both ends at same magnitude and rate Loading simulation performed by a) displace both the left and right boundaries, b) displace only the right boundary, and c) displace only the left boundary Effects of other loading simulations by displacing either left or right boundary while keeping the other fixed...62 x

12 4.15 Load-displacement response for loading simulations by applying horizontal body load for s/d=4 at a sliding depth of 3-m Maximum force as a function of s/d obtained by applying horizontal body load at a sliding depth of 3-m Schematic view of the progression of yielding for s/d=4 at end displacements of a) 6-mm, b) 1-mm, c) 18-mm, and d) 58-mm Bottom view of the yielding pattern for s/d=4 at the ultimate state Progression of yielding for s/d=2, K o =1., and µ=.577 at end displacements of a) 36-mm, b) 5-mm, c) 2-mm Contours of vertical stress for s/d=4, z s =3-m, K o =1., and µ=.577 at end displacements of 58-mm (9-kPa <σ z <295-kPa) Deformed shape of soil elements in proximity of the pile at the ultimate state Comparing the maximum lateral force on pile as a function of s/d at 3-m depth below ground from a) 2-D, plane-strain model, and b) 3-D horizontal slice model Comparison of failure mechanism for s/d of 2 from a) 2-D, plane-strain model, and b) 3-D horizontal slice model Comparing the effect of dilation angle on p-δ behavior on pile for s/d of 4 with rough interface at 3-m depth from a) 2-D, plane-strain model, and b) 3-D model Stress paths of elements near the pile for s/d of 4 with rough interface at 3-m depth from a) 2-D, plane-strain model, and b) 3-D model a) Plan and b) schematic view of 3-D model including the whole sliding zone for s/d= Contact normal and shear stresses at the ultimate state along the pile-soil interface at a) 1-m, b) 2-m, c) 3-m, and d) 4-m below ground surface Transverse distance in the y-direction from center of pile (plan view) Distributions of contact a) normal stress, and b) shear stress at the ultimate state along the length of pile on the upstream side of the pile...87 xi

13 6.5 Influence of upstream and downstream boundary conditions on P T -δ response for sliding depths of a) 4-m and b) 6-m Effect of initial states of stress and interface roughness on P T -δ response for s/d=8 and z s =2-m Distribution of limit lateral force per unit length of pile for sliding depths of a) z s =2-m, b) z s =4-m, and c) z s =6-m at s/d= P T,max as a function s/d, for sliding depths of a) 2-m, b) 4-m, and c) 6-m Sign convention of the pile inclination angle Effect of pile inclination on lateral load transfer behavior Effect of pile inclination on axial load transfer behavior Distribution of limit lateral force per unit length of pile at a) 1-mm displacement, and b) the ultimate state for s/d=4, K o =.5, and µ=.577 at z s =2-m Distribution of limit axial force and per unit length of pile at a) 1-mm displacement, and b) the ultimate state for s/d=4, K o =.5, and µ=.577 at z s =2-m Development of yielding pattern for s/d=2, z s =4-m, K o =.5, and µ=.577 at end displacements of a).2-mm, b) 2-mm, c) 2-mm, d) 5-mm, and e) 78-mm Comparing the yielding pattern of z s =4-m, K o =.5, and µ=.577 at the ultimate state for a) s/d=2, b) s/d=4, and c) s/d= Displacement vectors of z s =4-m, K o =.5, and µ=.577 at the ultimate state for a) s/d=2, b) s/d=4, and c) s/d= Comparing the yielding pattern for s/d=4, z s =2-m, K o =.5, and µ=.577 at the ultimate state for a) θ=+3 o, b) θ=+15 o, c) θ=, d) θ=-15 o, and 3) θ=-3 o Contours of vertical stress at the bottom layer of s/d=4, z s =2-m, K o =.5, and µ=.577 for a) θ=+3 o, b) θ= o, c) θ=-3 o at the ultimate state a) Plan and b) profile view of a complete 3-D model for s/d = Contact normal and shear stresses at the ultimate state along the pile-soil interface at a).5-m, b) 1-m, c) 1.5-m, and d) 2-m below ground surface Transverse distance in the y-direction from center of pile (plan view) xii

14 7.4 Distributions of contact a) normal stress, and b) shear stress at the ultimate state along the length of pile on the upstream side of the pile in the sliding zone The (a) total lateral load and (b) total axial load as a function of displacement for s/d=8, K o =.5, z s =2-m, and µ= The (a) soil pressure (b) total lateral load, and (c) bending moment as a function of depth for s/d=8, K o =.5, z s =2-m, and µ=.577 at the ultimate state The (a) soil pressure (b) total lateral load, and (c) bending moment as a function of depth for s/d=8, K o =.5, z s =4-m, and µ=.577 at the ultimate state The (a) lateral load response, and (b) axial load response for s/d=8, K o =.5, z s =2- m, and µ= The (a) soil pressure, and (b) total lateral load as a function of depth for s/d=8, K o =.5, z s =2-m, and µ=.577 at the ultimate state Development of yielding pattern for s/d=8, z s =4-m, K o =.5, and µ=.577 at end displacements of a) 9-mm, b) 5-mm, and c) 122-mm Distribution of soil pressure with depth for s/d=8, K o =.5, z s =4-m, and µ=.577 at end displacements of a) 9-mm, b) 5-mm, and c) 122-mm Comparison of the limit soil pressure in the sliding zone for s/d=8, K o =.5, and µ=.577 and for a) z s =2-m, and b) z s =4-m Comparison of the lateral load response in the sliding zone for s/d=8, K o =.5, and µ=.577 and for a) z s =2-m, and b) z s =4-m Comparison of the axial load response in the sliding zone for s/d=8, K o =.5, and µ=.577 and for a) z s =2-m, and b) z s =4-m Mobilized axial load in the sliding zone for s/d=8, K o =.5, µ=.577 and z s =4-m from a) to 1-m, b) 1 to 2-m, c) 2 to 3-m, and d) 3 to 4m below ground...14 xiii

15 CHAPTER 1: INTRODUCTION Prediction of loading on piles in slopes reinforced using piles (piled-slopes) is important in order to properly calculate the stability of piled-slopes. One of the major concerns in evaluating stability of piled-slopes is the limit soil pressure that can be developed at the pile-soil interface, or the pressure at which the soil will fail by flowing around or between piles. Most current methods used to calculate the stability of pileslopes are uncoupled analyses, where the limit soil pressure is obtained using an analytical, empirical or numerical method and, subsequently, the limit soil pressure obtained is used as an additional resistance in slope stability analysis using limit equilibrium methods. Unfortunately, the available methods produce inconsistent results. The loading on piles predicted using the currently available theories may differ significantly or may be similar depending on conditions as shown in Figure 1.1. Figure 1.1 shows the predicted maximum lateral force acting on the pile as a function of the pile spacing to pile diameter ratio (s/d) by two common methods: Ito and Matsui (1975) and Broms (1964). The magnitude of loading on a pile may be 3 to 4 percent off between these two theories depending on the s/d ratio used for the piled-slope. Coupled analysis of piled-slopes using the three-dimensional (3-D) elasto-plastic shear strength reduction finite element method was proposed by Cai and Ugai (2). Although the shear strength reduction method in which the pile response and slope stability are considered simultaneously is very powerful, this method requires extensive training in the use of finite element software and requires long analysis run-time (hours or days depending on the problem sizes) due to the 3-D nature of the problem and the nonlinearity of the pile-soil interaction problem. In addition, it requires a lot of training 1

16 in the 3-D finite element software prior to constructing a good 3-D finite element model for a piled-slope problem. Therefore, it is not very practical at this point in time to use this technique for the design and stability analysis of piled-slopes Broms 1 Force / Unit Length on Pile (kn/m) Ito & Matsui Force / Unit Length on Pile (kip/ft) s/d Figure 1.1 Maximum lateral force acting on pile as a function of s/d. The hypothesis of this research work is that 3-D analyses are required to accurately model the complex soil-structure interaction associated with stabilizing piles in slopes, and to accurately predict both working and limit loads on such piles. Much previous research work to investigate the 3-D aspects of pile-soil interaction has utilized simplified 2-D, plane strain analyses of a horizontal section, but that limitations of this approach are expected to produce significant errors especially under drained loading conditions. The primary objective of the research work is to use numerical analyses to evaluate the limit soil pressure on the pile, to evaluate the load transfer mechanism for vertical and inclined piles under fixed and free rotating conditions, and to evaluate 2

17 alternative means for estimating mobilized and limit loads on piles for piled-slope problems under drained conditions. ABAQUS, a commercial finite element software, was used for all the finite element analyses performed in this research work. Parametric studies were performed to investigate the effects of different variables on loaddisplacement behavior of piles in slopes reinforced with one row of piles. In Chapter 2, a summary of previous research on methodologies for uncouple analyses of piled-slope problems and methods of predicting the limit soil pressure on piles in piled-slopes is presented. The results from 2-D, plane strain finite element analyses and from analyses using a 3-D thin horizontal slice finite element model are presented in Chapter 3 and Chapter 4, respectively. In Chapter 5, the interpretation of the finite element analysis results from 2-D and thin 3-D models is discussed and conclusions about the significance of model dimensionality on the load transfer behavior in piledslope problems are presented. The results from the 3-D finite element analyses including only the whole sliding zone, but not the anchorage zone are presented in Chapter 6. In Chapter 7, the results from the complete 3-D finite element analyses of the piled-slope problems, including both the sliding and anchorage zones, are presented. The interpretation of the finite element analysis results from Chapters 6 and 7 is discussed and conclusions about the importance of using the complete 3-D finite element model in understanding the load transfer mechanism in piled-slopes are presented in Chapter 8. Finally, a summary of the work performed for this dissertation is presented in Chapter 9 along with recommendations for future work to further the understanding of load transfer mechanisms in piled-slopes. 3

18 CHAPTER 2: LITERATURE REVIEW 2.1 Introduction In this chapter, limit equilibrium analyses of earth slopes reinforced with piles are briefly discussed. In addition, different uncoupled analysis methods proposed in the literature for piled-slopes are reviewed followed by discussions of the available methods found in the literature to predict the limit soil pressure provided by stabilizing piles. Comparison of the limiting soil resistance using both Ito and Matsui (1975) and Broms (1964b) equations are made followed by conclusions at the end of this chapter. 2.2 Limit Equilibrium Analyses of Pile-Reinforced Slopes Inclusion of reinforcing members or piles in an unstable slope increases the overall stability of the slope by providing a direct resisting force to sliding. Figure 2.1 conceptually shows the contribution of reinforcement using piles. In limit equilibrium calculations, a limiting resistance force per unit width, F r, at the sliding surface provided by the reinforcing member is added to the other internal forces of the slice it is intersecting. This additional resistance is then included in the equilibrium equations to satisfy static equilibrium for the calculation of a factor of safety for the reinforced slope. It is commonly assumed that F r provides only transverse or shear resistance at the sliding surface. In reality, reinforcing members may provide not only a transverse resistance component at the sliding surface, but also an axial resistance component due to frictional resistance at the pile-soil interface when soil slides along a pile in the vertical direction. However, there is still uncertainty about the nature in which load is transferred to the reinforcing members and about the likelihood of simultaneously mobilizing both 4

19 the shear and axial resistance of reinforcing members. Since it is reasonable to assume that the main loading mechanism in a reinforcing member used for stabilizing a slope is lateral load, it is commonly assumed that the transverse force is mobilized first and is normally considered alone in the limit equilibrium analyses (Parra, 24). Reinforcing Member Slope Crest Slope Toe Assumed sliding surface resistance "Fr" provided by reinforcement Figure 2.1 Effect of adding reinforcing members to increase stability of a slope. 2.3 Uncoupled Analysis of Slope Reinforced with Piles Uncoupled analysis method can be classified into two categories. The first category is the pressure-based method as proposed by De Beer and Wallays (197), Ito et al. (1981), Hassiotis et al. (1997), and Loehr et al. (24). The second category is the displacement-based method proposed by Hull et al. (1991), Poulos (1995b), Lee et al. (1995), and Jeong et al. (23). Since methods proposed in each category are similar, the author selected only one method from each category for discussion in the following. The proposed methods by Loehr et al. (24) from the first category and Hull et al. (1991) from the second category were selected for the following discussion. These methods are developed using an 5

20 uncoupled approach in which the pile response and slope stability are considered separately Pressure-Based Method Loehr et al. (24) developed a basic analysis methodology for predicting the resistance provided by individual slender reinforcing members for slope stabilization using pressure-based method. The method uses a limit state design approach wherein a series of potential failure modes are considered to develop the distribution of limiting resistance forces along a reinforcing member. The three general limit states due to different failure modes that are considered include: 1) failure of soil around or between reinforcing members referred to as limit soil resistance, 2) failure of soil due to insufficient anchorage length members referred to as limit anchorage resistance, and 3) structural failure of the reinforcing members in bending or shear due to loads applied from the soil mass referred to as limit member resistance. In this method, separate limit resistance curves are developed for each limit state as illustrated in Figure 2.2. From these individual curves, a composite limit resistance curve that corresponds to the most critical component of resistance at each sliding depth is established by taking the component with the least resistance at each sliding depth. The resulting composite limit resistance curve obtained in this manner is shown by the shaded region in Figure

21 1 Limit Resistance Per Unit Width of Slope (lb/ft) Limit member resistance (Failure mode 3) Sliding Depth, z (ft) Composite limit resistance Limit anchorage resistance (Failure mode 2) Limit soil resistance (Failure mode 1) 8 Figure 2.2 Limit lateral resistance curves due to different failure modes. Calculation of the limit soil resistance requires that the lateral pressure on the pile at which failure of the soil will occur be known. This pressure is called the limit soil pressure, p u. Loehr et al. (24) and many others have selected the method proposed by Ito and Matsui (1975), because it is generally believed to be conservative, to obtain the limit soil pressure which in turn used to calculate the limit soil resistance. The development of the limit lateral soil resistance distribution proposed by Loehr et al. (24) is described in the following. The limit state considered for the limit soil resistance is failure of the soil above the sliding surface by flow around or between reinforcing members. The limit soil pressure is the pressure that will cause the soil to fail laterally at a particular depth. If it is assumed that this load can be simultaneously mobilized along the length of the reinforcing member above the sliding surface, the total limit resistance based on failure of soil above the sliding surface (deemed failure mode 1) is obtained by integrating the 7

22 computed limit soil pressure over the length of reinforcement above the sliding surface as shown in Figure 2.3. For stability analysis, this total limit resistance force is assumed to act at the sliding surface (Figure 2.3b). Since the sliding surface may in general pass through any point on the reinforcing member, additional points on the limit resistance curve are computed by repeating the integration for different sliding depths to establish a complete limit resistance curve describing the total resistance for failure mode 1 as a function of sliding depth as shown in Figure 2.2. For failure mode 1, the total resistance increases from a minimum value at the ground surface to a maximum value at the tip of the member. Since stability analyses are generally performed for cross-sections of unit width, the total resisting forces computed by integrating the limit soil pressure are divided by the longitudinal spacing to produce values of the limit force per unit width of slope suitable for stability analyses. total resistance Sliding surface P limiting lateral soil pressure, p u(z) equivalent force Sliding surface (a) z (b) Figure 2.3 Graphical illustration of method for computing limit soil resistance: (a) integral of limiting soil pressure, and (b) equivalent total resisting force. The second limit state considered is the one in which reinforcing members have insufficient anchorage length beyond the sliding surface to provide passive resistance that 8

23 is equal to or greater than that provided by the soil above the sliding surface. If passive failure of the soil below the sliding surface is assumed to be governed by the same limit soil pressure, p u, as used for failure mode 1, a similar procedure can be used to calculate the limiting anchorage resistance. The resisting force provided by the length of the reinforcing element extending below the sliding surface is obtained by integrating the limiting soil pressure, p u, over the length of the reinforcing element extending from the sliding surface to the tip of the member as shown by the shaded zone in Figure 2.4. It is again assumed that the full limiting soil pressure can be mobilized over the entire length of reinforcing member below the sliding surface. The total resisting force for a particular sliding depth is again replaced with an equivalent force for stability analysis (Figure 2.4b) and the complete limiting resistance distribution for the anchorage limit state is calculated by computing the total resisting force for different sliding depths (Figure 2.2). As shown in Figure 2.2, the limiting resistance for failure mode 2 increases from zero for a sliding surface passing through the lower end of the reinforcement to a maximum for very shallow sliding surfaces. (a) Sliding surface limiting lateral soil pressure below sliding surface total resistance (b) equivalent force due to limiting soil pressure below sliding surface Sliding surface Figure 2.4 Graphical illustration for computing limit anchorage resistance: (a) integral of limiting soil pressure, and (b) equivalent total resisting force. 9

24 The approach used to account for the potential of the reinforcing member to fail structurally, deemed failure mode 3, is to consider a factored lateral soil pressure of the form p' ( z) = α p u ( z) (2.1) where p (z) is a factored pressure and p u (z) is the limit soil pressure. The unknown factor α is the factor that will produce a distribution of shear or moment such that the maximum shear or moment just equals the shear or moment capacity of the reinforcing member, respectively. To establish the magnitude of α for a particular sliding depth, elastic analyses are first performed to establish the distribution of shear and moment in the member when subjected to the limit soil pressures either above or below the sliding surface. Considering moments, α is then approximated as M ult α = (2.2) M max where M max is the maximum moment determined from elastic analyses and M ult is the moment capacity of the member. While Equation 2.2 is strictly an approximation, results of analyses performed to date indicate that it produces acceptably precise values for α. The factor is then applied to the limiting soil pressures (Equation 2.1) to determine the factored pressures to avoid structural failure of the reinforcing member in bending and the limit resistance is computed using the factored pressure distribution in a manner similar to that used for the other limit states considered. Since the distribution of moment, and the maximum moment, are functions of the sliding depth, the factor α must also be a function of the sliding depth so the process is repeated for different sliding depths to develop a limit resistance curve for failure mode 3 as shown in Figure

25 2.3.2 Displacement-Based Method Hull et al. (1991) investigated the behavior of a pile in an unstable slope using a displacement-based method. The pile response, shear force at sliding depth, maximum bending moment in the pile, and deflection of the pile, when subjected to external lateral soil movements from slope instability is analyzed using a modified boundary element method developed by Poulos (1973). This method takes into consideration the relative displacements between the soil and the pile, the relative stiffness of the pile and soil, as well as the finite length of pile. Using the modified boundary element method, the computer program PALLAS has been developed where loading on a pile may be lateral head loading and loading generated by soil movements in an elastic soil possessing a non-uniform stiffness with depth and with non-linear pile-soil interface elements (Hull et al. 1991). From the analyses of pile response due to lateral soil movement at different sliding depths for a fixed pile length, four types of failure modes, shown in Figure 2.5, can be identified from the analyses as discussed by Hull et al. (1991). These include: 1) Flow mode of failure when the pile is strong enough and the sliding depth is shallow enough so that the unstable sliding soil merely passed around the pile as shown in Figure 2.5a. This is equivalent to failure mode 1 described by Loehr and Bowders (23), and Loehr et al. (24); 2) Short pile failure mode when the depth of sliding is deeper, of similar depth to the pile length, and again the pile is sufficiently strong to withstand the generated bending moments and shear forces as shown in Figure 2.5c. This is equivalent to failure mode 2 described by Loehr and Bowders (23), and Loehr et al. (24); 11

26 3) Intermediate mode of failure this failure mode occurs at sliding depths intermediate to those of the Short and Flow modes, where the soil strength in both the unstable and the stable soil is fully mobilized everywhere along the pile length as shown in Figure 2.5b; and 4) Long pile failure mode in which one or more position along the pile are found to have attained the yield moment and then developed so-called plastic hinges as shown in Figure 2.5d. This is equivalent to failure mode 3 described by Loehr and Bowders (23), and Loehr et al. (24). Soil Movement Soil Movement Soil and Pile Movement Soil and Pile Movement Pile Pile deflection Limiting Pressure Distributed load Distributed load Bending moment Bending moment Distributed load Bending moment Figure 2.5 Relative displacements between the pile and soil (top figures) and their corresponding distributions of bending moment, distributed load, and limiting pressure with depth generated (bottom figures) by due to different failure modes: a) Flow mode, b) Intermediate mode, c) Short Pile mode, and d) Long Pile mode. (from Hull et al. 1991). 12

27 From the computer program (PALLAS) the responses of the pile at any slide depth in a piled-slope can be determined. Figure 2.6 shows the shear force induced at sliding depth as a function of the normalized sliding depth by the pile length at the ultimate state. The shear force at the sliding depth increases at an increasing rate with increasing sliding depth up to approximately 45 percent of the pile length. For all sliding depths less than the 45 percent of the pile length, Zone I in Figure 2.6, the resistance of the soil in the sliding zone is fully mobilized and the soil in the sliding zone flows past the pile ( flow failure mode), while the pile remains fixed in the stable soil without fully mobilizing the resistance of the soil in the anchorage zone. Between the sliding depths of approximately 45 percent and 85 percent of the pile length, Zone II in Figure 2.6, the shear force remains relatively unchanged from 45 percent to 6 percent of the pile length, and then increases rapidly from 6 percent to 85 percent of the pile length. Within the Zone II, the soil resistance in both the stable and unstable zones is fully mobilized. It should be stresses that this intermediate failure mode may be accompanied by large rotations of the pile in order to achieve full mobilization of the soil resistance in both sliding and anchorage zones. Below the sliding depths of 85 percent of the pile length, Zone III in Figure 2.6, the shear force decreases rapidly to zero at the pile tip. Within the Zone III, the short pile failure mode exists with the soil resistance fully mobilized in the anchorage zone, without mobilizing its full resistance in the sliding zone. This mode is virtually an inversion of the flow failure mode. 13

28 Figure 2.6 Shear force induced at sliding depth as a function of the normalized sliding depth by the pile length at the ultimate state. (from Hull et al. 1991) Calculation of Factor of Safety of Piled-Slopes Once the overall limit lateral soil resistance distribution is developed for individual members, either by pressure-based methods or displacement-based methods, the mechanics of stability analysis for reinforced slopes is relatively straightforward and well established. Regardless of the method used to develop the limiting soil resistance, the key to limit equilibrium method for slopes reinforced with structural members is an accurate estimation of the limiting soil pressure acting against the stabilizing piles. This is because the reaction force generated in the reinforcing members is directly affected by the magnitude of limiting lateral pressure developed in the soil. Several alternative methods have been proposed for predicting the limit soil pressure for stabilizing piles as discussed in the following section. 14

29 2.4 Methods for Predicting Limit Soil Pressure At present, there are several methods to compute limit soil pressure. Many were developed for individual piles based on empirical relations from field lateral load tests, while others were based on theoretical models and numerical models. The theoretical limit soil pressure developed by Ito and Matsui (1975), the empirical limit soil pressure proposed by Broms (1964), and the limit soil pressure investigated by Pan et al. (22) using three-dimensional finite element analysis are selected for the following discussions Ito and Matsui s Theoretical Method Ito and Matsui (1975) present a method to predict the limit soil pressure on stabilizing piles aligned in a row based on the theory of plastic deformation. The method predicts the pressure at which the soil reaches a plastic limit state according to the Mohr- Coulomb yield criterion. The main assumptions of the method are: a) The soil becomes plastic in the shaded region AEBB E A around piles, as shown in Figure 2.7. b) No friction is assumed in surfaces EB and E B, and therefore stresses at pile-soil interface are considered principal stresses. c) Plain strain condition applies in the direction of depth. d) The sliding surfaces occur along the inner pile surfaces. e) Piles are rigid as compared to the soil, and aligned vertically. f) Stress distribution along failure surface is independent of friction on those surfaces. g) Horizontal ground surface lies behind and in front of the row of piles. 15

30 The method is based on calculation of the differential or net lateral pressure acting on a circular pile within a row of piles at constant center-to-center spacing. Net lateral pressures on a pile may vary from zero when there is no movement to limit pressure at large lateral deformations. Ito and Matsui s method predicts an intermediate value of net pressure between these two extremes, at the condition where the soil achieves a plastic state. The limit force per unit length of pile P(z) at any depth z where the soil is in a state of plastic equilibrium (Figure 2.7) is a function of the soil unit weight, γ, angle of internal friction, φ, cohesion intercept, c, pile center-to-center distance, D 1, and inner distance between pile faces, D 2. The equation that relates all these terms is: G ( φ) D 2N + 1 φ tanφ 1 D1 D2 ( ) G2 P( z) = cd 1 exp G3 φ 1 + D2 Nφ tanφ D2 G1 G 1( φ) γ z 2c + D1 D1 D2 D ( ) 1 1 exp G3 φ D2 Nφ 2 N D2 D φ 2 ( φ) ( φ) G2 c D1 G1 ( φ) ( φ) (2.3) where φ = tan 2 π + 4 φ = φ φ = 2 φ φ N, G ( φ ) N 2 tanφ + N 1, ( φ ) 2tanφ + N 2 + N 2 and ( φ ) = N tanφ tan π + G, G φ 3 φ. The limit soil pressure per unit area of pile face, p 8 4 u, is obtained by dividing the limit force computed from Equation 2.3 by the width of the pile. 16

31 Pile B Assumed Plastic Region A E D D 1 2 A' E' Direction of Deformation Pile B' Figure 2.7 Ito and Matsui s state of plastic deformation in the ground around the piles Broms Empirical Method Broms (1964a) and Broms (1964b) proposed equations to estimate the ultimate soil pressure on piles based on empirical data from lateral load tests for cohesive soil and cohesionless soil, respectively. On the basis of the measured and calculated lateral resistance of laterally loaded piles in cohesive soils (φ = ), Broms (1964a) assumed that the ultimate soil pressure is equal to zero from the ground surface to a depth of 1.5 pile diameters and equal to nine times the undrained shear strength of cohesive soil (p u = 9c u ) at greater depths. For the cohesionless soil (c = ), Broms (1964b) assumed that the ultimate soil pressure which develops at failure is equal to three times the passive Rankine earth pressure (p u = 3σ p ) at all depths Pan et al.3-d Numerical Analysis Pan et al. (22) performed 3-D finite element analysis to investigate the behavior of single piles subjected to lateral soil movements and to determine the ultimate soil pressure acting along the pile shaft. Their results indicated that the ultimate soil pressure 17

32 in cohesive soils, under undrained conditions, for a stiff pile is 1 times the undrained shear strength and for a flexible pile is 1.8 times the undrained shear strength of the soil. These values of ultimate soil pressures in cohesive soils agree well with those documented from the literature which is in the range between 9 to 12 times the undrained shear strength c u of the soil Comparison of Limit Soil Pressure from Different Methods As mentioned before, prediction of loading on piles in piled-slopes is important in order to properly calculate the stability of piled-slopes. There are many theories available to estimate the loading on piles in piled-slopes. Unfortunately, the methods produce inconsistent results (Loehr et al. 24). The loading on piles predicted using the currently available theories may differ significantly or may be similar depending on the problem as shown in Figure 2.8. Figure 2.8 shows the predicted maximum lateral force acting on the pile as a function of the pile spacing to pile diameter ratio (s/d) by the Ito and Matsui theory (1975) and the Broms (1964) empirical method. As shown, the magnitude of loading on the pile may differ by as much as 3 to 4 percent for these two theories depending on s/d ratio used for the piled-slope. Although 3-D finite element analysis to determine the ultimate soil pressure acting along the pile shaft in cohesive soils under undrained condition has been investigated by Pan et al. (22), there is no literature found, to the author s knowledge, on 3-D numerical analysis to determine the ultimate soil pressure for piled-slope in soil under drained conditions. 18

33 16 14 Broms 1 Force / Unit Length on Pile (kn/m) Ito & Matsui Force / Unit Length on Pile (kip/ft) s/d Figure 2.8 Maximum lateral force acting on pile as a function of s/d (Loehr et al. 24). 2.5 Summary and Conclusions Most current methods used to calculate the stability of pile-slopes are uncoupled analyses which can be classified into two categories: pressure-based methods and displacement-based methods. Both methods required prior knowledge of the limit soil pressures that can be developed at the pile-soil interface. The prediction of loading on piles in piled-slopes is important in order to properly calculate the stability of piledslopes. It is found that the loading on piles predicted using the currently available theories may differ significantly or may be similar depending on the problem. In the following chapters, 3-D displacement-based finite element analyses performed to investigate the load transfer mechanism and the ultimate soil pressure generated on the pile in piled-slopes under drained conditions are presented. It is believe that the finite element method will provide good insight into the 3-D nature of the complicated problem presented by slopes reinforced using structural members or piles. 19

34 CHAPTER 3: TWO-DIMENSIONAL, PLANE STRAIN FINITE ELEMENT MODEL 3.1 Introduction The first task of the research work is to investigate the complex interaction between piles and surrounding soils of piled-slope problems using a simplified twodimensional (2-D) displacement-based finite element (FE) model. It is assumed that plane strain conditions exist in the out-of-plane (vertical) direction where there are no movements of soil occurred in the vertical direction. The effects of initial in-situ stress condition, interface friction between piles and the soil surrounding piles, sliding depth, ratio of pile spacing to pile diameter (s/d), method of loading simulation, and constitutive model of the soil were studied as presented in the following sections. 3.2 Idealized Piled-Slope Model Figure 3.1 shows plan and profile views of an idealized prototype of piled-slope problems for a row of piles with diameter, d, and center-to-center spacing, s. It is assumed that sloping ground has little effect on the load-displacement behavior of the pile-soil system. Therefore, the prototype is simplified to a horizontal ground surface. In addition, uniform soil movement is assumed in the sliding zone as shown in Figure 3.1b. The problem shown in Figure 3.1 was modeled using a two-dimensional (2-D), plane strain FE model of a horizontal slice through piled-slopes in a manner similar to that used by Chen and Martin (22). In this model, symmetry boundary conditions were applied on the top and bottom boundaries of the mesh as shown in Figure 3.2. The left and right boundaries of the mesh were initially constrained in the x-direction to establish 2

35 initial states of stress under geostatic conditions. Two types of initial states of stress were considered: 1) hydrostatic states of stress with K o = 1., and 2) lateral stresses equal to one half of the vertical stress with K o =.5, where K o represents the coefficient of lateral earth pressure at rest. Loading on the piles was then applied by simultaneously moving both the left and right boundaries in the x-direction at the same magnitude and rate to simulate free-field soil movement; effects of moving only the left or only the right boundary were also evaluated. The pile-soil interface was modeled with contact elements using two different types of interface roughness conditions: 1) smooth with a coefficient of friction µ =., and 2) rough with µ =.577 (or tan 3 o ). These two interface conditions represent two extreme cases for the shear strength on the pile-soil interface: µ =. represents the lower bound case where is no available shear strength on the interface, and µ =.577 (or tan 3 o ) represents the upper bound case where the interface shear strength is equal to the shear strength of the soil (φ = 3 o ). Linear 4-node plane strain elements were used in the finite element analyses. Linear elements are recommended when dealing with contact problems (ABAQUS, 23). One method to model soil failure flowing through piles in piled-slope problem is by applying displacement on the soil elements and fixing the piles. In the 2-D, plane strain model, the piles were assumed rigid and constrained in all directions since the deformation of the piles was not taken into consideration in the model. The diameter of the piles, d, was chosen to be 1-m (3.28-ft). The width of the model from left to right was chosen to be 24-m (78.7-ft). Chen and Martin (22) verified that the boundary space of 1 times the pile diameter (1*d) was sufficient to represent an isolated pile without inducing group effects. 21

36 soil movement ground surface pile diameter, d pile spacing, s sliding depth, z s piles sliding surface 24-m upstream side pile downstream side (a) plan view (b) profile view Figure 3.1 (a) Plan, and (b) profile views of a slope reinforced with one row of piles. Left boundary Top boundary Right boundary y x Bottom boundary Pile (d=1-m) 24.-m Figure 3.2 Two-dimensional, plane strain finite element model. ABAQUS, a commercial FE software, was used to perform all analyses presented throughout this document. The soil constitutive model used for the FE analyses was the elasto-plastic model with a Mohr-Coulomb failure criterion unless otherwise specified. Young s modulus and Poisson s ratio of the soil were chosen to be 35.8-MPa (747.7-ksf) and.42, respectively. The soil was assumed to have a unit weight of 21-kN/m 3 (133.6-pcf) with an angle of internal friction, φ, and cohesion intercept, c, of 3 degrees and.1-kpa (2.1-psf), respectively. A small cohesion intercept of.1-kpa was chosen in order to make the problem stable numerically (cohesion intercept of zero is not allowed in the FE software used). Furthermore, it was observed that a cohesion intercept of.1-kpa (2.1-psf) or less basically resulted in the same load-displacement behavior while 22

37 a cohesion intercept of 1-kPa (21-psf) or greater did affect the load-displacement behavior at large displacements. A dilation angle, ψ, of zero was assumed for the Mohr- Coulomb model unless otherwise specified. The soil shear strength parameters used in the FE analyses are typical shear strength of soils under drained loading condition. The soil properties used are summarized in Table 3.1. Table 3.1 Soil properties used for the Mohr-Coulomb failure criterion. Description Symbol Magnitude and Unit Cohesion intercept c'.1-kpa Internal angle of friction φ ' 3 o Unit weight γ 21-kN/m 3 Young's modulus E 35.8-MPa Poisson's ratio ν.42 Dilation angle ψ o Table 3.2 shows the variables depth of sliding z s, initial stress condition K o, and interface friction coefficient µ, selected for the parametric studies of piled-slope problems for each s/d ratio. In addition, center-to-center spacing of pile to pile diameter ratios (s/d) of 2, 4, 8, and 2 were chosen to evaluate the influence of pile spacing on the lateral force induced on piles. The range of s/d evaluated is typical range of interest in practice for piled-slope problems. 23

38 Table 3.2 Variables selected for parametric studies for each s/d ratio. z s (m) K o µ Mesh Convergence Prior to all finite element analyses, mesh convergence studies were performed to make sure that the finite element mesh generated was appropriately refined so that the solution obtained would be acceptable. Figure 3.3 shows two meshes generated in the area close to the piles to study the convergence of the finite element mesh for s/d = 4, K o = 1., and µ =.577 at a sliding depth of 3-m (9.8-ft). Figure 3a has eighty soil elements in the region just next to the piles whereas Figure 3b has thirty two soil elements. The number of elements is distributed equally on both upstream and downstream sides of the pile. 24

39 upstream side downstream side upstream side downstream side (a) finer mesh (b) coarse mesh Figure 3.3 (a) Finer mesh, and (b) coarser mesh generated to study mesh convergence. Figure 3.4 shows a typical lateral force per unit length of pile, p, induced on the rigid piles versus displacement, δ, of a node between the piles from the two different meshes shown in Figure 3.3 while other variables remain constant. The p-δ response of the coarser mesh is plotted using a solid line while the finer mesh is plotted using a dashed line. It is shown in Figure 3.4 that the coarser mesh resulted in a slightly higher lateral force on pile but the difference is very small (about 2.5 % difference) and not significant. However, the computing time savings for the coarser mesh is significant compared to the finer mesh. The CPU time for the coarser mesh required only about 18 minutes while the finer mesh required about 64 minutes to complete the finite element analysis. Therefore, due to the advantages of CPU time savings of the coarser mesh over the finer mesh, it was decided that further analyses be done using the coarser mesh (Figure 3.3b) and at the same time the accuracy of the finite element results were not affected significantly as shown in Figure

40 Force / Unit Depth on Rigid Pile (kn/m) Coarser mesh Finer mesh Displacement at midpoint between piles δ (mm) Figure 3.4 Lateral Force induced versus displacement at midpoint between piles. 3.4 Distributions of Contact Stresses and Lateral Force on Piles Figure 3.5 shows typical contact normal stress (in directions normal to the pilesoil interface) and contact shear stress (in directions tangent to the pile-soil interface) at the pile-soil interface on the upstream and downstream sides of the pile at the ultimate state as a function of transverse distance from the center of the pile. Figure 3.5 shows the results for s/d = 4 at a depth of 3-m (9.8-ft) below ground with coefficient of lateral earth pressure at rest K o = 1. and interface friction µ =.577 (tan 3 o ). The transverse distance is the distance in the y-direction as shown on upper right corner of Figure 3.5. Contact normal stress is the greatest (about 46-kN/m 2 ) at the upstream side at y =.-m and it remains relatively constant up to y =.15-m from the pile center on the upstream side of pile where soil elements are moving towards the pile. The normal stress then decreases rapidly beyond y =.15-m to about a value of 15-kN/m 2 at 26

41 the edge of the pile at y =.5-m. On the other hand, shear stress on the upstream side is zero at y =.-m and increases slowly to its greatest value of about 135-kN/m 2 at y =.32-m. Then the shear stress decreases rapidly to a value of 8-kN/m 2 at the edge of the pile at y =.5-m. On the downstream side of the pile, normal and shear stress are close to zero from y =.-m to y =.45-m since soil elements are moving away the pile on the downstream side of the pile. The lateral force per unit length of pile, p, is the resultant force due to the contact normal and shear stresses at the pile-soil interface in the positive x-direction as shown in Figure 3.2. The lateral force is computed as two times the integral of the x-component of contact normal and shear forces on both the upstream and downstream sides of pile along the pile-soil interface. The two times is required since the resultant lateral force calculated is only half of its actual value due to the symmetry condition analyzed at the center line of the pile..6.5 Normal and shear stresses on "downstream" side "soil movement" Transverse distance, y Transverse Distance, y (m) "upstream" Normal stress on "upstream" side "downstream" y =.5 y =..1 Shear stress on "upstream" side Distribution of Contact Stresses (kn/m 2 ) Figure 3.5 Contact normal and shear stresses at the interface at ultimate state. 27

42 3.5 Results from 2-D, Plane Strain Finite Element Analyses Results from 2-D, plane strain FE analyses are presented in this section. The effect of initial states of stress and interface roughness on the p-δ response is first shown followed by the effect of sliding depth and s/d ratio on the maximum lateral force induced on the pile. After that, the effect of dilation angle, soil constitutive model, and loading simulation on the p-δ response is presented Effect of Initial States of Stress and Interface Roughness Figure 3.6 shows typical lateral force-displacement (p-δ) behavior of the pile-soil system for s/d = 4 at a sliding depth, z s, of 3-m (9.8-ft) below ground. The p-δ curve shown in Figure 3.6 is linear at small displacements. The rate of increase of lateral force on the pile then decreases at intermediate displacements. Eventually, the reaction force on pile reaches a peak value and remains unchanged at large displacements when the frictional strength of the soil is fully mobilized. Also shown in Figure 3.6 are the effects of interface roughness and initial states of stress on the p-δ response. It can be seen that interface roughness and initial stress conditions do not affect the initial stiffness of the p-δ curve at small displacements. However, the maximum lateral load on a pile is affected by both interface roughness and initial stress condition. The rough interface condition with K o = 1. (rough, K o = 1.) yields the largest ultimate lateral force on pile while the smooth interface condition with K o =.5 (smooth, K o =.5) yields the smallest ultimate lateral force on pile. The ratio of largest (rough, K o = 1.) to smallest (smooth, K o =.5) maximum lateral force on the pile 28

43 in this case is about 2. Similar behavior was observed for other sliding depths as well as other s/d ratios. 5 Force / Unit Length on Pile (kn/m) Rough, K o =1. Smooth, K o =1. Rough, K o =.5 Smooth, K o = Force / Unit Length on Pile (kip/ft) Displacement at midpoint between piles (mm) Figure 3.6 Lateral forces on pile versus displacement for s/d = 4 at z s = 3-m Effect of Sliding Depth The maximum lateral force on the pile for sliding depths of 3-m (9.8-ft), 6-m (19.7-ft) and 9-m (29.5-ft) below ground are plotted in Figure 3.7 for the case of s/d = 4. Also shown in Figure 3.7 are the limiting lateral forces predicted using the Ito-Matsui theory (Ito and Matsui, 1975) and the ultimate lateral force predicted using three times the Rankine passive pressure (Broms, 1964). The maximum lateral force on the pile induced by soil movement increases linearly with sliding depth. The maximum lateral force on the pile for the rough interface condition is about 25 percent higher than the maximum force on the pile with smooth interface condition for all initial stress conditions 29

44 shown. The maximum force on pile with initial stress condition of K o = 1. is about 7 percent higher than the maximum force on pile with initial stress condition of K o =.5 for all interface coefficients of friction. In this case of s/d = 4, using three times the passive pressure results in a maximum force that exceeds all computed results. Ito-Matsui s prediction falls between the results from the rough and smooth interfaces with initial stress condition of K o =.5. Sliding Depth, zs (m) Maximum Force on Pile/Unit Length (kn/m) Ito & Mats ui Rough, Ko=1. Smooth, Ko=1. Rough, Ko=.5 Smooth, Ko=.5 Broms Maximum Force on Pile/Unit Length (kip/ft) Figure 3.7 Maximum force on pile as a function of depth for s/d = Effect of s/d Ratio Figure 3.8 shows the maximum force that can be developed on the pile as a function of s/d ratio at a depth of 3-m (9.8-ft) below ground. For the rough interface condition, the maximum calculated force on the pile increases with increasing s/d from s/d = 2 to s/d = 8, and then remains unchanged for s/d > 8. For the smooth interface condition, the maximum force on the pile increases with increasing ratio from s/d = 2 to 3

45 s/d = 4, and then remains relatively unchanged for s/d > 4. For s/d 4, the maximum lateral force on the piles obtained from the FE model using 2-D, plane strain assumptions generally fall between the results predicted by the Ito and Matsui and Broms methods. Furthermore, the lateral force resulting from the rough interface condition with K o = 1. is closer to the ultimate lateral force predicted by the Broms method whereas the smooth interface with K o =.5 is closer to the Ito-Matsui theory for s/d > 4. Conversely, for s/d < 4, the maximum lateral forces on piles predicted by the Ito and Matsui and Broms methods are always greater than the results obtained from 2-D finite element analyses. The difference is more significant (greater than 2 percent) with decreasing s/d ratio. Similar behavior was observed for depths of 6-m (19.7-ft) and 9-m (29.5-ft). Force / Unit Length on Pile (kn/m) Broms Rough, Ko=1. Smooth, Ko=1. Rough, Ko=.5 Smooth, Ko=.5 3K Ito & Matsui s/d Force / Unit Length on Pile (kip/ft) Figure 3.8 Maximum force on pile as a function of s/d at z s = 3-m. 31

46 3.5.4 Effect of Dilation Angle The influence of dilation on the pile reaction force is shown in Figure 3.9, which shows the p-δ behavior for dilation angles, ψ, of and 3 degrees applied to the Mohr- Coulomb failure criterion. For the case where ψ =, the reaction force reaches a peak value after some displacement as previously shown in Figure 3.6. However, for ψ = 3 o (associated flow rule) the p-δ response is a strain hardening type with no discernable peak reaction force observed even at large displacements. Thus, dilation has a substantial effect on the maximum lateral force on the pile from the 2-D, plane strain finite element analyses; the ultimate force for the soil model with dilation angle ψ = 3 o is more than 3 percent greater than that from the soil model with ψ =. Similar behavior was observed for K o =.5 with smooth and rough pile-soil interfaces Rough, K o =1., ψ =3 o 9 8 Force / Unit Length on Pile (kn/m) Smooth, K o =1., ψ =3 o Rough, K o =1., ψ = Smooth, K o =1., ψ = Force / Unit Length on Pile (kip/ft) Displacement at midpoint between piles (mm) Figure 3.9 Influence of dilation angle on reaction force on pile for s/d = 4. 32

47 3.5.5 Effect of Soil Constitutive Model The dilatancy predicted using the Mohr-Coulomb yield criterion with an associated flow rule (ψ = φ) is generally unrealistic because it over predicts observed dilatancy for actual soils. Overconsolidated soils do dilate but not so much as predicted by the associated flow rule (Naylor and Pande, 1981). Therefore, a more realistic soil model that can better model the dilative behavior of soil the Modified Cam-clay model was used to evaluate whether similar response would be obtained. The parameters for the Modified Cam-clay model used are M = 1.2 (or φ' critical = 3 o ), λ =.87, and κ =.26, where M is the ratio of shear stress, q, to mean effective stress, p, at the critical state; λ and κ are, respectively, the slopes of the isotropic normal compression line and the recompression line. The isotropic normal compression line is anchored at e 1 = 1.36, where e 1 is the void ratio at the intercept of the isotropic normal compression line λ at lnp =. (or p = 1.). The soil parameters used for the extended Cam-Clay model are summarized in Table 3.3. Table 3.3 Parameters used for the Extended Cam-Clay model. Description Symbol Value Critical stress ratio M 1.2 slope of isotropic normal compression λ.87 slope of isotropic recompression compression κ.26 Void ratio at lnp' = e Figure 3.1 shows the comparison of p-δ response obtained from FE analyses using different soil constitutive models for s/d = 4, K o = 1., and µ =.577 (rough interface) at a sliding depth of 3-m. It is shown that only the p-δ curve using Mohr- Coulomb with ψ = exhibits an obvious peak lateral force. In contrast, the p-δ curves 33

48 obtained using other constitutive models show no clear peak values for the lateral force on the pile with increasing displacement even at very large displacements. The rate of increase of the lateral force on the pile increases with increasing overconsolidation ratio. Figure 3.1 also shows that all final lateral forces obtained using the Modified Cam-clay model are less than the peak lateral force obtained from the Mohr-Coulomb failure criterion with ψ =. This is believed to be a result of convergence issues when using the Modified Cam-Clay model rather than actual physical behavior. At this stage, FE analyses performed using Modified Cam-clay soil constitutive model were terminated due to difficulties in satisfying the convergence criteria caused by relatively large distortion of some elements at large displacements. If the analyses are allowed to continue, the lateral forces induced from the Modified Cam-clay model are believed likely to surpass the peak lateral force induced from the Mohr-Coulomb soil model with no dilation. The initial stiffness of the p-δ curves for the Modified Cam-clay failure criterion is less than the initial stiffness of the p-δ curves for the Mohr-Coulomb failure criterion. The Young s modulus is constant for the Mohr-Coulomb failure criterion whereas the bulk modulus for the Modified Cam-clay failure criterion is dependent on the mean effective confining stress in the soil mass. The difference in the initial stiffness of the two failure criteria is thus attributed to that fact that the bulk modulus used is different. 34

49 Force / Unit Length on Pile (kn/m) M-C, ψ =3 o M-C, ψ = MCC, OCR =8 MCC, OCR =2 MCC, NC Displacement at midpoint between piles (mm) Figure 3.1 Effects of soil constitutive model on maximum lateral force on pile Effect of Loading Simulation So far, all results in previous figures were obtained by displacing both the right ( downstream ) and left ( upstream ) model boundaries from left to right at the same magnitude and rate. The p-δ behavior for other loading conditions are shown in Figure 3.11, where loading is simulated by displacing only the left or the right boundary in the horizontal direction (or x-direction) while keeping the other end fixed. Figure 3.11 shows the results for s/d = 4, K o = 1., and µ =.577 (rough interface) at a sliding depth of 3-m. Figure 3.11 shows that if the loading is simulated by only displacing the right ( downstream ) boundary while fixing the left ( upstream ) boundary, the maximum lateral force induced on the pile is substantially less than the maximum lateral force induced by moving both boundaries at the same time. For loading simulated by displacing only the left ( upstream ) boundary while fixing the right ( downstream ) boundary, the lateral force increases without reaching a peak value with increasing 35

50 displacement. The results show that the p-δ behavior is significantly affected by the boundary conditions applied in the 2-D, plane-strain model. It is believed that movement of both boundaries simultaneously is the most representative of actual field loading condition. 25 Force / Unit Length on Pile (kn/m) Right, Rough, K o =1. Left, Rough, K o =1. Both, Rough, K o = Displacement at midpoint between piles (mm) Figure 3.11 Effects of other loading simulations on maximum lateral force on pile. 3.6 Loading and Failure Patterns In this section, loading and failure patterns from the 2-D, plane strain FE model are presented. The development of yielding patterns in the soil mass is shown followed by the changes in contour plots of vertical stress as the soil mass is moving through the piles in piled-slope problems Development of Yielding Patterns Figure 3.12 shows the development of yielding in the soil mass, when the Mohr- Coulomb failure criterion is used as the constitutive model for the soil elements with ψ = 36

51 , for end displacements of 1-mm (.4-in) to 19.1-mm (.75-in). Figure 3.12 shows the graphical yield pattern for s/d = 4, K o =.5, and µ =.577 at a sliding depth of 3-m (9.8- ft) when both the right ( downstream ) and left ( upstream ) model boundaries are moving simultaneously from left to right at the same rate and magnitude. Figure 3.12a shows that at very small end displacements of 1-mm some soil elements on the downstream side of the pile have begun to yield (shown by dark region) due to an active loading condition as the soil mass on the downstream side is moving away from the pile. At end displacement of 3-mm, soil elements on the upstream side begin to yield and the yielding in the downstream side spreads to a larger zone as shown in Figure 3.12b. At end displacement of 8-mm the whole downstream side of pile has yielded and the yielding on the upstream side spreads further due to the arching effect transferring pressure from the yielding mass of soil onto adjacent stationary parts (Terzaghi, 1943), as shown in Figure 3.12c. Figure 3.12d shows that the strength of the arch in the upstream side has been completely mobilized at end displacement of 11-mm. There is also an elastic zone, named arching foothold by Chen and Martin (22), on the upstream side of the pile as shown in Figure 3.12d. Once the strength along the arch is fully mobilized, most of the soil elements between the piles that previously yielded come back to the elastic region as shown in Figure 3.12e. This region is called elastic arching zone by Chen and Martin (22). Similar behaviors were observed for other s/d ratios, other sliding depths, as well as other initial stress conditions with rough or smooth pile-soil interfaces. However, for the smooth pile-soil interface condition, smaller influence zone due to arching and no arching foothold were observed at large displacements as shown in Figure

52 a) at end displacement of 1-mm b) at end displacement of 3-mm c) at end displacement of 8-mm Arching foothold d) at end displacement of 11-mm Elastic arching zone Arching foothold e) at end displacement of 19.1-mm Figure 3.12 Development of yielding pattern for s/d=4, z s =3-m, K o =.5, and µ=.577 at end displacements of a) 1-mm, b) 3-mm, c) 8-mm, d) 11-mm, and e) 19.1-mm. 38

53 a) at end displacement of 1-mm b) at end displacement of 4-mm c) at end displacement of 8-mm c) at end displacement of 11-mm e) at end displacement of 18-mm Figure 3.13 Development of yielding pattern for s/d=4, z s =3-m, K o =.5, and µ=. at end displacements of a) 1-mm, b) 4-mm, c) 8-mm, d) 11-mm, and e) 18-mm. 39

54 3.6.2 Contours of Vertical Stress (Out-of-Plane Stress) Figure 3.14 shows contours of vertical normal stress σ z (out-of-plane stress) on the soil mass for end displacements of -mm, 1-mm and 19.1-mm. The contour plots are for s/d = 4, K o =.5, and µ =.577 at sliding depth of 3-m. Both right and left ends are moving simultaneously at the same rate and magnitude from left to right. Prior to any movement, the vertical stress is equal to 63-kPa everywhere in the soil mass at 3-m below ground surface as shown in Figure 3.14a (constant σ z everywhere). The sign convention is positive for compressive stress and negative for tensile stress. The darker zones on the left ( upstream ) side of the pile indicate higher compressive stress. Figure 3.14b shows only the contours ranged from 63-kPa to 74-kPa, however, the actual range is from 42-kPa to 74-kPa. Therefore, in Figure 3.14b, the white area represents vertical stresses ranging from 42-kPa to 63-kPa. The intention of the contour plots is to show that as the soil mass begins to move, the soil elements on the downstream side of pile become less compressive (less than 63-kPa) while the soil elements on the upstream side of pile become more compressive (more than 63-kPa). Figure 3.14c shows the contours ranged from 63-kPa to 28-kPa and the actual range is from to 28-kPa. At large displacements the vertical stresses of a portion of soil elements on the upstream side also become less compressive than the initial compressive stress as shown in Figure 3.14c (white area on upstream side). Furthermore, the vertical compressive stresses on the downstream side of pile become almost zero. If we compare Figure 3.14c with Figure 3.12e, at the same state, it is noticed that generally to the right of the arch on the upstream side there is a tendency for reduction in vertical compressive stresses; on contrary, the vertical compressive 4

55 stresses on the left hand side of the arch, on the upstream side, tend to increase. Similar behaviors were observed for other cases. a) at beginning (no displacement), σ z =63-kPa b) at end displacement of 1-mm, 42-kPa< σ z <74-kPa c) at end displacement of 19.1-mm, -.5-kPa< σ z <28-kPa Figure 3.14 Contours of vertical stress for s/d=4, z s =3-m, K o =.5, and µ=.577 at end displacements of a) -mm, b) 1-mm, and c) 19.1-mm. 3.7 Summary and Conclusions A series of 2-D, plane strain, displacement-based FE analyses was performed to investigate the lateral load transferred to piles in slopes reinforced using piles. Parametric studies were performed to study the effects of initial stress condition, interface roughness, depth of sliding, and pile-spacing to pile-diameter ratio (s/d) on the load 41

56 transfer mechanism in pile-reinforced slopes. In addition, the effects of soil constitutive model and loading simulation by displacing different boundaries on the maximum lateral load on pile in piled-slopes were also investigated. It is found that initial stress condition has more effect on the maximum lateral force on the pile than the interface roughness for the 2-D, plane-strain finite element model. The maximum lateral force per unit length of pile increases linearly with increasing depth of sliding. The maximum lateral force on the pile increases with increasing s/d up to s/d =8 for the rough interface condition, and up to s/d = 4 for the smooth interface condition, and then remains unchanged with increasing s/d ratio. The loading condition simulated by applying displacements at different boundaries of the model was found to have significant effects on the load transfer mechanism of piledslopes. In addition, it was shown that the vertical stresses σ z changed significantly due to soil movements in the 2-D, plane strain FE model. At the ultimate state, the vertical stresses σ z increased significantly on the upstream side of the pile, whereas the vertical stresses σ z decreased significantly on the downstream side of the pile. In actual field condition, the vertical stresses σ z should remain relatively constant as the soil mass moves in the sliding zone. Therefore, it is an appropriate question to ask if the 2-D, plane-strain finite element model in the horizontal direction can actually be used to accurately model the pile-soil interaction in slopes reinforced with piles. This question is investigated in more detail in Chapter 4 by using a more complicated 3-D finite element model. 42

57 CHAPTER 4: THREE-DIMENSIONAL THIN HORIZONTAL SLICE FINITE ELEMENT MODEL 4.1 Introduction The second task of the research work was to investigate the complex interaction between piles and surrounding soils of piled-slope problems using a 3-D horizontal slice of unit thickness with three-dimensional (3-D) displacement-based finite element (FE) models. For these analyses, it is assumed that a constant and uniform pressure from the weight of the soil mass above the sliding depth is applied at the top surface of the thin 3- D FE model. The effects of initial in-situ stress condition, interface friction between piles and soil, ratio of pile spacing to pile diameter (s/d), method of loading simulation, and constitutive model of the soil were studied as presented in the following sections D Model of a Horizontal Slice A 3-D model of a horizontal slice of unit thickness in the vertical- or z-direction (1-m thick) is shown in Figure 4.1 for an s/d ratio of 4 in plan and schematic views. The size of the FE model from the left to the right boundaries, 2x b, was chosen to be 24-m, where x b is the distance from the center of the pile to either the right or the left boundary. The chosen size, x b = 12-m, is far enough to represent an isolated pile without inducing group effects for the thin horizontal slice model. The left ( upstream ) and right ( downstream ) boundary conditions were constrained in the x-direction at initial state and then prescribed displacements were applied in the positive x-direction, from the left to the right, to simulate mass movement of soil in an unstable slope condition. Symmetry boundary conditions were applied at the upper and lower symmetry planes of the FE 43

58 model as shown in Figure 4.1. In the plane strain case described in Chapter 3, the top and bottom surfaces were constrained such that no expansion or contraction (no strains) is possible. However, in the 3-D model, an appropriate pressure was applied on the top surface of the 3-D model to simulate the vertical stress at a particular depth in the field. The vertical stress was kept constant throughout the whole analysis; the bottom surface was constrained in the vertical direction (z-direction). The pile was modeled as rigid and constrained in all directions. Left ( upstream ) boundary Upper symmetry boundary Right ( downstream ) boundary Lower symmetry boundary Pile (d=1-m) y x 2*x b = 24.-m a) plan view Left boundary ( upstream ) Top surface Upper symmetry plane z y Right boundary ( downstream ) x Lower symmetry plane Pile (d=1-m) Bottom b) schematic view Figure 4.1 a) Plan and b) schematic view of 3-D horizontal slice model for s/d=4. The soil constitutive model used for the FE analyses was the elasto-plastic model with a Mohr-Coulomb failure criterion and zero dilation angle (ψ = ) unless otherwise specified. Soil elements are assumed to be homogeneous and isotropic. The soil 44

59 properties used is the same as described in Chapter 3 and is summarized in Table 4.1. The pile-soil interface was modeled with contact elements using two different types of interface roughness conditions: 1) smooth with a coefficient of friction µ =., and 2) rough with µ =.577 (or tan 3 o ). Table 4.1 Soil properties used for the Mohr-Coulomb failure criterion. Description Symbol Magnitude and Unit Cohesion intercept c'.1-kpa Internal angle of friction φ ' 3 o Unit weight γ 21-kN/m 3 Young's modulus E 35.8-MPa Poisson's ratio ν.42 Dilation angle ψ o 3-D, 8-node linear solid (continuum) elements were used with reduced integration scheme for the 3-D finite element model to analyze piled-slope problems. Figure 4.2 shows the p-δ response resulting from using full and reduced integration formulations to model the soil elements. It is shown in Figure 4.2 that there is no significant difference between the results obtained using the full and reduced integration formulations. Therefore, the reduced integration formulation was selected in order to save computation time without significantly affecting the analysis results. 45

60 Force / Unit Depth on Rigid Pile (kn/m) Full Integration Reduced Integration Displacement at M idpoint Between Piles (mm) Figure 4.2 Comparison of results with full and reduced integration formulations. 4.3 Distribution of Contact Normal and Shear Stresses Figure 4.3 shows typical contact normal and shear stresses at the ultimate state for horizontal sections along the top edge, along the middle, and along the bottom edge of the models, where the contact normal stress is the stress acting perpendicular to the pilesoil interface and the contact shear stress is the total tangential stress at the interface. The direction of the resultant normal force is always perpendicular to the pile-soil interface, but the direction of the resultant shear force varies depending on the slip direction of the soil and pile surfaces. The contact normal and shear stresses shown in Figure 4.3 are plotted as a function of transverse distance from the center of the pile in the y-direction, as shown in Figure 4.4. Figure 4.3 shows the results for s/d = 4 at a depth 46

61 of 3-m (9.8-ft) below ground surface with coefficient of lateral earth pressure at rest, K o = 1., and interface friction µ =.577 (or tan3 o ). The contact normal stress on the upstream side of pile at all levels (top, middle, and bottom) is the greatest at y =.-m (at the symmetry plane) and it remains relatively constant up to about y =.18-m from the pile center. The normal stress, then, decreases rapidly beyond y =.18-m. Figures 4.3a and 4.3b also show that the shear stress along the top edge and middle of the pile-soil interface remains relative constant up to y =.35-m for the top edge and y =.45-m for the middle portion, before a sharp drop in its value. On the contrary, the shear stress along the bottom edge of the interface is zero at y =. and then increases rapidly from y =. to y =.1-m. It then remains relatively constant up to y =.45-m before dropping off. This is because at the bottom surface of the model the nodes are constrained in the vertical- or z-direction; on the top surface the node are not constrained in any direction but there is a constant pressure from the overburden pressure of the soil mass applied in the vertical direction. Therefore, all nodes except the bottom edge nodes on the interface are allowed to slide relative to the rigid pile in the vertical direction, and consequently shear stresses are induced. The contact normal and shear stresses at the interface on the downstream side are negligible as compared to the upstream side. 47

62 Transverse Distance, y (m) Transverse Distance, y (m) Transverse Distance, y (m) Shear stress "downstream" side Normal stress "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (a) along the top edge of the interface Normal and shear stresses "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (b) along the middle of the interface Normal and shear stresses "downstream" side Shear stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (c) along the bottom edge of the interface Normal stress "upstream" side Figure 4.3 Contact normal and shear stresses at ultimate state along the a) top edge, b) middle, and c) bottom edge of the pile-soil interface. 48

63 Transverse distance, y y =.5 y =. Figure 4.4 Transverse distance in the y-direction from the center of the pile (plan view). 4.4 Results from 3-D Model of a Horizontal Slice The results from the 3-D horizontal slice FE analyses are presented in this section. The effect of initial states of stress and interface roughness on the load-displacement response is first shown followed by the effect of sliding depth (z s ), and s/d ratio on the maximum lateral force induced on the pile. After that, the effect of dilation angle, soil constitutive model, and loading simulation on the p-δ response is presented Effect of Initial States of Stress and Interface Roughness Figure 4.5 shows the p-δ response obtained for the 3-D model of a horizontal slice for s/d = 4 at a depth of 3-m (9.8-ft) below ground surface, where p is the lateral force per unit length of the pile and δ is the displacement at mid-point between two piles. It can be seen in Figure 4.5 that initial stress condition affects the stiffness of the p-δ curves at small displacements. However, the initial stress condition does not affect the maximum lateral force generated on the pile at large displacements. The effect of initial stress condition on the stiffness is more pronounce when the interface roughness is smooth (µ = ); in contrast, when the interface is rough, the initial states of stress in the soil has no significant effect on the load-displacement response, whereas the initial states of stress 49

64 has significant effect on the load-displacement response for the 2-D, plane strain model (see Figure 3.6). The maximum lateral force induced on the pile is affected by the interface roughness as shown in Figure 4.5. The maximum lateral force is about 25 percent higher for piles with the rough interface condition compared to piles with the smooth interface condition. Figure 4.5 also shows that p continues to increase at a small rate with increasing soil displacement at large displacements. Similar behavior was observed for s/d 4 at sliding depths of 6-m (19.7-ft) and 9-m (29.5-ft). On the contrary, different load-displacement behavior was observed for s/d = 2 as shown in Figure 4.6. Force / Unit Length on Pile (kn/m) Rough, Ko = 1. Rough, Ko =.5 Smooth, Ko = 1. Smooth, Ko = Displacement at midpoint between piles (mm) Force / Unit Length on Pile (kip/ft) Figure 4.5 Load-displacement response for s/d=4 at depth of 3-m below ground. Figures 4.6 shows the p-δ response for s/d = 2 at 3-m (9.8-ft) depth below ground surface of the 3-D model. Figure 4.6 shows that the initial stress conditions do not affect the p-δ response of the pile-soil system when the piles are closely spaced. The pile-soil 5

65 interface roughness also has no significant effect on the maximum lateral force generated on piles. For s/d = 2, the p-δ response also levels out and remains unchanged at large displacements for piles with the rough and smooth interface conditions. The maximum lateral force generated on the pile for the rough pile-soil interface condition is only about 6 percent larger than the maximum force on pile generated for the smooth pile-soil interface condition. Similar behavior was observed for sliding depths of 6-m (19.7-ft) and 9-m (29.5-ft) in the case of s/d = 2. 5 Force / Unit Length on Pile (kn/m) Rough, Ko = 1. Rough, Ko =.5 Smooth, Ko =.5 Smooth, Ko = Displacement at midpoint between piles (mm) Force / Unit Length on Pile (kip/ft) Figure 4.6 Load-displacement response for s/d=2 at depth of 3-m below ground. Results obtained using the 3-D model of a horizontal slice for piled-slope problems indicate that not only lateral force is generated while soil is flowing through the piles but that axial force can also be generated at the pile-soil interface. Since nodes on the soil surface are allowed to slide in the vertical direction, significant axial force is induced at the pile-soil interface for the rough interface condition as shown in Figure

66 No axial force is induced at the pile-soil interface for the smooth interface condition. Figure 4.7 shows that the axial force-displacement response is similar to that of the lateral force-displacement response where the axial force increases rapidly at small lateral soil movements and remains relatively constant at large soil movements. However, the peak axial load is mobilized at about 5-mm, which is slightly after the peak lateral load is mobilized at about 25-mm (see Figure 4.5). The initial stress condition also does not have any noticeable affect on the axial force-displacement response. Similar behavior was observed for sliding depths of 6-m (19.7-ft) and 9-m (29.5-ft) as well as for other s/d ratios. 8 Axial Force / Unit Length on Pile (kn/m) Rough, Ko = 1. Rough, Ko = Axial Force / Unit Length on Pile (kip/ft) Displacement at midpoint between piles (mm) Figure 4.7 Axial force-displacement response for s/d=4 at a depth of 3-m below ground Effect of Sliding Depth Figure 4.8 shows the computed maximum lateral force induced on the pile at the ultimate state from soil movement passing through the piles at sliding depths of 3-m 52

67 (9.8-ft), 6-m (19.6-ft) and 9-m (29.5-ft) below ground for the case of s/d = 4. Also shown in Figure 4.8 are the limiting lateral forces predicted using the Ito-Matsui method (Ito and Matsui, 1975) and the ultimate lateral force predicted using 3 times the Rankine passive pressure, 3σ p (Broms, 1964). The maximum lateral force on the pile increases linearly with increasing sliding depth as predicted by available theories. The maximum lateral force on the pile for the rough interface condition is about 25 percent higher than the maximum force on the pile with the smooth interface condition for all initial stress conditions shown. It is shown in Figure 4.8 that the initial stress condition in the soil has no influence of the maximum lateral force generated on the pile. In the case of s/d = 4, using Broms method (3σ p ) yielded the highest maximum lateral force on pile while Ito- Matsui s prediction yielded the lowest maximum lateral force on pile. The maximum lateral force generated from FE models falls between these two extreme predictions. Sliding Depth, zs (m) Maximum Lateral Force on Pile/Unit Length (kn/m) Ito & Mats ui Rough, Ko=1. Smooth, Ko=1. Rough, Ko=.5 Smooth, Ko=.5 Broms Maximum Force on Pile/Unit Length (kip/ft) Figure 4.8 Maximum lateral force on pile as a function of depth for s/d=4. 53

68 4.4.3 Effect of s/d Ratio Figure 4.9 shows the maximum lateral force acting on the pile as a function of s/d at 3-m (9.8-ft) below ground. The maximum lateral force induced increases with increasing s/d ratios up to s/d = 4; the maximum force then remains unchanged with increasing s/d ratio for both the rough and smooth interface conditions. The initial states of stress do not affect the maximum lateral force induced on the pile for all s/d ratios. Figure 4.9 also shows that the interface roughness has more pronounced effect on the maximum lateral force for s/d 4. However, the interface roughness has little effect on the maximum lateral force for s/d = 2. For s/d 4, the maximum lateral forces obtained from all FE analyses fall between these the Ito-Matsui theory and 3 times passive pressure. For s/d < 4, both methods tend to over predict the maximum lateral force acting on the pile. This observation is attributed to differences in failure modes for closely spaced piles as compared to more widely spaced piles as discussed in Section 4.5. Similar behavior was observed for sliding depths of 6-m (19.7-ft) and 9-m (29.5-ft) below ground. 54

69 Force / Unit Length on Pile (kn/m) Broms Ito & Matsui Rough, Ko=1. Rough, Ko=.5 Smooth, Ko=1. Smooth, Ko= s /d Force / Unit Length on Pile (kip/ft) Figure 4.9 Maximum lateral force as a function of s/d at 3-m depth below ground Effect of Dilation Angle The effect of dilative soil behavior on the mobilized lateral force on the pile is shown in Figure 4.1 for s/d = 4 for models with rough pile-soil interfaces, hydrostatic initial stresses (K o = 1.), and a sliding depth of 3-m (9.8-ft). Figure 4.1 compares the p-δ response when dilation angles of (ψ = ) and 3 degrees (ψ = φ) are applied for the Mohr-Coulomb failure criterion. For ψ = 3 ο, the lateral force initially increases rapidly at small displacements and then keeps increasing at a small rate even at very large displacements. Figure 4.1 also shows that the dilation angle does not affect the initial stiffness of the p-δ curve at small displacements; however, it has a noticeable effect on the maximum lateral force induced on the pile at large displacements. 55

70 Force / Unit Length on Pile (kn/m) Rough, Ko = 1., ψ = 3 o Rough, Ko = 1., ψ = Displacement at midpoint between piles (mm) Force / Unit Length on Pile (kip/ft) Figure 4.1 Influence of dilation angle on lateral force on pile for s/d= Effect of Soil Constitutive Model The constitutive model used to model the soil behavior for all analyses presented previously has been an elasto-plastic model with a Mohr-Coulomb failure criterion and associated (ψ = φ) and non-associated (ψ = ) flow rules. As mentioned in Section 3.5.5, the dilatancy predicted by using the Mohr-Coulomb yield criterion with associated flow rule is unrealistic because it over predicts the dilatancy behavior of actual soil. The extended Cam-clay model, which more accurately models the dilatancy of actual soils, was therefore used to investigate the effect of constitutive model of soil on the lateral force generated on piles in piled-slopes. The parameters for the Modified Cam-clay model used are M = 1.2 (or φ' critical = 3 o ), λ =.87, and κ =.26, where M is the ratio of shear stress, q, to mean effective stress, p, at the critical state; λ and κ are, 56

71 respectively, the slopes of the isotropic normal compression line and the recompression line. The isotropic normal compression line is anchored at e 1 = 1.36, where e 1 is the void ratio at the intercept of the isotropic normal compression line λ at lnp =. (or p = 1.). The soil parameters used for the extended Cam-Clay model are summarized in Table 4.2 and they are identical to the extended Cam-Clay model parameters used for the 2-D, plane strain model. The results shown in the following were obtained by moving both the left and right model boundaries at the same magnitude and same rate. Table 4.2 Parameters used for the Extended Cam-Clay model. Description Symbol Value Critical stress ratio M 1.2 slope of isotropic normal compression λ.87 slope of isotropic recompression compression κ.26 Void ratio at lnp' = e Figure 4.11 shows the p-δ response obtained from FE analyses using the extended Cam-Clay model for s/d = 4 and K o = 1. at a sliding depths of 3-m, 6-m, and 9-m. Figure 4.11 shows that for the smooth interface condition, p increases at decreasing rate with increasing displacement, and levels out at large displacements. On the contrary, for the rough interface condition, p increases at decreasing rate with increasing displacement up to some displacement depending on the sliding depth and then keeps increasing at a constant rate even at large displacements with no peak values for the lateral force at the end of the FE analyses. Figure 4.11 also shows that the interface roughness does not affect the initial slope of the p-δ curve at small displacements. 57

72 18 16 Force / Unit Length on Pile (kn/m) Rough, K o = 1. Smooth, K o = Displacement at midpoint between piles (mm) a) z s = 3-m Force / Unit Length on Pile (kn/m) Rough, K o = 1. Smooth, K o = Displacement at midpoint between piles (mm) b) z s = 6-m 18 Force / Unit Length on Pile (kn/m) Rough, K o = Displacement at midpoint between piles (mm) c) z s = 9-m Smooth, K o = 1. Figure 4.11 Load-displacement response for s/d = 4 and K o = 1. at sliding depths of a) 3-m, b) 6-m, and c) 9-m below ground. 58

73 The computed maximum lateral force per unit length of the pile is plotted as a function of s/d ratio at a sliding depth of 6-m (19.7-ft) for the 3-D model of a horizontal slice using the extended Cam-clay plasticity model in Figure The computed maximum lateral force is taken as the peak value or the last point on the p-δ curve if the curve exhibits no peak value as shown in Figure Figure 4.12 shows that the initial state of stress has little effect on the maximum force on the pile as was the case for the previous 3-D models using the Mohr-Coulomb failure criterion. In contrast, the interface roughness was found to have an even greater influence, about 2 percent difference, on the maximum force for s/d 4 than was observed for models using the Mohr-Coulomb model with no dilation (Figure 4.9). For s/d = 2, the maximum force was again found to be essentially constant regardless of the initial state of stress and interface roughness used in the model, an observation that is attributed to a passive type of failure mode for closely spaced piles as discussed in more detail in Section Results of analyses using the extended Cam-Clay model also produced a different trend for the effect of s/d when the pile-soil interface is smooth. For smooth interfaces, the maximum force on the pile decreases from s/d of 2 to 4 and then remains unchanged with increasing s/d. This trend is contradictory to trends observed for other models, which show increases in the maximum force with increasing s/d between 2 to 4. It is also notable that, when the pile-soil interface is taken as rough, the maximum force obtained from finite element analyses compares closely with the ultimate forces predicted by Broms method. Conversely, when the pile-soil interface is smooth, the maximum force obtained from finite element analyses is closer to the limit force predicted using the Ito- Matsui theory, especially at s/d = 4. 59

74 Force / Unit Length on Pile (kn/m) Broms Ito & Mats ui Rough, Ko=1. Rough, Ko=.5 Smooth, Ko=1. Smooth, Ko= s /d Force / Unit Length on Pile (kip/ft) Figure 4.12 Maximum lateral force as a function of s/d at 6-m below ground with extended Cam-clay model when moving both ends at same magnitude and rate Effect of Loading Simulation by Applying Displacements at Different Boundaries All results shown in previous figures were obtained from simulations performed by displacing both the right ( downstream ) and left ( upstream ) model boundaries, from left to right, at the same magnitude and the same rate as shown in Figure 4.13a. Other loading methods were also evaluated. One loading method was simulated by displacing the right ( downstream ) model boundary while keeping the left ( upstream ) boundary fixed as shown in Figure 4.13b. Another method for loading was to displace the left ( upstream ) model boundary while keeping the right ( downstream ) model boundary fixed as shown in Figure 4.13c. The results of these analyses are shown in Figure Figure 4.14 shows the results for s/d = 4, K o = 1., and µ =.577 at a sliding depth of 3-m (9.8-ft). The figure shows that for the loading simulated by displacing the 6

75 right ( downstream ) model boundary while fixing the left ( upstream ) model boundary, the maximum lateral force induced on pile is substantially less than the maximum lateral force induced by moving both the left ( upstream ) and right ( downstream ) model boundaries at the same time. For loading simulated by displacing the left ( upstream ) model boundary while fixing the right ( downstream ) model boundary, the lateral force increases to a peak value, close to the maximum lateral force generated when displacements are applied at both boundaries, prior to declining to a final value that is approximately 16 percent less than its peak value. This phenomenon is discussed in more detail in Section y x Prescribed Displacement Prescribed Displacement a) displace both the left and right model boundaries Fixed x-direction y x Prescribed Displacement b) displace only the right model boundaries y x Fixed x-direction Prescribed Displacement c) displace only the left model boundaries Figure 4.13 Loading simulation performed by a) displace both the left and right boundaries, b) displace only the right boundary, and c) displace only the left boundary. 61

76 Force / Unit Length on Pile (kn/m) Move Left, Fixed Right, Rough, Ko=1 Move Right, Fixed Left, Rough, Ko=1 Move Both, Rough, Ko= Force / Unit Length on Pile (kip/ft) Displacement at midpoint between piles (mm) Figure 4.14 Effects of other loading simulations by displacing either left or right boundary while keeping the other fixed Effect of Loading Simulation by Applying Horizontal Body Load Another loading method considered was the application of horizontal body load in the soil mass ( load controlled method). The body load was applied in the positive x- direction to simulate gravitational loading on the pile in a sloping ground condition. The constraints at both the left ( upstream ) and right ( downstream ) boundaries were released after obtaining the desired geostatic stress conditions while an appropriate pressure, which was equal to the initial lateral stress conditions in the x-direction, was applied on both the upstream and downstream model boundaries. The results of finite element analyses for this type of loading are presented in the following. Figure 4.15 shows the p-δ response obtained for s/d = 4 at a depth of 3-m (9.8-ft) below ground surface. It is shown in the figure that p increases at decreasing rate with increasing displacement and remains unchanged at large displacements. The initial stress condition and interface roughness do not affect the initial stiffness of the p-δ curves at 62

77 small displacements. On the contrary, the initial stress condition has significant effect on the maximum lateral force generate on the pile. Figure 4.15 also shows that for K o =.5, the interface roughness affects the stiffness of the p-δ curve at larger displacement but does not have significant effect on the maximum lateral force on the pile. For K o = 1., the interface roughness has effect on both the stiffness at larger displacement and the maximum lateral force on the pile. 4 Force / Unit Length on Pile (kn/m) Rough, Ko=1. Smooth, Ko=1. Rough, Ko=.5 Smooth, Ko= Displacement at midpoint between piles (mm) Figure 4.15 Load-displacement response for loading simulations by applying horizontal body load for s/d=4 at a sliding depth of 3-m. Figure 4.16 shows the maximum force on the pile as a function of s/d ratio for sliding depth of 3-m (9.8-ft) below ground. The figure shows that the initial stress condition has a significant effect on the maximum lateral force on the pile. For K o = 1., interface roughness has an effect on the maximum lateral force on the piles for all s/d ratios except for s/d = 2. On the contrary, for K o =.5, interface roughness has little effect on the maximum lateral force on the pile. 63

78 Figure 4.16 also shows that for K o = 1., the maximum lateral force on the pile increases with increasing s/d ratio from s/d = 2 to s/d = 8; the maximum lateral force then remains unchanged for s/d 8. For K o =.5, the maximum lateral force on piles increases over the range of s/d evaluated. 8 Force / Unit Length on Pile (kn/m) Broms Rough, Ko=1. Smooth, Ko=1. Rough, Ko=.5 Smooth, Ko=.5 Ito & Matsui Force / Unit Length on Pile (kip/ft) s /d Figure 4.16 Maximum force as a function of s/d obtained by applying horizontal body load at a sliding depth of 3-m. 4.5 Loading and Failure Patterns In this section, the development of yielding patterns in the soil mass is first presented followed by the changes in contour plots of vertical stress as the soil mass is moving through the piles in piled-slope problems. The deformed shape at the ultimate state is presented at the end of this section. 64

79 4.5.1 Development of Yielding Patterns Figure 4.17 shows the development of yielding in the soil mass at end displacements of 6-mm, 1-mm, 18-mm, and 58-mm (at the ultimate state) for s/d = 4, K o = 1., and µ =.577 at sliding depth of 3-m (9.8-ft). The results shown were determined by moving both the right and left model boundaries simultaneously at the same rate and magnitude in the positive x-direction (or 1-direction) as shown in Figure 4.13a. The progression of yielding from the 3-D thin horizontal slice model shown in Figure 4.17 is similar to that shown in Figure 3.12 for the 2-D, plane-strain model as discussed in Section However, the elastic arching zone (an elastic region between piles) and the arching foothold (a small elastic region on the upstream side of the pile), as shown in Figure 4.17d, are not as clear as those shown in Figure 3.12 for the 2-D, plane-strain model. However, if the yielding pattern shown in Figure 4.17d is viewed from the bottom as shown in Figure 4.18, the elastic arching zone and the arching foothold are clearer. This is because the nodes on the bottom surface are constrained in the z- or 3-direction (vertical-direction); thus the yielding pattern on the bottom surface for the 3-D thin horizontal slice model more closely resembles the 2-D, plane-strain model. Furthermore, since all nodes above the bottom surface are not constrained, they are allowed to displace in the any directions including the z-direction as a result the yielding pattern is not the same compared to the 2-D, plane-strain model. The difference is discussed in more detail in Section by examining the stress paths for the two models. Similar behavior was observed for other cases except for the smooth interface where no arching foothold was developed on the upstream side of the pile. 65

80 Top surface Direction of Soil movement Direction of Soil movement a) at end displacement = 6-mm b) at end displacement = 1-mm Elastic arching zone Arching Direction of Soil movement Direction of Soil movement Arching foothold c) at end displacement = 18-mm d) at end displacement = 58-mm Figure 4.17 Schematic view of the progression of yielding for s/d=4 at end displacements of a) 6-mm, b) 1-mm, c) 18-mm, and d) 58-mm. Direction of Soil movement Elastic arching zone Arching foothold Bottom view Figure 4.18 Bottom view of the yielding pattern for s/d=4 at the ultimate state. Figure 4.19 shows the progression of yielding of the 3-D thin horizontal slice models for s/d = 2 at end displacement of 36-mm, 5-mm, and 2-mm (at the ultimate state). Figure 4.19a shows that at small displacements, a passive wedge failure condition is observed for the soil elements on the left hand ( upstream ) side of the pile. 66

81 Figure 4.19b shows that with increasing end displacements, the yielding spreads to a larger extend on the upstream side, and the passive wedge failure pattern is also observed. At large displacements (at the ultimate state), although the passive wedge failure pattern is not obvious in region close to the pile on the upstream side, it is observed that the yielding of the soil extends further in the upstream direction as shown in Figure 4.19c. Typically, the yielding pattern for s/d 4 is the arching type as shown in Figure 4.18d. Whereas, the yielding pattern observed for the s/d = 2, closely spaced piles, is the wedge type on the upstream side of pile as shown in Figure It is observed that for closely spaced piles, it basically behaves like a continuous wall with passive wedge failure conditions for soil on the upstream of the piles Contours of Vertical Stress Figure 4.2 shows a contour plot of the vertical stress, σ z, ranging from 9-kPa to 295-kPa for s/d = 4, z s = 3-m, K o = 1., and µ=.577 (rough interface condition) at end displacements of 58-mm (at the ultimate state). The sign convention is positive for compressive stress and negative for tensile stress. For soil elements away from the pile, the vertical stress, σ z, generally remains relatively constant at 63-kPa (the initial σ z value). However, for the soil elements in the vicinity of piles, the vertical stress changes significantly. For the soil elements next the pile on the downstream side of the pile, the vertical stress reduces substantially at the ultimate state, while the vertical stress increases substantially for the soil on the upstream side of the pile. The soil elements on the lower left hand ( upstream ) side of the pile show increase in vertical stress from 63-kPa at the initial state to about 295-kPa at the ultimate state; on the lower right hand 67

82 ( downstream ) side of the pile the vertical stress decreases from 63-kPa at the initial state to about 9-kPa at the ultimate state. Similar behavior was observed for all other analyses. Direction of Soil movement Passive wedge failure Passive wedge failure a) at end displacement = 36-mm Direction of Soil movement Passive wedge failure b) at end displacement = 5-mm Direction of Soil movement Passive wedge failure c) at end displacement = 2-mm Figure 4.19 Progression of yielding for s/d=2, K o =1., and µ=.577 at end displacements of a) 36-mm, b) 5-mm, c) 2-mm. 68

83 Top surface Direction of Soil movement Figure 4.2 Contours of vertical stress for s/d=4, z s =3-m, K o =1., and µ=.577 at end displacements of 58-mm (9-kPa <σ z <295-kPa) Deformed Shape Figure 4.21 shows the deformed shape of the model in the proximity of the pile at the ultimate state for s/d = 4, z s = 3-m, K o = 1., and µ =.577. It can be seen that the soil elements on the left hand ( upstream ) side of the pile tend to move upward and expand while the soil elements on the right hand ( downstream ) side of the pile tend to move downward and contract. There was no gap formed in the pile-soil interface even at very large displacements. Similar behavior was observed for all other analyses. This deformation behavior shown in Figure 4.2 is prevented in the 2-D, plane strain model because the strain or movement in the vertical direction is not allowed for the 2-D model. 69

84 Direction of Soil movement Figure 4.21 Deformed shape of soil elements in proximity of the pile at the ultimate state. 4.6 Summary and Conclusions A series of 3-D, displacement based finite element analyses was performed on models of a thin horizontal slice to investigate the load transferred to piles in slopes reinforced using piles. Parametric studies were performed to study the effects of the initial stress condition in the soil mass, interface roughness between the pile and soil, depth of sliding, and pile-spacing to pile-diameter ratio on the load transfer mechanism in pile-reinforced slopes. In addition, the effects of the constitutive model of soil, and loading simulation by displacing different boundaries, and by horizontal body load were also investigated. It was found that the initial stress condition has less effect on the maximum lateral force on the pile than the interface roughness for most analyses performed using the 3-D thin horizontal slice model except for those analyses when loading is applied by horizontal body load. Changing the interface roughness from rough to smooth at the pile- 7

85 soil interface reduces the maximum lateral force by about 2 percent. Maximum lateral force per unit length of pile increases linearly with increasing depth of sliding. For both rough and smooth pile-soil interfaces, the maximum lateral force on pile increases from s/d = 2 to s/d = 4 and remains unchanged after that; the increase is more significant for the rough interface condition. It was also found that the constitutive model of the soil has a significant effect on the maximum lateral load on pile as well as the load-displacement response. Load simulation by applying displacement at different boundaries and by applying horizontal body force of the model has significant effects on the load transfer mechanism of piledslopes due to different the failure mechanisms observed. Therefore it is concluded that understanding of the loading condition, and the relative displacement of the pile and the soil mass in piled-slopes is crucial in determining the limiting lateral force acting on the piles. 71

86 CHAPTER 5: INTERPRETATION OF RESULTS FOR 2-D, PLANE-STRAIN AND 3-D THIN HORIZONTAL SLICE FINITE ELEMENT MODELS 5.1 Introduction Displacement-based finite element analyses have been performed using 2-D, plane-strain (Chapter 3) and 3-D thin horizontal slice (Chapter 4) models to investigate the load-transfer behavior and failure mechanisms of piled-slope problems. Piles were assumed to be rigid and the soil was forced to flow through the rigid piles under drained conditions. It was observed in Chapter 3 and Chapter 4 that there were many differences in the results obtained using these two models. In this chapter, the results from the two models are compared and discussed. 5.2 Interpretation of Results for 2-D, Plane-Strain and 3-D Thin Horizontal Slice The results obtained using the 2-D, plane strain and 3-D thin horizontal slice model are compared and discussed in this section. The differences in the results obtained using these two models from the parametric studies are presented and discussed in Sections and In Sections and 5.2.4, interpretation of the results for these two models is discussed by investigating the failure mechanisms observed and the change in confining stresses through stress path plots for the two models Effect of Initial Stress Condition, Interface Roughness and s/d Ratio Figure 5.1 shows the calculated maximum force that can be developed on the pile as a function of s/d ratio at a sliding depth of 3-m (9.8-ft) from 2-D, plane strain and 3-D analyses, respectively. Figure 5.1 shows the maximum lateral force on the pile obtained when displacements are applied on both left and right boundaries simultaneously. The 72

87 figure shows that the initial stress condition has a significant effect for the 2-D, plane strain model, and has no effect on the results from the 3-D thin horizontal model for all s/d ratios. As shown in Figure 5.1b, for the 3-D thin horizontal slice model, the interface roughness has no significant effect on the maximum lateral force on the pile for s/d ratio less than 4. On the contrary, the interface roughness has noticeable effect on the maximum lateral force on the pile for all s/d ratios for the 2-D, plane strain model. It is also shown in Figure 5.1 that the maximum lateral force on the pile obtained using the 2-D, plane-strain model with K o = 1. for s/d 4 is close to the maximum lateral force on the pile obtained using the 3-D thin horizontal slice for both the rough interface and smooth interface conditions. The fact that the effect of the interface roughness diminishes at s/d = 2 for the 3-D model is due to the passive wedge failure that occurred at small s/d ratios as shown in Figure 5.2b as well as in Figure This is because for the passive wedge failure condition, the failure surface, where the soil resistance is fully mobilized, lies in the soil mass on the upstream side of the pile and does not intersect the pile-soil interface. Thus, the interface roughness has little influence on the maximum force generated on the pile. However, for the 2-D, plane strain model since vertical strain is prevented, the only possible failure pattern is the arching type as shown in Figure 5.2a. For the arching type of failure, one of the failure surface lies on the pile-soil interface on the upstream side of the pile as shown in Figure 5.2a. Therefore, the shear strength at the pile-soil interface has a significant effect on the maximum lateral force on the pile, and so does the interface roughness since the interface shear strength depends on the interface roughness. 73

88 Force / Unit Length on Pile (kn/m) Broms Rough, Ko=1. Smooth, Ko=1. Rough, Ko=.5 Smooth, Ko=.5 3K Ito & Matsui s/d (a) 2-D, plane strain Force / Unit Length on Pile (kip/ft) Force / Unit Length on Pile (kn/m) Broms Ito & Mats ui Rough, Ko=1. Rough, Ko=.5 Smooth, Ko=1. Smooth, Ko= s /d Force / Unit Length on Pile (kip/ft) (b) 3-D horizontal slice Figure 5.1 Comparing the maximum lateral force on pile as a function of s/d at 3-m depth below ground from a) 2-D, plane-strain model, and b) 3-D horizontal slice model. 74

89 Failure surfaces ( arching type) Direction of soil movement upstream downstream Failure surfaces ( arching type) (a) 2-D, plane strain upstream downstream Direction of soil movement Possible Failure surfaces (passive wedge type) (b) 3-D horizontal slice Figure 5.2 Comparison of failure mechanism for s/d of 2 from a) 2-D, plane-strain model, and b) 3-D horizontal slice model Effect of Dilation Angle with the Mohr-Coulomb Failure Criterion The effect of dilative soil behavior on the pile reaction force is shown in Figure 5.3 for s/d = 4, K o = 1., µ =.577 at a sliding depth of 3-m. Figures 5.3a and 5.3b show the p-δ response from 2-D, plane strain and 3-D FE analyses, respectively, when a dilation angle of 3 degrees (φ=ψ) is applied for the Mohr-Coulomb failure criterion with an associated flow rule. The effect of dilation is not as pronounced for the 3-D models (Figure 5.3b) as for the 2-D models (Figure 5.3a). The lesser effect exhibited in the 3-D models is attributed to the fact that vertical deformations are not constrained as is the case in the 2-D, plane strain models. This is discussed in more detail in next section by investigating the stress paths of the soil elements on both the upstream and downstream sides of the pile. 75

90 Force / Unit Length on Pile (kn/m) Rough, Ko = 1., dilation angle of 3 o Rough, Ko = 1., no dilation Force / Unit Length on Pile (kip/ft) Displacement at midpoint between piles (mm) (a) 2-D, plane strain Force / Unit Length on Pile (kn/m) Rough, Ko = 1., dilation angle of 3 o Rough, Ko = 1., no dilation Force / Unit Length on Pile (kip/ft) Displacement at midpoint between piles (mm) (b) 3-D Figure 5.3 Comparing the effect of dilation angle on p-δ behavior on pile for s/d of 4 with rough interface at 3-m depth from a) 2-D, plane-strain model, and b) 3-D model. 76

91 5.2.3 Stress Paths The effect of the unreasonable changes in vertical stress near the pile for 2-D, plane strain models is further illustrated by comparing the stress paths for elements near the pile. Figure 5.4 shows stress paths for elements adjacent to the pile for s/d = 4 with rough pile-soil interfaces, where p is the mean effective confining stress (p = (σ 1 +σ 3 )/2) and q is the effective principal stress difference (q = (σ 1 -σ 3 )/2). σ 1 and σ 3 are the maximum and minimum principal effective stresses, respectively. The stress paths plotted are for two elements - one on the upstream side of the pile and one on the downstream side, as indicated by the arrows in the figures. The element on the upstream side of the pile for the 2-D, plane strain model remains in the elastic region because of dramatic increases in mean confining stress attributed to increases in the vertical stress near the pile resulting from the vertical constraint. In contrast, the element on the upstream side of the pile for the 3-D model is observed to yield at much lower confining stress because the vertical stress is maintained by the (appropriate) boundary conditions. On the downstream side of the pile, the confining stress from the 2-D, plane strain model is observed to approach zero due to significant decreases in the vertical stress as a result of the plane strain condition, while the confining stress for the 3-D model remains relatively close to the active failure state. The inset figure in the bottom right corner of Figure 5.4b shows that elements on the upstream side of the pile in the 3-D model dilate while elements on the downstream side contract. These deformations are prevented in the 2-D, plane strain model, which leads to substantial changes in mean confining stress on the upstream and downstream sides of the pile. 77

92 q (KN/m 2 ) Direction of soil movement "downstream" "upstream" Mohr-Coulomb Envelope p' (KN/m 2 ) (a) 2-D, plane strain q (KN/m 2 ) Mohr-Coulomb Envelope "upstream" Direction of soil movement 4 2 "downstream" "downstream" p' (KN/m 2 ) (b) 3-D Figure 5.4 Stress paths of elements near the pile for s/d of 4 with rough interface at 3-m depth from a) 2-D, plane-strain model, and b) 3-D model. 78

93 5.3 Summary and Conclusions Modeling of piled-slope problems using the finite element method is a complicated task due to the complexity of the problem and the multitude of potential failure modes. Table 5.1 summarizes the factors found to affect the computed limit force on piles in piled-slopes for 2-D, plane strain and 3-D analyses. The table shows that interface roughness, pile spacing, and soil dilation were consistently found to affect the computed limit loads on the piles for both 2-D, plane strain and 3-D models. In contrast, the initial state of stress in the soil was found to significantly affect computed limit loads for 2-D, plane strain analyses, but to have little effect on computed limit loads for 3-D analyses, regardless of the constitutive model used. This observation is attributed to the high degree of vertical constraint imposed in 2-D, plane strain models, which induces unreasonable changes in vertical stress both upstream and downstream of the piles that, in turn, can prevent development of potential failure modes in the models and otherwise corrupt the results. The 3-D models are not subjected to the vertical constraint, but instead are subjected to a constant vertical stress that is more representative of reality for piled slopes. Table 5.1 Summary of factors affecting limit lateral force on piles. Model 2-D (plane strain) 3-D (thin slice) Factor affecting limit lateral force on pile in piled-slopes Initial Stress in Soil Interface roughness s/d ratio Dilation angle X X X X X indicates factor has significant effect on calculated limit force. O indicated factor has little effect on limit force. X X 79

94 The following conclusions are reached based on the results presented: 1. Limit forces predicted using 2-D, plane strain and 3-D analyses differ substantially as a result of differences in vertical stresses developed in the soil adjacent to the pile. 2. Modeling of the piled-slope problem using 2-D, plane strain analyses can prevent development of potential failure modes because of the unreasonable constraints enforced adjacent to the pile. 3. Initial state of stress has an effect on predicted limit loads when they are evaluated using 2-D, plane strain models, but not when evaluated using 3-D models. This is a result of unreasonable constraints imposed by plane strain condition for 2-D analyses. 4. Accurate prediction of the limit loads on piles in piled slopes requires modeling of the problem in 3-D. 5. The computed limit force on piles in piled-slopes is sensitive to a number of factors including the pile-soil interface roughness, pile spacing, modeling techniques (2-D or 3-D), and constitutive model, provided that soil strength parameters and unit weight remain constant. 8

95 CHAPTER 6: THREE-DIMENSIONAL FINITE ELEMENT MODEL INCLUDING THE WHOLE SLIDING ZONE 6.1 Introduction It has been shown in previous chapters that accurate prediction of the limit loads on piles in piled-slopes requires modeling of the problem in 3-D. In this chapter, a further step is taken to model the piled-slopes including the whole sliding zone instead of a thin horizontal slice in 3-D finite element (FE) model. It is believed that the 3-D piledslope model including the whole sliding zone is a better representation of the actual field condition, therefore, providing a better insight into the load transfer mechanisms of piledslopes. The effects of initial in-situ stress condition, interface friction between piles and the soil surrounding piles, sliding depth, and ratio of pile spacing to pile diameter (s/d) are investigated. Furthermore, the effect of the inclination of piles on the load transfer behavior is described. The results of the 3-D FE model including the whole sliding zone are presented in the following sections D Model Including the Whole Sliding Zone Figure 6.1 shows the plan and schematic views of a 3-D model including the whole sliding zone of 4-m (13.1-ft) thick for s/d ratio = 4. The size of the FE model from the left to the right boundaries, 2x b, was chosen to be 24-m, where x b is the distance from the center of the pile to either the right or the left boundary. This chosen size, 2x b = 24- m, is typically wide enough for shallower sliding depths to be considered as an isolated pile without inducing group effect; however, for deeper sliding depths this might not be wide enough to represent an isolated pile as shown in Section for the effect of 81

96 boundary condition on load transfer behavior. The left ( upstream ) and right ( downstream ) boundary conditions were constrained in the x- or 1-direction at the initial state and then prescribed displacements were applied in the positive x-direction to simulate mass movement of soil in an unstable slope condition. Symmetry boundary conditions were applied at the upper and lower boundaries of the FE model as shown in Figure 6.1a. In this 3-D model, the top surface is representing the ground surface of a slope. Thus, it is a free surface with no prescribed boundary condition. The bottom surface, the sliding surface, was assumed to be smooth and vertical movements in the z- or 3- direction were not allowed as shown in Figure 6.1b. In addition, a vertical body load of 21-kN/m 3 in the z- or 3-direction was applied on all soil elements in the sliding zone to represent the weight of the soil in the field. The vertical body load was kept constant throughout the whole analysis. The pile, 1-m in diameter, was assumed to be rigid and constrained in all directions. The elements used were 3-D, 8-node linear solid elements with reduced integration. The soil constitutive model used for the FE analyses was the elasto-plastic model with a Mohr-Coulomb failure criterion and zero dilation angle (ψ = ) unless otherwise specified. Soil elements are assumed to be homogeneous and isotropic. The soil properties used are the same as those described in Chapters 3 and 4 and summarized in Table 6.1. The pile-soil interface was modeled with contact elements using two different types of interface roughness conditions: 1) smooth with a coefficient of friction µ =., and 2) rough with µ =.577 (or tan 3 o ). Table 6.2 shows the variables depth of sliding z s, initial stress condition K o, interface friction coefficient µ, and s/d ratio selected for the parametric studies of piled- 82

97 slope problems with vertical piles. For the piled-slope problems with inclined piles, the variables selected are shown in Table 6.3. Left ( upstream ) boundary (Rollers) Upper symmetry boundary (Rollers) Right ( downstream ) boundary (Rollers) Lower symmetry boundary (Rollers) Pile (d=1-m) y x 2*x b = 24.-m a) plan view Top surface Right boundary ( downstream ) Left boundary ( upstream ) Lower symmetry plane Sliding depth, z s =4-m Bottom surface (Rollers) b) schematic view Figure 6.1 a) Plan and b) schematic view of 3-D model including the whole sliding zone for s/d=4. Table 6.1 Soil properties used for the Mohr-Coulomb failure criterion. Description Symbol Magnitude and Unit Cohesion intercept c'.1-kpa Internal angle of friction φ ' 3 o Unit weight γ 21-kN/m 3 Young's modulus E 35.8-MPa Poisson's ratio ν.42 Dilation angle ψ o 83

98 Table 6.2 Variables selected for parametric studies for vertical piles. z s (m) K o µ s /d ,4,8, ,4,8, ,4,8, ,4,8, Table 6.3 Variables selected for parametric studies for inclined piles. z s (m) K o µ s /d Distribution of Contact Normal and Shear Stresses Figure 6.2 shows typical contact normal and shear stresses at the ultimate state along the pile-soil interface of horizontal sections at 1-m, 2-m, 3-m, and 4-m under the ground surface for vertical piles, where the contact normal stress is the stress acting normal to the pile-soil interface and the contact shear stress is the total tangential stress at the interface. The direction of the resultant normal force is always perpendicular to the pile-soil interface, but the direction of the resultant shear force varies depending on the slip direction of the soil and pile surfaces. The contact normal and shear stresses shown in Figure 6.2 are plotted as a function of transverse distance in the y- or 2-direction, as shown in Figure 6.3, from the center of the pile. Figure 6.2 shows the results for s/d = 4 84

99 in a 4-m thick sliding zone with K o =.5 and µ =.577 (or tan 3 o ). The contact normal stress on the upstream side of pile at all depths is the greatest at y =.-m (at the pilesymmetry plane), decreases at a lower rate up to about y =.31-m, and then decreases rapidly for y.31-m. The contact normal stress on the downstream side is negligible compared to the upstream side at all depths. Figure 6.2 also shows that the shear stress at the pile-soil interface on the upstream side increases from a lower value to its maximum value at about y =.31-m. At z s = 4-m, the shear stress distribution is close to that of the 2-D, plane strain model (Figure 3.5) since nodes on the bottom surface of the 3-D model are constrained in the z- or 3- direction. The contact shear stress distribution above the bottom surface deviates from the 2-D, plane strain behavior because these nodes are allowed to slip at the pile-soil interface. The contact shear stress on the downstream side is negligible as shown in Figure 6.2 compared to the upstream side. The contact stresses on the upstream side from Figure 6.2 are plotted as a function of depth along the length of the pile as shown in Figure 6.4 for different transverse distances, y. Figure 6.4 shows that the contact normal and shear stresses, generally, increase with increasing depth along the pile on the upstream side, except for the contact shear stress at y =, where the shear stress becomes zero at the sliding depth due to the constraints specified at this node. 85

100 Transverse Distance, y (m) Transverse Distance, y (m) Transverse Distance, y (m) Transverse Distance, y (m) Normal and shear stresses "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (a) at 1-m below ground surface Normal and shear stresses "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (b) at 2-m below ground surface Normal and shear stresses "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (c) at 3-m below ground surface Normal and shear stresses "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (d) at 4-m below ground surface Figure 6.2 Contact normal and shear stresses at the ultimate state along the pile-soil interface at a) 1-m, b) 2-m, c) 3-m, and d) 4-m below ground surface. 86

101 Soil moving direction Transverse distance, y upstream side downstream side y =.5 y =.31 y =. Figure 6.3 Transverse distance in the y-direction from center of pile (plan view). Depth, z (m) Distribution of Normal Stress - "upstream" (kn/m 2 ) at y=. at y=.111 at y=.216 at y=.31 at y=.389 at y=.449 at y=.486 at y= a) distribution of contact normal stress along the pile length Depth, z (m) Distribution of Shear Stress - "upstream" (kn/m 2 ) at y=. at y=.111 at y=.216 at y=.31 at y=.389 at y=.449 at y=.486 at y= b) distribution of contact shear stress along the pile length Figure 6.4 Distributions of contact a) normal stress, and b) shear stress at the ultimate state along the length of pile on the upstream side of the pile. 87

102 6.4 Results from 3-D Model Including the Whole Sliding Zone The results from 3-D model including the whole sliding zone are presented in this section. All results shown in this section are from loading simulated by displacing both the right and left model boundaries at the same rate and magnitude. The effect of model boundary conditions on the load transfer response is first presented followed by the effect of initial states of stress and interface roughness on the load transfer behavior. In addition, the effect of sliding depth and s/d ratio and the effect of inclination of the pile on the load transfer behavior are discussed Effect of Boundary Conditions In previous analyses presented in Chapters 3 and 4, the distance from the center of the pile to either end of the model boundary x b = 12-m was used and it was appropriate since unit thickness of soil was being model. However, when thickness (or depth of sliding), z s, increases the model boundaries need to be increased to avoid boundary effect on the load transfer response. Figure 6.5 shows the effect of boundary conditions on the response of the total lateral force (or the integral of the mobilized soil pressure) on the pile, P T, as a function of displacement at midpoint between adjacent piles, δ. The figure shows the results for s/d = 8, K o =.5 and µ =.577 at sliding depths of 4-m and 6-m. Figure 6.5a shows that for the ratio of x b / z s 3, the effect of boundary condition on the load transfer response is negligible. On the contrary, the boundary condition has significant effect on the load transfer behavior if x b / z s = 2 is used as shown in Figure 6.5b. Therefore, it is recommended that x b / z s 3 is used to reduce the boundary effect on the load transfer behavior. 88

103 5 Total Lateral Force (kn) Distance, x b x b /z =3 Distance, x b x b /z = Displacement at midpoint between piles (mm) a) z s = 4-m Total Lateral Force (kips) Total Lateral Force (kn) x b /z= Displacement at midpoint between piles (mm) b) z s = 6-m x b /z= Total Lateral Force (kips) Figure 6.5 Influence of upstream and downstream boundary conditions on P T -δ response for sliding depths of a) 4-m and b) 6-m Effect of Initial States of Stress and Interface Roughness Figure 6.6 shows the response of the total lateral force on the pile, P T, as a function of displacement at the midpoint between adjacent piles, δ, for s/d = 8 at z s = 2-m 89

104 (6.6-ft). The figure shows that the P T -δ response is similar to the p-δ response for the 3-D thin horizontal slice model (Figure 4.5). Figure 6.6 shows that the initial stress condition affects the initial stiffness of P T -δ response, but has little effect on the maximum total lateral force, P T,max. The interface roughness condition also has a noticeable effect on P T,max as shown in Figure 6.6. Similar behavior was observed for other sliding depths Rough, Ko =1. 1 Total Lateral Force (kn) Rough, Ko =.5 Smooth, Ko =1. Smooth, Ko = Total Lateral Force (kips) Displacement at midpoint between piles (mm) Figure 6.6 Effect of initial states of stress and interface roughness on P T -δ response for s/d=8 and z s =2-m Effect of Sliding Depth and s/d Ratio Figure 6.7 shows the distribution of the limit lateral force per unit length, p u, with depth for z s = 2-m, z s = 4-m, and z s = 8-m with K o =.5 at s/d = 8. Also shown in Figure 6.7 are p u predicted using the Ito-Matsui theory (Ito and Matsui, 1975) and the three times the Rankine passive pressure, 3σ p (Broms, 1964b), where σ p is the Rankine passive 9

105 pressure. The figure shows that the distribution of pressure calculated at the limit condition is reasonable close to 3σ p for all conditions. For z s 4-m, the limit pressure calculated at times exceeds the 3σ p as shown in Figures 6.7b and 6.7c. It is also observed in the figure that limit pressure calculated for the smooth interface condition is consistently less than that for the rough interface condition. Furthermore, near the sliding surface, the mobilized pressure drops as shown in Figure 6.7. Figure 6.8 shows the maximum total lateral force, P T,max, on the pile at the ultimate state as a function of s/d for sliding depths of 2-m, 4-m and 6-m with K o =.5. For a sliding depth of 2-m, P T,max increases with increasing s/d from s/d = 2 to s/d = 4, and then remains unchanged with increasing s/d ratio. For sliding depth of 4-m, P T,max increases with increasing s/d from s/d =2 to s/d = 8, and then remains unchanged with increasing s/d ratio. It is also observed in Figure 6.8 that for s/d = 2, the calculated values of P T,max from FE analyses are always lower that those predicted by Ito and Matsui (1975) and Broms (1964b) for all sliding depths. However, for s/d 4 and for shallow sliding depths, the calculated values of P T,max from FE analyses are close to those predicted by Broms (1964b) equation as shown in Figure 6.8a. As shown in Figures 6.8b and 6.8c. for s/d 4 and for deeper sliding depths, z s = 4-m and z s = 6-m, the calculated values of P T,max from FE analyses exceed those predicted by Broms (1964b) equation. Also shown in Figure 6.8 that P T,max is essentially the same at s/d = 2 regardless of interface roughness condition for all sliding depths. The effect of interface roughness on P T,max increase is more significant at larger s/d ratios. This implies that different failure mechanisms might exist for small s/d ratios (i.e. s/d < 4) in such a way that the interface roughness does not affect the failure behavior. 91

106 1 Limit Lateral Force per Unit Length (kn/m) Rough, Ko=.5 Smooth, Ko=.5 2 Depth (m) 3 4 Sliding surface 5 6 Ito & Mats ui Broms 1 a) z s = 2-m Limit Lateral Force per Unit Length (kn/m) Rough, Ko=.5 Smooth, Ko=.5 2 Depth (m) 3 4 Sliding surface 5 Ito & Matsui Broms 6 1 b) z s = 4-m Limit Lateral Force per Unit Length (kn/m) Rough, Ko=.5 2 Depth (m) 3 4 Ito & Mats ui 5 6 Sliding surface Broms c) z s = 6-m Figure 6.7 Distribution of limit lateral force per unit length of pile for sliding depths of a) z s =2-m, b) z s =4-m, and c) z s =6-m at s/d=8. 92

107 Rough, Ko=.5 Smooth, Ko=.5 b h h K 1 1 Total Force on Pile (kn) Broms Ito & Matsui Total Force on Pile (kips) Total Force on Pile (kn) Total Force on Pile (kn) s /d a) z s = 2-m Ito & Mats ui Rough, Ko=.5 Smooth, Ko=.5 I M i Broms s /d b) z s = 4-m Rough, Ko=.5 It M t i Ito & Matsui Broms Total Force on Pile (kips) Total Force on Pile (kips) s /d c) z s = 6-m Figure 6.8 P T,max as a function s/d, for sliding depths of a) 2-m, b) 4-m, and c) 6-m. 93

108 6.4.4 Effect of Pile Inclination So far, only piles installed perpendicular to the slope surface were considered for all previous FE analyses. Since not all piles in the field are installed perpendicular to the slope surface, it is important to investigate the effect of pile inclination other than that installed perpendicular to the slope surface. The sign convention for the inclination angle, θ, is taken to be positive if the pile is installed clockwise from the vertical line and is considered negative if the pile is installed counterclockwise from the vertical line as shown in Figure 6.9. As with previous analyses, the pile is assumed to be rigid and is constrained in all directions. The soil movement is assumed to be uniform and is simulated in the FE model by displacing both the right and left model boundaries at the same rate and magnitude. Soil movement Sliding depth, z s z x Inclined Pile Positive θ +θ Vertical line Sliding surface Figure 6.9 Sign convention of the pile inclination angle. Figure 6.1 shows the effect of pile inclination on the P T -δ behavior for s/d = 4, K o =.5, and µ =.577 (tan 3 o ) at sliding depth of 2-m. The figure shows that the maximum total lateral force on the pile, P T,max, increases with decreasing pile inclination angle, θ, where the lateral force is the force perpendicular to the longitudinal axis of the pile in the x-z plane That is to say that if the pile is installed at an angle rotated clockwise from the vertical line, which would tend to promote tension in the pile, the maximum 94

109 total lateral force induced on the pile is less than if the pile is installed at an angle rotated counterclockwise from the vertical line, which would tend to promote compression in the pile. Figure 6.1 also shows that the initial stiffness of the P T -δ curve is not affected by the pile inclination. 6 5 θ = -3 o (counterclockwise) 12 Total Lateral Force (kn) θ = +3 o (clockwise) θ = -15 o (counterclockwise) θ = +15 o (clockwise) θ = o (vertical) Total Lateral Force (kips) Displacement at midpoint between piles (mm) Figure 6.1 Effect of pile inclination on lateral load transfer behavior. The inclination of the pile affects not only the lateral load transfer behavior but also the axial load transfer behavior as shown in Figure 6.11 under the same loading condition, where axial force is defined as the force parallel to the longitudinal axis of the pile. Figure 6.11 shows that for θ, an upward resultant axial force at the pile-soil interface is induced and as a consequence the pile is placed under tensile stress (positive region). On the contrary, for θ = -15 o, the axial force decreases (goes into compression) at small displacements and then increases to near zero at about 5-mm displacement before leveling out at large displacements. For θ = -3 o, the axial force decreases 95

110 (increasing compression) with increasing displacements, then it levels out at large displacements. 15 Total Axial Force (kn) θ = +3 o (clockwise) θ = +15 o (clockwise) θ = o Tension θ = -15 o Compression (counterclockwise) Total Axial Force (kips) -1 θ = -3 o (counterclockwise) Displacement at midpoint between piles (mm) Figure 6.11 Effect of pile inclination on axial load transfer behavior. -22 Figure 6.12 shows the distribution of the limit lateral force per unit length, p u, with depth for different pile inclination angles for z s = 2-m, K o =.5, and µ =.577 at 1- mm displacement and at the ultimate state condition. Also shown in Figure 6.12 is p u predicted using Broms (1964b) method. Figure 6.12 shows that the distribution of lateral force per unit length of pile with depth for all pile inclination angles exhibit similar response. The figure shows that the distribution of lateral load increases with decreasing inclination angle from θ = +3 o to θ = -3 o. Figure 6.12b shows that for θ, the limit lateral force is generally close to or slightly greater than the ultimate lateral force predicted using the Broms method. On the contrary, for θ <, the limit lateral force is less than the ultimate lateral force predicted using the Broms method. 96

111 Depth (m) Limit Lateral Force per Unit Length (kn/m) θ = 3 θ = 15 θ = θ = +15 θ = Broms 2. a) at 1-mm displacement Depth (m) Limit Lateral Force per Unit Length (kn/m) θ = 3 θ = 15 θ = θ = +15 θ = Broms 2. b) at ultimate state Figure 6.12 Distribution of limit lateral force per unit length of pile at a) 1-mm displacement, and b) the ultimate state for s/d=4, K o =.5, and µ=.577 at z s =2-m. The distribution of the limit axial force per unit length, t u, with depth for different pile inclination angles for z s = 2-m, K o =.5, and µ =.577 at 1-mm displacement and at the ultimate state condition is shown in Figure The figure shows that θ, the distribution of axial force increases with increasing depth and then drops at depths close to the sliding depths. On the contrary, for θ <, the distribution of axial force decreases 97

112 with increasing depth and then increases at depths close to the sliding surface. In addition, for θ = -15 o, positive (or upward) axial force is generate in the upper half portion of the pile while negative (or downward) axial force is generate in the lower half portion of the pile. For θ = -3 o, only.3-m at the upper portion of the pile where the axial force is positive (acting upward) while the remaining portion of the pile the axial force is negative (acting downward). Limit Axial Force per Unit Length (kn/m) Depth (m) θ = 3 θ = 15 θ = θ = +15 θ = +3 a) at 1-mm displacement Limit Axial Force per Unit Length (kn/m) Depth (m) θ = 3 θ = 15 θ = θ = +15 θ = +3 b) at the ultimate state Figure 6.13 Distribution of limit axial force and per unit length of pile at a) 1-mm displacement, and b) the ultimate state for s/d=4, K o =.5, and µ=.577 at z s =2-m. 98

113 6.5 Failure Patterns from 3-D Model Including the Whole Sliding Zone In this section, the development of yielding patterns in the soil mass is first presented and discussed. Then, contour plots of vertical stress as the soil mass is moving through the piles in piled-slope problems is presented Development of Yielding in the Soil Mass The development of yielding pattern in the soil mass at end displacements of.2-mm, 2-mm, 2-mm, 5-mm, and 78-mm for s/d = 2, K o =.5, and µ =.577 at sliding depth of 4-m is shown in Figure The figure shows the results for both the right and left ends moving simultaneously at the same rate and magnitude from left to right in the positive x- or 1-direction. It can be seen that the development of the active failure condition on the downstream side of the pile develops relatively quickly at smaller displacements (approximately at 2-mm end displacements) compared to the development of the complete passive failure condition on the upstream side of the pile at larger displacement (approximately at 5-mm end displacements). At s/d = 2 the rigid piles are close enough to act as a continuous wall. Since the piles are rigid and are not allowed to rotate and displace, the failure surface observed caused by displacing soil towards the rigid piles is a passive wedge type as shown in Figure 6.14e at the ultimate state. It is also observed that under the passive failure wedge there is an elastic region where soil always remains elastic throughout the whole analyses. 99

114 Soil moving direction a) at end displacement of.2-mm Soil moving direction b) at end displacement of 2-mm Soil moving direction c) at end displacement of 2-mm Soil moving direction d) at end displacement of 5-mm Soil moving direction Passive wedge failure Elastic region e) at end displacement of 78-mm Figure 6.14 Development of yielding pattern for s/d=2, z s =4-m, K o =.5, and µ=.577 at end displacements of a).2-mm, b) 2-mm, c) 2-mm, d) 5-mm, and e) 78-mm. 1

115 Figure 6.15 shows the comparison of the yielding pattern for z s = 4-m, K o =.5, and µ =.577 at ultimate state for s/d = 2, s/d = 4, and s/d = 8. The figure shows that there are different yielding patterns shown for different s/d up to s/d = 8. As s/d increases the yielding pattern tends to shift from the passive wedge failure type to the arching type at deeper depths. However, at shallower depths the failure pattern is predominately the passive wedge failure type. This is because there is little confining stress at the ground surface; therefore, the soil in the vicinity of the piles close to the surface tends to expand (move upward) when the soil moves past the piles as shown in Figure In contrast, at deeper depths the soil elements are confined and relatively less upward movements are noticed as shown in Figure Therefore, at deeper depth, the failure pattern is predominately the arching type. The comparison of yield pattern for s/d = 4, z s = 2-m, K o =.5, and µ =.577 at the ultimate state for the piles inclined at different angles, θ = +3 o, θ = +15 o, θ =, θ = - 15 o, and θ = -3 o is shown in Figure Figure 6.17 shows that not only does the distance from the pile center to the edge of yield zone, x y, increase with decreasing inclination angle, from positive 3 degrees to negative 3 degrees, but also that the distance from the ground surface to the depth of yielding zone, z y, increases. In other words, the figure shows that the volume of soil yielding increases with decreasing inclination angle of the pile from positive 3 degrees to negative 3 degrees. 11

116 Passive wedge failure Elastic region Soil moving direction a) s/d = 2 Passive wedge failure Arching failure Soil moving direction b) s/d = 4 Arching failure Soil moving direction c) s/d = 8 Figure 6.15 Comparing the yielding pattern of z s =4-m, K o =.5, and µ=.577 at the ultimate state for a) s/d=2, b) s/d=4, and c) s/d=8. 12

117 Soil moving direction a) s/d = 2 Soil moving direction b) s/d = 4 Soil moving direction c) s/d = 8 Figure 6.16 Displacement vectors of z s =4-m, K o =.5, and µ=.577 at the ultimate state for a) s/d=2, b) s/d=4, and c) s/d=8. 13

118 z y x y a) θ = +3 o z y x y b) θ = +15 o z y x y c) θ = z y x y d) θ = -15 o z y x y e) θ = -3 o Figure 6.17 Comparing the yielding pattern for s/d=4, z s =2-m, K o =.5, and µ=.577 at the ultimate state for a) θ=+3 o, b) θ=+15 o, c) θ=, d) θ=-15 o, and 3) θ=-3 o. 14

119 6.5.2 Contours of Vertical Stress, σ z Figure 6.18 shows contour plots of the vertical stress, σ z, at the bottom layer of the 3-D model for s/d = 4, z s = 2-m, K o =.5, and µ =.577 (rough pile-soil interface) for the piles inclined at different angles, θ = +3 o, θ = o, and θ = -3 o, at the ultimate state. The sign convention is positive for compressive stress and negative for tensile stress. At the initial condition, the initial vertical stress is approximately 35-kPa to 4-kPa on the bottom layer. At the ultimate state, the compressive stresses on the downstream side of the pile decrease whereas the compressive stress on the upstream side of the pile increases. It is also observed that the zone and the magnitude of the increase in the compressive stress increases with decreasing inclination angle. The increase in the vertical confining stress leads to the increase in shear strength of the soil, and thus increases the load on the pile as shown in Figure 6.11 for the pile inclined at an angle, θ, of negative 3 degrees from the vertical. This may be the reason why the extent of the yielding zone increases with decreasing inclination angle as shown in Figure 6.15 due to additional constraint provided by the pile preventing the soil for expanding on the upstream side of the pile. 15

120 Direction of Soil movement a) θ = +3 o (4-kPa <σ z <115-kPa) Direction of Soil movement 14 3 b) θ = o (4-kPa <σ z <15-kPa) Bottom layer Direction of Soil movement c) θ = -3 o (8-kPa <σ z <17-kPa). Figure 6.18 Contours of vertical stress at the bottom layer of s/d=4, z s =2-m, K o =.5, and µ=.577 for a) θ=+3 o, b) θ= o, c) θ=-3 o at the ultimate state. 16

121 6.6 Summary and Conclusions A series of 3-D models including the whole sliding zone were analyzed using the displacement based finite element method to investigate the ultimate load transferred to piles in slopes reinforced with piles. Parametric studies were performed to study the effects of initial stress condition in the soil mass, interface roughness between the pile and soil, depth of sliding surface, and pile-spacing to pile-diameter ratio on the load transfer mechanism in pile-reinforced slopes. In addition, the effect of the inclination of the pile and the boundary effect on the load transfer behavior in piled-slopes were also investigated. From the FE analyses performed using the 3-D model including the whole sliding zone, the following conclusions is reached: The initial stress condition has no effect on the limit load on the pile. The maximum total lateral force induced on the pile increases with increasing interface roughness for s/d > 2. The distribution of limit lateral force is similar to greater than those predicted using Broms empirical equation. For s/d = 2, different failure mechanism is observed compared to the failure mechanism for s/d 2. The failure pattern for s/d = 2 is generally the passive wedge type, and the failure patterns for s/d 2 are the combination of the passive wedge type and the arching type depending on the sliding depths. The inclination of the pile is found to have significant effect on the load transfer behavior. The maximum total lateral load induced on the pile generally increases with decreasing the inclination angle from positive 3 degrees to negative 3 degrees. 17

122 CHAPTER 7: COMPLETE THREE-DIMENSIONAL FINITE ELEMENT MODEL 7.1 Introduction Analyses presented in Chapters 3, 4, and 6 were based on the assumption that the anchorage zone beneath the sliding surface is stable and not moving, thus the anchorage zone is neglected in all previous finite element (FE) analyses by considering only the load transfer mechanisms of the pile-soil interaction in the sliding zone. In practice, generally, piles in the anchorage zone are not stationary except in a very stiff layer. Therefore, the effect of pile rotation and displacement needs to be taken into account in the FE model to simulate more realistic load transfer due to the pile-soil interaction in piles-slope. In this chapter, the load transfer mechanism in both the sliding and the anchorage zones is investigated. The results of the complete 3-D FE model analyses are presented in the following. 7.2 Complete 3-D Finite Element Model of Piled-Slope Figure 7.1 shows the plan and profile views of a complete 3-D FE model for s/d = 4. The size of the FE model from the left to the right boundaries, 2x b, was chosen to be 48-m, where x b is the distance from the center of the pile to either the right or the left boundary. The left ( upstream ) and right ( downstream ) boundary conditions were constrained in the x- or 1-direction to establish the initial state and then either uniform prescribed displacements or body loads were applied in the positive x-direction to simulate mass movement of soil in sliding zone. Symmetry boundary conditions were applied at the upper and lower boundaries of the FE model as shown in Figure 7.1a. The 18

123 top surface is representing the ground surface of a slope. Thus, it is a free surface with no prescribed boundary condition. The bottom surface, at the base of the anchorage zone, was assumed to be smooth and vertical movements in the z- or 3-direction were not allowed. In addition, a vertical body load of 21-kN/m 3 in the z- or 3-direction was applied on all soil elements to represent the weight of the soil in the field. The vertical body load was kept constant throughout the whole analysis. A commercial FE software, ABAQUS, was used to perform all FE analyses described in this chapter. Left ( upstream ) boundary (Rollers) Upper symmetry boundary (Rollers) Right ( downstream ) boundary (Rollers) Lower symmetry boundary (Rollers) Pile (d=1-m) y x 2*x b = 48.-m a) plan view 2*x b = 48-m Sliding zone L s Applied uniform soil movement or body load Soil-soil interface (sliding surface) L a =12-m Anchorage zone Rigid Pile (d=1-m) Bottom of anchorage 4-m b) profile view Figure 7.1 a) Plan and b) profile view of a complete 3-D model for s/d = 4. 19

124 The pile, 1-m in diameter, was assumed to be rigid and allowed to rotate unless otherwise specified. The pile was taken to be vertical. The length of the pile was assumed to have a finite length with a fixed pile length in the anchorage zone (called anchorage length), L a, of 12-m, and a variable pile length in the sliding zone (called sliding length), L s, as shown in Figure 7.1b. The total length of the pile is the anchorage length, L a, plus the length of the pile in the sliding zone, L s. The distance from the bottom of the pile to the bottom of the anchorage zone was 4-m. The 3-D solid (continuum), 8-node linear elements with reduced integration were used. The soil constitutive model used for the FE analyses was the elasto-plastic model with a Mohr-Coulomb failure criterion and zero dilation (ψ = ) unless otherwise specified. Soil elements were taken to be homogeneous and isotropic. The soil properties used, the same as those used in previous chapters, are summarized in Table 7.1. The pile-soil interface along the pile length and soil-soil interface between the sliding zone and anchorage zone were modeled with contact elements using two different types of interface roughness conditions: 1) smooth with a coefficient of friction µ =., and 2) rough with µ =.577 (or tan 3 o ). The pile-soil interface between the base of the pile and the soil was taken to be smooth. Table 7.1 Soil properties used for the Mohr-Coulomb failure criterion. Description Symbol Magnitude and Unit Cohesion intercept c'.1-kpa Internal angle of friction φ ' 3 o Unit weight γ 21-kN/m 3 Young's modulus E 35.8-MPa Poisson's ratio ν.42 Dilation angle ψ o 11

125 Table 7.2 shows the variables depth of sliding z s, initial stress condition K o, interface friction coefficient µ, and s/d ratio selected for the parametric studies of piledslope problems with vertical piles for loading simulated by prescribed displacements. For loading simulated by applying horizontal load, the variables selected are shown in Table 7.3. Table 7.2 Variables selected for loading simulated by prescribed displacements. z s (m) K o µ s /d Table 7.3 Variables selected for loading simulated by applying horizontal body load. z s (m) K o µ s /d ,4.5. 2,4, ,4, Distribution of Contact Normal and Shear Stresses Figure 7.2 shows typical contact normal and shear stresses at the ultimate state along the pile-soil interface of horizontal sections at.5-m, 1-m, 1.5-m, and 2-m under the ground surface for vertical piles, where the contact normal stress is the stress acting normal to the pile-soil interface and the contact shear stress is the total tangential stress at the interface. The direction of the resultant normal force is always perpendicular to the pile-soil interface, but the direction of the resultant shear force varies depending on the slip direction of the soil and pile surfaces. The contact and shear stresses shown in 111

126 Figure 7.2 are plotted as a function of transverse distance in the y- or 2-direction, as shown in Figure 7.3, from the center of the pile. Figure 7.2 shows the results for s/d = 8 in a 2-m thick sliding zone with K o =.5 and µ =.577 (or tan 3 o ). Figure 7.2 shows that the location of the maximum contact normal stress and the behavior of the contact normal stress on the upstream side vary with depth. Generally, the location of the maximum contact normal stress is at some distance from the center of the pile (at y > ). In contrast, Figure 6.2 shows that the location of maximum contact normal stress on the upstream side of the pile is always at y = for the 3-D model including the sliding zone only. Figure 7.2 also shows that the contact normal stress on the downstream side is negligible compared to the contact normal stress on the upstream side for all depths except at the sliding surface where it becomes greater than the contact normal stress on the upstream side. On the contrary, the behavior of contact shear stress on the upstream side along the interface is relatively consistent where it increases or remains unchanged from y = to y =.4 and then decreases rapidly except at the sliding surface where the contact shear stress is zero. 112

127 Transverse Distance, y (m) Transverse Distance, y (m) Transverse Distance, y (m) Transverse Distance, y (m) Normal and shear stresses "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (a) at.5-m below ground surface Normal and shear stresses "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (b) at 1-m below ground surface Normal and shear stresses "downstream" side Shear stress "upstream" side Normal stress "upstream" side Distribution of Contact Stresses (kn/m 2 ) (c) at 1.5-m below ground surface Normal stress "upstream" side Shear stress both sides Normal stress "downstream" side Distribution of Contact Stresses (kn/m 2 ) (d) at 2-m below ground surface Figure 7.2 Contact normal and shear stresses at the ultimate state along the pile-soil interface at a).5-m, b) 1-m, c) 1.5-m, and d) 2-m below ground surface. 113

128 Soil movement direction Transverse distance, y y =.5 upstream side downstream side y =. Figure 7.3 Transverse distance in the y-direction from center of pile (plan view). Figure 7.4 shows typical distributions of contact stresses on the upstream side of the pile at the ultimate state along the pile length for a 2-m thick sliding zone at different transverse distances, y, from the center of the pile for s/d = 8, K o =.5 and µ =.577 (or tan 3 o ). The transverse distance is the distance in the y-direction from the center of the pile as shown in Figure 7.3. It is shown in Figure 7.4 that the contact normal and shear stresses increase with increasing depth in the sliding zone from the ground surface to about 1.5-m below the ground surface. Then the contact normal and shear stresses decrease to zero at the sliding surface (z = 2-m), except for the contact normal stress at y >.46 which decrease below z = 1.5-m but remain slightly greater than zero. The contact stresses on the downstream side of the pile are relatively small compared to the contact stresses on the upstream side in the sliding zone, and therefore they are not shown. 114

129 Depth, z (m) Distribution of Normal Stress - "upstream" (kn/m 2 ) at y=. at y=.98 at y=.191 at y=.278 at y=.354 at y=.416 at y=.462 at y=.49 at y= a) distribution of contact normal stress along the pile length Depth, z (m) Distribution of Shear Stress - "upstream" (kn/m 2 ) at y=. at y=.98 at y=.191 at y=.278 at y=.354 at y=.416 at y=.462 at y=.49 at y= b) distribution of contact shear stress along the pile length Figure 7.4 Distributions of contact a) normal stress, and b) shear stress at the ultimate state along the length of pile on the upstream side of the pile in the sliding zone. 115

130 7.4 Results from the Complete 3-D Finite Element Model Results from the complete 3-D finite element model are presented in this section. The loading condition is simulated by displacing both the right and left model boundaries at the same rate and magnitude when the soil-soil interface between the sliding zone and the anchorage zone is taken as smooth. When the soil-soil interface between the base of the sliding zone and the top of the anchorage zone is taken as rough, the loading condition is simulated by applying horizontal body load in the positive x-direction Results for Loading by Displacing Both the Right and Left Boundaries Figure 7.5 shows the total load transfer responses for s/d = 8, K o =.5, z s = 2-m, and µ =.577 in both the sliding and the anchorage zone. Figure 7.5a shows the total lateral load and the Figure 7.5b shows the total axial load versus displacement for both the sliding and the anchorage zones. Note that the scale of the y-axes plotted in Figures 7.5a and 7.5b is different. The total lateral load and the total axial load on the pile is the integral of the stresses at the pile-soil interface along the pile length in the x-direction and z-direction, respectively. As shown in Figure 7.5a, the total lateral load on the pile in the anchorage zone is equal and opposite to the total lateral on the pile in the sliding zone. On the contrary, the total axial load on the pile in the sliding zone is equal and opposite to the total axial load on the pile in the anchorage zone plus the reaction force at the pile base as shown in Figure 7.5b. The sign convention is positive if the direction of the force is in the positive direction of the Cartesian coordinate system and vice versa. In other words, the positive lateral force is the resultant force in the positive x-direction, and the positive axial force is the resultant force in the positive z-direction. 116

131 The total lateral load on the pile increases rapidly at small displacements and then levels out with increasing displacement. At large displacements, the total lateral force reaches a maximum value of approximately 4-kN (9-kips) as shown in Figure 7.5a. On the contrary, the total axial load on the pile in the sliding zone becomes negative at small displacements (pushing the pile downward into the anchorage zone). For displacement less than 1-mm, all the downward force is resisted by the side friction in the anchorage zone and no force is transmitted to the pile base. For displacement greater than 1-mm, the total axial force on the pile in both the sliding and the anchorage zones become less negative in the sliding zone and more negative in the anchorage zone, and some total axial force begins to transmit to the pile base. At large displacement, the axial forces in both the sliding and anchorage zones level out with increasing displacement as shown in Figure 7.5b. It is also observed that the axial force changes direction in the sliding zone from a downward force at initial state to an upward force at ultimate state, and vice versa in the anchorage zone. 117

132 Total Lateral Force (kn) Sliding Zone Anchorage Zone Total Lateral Force (kips) -5 Displacement at midpoint between piles (mm) -112 a) lateral load response 5 Total Axial Force (kn) Pile base Sliding Zone Anchorage Zone Total Lateral Force (kips) -5 Displacement at midpoint between piles (mm) -11 b) axial load response Figure 7.5 The (a) total lateral load and (b) total axial load as a function of displacement for s/d=8, K o =.5, z s =2-m, and µ=

133 Figure 7.6 shows the distributions of soil pressure, total lateral force (or shear force) and moment produced by the moving soil in both the sliding and anchorage zones as a function of depth for s/d = 8, K o =.5, z s = 2-m, and µ =.577 at the ultimate state. Figure 7.6a shows that the soil pressure generated in the sliding zone due to the applied uniform soil movement is resisted by the soil in the anchorage zone due to the rotation of the pile. The point of rotation is located at approximately 1-m below the ground surface in this case, where zero soil pressure occurred in the anchorage zone. Also shown in Figure 7.6a is the limit soil pressure predicted as three times the Rankine passive soil pressure, 3σ p, (Broms, 1964b). At the ultimate state, the ultimate soil pressure calculated from the FE analysis due to uniform soil movement in the sliding zone is generally slightly higher than 3σ p except in at depths close to the sliding surface. Figures 7.6b and 7.6c show the shear force and bending moment induced on the pile, respectively. The maximum shear force induced on the pile is generally located at the sliding depth for uniform soil layer as shown in Figure 7.6b. The maximum bending moment is located at approximately 5.7-m below the ground surface in this case, where the shear force on the pile is zero, as shown in Figure 7.6c. Similar behavior was observed for sliding depth of 4-m as shown in Figure

134 Soil Pressure (kn/m 2 ) Sliding surface 2 Broms 4 6 Broms Depth (m) "point of rotation" 16 a) soil pressure Total Lateral Force (kn) Sliding surface 4 6 Depth (m) b) total lateral force Bending Moment (kn-m) Sliding surface Depth (m) c) bending moment Figure 7.6 The (a) soil pressure (b) total lateral load, and (c) bending moment as a function of depth for s/d=8, K o =.5, z s =2-m, and µ=.577 at the ultimate state. 12

135 Soil Pressure (kn/m 2 ) Sliding surface 6 Depth (m) Broms Broms "point of rotation" a) soil pressure Total Lateral Force (kn) Sliding surface Depth (m) b) total lateral force Bending Moment (kn-m) Sliding surface Depth (m) c) bending moment Figure 7.7 The (a) soil pressure (b) total lateral load, and (c) bending moment as a function of depth for s/d=8, K o =.5, z s =4-m, and µ=.577 at the ultimate state. 121

136 7.4.2 Results for Loading by Applying Horizontal Body Load Figure 7.8 shows the total lateral and axial load responses in the sliding zone as a function of displacement at the midpoint between adjacent piles at the ground surface, δ, for s/d = 8, K o =.5, z s = 2-m, and µ =.577. The loading on the pile is simulated by applying horizontal body load in the positive x-direction. Note that the scale of the y- axes plotted in Figure 7.8a and 7.8b is different. Two cases are compared in Figure 7.8. The first case is that the pile is taken to be fixed (constrained in all directions), and the second case is that the pile is free to rotate (the only constraint on the pile is the symmetry boundary condition). For the fixed pile case, the total lateral force increases with increasing displacement up to a peak value, and then decreases slightly with additional displacement. For the free pile case, the total lateral force increases almost linearly with increasing displacement up to a maximum value, and then remains unchanged with additional displacement. It is also shown in Figure 7.8a that the initial stiffness and the maximum force for the fixed pile case, is higher than that for the free pile case. Figure 7.8b shows that opposite axial load response is observed for the fixed and the free pile cases. For the fixed pile case, the axial force always stays positive, indicating an upward tensile resultant force, whereas, the axial force always stays negative, indicating a downward compressive resultant force for the free pile case. In addition, the axial force induced for the free pile case continues to decrease with increasing displacement, whereas, the axial force induced for the fixed pile case increases to a peak value and then decreases with increasing displacement. 122

137 Total Lateral Force (kn) Fixed Pile Free Pile Total Lateral Force (kips) Displacement at midpoint between piles (mm) a) lateral load response 15 1 Fixed Pile 2.6 Total Axial Force (kn) Tension Compression -1.4 Free Pile -2.4 Total Axial Force (kips) -15 Displacement at midpoint between piles (mm) -3.4 b) axial load response Figure 7.8 The (a) lateral load response, and (b) axial load response for s/d=8, K o =.5, z s =2-m, and µ=

138 Figure 7.9 shows the distribution of soil pressure and total lateral force with depth induced on the pile by the horizontal body load applied in the sliding zone at the ultimate state for s/d = 8, K o =.5, z s = 2-m, and µ =.577. Figure 7.9a shows that at shallow depths in the sliding zone, the soil pressure generated for both the free pile and the fixed pile cases are similar, and the ultimate soil pressure calculated from FE analysis at these shallow depths is close to Broms empirical equation. For the free pile case, the soil pressure reduced to zero at the sliding depth and the lateral force generated in the sliding zone is resisted by the soil resistance in the anchorage zone due to the rotation of the pile. On the contrary, for the fixed pile case, the lateral force generated in the sliding zone is not resisted by the soil resistance in the anchorage zone, but instead more lateral force is induced in the anchorage zone. Figure 7.9b show the total lateral force (or shear force) response on the pile. For the free pile case, the maximum shear force on the pile is located at the sliding depth. However, for the fixed pile case, the maximum shear force on the pile is located at the base of the pile. 124

139 Soil Pressure (kn/m 2 ) Sliding surface 2 4 Broms 6 Broms Depth (m) Free pile "point of rotation" Fixed pile a) soil pressure Total Lateral Force (kn) Sliding surface Depth (m) b) total lateral force Figure 7.9 The (a) soil pressure, and (b) total lateral load as a function of depth for s/d=8, K o =.5, z s =2-m, and µ=.577 at the ultimate state. 125

140 7.5 Failure Patterns from 3-D Model Including the Whole Sliding Zone The development of yielding in the soil mass for the free pile at end displacements of 9-mm, 5-mm, and 121-mm for s/d = 8, K o =.5, and µ =.577 and a sliding depth of 4-m is shown in Figure 7.1. The figure shows the results for both the right and left ends moving simultaneously at the same rate and magnitude from left to right, as shown in the figure, in the positive x- or 1-direction. The interface between the base of sliding zone and the top and the anchorage zone is smooth. It can be seen in Figure 7.1a that the development of the active failure condition occurred at small displacements on the downstream side of the pile in the sliding zone where soil is moving away from the pile, and on the upstream side of the pile in the upper anchorage zone where the pile is moving away from the soil due to pile rotation. The pile rotates in the clockwise direction due to the soil movement in the sliding zone. On the contrary, passive failure condition is observed on the upstream side of the pile in the upper sliding zone where soil is moving towards the pile as show in Figure 7.1a. Figure 7.1b shows that with increasing movement, both the passive failure zone on the upstream side of the pile in the sliding zone and the active failure zone expand in the horizontal direction. The active failure on the upstream (left hand) side of the pile in the upper part of the anchorage zone extends downward as the pile continues to rotate, and active failure condition begins to form on the downstream side of the pile at the tip of the pile. In addition, the passive failure condition is observed on the downstream (right hand) side of the pile in the upper portion of the anchorage zone. Figure 7.1c shows the soil yield pattern at the ultimate state where the soil strength in the sliding zone is fully mobilized. However, only part of the soil resistance 126

141 in the upper portion of the anchorage zone is fully mobilized. This is the flow failure mode as described by both Loehr et al. (24), and Hull et al. (1991). The point of rotation is also shown in Figure 7.1c where zero soil pressure occurred as shown in Figure 7.6a. Above the point of rotation the active failure condition occurred on the upstream side of the pile, whereas below the point of rotation the active failure condition occurred on the downstream side of the pile in the anchorage zone. Figure 7.11 shows the soil pressure distribution with depth for end displacements of 9-mm, 5-mm, and 122-mm. Also shown in Figure 7.11 is the Rankine passive pressure, and the soil pressure from Broms equation. It is shown in Figure 7.11 that the soil pressure at end displacement of 9-mm in the upper 1-m of the sliding zone is near the Rankine passive pressure, σ p, and the corresponding failure pattern is a wedge failure type as shown in Figure 7.1a. At end displacement of 5-mm, the soil pressure distribution in the sliding zone exceeds the Rankine passive pressure but is slightly less than the Broms empirical prediction, 3σ p. On the contrary, in the upper portion of the anchorage zone, the soil pressure is approximately equal to σ p as shown in Figure 7.11b, and the corresponding failure pattern is shown in Figure 7.1b where the passive wedge failure is observed on the downstream side of the pile in the upper anchorage zone. Finally, at the ultimate state, the soil pressure distribution in the sliding zone is for the most part equal to 3σ p as shown in Figure 7.11c except for the region close to the sliding surface. It is also shown in Figure 7.11c that the limit soil pressure (the difference between the passive and active pressures at the limit state) just below the sliding surface is approximately 1.4 times the passive soil pressure, about half of the Broms empirical prediction. 127

142 Soil moving direction Passive failure Pile Active failure Sliding surface a) at end displacement of 9-mm Soil moving direction Passive failure Active failure Pile Passive failure Active failure Sliding surface b) at end displacement of 5-mm Soil moving direction Passive failure Active failure Passive failure Point of rotation Active failure Sliding surface c) at end displacement of 122-mm Figure 7.1 Development of yielding pattern for s/d=8, z s =4-m, K o =.5, and µ=.577 at end displacements of a) 9-mm, b) 5-mm, and c) 122-mm. 128