《高等土力学》课程教学资源（书籍文献）United States Steel - Steel Sheet Piling Design Manual

Steel Sheet PilingUssDesign ManualUnitedStatesSteelUpdated and reprinted by U.S.Department of Transportation/FHWA withpermission.July1984

Steel Sheet Piling Design Manual United States Steel Updated and reprinted by U. S. Department of Transportation /FHWA with permission. July 1984

SteelSheet PilingSectionsSectonMomentoDrnv.AreaWeightInertiaModulusBWebPerPerPerThickSectionDistriPerPerFootProfiletancePerSquarePerFootnessIndexRolleperPileofofFootPilePileFootPileWallW2!!of WallIn,In.In.In.In.Lbs.In.In.?Lbs.-5.13.312.943.72.4PSX32H.16%44.032.029/64mwOKd2.92.411.773.6PS32*H.S.1540.032.0V21.9292.8153.5PS28H.S.35.028.03%2.41.910.30a6.04.51637.33.32.510.98H.28.0y2PSA28*P9rK5.54.13.22.48.99H.S.1630.723.03%PSA235'H.S.53.039.8PDA271636.027.03%g14.310.710.59oaR313.7PMA2222.03/g.5.410.5922.4H.S.195%36.08.83%*LT'%12PZ381857.038.03%70.246.816.77421.2280.8H.5PZ322156.0H.32.03%67.038.316.47385.7220.402115'ISSsSR'A1840.527.045.330.211.91276.3PZ27H.3g184.212".3/8PZ 222222.03/34.891.1H.0.39.011.91679%*SectionsPS32 and PSA28ate infre-quentlyrolled andwedo notadvisetheirSuggestedAllowableDesignStresses-SheetPilinguse in adesign unless an adequateton-Minimum YieldAllowable Design Stress,nage can beorderedatonetimeto assureSteel Brand or GradePoint, psipsi*a minimum rolling55.00035,000USS-EX-TEN55(ASTMA572GR55Complete data regarding these sections50,00032.000USS EX-TEN50(ASTMA572GR50)will be found in a separate publication50,00032,000USSMARINERSTEELentitled "USS Steel Sheet Piling?45,00029,000USS EX-TEN 45 (ASTM A572 GR 45)H-Homestead, Pa.(Pittsburgh District)25,00038,500RegularCarbonGrade(ASTMA328)S-South Chicago (Chicago District)*Based on 65% of minimum yield point. Some increase for temporary Overstressesgenerally permissible

Complete data regarding these sections will be found in a separate publication entitled “USS Steel Sheet Piling? H-Homestead, Pa. (Pittsburgh District) S-South Chicago (Chicago District) Sections PS32 and PSA28 ate infre￾quently rolled and we do not advise their use in a design unless an adequate ton￾nage can be ordered at one time to assure a minimum rolling. Suggested Allowable Design Stresses-Sheet Piling Steel Brand or Grade Minimum Yield Point, psi Allowable Design Stress, psi* USS-EX-TEN 55 (ASTM A572 GR 55) USS EX-TEN 50 (ASTM A572 GR 50) USS MARINER STEEL USS EX-TEN 45 (ASTM A572 GR 45) Regular Carbon Grade (ASTM A 328) *Based on 65% of minimum yield point. Some increase for temporary Overstresses generally permissible

Steel Sheet PilingUssDesign ManualNotice"The information, including technical and engineering data, figures, tablesdesigns, drawings, details, suggested procedures, and suggested specificationspresented in this publication are for general information only.Whileevery effort has been madeto insure its accuracy,this information should not be usedor relied upon for any specific application without independent competentprofessional examination and verification of its accuracy,suitability andapplicability.Anyonemakinguse of thematerialdoes soathisown riskandassumes anyand all liabilityresultingfromsuchuse.UNITED STATEs STEELCORPORATIONDISCLAIMSANYANDALLEXPRESSORIMPLIEDWARRANTIESOF MERCHANTABILITYFITNESSFOR ANYGENERAL OR PARTICULARPURPOSEOR FREEDOM FROM INFRINGEMENT OF ANY PATENT,TRADEMARK,OR COPY-RIGHT INREGARD TOINFORMATION OR PRODUCTS CONTAINED OR REFERREDTo HEREIN. Nothing herein contained shall be construed as granting a license,expressorimplied,underanypatents.Notice by U.S. DOTThis document is disseminated under the sponsorship of the Department ofTransportation in the interest of information exchange. The United StatesGovernment assumes no liabilityfor its contents oruse thereof.The contentsof thisreport reflecttheviews of the UsS,whois responsibleforthe accuracyof thedatapresentedherein.Thecontentsdonotnecessarilyreflecttheofficial views or policy of the Department of Transportation. This report does notconstitute a standard, specification, or regulation.The United States Government does not endorse products or manufacturers.Trade or manufacturers'names appear herein onlybecausetheyareconsid-ered essential totheobjectofthisdocument.Updated and reprinted byFHWAwith permission, July, 1984USSandEX-TENareregisteredtrademarks

Steel Sheet Piling Design Manual Notice “The information, including technical and engineering data, figures, tables, designs, drawings, details, suggested procedures, and suggested specifications, presented in this publication are for general information only. While every ef￾fort has been made to insure its accuracy, this information should not be used or relied upon for any specific application without independent competent professional examination and verification of its accuracy, suitability and applicability. Anyone making use of the material does so at his own risk and assumes any and all liability resulting from such use. UNITED STATES STEEL CORPORATION DISCLAIMS ANY AND ALL EXPRESS OR IMPLIED WARRANTIES OF MERCHANTABILITY FITNESS FOR ANY GENERAL OR PARTICULAR PURPOSE OR FREEDOM FROM INFRINGEMENT OF ANY PATENT, TRADEMARK, OR COPY￾RIGHT IN REGARD TO INFORMATION OR PRODUCTS CONTAINED OR REFERRED TO HEREIN. Nothing herein contained shall be construed as granting a license, express or implied, under any patents.” Notice by U.S. DOT This document is disseminated under the sponsorship of the Department of Transportation in the interest of information exchange. The United States Government assumes no liability for its contents or use thereof. The contents of this report reflect the views of the USS, who is responsible for the accuracy of the data presented herein. The contents do not necessarily reflect the offi￾cial views or policy of the Department of Transportation. This report does not constitute a standard, specification, or regulation. The United States Government does not endorse products or manufacturers. Trade or manufacturers’ names appear herein only because they are consid￾ered essential to the object of this document. Updated and reprinted by FHWA with permission, July, 1984 USS and EX-TEN are registered trademarks

TABLEOFCONTENTSUS Steel Sheet Piling SectionsProfiles and PropertiesInside Front CoverAllowableDesignStressesand Rankine CoefficientsInside Back Cover4FOREWORD5PILEWALLSLATERAL PRESSURES ON SHEET5EarthPressureTheories5Rankine Theory7Coulomb Theory9Log-Spiral Theory12Soil Properties14Surcharge Loads14Uniform Surcharge15Point Loads15Line Loads16Strip Loads17Effects on Unbalanced Hydrostatic and Seepage Forces18OtherLateral Loads18Ice Thrust18Wave Forces18Ship Impact18Mooring Pull18Earthquake Forces19DESIGN OFSHEET PILE RETAININGWALLS19General Considerations19Cantilever Walls20Cantilever Sheet Piling in Granular Soils23Cantilever Sheet Piling in Cohesive Soils26Anchored Walls26General.27Free Earth Support30Rowe'sMoment ReductionTheory33Fixed Earth Support35Graphical Methods37DanishRules39HighSheet Pile Walls40Stabilityof Sheet PileWalls42DESIGN OF ANCHORAGESSYSTEMSFORSHEETPILEWALLS42Tie Rods42Wales

TABLE OF CONTENTS USS Steel Sheet Piling Sections Profiles and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . Inside Front Cover Allowable Design Stresses and Rankine Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . Inside Back Cover FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 LATERAL PRESSURES ON SHEET PILE WALLS . . . . . . . . . l . . . . . . 5 Earth Pressure Theories . . 5 Rankine Theory . . 5 Coulomb Theory . . 7 Log-Spiral Theory . . 9 Soil Properties . . 12 Surcharge Loads . . Uniform Surcharge . . Point Loads . . Line Loads . . Strip Loads . . Effects on Unbalanced Hydrostatic and Seepage Forces . . . . . . . . . . . . . 17 Other Lateral Loads . . 18 Ice Thrust . . 18 Wave Forces . . 18 Ship Impact . . 18 Mooring Pull . . 18 Earthquake Forces . . 18 DESIGN OF SHEET PILE RETAINING WALLS . . . . . . . . . . . . . . . . . . 19 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Cantilever Walls . . 19 Cantilever Sheet Piling in Granular Soils . . 20 Cantilever Sheet Piling in Cohesive Soils . . 23 Anchored Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General . . Free Earth Support . . Rowe’s Moment Reduction Theory . . Fixed Earth Support . . Graphical Methods . . Danish Rules . . High Sheet Pile Walls . . Stability of Sheet Pile Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . DESIGN OF ANCHORAGE SYSTEMS FOR SHEET PILE WALLS . . . . . . . . Tie Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 14 15 15 16 26 26 27 30 33 35 37 39 40 42 42 42

44Anchors44Location of Anchorage45Sheet Pile AnchorWalls45DeadmenAnchors47AnchorSlab Design Based on Model Tests47General Case in Granular Soils55AnchorSlab in Cohesive Soils55H-Pile A-Frames55H-Pile Tension Ties56DESIGNOF COFFERDAMSFORDEEPEXCAVATIONS56General57Lateral Pressure Distribution59Sizing of Braced Cofferdam Components59Sheet Piling60Wales60Struts61Raking Braces62Circular Bracing62PrestressedTiebacks63Stability of Braced Cofferdams.63Heaving in Soft Clay64Piping inSand-68CELLULARCOFFERDAMS.68General.68Circular Type.69Diaphragm Type.69Cloverleaf Type.69Modified Types69Components of Cellular Cofferdams70General DesignConcepts72Stability of Cofferdams on Rock72Sliding on Foundation73Slipping Between Sheeting and Cell Fill73ShearFailureonCenterline of Cell75Horizontal Shear (Cummings'Method)76Interlock Tension77Cofferdams on Deep Soil Foundations77General77Stability79Underseepage80Pull-Out ofOuterFace Sheeting80Hansen's Theory.83BIBLIOGRAPHY85DESIGNEXAMPLES

Anchors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Location of Anchorage . . 44 Sheet Pile Anchor Walls . . 45 Deadmen Anchors . . 45 Anchor Slab Design Based on Model Tests . . . . . . . . . . . . . . . . . . . 47 General Case in Granular Soils . . 47 Anchor Slab in Cohesive Soils . . 55 H-Pile A-Frames . . 55 H-Pile Tension Ties . . 55 DESIGN OF COFFERDAMS FOR DEEP EXCAVATIONS . . . . . . . . . . . . . 56 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Lateral Pressure Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Sizing of Braced Cofferdam Components . . 59 Sheet Piling . . 59 Wales . . 60 Struts . . 60 Raking Braces . . 61 Circular Bracing . . 62 Prestressed Tiebacks . . 62 Stability of Braced Cofferdams . . 63 Heaving in Soft Clay . . 63 Piping in Sand . . 64 CELLULAR COFFERDAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Circular Type . . 68 Diaphragm Type . . 69 Cloverleaf Type . . 69 Modified Types . . 69 Components of Cellular Cofferdams . . 69 General Design Concepts . . 70 Stability of Cofferdams on Rock . . 72 Sliding on Foundation . . 72 Slipping Between Sheeting and Cell Fill . . 73 Shear Failure on Centerline of Cell . . 73 Horizontal Shear (Cummings’ Method) . . 75 Interlock Tension . . 76 Stability Cofferdams on Deep Soil Foundations . . 77 General . . 77 Underseepage . . 79 Pull-Out of Outer Face Sheeting . . 80 . . 77 Hansen’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 DESIGN EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

FOREWORDThis manual is directed to the practicing engineer concerned with safe, economicaldesigns of steel sheetpileretaining structures.The content isdirected basically towardthe designer'stwo primary objectives:overall stability ofthe structural systemandtheintegrityofitsvariouscomponents.Emphasis isplaced on step-by-stepproceduresfor estimating the external forces on thestructure,evaluatingtheoverall stability,andsizingthesheetpilingandotherstructuralelements.Graphs and tablesareincluded to aid the designer inarriving at quicksolutionsThree basic types of sheet pile structures are considered: (3) cantilevered and anchoredretaining walls,(2)braced cofferdamsand(3)cellular cofferdams.Consideration is alsogiventothedesignofanchoragesystemsforwallsandbracingsystemsforcofferdamsThe design procedures included in this manual are in common use today by mostengineers involved in the design of sheet pile retaining structures.These methods haveconsistentlyprovided successful retaining structuresthathaveperformedwell in service.However, in using these procedures, one should not be lulled into a false sense of securityabout the accuracy of the computed results.This is especially true with regard to lateralearthpressuresonretainingstructures.The simplifyingassumptionsinherentinanyofthese procedures and their dependence on the strength properties of the soil provide onlyapproximationsto realityIt is assumed throughout that the reader has a fundamental knowledge of soilmechanics andaworkingknowledgeof structural steel design.It isfurtherassumedthatthesubsurface conditions and soil properties at the site of theproposed construction havebeen satisfactorily established andthedesignerhas-chosenthetypeof sheetpilestructurebestsuitedtothesite

FOREWORD This manual is directed to the practicing engineer concerned with safe, economical designs of steel sheet pile retaining structures. The content is directed basically toward the designer’s two primary objectives: overall stability of the structural system and the integrity of its various components. Emphasis is placed on step-by-step procedures for estimating the external forces on the structure, evaluating the overall stability, and sizing the sheet piling and other structural elements. Graphs and tables are included to aid the designer in arriving at quick solutions. Three basic types of sheet pile structures are considered: (3) cantilevered and anchored retaining walls, (2) braced cofferdams and (3) cellular cofferdams. Consideration is also given to the design of anchorage systems for walls and bracing systems for cofferdams. The design procedures included in this manual are in common use today by most engineers involved in the design of sheet pile retaining structures. These methods have consistently provided successful retaining structures that have performed well in service. However, in using these procedures, one should not be lulled into a false sense of security about the accuracy of the computed results. This is especially true with regard to lateral earth pressures on retaining structures. The simplifying assumptions inherent in any of these procedures and their dependence on the strength properties of the soil provide only approximations to reality. It is assumed throughout that the reader has a fundamental knowledge of soil mechanics and a working knowledge of structural steel design. It is further assumed that the subsurface conditions and soil properties at the site of the proposed construction have been satisfactorily established and the designer has-chosen the type of sheet pile structure best suited to the site

LATERALPRESSURESONSHEETPILEWALLSEARTHPRESSURETHEORIESEarth pressure is the force per unit area exerted by the soil on the sheet pile structure.Themagnitudeoftheearthpressuredependsuponthephysical propertiesofthesoil,theinteraction at the soil-structure interface and the magnitude and character of thedeformations in the soil-structure system. Earthpressure is also influenced by thetime-dependent nature of soil strength,whichvaries due to creep effects and chemicalchanges in the soil.Earth pressureagainst a sheet pile structure is nota unique function for each soil,butratherafunctionofthesoil-structure system.Accordingly,movementsofthestructureare a primary factor in developing earth pressures. The problem, therefore, is highlyindeterminate.Two stages of stress in the soil are of particular interest in the design of sheet pilestructures,namely theactiveand-passivestates.When avertical plane,such as a flexibleretainingwall,deflects undertheactionof lateral earthpressure,eachelementof soiladjacent to the wall expands laterally,mobilizing shear resistance in the soil and causing acorresponding reduction in the lateral earth pressure.One might say that the soil tends tohold itself upby its boot straps;that is, by its inherent shear strength.The lowest state oflateralstress,whichisproducedwhenthefull strengthofthesoil isactivated(astateofshearfailure exists),is called the active state.Theactive state accompanies outwardmovement of the wall. On the other hand, if the vertical plane moves toward the soil,such as the lower embedded portion of a sheet pile wall, lateral pressure will increase asthe shearing resistance of the soil is mobilized.When the full strength of the soil ismobilized,thepassivestateof stress exists.Passive stresstends toresistwall movementsand failure.Thereare two well-known classical earth pressure theories;the Rankine Theory and theCoulomb Theory.Each furnishes expressions for active and passive pressures fora soilmassatthestateoffailure.Rankine Theory-The Rankine Theory is based on the assumption that the wallintroduces no changes intheshearing stresses at the surface of contact between the walland thesoil.Itis alsoassumedthatthegroundsurfaceisastraight line(horizontal orslopingsurface)andthataplanefailuresurfacedevelopsWhentheRankinestateoffailurehasbeenreached,activeand passivefailurezoneswilldevelopasshowninFigure1.AARailureZoneailureZoneWallWallMovementMovement0-045*+@/2FAFFEFFAFACTIVECASEPASSIVE CASEof soil (degreesFig.1-Rankinefailurezones

LATERAL PRESSURES ON SHEET PILE WALLS EARTH PRESSURE THEORIES Earth pressure is the force per unit area exerted by the soil on the sheet pile structure. The magnitude of the earth pressure depends upon the physical properties of the soil, the interaction at the soil-structure interface and the magnitude and character of the deformations in the soil-structure system. Earth pressure is also influenced by the time-dependent nature of soil strength, which varies due to creep effects and chemical changes in the soil. Earth pressure against a sheet pile structure is not a unique function for each soil, but rather a function of the soil-structure system. Accordingly, movements of the structure are a primary factor in developing earth pressures. The problem, therefore, is highly indeterminate. Two stages of stress in the soil are of particular interest in the design of sheet pile structures, namely the active and-passive states. When a vertical plane, such as a flexible retaining wall, deflects under the action of lateral earth pressure, each element of soil adjacent to the wall expands laterally, mobilizing shear resistance in the soil and causing a corresponding reduction in the lateral earth pressure. One might say that the soil tends to hold itself up by its boot straps; that is, by its inherent shear strength. The lowest state of lateral stress, which is produced when the full strength of the soil is activated (a state of shear failure exists), is called the active state. The active state accompanies outward movement of the wall. On the other hand, if the vertical plane moves toward the soil, such as the lower embedded portion of a sheet pile wall, lateral pressure will increase as the shearing resistance of the soil is mobilized. When the full strength of the soil is mobilized, the passive state of stress exists. Passive stress tends to resist wall movements and failure. There are two well-known classical earth pressure theories; the Rankine Theory and the Coulomb Theory. Each furnishes expressions for active and passive pressures for a soil mass at the state of failure. Rankine Theory - The Rankine Theory is based on the assumption that the wall introduces no changes in the shearing stresses at the surface of contact between the wall and the soil. It is also assumed that the ground surface is a straight line (horizontal or sloping surface) and that a plane failure surface develops. When the Rankine state of failure has been reached, active and passive failure zones will develop as shown in Figure 1. ACTIVE CASE PASSIVE CASE where = angle of internal friction of soil (degrees) Fig. 1 - Rankine failure zones

The active and passive earth pressures for these states are expressed by the followingequations:Pa=ZKa-2cVKaPp=KZp+2c VKpwhere Paandpp=unitactiveand passiveearthpressure,respectively,atadepthzbelowtheground surfaceZ =vertical pressure at a depth Z due to the weight of soilabove,usingsubmergedweightforthesoilbelowgroundwater levelc =unit cohesive strength of soilKa and Kp=coefficientsofactiveand passiveearth pressures,respec-tivelyThe coefficients Ka and Kp,according to the Rankine Theory,are functions of theβ--angle of the soil and the slope of the backill, .They are given bythe expressionscosβ-Vcos"β-cos"gKa=cosβcosβ+cos?β-cos?gcosβ+cos"β-cos"sKp=cosβcosβ-Vcos"β.-.cos"pNotethatforthecaseof a level backfill,theseequationsreduceto1 sing= tan2 (45Φ/2)Ka "1+ sing1+sing= tan (45+Φ/2)Kp =1- sindThe triangular pressure distributions for a level backfill are shown in Figure 2.For variousslope conditions, referto Mechanics of Soils by A.Jumikis.16cosp-cos"β-cos"o3Ka"cosβ-Pa=YZ tan (45-号）Pa=YZKacosβ+Jcosβ-cos"sPp=Z tan (45+g)P,"TZKp-Ko-cop cono+/0-c.7-unit weightcosB-/cos'β-cosof soil, (pef)Poal.saC saLevel BackillSloping BackfillFig. 2 Rankine earth pressure (after Teng")

The active and passive earth pressures for these states are expressed by the following equations: where and = unit active and passive earth pressure, respectively, at a depth Z below the ground surface = vertical pressure at a depth Z due to the weight of soil above, using submerged weight for the soil below ground water level = unit cohesive strength of soil and = coefficients of active and passive earth pressures, respec￾tively The coefficients Ka and Kp, according to the Rankine Theory, are functions of the -angle of the soil and the slope of the backfill, . They are given by the expressions Note that for the case of a level backfill, these equations reduce to The triangular pressure distributions for a level backfill are shown in Figure 2. For various slope conditions, refer to Mechanics of Soils by A. Jumikis.16 (a) Granular Soil Level Backfill Granular Soil Sloping Backfill Fig. 2 - Rankine earth pressure (after Teng1 )

can (45+)PpZtan （45号）+2ctan (45+号tan(45-Cohesive Soil-Active PressureCohesive Soil -Passive PressureFig. 2 - (Continued)Coulomb Theory-An inherentassumption of theRankineTheory is that thepresenceof thewall does not affectthe shearing stressesatthe surface ofwall contact.However,since the friction between the retaining wall and the soil has a significant effect on thevertical shear stresses in the soil,thelateral stressesonthe wall are actuallydifferentthanthose assumed by the Rankine Theory. Most of this error can be avoided by using theCoulomb Theory,which considers the changes in tangential stress along the contactsurface due to wall friction.As the wall yields,thefailurewedge tends to move downward for the active case.Forthe passive case, where the wall is forced against the soil, the wedge slides upward alongthe failure plane.These differential movements involve vertical displacements betweenthe wall and backfill and create tangential stresses on theback of the wall due to soilfriction and adhesion. The resulting force on the wall is,therefore,inclined at an angle tothe normal to thewall.This angleis known asthe angle of wall friction,.For theactivecase,whentheactivewedge slides downward relative to thewall,is taken as positiveForthepassive case,whenthepassivewedge slidesupwardrelativetothewall,istakenas negative.If the angleof wall friction is known,the following analytical expressionsforK, and K, in the horizontal direction for a vertical wall are:cos'SKasin(+の）sin(@-β)cosScoso cosβcos"dFsin(@+8)sin(@+β)COSOcosScosβwhereΦ=angle of internal friction of soil=angle of wall frictionβ =angle of the backfill with respect to horizontalFigure 3(a) is included for ease in obtaining Ka and KpAsintheRankine Theory,theCoulombTheoryalso assumesaplanesurfaceoffailure.However,theposition of thefailureplaneisafunctionofboththe-angleofthesoilandthe angle of wall friction,.The position of the failure plane for the active and passivecases fora level backfill isgiven by:tang+tanp(tanp+cotp)(1+tang.cotp)αa = 90°--arctan1+tang (tanp+cotp)tang+tanp(tanp+cot$)(1+tan.cotp)αp=90°+α-arctan1+tano (tanp+cot)

Cohesive Soil - Active Pressure Cohesive Soil - Passive Pressure Fig. 2 - (Continued) Coulomb Theory - An inherent assumption of the Rankine Theory is that the presence of the wall does not affect the shearing stresses at the surface of wall contact. However, since the friction between the retaining wall and the soil has a significant effect on the vertical shear stresses in the soil, the lateral stresses on the wall are actually different than those assumed by the Rankine Theory. Most of this error can be avoided by using the Coulomb Theory, which considers the changes in tangential stress along the contact surface due to wall friction. As the wall yields, the failure wedge tends to move downward for the active case. For the passive case, where the wall is forced against the soil, the wedge slides upward along the failure plane. These differential movements involve vertical displacements between the wall and backfill and create tangential stresses on the back of the wall due to soil friction and adhesion. The resulting force on the wall is, therefore, inclined at an angle to the normal to the wall. This angle is known as the angle of wall friction, For the active case, when the active wedge slides downward relative to the wall, is taken as positive. For the passive case, when the passive wedge slides upward relative to the wall, is taken as negative. If the angle of wall friction is known, the following analytical expressions for Ka and Kp in the horizontal direction for a vertical wall are: where = angle of internal friction of soil = angle of wall friction = angle of the backfill with respect to horizontal Figure 3(a) is included for ease in obtaining and As in the Rankine Theory, the Coulomb Theory also assumes a plane surface of failure. However, the position of the failure plane is a function of both the -angle of the soil and the angle of wall friction, The position of the failure plane for the active and passive cases for a level backfill is given by:

1008.01407.01206.0--1005.0EE--802-/I#/oo-Kto1Dy/oy-V804.00-----603.01312l1/KpS010/n/07S402.04一Ka/201.0C0=005=3002Ka-1050203040ANGLEOFINTERNALFRICTION,INDEGREESFig.3(a)-Coulomb earth pressure coefficient vs.Φ-angle for level backfll and dredge linewhereαaandαp=anglebetweenthefailureplaneandtheverticalfortheactiveandpassivecases,respectivelyFigure 3(b)shows theCoulombactiveandpassivefailure wedges togetherwiththeCorrespondingpressuredistributions.8

0 10 2 0 3 0 40 50 ANGLE OF INTERNAL FRICTION, IN DEGREES Fig. 3(a) - Coulomb earth pressure coefficient vs. -angle for level backfill and dredge line where and =angle between the failure plane and the vertical for the active and passive cases, respectively Figure 3(b) shows the Coulomb active and passive failure wedges together with the Corresponding pressure distributions. 8