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CHAPTER FOURClassic methods of slope stability analysisOverviewHistorically, the stability of slopes has been assessed by methods ofcalculation `by hand' before the advent of computers, spreadsheetsand bespoke software packages. The most widely accepted of theseclassic methods are described below. Principally, these are by Bishop(1955) and Janbu (1957). The chore of calculation `by hand' was speededup by the introduction of stability coefcients and design charts by Bishopand Morgenstern (1960) and Hoek and Bray (1977). Computer programshave automated the methods of Bishop and Janbu and others, but notreplaced them. On the contrary, these classic methods continue, exceptnow they are in computer-automated form so that numerous trial slipand slope geometries can be rapidly evaluated.Wider accessibility to (`main frame') computing in the late 1950s to theearly 1960s gave rise to methods of analysis that could only be solved bycomputer-automated numerical methods, principally by Little and Price(1958) at that time, this was thought to be the most complicated programever written for a computer of British origin (Bromhead, 1985) andMorgenstern and Price (1965, 1967). Now commercial computer programsfor slope stability analysis are widely available and some can link withother programs for stress distribution and for seepage (e.g. see SLOPE/W Student Edition CD supplied with this book).While classic methods have been automated by computer andnew computer methods have been devised, it is still the responsibility ofthe design engineer to ensure that ground properties (inputs) anddesign predictions (outputs) pass the sanity test! Indeed, it is theexplicit duty of the engineer to check the outputs of computer programsby `hand' calculation, i.e. in an immediately obvious and transparentway without the aid of a `black box' (or mysterious) computer program although perhaps a computer spreadsheet may be permitted! Thesechecks `by hand' will use the same classic methods describedbelow, many of which are still used to this day by commercial softwarepackages.77Chronology`Following the success of Bishop's theory [of slope stability analysis],and with good results obtainable from the routine method [Bishop-simplied], efforts were made to develop a similar theory applicableto slides with any shape of slip surface. The chronology of the earlydevelopments is confused, but appears to have been as follows.Bishop's paper was presented at a conference a year before itappeared in print. Separate researches led Janbu and Kenney tothe same result (Kenney working under Bishop at Imperial College).Janbu published rst, in 1955, but in an incorrect form, and Kenney'sthesis appeared the following year. Subsequently, Janbu repub-lished a corrected form of the equations, but in a form relativelyinaccessible to English readers. In the meantime, Bishop hadpersuaded Price, who had been one of the authors of the rst stabilityanalysis computer program (see Little and Price, 1958), to try to pro-gram Kenney's equations for non-circular slips. It was then foundthat the basic equations could give rise to numerical problemswhen evaluated to high precision that would not appear at slide-rule accuracy [i.e. to within about 5% at best]: he and Morgenstern(Morgenstern and Price, 1965, 1967) then developed a more sophis-ticated method (again at Imperial College). This time, they weresecure in the knowledge that complexity in computation was nolonger a bar to widespread use of a method because of the growingavailability of computers. Janbu developed his method further, andpublished his generalized procedure of slices in 1973, and a numberof other methods also appeared in print throughout the late 1960sand 1970s.'Bromhead (1985)There follows a summary in lecture-note form of the methods of slopestability analysis common throughout the geotechnical literature.Simple casesDry slope in sandReferring to Fig. 4.1, assume that the stresses on the vertical sides of theslice are equal and opposite. Resolving forces parallel to the slope givethe disturbing force W sinu. Resolving forces normal to the slope givethe resultant normal force W cos u.The factor of safety against sliding is thenF = Resisting forceDisturbing force = W cos u tanc/W sinu = tanc/tanuSHORT COURSE IN SOIL AND ROCK SLOPE ENGINEERING78whence, for limiting equilibrium,umax = c/Fully submerged slope in sandReferring to Fig. 4.2, consider a slice of width a and depth d. The buoyantweight of the slice is:W/ = ad/ (/ = w)The effective stress normal to the base of the slice is:o/n = ad/cos ucos uaThe shear stress along the sloping base of the slice is:t = ad/sinucos uaThe factor of safety against sliding may be derived asF = d/cos2u tanc/d sinu cos u . whenceF = tanc/tanuNote that the expression for factor of safety for the fully submerged slopeis identical to that for a dry slope.WFig. 4.1 Denition sketch for a dry slope in sandW adFig. 4.2 Denition sketch for a fully submerged slope in sandCHAPTER 4 CLASSIC METHODS OF SLOPE STABILITY ANALYSIS79Semi-innite slope: = 0 analysisReferring to Fig. 4.3, consider a slice of width b and depth z. The disturb-ing force is zbsinu. The restoring force issuacos uThe factor of safety against sliding is thereforeF = suz cos u sinuaz = bulk unit weightFig. 4.3 Denition sketch for a semi-innite slope, c = 0 analysisWORKED EXAMPLEConsider a natural slope in over-consolidated London clay near limitingequilibrium.Take z = 7 m, su = 100kPa, = 20 kN/m3, u = 108 which are typicalvalues. ThenF = suz cos u sinu= 10020 7 cos 108 sin108= 4.2This is a nonsense for London clay slopes which can fail at this inclina-tion! This calculation serves to illustrate how inappropriate a c = 0analysis is when applied to natural slopes.Nowconsider a natural slope in soft normally consolidated Norwegianclay.Take z = 3 m, su = 10 kPa, = 18 kN/m3, u = 158, as typical valuesfor a slope about to fail. Now F = 0.74.This calculation again conrms that a c =0 analysis should not beapplied to assess the stability of a natural slope whether in an over-consolidated ssured clay or a normally-consolidated intact clay.SHORT COURSE IN SOIL AND ROCK SLOPE ENGINEERING80 = 0 analysis for vertical cut: no tension crackIn Fig. 4.4, the disturbing force is 1,2H2sinc, tanc. The restoring force issuH,sinc.Taking the critical angle of sliding as ccrit = 458, then Hc = 4su,.Alternatively, assuming a slip circle, the corresponding result would beHc = 3.85su,. These predictions would be unsafe in practice becausereal clay soils are weak in tension. = 0 analysis for vertical cut: with tension crackRefer to Fig. 4.5. Assume the depth of tension crack D = Hs,2, followingTerzaghi (1943).The disturbing force is34Hc Hc2 sin458The restoring force issuHc2 1sin458At failure when F = 1, the critical height is Hc = 2.67su,. This gives amore realistic prediction for the height of a temporary vertical cut in clay.HHtan Fig. 4.4 Denition sketch for a vertical bank, c = 0 analysis, no tensioncrackHc45D = Hc/2 = bulk unit weightsu = undrained shear strengthFig. 4.5 Denition sketch for a vertical bank, c = 0 analysis, with tensioncrackCHAPTER 4 CLASSIC METHODS OF SLOPE STABILITY ANALYSIS81WORKED EXAMPLEConsider the different approaches to analysing the short term stabilityof a vertical bank illustrated in Fig. 4.6.Case (a) may be summarized as follows.608: Disturbing = 12 10 5.774 20 sin608= 500kNRestoring = 60 11.55 = 693kNF = 693,500 = 1.30458: Disturbing = 12 10 10 20 sin458 = 707kNRestoring = 14.14 60 = 848kNF = 848/707 = 1.2308: Disturbing = 12 10 17.3 20 sin308 = 865kNRestoring = 60 20 = 1200kNF = 1200,865 = 1.39For case (b) we have:For a dry slope:Disturbing = 5 7.5 20 sin458 = 530kNRestoring = 7.07 60 = 4.24kNFdry = 424/530 = 0.804545306010 m10 m10 m10 m5 m5 mQw424 m(a)(b)(c) = 20 kN/m3su = 60 kN/m2Fig. 4.6 Denition sketches for worked example (a) assuming differentangles of slip planes, (b) taking a 458 slip plane and a water-lledtension crack, (c) assuming slip plane is the quadrant of a circleSHORT COURSE IN SOIL AND ROCK SLOPE ENGINEERING82Semi-innite slope: eective stress analysisRefer to Fig. 4.7. For ow parallel to slope, an effective stress analysisgives the following.Soil strength is s = c/ o/tanc/.Shear stress = t = z sinu(1,cos u)= z sinu cos uNormal stress = o = z cos u(1,cos u)= z cos2uPore water pressure = u = mzwcos u cos u= mzwcos2uThe normal effective stress on the base of the slice is:o/ = ( mw)z cos2uWhence the factor of safety against sliding isF = c/ ( mw)z cos2u tanc/z sinu cos uFor a wet slope (tension crack lled with water)Water force = 12 5 5 10 = 125kNExtra disturbing = 125 sin458 = 88.4kNFwet = 424,530 88.4 = 0.69It can be seen that the lling of the tension crack with water has asignicant effect on stability.For case (c) we have (noting the centroid of a quadrant of a circleradius R is located 4R,3 from the centre of