Stabilization of Slopes Using Piles_ Interim Repor

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Purdue University

Purdue e-Pubs

JTRP Technical Reports Joint Transportation Research Program

1984

Stabilization of Slopes Using Piles : Interim ReportSophia Hassiotis

Jean-Lou Chameau

is document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for

additional information.

Recommended CitationHassiotis, S., and J. Chameau. Stabilization of Slopes Using Piles : Interim Report. Publication FHWA/IN/JHRP-84/08. Joint Highway Research Project, Indiana Department of Transportation andPurdue University, West Lafayee, Indiana, 1984. doi: 10.5703/1288284314072.
http://docs.lib.purdue.edu/http://docs.lib.purdue.edu/jtrphttp://docs.lib.purdue.edu/jtrprogramhttp://docs.lib.purdue.edu/jtrprogramhttp://docs.lib.purdue.edu/jtrphttp://docs.lib.purdue.edu/


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JOINT HIGHWAY RESEARCH PROJECTJHRP-84-8STABILIZATION OF SLOPES USINGPILESFINAL REPORTS. Hassiotis and J. L. Chameau


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Digitized by the Internet Archivein 2011 with funding from

LYRASIS members and Sloan Foundation; Indiana Department of Transportation

http://www.archive.org/details/stabilizationofsOOhass


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Interim ReportStabilization of Slopes Using Piles

May 1, 1984Project: 6-36-360

From: J.L. Chameau, Research AssociateJoint Highway Research Project File: 6-14-15

To: H.L. Michael, DirectorJoint Highway Research Project

Attached is an Interim Report on the HPR Part II studytitled Design of Laterally Loaded Drilled-In-Piers for LandslideCorrection Anchored Within Sedimentary Rocks . The report isentitled Stabilization of Slopes Using Piles . It is authoredby S. Hassiotis and J.L. Chameau of our staff.

The report presents a methodology for the design of piles orpiers used to improve the stability of a slope. A step-by-stepprocedure is proposed to select design parameters such as pilediameter, spacing, and location which will provide an appropriatefactor of safety for the slope and insure the integrity of thepiles. Two computer programs are provided to perform the neces-sary operations. The first computer program calculates the fac-tor of safety of the reinforced slope; the second program deter-mines shear force, bending moment, and displacement profilesalong the pile. These programs can be used iteratively toachieve an optimum design solution.

The report Is submitted as partial fulfillment of the objec-tives of the study.

Respectfully submitted,

J.L. ChameauResearch Associate

cc: A.G. Altschaeffl W.H. Goetz C.F. ScholerJ.M. Bell G.K. Hallock R.M. ShanteauW.F. Chen J.F. McLaughlin K.C. SInhaW.L. Dolch R.D. Miles C.A. VenableR.L. Eskew P.L. Owens L.E. WoodJ.D. Fricker B.K. Partridge S.R. YoderG.D. Gibson G.T. Satterly


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IllI. Report No.FHWA/IN/JHRP-84/8

2. Government Accession No

4. Title and SubtitU

Stabilization of Slopes Using Piles7. Author(s)

S. Hassiotis and J. L. Chameau,8. Performing Organizotlon Report No.

JHRP-84-89. Performing Organization Name and AddressJoint Highway Research ProjectCivil Engineering BuildingPurdue UniversityWest Lafayette, IN 47907

12. Sponsoring Agency Nome ond Addres*Indiana Department of HighwaysState Office Building100 North Senate AvenueIndianapolis, IN 46204

TECHNICAL REPORT STANDARD TITLE PAG3. Recipient's Catolog No.

5. Report DatMay 1, 1984

6. Performing Oryonizotion Code

10. WorW Unit No.

11. Contract or Grant No.HPR-K2 1) Part II13, Type of Report and Period Covered

Interim Report4. Sponsoring Agency CodeCA 359

15. Supplementary NotesConducted in cooperation with U.S. Department of Transportation, Federal HighwayAdministration under a research study entitled, Design of Laterally Loaded Drilled-In-Piers for Landslide Correction Anchored Within Sedimentary Rocks16. Abstract ~ ~ '

This study is part of a project undertaken at Purdue University to develop amethodology for the design and analysis of slopes stabilized with piles. Differentaspects of this problem are considered: (1) the determination of the force exertedon the piles by the slope; (2) the effect of a row of piles on the stability of aslope; and (3) simultaneous slope stability analysis and pile design to meetminimum safety requirements for both the slope and the piles.It is suggested to compute the force exerted by the piles on the slope bydividing the maximum value calculated using the theory of plastic deformation bythe factor of safety of the slope. The Friction Circle Method is extended toincorporate the reaction force provided by the piles and calculate the safetyfactor of the reinforced slope. The displacement, bending moment, and shear forceprofiles along the piles are also determined. A step-by-step procedure isproposed to select the pile dimensions and reinforcement which will provide anappropriate factor of safety for the slope and insure the integrity of the piles.These results are incorporated in two computer programs which can be usediteratively to provide an optimum design solution.

17. Key WordsSlope Stability, Pile, Pier ,Factor of Safety, Stabilization,Reinforcement

19. Security Classif. (of this report)

Unclassified

18. Distribution StatementNo restrictions. This document isavailable to the public through theNational Technical Information Service,Springfield, VA 2216120. Security Classif. (of this poge)

UnclassifiedForm DOT F 1700.7 (b-69)

21. No. of Pages 22. Prie

181


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iv

ACKNOWLEDGEMENTS

The financial support for this research was pro-vided by the Indiana Department of Highways and theFederal Highway Administration. The research wasadministered through the Joint Highway Research Pro-ject, Purdue University, West Lafayette, Indiana.

Special thanks are extended to Ms. Cathy Ralstonfor typing this report. The help and recommendationsprovided by Messrs. Mike Oakland and Frank Adams aregreatly appreciated.


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TABLE OF CONTENTS

Page

LIST OF TABLES v11

LIST OF FIGURES viiiLIST OF ABBREVIATIONS AND SYMBOLS xlHIGHLIGHT SUMMARY CHAPTER I. - INTRODUCTION 1CHAPTER II . - SLOPE REINFORCEMENT 4CHAPTER III. - SLOPE STABILIZATION USING

PILES 10

Piles in Soil Undergoing LateralMovement ' \Theoretical Solutions 12

Subgrade Reaction Method 12Elastic Plastic Material 15Finite Element Method 16Other Approaches 16

The Theory of Plastic Deformation ... 18Assumptions 1Derivation 24Parametric Studies 32Field and Laboratory Measurements. 33

Safety Factor of the Stabilized Slope. ... 38The Friction Circle Method 40Parametric Studies 47

Summary 5

CHAPTER IV. - LATERALLY LOADED PILES 66The Concept of Subgrade Reaction 67Governing Equations 67Solution Techniques 72

Closed Form Solution of Equation (4-4a) 72


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Table

LIST OF TABLES

PageValues of dimensionless coefficientA, to calculate k in tons /ftof a pile embedded in moist or sub-merged sand 8 '

3Values of k gl in tons/ftfor a square plate, lxl ft and forlong strips, 1 ft wide, resting onprecompressed clay

Young's modulus for vertical staticcompression of sand from standardpenetration test and static coneresistanceModulus of elasticity of rocks 102Correlation between def ormabilityand RQD 104


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viii

LIST OF FIGURES

Figure Page1 Plastically Deforming Ground AroundStabilizing Piles 202 Stresses Acting on Elements inPlastically Deforming Ground 223 Mohr Circles of Elements in Plasti-

cally Deforming Ground 234 Stresses on Elements in Plastically

Deforming Ground 255 Force Acting on Piles Versus Ratio

D2 /D 1 for Different Cohesive Soils . . 346 Force Acting on Piles Versus RatioD-/D. for Different Cohesionless

Soils 357 Force Acting on Piles Versus Ratio

D2 /D i for Different Pile Diameters . . 368 Forces on Unreinforced Slope 429 Forces on Slopes Reinforced with Piles 45

10 Critical Surfaces of a Shallow Slopeas a Function of Pile Row Location . . 4911 Effect of Pile Location on the Factor

of Safety of a Shallow Slope 5112 Effect of Pile Location on the Reac-

tion Force Provided by the Piles ... 5313 Effect of Pile Location on the Factor

of Safety of a Steep Slope 54


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14 Critical Surfaces of a Steep Slope asa Function of Pile Row Location. ... 55

15 Normalized Curves of the Safety FactorVersus Distance S for Different Slopesof Same Height 56

16 Effect of the Degree of Mobilization ofF on the Factor of Safety 59P

17 Safety Factor Versus Ratio D~/D ... 6118 Safety Factor Versus Location of the

Piles Upslope for Different Ratios ofD2 /D 1 62

19 Factor of Safety Versus Position ofthe Piles Upslope for Different Valuesof Friction Angle 63

20 Safety Factor Versus Position of thePiles Upslope for Different Valuesof Cohesion 64

21(a) Beam on Elastic Foundation 6921(b) Cross-Section of a Beam on Elastic

Foundation 6922 Stabilizing Piles Embedded in Bedrock. 7023 Finite Difference Solution for a Pile. 7424 Load Deformation Curve from Plate

Jack. Test 10025 Comparison of RQD and the Modulus

Ratio \fet50 10326 Slope Configuration of the Example

Problem 10727 Safety Factor Versus Ratio D /b. . . . 11028 Displacement Along the Pile for Four

Boundary Conditions 11329 Bending Moment Along the Pile - Four

Boundary Conditions 11430 Shear Force Along the Pile - Four

Boundary Conditions 115


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31 Displacement Along the Fixed Head Pile 11832 Bending Moment Along the Fixed Head

Pile 11933 Shear Force Along the Fixed HeadPile 121

AppendixFigure34 Toe Failure 14535 Failure Below the Toe 146


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xi

LIST OF ABBREVIATIONS AND SYMBOLS

A dimensionless coefficient dependingon the density of sand

AB chord lengthAB arc length

a , a., a-, a- constants of integrationo 1 2 3b pile diameter

BC boundary conditionBD distance that the pile penetrates

into the bedrockcohesion Intercept

c available unit cohesionaCE length of the pile from the ground

surface to the critical surfaceCEO angle between F and the

horizontal pc unit cohesion required for equilibriumC resistance due to cohesionr

C. , C, C_, C, constants of integrationD distance in the direction of the

pile rowD. center to center distance between the

pilesD_ clear distance between the piles


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xiii

K coefficient of subgrade reaction atthe end of the first layer

K coefficient of subgrade reaction atthe beginning of the second layer

K_ coefficient of subgrade reaction atend of the second layer

Kg. coefficient of vertical subgradereaction of a 1-ft plate

M bending moment on the pileMT total number of equally spaced intervalsm modulus of volume decreasev

n,n ,n_ empirical indexes equal toor greater than zero

OG moment arm of FPP resultant of normal frictional forcesp intensity of loading, proportional to

deflection of the beamq, force per unit length on the pile

at the ground surfaceq2 force per unit length on the pileat the critical surfaceq intensity of distributed load applied on

the beamR radius of the critical circleS horizontal distance from the toe of

the slope to the pile rowV shear force on the pile

V shear strength provided bythe concrete

W weight of the critical mass

x,y angles that define the criticalsurface


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y pile deflection at point my horizontal displacement of the

pile at depth zy horizontal displacement that thesoil would undergo without the pilesy, pile deflection above the criticalsurfacey 2 pile deflection below the criticalsurface without the pilesz depth along the pile measured from

the ground surfacez depth along the pile measured from

the critical surfacea tt/4 + ij>/2Y unit weight of the soil9 slope of the pileA length of the intervals in the

discretized pileV Poisson's ratiosa normal stress at the surface

perpendicular to the x-axiso normal stress at the failure

surfacea normal stress at the surface

inclined an angle a from thex-axisangle of internal friction of thesoil

available friction angle$ friction angle required forequilibrium


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XV

HIGHLIGHT SUMMARY

Drilled piers or bored piles can be an efficientmeans to control slope movements where other correctivemeasures fail to insure stability or when their use isprohibited due to space limitations. A design metho-dology is proposed herein to assess both the forcesacting on the piles (or piers) and the influence of thepile row on the stability of the slope. The forcesacting on the pile sections above and below the criti-cal surface are calculated using the theories of plas-tic deformation and subgrade reaction, respectively.

The theory of plastic deformation is based on theassumption that the soil surrounding the piles is in astate of plastic equilibrium. Under this assumption,the force acting on the passive piles is expressed as afunction of the soil strength parameters and of thepile diameter, spacing and position. The actual reac-tion force exerted by the piles on the slope is assumedto be a fraction of the force corresponding to theplastic state condition. The Friction Circle Method of


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slope stability analysis is extended to incorporate thereaction force and determine the critical surface andsafety factor of the reinforced slope. A computer pro-gram is also developed to obtain the displacement,bending moment, and shear force profiles along thepiles. A step-by-step procedure is proposed to achievean optimum design solution which provides a requiredfactor of safety for the slope and insures theintegrity of the piles.


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CHAPTER I. INTRODUCTION

Insuring the stability of both natural and man-

made slopes continues to be an important problem ingeotechnical engineering. Slides account for manycivil engineering failures and often result in exten-sive property damage and sometimes loss of human life.There is no universally accepted method for the preven-tion and/or correction of landslides. Each slide isunique and should be considered on the basis of its ownindividual characteristics. Avoidance of a potentialslide area can be a primary consideration when select-ing a new site. Otherwise, corrective measures can betaken which include: (1) improving the slope geometryby changing the slope angle, excavating the soil at thehead, or increasing the load at the toe; (2) construct-ing a compound slope; and (3) providing surface andsubsurface drainage. However, where such correctivemeasures fail to insure stability or when their use isprohibited due to space limitations, retaining struc-

tures may be necessary. Piles carried across theactive or potential failure surface can be used


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In this report, the pressure acting on the pilesis calculated using the theory of plastic deformationdeveloped by Ito and Matsui (1975). The major assump-tion in this theory is that the soil in the area sur-rounding the piles is in a state of plastic equili-brium. With this assumption, the force acting on thepiles can be expressed as a function of the soilstrength, and the pile diameter, spacing and location.The safety factor of the slope after the placement ofthe piles is calculated assuming that a portion of thatforce is counteracting the driving forces of the slope.Finally, a step-by-step procedure is proposed to selectdesign parameters, which will achieve the stability ofthe slope and adequate dimensioning of the piles.

The concepts and methods presented herein areincorporated in two computer programs, which shouldprovide efficient design tools to practicing engineers.The first computer program calculates the safety factorof the reinforced slope as a function of pile diameter,location and spacing. The second program calculatesthe shear force, bending moment and displacement alongthe length of the pile as a function of the soil pres-sure, and the pile diameter, stiffness, location, spac-ing, and boundary conditions. These programs can beused iteratlvely to provide an optimum design solution.


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groups of 2 or 3. Initially the soil flowed around thecaissons, but these were eventually broken by the massmovement. Since the original slide had remolded thesoil to its residual strength, the small relativeincrease in shear strength provided by the caissons hadan insignificant effect on the slope stability.

Since then, more effective walls have been con-structed. Shallow slides, up to 20-ft deep, have beenstabilized by the use of sheet piling (Toms andBartlett, 1962) and closely spaced driven cantileverpiles (Zaruba and Mencl,1967). A root pile wall wasused to correct a landslide near Monessen, Pennsylvania(Dash and Jovino, 1980). Stone columns have been used

effectively in Europe (Goughnour and DiMaggio, 1978).Drilled cantilever pier walls have been used success-fully in the Ohio River Valley (Nethero,1982). Satis-factory results were achieved with 18-in. to 30-in.diameter piers spaced 5 to 7-ft on center. Actual instal-lations of such piers include the use of cast-in-placeconcrete cylindrical columns in an effort to stop soilmovements, which had been sufficient to cause closingof a roadway. In a different application, piers placedat the toe of a slope stopped a slide which had beentriggered by an unbraced cut.

Deep-seated slides have also been stabilized suc-cessfully by large diameter cylinder piles, anchored


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sheet pile walls, or rock anchored cylinder piles. Someof these cases will be discussed below.

Large diameter cast-in-place concrete cylinderswere used for a sidehill stabilization during the con-struction of the Seattle freeway in Washington (Andrewsand Klassel, 1964). During excavation for the footingof a bridge structure, a large deep-seated slidestarted to develop, immediately adjacent to a seven-story apartment building. A retaining wall, which hadbeen planned between the bridge footing excavation andthe building, could not be built because further exca-vation for the wall would cause the slide to progressand endanger the building. To avoid this, cast-in-place cylinder piles were used to form the wall. Todevelop the resistance required to withstand theassumed loads, the design called for 4.75 ft cylinderson 6-ft centers. The cylinders were designed topenetrate the slip plane and other potential slidingplanes. Welded steel beams were employed to reinforcethe concrete cylinders.

A sheet pile wall anchored with pre-tensioned soilanchors helped to stabilize a potentially dangerousslope during construction in the Ohio River valley.According to D'Appolonia et al. (1967), the wall pro-vided short term stability by preventing progressivefailure, while drainage assured long-term stability.


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Cylinder piles were used in an effort to stopmovement during construction for the freeway on PotreroHill in San Francisco (Nicoletti and Keith, 1969). A

rock bolted retaining wall was installed to restrain anunbraced cut. The top of a railroad tunnel was located25-ft below the freeway and near the top of the wall.The overall stability of the tunnel, wall, and theslope rising above it, had been unbalanced by the cut.

Towards the end of the construction, large lateralmovements were observed in the tunnel. In order tostop the movements, a wall was constructed of a seriesof heavy steel piles, placed in drilled holes on eachside of the tunnel, and connected by steel strutsacross the top of the tunnel. This technique proved tobe efficient in preventing further movement.

A double row of anchored piles became necessaryduring railroad work in Belgium (DeBeer and Wal-lays,1970). In order to enlarge the railroad bed, anexisting slope, nearly at its equilibrium, was cutback, maintaining the original angle. The excavationsreactivated slips which had previously occurred in acrushed schist mass, and a series of retrogressiveslips was initiated. Since the slope was located in acity, only limited flattening was possible.


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Hence, the stability was insured by the resistance pro-vided by 3.5 to 5-ft diameter piles spaced 6.5 to 10-fton centers.

An anchored cylinder pile wall was installed dur-ing construction of interchanges to connect 1-471 withlocal traffic arteries in Cincinnati, Ohio (Offen-berger, 1981). Excavations on the southwest side ofMt. Adams triggered large ground movements in an areaknown to have experienced small movements in the past.Data from slope indicators showed that the earth move-ment was in a deep seated weak clay layer near the rocksurface, and well below the elevation of retaining wallfootings. A cantilever wall was determined to be inap-propriate, because considerable embedment into the rockwould be necessary. In addition, undesirable downslopedisplacement would be needed to develop the requiredlateral resistance. Thus, a cylinder pile wall withtie-back tendons was recommended. The cylinder pileswere drilled through 50 ft of overburden and socketed

into shale and limestone.

Recent applications of slope stabilization usingsteel piles have been described by Ito and Matsui(1981). Several landslides have been stabilized inJapan using such piles. In the United States, pierswith tie-back rock anchors were drilled through alandslide and embedded in an unweathered schist during


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construction at the Geyser Power Plant in California(Hovland et al. , 1982).

These case histories indicate that the use ofpiles for stabilizing slopes is becoming more common.However, little information is currently availableregarding (1) the pile behavior under lateral loadinginduced by the movement of the slide and (2) the effectof the pile row on the overall stability of the rein-forced slope. The research reported herein is a steptowards solving these two problems and developing amethodology for the analysis and design of theslope/pile system.


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10

CHAPTER III. SLOPE STABILIZATION USING PILES

The analysis of a slope reinforced with pilesrequires that the force applied to the piles by thefailing mass, and in turn the reaction from the pilesto the slope, be known. In addition, a slope stabilityanalysis that takes into account the reaction force isnecessary. In this chapter, a theoretical method forthe calculation of pressures acting on passive piles(piles subjected to lateral loading by horizontal move-ments of the surrounding soil) is discussed. Then, theresulting pile reaction is incorporated into a slopestability analysis to determine the factor of safety ofa slope reinforced with a row of piles. Finally,parametric studies are performed to assess the influ-ence of the location and spacing of the piles on thestability of the slope.


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11

PILES IN SOIL UNDERGOING LATERAL MOVEMENT

The problems of active piles (piles subjected to ahorizontal load at the head and transmitting this loadto the soil) embedded in clay or sand and subjected tohorizontal static loads of short or long duration havebeen treated by several authors (Matlock and Reese,1960; Davisson and Gill, 1963; Broms, 1964; Vesic,1965; Davisson, 1970; Poulos, 1973; Jamiolkowski andGarassino, 1977; Kishida and Nakai, 1977; Coyle et al.

,

1983). By making simplifying assumptions with regardto the deformation characteristics of the soil layersurrounding the piles, these authors arrived at accept-able solutions of piles subjected to horizontal loadsat the top. In most cases, the problem of a singlepile has been solved, and some correction factors forgroup effects have been introduced.

For the case of passive piles (piles subjected tolateral soil movements), the problems are more compli-cated because the lateral forces acting on the pilesare now dependent on the soil movements, and these arein turn affected by the presence of the piles. Forexample, it is possible for a pile group to stop thesemovements, creating a completely different situation

than that for a single pile. Hence, the solution for asingle pile cannot be easily adapted for the situation


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of a pile group, although several authors have sug-gested such an approach (Poulos,1973; Baguelln etal. ,1976; Viggiani.,1981). Other researchers have con-sidered the problem from the fundamental standpoint ofgroup (row) action (Ito and Matsui, 1975; Winter etal., 1983). Several theoretical approaches for the cal-culation of pressures on piles placed in deformingsoils will be presented here.

Theoretical Solutions

Analytical models that have been used to obtain atheoretical solution for piles placed in deformingsoils include: 1) a soil characterized by a modulus ofsubgrade reaction; 2) a soil considered to be anelastic-plastic material; 3) the finite element methodused with bilinear or hyperbolic approximations of thesoil behavior; and 4) other approaches.

Subgrade Reaction Method

In the modulus of subgrade reaction method, thepile is considered to be a beam on an elastic founda-tion, and the equilibrium equation is written as:

dz r

where EI pile stiffness


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v = horizontal displacement of the pile atPdepth z

v = horizontal displacement the soil'swould undergo without the pile

K subgrade reaction modulus, function of zsand (yp - yg )

Computer programs are available to solve this

equation (Baguelin et.al., 1976), but problems existin

the evaluation of y . When measured values of y g areintroduced, the solution of this equation agrees well

with experimental results. When theoretical values of

v are introduced, the solution of the equation is1 sunrealistic (Theoretical values of y g are discussed by

Poulos, 1967 and Canizo and Merino, 1977; Tables canbe used to determine the horizontal displacements of an

elastic layer subjected to various vertical load condi-

tions). In addition, the applicability of the solution

to a row of piles is uncertain because the group action

modifies the initial soil deformation values that are

used in the calculations (DeBeer, 1977).

Other researchers have used the subgrade reaction

concept for the calculation of the horizontal force.

Fukuoka (1977) studied the pile-soil interaction in

slopes subject to creep by means of a Winkler type

model. He assumed that the unbalanced force that


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causes the creep is a function of the velocity of thecreeping mass. Measurements are needed along thelength of the pile for the determination of such avelocity, and in turn, for the calculation of theunbalanced force. A reaction force provided by thepiles and equal to the unbalanced force should be ade-quate to stop the landslide. To calculate the lateralresistance of the piles against the landslide, Fukuokautilized the modulus of subgrade reaction method, andgave equations for the calculation of the deflectioncurve of the pile and for the bending moments and hor-izontal reactions. However, when the soil is moving,it is very difficult to define a coefficient ofsubgrade reaction to be used in the equations (DeBeer,

1977).

Viggiani (1981) used the subgrade reaction theoryto analyze the interaction between a sliding mass and astabilizing pile. His approach is based on conceptsdeveloped by Broms (1964) who evaluated the ultimatecapacity of a vertical pile acted upon by a horizontalload, using an estimated subgrade reaction coefficient.Viggiani evaluated the maximum shear force that a pilecan receive from the sliding mass and transmit to theunderlying soil, by analyzing six possible failuremechanisms for the pile. This theory can be used forthe design of the piles against an ultimate pressure.


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However, it does not provide a way to calculate a valueof the pressure at the initial stage of the landslide,which could be used to determine the effect of thepiles on the factor of safety of the slope.

Elastic-Plastic Material

Calculations based on the assumption of anelastic-plastic material are described by Poulos(1973). The solution of the problem is obtained byimposing compatibility of displacements between thepile and the adjacent soil. The pile displacement isevaluated from the bending equation of a thin strip.The soil displacement is evaluated from the Mindlinequation (Mindlin, 1936) for horizontal displacementcaused by horizontal loads within a semi-infinite mass.By solving a system of equations, the displacement andhorizontal pressure on the pile can be evaluated. Thismethod is applicable only when dealing with an idealsoil and a single pile.

More recently, Oteo (1977) refers to a methodapplied by Begemann-DeLeeuw (1972). The method hasbeen derived for the analysis of pressures on pileswhen the surrounding soil undergoes horizontal dis-placement due to a surficial surcharge in the vicinity.It considers the soil to be a linearly elastic materialand is based on stress and displacement fields given by


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the Boussinesq theory. This technique is only applica-ble to the case of a known surcharge applied to theslope.

Finite Element Method

The applicability of finite-element techniques tothe analysis of slope stabilization has been discussedby Rowe and Poulos (1979). They employed a two-

dimensional finite element model for soil-structureinteraction proposed earlier by Rowe et al.(1978). Itis recognized that a finite element analysis of thestabilization problem should make allowance for thesoil-structure interaction effects, but also forthree-dimensional effects such as arching betweenpiers. A computer program which can model theseeffects is being developed at Purdue University (Oak-land and Chameau, 1983). The program can also evaluatethe influence of piles position, size, spacing andstiffness on slope movements. When completed, thisprogram will be added to the design methodology pro-posed herein to form a complete design-analysis pack-age.

Other Approaches

Several other methods have also been used in thepast. Baker and Yonder (1958) suggested calculatingthe thrust against the piles by the procedure of slices


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and considering the piles as cantilever beams, providedthat they penetrate into a stable soil layer for onethird of their total length. Andrews and Klassel(1964) and Nethero (1982) calculated the pressure onthe pile by using the concept of passive and activepressures assuming that the piles form a cantileverwall. However, analyzing the pile group as a retainingwall can lead to very conservative designs, since thesoil arching between the piles is not taken intoaccount.

DeBeer and Wallays (1970) reported a method fordetermining the forces and bending moments on a pile inthe case of an unsymmetrical surcharge placed aroundthe pile. The method is based on concepts introducedby Brinch Hansen (1961) on the ultimate resistance ofrigid piles against transverse forces.

Wang et.al. (1979) proposed a semi-empirical tech-nique to calculate the pressure on piles embedded indeforming soils using inclinometers to determine thesoil movements. The soil pressures required to causethose movements where then estimated using the modulusof subgrade reaction theory.

Winter et.al. (1983) described a method for slopestabilization using piles based on the viscosity law ofcohesive soils. It uses a solution derived for the


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horizontal pressure against piles in viscous soils.Two requirements, a desired reduction of the slidingvelocity and a safe maximum bending moment on thepiles, are satisfied by the correct choice of pilespacing and pile diameter. This method is only appli-cable for the case of a clay mass subject to creep.

The Theory of Plastic Deformation

In most of the methods discussed so far, atheoretical solution for the passive piles was obtainedby either solving the problem of a single pile or byanalyzing the pile group as a retaining wall. Themethod proposed by Winter et.al.(1983) considers the

solution of piles placed in a row and takes intoaccount the spacing between the piles at the beginningof the analysis. However, this method can only be usedin purely cohesive slopes undergoing creep. A theoreti-cal method has been proposed by Ito and Matsui (1975)to analyze the growth mechanism of lateral forces act-ing on stabilizing piles when the soil is forced tosqueeze between the piles. In this analysis, the plas-tic state which satisfies the Mohr-Coulomb yield cri-terion is assumed to occur in the soil just around thepiles. The method was developed specifically to calcu-late pressures that act on passive piles in a row. Theassumptions of the theory of plastic deformation are


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following assumptions:1) When the soil layer deforms, two sliding sur-

faces, AEB and A'E'B', occur, making an angle of

(tt/4 +


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Element I

Figure 2 Stresses Acting On Elements in PlasticallyDeforming Ground (After Ito and Matsui, 1975),


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Element I ^ Element II

Ca) Cohesionless Soil

(b) Frictionless Soil

Figure 3 Mohr Circles of Elements in Plastically DeformingGround (After Ito and Matsui, I 979).


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ments are considered, advancing from the center towardsthe edge of area GHH'G' (i.e., in Figure 3, point Nwill move along the failure envelope to point M). How-ever, it would be difficult to analyze such a complexdistribution of stresses. For simplicity, every pointon area GHH'G', except planes GH and G'H', is assumedto be under the same stresses as element I. For planesGH and G'H', it is assumed that the normal stress act-ing on element II, o -._, is equal to the normal stressacting on element I, o _ (Figure 3). It is clear thatthe assumption of a uniform stress distribution forAEBB'E'A' does not consider planes EB and E'B' to beprincipal planes. The stresses acting on those planesare represented by points M and M', and are taken intoaccount in the equilibrium equations written for areaGHH'G' (Equation 3-2 below).

Derivation

The successive steps necessary to derive the forceacting on the pile per unit depth are presented herein.Expressions of this force are given for c-J>, , and cmaterials.

First, the equilibrium of the differential elementGHH'G' surrounded by solid lines in Figure 4a is con-sidered. Summation of forces in the x direction gives:


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25

-co**

(a) /c + ^aTAN 4>

Qo+

iOaJL^ c+

1

x

. d * r

CMQ

|

CTaTANc^)

^x + do;'

(b) T

Figure 4 Stresses on Elements in PlasticallyDeforming Ground,


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where C. is a constant of integration.

Similarly, the equilibrium of the differentialelement shown in Figure 4b will give the state ofstress in zone AEE'A' (Figure 1). Summing all forcesin the x direction:

Dda = 2(o tan + c)dx (3-8)2 x a

Substituting Equation (3-3) into Equation (3-8) andintegrating it:

2N tan$ ,,-C exp ( x) - c(2N / tan

- Z M - . (3-9)x N , tan


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ing the above value in Equation (3-7):

PBB' VScW A [FT^ Tf n*+c)1 9

D 1~D 2 IT A 1/2exp(-5-N^tan* tan(^-|)) - c(2N7 tanf + 1)}

2tan + 2N 1/ 2 + N~ 1/2 2tan* + 2N 1/ 2 + N~ 1/2+c J. f l _ C * D i $1/2 J 1 1/2N ' tan9+N-l N^' tan9 + N^l

where

D, (N 1/ 2 tan9 + N -1)

(3-15)

The lateral force q acting on the pile per unit thick-ness of layer is the difference between the forces act-ing on planes BB' and AA' (Equations (3-10) and (3-15)):

1 VD2 PBB'-D 2(ax) x-0 A C [FTa^{eXP (-D7-%tan *9 *1/2 -1/2

. /0 2tan9 + 2NV + N . 'tan(|f|)) - 2N 1/2tan9-l}H m * *v N/ tan* + N -19 91/2 -1/2

2 tan*+ 2N7 + N . ' . /0-c{D. m x * 2D 2N:1/2 }1 N 1/ 2tan9 + N -1 2 *9 9


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4^2- {A exp(^- N^tan* tan(|f|)) - D2 } (3-16)* 2 2

(3-17)

For a purely cohesive soil the angle of internalfriction is zero and several of the above equations aremodified, Equation (3-6) becomes:

,n da

The state of stress in EBB'E' is obtained by integrat-ing this equation:

o = 3c logD + C3 (3-19)

where C, is a constant of integration. After substi-tuting Equation (3-3) into Equation (3-8) the followingexpression is obtained:

do j^ = jT-dx (3-20)c D2the state of


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stress in AEE'A'

Ox =^x + C4 (3-21)where C, is the constant of integration. From assump-tion 3 the state of stress in plane AA' is:

|ox'x-0 = *~ 2c (3_22)If Equation (3-22) is considered as a boundary condi-tion, substitution in the above expression gives thefollowing for C,

C4 - Y2 - 2c (3-23)

Then the normal stress acting on plane EE' is:

D 1~D2laxlx={(DrD2 )/2} tan(w/8) = c(-^ tan -2)+^ (3-24)

Considering Equation (3-24) as a boundary condition, C_is obtained as:

D-DC3 - c (-g tan| - 3logD2 - 2) + yz (3-25)

Using Equations (3-25) and (3-19), the lateral force,PB ,, on the plane BB' is:


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linearly as y or z increases. The effects of theremaining parameters are more difficult to detect. The

lateral force, F , is plotted versus the ratio ^/D^for different values of c, $, and pile diameter, b, inFigures 5, 6, and 7, respectively. In general, theseresults indicate that the lateral force decreases asthe ratio D 2 /D i increases. It is also apparent thatthe force increases with an increase in 4> or c. Thisis to be expected, since it is harder for a stiffersoil to squeeze between the piles. Finally, thelateral force increases as the pile diameter, D. - D,increases (Figure 7).

Field and Laboratory Measurements

In order to test the validity of the theoreticalapproach for the estimation of the lateral forces onpiles, Ito and Matsui (1975) compared observed fieldvalues to calculated ones. The observations were madein three different locations in Japan, at Katamachi,Higashitono and Kaniyama. Reinforced concrete pilesabout 1 ft in diameter were used in Katamachi and steelpipe piles of about the same diameter were used at theother two locations for the stabilization oflandslides.

The field lateral forces were deduced from meas-ured strains induced in the piles by the sliding soil


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toCL

oIIo

tg 00o

o

COo

o

COc_


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37

masses. The theoretical forces were estimated using cand values obtained from shear and standard penetra-tion tests. The experimental and theoretical resultswere compared and several conclusions were reached.Since the computed and measured forces were found to bewithin the same order of magnitude, it was concludedthat the theory of plastic deformation can be used topredict the force acting on stabilizing piles. Whenthe pile head was restricted, the distribution of theforce was trapezoidal, which agrees with the theory.When the pile head was allowed to deflect, the distri-bution of the force was triangular. In most applica-tions, the condition of restrained pile head will beused and the force on the pile as given by this methodwill give reasonable solutions. As will be seen in theremaining of this work, the bending moments and shearforces acting on a free head pile are much larger thanthose acting on a restrained head pile, assuming thatthe forces given by the theory of plastic deformationapplies to both piles.

In addition to the field tests, Ito and Matsui(1982) performed a series of model tests on pilesplaced in sand or clay. The test apparatus consistedof a soil container with the model piles in a row and alateral loading system. The soil was pushed throughthe piles slowly and its behavior was observed. A


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series of marks were placed at the top of the soillayer to indicate the shape of the flow lines as thesoil moved between the piles. It was observed that thedisturbance of the flow lines appeared to occur mainlywithin the zone of plastic deformation that was origi-nally assumed. This validates the assumption that thesoil is in a plastic condition around the piles.

The relationship between the soil displacement andthe lateral force on the pile was also investigated.It was found that the lateral force increases to ayield value, and then reaches an ultimate value withIncreasing soil displacement. In a logarithmic scalethis relationship can be represented by a bilinearcurve with an Inflection point. The theoretical lateralforces given by Equations 3-16, 3-17, and 3-27 are veryclose to the experimental values at the inflectionpoint (i.e., the yield value). The ultimate lateralforce can be approximately estimated as 1.6 times thetheoretical lateral yield force.

SAFETY FACTOR OF THE STABILIZED SLOPE

The stability of a slope can be investigated by anumber of limit equilibrium methods, including the log-arithmic spiral method, the friction circle method, and


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39

the method of slices. Of these, the friction circlemethod was found to be the most convenient to take intoaccount the force exerted on the slope by the piles inthe limit equilibrium analysis. Parametric studieswere performed to observe the change in the potentialfailure surface with the addition of the piles, and toobtain the relationships between the safety factor ofthe slope and the location and spacing of the piles.

The limit equilibrium calculations are based on anassumed shape of the rupture surface. The safety fac-tor, FS, is defined as the ratio of the shear strengthavailable to the shear strength required to maintainthe slope in a state of limit equilibrium. Assumingthe Mohr-Coulomb failure criterion, the factor ofsafety is given by:

c + o tan FS = a n -2. (3-28)c + a tan


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42

RSIN


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43

The force P is the resultant of the normal andfrictlonal forces concentrated at one point. It isalmost tangent to a circle of radius Rsin as shown inFigure 8.

As a result of the above assumptions, a forcepolygon can be constructed (Figure 8). A detailedmathematical solution for the equilibrium equations ofthe slope is given by Taylor (1937). The following twoexpressions for the stability number were derived byTaylor for a toe failure and a failure below the toe,respectively:

12 2c -7? esc x (ycsc y - coty) + cotx - cotla - - (3-33)F tfl 2cotx cotv + 2cand

ca12 2sr csc x (ycsc y - coty) + cotx - coti - 2r

F 1& 2cotx cotv + 2c-(3-34)

wherex,y = angles describing the


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n = ratio of the distance between the toeof the slope and point A (Figure 8)to the height of the slope.

The safety factor with respect to cohesion on any sur-face defined by angles x and y can be obtained usingEquations 3-33 and 3-34. The true safety factor of anyof these surfaces is obtained through successive itera-tion until F is equal to F , . The critical surface isthe one which minimizes the factor of safety. Thisminimum value is the safety factor of the slope.

When a row of piles is inserted in the slope, theadditional resistance provided by the piles changesboth the safety factor and the critical surface. Thenew slope to be analyzed is shown in Figure 9. Theforces applied to the slope are identical to the onesin Figure 8, with the exception of the force exerted onthe slope by the piles, F . To obtain this force perunit width of the sliding mass, either Equation 3-16,3-17 or 3-27 is integrated along the depth of the pile,and the result is divided by the center to center dis-

Fttance, D. (i.e., F =).1 ' p D '

The resistance force, F , can be incorporated inPthe force polygon (Figure 9) resulting in two newexpressions for the stability number for a toe failureand a failure below the toe, respectively:


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45

Figure 9 Forces on Slopes Reinforced with Piles.


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E -a yH

46

12Fp f cos(CEO) H*- { tt cscx cscy sin* + OG}sin v 2 ' TF ^H , 2 r cos x , , ,. / \ic o esc x cxcy sin {

:

+ csc(u-v) cosCx-u)}T sin v(3-35)

where

2E=l-2(cot i + 3coti cotx - 3coti coty + 3cotx coty)

and

12Fa _ 7H

n{E + 6n - 6nsin


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Equations 3-16, 3-17 and 3-27. This assumption is rea-sonable since each individual pile is preventing move-ment essentially by cantilever action. The directionof F is assumed to be parallel to the line tangent tothe failure surface at the point of intersection ofthat surface with the piles.

The magnitude of F is a function of the length of

the pile from the ground surface to the failure sur-face, shown by distance CE in Figure 9. For each setof angles, x and y, selected to investigate a particu-lar surface in the slope, the distance CE changes and,consequently, the magnitude of the force changes. Totake this into account, the length of the pile abovethe failure surface was expressed as a function of boththe ^circle parameters, angles x and y, and the loca-tion of the pile. For each set of values, x and y,chosen to determine a failure surface, a new pilelength, and thus a new F , is calculated. This forcePis then used in Equations 3-35 or 3-36 to determine anew stability number. The calculations necessary todetermine the length CE are discussed in Appendix B.

Parametric Studies

A computer program was developed to perform thetwo-dimensional slope stability analysis presented in


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48

for this program are given in Appendix C. It can beused to calculate the factor of safety of a slope madeof a homogenous material and reinforced by a single rowof piles. The soil properties are defined by the unitweight, y. cohesion, c, and friction angle, . Theprogram was used to perform parametric studies toinvestigate the effect of the location, spacing, andsize of the piles on the stability of the slope.

When piles are inserted in the slope, the locationof the critical surface changes since an additionalforce, F , is introduced in the limit equilibrium equa-tions. This is illustrated in Figure 10 for a slope ofheight 45 ft and angle 30 degrees with material proper-ties c, , and y equal to 500 psf, 10 degrees and 125pcf, respectively. The original factor of safety ofthe slope (without the pile reinforcement) was 1.08,obtained for the critical surface OAB. After insertionof a row of piles with diameter ratio D2 /D. of 0.6,placed 45 feet upslope, the factor of safety increasedto 1.82 and the critical surface changed to O'AB'.Insertion of piles 70 feet upslope resulted in a factorof safety of 1.64 and the critical surface 0 AB .

Without the piles, any surface below or above thecritical one gives a higher safety factor. With the

addition of the piles, every possible surface in theslope is reexamined. To obtain a factor of safety for


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o u. L. (0IO CO OHO ao a u. oIIs o CM m Qm >. X Q

uo_l>oa.

ll

3

noCO*o


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50

each surface, the weight of the failure wedge, the soilresistance, and the reaction from the piles, F , arebalanced. F is a function of the pile length from theface of the slope to the surface under investigation.As an example, when the surface AEB, which was thecritical surface before the piles were inserted, isexamined, the distance CE is used to calculate thereaction force F . As the surface becomes shallower,P 'this distance decreases and the force F decreases.PAs F decreases, both the effect of F on the safetyP Pfactor and the rate of change of the safety factordecrease. When F is large, it has a predominant effecton the safety factor whereas the other resisting forcesare negligible. As F becomes smaller, the effect ofthese forces becomes more important. Consequently, theintersection of the resulting critical surface with thepiles is located above the original surface (points E'and E in Figure 10).

Figure 11 shows how the location of the piles,indicated by the distance S from the toe of the slope,influences the factor of safety of the same slope for agiven ratio D^/D.. For each value of S the safety fac-tor was computed for both the original critical surface(solid curve in Figure 11) and the new critical surface

which was found after the addition of the piles (dashedline in Figure 11).


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o>exo GO00 *oCO

O GOh-Q>o 4-

COcd00Ct-o Oto o

r\ o+J frtU_o rn o>* or

05-J00 - - -T ro

o oro +Joo_l

o a>OJ CL

_o oLJ

OCJ CVJ

SJ iZ


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52

These results indicate that the safety factor com-puted on the original surface becomes very large for S

between 45 and 75 feet. For this range of values of S,the force exerted by the piles on the original surfaceis so large that the components of soil strength arenot required for equilibrium. For the actual criticalsurface, the reaction force F is smaller and thePactual factor of safety is less than computed for theoriginal surface. The force F is plotted against dis-Ptance S for both the original and the new surface inFigure 12. As an example, for S-45 feet, the forceexerted on the slope (original surface) is equal to49,000 lb/ft, and the factor of safety 2.10, while theactual factor of safety (critical surface) is only1.82, with a force of 30,000 lb/ft.

The safety factor of a steep slope reinforced withpiles reaches a maximum when the piles are placed veryclose to the top of the slope (Figure 13). The criti-cal surfaces of a steep slope remain deep and the fac-tor of safety keeps increasing as S increases until thepiles are placed very close to the top of the slope(Figure 14). Then, the surfaces become shallower, thedistance CE' is very small, and the safety factorstarts decreasing. A comparison of the behavior ofsteep and shallow slopes is given in Figure 15. Thenormalized safety factor, FS/FS , where FS is themax max


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llio< a: wcr->

UJ UJi 1w ^

,. i . . w< Id xo %-1- U. L.rr a: oo_i< < 2z o t*Je> E3IT tr -jo OCL

-8

oCO

8-8 b

CO

>eaot~oU.

u

9

ofO

oCM

.o

co

E

UJC\J

3

UJ/#) J


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54

uiI

UJIcc toUJ UJI- =/%-UJo*CO


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56

U. U. ^ q

COenof-o< mro oo cvj

ro o oo oUJ II II II5f= o o o^ CM W *CO CO CO CO

Li. U. Ll.C\J

COo>uctdcoQCO=>COt_CO>

O)

COoCOO

- oCVJ

o

ocd coX>-

coEco c_

a.oCO

o-o 0>c_a> a>N


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57

maximum possible safety factor for a constant ratioD./D. , is plotted versus the normalized distance, S/L,

where L is the horizontal distance from the toe to thetop to the slope. The figure shows that the piles needto be placed closer to the top of a steep slope thanthat of a shallow slope for a maximum safety factor tobe achieved.

According to the assumptions made by Ito andMatsui (1975), the force acting on the slope is equalto F regardless of the state of equilibrium of theslope. Based on that assumption, the stability numbercan be expressed as:

Cfl/Fc YH - f(F ) (3-37)

However, an overestimation of the force F can lead toPunconservative results in the design of the slope. Amore practical approach for design is to introduce thenotion of a mobilized lateral force, F , where:m

F - F /a (3-38)m pwith a being greater than 1.0. The mobilized force isused to analyze the slope, but the total force per unitlength will be used to design the pile. This results

in a conservative design for both the piles and theslope.


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It is proposed herein to scale the force F by thefactor of safety with respect to cohesion of the rein-forced slope (i.e. a - F ). The resulting stabilitycnumber is:

c /F 1H - f(F /F ) (3-39)a c p c

This equation can be solved by iteration, until F isequal to F. (for the critical surface, F - F, - FS).


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59

Q. LLu.O oz oo 1- IT) ^r n* a.o II II II II

CO e- o >^ I o

fooj

- - ICD Is- CD in

-r~ro CJ

OCD

O

oCD

olO

o

oro

oCVJ

_ o

o

CO

>>o>

CO

oU.

Q.

coN

o0>l_en4>Q

oUJ

CD

a>

en

SJ


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60

safety is illustrated in Figure 17 for a slope ofheight 30 ft, slope angle 49 degrees, and a distance Sof 7 ft. As expected the factor of safety decreasessignificantly as D_/D. increases.

Figure 18 shows the relation between the safetyfactor and the distance S for a steep slope for ratiosD2 /D, of 0.538 (dashed line), and 0.6 (solid line).The force per unit length of the pile given by Equa-tions (3-16) and (3-27) decreases with an increase ofthe ratio D_/D. , regardless of the position of the pileupslope. However, to find the force F , the force perunit length diagram has to be integrated over the dis-tance CE which increases as S increases. Hence, thedifference between the curves becomes larger as Sincreases due to the nonlinear increase of F with S.PAfter the maximum safety factor is reached, the twocurves become asymptotic for the same reason.

The variation of the safety factor with S for dif-ferent values of the friction angle, $, and cohesion,c, was investigated for a slope of height 18 ft, angle30 degrees and a ratio Do/ ] ^ 0.716. The shapes ofthe curves (Figures 19 and 20) are the same asdescribed before. Since F is an increasing functionof $, c and S, its influence on the safety factor alsoincreases with c and $.


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61

1.6 -

.5 H

,y*WASLOPE =49*

C = 500 PSFr=l25 PCFH=30 FTS=7 FT

.4 ~

I .3 -

1.2 -

I.I I i0.50 0.55

i 10.60 0.65 0.70 0.75

D2/D|

Figure I 7 Safety Factor versus Ratio D2/D|.


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62

COroinoII

Q

CO o0_ 0. (-o o o lO Ll.en O o CJ Ocdor

CO 4->CJ cCOc_09CO *-CM *-Q

-CJ 03CJ Cl.O

COo Q.CJ 3CO_a>co~ aCO

CO ^ -=H +*Ou

Jt Ll.>M

CJ CDJCO

lOI-

TO CJ CO

SJa>

o>


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o*~i r-CD CD

CJ CM CM

r-CVJCM

OCM

CO CO-iCM

coCD

3cd>+-ct_a>>> + CDCU COt- co cd - oo -c_ oo c_

m +o u_cd

SJ


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Ll.COQ_CO0.

CDCVJ

COCO

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Ll.CO

O ro

Oto

incvj

OC\J

_ io

CO

CO-r~Cvj

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r~O CO_ _ _ - - oSd

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O-ocoCl

COCD

c o C+-> oW COO CDQ. JZoCO O3CO > COCOc_ =Jo *-> cdo >rtL_ *->c> CD+-> t_CO CDl d-JCO QOCVJCD_V.


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65

SUMMARY

A theoretical approach to the calculation of thelateral forces induced on piles placed in moving soilshas been discussed. Field tests indicate the the pro-posed equations can be used to predict the forces onpiles used in the stabilization of slopes. Laboratory

tests also validate the major assumptions behind thetheoretical derivation. The Friction Circle Method wasadapted to incorporate the resistance provided by therow of piles into the slope stability analysis. Com-puter programs were developed to compute both thelateral force and the factor of safety of the slope.Parametric studies were performed to investigate theeffects of the geometric and material parameters on thefailure surfaces and factors of safety of typicalslopes.


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66

CHAPTER IV. LATERALLY LOADED PILES

In designing the piles to resist lateral loads,the deflection, bending moment and shear force profilesalong the piles are required. In this chapter, thesevalues are obtained by solving the differential equa-tion (beam equation) governing the pile displacements.A closed form solution of this equation is used toanalyze the pile section which extends above the criti-cal surface. The force Intensity on that section iscalculated using the principle of plastic deformationderived in Chapter 3 (Equations 3-16, 3-17, and 3-27).A finite difference method is proposed to analyze thepile section which is embedded below the critical sur-face. The forces acting on this section are calculatedusing the subgrade reaction model. Several techniquesto obtain the modulus of subgrade reaction for the soilor rock below the critical surface are discussed, andrecommendations are made for their use in the pileanalysis calculations.


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THE CONCEPT OF SUBGRADE REACTION

The subgrade reaction model, which was originallyproposed by Winkler in 1867, characterizes the soil asa series of unconnected linearly elastic springs(Hetenyi, 1946). The beam (or pile) reaction at apoint is simply related to the deflection at that

point. One disadvantage of this soil model is the lackof continuity. It is obvious that the displacements ata point are influenced by stresses and forces at otherpoints within the soil. However, the subgrade reactionapproach provides a relatively simple and efficientanalysis and enables factors such as variation of soilstiffness with depth and layering of the soil profileto be taken into account. It is used here to determinethe displacements, bending moments and shear forces ina pile stabilizing a slope.

GOVERNING EQUATIONS

A beam (or pile) supported along its entire lengthby an elastic medium, and subjected to a system of con-centrated forces and distributed loads, deflects andcreates continuously distributed reaction forces In the

that the intensity


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of these forces is proportional to the deflection y ofthe beam (Figure 21a):

P - Ky (4-1)

For a pile of width b, the elastic constant K isrelated to the modulus of subgrade reaction K by:

K = b Kg (4-2)

The differential equation governing the beam dis-placements, y, is obtained by satisfying the equili-brium of an element of length dz (Figure 21b):

.4

HF--dzEI *-f = -Ky + q (4-3)where: E = Modulus of elasticity of the pile

I = Moment of inertia of the pileq = Intensity of the distributed load

The geometry of the pile used to stabilize a slopeis shown in Figure 22. The following notation isadopted in this figure and In the sequel:

z = depth from the ground surfacez = depth from the sliding surface

q. = force/unit length acting on the pileat the ground surface (z = -CE)

q_ = force/unit length acting on the pile at


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y -

Figure 2 I (a) Beam on Elastic Foundation(After Hetenyi, I 946).

M/^ ^ S*kydZ *: 2 qd^-< >- -*

S + dSM + dM

Figure 2 I (b) Cross Section of a Beam on Elastic Foundation(After Hetenyi, I 946).


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Ground Surface;;s>\\sj;;\\\s//

Figure 22 Stabilizing Piles Embedded in Bedrock,


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the failure surface (z = 0)CE length of pile from ground surface to failure

surfaceED length of pile from failure surface to the pile

tipEB = length of pile from failure surface to bedrock

surfaceBD = length of pile from bedrock surface to the

pile tip

It is convenient to decompose the pile equationinto two equations governing the pile deflection aboveand below the failure surface, respectively:

A,EI r=- = q(z) (-CE < z < 0) (4-4a)dz

A2EI ^- = -K y (z > 0) (4-4b)dz

where y. and y are the pile deflections above andbelow the sliding surface respectively, and q(z) is theapplied lateral pressure given by:

q2~q lq(z) = q 2 +-e-L z (*-5>

where q. and q~ are obtained according to the methodol-ogy developed in the previous chapter.


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SOLUTION TECHNIQUES

Equation (4-4a) is a simple beam equation and aclosed form solution can be readily obtained by directintegration. A finite difference scheme was chosen tosolve Equation (4-4b), because closed form solutions tothis equation are difficult to develop when the elasticconstant is varying with depth.

Closed Form Solution of Equation (4-4a)

The solution to Equation (4-4a) is obtained bydirect integration:

f f z 52 3 14 2y x = aQ + 8l z + a^ + a3 z + j^^ z + y^^-gj (4-6a)where

f, = q.

.q 2q i

2 CEand an , a., a > and a_ are constants of integration.The slope, bending moment, and shear force can beobtained by:

dy xSlope = Ii-

d 2y xMoment = -EI ir- (4-6b)


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Shear = -EI r-dz

where

dYi 2 f l 3 , f 2 4I = &1 + 2a2z + 3a3z + ^-gj z + ^Til Zdz

A 1 f fd y, f l 2 2 3T = 2a2 + 6a3 z + 2-EI z + 6H Zdzand

3d y x f x f 2 2T = 6a3 + EI 2 + 2~EI ZdzSign conventions are given in Figure 21b.

Finite Difference Solution of Equation (4-4b)

A finite difference scheme is used to solve Equa-

tion (4-4b). The embedded length below the failuresurface is discretized in MT equally spaced intervalsof length X (Figure 23). The interval length is:

* i (W>and the location of any node m is defined by:

z = mX (4-8)

(4-6c)


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74

ED

1

m-2 | J-rIm- I |_.

Im = |

m = MTI

mt+i pMT + 2 L-

L I_J

Sliding Surface

m-2m- Imm+ Im + 2

Figure 23 Finite Difference Solution for a Pile,


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The discretized form of Equation (4-4b) is:

AEI ( &>m = b(Vm ym (4_9),4m o m mdz

where EI is the pile stiffness, b is the pile diameter,(K) is the modulus of subgrade reaction of the soilo m At node m, y is the deflection at point m, and ( j-)m dz*is the fourth derivative of the displacement at node m.

In general, the modulus of subgrade reactionvaries with depth, and the value (Kc ) in Equation (4-b m9) is different for each node m. This can be takeninto account either by inputting a value for themodulus at each node or by introducing an empiricalvariation with depth. Palmer and Thomson (1948) pro-posed the following expression for the variation of Kwith depth:

KS - KSL ( E>n (4-10a)

where K is the value of Kg at the pile tip (z = ED)and n is an empirical constant equal to or greater thanzero. According to Poulos and Davis (1980), this equa-tion Is widely used to predict pile behavior using thesubgrade reaction approach. It is proposed herein tomodify this expression to account for soil resistanceat the point z = (i.e., at the failure surface).


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Also, two different equations will be used to representthe material above and below the soil rock interface,

respectively:

h KS0 + 1 * ' EB (4-'b)

Ks So + (,Sl- V EB (4-10c>where K = K at z =

KSL KS at Z EB1C = coefficient of subgrade reaction of the

material below z = EBK^ = KR at z = EBKRL

= KRatZ = ED

n. and n are empirical indices greater than or equalto zero. Typical values for K , 1C, n. and n will bepresented in later sections.

These expressions for the soil constant can beintroduced in the finite difference equations to pro-vide a set of simultaneous equations with the nodaldisplacements as unknowns. Finite difference approxi-mations for the first, second, third and fourth deriva-tives of the displacement below the critical surfaceare:

*[2. ym+l ym-lC dz V = 2 A


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( h - -2i _E 5_L (4-H)dz m .2

,d3y2, ym+2 - 2ynH-l + 2ym-l ym-2dz 3 m 2X3

and

/y 2, ym-2 I 4ym-l + 6ym 4ynH-l + ym+2\ 4 ;m Adz A

Substituting equations (4-11), (4-10b) and (4-10c) inthe finite difference equation (4-9), the followingexpressions are obtained for each node m

for z < EB

ym-2-4ym-l+6ym-4ym+l^nH-2 = a b[KS0+(KSL-KSO )( ET)ni]y

m(4-12a)

and for z > EB

ym-2-4ym-l+6ym-4ym+ l4ynH-2 = a ^^O^hL^RO^^^K(4-12b)

where

a = -^ (4-12c)


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The number of equations that the finite differencescheme generates is equal to the number of nodes, MT +1. Four additional imaginary points (Figure 23) areneeded for the complete solution of equation (4-9),which brings the number of unknowns to MT + 5. Inaddition to the MT + 5 unknown displacements, four con-stants of integration, an , a,, a-, and a, are unknowns.This brings the total number of unknowns to MT + 9.

The total number of equations includes the MT + 1 fin-ite difference equations, two boundary conditions atthe pile top, two boundary conditions at the pile endand four continuity equations (at the failure surface),which brings the total number of equations to MT + 9,equal to the number of unknowns. The boundary condi-tions and continuity equations are discussed in thefollowing section.

Boundary Conditions

To ensure continuity of the pile at the slidingsurface, the following continuity relations are neces-sary:

[ y ] z=o = [yi ] z=o = [y2 ] z=o

dy x dy2t6] z=0 = t dz_1 z=0 = [ dz~ ] z=0

a 1 a 1d y, d y,


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If the pile is driven in soft rock the moment andshear at the end of the pile can be assumed to be zero.In this case, the two boundary conditions become:

MT+2 yMT-2 2yMT+l + 2yMT-l

v - 2v + V , = (4-15b)yMT+l yMT yMT-l V

At the top of the pile (z = -CE), the boundaryconditions depend on the type of restraint. One of thefollowing four conditions should model closely the res-traint of the pile in the field.

Condition 1 - Free Head

The moment and the shearing force at the pile head(z = -CE) are zero:

'M1 Z~CE EI t-rU-CE dzand, (4-16)

*\m z~CE--EI l-T-'z-CE-dzSubstituting equation (4-6c) into the above equationsgives


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2a 2 - 6a 3CE + ^ CE* - ^| CE 3 =and, (4-17)

f l f 2 26a3 EI CE + 2ET CE

The following relations are obtained for the moment andshear by substituting equation (4-14) in equation (4-17):

(CE)y_2 + (2A - 2CE)y_ 1 - (4A)yQ + (2 A + 2CE)y x - (CE)y 2

3 2 3 12= 2XJ [gg| CEJ - ^eY CE^] (4-18a)and

f f-y_ 2 + ( 2 )y_r ( 2 )y + y 2 2x3 tfl CE HI ce2] ( 4_18b >

Condition 2 - Unrotated Head

The slope and the shearing force at the pile head(z = -CE) are zero.

[6] z=-ce = i^r-WcE- (4 19a)

[V] Z=-CE = EI [TT- ] z=-CE = < 4 19b >dz

The final expression for the shear is the same as


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the one for the free head condition (Equation 4-18b).The expression for the slope is obtained by substitut-ing equation (4-6c) into equation (4-19a):

a^ - 2a2 CE + 3a3 CE - ~: CE + ^ |= CE - (4-20a)Introducing the values of a., a , and a~ given in equa-tion (4-14), the following relation is obtained:

(-CE 2 )y_2+(2CE2-4 ACE-2 A2 )y_ 1+(8 XCE)yQ+(2 \

2-4 ACE-2CE2 )yi

+ (CE2 )y 2 = 4 A3 [^gi CE 3 - jjL. CE 4 ] (4-20b)

Condition 3 - Hinged Head

In the hinged head condition, the deflection andthe moment at the pile head (z = -CE) are zero:

^z=-CE = frll,-CB = (42U)

[Ml z=-CE = EI hA-CE = (421b)The final expression for the moment is the same as

the one for free head condition of equation (4-18a).The expression for the deflection is obtained by sub-stituting equation (4-6c) into equation (4-2 la):


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a -aiCE + a 2 CE2 - a3 CE

3 + ^T CE T20~EI CE * (4-22a)

Introducing aQ , a^ a 2 , and a 3 given in Equation (4-14), the following relation is obtained:

(CE3 )y_2 + (6X2CE + 6XCE 2 - 2CE 3 )y_ 1 + (12X

3 - 12XCE 2 )y

+(6XCE2 -6A2CE + 2CE3 )yi - (CE3 )y 2 - 12

r L_ CE4 + * CE 5 ] (4-22b)1 24EI 120 EI J

Condition 4 - Fixed Head

The deflection and the slope at the pile head (z =-CE) are zero:

fvl = [v 1 =0 (4-23a)lyJ z=-CE iy l J z=-CE

t^U-cE-^rWcE- (4-23b>

The final expression for the deflection is thesame as the one for the hinged head condition given byequation (4-22b), and the expression for the slope isthe same as the one for the unrotated head conditiongiven by equation (4-20b).


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COEFFICIENTS OF SUBGRADE REACTION

Coefficients of Subgrade Reaction for Soils

The theory of subgrade reaction Is used to modelthe pile-soil interaction below the failure surface.Estimates of the coefficient of subgrade reaction, K ,are necessary to evaluate the pile displacements,moments and shear forces. In this section, publishedcriteria and data on K are reviewed and recommenda-tions are made for their use in design applications.

Two fundamental assumptions are made in thistheory: (1) The coefficient of subgrade reaction, K ,at every point is independent of the contact pressure,and (2) it has the same value at every point along thecontact face. Both of these assumptions are approxima-tions of the true conditions. Loading tests performedon actual subgrades show that the settlement does notincrease linearly with increasing pressure. The assump-tion of linearity is usually not valid for pressureslarger than about half the ultimate bearing capacity ofthe soil. This limit should always be taken into con-sideration in problems involving coefficients ofsubgrade reaction. The second assumption is also anapproximation. The subgrade reaction does not have the


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85

Depending on the elastic properties of the subgrade,the pressure at the rim is either greater or smallerthan at the center. However, the errors resulting fromthis assumption can be neglected in practical problems(Terzaghi, 1955).

The coefficient of subgrade reaction is generallydetermined by one of the following three methods: (1)

full scale lateral-loading tests; (2) in situ testssuch as plate loading tests, pressuremeter and flatdilatometer tests; and, (3) empirical correlations withother soil properties.

There are several techniques to perform lateralloading tests on piles. One method is to instrumentthe pile so that the soil pressures and the piledeflections are measured directly (Matlock and Ripper-berger, 1958; Bishop and Mason, 1954). This methodprovides a direct evaluation of K , but it is time con-suming and relatively expensive. A more convenientprocedure is to measure the ground line deflection androtation and back-calculate K assuming an approximatedistribution with depth (Reese and Cox, 1969; Welch,1972). A method that has been used successfully in thepast is the placement of strain gages at points alongthe pile. The strain readings are converted to momentvalues by the use of calibration curves. Then, deflec-


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86

The use of plate loading tests has been discussedby Terzaghi (1955). The main problem with this

approach is the extrapolation of the results. Terzaghiassumed that the coefficient of horizontal reaction fora vertical pile surrounded by sand is a function of thedepth below the surface, z, the width of the pile, b,the effective unit weight of the sand, y', and therelative density of the sand. At a depth z below thesurface, the modulus of elasticity of sand, E , issgiven by:

E = y'zA (4-24)s

where A is a dimensionless coefficient depending onlyon the relative density of the sand (Table 1). Ter-zaghi proposed the following relationship between Kand E :s

Ks = u33b (425a)

Substitution of equation (4-25a) into equation (4-24)gives:

Ks - TTH F (425b >

For a pile embedded in stiff clay, Terzaghi recom-mended a constant value of K with depth:


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Table 1Values of dimensionless coefficient A, to calculate kg(tons/ft ) for a pile embedded in moist or submergedsand (Terzaghi, 1955).

Relative density of sand Loose Medium Dense

Range of values of A 100-300 300-1000 1000-2000Recommended values of A 200 600 1500


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KS = d~b KS1 (426)where K,,. is the coefficient of vertical subgrade reac-tion of a plate of width equal to one foot. In thisequation b is in feet, and the value of K is obtained

3 in tons/ft . Recommended values of K are given inTable 2.

Additional suggestions for the calculation of Kinclude the equations proposed by Chen (1978), whichare based on the pressuremeter modulus, E . In cohe-sionless soil:

E,Kg = 3.3 f- (4-27a)

and in cohesive soil:

E,Kg = 1.6 ^ (4-27b)

2where E, is the pressuremeter modulus in tons /ft andd3K is given in tons /ft .

Another relation that uses the pressuremetermodulus was derived by Yoshida and Yoshinaka (1972):

Ks = ^-Ed (b) 1/4 (4-28)Empirical correlations between K and the Young's

Modulus, E , have been suggested by several authors,s

Vesic (1961) extended the work of Biot (1937) for a


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Table 2 3Values of k . in tons/ft for a square plate (lxl ft)resting on pre compressed clay (Terzaghi, 1955).

Consistency of Clay Stiff Very Stiff Hard2Values of q (tons/ft )u

Range for k . (square plates)Recomended values (square plates)

1-2 2-4 >450-100 100-200 >200

75 150 300


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beam placed on an infinite elastic foundation and sub-ject to a concentrated load. His results indicated

that the coefficient of subgrade reaction for a longbeam can be expressed as:

1 - s

where v is the Poisson ratio and E is the Young'ss smodulus of the subgrade. Broms (1964) and Francis(1964) suggested that this equation can be applied topiles, and they used it to estimate the modulus K forthe lateral resistance of piles. Francis, however,considered that the medium extends on both sides of thepile, and concluded that the above expression should bedoubled:

1 s

Equation (4-29b) can be used for either sand orclay subgrades. For long piles in soft clay, Gibson,in an unpublished report, suggested that a soil stiff-ness E be assessed for two different conditions,sinstantaneous and long term loading. Following thissuggestion, E may be calculated as either: (1) a5secant modulus measured at 50% of the ultimateundrained triaxial compression test for instantaneous


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91

loading (u should be taken as 0.5 for this condition);sor (2) as for the long term condition (um sv0.40), where m is the modulus of volume decrease

vobtained from oedometer or drained triaxial tests. Forpiles in sand, any of the equations given in Table 3can be used to calculate the Young's modulus. Theseequations have been used to predict vertical staticcompression and can be used in the case of lateralloading under the assumption that the soil is isotropicand homogeneous. Correlations between E and the coneresistance, q , need some additional experimentalverification and should be used with caution (Jamiol-kowski and Garassino, 1977).

In general, it is expected that the coefficient ofsubgrade reaction will increase with depth in sand andwill either increase or remain constant with depth inclay. Terzaghi (1955) recommends values of nl = 1 (tobe used in Equation 4-10) in sand and nl = in clay.Davisson and Prakash (1967) suggested that nl = 0.15 isa more realistic value for clay under undrained condi-tions. Broms (1974) stated that the coefficient ofsubgrade reaction for cohesive soils is approximatelyproportional to the unconfined compressive strength.As the unconfined compressive strength of normally con-solidated clay increases linearly with depth, the coef-ficient of subgrade reaction should increase in a simi-


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Documents

Stabilization of Slopes Using Piles_ Interim Repor