A simple approach to slope stabilityby J. R. GREENWOOD*, BSc(Eng), MSc, MICE

IntroductionSLOPE STABILITY analysis has for someyears been regarded as an area wherecomputing techniques can readily assistthe engineer. The computer enables hun-dreds of potential slip surfaces to be ana-lysed in a very short space of time andcan subsequently search for the minimumfactor of safety —all at very reasonablecost.

Unfortunately the ease with which datacan be fed in to commercially availableprograms with the satisfaction of almostimmediate solutions being churned out hasled in certain cases to the engineer nolonger appreciating the significance of theanalysis being undertaken.

In all applications of slope stability ana-lysis a 'hand'alculation should be carriedout preferably before the computer is used.In this way the effects of slope geometry,assumed water levels and properties as-signed to the strata can be studied notonly in relation to the overall factor ofsafety but also as they affect the forceson individual 'slices'ithin the analysis.For example the soil parameters andgroundwater table assumed at the toe of theslope may well be shown to be the majorfactor in controlling the stability of thatslope. The more complex the soil stratathe more important it is that a hand cal-culation is carried out since the computer

*Pnncipal Soils Engineer, Departments of Environ-ment and Transport, Eastern Regional Office(Transport) Bedford.

program often makes over-riding assump-tions not fully appreciated by the operator.

Simple solutionAn effective stress solution for slope

stability which by its simplicity lends it-self to hand calculation for both circularand non circular slips is developed in theAppendix. The solution is based on con-ventional shear strength theory. Wherehorizontal stresses in the ground can beestimated or measured the equation maybe applied in its full form (equation 7).The horizontal earth pressure coefficient(K) is dependent on the previous stressand geological history of the strata andthe method of construction of the slope;it may also be time-dependent. In a nor-mally consolidated soil slope K might varytypically from a value corresponding tothe 'active'tate near the crest of theslope (reducing to zero if a tension crackis present) to something near the 'at

rest'tate

below the toe. However, the equa-tion is not particularly sensitive to thevalues of K selected (the factor of safetyassuming K=O is approximately 5 —15%less than that calculated for K=0.5) andfor routine calculations it may be conser-vatively assumed that K=O. This has theeffect of ignoring the contribution to theshearing resistance made by horizontalforces on the slip surface and gives thesimple equation:

~'c' sec z + (W-ub) cos x tan tt>']

F =< W sin x

F

I g

oTe

Notationwidth of soil slice (m)effective cohesion of soil atslip surface (kN/m')factor of safety

average height of soil slice

ratio of horizontal to verticaleffective stresseslength of slip surface of soilsliceassumed b = I cos x (m)effective normal force (kN)pore pressure ratio at slip sur-

pore water pressureface=

total overburden pressure

shearing resistance of soil slice

pore water pressure at slipsurface (kN/m')total weight of soil slice (kN)base inclination of soil slice(negative at toe)total density

submerged density

effective normal stress on baseof soil slice

effective vertical stress onbase of soil sliceeffective friction angle of soilat slip surface

Typical homogeneous slope considered

7.5m

0.5m hack scartt

15-

Pore pr

Remedial toe we

10—

c0

0lA

gl

CL

E

E0giuceIll

ciaa

—10-

—15-

Simple solution(Greenwood k = 0)and Swedish r„=0

7

/

Janhu

i Swedish

I

ReconstitutedToe heave Gault clay

L J

face

Gault clay= 1 5kNim= 15'residuall=

20kN!m'ig.

1 (left J. Comparison between "simple" solution and othersolutions for a typical slope with circles of varying depth

0.1 0.2tflL ratio

0.3 Fig. 2 (above). Application of simple analysis. Foundation failurebeneath a marly chalk embankment near Cambridge

May, .1983 fl5

or in terms of the pore pressure ratio, r

~'c' sec x + W (1 —r„) cos x tan fi']F =

< W sin x

ubwhere ra

W

Comparison with other solutionsThis equation is identical to the Swedish

equation (Fellenius, 1936) modified as sug-gested by Turnbull & Hvorslev (1967) toresolve only the effective vertical force oneach slice of the analysis.

The simple solution is compared in Fig.1 with the full equation (assuming K0.5) and with the Bishop Simplified (Bis-hop, 1955), Swedish and Janbu (Janbuet al, 1956) solutions for a typical slopewith circles of varying depth and for bothra = 0 and r„= 0.5.

The effect of assuming K = 0 is conser-vative reducing the calculated factor ofsafety by about 10%. The difference be-tween the author's simple solution and theBishop Simplified is generally less than15% with the simple solution giving highervalues for shallow slips and Bishop givinghigher values for deeper slip circles. Theauthor's 'full'nd 'simple'quations giveconsistent results for both shallow anddeep slip surfaces whereas the Bishopand Swedish equations tend to become'unstable'or deep slip surfaces.

Practical applicationIn practice, when the geotechnical en-

gineer is presented with a stability prob-lem, the soil strata, strength parametersand pore water pressures are rarely knownwith any accuracy.

The application of a simple equationsuch as that described enables the signi-ficance of the variables and their effecton the factor of safety to be studied. Thereis little point in applying more sophistica-ted methods of analysis when the differ-ences in calculated factor of safety dueto the various assumptions far exceed thedifferences due to the method used andindeed the more complex solutions mayobscure the effects of changed parameters.

Many stability problems involve rein-statement of slipped soil masses. Thelogical approach is to analyse the failureand compare the existing factor of safetywith that after the proposed remedial mea-sures have been taken. It is the increasein factor of safety that is usually morerelevant than the absolute values

Example of applicationIn this example the above philosophy

was applied to a slip which occurred ap-proximately 18 months after constructionof a Bm high marly chalk embankment on

a Gault Clay foundation (Fig. 2). The GaultClay had been excavated and recompactedto a depth of 3m beneath the toe priorto construction of the bank in an attemptto destroy existing shear surfaces knownto be present in the Gault in this region.

The top of the embankment settled byabout 0.5m at the edge of the road andheave occurred at and beyond the toeindicating a foundation failure. The geo-metry of the slip suggested that it passedalong the base of the dig-out zone.

A rapid assessment of remedial mea-sures was required because in additionto the road itself main trunk services wereat risk.

Typical parameters were assumed asindicated on Fig. 2 based on previous de-sign testing and analysis. Residual strengthparameters were assumed for the Gaultas movement had already occurred. Initial-ly a high water table was assumed withinthe bank because water was observed in

the service manholes at the top of thebank. The resulting factor of safety wasunrealistically low (0.76). The water tablewas then assumed 0.5m below the originalground surface giving a more sensible fac-tor of safety of 0.86. (Subsequent piezo-meter installations confirmed the lowerwater table).

Remedial measures were then consider-ed. With little scope for lowering of thewater table in the foundation, toe weight-ing appeared the most feasible solutionand analysis of the slip with a 3m highunderdrained berm showed a factor ofsafety of 1.28. A reassuring 50% increaseon this particular assumed slip surface.Remedial works were undertaken on thisbasis and the movement on the slope suc-cessfully arrested.

The revised slope geometry was check-ed for other possible critical circles usingthe commercial programs, SLIPSYST andCIRCA. The minimum factor of safety wasfound to be a satisfactory 1,5 by the Bis-hop method and 1.1 by the Swedish met-hod.

ConclusionsWhen analysing the stability of a soil

slope a hand calculation is beneficial in

understanding the potential forces actingon the slope and for appreciating thesensitivity of the calculated factor of safe-ty to the parameters selected. The com-mercial computer program is a valuableaid in determining the most critical slipsurfaces once there is a basic understand-ing of the slope and parameters.

A simple equation for slope stabilityanalysis has been developed. Its applica-tion has been demonstrated for routinecalculations where often the strata, stren-gth parameters and water pressures can-not be precisely defined and where the

effects of variations in the parameters areto be studied.

The simple equation is readily appliedby hand and results are shown to be com-parable with other accepted, but morecomplex, methods of analysis.

AcknowledgementsThe author is grateful to Mr. J. Tiplady,

Director of the Eastern Road ConstructionUnit, for permission to publish this workand would particularly like to thank Mes-srs. C. Garrett, R. Grafton, R. Locker andother colleagues who have assisted withcalculations and provided helpful discus-sion on this topic, The views expressed arenot necessarily those of the Department.

ReferencesBishop, A. W. (1955): "The use of the slip circlein the stability analysis of earth slopes". Geo-technique Vol, 5 pp. 7-17.Bishop, A. W. 8i Motgenstern, N. (1960): "Stabil-ity coefficients for earth slopes". GeotechniqueVol. 10 pp, 129-150.

Fettenius, W. (1936): "Calculations of the stabilityof earth dame" Trans. 2nd Congress on LargeDams (Washington) Vol. 4. p, 445.

Janbu, Bjertum, Kjaernsli (1956): Norwegian Geo-technical Institute Publication No 16,Lambe, T W. 8i Whitman, ff V (1969): SoilMechanics, Wiley, New York, pp 363-365.

Turnbull, W. J fk.Hvorslev, M, J, (1967): "Spec-ial problems in slope stability". Proc. ASCE Vol.93, No. SM4, pp. 499-528.

Whitman, ff, V. (k Bailey, W. A. (1967): "Use ofcomputers for slope stability analysis". ProcASCE Vol. 93, No. SM4, pp. 475-498.

APPENDIXDevelopment. of simple equation forslope stability

The factor of safety of a slip surface ismost simply defined as the shear resis-tance available along the slip surfacedivided by the shear stress along that slipsurface (Lambe 8t Whitman, 1969) . Bysummation for vertical soil slices, as shownin Fig. 3, the following expression is ob-tained for the factor of safety of a slipsurface in a soil having both c'nd i'' val-ues

(c' + P'an y')F

~' sin x

This expression may be applied to bothcircular and non circular analysis.

Proposed effective stress stabilityequation

It is necessary to determine appropriatevalues of the effective normal stress onthe slip surface, P', for each soil slice asshown in Fig. 4.P'ay be determined by considera-tion of a Mohr's circle diagram for thesoil element at the base of the slice, asshown in Fig. 4. The horizontal effectivesoil pressure is given by K<r„'here K is

Shearstl'ess

Originof

planes

'=/a„'

a„'n aV

Normaleffective

stress

Fig. 3. Sli p circle analysis by considerationof soil slices

(a) Slip surface (b) Moht circle of effective stress

Fig. 4. Consideration of soil element on potential slip surface at base of slice

May, 1983 47

the ratio of the horizontal to vertical ef-fective stresses at the base of the slice.Possible rotation of the principal stressesdue to the inbalance of forces within theslope is ignored. This may be comparedto ignoring inter-slice forces in other deri-vations of equations for slope stability,and this has been shown to be generallyinsignificant on circular slip surfaces (Bis-hop, 1954; Whitman 8< Bailey, 1967; Bishop8< Morgenstern, 1960).

From the Mohr circle of effective stress(Fig. 4):

<r„' K<r„' (<r,,' K<r„') cos'-'... (2)

re-arranging

andW —ub

(neglecting inter-slice forces)

therefore

I (W —ub)P'cos- x + K sin'-')b

(4)

Providing the width of slice is not largeit is usually assumed, without significantloss in accuracy, that

OI''(W —ub) cos x (1 + K tan'-')

(6)

substituting for P'nd I (eqs. 5 and 6) ineqn. 1, the factor of safety of the slip sur-face is obtained:

1F = ~'c'b sec x+

~' sin x

(W —ub) (1 + K tan'-') cos x tan <I>']

(7)or in terms of the pore pressure ratio

u ub

,r„' —<r,,'cos'-'x + K [1 —cos'-'])

<T„' —o-„'cos'-' + K sin'-') ... (3)

cos x

substituting for I

(5)

F

yh W

1~'c'bsecx+

~ Wsin xbut

P' I

W —ub(cos'-' + K sin'-')

cosW (1 —r„) (1 + K tan'-'<x) cos x tan <t/]

... (7a)

down the bar (similar to the ram's imagemoving down the slit), a continuous traceis plotted on the paper.

The result of using the attachment istherefore a polaroid print with a continuoustrace of the ram-displacement against time.If suitable scales can be included for thetwo axes, the impact velocity may be de-duced from the gradient of the trace atimpact.

Practical applicationA small mark (20mm x 40mm) is print-

ed on the ram as a reference point formonitoring the ram movement. A sequenceof marks, at regular intervals, is printed onthe hammer leaders or guide frame. Bysuitable positioning of the camera, the rammark is aligned to run adjacent to theframe marks, and their image is positionedin the fixed slit (Plate 1).

During the ram's descent the d.c. motoris manually triggered to move the polaroid

print across the slit. The ram's movementis recorded for 0.4 seconds, and duringthis time flashes of light from the LEDare exposed to the print. These give aprecise time scale for the displacement/time graph, and can be clearly seen inFigs. 4 and 5.

The image of the frame marks is a ser-ies of stationary points in the slit. Theirtrace on the polaroid print is therefore asequence of horizontal lines. Since thespacing of the marks on the hammer frameis known, these lines give the precise displacement scale for the ram movement.

The image of the ram mark is a singlepoint in the slit which moves down as theram descends. Its trace is therefore asmooth curve as it accelerates with time.This can be seen in Fig. 5.

ResultsTwo results are presented in Figs. 4 and

5. Fig 5 shows a hammer impact recorded

A new method for measuring the impact energyOf a piling hammer (continued from page 38)

in the field. The camera was positioned60m from the hammer, and the framemarks were spaced at 100mm. The calcu-lated impact velocity was 4.6 metres/sec-ond.

Fig. 4 shows the result of a laboratorytest used to assess the accuracy of themethod. A signal generator was used tooscillate the spot of an oscilloscope in avertical line, at a frequency of 2.86Hz, andan amplitude of 69mm (a constant veloc-ity of 0.394m/sec). Using the streak cam-era, the spot's velocity was measured at0.408m/sec, giving an error of 3.5%. Thiserror was due to the inaccuracy of mea-suring directly from the polaroid print. High-er resolution may be achieved by re-photographing and enlarging the print.

ConclusionsA reliable, accurate and remote method

of measuring the working efficiency of apiling hammer has been described. Theonly requirement is a clear view of theram. The method is as accurate as alter-native methods; it is however quicker andeasier to use.

8 Trade Literature

Pile Hammer Brochures. Two new bro-chures have been published by MKT Geo-technical Systems, Box 793., 100 RichardsAvenue, Dover, NJ 07801, USA. A 4-pagebrochure gives details of four models ofconvertible diesel pile hammers which arebasically of double-acting design but witha few modifications can be changed tosingle-acting. These range from the DA-15C with an energy rating of 913-1 134kgmto the DA-55C with a rating of 4314 to5 282kgm.

Two new compressed-air or steam oper-ated single-acting pile hammers are des-

48 Ground Engineering

cribed in a 2-page brochure. These are theMS-350, with a ram weight of 3.5tonnesand stroke variable between 300mm and1.2m, and the MS-500, with a ram weightof 5tonnes and stroke variable between600mm and 1.2m. A claimed feature ofthese hammers is that the ram weightrepresents about 65-70% of the totalweight, resulting in a lower overall weightrelative to the energy output.

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