stability methods of analysis

8/12/2019 Stability Methods of Analysis

http://slidepdf.com/reader/full/stability-methods-of-analysis 1/11

Comparison of slope stability methods of analysis

Received October 12, 1976

Accepted April 4 , 1977

The papercompares six methods ofslicescommonly ~ ~ s e dor slopestability analysis. The factorof safety equations are written in the same form, recognizing whether moment and (or ) forceequilibrium is explicitly satisfied. The normal force equation is of the same form for all methodswith the exception of the ordinary method. The method of handling the interslice forces differen-tiates the normal force equations.

A new derivation for the Morgenstern-Price method is presented and is called the 'best-fitregression' solution. It involves the independent solution of the force and moment equilibriumfactors of safety for various values of A The best-fit regression solution gives the same factor ofsafety as the 'Newton-Raphson' solution. The best-fit regression solution is readily com-prehended, giving a complete understanding of the variation of the factor of safety with A

L'article presente Line comparaison des six methodes de tt mch es ~ ~t il is ee soummment pourI'analyse de la stabilite des pentes. Les eq~~ationses facteurs de sec~~ rit eont ecrites selon lam h e forme, en montrant si les conditions d'equilibre de moment ou de force sont satisfaitesexplicitement. L'Cquation de la force normale est de la m h e orme pour toutes les methodesI'exception de la methode des tranches ordinaires. La fa ~ o ne traiter les forces intertranches estce qui diffirencie les equations de force normale.

Une nouvelle derivation appelee solution best-fit regression de la mkthode de Morgens-tern-Price est presentee . Elle consiste determiner independemment les facteurs de secu-rite satisfaisant aux equilibres de force et de moment ~ O L I ~iffirentes valeurs de A La solu-tion best-fit regression donne les m h e s acteurs de securiti de la solution de Newton-Raph-son. La solution best-fit regression est plus facile a comprendre et donne Line image detailleede la variation du facteur de securite avec A

C a n . Geotech J . . 14 429 (1977)[Traduit par la revue]

Introduction

The geotechnical engineer frequently useslimit equilibrium methods of analysis whenstudying slope stability problems. T he m ethodsof slices have become the most commonmethods due to their ability to accommodatecomplex geometrics and variable soil andwater pressure conditions (Terzaghi and Peck19 67 ). During the past three decades ap-proximately one dozen methods of slices havebeen developed (Wright 19 69 ). They differin (i) the statics employed in deriving thefactor of safety equation and (ii) the assump-tion used to render the problem determinate(F re d lund 1975) .

This paper is primarily concerned with sixof the most commonly used methods:

( i ) Ordinary or Fellenius method (some-times referred to as the Swedish circle methodor the conventiona l method)

(ii ) Simplified Bishop metho d

'Presented at the 29th Canad ian GeotechnicalConference, Vancouver, B.C., October 13-15, 1976.

(iii) Spencer s method

(iv ) Janbu s simplified method(v ) Janbu s rigorous method(vi) Morgenstern-Price methodTh e objectives of this paper a re:(1) to compare the various methods of

slices in terms of consistent procedures forderiving the factor of safety equations. Allequations are extended to the case of a com-posite failure surface and also consider partialsubmergence, line loadings, and earthquakeloadings.

2) to present a new derivation for theMorgenstern-Price method. Th e proposedderivation is more consistent with that used

for the other methods of analysis but utilizesthe elemen ts of statics and the assumptionproposed by Morgenstern and Price 1965Th e Newton-Raphson numerical technique isnot used to compute the factor of safetyand A

3 ) to compare the factors of safety ob-tained by each of the methods for severalexample problems. The University of Sas-



C A N . G E O T E C H . J. VOL . 14, 977

WATERA R

FIG.1. Forces acting for the method of slices applied to a composite sliding surface.

katchewan SLOPE computer program wasused for all com puter analyses Fred lund1 9 7 4 ) .(4) to compare the relative computational

costs involved in using the various methodsof analysis.

Definition of Problem

Figure 1 shows the forces that must be de-fined for a general slope stability problem.

The variables associated with each slice aredefined as follows:

W = total weight of the slice of width b andheight ?

P = total normal force on the base of the sliceover a length

Sm shear force mobilized on the base of theslice. It is a percentage of the shearstrength as defined by the Mohr-Coulombequation. That is, S = {c [PI1 u ]tan + )IF where c effective cohesionparameter, 4 effective angle of internalfriction, = factor of safety, andporewater pressure

R radius or the mom ent arm associated withthe mobilized shear force Smperpendicular offset of the normal forcefrom the center of rotation

x horizontal distance from the slice to thecenter of rotationangle between the tangent to the center ofthe base of each slice and the horizontal

E = horiz onta l interslice forcesL subscript design ating left sideR = subscript designating right sideX = vertical interslice forcesk = seismic coefficient to acc ount for a dynam-

ic horizontal forcee = vertical distance from the centroid of each

slice to the center of rotationA uniform load on the surface can be taken

into account as a soil layer of suitable unit

weight and density. The following variables arerequired to define a line load :L = line load force per unit width)

= angle of the line load from the horizontal= perpendicular distance from the line load

to the center of rotationThe effect of partial submergence of the slope

or tension cracks in water requires the definitionof addition al variables:A = resultant water forcesa perpendicular distance from the resultant

water force to the center of rotation

Derivations for Factor of Safety

T he elements of statics that can be used t oderive the factor of safety are summations offorces in two directions and the summationof moments. These, along with the failurecriteria, are insufficient to make the problemdeterminate. More information must be knownabout either the normal force distr ibution orthe interslice force distribution. Either addi-



FREDLUND AND KRAHN

Q = RESULTANT INTERSLICE1 FORCE

O P P O S I T E T O1L6

FIG. Interslice forces for the ordinary method

NOTE: QR51S NOT EQUAL

A N D

Q ~ 6

tional elements of physics or an assumptionmust be invoked to render the problem deter-

minate. All methods considered in this paperuse the latter procedure, each assumptiongiving rise to a different method of analysis.For comparison purposes, each equation isderived using a consistent utilization of theequations of statics.

Ordinary o r Fellenius M ethodThe ordinary method is considered the

simplest of the methods of slices since it isthe only procedure that results in a linearfacto r of safe ty equation. It is generally statedthat the interslice forces can be neglected be-cause they are parallel to the base of eachslice (Fellenius 1936). However, Newton's

principle of 'action equals reaction' is not satis-fied between slices (Fig. 2) . Th e indiscriminatechange in direction of the resultant intersliceforce from one slice to the next results infactor of safety errors that may be as muchas 60 (Whitman and Bailey 19 67 ).

Th e normal force on the base of each sliceis derived either from summation of forcesperpendicular to the base or from the sum-mation of forces in the vertical and horizontaldirections.

C11 v = 0

W - P c o s a S , s i n a=

0

C21 H 0

S , c o s a P s i n a k W = 0

Substituting [2] into [I] and solving for thenormal force gives

Th e factor of safety is derived from th esummation o f moments abo ut a common point

(i.e. either a fictitious or real center of rotatio nfor the entire mas s).

Introducing the failure criteria and thenormal force from [3] and solving for thefactor of safety gives

X c f l R (P u1)R tan 4 )[5] F

C W x C P ~C k ~ et An Ld

Simplified Bishop Meth odThe simplified Bishop method neglects the

interslice shear forces and thus assumes that anormal or horizontal force adequately definesthe interslice forces (Bishop 1955). The nor-mal force on the base of each slice is derivedby summ ing forces in a vertical direction (as in[I]). Substituting the failure criteria andsolving for the norm al force gives

where nz = cos Y (sin Y tan + ) /F

The factor of safety is derived from thesummation of moments about a commonpoint. This equation is the same as [4] sincethe interslice forces cancel out. Therefore, thefactor of safety equ ation is the same as for theordinary method ([5 ]) . However, the defini-tion of the normal force is different.



432 C A N . G E O T E C H . J . VOL. 14, 1977

Spencer s Metho dSpencer s metho d assumes there is a con-

stant relationship between the m agnitude of theinterslice shear and normal forces (Spencer1 9 6 7 ) .

where 0 = angle of the resultant intersliceforce from the horizontal.

Spencer 19 67 ) summed forces perpendic-ular to the interslice forces to derive thenormal force. The same result can be ob-tained by summing forces in a vertical andhorizontal direction.

W X R X,) P cos S sin a = 0

The normal force can be derived from 181and then the horizontal interslice force is ob-tained from [9].

[lo] P = W (ER EL) tan 0[c l sin a: ul tan ql sin a

F+

F 1 1

Spencer (1967) derived two factor of safetyequations. One is based on the summation ofmoments about a common point and the otheron the summation of forces in a directionparallel to the interslice forces. The momentequation is the same as for the ordinary andthe simplified Bishop methods (i.e. [ 4] ). Thefactor of safety equation is the same as [5].

The factor of safety equation based onforce equilibrium can also be derived bysumming forces in a horizontal direction.

C(E, ER)+ C P sin C S , , cos

Th e interslice forces Er, El,) must cancelout an d the facto r of safety equation withrespect to force equilibrium reduces to

C { d l c o s a + ( P zil) tan 4 cos a )[I21 F , =

X P s i n a + X ~ W + - L c o s w

Spencer s met hod yields two facto rs ofsafety for each angle of side forces. However,at some angle of the interslice forces, the two

factors of safety are equal (Fig. 3 and bothmoment and force equilibrium are satisfied.

The Corps of Engineers method, sometimesreferred to as Taylor s modified Swedishmethod, is equivalent to the force equilibriumportion of Spen cer s metho d in which the direc-tion of th e interslice forces is assu med , generallyat an angle equal to the average surface slope.

Janb u s Simplified MetlzodJanbu s simplified m ethod uses a correction

factor f o to account for theeffect of the inter-slice shear forces. The correction factor isrelated to cohesion, angle of internal friction,

and the shape of the failure surface (Janbuet nl. 1 95 6) . Th e normal force is derived fromthe summation of vertical forces ( [a ] ) , withthe interslice shear forces ignored.

ul tan ql sin r+F

The horizontal force equilibrium equationis used to derive the factor of safety ( i .e11 11 ). T h e sum of t he interslice forces mustcancel and the factor of safety equationbecomes

C{c l cos + (P zrl) tan 4 cos a

[14] F , = C P s i n a + CIc W+ A L cos w

F,, is used t o designate the fa cto r of safetyuncorrected for the interslice shear forces.

0 9 0 1 I I I I0 5 1 15 2 0 25 3 0

SIDE FORCE ANG LE, DEGREES)

FIG. Variation of the factor of safety witrespect to moment and force equilibrium 1 5. thangle of the side forces. Soil properties c'/-yk = 0.02+ = 40 ; I. = 0.5. Geometry: slope = 26.5 height = 100 ft (30 m).



F R E D L U N D A N D K R A H N 433

The corrected factor of safety is

Janbu s Rigorous M ethodJanbu s rigorous method assumes that the

point at which the interslice forces act canbe defined by a line of thrust . New te rmsused are defined as follows (see Fig. 4 :

t , , tn = vertical distance from the base of theslice to the line of thrust on the left and rightsides of the slice, respectively; a t = anglebetween the line of thrust on the right side ofa slice and the horizontal.

T he normal force on th e base of t he s liceis derived from the summation of verticalforces.

d l sin a ul tan 4 sin a

F F

T he factor of safety equation is derivedfrom the summation of horizontal forces ( i .e .[ I ) . Janb u s rigorous analysis differs fromthe simplified analysis in that the shear forcesare kept in the derivation of the normal force.Th e factor of safety equation is the same asSpencer s equ ation based on fo rce equilibrium( i. e. [ 1 2 ] ) .

In o rder to solve the factor of safety equa-

tion, the interslice shear forces must beevaluated. For the first iteration, the shears

Frc 4. Forces act ing on each sl ice for Janbu srigorous method.

are set to zero. For subsequent iterations,the interslice forces are computed from thesum of the moments about the center of the

base of each slice.

[17] C M , = 0

X L b / 2 X R b / 2 E L [ t ( b / 2 ) an a]

E R t ( b / 2 ) an a b tan a ] W11/2=0

After rearranging [ I 71, several terms becomenegligible as the width b of the slice is reducedto a width dx. These terms are X,:XT. / 2 ,( E , : E , ) ( b / 2 ) tan and El : E,)btan 0 .Eliminating these terms and dividing bythe slice width, the she ar force on th e right sideof a slice is

[18] X R = R a n a , ( E R E L ) R / b

( k W / b ) h / 2 )

The horizontal interslice forces, requiredfor solving [18], are obtained by combiningthe summation of vertical and horizontalforces on each slice.

[19] ( E R E L ) = [ W - ( X R X L ) ] a n a

The horizontal interslice forces are obtainedby integration from left to right across theslope. The m agnitude of the interslice shear

forces in [I91 lag by one iteration. Eachiteration gives a new set of shear forces. Thevertical and horizontal components of lineloads must also be taken into account whenthey are encountered.

Morgenstern-Price MethodTh e Morgenstern-Price method assumes an

arbitrary mathematical function to describethe direction of the interslice forces.

where \ = a constant to be evaluated insolving for the factor of safety, and f ( x ) =

functional variation with respect to x . Figure5 shows typical functions ( i .e . s ) . F or aconstant function, the Morgenstern-Pricemethod is the same as the Spencer method.Figure 6 shows how the half sine functionand x are used to designate the direction ofthe interslice forces.

Morgenstern and Price 1 9 6 5 ) based theirsolution on the summation of tangential and



4 4 C A N . GE OT E C H

f x)=CONSTANT f x)=HALF-SINE

I f x)=CLIPPED-SINE f x) TRAPEZOID

I f x )=SPECIFIED

FIG.5. Functional variation of the direction of theside force with respect to the x direction.

SINE CURVE

FIG. 6. Side force designation for the Morgenstern-Price method.

normal forces to each slice. The force equilib-rium equations were combined and then the

Newton-Raphson numerical technique wasused to solve the moment and force equationsfor the facto r of safety and A

In this paper, an alternate derivation forthe Morgenstern-Price method is proposed.The solution satisfies the same elements ofstatics but the derivation is more consistentwith that used in the other methods of slices.It also presents a complete description of the

J . VOL. 14 , 1977

variation of the factor of safety with respecto A

The normal force is derived from the

vertical force equilibrium e quation [I 61). Twofactor of safety equations are computed, onewith respect to moment equilibrium and onewith respect to force equilibrium. The momenequilibrium equation is taken with respect toa com mon point. Ev en if the sliding surfaceis composite, a fictitious common center canbe used. The equation is the same as that ob-tained for the ordinary method, the simplifiedBishop method, and Spencer's method [4

and [5]). The factor of safety with respect toforce equilibrium is the same as that derivedfor Spencer's method [ I21) The intersliceshear forces are computed in a manner similar

to that presented for Janbu's rigorous methodOn the first iteration, the vertical shear forcesare set to zero. On subsequent iterations, thehorizontal interslice forces are first computed[19]) and then the vertical shear forces are

compu ted using an assumed value and sideforce function.

[211

T h e

1.20

1.15

* 1.10

ILL' 1.05

[

1.00

0.95

0.9C

XI: = Edf x)

side forces are recomputed for each

FIG. 7. Variation of the factor of safety with re-spect to moment and force equi l ibrium vs. X for theMorgenstern-Price method . Soil properties: c1/ fl1=0.02; ' = 40 ; r = 0.5. Geometry: s lope = 26.5 ;height = 100 f t ( 3 0 m ) .



F R E D L U N D A N D K R A H N

I CONDITION 2 (weak layer)

13-

BEDROCK

FIG. 8 Example problem.

i teration. The moment and force equilibriumfact ors of safety a re solved for a ran ge of h

values and a specified side force function.These factors of safety are plotted in a mannersimilar to th at used for Spencer s me thod(Fig. 7) . The factors of safety vs h are fit bya second order polynomial regression and thepoint of intersection satisfies both force andmoment equilibrium.

Comparison of Methods of Analysis

All methods of slices satisfying overallmoment equilibrium can be written in thesame form.

C d l ~ C( P ul)R tan 4[22] F =

CWx CPf + CkWe Aa + Ld

All methods satisfying overall force equilib-rium have the following form for the factor ofsafety equation:

Cc l cos a + C (P u l) t an 4 cos

1 F f = x p s i n c t + x k ~ + ~ - ~ c o s o

Th e factor of safety equations can bevisualized as consisting of the following com-ponents :

Moment Forceeauilibriuni equilibrium

CohesionFrictionWeightNormalEarthquakePartial

submergenceLine loading

Zc lR Zc l cosC(P-rd)R tan + C(P-ul) tan + cosC wsCPf CP sin i

Ck We C k W

Ld L cos o

From a theoretical s tandpoint, the derivedfactor of safety equations differ in (i) theequations of statics satisfied explicitly for theoverall s lope and (ii) the assumption to makethe problem determinate. Th e assumption usedchanges the evaluation of the interslice forcesin the normal force equation (Table 1 . Allmethods, with the exception of the ordinarymethod, have the same form of equation forthe normal force.

c l sin a 111 ta n4

sin a-- --F F

where rn, = cos + (s in a tan (p ) /F.

It is possible to view the analytical aspectsof slope stability in terms of one factor ofsafety equation satisfying overall momentequilibrium and another satisfying overallforce equilibrium. Then each method becomesa special case of the best-fit regression solu-tion to the Morgenstern-Price method.

TABLE. Comparison of factor of safety equations

Factor of safety based on

Moment Force Normalequili- equili- force

Method brium brium equation

Ordinary or Fellenius x PISimplified Bishop x [61Spencer s x x D O 1Janb u s simplified x U31Janbu s r igorous x [I61Morgenstern-Price x x [241



436 C AN. GE OT E C H. J. VOL. 14, 1977

TABLE. Comparison of factors of safety for example problem

Morgenstern-

Price methodSimplified Spencer s method Janbu s Janbu s f x) constan

Case Ordinary Bishop simplified rigorousno. Example problem* method method F method method** F

1 Simple 2:1 slope, 40 1 .928 2 .080 2 .073 14 .81 0 .237 2 .041 2 .008 2 .076 0 .254ft (12 m) high,4 20°, c 600psf (29 kPa)

2 Same as 1 with a thin, 1.288 1 .377 1 .373 10 .49 0 .185 1 .448 1 .432 1 .378 0 .159weak layer with4 lo0, c 0

3 Same as 1 e x c e ~ t i th 1.607 1 .766 1 .761 14 .33 0 .255 1 .735 1 .708 1 .765 0 .244r 0.25

4 Same as except withr., 0.25 for bothmaterials

5 Same as 1 except with 1.693 1 .834 1 .830 13 .87 0 .247 1 .827 1 .776 1 .833 0 .234a piezometric line

6 Same as 2 except with 1.171 1 .248 1 .245 6 .88 0 .121 1 .333 1 .298 1 .250 0 .097a piezometric linefor both m aterials

*Width of slice is 0 5 ft 0.3m ) and the tolerance on the nonlinear**The line of t h rus t is assumed at 0 333

Figure 8 shows an example problem in-volving both circular and composite failuresurfaces. The results of six possible combina-tions of geometry, soil properties, and waterconditions are presented in Table 2. Thisis not meant to be a complete study ofthe quantitative relationship between various

methods but rather a typical example.The various methods (with the exceptionof the ordinary method) , can be compared byplotting factor of safety vs. A The simplifiedBishop method satisfies overall moment equi-librium with h = 0. Spencer s method hasequal to the tangent of the angle between thehorizontal and the resultant interslice force.Janbu s factors of safety can be placed alongthe force equilibrium line to give an indicationof an equivalent h value. Figures 9 and 10

show comparative plots for the first two casesshown in Table 2.

The results in Table 2 along with tho~e

from other comparative studies show that thefactor of safety with respect to moment equi-librium is relatively insensitive to the inter-slice force assumption. Therefore, the factorsof safety obtained by the Spencer andMorgenstern-Price methods are generallysimilar to those computed by the simplifiedBishop method. On the other hand , the factors

solutions is 0 001

of safety based on overall force equilibriumare far more sensitive to the side forceassumption.

The relationship between the factors of

B I SHOP

v- S P E N CE R

95 / M OR GENS TER N PRICE

19 v

f ( x ) = CONSTANT

8 ORDINARY = 928

FIG.9. Comparison of factors of safety for case 1.




TABLE. Comparison of two solutions to the Morgenstern-Price method**

University of University of Saskatchew an

Alberta program SLOP E programside force function side force function

Constant Half s ine Constant Half sine Clipped sine*Caseno. Example problem X X F X X h

S im p le2 :1 s lo p e , 4 0 f t (1 2 m ) 2.085 0.257 2.085 0.314 2.076 0.254 2.076 0.318 2.083 0.390high, 4 = 20°, c = 600 psf(29 k Pa)

Sa m e a s l w i t h a t h i n , w e a k 1 . 3 9 4 0 . 1 8 2 1 . 38 6 0 . 2 1 8 1 . 3 7 8 0 . 1 5 9 1 . 3 7 0 0 . 1 8 7 1 . 3 6 4 0 . 20 3laver with 4' = 10 . c = 0

Sam eas 1 except withr , 0.25 1.772 0.351 1.770 0.432 1.765 0 .244 1 . 7 6 4 0.304 1 .779 0.417Sa m e a s 2 e x c e p t wi t h r U= 0.25 1.137 0.334 1 .117 0 .441 1 .124 0 .116 1.118 0 .130 1.113 0 .138

for both materialsSa m e a s l e x c e p t w i t h a p i e z o - 1 . 8 3 8 0 . 2 7 0 1 . 8 3 7 0 . 3 3 1 1 . 8 3 3 0 . 2 3 4 1 . 8 3 2 0 . 2 9 0 1 . 8 3 2 0 . 3 0 0

metric line

Same as 2 except with a piezo-1

265 0.159 Not converging 1 250 0.097 1 245 0.101 1 242 0.104metric line for both m aterials

*Coord inates 0, 0.5, and 1.0, 0.25.**Tolerance on both Morgenstern-Price solutions is 0.001.

MO R G E N S T E R N - PRICE

x) CONSTANT

25

200 2 4 6

FIG 0. C omp arison of fac tors of safety for case 2.

safety by the various methods remains similarwhether the failure surface is circular or com-posite. For example, the simplified Bishopmetho d gives factors of safety that are alwaysvery similar in magnitude to the Spencer andMorgenstern-Price metho d. This is due to thesmall influence that the side force fun ction ha son the moment equilibrium factor of safety

equation. In the six example cases, the aver-age difference in the factor of safety wasapproximately 0.1 .

Comparison of Two Solutions to theMorgenstern-Price Metho d

Morgenstern and Price 19 65) originallysolved their method using the Newton-Raphsonnumerical technique. This paper has presentedan alternate procedure that has been referredto as the 'best-fit regression' method. The twomethods of solution were compared using theUniversity of Alberta computer program( K r a h n t 01 1971 ) for the original methodand the University of Saskatchewan computerprogram (Fredlund 1974 ) for the a l ternatesolution. In addition, it is possible to comparethe above solutions with Spencer's method.

Table 3 shows a comparison of the twosolutions for the example problems (Fig. 8 ) .Figures 11 and 1 2 graphically display thecomparisons. Although the com puter programsuse different methods for the input of the

geometry and side force function and differenttechniques for solving the equations, thefactors of safety are essentially the same.

The example cases show that when the s ideforce function is either a constant or a halfsine, the average factor of safety from theUniversity of Alberta computer programdiffers from the University of Saskatchewan



438 C A N . G E O T E CH

MORGENSTERN PRICE /95 f ( x ) = CONSTANT

2 00

x- f (x) = SINEA f ( x ) = C L I PPE D S lNE

190 /

I I I I ISPENCER /

FIG 11 Effect o f s ide fo rce funct ion on facto r o fsafety for case 3 .

MORGENSTERN-PRICE- f ( x ) = CONSTANT

x - f ( x ) = S l N E

-A- f (X I = CLIPPED S I N E

FIG 12. Effect o f s ide fo rce funct ion on facto r o fsafety fo r case 4

computer program by less than 0 .7 . Usingthe University of Saskatchewan program, theSpencer method and the Morgenstern-Pricemethod (for a constant side force function)differ by less than 0. 2 . T he average h valuescomputed by the two programs differ by ap-

1. VOL 14 1977

0 14

ORDINARY SIMPLIFIED JANBU S JANBU S SPENCERS MOW ENST EBISHOP SIMPLIFIED RIGOROUS -PRIC E

METHOD OF ANALYSIS

FIG 1 3 . Time per stabili ty analysis trial for allmethods of analysis.

proximately 9 . However, as shown above,this difference does not significantly affect thefinal factor of safety.

Comparison of Computing Costs

A simple 2 : 1 slope was selected to comparethe computer costs (i.e. CPU time) associatedwith the various methods of analysis. The

slope was 440 ft long and was divided into5-ft slices. The results shown in Fig. 13 wereobtained using the U niveristy of SaskatchewanSLOPE program run on an IBM 370 model158 comp uter.

The simplified Bishop method required0.012 min for each stability analysis. Theordinary method required approximately 60as much time. Th e fact or of safety by Spencer'smethod was computed using four side forceangles. The calculations associated with cachside force angle required 0.0 24 min. Th e factorof safety by the Mo rgenstern-Price metho d wascomp uted using six h values. Each trial required

0.021 min. At least three estimates of the sideforce angle or value are required to obtainthe factor of safety. Therefore, the Spencer orMorgenstern-Price metho ds are at least sixtimes as costly to run as the simplified Bishopmethod. The above relative costs are slightlyaffected by the width of slice and the toleranceused in solving the nonlinear factor of safetyequations.




Conclusions

( 1 ) The factor of safety equations for allmethods of slices considered can be writtenin the same form if it is recognized wh ethermoment and (or) force equilibrium is ex-plicitly satisfied. The normal force equation isof the same form for all methods with theexception of the ordinary method. The methodof handling the interslice forces differentiatesthe normal force equations.

2) The analytical aspects of slope stabilitycan be viewed in terms of on e factor of safetvequation satisfying overall m omen t equilibriumand ano ther satisfying overall force equilibriumfor various h values. Then each method be-comes a sp ecial case of the best-fit facto r of

safety lines.3 ) Th e best-fit regression solution andthe Newton-Raphson solution give the samefactors of safety. They differ only in themanner in which the equations of statics areutilized.

4) Th e best-fit regression solution is readilycomprehended. It also gives a complete under-standing of the variation of factor of safetywith respect to h

cknowledgements

The Department of Highways, Government

of Saskatch ewan, provided m uch of thefinancial assistance required for the develop-ment of the University of SaskatchewanSLOPE computer program. This program

contains all the methods of analysis used forthe comparative study.

The authors also wish to thank Mr. R.

Johnson, Manager, Ground Engineering Ltd. ,Saskatoon, Sask. , for the comparative com-puter time study.

B I S H O P ,A. W. 1955. The use of the s l ip c irc le in thestabi l i ty analysis of s lopes . G eotechnique , 5, pp. 7-17.

FE LL E NI US , . 1936. Calculat ion of the s tabi li ty of ear thdams. Proceedings of the Second Congress on LargeDams , 4 pp. 445-463.

F R E D L U N D ,. G. 1974. Slope stability analysis. U ser sManual CD -4, Departm ent of Civil Engineer ing, U niver-sity of aska at chew an Saska toon , Sask .

1975. A comprehensive and flexible slope stabilityprogram. Presented a t the Roads and Transporta t ionAssociat ion of Canada Meet ing, Calgary, Aka.

J A N B U ,N . , B J E R R U M , ., a n d K J A E K N S L I ,. 1956.Stabilitetsbe regning for fyllinger skjaeringer og naturligeskraninger . Norwegian Geotechnical Publ icat ion No.16, Oslo , N orway.

K R A H N .. , P I U C E , . E. , a n d M O RG EN SI EK N ,. R. 1971.Slope s tabil i ty compu ter program for Morgenstern-Price method of analysis . User s Mantia l No. 14, Uni-vers i ty of Alber ta , Edm onton, Alta .

M O K G E N S I ~ E I I N ,. R and PI I ICE, . E. 1965. The analy sisof the s tabi l i tv of genera s l ip s~t l -h~ces .eo techn ique ,15, pp. 70-93.

SP EN CE R. . 1967. A method of analvsis of the s tabi li tv ofembank ments assuming pa rdl e i inter-s l ice forEes.Geo techn ique , 17 pp. 11-26.

T E K Z A G H I ,. , and PECK.R . B. 1967. Soil mechanics inengineerin g practice. (211d etl.) . John W iley and S on s,Inc . , New York , N.Y.

W H I T M A N ,. V . and BA IL EY , . A. 1967. Use o f compu-ter for s lope s tabi l ity analysis . ASC E Journal of the Soi lMech anics and Foundat ion Divis ion, 93(SM4).

WR IG HT , . 1969. A s tudy of s lope s tabi li ty and the un-drained sh ear s t rength of c lay shales . Ph D thesis , Uni-vers i ty of Cal i fornia , Berkeley, CA .

stability methods of analysis

Documents