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1、What is the Future for Slope Stability Analysis?(Are We Approaching the Limits of Limit Equilibrium Analyses?)Dr. Delwyn G. FredlundUniversity of Saskatchewan, CanadaUniversity of Saskatchewan, CanadaSecond Symposium and Short Course on Unsaturated Soils and Environmental GeotechnicsBudapest, Hungar
2、yNovember 4-5, 2003溜仪茶珐烯涟陀剃摆氯痔故微尔辉酱说爽全策磅呸围桩困奶烟糊巧嗓堑戏FREDLUND边坡稳定未来讲义（精品）Japan ConferenceIntroduction qqLimit Equilibrium methods of slices have been “Good” for the geotechnical engineering profession since the methods have produced financial benefit qqEngineers are often surprised at the results they
3、 are able to obtain from Limit Equilibrium methodsSo Why Change?胃醚炎孵彭捆肇殊雹频瘪镜帚疟拢搽臣右峭碱盂寓渤病矽白首铬液女薄姥FREDLUND边坡稳定未来讲义（精品）Japan ConferenceThere are Fundamental Limitations with Limit Equilibrium Methods of Slices?The boundaries for a FREE BODY DIAGRAM are not known-The SHAPE for the slip surface must be a
4、ssumed-The LOCATION of the critical slip surface must be found by TRIAL and ERROR浩魂礁砾洞娜眉目膳界丑夷沸遣蹬斜圣畴挫免遍臭敲阴睛渔慷逝碰穷纱嘘FREDLUND边坡稳定未来讲义（精品）Japan ConferenceSHAPE and LOCATION are the driving force for a paradigm shiftObjectives of this Presentation:qTo show the gradual change that is emerging in the way th
5、at slope stability analyses can be undertakenqTo illustrate the benefits associated with improved procedures for the assessment of stresses in a slope坷紊枫割虞叹宵项赦穴莽仙夸捍高差纸墓壬剔薛溢牧侵撮氟夸痪趴蓖漫榷FREDLUND边坡稳定未来讲义（精品）Japan ConferenceOutline of PresentationqqProvide a brief Summary of common Limit Equilibrium metho
6、ds along with their limitations (2-D & 3-D)qqTake the FIRST step forward through use of an independent stress analysisqqTake the SECOND step forward through use of Optimization Techniques哇周谁凑替狡章堕室酪拔笛楚示瞻误菲皇谷架阉喻鬼盅撮醚弱霉檬沏秒捎FREDLUND边坡稳定未来讲义（精品）Japan ConferenceIs a Limit Equilibrium Analysis an Upper Boun
7、d or Lower Bound Solution?qqLimit Equilibrium Methods primarily satisfy the requirements of an upper bound type of solutionReason: the shape of slip surface is selected by the analyst and thereby a displacement boundary condition is imposed痔冈充欧戈酥鹊缓杯腐烁唤英蛛屹单欣株惹僚垢活给珊痘柿峭毅孵师午扣FREDLUND边坡稳定未来讲义（精品）Japan Co
8、nferenceLimit Equilibrium and Finite Element Based Methods of AnalysesWWWWWWWWNLimit EquilibriumMethod of AnalysisSm = t ta dldls sndlFinite Element Based Method of Analysisldt ta dlQUESTION: How can the Normal Stress at the base of a slice be most accurately computed?Consider the Free Body Diagrams
9、 used to calculate the Normal Stress?奢秃合受祟陨炙滁踞勺侦铝路倾闻任亦氛纺斑龋疏粕五嗅栓挫祟镀校愈咏FREDLUND边坡稳定未来讲义（精品）Japan ConferenceAssumption for all Limit Equilibrium AnalysisqqSoils behave as Mohr-Coulomb materials (i.e., friction, (i.e., friction, , and cohesion, c) , and cohesion, c) qqFactor of safety, Fs, for the cohes
10、ive component is equal to the factor of safety for the frictional componentqqFactor of safety is the same for all slices( () ) msnsSFuFc= =- -+ +b b s sb btan飞苹河晓他嘱躲悉抿渊逃晶琢筷讫梧读片彝潞俞蜡殃送辱牛膘沁残霍耽菱FREDLUND边坡稳定未来讲义（精品）Japan ConferenceSummary of Available Equations Associated with a Limit Equilibrium Analysi
11、sqqEquations (knowns): Quantityl lMoment equilibriumnl lVertical force equilibriumnl lHorizontal force equilibriumnl lMohr-Coulomb failure criterionn4n坊戊颠发枯罚档复索鹅伶陕熙猜蹲大恒种斌晋茄善茸酿哗付衍顾势狈阮秧FREDLUND边坡稳定未来讲义（精品）Japan ConferenceqqUnknowns:Unknowns: Quantity Quantityl lTotal normal force at base of slice Tota
12、l normal force at base of slice n nl lShear force at the base of slice, SShear force at the base of slice, Smmn nl lInterslice normal force, EInterslice normal force, En-1n-1l lInterslice shear force, XInterslice shear force, Xn-1n-1l lPoint of application of interslice force, EPoint of application
13、of interslice force, E n-1n-1l lPoint of application of normal forcePoint of application of normal forcen nl lFactor of safety, FFactor of safety, Fs s1 1 6n-26n-2Summary of Unknowns Associated with a Limit Equilibrium AnalysisOne FOne Fs s per sliding mass per sliding mass慕紊恨讽堑给炊捅看波旷臂滦攒雪借薯泳些禽耘绍谰饺鼠移
14、伍鳞寒足驮淄FREDLUND边坡稳定未来讲义（精品）Japan ConferenceForces Acting on Each SliceFocus on SmbyxSmXREREXLSlip surfaceGround Ground surfacesurfaceWhRN = s snb bb bfNLPhreatic linePhreatic line姨杭饺炯壬洱浩府断助国磅抗边答阅喇晰宝撤闹峡人倔诊欠皮瑚旅渊裤烽FREDLUND边坡稳定未来讲义（精品）Japan ConferenceMobilized Shear Force, Sm forSaturated-Unsaturated Soi
15、ls( () ) ( () )swasansmFuuFuFcSb b b b s sb bbtantan- -+ +- -+ += =Only new variable required for solving saturated-unsaturated soils problems is the shear force mobilized b b = Friction angle with respect to matric suction= Friction angle with respect to matric suctionu ua a = Pore-air pressure = P
16、ore-air pressureu uww = Pore-water pressure= Pore-water pressure 囚较掠傅妆故袱预滇成锚凉安碳慷佛鳞钝癌簇汽殖灭承俐借圆拖篓煤婿戌FREDLUND边坡稳定未来讲义（精品）Japan ConferenceqMoment equilibrium, Fm:qqForce equilibrium , Ff: - - - -+ += =NfWxtanRtantanuNR cFbwm b bb b - -+ += =a aa a b ba ab bsincostantantancosNuNcFbwfPore-air pressures are
17、 assumed to be zero gauge碾谦晒唱厘吝多击影突哼厦羽泥叠异俊许痈惯拖叙琴烷猖反瞩鸟壮闺稳秩FREDLUND边坡稳定未来讲义（精品）Japan ConferenceqqNormal force at base of slice:qqLimit Equilibrium methods differ in terms of how (XR-XL) is computed and overall statics satisfiedqqLimit Equilibrium problem is indeterminate:l lCan apply an assumption Can
18、 apply an assumption (Historical solution)(Historical solution)l lCan utilize additional physics Can utilize additional physics ( (Future solution)Future solution)( () )FFuFcXXWNbwLRtansincostansinsin a aa a a ab ba ab b+ + +- - - -= =逸喊游陛乍盆漾匣冲犬修严铭羞八跃牌佐阶迂衷祸借挨利摔蓄却虚束誓晌FREDLUND边坡稳定未来讲义（精品）Japan Confere
19、nces sxbaArea = Interslice normal force (E)width of slice, b bs sxt txys syDistance (m)Elevation (m)t txybaArea = Interslice shear force (X)Vertical sliceDistance (m)Elevation (m) = =baxydyXt t = =baxdyEs sStresses on the Boundary Between SlicesMorgenstern & Price, 1965抖故逐中眩饭亏舵贞攻悟你详凑渣兆造同闹匙安戈闪阳腮最底突桃时
20、唐邱FREDLUND边坡稳定未来讲义（精品）Japan ConferenceSummary of Limit Equilibrium Methods and AssumptionsMethod Equilibrium Satisfied Assumptions Ordinary Moment, to base E and X = 0 Bishops Simplified Vertical, Moment E is horizontal, X = 0 Janbus Simplified Vertical, Horizontal E is horizontal, X = 0, empirical
21、correction factor, f0, accounts for interslice shear forces Janbus Generalized Vertical, Horizontal E is located by an assumed line of thrust Spencer Vertical, Horizontal, Moment Resultant of E and X are of constant slope 弛阶罚改踢闭叔呈侄拯爪晾蔡俐怎亿橇刷案特腾锻虐三麓肝论婴盯匿傻蚂FREDLUND边坡稳定未来讲义（精品）Japan ConferenceForces Act
22、ing on One Slice in Ordinary or Conventional MethodhWbb ba aN = s bN = s b nN NSm仙挚宝龄牧好欲硝讶赔四迭逻妖威中得霜芳制涡脂拘荡岗哟髓慈弧颐杆紫FREDLUND边坡稳定未来讲义（精品）Japan ConferenceForces Acting on One Slice in Bishops Simplified and Janbus Simplified Methods hWbb ba aN = s bN = s b nN NSmEREL跨寡烩淘庐毡溃塞妮维寸雅焊胁磷肪乃挖卉砸岳我样昂吏抑笋寇震氦友丑FREDLU
23、ND边坡稳定未来讲义（精品）Japan ConferenceSummary of Limit Equilibrium Methods and AssumptionsDirection of X and E is the average of the ground surface slope and the slope at the base of a sliceVerticalHorizontalLowe and KarafiathDirection of X and E is parallel to the groundVerticalHorizontalCorps of Engineers
24、Direction of E and X is defined by an arbitrary function. Percent of the function required to satisfy moment and force equilibrium is called VerticalHorizontalMomentMorgenstern-Price, GLEAssumptionsEquilibriumSatisfiedMethod 瑞偏土卑颇剧伪彼娄度不赦偷讣缝代慢央惨御暂新菲穷丈谭畅侠否垦垫足FREDLUND边坡稳定未来讲义（精品）Japan ConferenceForces
25、Acting on One Slice in Spencers, Morgenstern-Price, andGLE Methods hWbb ba aN = s bN = s b nN NSmERELXRXL舟竖积梭头鸽谆凭褂险挖冗勃耸掘赡贵拼户着更阉少绍机新陀德坐颖桓鸟FREDLUND边坡稳定未来讲义（精品）Japan ConferenceVarious Interslice Force FunctionsProposed byMorgenstern & Price (1965)Spencers酌柏蛹饱切已捣媳吵悠由崭舆而阮芜荡俯严涩纸镀嫉镜冀粉叭以羚痰椽凌FREDLUND边坡稳定未来讲义
26、（精品）Japan ConferenceWilson and Fredlund (1983)Used a finite element stress analysis (with gravity switched on) to determine a shape for the Interslice Force FunctionInterslice Force Function for a Deep-seated Slip Surface Through a 1:2 SlopeX = E f(x)痛盔搬劫崎磷涸者见冉十射竭施锨榜仪诗焉粉自哺缀霍啸辕铆促七锹胺携FREDLUND边坡稳定未来讲义（
27、精品）Japan ConferenceDefinition of Dimensionless Distance f(x) is largest at mid-pointInflection points near crest & toeGeneralizedGeneralizedFunctionalFunctionalShapeShape巫甜禹畜巍姬撑庭块烘酿垮讥挪扬擎熙闷鞘住梗客肖谋沼指支述庚硼很冰FREDLUND边坡稳定未来讲义（精品）Japan Conferencewhere:K = magnitude of function at mid-slopee = base of natura
28、l logC = variable to define inflection pointn = variable to define steepness = dimensionless x-position( () )2/)(nnCKexfw w- -= =Wilson and Fredlund, 1983X = E f(x)Dimensionless DistanceDimensionless Distance镭疾永蛮瞎奈脉院拍航伊吴棺焚赢拓兹捕涂回玻裸票良埠剪挎肠汹站凋避FREDLUND边坡稳定未来讲义（精品）Japan ConferenceUnique function of “slop
29、e angle” for all slip surfaces“C” coefficient versus slope angle坯芒砚糯尺诡姚掸吊袍佯父尽楷葵麓纂钝巳旅酮箱墓疯麦抖忌厢谈暑蝗协FREDLUND边坡稳定未来讲义（精品）Japan ConferenceUnique function of “slope angle” for all slip surfaces“n” coefficient versus tangent of slope angle福眩右夸帘渝毕瞳熊恃隙诱蔽陨屋簿贿奈腊峪盟袭脓纱迷稗谓近导啄塞蛛FREDLUND边坡稳定未来讲义（精品）Japan ConferenceC
30、omparison of Factors of Safety Circular Slip Surface00.20.40.61.801.851.901.952.002.052.102.152.202.25l lJanbus GeneralizedSimplified BishopSpencerMorgenstern-Pricef(x) = constantOrdinary = 1.928FfFmFredlund and Krahn1975Factor of safety醋僚赘宿五绷伶包毅盏汕泣加浆锻窖宁丙雀阜伊婴湖排孩捧卞垛蓖菜灾没FREDLUND边坡稳定未来讲义（精品）Japan Confe
31、renceMoment and Force Limit Equilibrium Factors of SafetyFor a Circular typeslip surfaceMoment limit equilibrium analysisForce limit equilibrium analysisFredlund and Krahn, 1975Lambda, l l Factor of safety垦咽苔举免净难炭做爷杖瞄牙债烽蓟亿社丢说戍览倔沸捂蔓汤军抱让吹楞FREDLUND边坡稳定未来讲义（精品）Japan ConferenceForce and MomentLimit equil
32、ibriumFactors of Safety for a planar toe slip surfaceForce limit equilibrium analysisMoment limit equilibrium analysisLambda, l l Factor of safetyKrahn 2003咯刘愈鄂很凝爱历渡克剂埔哼泰镁词陪趾禄身圃拴涯拄兽羔腰郡捧傻恭利FREDLUND边坡稳定未来讲义（精品）Japan ConferenceForce and MomentLimit equilibrium Factors of Safety for a composite slip sur
33、faceMoment limit equilibrium analysisForce limit equilibrium analysisLambda, l l Factor of safetyFredlund and Krahn 1975纽录樊捏现议释腆祟琉辞日槛驼扇菩靛崎脏诵道庸驰喀疯那芍耀卖缀历鼓FREDLUND边坡稳定未来讲义（精品）Japan ConferenceForce and MomentLimit equilibrium Factors of Safety for a “Sliding Block” type slip surfaceMoment limit equilibr
34、ium analysisForce limit equilibrium analysisLambda, l l Factor of safetyKrahn 2003澳洒赠症爹碎牧箩胚掘影易邮激饲吵逞轰八陡行衡舀仟兰玩悄蔑磨蓖萨煎FREDLUND边坡稳定未来讲义（精品）Japan ConferenceExtensions of Methods of Slices toThree-dimensional Methods of ColumnsqqHovland (1977) 3-D of OrdinaryqqChen and Chameau (1982) 3-D of Spencer qqCavou
35、nidis (1987) 3-D Fs 2-D FsqqHungr (1987) 3-D of Bishop SimplifiedqqLam and Fredlund (1993) 3-D with f(x) on all 3 planes; 3-D of GLE严恕崇栓墨才颖天训绕柿茹叭跟类审酚涯孔班冗竿吉湛椒惹槛蚕育总佩署FREDLUND边坡稳定未来讲义（精品）Japan ConferenceShape and Location Become Even More Difficult to Define in 3-D貌嘎娇拦给稀串亏悯劝捕乒铝们抢扼酌赚载祝崖概掸惧铅粉迪瞎声育小黍FREDLU
36、ND边坡稳定未来讲义（精品）Japan ConferenceTwo Perpendicular Sections Through a 3-D Sliding MassSection Parallel to MovementSection Perpendicular to Movement霄版逗棒宰解氨辞珠杏狄蜗清名獭革死榷曹凭苯淘订顾型唉翻坦堆瞪围扶FREDLUND边坡稳定未来讲义（精品）Japan ConferenceFree Body Diagram of a Column with All Interslice ForcesParallelPerpendicularBase啃班启纠沸灭写
37、叫盲刀篆收攘吃卯劈挫哆痴峪掂亦沈六份斟整远沂捕徐薛FREDLUND边坡稳定未来讲义（精品）Japan ConferenceInterslice Force Functions for Two of the DirectionsX/EV/P盅汀边砾洞淳稽澡迎抉疙帛峪修霓纯徐态凳友羹卿屹阵募洁窿旬轮但诣列FREDLUND边坡稳定未来讲义（精品）Japan ConferenceFirst Step ForwardQuestion:qIs the Normal Stress at the base of each slice as accurate as can be obtained?qIs the
38、 Normal Stress only dependent upon the forces on a vertical slice?Improvement of Normal Stress ComputationsFredlund and Scoular1999酬梦轨人桂纱估缓左荡忌翌肿手茧侍僻权此射唆临据原厚戴啥愤赃胚郸邓FREDLUND边坡稳定未来讲义（精品）Japan ConferenceLimit equilibrium and finite element normal stresses for a toe slip surfaceFrom limit equilibrium ana
39、lysisFrom finite element analysis钒砂痰队蝗派坠戮艘惶叔牙愈妒渣欺牺莱租捆俗傻髓受付棕梧咸浇婚蹭瓤FREDLUND边坡稳定未来讲义（精品）Japan ConferenceLimit equilibrium and finite element normal stresses for a deep-seated slip surfaceFrom finite element analysisFrom limit equilibrium analysis阴恕夜践捶惟蹿涩挥光佐舀跟意萝包旨倦尺销豫囊葬惧石醚明仅辟硝扣梦FREDLUND边坡稳定未来讲义（精品）Japan
40、 ConferenceLimit equilibrium and finite element normal stresses for an anchored slopeFrom finite element analysisFrom limit equilibrium analysis轨挫吼壁位哮晾兑拳苯滴氓炙挝芋萍脾华圃马大未登落凉己潜界绩异痴腕FREDLUND边坡稳定未来讲义（精品）Japan ConferenceqqTo illustrate procedures for combining a finite element stress analysis with concepts
41、of limiting equilibrium. (i.e., finite element method of slope stability analysis) qqTo compare results of a finite element slope stability analysis and conventional limit equilibrium methods Using Limit Equilibrium Concepts in a Finite Element Slope Stability AnalysisObjective:臀救抱枝蚀饭毯杂驻糙苹膛嗽衔诉驮掉眠宝寺环
42、饶拽饱肪薛默际淆拖煌君FREDLUND边坡稳定未来讲义（精品）Japan ConferenceqqThe complete stress state from a finite element analysis can be “imported” into a limit equilibrium framework where the normal stress and the actuating shear stress are computed for any selected slip surface HypothesisAssumption: The stresses computed
43、 from“switching-on” gravity are more reasonable than the stresses computed on a vertical slice幂达戏明俱伎世抠腻济堰雹扦应疾差役恿殖锅婿遂悬问黔终丁俄谦洪臼劝FREDLUND边坡稳定未来讲义（精品）Japan ConferenceManner of “Importing Stresses” from a Finite Element Analysis into a Limit Equilibrium Analysis s snFinite Element Analysis for StressesLi
44、mit Equilibrium Analysiss snt tmMohr Circlet tmIMPORT:Acting Normal StressActuating Shear StressLimit Equilibrium AnalysisFinite Element Analysis for Stresses上楼隶煽媚砒址纲菏貌赶鼠搂勿瘴悄灰毒匿耙棉畴碌崖愚镊稚织炎敲检俊FREDLUND边坡稳定未来讲义（精品）Japan ConferenceqqBishop (1952) - stresses from Limit Equilibrium methods do not agree wit
45、h actual soil stresses qqClough and Woodward (1967) - “meaningful stability analysis can be made only if the stress distribution within the structure can be predicted reliably” qqKulhawy (1969) - used normal and shear stresses from a linear elastic analysis to compute factor of safety “Enhanced Limi
46、t Strength Method” Background to Using Stress Analyses in Slope Stability习墒臭秤六酥礼栏蛊占洲荔啤蛮弟贯骇缀瞧骸汽芒团妄犊钓要杯舞舀喇侩FREDLUND边坡稳定未来讲义（精品）Japan ConferenceStress LevelRezendiz 1972Zienkiewicz et al 1975Strength & Stress LevelAdikari and Cummins 1985Enhanced limit methods(finite element analysiswith a limit equili
47、briumFinite Element Slope Stability MethodsDirect methods(finite element analysis only)Strength LevelKulhawy 1969F - - Z = 1313 D DD DLLfs ss ss ss s F=(c + tan ) - - c + tanA 1313 s s s ss ss ss ss s D D D DL Lf*F = (c + tan ) K s s t tD D D DLLDefinition of Factor of SafetyLoad increaseto failureS
48、trength decreaseto failureanalysis)挞涣耶暑爹獭镀挡何项每寻瘪棚挪侨氢间撑储示霄荆致柔续局慌翻肌阮践FREDLUND边坡稳定未来讲义（精品）Japan ConferenceDifferences and Similarities Between the Finite Element Slope Stability and Conventional Limit Equilibrium qqDifferencesl lSolution is determinatel lFactor of safety equation is linearqqSimilaritie
49、sl lStill necessary to assume the shape of the slip surface and search by trial and error to locate the critical slip surface窘镣成臭鹃恫碰稀颖革巨病丽披铂里梆僻窍翰区曝阔液浇瓷忍沏股脖迎蟹FREDLUND边坡稳定未来讲义（精品）Japan ConferenceWhy hasnt Finite Element Slope Stability Method been extensively used? qqDifficulties and perceptions relat
50、ed to the stress analysisqqInability to transfer large amounts of data and find needed information Now: Microcomputer have dramatically changed our ability to combine Finite Element and Limit Equilibrium analyses揪垛句榔裕故程腔补辱包乖厚寻俩惶纯躬寨在愉淖弓乳殃摧摩鼓驰列臃诸FREDLUND边坡稳定未来讲义（精品）Japan ConferenceDefinition of Factor
51、 of Safety qKulhawy (1969)qwhere:l lSr = resisting shear strength or l lSm = mobilized shear force = =mrFEMSSFb b s stan)u( cSwnr- -+ += =Actuating ShearActuating ShearNormal StressNormal Stress烷艰刮仅健医罗籍功捷洲枝绸翱糖娃外阉序怕脸恍辐陨换猴钥药泄誓垄辐FREDLUND边坡稳定未来讲义（精品）Japan ConferenceAnalysis Study Undertaken by Fredlund
52、and Scoular (1999) qqAdopted the Adopted the Kulhawy (1969)Kulhawy (1969) procedure procedureqqUsed Sigma/W and Slope/W Used Sigma/W and Slope/W qqPoissons ratio range = 0.33 to 0.48Poissons ratio range = 0.33 to 0.48qqElastic modulus, E = 20,000 to 200,000 kPaElastic modulus, E = 20,000 to 200,000
53、kPaqqCohesion, c = 10 to 40 kPaCohesion, c = 10 to 40 kPaqqFriction, Friction, = 10 to 30 degrees = 10 to 30 degreesqqCompared Compared conventional Limit Equilibrium conventional Limit Equilibrium results with Finite Element slope stability results with Finite Element slope stability results result
54、s 淮崎畜矫凝逢舅野肇吧掉腿宴捎赴尤奥更刊热睁鉴胆兔肯留卒吓颅乍闺读FREDLUND边坡稳定未来讲义（精品）Japan ConferenceLocation of Center of a Section along the Slip Surface within a Finite Element Analysisxyx-Coordinatey-CoordinateSlip SurfaceFinite Element(r, s)srFictitious slice defined withthe Limit Equilibrium analysisCenter of the base of a
55、slice (x, y)酬牺隔祥仲枚旗婚肮碘烃粮骑姆乖矾妄蚜兼迪锑云褐羞询昌商瑟揭僚吭湃FREDLUND边坡稳定未来讲义（精品）Japan ConferencePresentation of Finite Element Slope Stability Results qqConditions Analyzed:l l Dry slopel l Piezometric line at 3/4 height, exiting at toel l Dry slope, partially submergedl l Piezometric line at 1/2 height and submerg
56、ed to mid-height览秘柱潞摧驻常座帮咖贸幕炬瓣艰窗迪左撤缩岁摇伦熬攀吭低盏殉苟蔗致FREDLUND边坡稳定未来讲义（精品）Japan ConferenceSelected 2:1 Free-Standing Slope with a Piezometric Line Exiting at the Toe of the Slope20406080100120204060800CrestPiezometric LineToe21x - Coordinate (m)Note: Dry slope with & without piezometric liney - Coordinate
57、 (m)蓬载播雨迪裤夹虚歪跪空斑畦初资症醒螺巳合刨擂车岳刻墓莲坐蟹苯筑相FREDLUND边坡稳定未来讲义（精品）Japan ConferenceSelected 2:1 Partially Submerged Slope with a Horizontal Piezometric Line at Mid-Slope20406080100120204060800CrestToe21x - Coordinate (m)WaterPiezometric Liney - Coordinate (m)Note: Dry slope with & without piezometric line琶造卑尘梨
58、奸睹抢溢挠帚柒盼焉葵旁彦村吉级静雁洗咽抒链梅田写涸嫁账FREDLUND边坡稳定未来讲义（精品）Japan Conference050100150200250300203040506070x-Coordinate (m)Acting and restricting shear stress (kPa)CrestToeShear StrengthShear ForcePoisson Ratio, m m = 0.33Shear Strength and Shear Force for a 2:1 Dry Slope Calculated Using the Finite Element Slope
59、 Stability Method歌杖吼拌录堕荫岳势梧目沧盎斋宴糟耗茫浩讽酸纬躇柱击喻庸阵嗣利怎糟FREDLUND边坡稳定未来讲义（精品）Japan ConferenceLocal and Global Factors of Safety for a 2:1 Dry Slope012345672025303540455055606570x-CoordinateFactor of SafetyCrestToeLocal F (m mLocal F (m m= 0.33)Bishop Method, F = 2.360 = 2.173Global Factors of SafetyBishop 2
60、.360Janbu 2.173GLE (F.E. function) 2.356Fs (m m = 0.33 ) 2.342Fs (m m = 0.48 ) 2.339Ordinary 2.226sJanbu Method, FsssFs = 2.342Fs = 2.339 = 0.48)克奈倔彻跟纂乞魔箭绪引玄邻紧状雨违雄镣乔衍跳邯碱依声楚移裳辩瞥湖FREDLUND边坡稳定未来讲义（精品）Japan ConferenceFactors of Safety Versus Stability Number for a 2:1 Dry Slope as a Function of c 0.00.5
61、1.01.52.02.50510152025Stability Number, g g Htan /c Factor of Safetyc = 20kPac = 10kPac = 40kPaFs ( m m = 0.33)Fs ( m m = 0.48)Fs (GLE)2:1 Dry Slope 泊惨掺绅狼车霍土枷边弃剃非昨溜臆裤姚标吸遂培攒存厢力顿廓幕捷渝乐FREDLUND边坡稳定未来讲义（精品）Japan ConferenceFactor of Safety Versus Stability Coefficient for a 2:1 Dry Slope as a Function of
62、0.00.51.01.52.02.50.000.020.040.060.080.100.12Stability Coefficient, c /g g H Factor of Safety = 30 = 10 = 202:1 Dry SlopesFs ( m m = 0.33)F (m m = 0.48)Fs (GLE)s扑炎班家挺珠溃诱桓佃快麻婚盆卢阐利此嫌酮决姜赊宋块巨禽剁爪桓翟弛FREDLUND边坡稳定未来讲义（精品）Japan ConferenceFactor of Safety Versus Stability Coefficient as a Function of for 2:1
63、 Slope with a Piezometric Line 0.00.40.81.21.62.00.000.020.040.060.080.100.12Stability Coefficient, c/ g g H Factor of safety = 30 = 20 = 102:1 Slope with piezometric lineFs ( m m = 0.33)Fs ( m m = 0.48)Fs (GLE)住咎陨舞模肯妹弟厂踞稳剧厦洛刮钻唁窜坏汽柏抵亨黑赃锭放炎哨视证磨FREDLUND边坡稳定未来讲义（精品）Japan ConferenceLocation of the Criti
64、cal Slip Surface for a Slope with a Piezometric Line with Soil Properties of c = 40 kPa and = 3070102010060504030908070110506040102030x - Coordinate (m)80GLE (F.E. function)Fs (m m = 0.33)Fs (m m = 0.48)MethodXYRFactor of safetyGLE (F.E. Function) 58.556.037.91.741Fs (m m = 0.33)57.549.534.71.627Fs
65、(m m = 0.48)57.553.037.81.661Y- Coordinate (m) 舱疏貉津闷瞬浑毛绸兄荐墒右总皮包博你把角技旗剔火翟莲国坍斩渔泥妓FREDLUND边坡稳定未来讲义（精品）Japan ConferenceLocation of the Critical Slip Surface for a Slope with a Piezometric Line where the Factor of Safety is Closest to 1.070102010060504030908070506040102030110x - Coordinate (m)80Fs (m m =
66、 0.4 8 )Fs (m m = 0.33)GLE (F.E. function )sMethodXYR Factor of safetyGLE (F. E Function .63.559.039.61.102Fs (m m = 0.33)63.059.041.51.076F (m m = 0.48)61.559.542.31.100y- Coordinate (m)裤闯魂匙钠氦斤痰巩拧酋瞳灸坞哨虞撰馈农屏买秋酸歇霓媚稻蚁帧氧敖挣FREDLUND边坡稳定未来讲义（精品）Japan ConferenceFactor of Safety Versus Stability Coefficient
67、 as a Function of for 2:1 Dry Slope, 1/2 Submerged0.00.51.01.52.02.53.03.50.000.020.040.060.080.100.12Stability Coefficient, c / g g HFactor of Safety = 20 = 102:1 Dry slope, one-half submerged = 30Fs ( m m = 0.33)Fs ( m m = 0.48)Fs (GLE)内驳冷水浅佩荐泪七积葱倪忽币哼疆遭绩蓉岁至哮苇乏钱搅苞稀雌盅睡喻FREDLUND边坡稳定未来讲义（精品）Japan Conf
68、erenceFactor of Safety Versus Stability Coefficient as a Function of for 2:1 Slope Half Submerged with Piezometric Line0.00.51.01.52.02.50.000.020.040.060.080.100.12Stability Coefficient , c /H Factor of Safety = 30 = 10 = 202:1 Slope, one-half submergedg gFs ( m m = 0.33)Fs ( m m = 0.48)Fs (GLE)移汹抢
69、颤孜拌套原听宅烬蝶跃椎砂稠麦猜钾赂陈狙闸灰蔑秀歌囚吼撕撅筋FREDLUND边坡稳定未来讲义（精品）Japan Conference1020100605040309080701107050604010203080Fs (m = m = 0.33)Fs (m m = 0.48)GLE (F.E. Function)sMethodXYR Factor of safetyGLE (F.E. Function 58.058.5 40.22.303Fs (m m = 0.33)52.550.5 31.82.259F (m m = 0.48)51.551.5 31.02.273Location of the
70、 Critical Slip Surface for a Half Submerged Slope where the Soil Properties are c = 40 kPa and = 30x - Coordinate (m) y - Coordinate (m) 来伙脑鞋尼丙邵棒彼挽九倾槐惩硬痈耙存趟谤班芬蛙恨徐辊秤垄蓉屏吕箍FREDLUND边坡稳定未来讲义（精品）Japan ConferenceConclusions from Step 1 ForwardqqNormal and Actuating Shear stresses from a finite element anal
71、ysis appear to provide a more reasonable representation of the stress state in a slopeqqThe Enhanced Limit method by Kulhawy (1969) appears to open the way to simulate more complex slope stability problemsqqEnhanced Limit methods can readily be used in routine engineering practice 详摘幕蹲领厢抚宗允施铡锻博淡舆冕榴旺
72、睡涤蛇禄侮杏撒立水筷韧比钉岁FREDLUND边坡稳定未来讲义（精品）Japan ConferenceGlobal factors of safety appear to be essentially Global factors of safety appear to be essentially the same for most simple slopesthe same for most simple slopesqqSelection of Selection of Poissons ratioPoissons ratio has some effect has some effect
73、 on the Enhanced Limit factor of safetyon the Enhanced Limit factor of safetyqqFactors of Safety appear to differ slightly for:Factors of Safety appear to differ slightly for:l lLow cohesion valuesLow cohesion valuesl lHigh angles of internal frictionHigh angles of internal frictionHow do the Result
74、s from Enhanced Limit Methods Compare to Limit Equilibrium Methods? Local Factors of Safety can also be computed by the Enhanced Limit Method缉沽剑骤猾录企袒回劈固敖铭酱翠抠诌诌盘缸锗讨拴瑰午嫡匿岂躇夷髓勺FREDLUND边坡稳定未来讲义（精品）Japan ConferenceSecond Step ForwardQuestion:qqIs it possible for the computer to determine the Shape of the
75、 critical slip surface?qqIs it possible for the computer to determine the Location of the critical slip surface?Improvement on Shape and Location Ha and Fredlund2002宛剪豫雾岿痹蜕扰藤侣祁撤传曝秋硬杯扬温秀翱氛刷催酵施蓟磅唯想茂冈FREDLUND边坡稳定未来讲义（精品）Japan ConferenceqqOptimization Techniques (i.e., Dynamic Programming) can be used t
76、o find the pathway which minimizes a function of the shear strength available to the actuating shear stress within a soil mass HypothesisAssumption: The stresses computed from“switching-on” gravity can be used to represent the stress state in the soil mass诧内情让乡痉视农蚌含冯予代翅疲葛窜民男痴壹荚蚤沟裴现萤竞甄密锁乡FREDLUND边坡稳定
77、未来讲义（精品）Japan ConferenceSlope Stability Analysis Using Dynamic Programming Combined with a Finite Element Stress AnalysisqqDynamic Programming (DP) optimization techniques for slope stability analysis (Spencers Method) was introduced by Baker (1980)qqYamagami & Ueta (1988) and Zou et al. (1995) impr
78、oved on the Baker (1980) solution by coupling Dynamic Programming with a Finite Element stress analysis叛耽贬切袖呼美蚜恬扼罐症憾早八吟谜扒啥佛醇捅型馅原疹霄捏偏晚物媚FREDLUND边坡稳定未来讲义（精品）Japan ConferenceDefinition of Factor of SafetySmooth curveDiscrete form(1)(2)StageBState pointn+1AYi1Riii+1kjSijk.i i+1.Fs = ( Shear Strength) /
79、(Actuating Shear Stress) = =BABAfsdLdLFt tt t = = =D DD D= =niiiniiifsLLF11t tt t衣亏既掏式嚣躁冯两洋膜地碑乙媚澈心酵丑幌了段底族扎刻字翼鸽冶撇谆FREDLUND边坡稳定未来讲义（精品）Japan ConferenceDefinition of “Return Function“; Gstage i+1stage ilijlijfttfijijjs sijtijqkijsijtElement (ij)Element (ij)R = Resisting Shear Strength: S = Actuating Sh
80、ear StressFs = ( Shear Strength) / (Actuating Shear Stress)Difficult to Difficult to minimize !minimize ! = =D D- -= =niiisfiLFG1)(t tt tdLFGsBAf)(t tt t- -= = = =- -= =niisiSFRG1)(感狄落淘傲搂愚绿定扔鬃捍男杭矮辗埋痕了汲癸赁店照角拔撬悔锅谤虑碰FREDLUND边坡稳定未来讲义（精品）Japan ConferenceActuating Shear Forces and Resisting Shear S = Actu
81、ating Shear StressR = Resisting Shear Strength = = = = =D D= =neijijijneijijiiil SLS11t tt t = = = = =D D= =neijijfneijijifilRLRiji11t tt tijbijwaijaijneijijiluuucRtan)(tan)(1 s s- -+ +- -+ += = = =来娶匡便纹话钝叉柒仲呈盛戚垦履惋挥锅拂缎烤筒董鳞夜鹃蛮悬曝络裂号FREDLUND边坡稳定未来讲义（精品）Japan ConferenceDefinition of “Optimal Function“ :
82、Minimum Value of “Return Function“ = the optimal function obtained at point k of stage i+1, = the optimal function obtained at point j in stage i, and = the return function calculated when passing from the state point j in stage i to the state point k in stage i+1. where: Introduce an “optimal funct
83、ion”, H = Optimal FunctionG = Return Function = =- -= = =niisiSFRGG1min)(minmin )( jHi),()()(1kjGjHkHiii+ += =+ +)(1kHi+ +)(jHi),( kjGi仕臆凯妻潘搁掏焉歌沙胞苛毯孔虑豺擦匿喉瑰虽埂胎酣停裴姚库惨垣妄帆FREDLUND边坡稳定未来讲义（精品）Japan ConferenceBoundary Conditions of “Optimal Function“At the initial stage, (i=1) : At the final stage, ( i =
84、n+1) : where: = the number of state points in the final stage H = Optimal Function0)(1= =jH 1.1 NPj= =),()()(1kjGjHkHnnn+ += =+ + = =+ +- -= = =niisimnSFRGkH11).()(.1= = n+1 NPk1+ +nNP苛棠汁菲镊募棠蛇敖焉颐啊诸叔迎走默饶襟蹿爷耳譬邹庸呈佃耿姐喇檬呐FREDLUND边坡稳定未来讲义（精品）Japan ConferenceThe Minimum (or Optimal) Travelling Time Problem
85、DYNAMIC PROGRAMMING SOLUTION116487511114121H1 (1) = 092747H1(1) =13AH (2)= 812310B5674STAGE NUMBER1234567d=(4, 2)3G (1,2) = 33105243252827224415532BATHE MINIMUM TRAVELLING TIME PROBLEM针呜胃江贩卿甫硫邀衍庇歧淤臂曼辞砷拌寄蛰喳忽酚谢涸谎润脚耿贪镐蚀FREDLUND边坡稳定未来讲义（精品）Japan ConferenceAnalytical Scheme of the Dynamic Programming Met
86、hodEntry point1InitialABpointYState point.i i+1.XBBn+1X.Stage No.Exit pointSiGrid elementboundarySearchingii+1kSearching gridjRiFinal pointjk幢彩贬盘辈台贿屏乡诀速祥鳞啪疏链护稀旅棺钻萌豆粗聊客炬判考叭斗泻FREDLUND边坡稳定未来讲义（精品）Japan ConferenceKinematical Restriction5S6RSR3SR544RS223BR6SRS11AXYR11SR22SiSiRRnSn.Kinematical Restriction
87、Riii+1kjSiEliminated 涡细艺胰年振茨劝您壹窥颜幕杀卷黎贝蔬民腆您肿朽傀损源狙屿识妥这测FREDLUND边坡稳定未来讲义（精品）Japan Conference = 0.33DYNPROG = 1.02Enhanced = 1.13Bishop; M-P = 1.17Distance, mElevation, mExample of a Homogeneous Slope景婿贴宿糊仙赘惋榆眯源垦拣案船朱雍篙彝虹陡峙烫信悯私刚席辖求丈怒FREDLUND边坡稳定未来讲义（精品）Japan ConferenceExample of a Homogeneous Slope = 0.3
88、3DYNPROG = 1.02Bishop; M-P = 1.17Enhanced = 1.13饼佩豹灼炬睫湖像臂蔚赁橇忙艺悠墟添惧炸侥院胀腑雨逆丫示棠莱过谷披FREDLUND边坡稳定未来讲义（精品）Japan ConferenceExample of a Homogeneous Slope = 0.33Factor of SafetyFactor of SafetyStability Coefficient, C/Stability Coefficient, C/g g g g H H召帐骚怔黄话太德定箔葡城袋藉蚜敌妄炮诽凳恤惫坝始掺寻厩郎囊歼统带FREDLUND边坡稳定未来讲义（精品）Ja
89、pan Conference = 0.48Factor of SafetyFactor of SafetyStability Coefficient, C/Stability Coefficient, C/g g g g H HExample of a Homogeneous Slope弘铃蒲垛遮歼骂恐激墅狞悼偏尺低腾溪状殆驮汰硅虫轴电俱女兰诗妒佬扑FREDLUND边坡稳定未来讲义（精品）Japan ConferenceExample of a Homogeneous Slope = 0.33 = 0.48Factor of Safety,Factor of Safety, DYNPROGDY
90、NPROGFactor of Safety,Factor of Safety, Morgenstern-PriceMorgenstern-Price 糟勇驯旅滑古兔癸灯呐础耻算掐霞荡脐衙亿尹箕狼寓萍揖虎庄仅筛尸忍恰FREDLUND边坡稳定未来讲义（精品）Japan Conference = 0.33Distance, mElevation, mBishop; M-P = 1.64Enhanced = 1.62DYNPROG = 1.49Example of a Partially Submerged Slope祝喧尿仔耪船沁袜倒堤转蛊六唾真绢榔荔焚牺借蒙驳见琵桌将棒先淖蹬庆FREDLUND边坡
91、稳定未来讲义（精品）Japan ConferenceExample of a Partially Submerged Slope = 0.33Enhanced = 1.62Bishop; M-P = 1.64DYNPROG = 1.49富坛桂迷矛刀伸蕉申阐牺媒坑抽巢淫哆龟默锑蒙掖祖哈臣致架泊衅秀武录FREDLUND边坡稳定未来讲义（精品）Japan ConferenceExample of a Multilayered SlopeEnhanced = 1.10M-P = 1.14DYNPROG = 0.96Distance, mElevation, m撼阐浦绦才驴岛假花遂寡扔龋撰肌躁子针突裂撬
92、圾跺惨炒棱稗茬蒲皆秦乏FREDLUND边坡稳定未来讲义（精品）Japan ConferenceThe Re-Analysis of the Lodalen Slide = 0.48DYNPROG = 0.975Bishop = 1.00ActualActualEnhanced = 0.997绵秸揭摇刹戈羔爵炙鲍堑慰某傅蒸政憾料垦刚饺憋粱鹏邓黔屁煤浴爆嫁茶FREDLUND边坡稳定未来讲义（精品）Japan ConferenceThe Re-Analysis of the Lodalen Slide = 0.38Enhanced = 1.02Bishop = 1.00DYNPROG = 0.997
93、ActualDistance, mElevation, mActual旱宁贵蒙和席儿佣世参浑敛捅悄篓亨净堑斡晒职莹豢袋错曼侵锰侠功猾直FREDLUND边坡稳定未来讲义（精品）Japan ConferenceDYNPROG = 1.18Distance, mElevation, mExample Problem Involving the Search for a Convex Critical Slip Surface Along a Weak Clay Layer公萧距铆隧怒悦尤弛恨群畴蝎寥回逐枫殉糯仁娇婚绷魄骂纵揖弃啊掣功血FREDLUND边坡稳定未来讲义（精品）Japan Confere
94、nceSolution of the Concave Slip Surface Problem Using Slope/W Once the Critical Slip Surface has been DefinedElevation, mDistance, mSlope / W = 1.196厌粘尼域邦冀捉退牛滞燃罪鄂兑绰坊贸章烂予斋剩追帚很痰趁谜甩臂牧娃FREDLUND边坡稳定未来讲义（精品）Japan ConferenceConclusions from Step 2 ForwardqThe Shape of the critical slip surface can be made
95、part of the solutionqThe critical slip surface can be irregular in shape but must be kinematically admissibleqNo assumptions is required regarding the Location of the critical slip surface which is defined as an assemblage of linear segmentsqForce and moment equilibrium equations are satisfied throu
96、gh the stress analysis嗓嘘磕辈荡奈塞遇胀何庶番堂事胶官劲葵帘俭茄臭选斟翻衰彻斑裔常提因FREDLUND边坡稳定未来讲义（精品）Japan ConferenceRecommendations for the FutureqqThe normal and shear stresses should be studied using more sophisticated stress-strain nonlinear and elasto-plastic models including Poissons ratio effectsqqStudy of “true“ 3-dim
97、ensional modelling of slopes and past Case HistoriesqqDynamic Programming should be applied to Lateral Earth Pressure and Bearing Capacity problems主蹿趋络杜迎蛹烩六冻浅墨颁寸臣饿实征永倒束逾掘现滥情倪昏馈邱秀蜀FREDLUND边坡稳定未来讲义（精品）Japan ConferenceDelwyn G. Fredlund庚眠韭歇霄霓碱彤沼戒膛魁毡捏图满氦门开甫葡鬃帝仑菏曰均额窍筒样拘FREDLUND边坡稳定未来讲义（精品）Japan Conference
