Comparison Of Slope Stability Methods Of Analysis

Comparison of slope stability methods of analysisr D . G . F n p o l u N D A N DJ . K n e s N (Jnit'ersity o.fSaskatchev'an, Saskatoon, Sask., Canada 57N 0W0 Received October 12, 1976 Accepted April 4,1977 T h e p a p e r c o m p a r es s i x m e t h o d s o f s l i c es c o m m o n l y u s e df o r s l o p e s t a b i l i t y a n a l y s i s .T h e f a c t o r of safety equations are written in the same form, recognizing whether moment and (or) force e q u i l i b r i u m i s e x p l i c i t l y s a t i s f i e d .T h e n o r m a l f o r c e e q u a t i o n i s o f t h e s a m e f o r m f o r a l l m e t h o d s w i t h t h e e x c e p t i o n o f t h e o r d i n a r y m e t h o d . T h e m e t h o d o f h a n d l i n g t h e i n t e r s l i c ef o r c e s d i f f e r e n t i a t e st h e n o r m a l f o r c e e q u a t i o n s . A n e w d e r i v a t i o n f o r t h e M o r g e n s t e r n - h i c e m e t h o d i s p r e s e n t e da n d i s c a l l e d t h e ' b e s t - f i t r e g r e s s i o n ' s o l u t i o n . I t i n v o l v e s t h e i n d e p e n d e n ts o l u t i o n o f t h e f o r c e a n d m o m e n t e q u i l i b r i u m _ faitors of safety for various values of tr. The be st-fit regression solution gives the same factor of The best-fit regression solution is readily comsafety as the'Newton-Raphson'solution. p r e h e n c l e dg, i v i n g a c o m p l e t e u n d e r s t a n d i n go f t h e v a r i a t i o n o f t h e f a c t o r o f s a f e t y w i t h t r . L ' a r t i c l e p r 6 s e n t e u n e c o m p a r a i s o n d e s s i x m 6 t h o d e s d e t r a n c h e s u t i l i s 6 e sc o u r a m m e n t p o u l ' I ' a n a l y s e d e l a s t a b i l i t 6d e s p e n t e s . L e s 6 q u a t i o n sd e s f a c t e u r s d e s 6 c u r i t 6 s o n t 6 c r i t e s s e l o n l a m6me forme, en montrant si les conditions d'dquilibre de moment ou de force sont satisfaites e x p l i c i t e m e n t . L ' d q u a t i o n d e l a f o r c e n o r m a l e e s t d e l a m 6 m e f o r m e p o u r t o u t e s l e s m d t h o d e si r I ' e x c e p t i o n d e l a m 6 t h o d e d e s t r a n c h e so r d i n a i r e s . L a f a g o n d e t r a i t e r l e s f o r c e s i n t e r t r a n c h e se s t c e q u i d i f f 6 r e n c i e l e s 6 q u a t i o n sd e f o r c e n o r m a l e . "best-fit regression" de la m6thode de MorgensUne nouvelle d6rivation appel6e solution i r d 6terminer ind6pendemment les tacteuls tle secutern-Price est pr6sent6e. Elle consiste r i t € s a t i s f a i s a n ta u x 6 q u i l i b r e s d e f o r c e e t d e m o m e n t p o u r d i f f 6 r e n t e s v a l e u r s d e t r . L a s o l u "best-fit r e g r e s s i o n " d o n n e l e s m e m e s f a c t e u r s d e s 6 c u r i t 6d e l a s o l u t i o n d e N e w t o n - R a p h tion "best-fit regression" est plus facile ir comprendre et donne une imaged6taill6e son. l-a solution d e l a v a r i a t i o n d u f a c t e u r d e s 6 c u r i t 6a v e c I " . lrraduit parla revuel c a n . G e o t e c hJ.. . r 4 . 4 2 g ( | g 1 7 \

Introduction The geotechnical engineer frequently uses limit equilibrium methods of analysis when studying slope stability problems. The methods of slices have become the most common methods due to their ability to accommodate complex geometrics and variable soil and water pressure conditions (Terzaghi and Peck 1967). During the past three decades approximately one dozen methods of slices have been developed (Wright 1969). They difier in (i) the statics employed in deriving the factor of safety equation and (ii) the assumption used to render the problem determinate (Fredlund 1975). This paper is primarily concerned with six of the most commonly used methods: (i) Ordinary or Fellenius method (sometimes referred to as the Swedish circle method or the conventional method) (ii) Simplified Bishop method 'Presented at rhe 29th Canadian Conference, Vancouver, B.C., October

Geotechnical 13-15, 1976.

(iii) Spencer'smethod (iv) Janbu'ssimplifiedmethod (v) Janbu'srigorousmethod (vi) Morgenstern-Price method The objectives of this paper are: ( 1) to compare the various methods of slices in terms of consistent procedures for deriving the factor of safety equations. All equations are extended to the case of a composite failure surface and also consider partial submergence, line loadings, and earthquake loadings. (2) to present a new derivation for the Morgenstern-Price method. The proposed derivation is more consistent with that used for the other methods of analysis but utilizes the elements of statics and the assumption proposed by Morgenstern and Price (1965). The Newton-Raphson numerical technique is not used to compute the factor of safety and ).. (3) to compare the factors of safety obtained by each of the methods for several example problems. The University of Sas-

CAN. GEOTECH.J. VOL. 14,1977

.-?,0 WATER

WATER, Ap

\-BEDROCK Frc. 1. Forces acting for the method of slices applied to a composite sliding surface.

katchewan SLOPE computer program was E used for all computer analyses (Fredlund L R 1 9 74 ) . (4) to compare the relative computational X costs involved in usine the various methods tof analysis. e Definition of Problem

: : : :

horizontal intersliceforces subscriptdesignatingleft side subscriptdesignatingright side vertical intersliceforces : S eismic ientto accountfor a dynaml S m l C coeffici ic horizontal force : vertical distancefrom the centroid ofeach sliceto the center of rotation A uniform load on the surface can be taken Figure I shows the forces that must be defined for a general slope stability problem. into account as a soil layer of suitable unit The variables associated with each slice are weight and density. The following variables are required to definea line load: defined as follows: W : total weight of the slice of width 6 and L : line load (force per unit width) u) : angle of the line load from the horizontal height ft P : total normal force on the base of the slice d : perpendicular distance from the line load to the centerofrotation over a length / The effect of partial submergenceof the slope : on the base of the shear force mobilized Sslice. It is a percentage of the shear or tension cracks in water requires the definition o f a d d i t i o n a vl a r i a b l e :s strength as defined by the Mohr-Coulomb : ul A : resultant water forces / {c' + lPll equation.That is, Sa : perpendicular distance from the resultant : cohesion where effective tan $'|lF c' water force to the center ofrotation parameter, 0' : effectiveangle of internal friction, -F : factor of safety, and u : Derivations for Factor of Safety porewater pressure : of statics that can be used to The elements radius or the moment arm associatedwith R derive the factor of safety are summations of the mobilized shear force S. : perpendicular offset of the normal force forces in two directions and the summation f of moments. These, along with the failure from the centerofrotation x : horizontal distance from the slice to the criteria, are insufficient to make the problem determinate. More information must be known center of rotation c( : angle betweenthe tangent to the center of about either the normal force distribution or the interslice force distribution. Either addithe baseofeach sliceand the horizontal

rr, r1*fffififiiiilffi|#ffiffi#l$,

tional elen must be in minate. Al use the I giving rise For comp derived u: equations Ordinary , The or simplestc the onll' factor of t that the it cause the slice (Fe principler fied betsc changein force frot factor of as 60% ( The no is derive perpendi mation ol direction

tI] IF wT2] IF

S-(

Substi normal fr

t3l

431

FREDLUNDAND KRAHN

O = R E S U L T A N TI N T E R S L I C E FORCE NOTE:Q^.IS NOT EQUAL A N O O P P O S I T ET O

o L,O ^

P o

Ftc. 2 Intersliceforces for the ordinary method.

tional elementsof physics or an assumption The factor of safety is derived from the must be invoked to render the problem deter- summation of moments about a common point minate. All methodsconsideredin this paper (i.e. either a fictitious or real center of rotation use the latter procedure, each assumption for the entire mass). giving rise to a different method of analysis. For comparisonpurposes,each equation is l4l l,vro : o derived using a consistentutilization of the Lw*- Is-R -Lpf +lkwe +Aa equationsof statics. Ordinary or FelleniusMethod The ordinary method is considered the simplestof the methods of slices since it is the only procedure that results in a linear factor of safetyequation.It is generallystated that the intersliceforces can be neglectedbecausethey are parallel to the base of each slice (Fellenius 1936). However, Newton's principleof 'actionequalsreaction'is not satisfied betweenslices(Fig. 2). The indiscriminate changein direction of the resultantinterslice force from one slice to the next results in factor of safety errors that may be as much as 60Vo (Whitmanand Bailey 1967). The normal force on the baseof each slice is derived either from summation of forces perpendicularto the base or from the summation of forcesin the verticaland horizontal directions.

tll lru:

o

W-Pcosd-S-sina:0

1 2 )I F ' :

o

S-cosa-Psina-kW:0 Substituting[2] into [1] and solvingfor the normal force gives t3l

P:Wcoso.-kWsina

+Ld--o Introducingthe failure criteria and the normal force from [3] and solving for the factor of safety gives

f{c'lR + (P - ul)R tan $'} -r Aa + Ld -Lpf + Lw* \kwe

r < - l IE _ - _-

LJr

SimplifiedBishopM ethod The simplifiedBishop method neglectsthe intersliceshearforcesand thus assumesthat a normal or horizontalforce adequatelydefines the intersliceforces(Bishop 1955). The normal force on the baseof each slice is derived by summingforcesin a verticaldirection(as in tll).Substituting the failure criteria and solvingfor the normal force gives

c'l sina 4i1l {' sin!lt^,, _ :f wI +F i6l p F I ' wheremo= "o, o * (sin a lan6') /F The factor of safety is derived from the summation of moments about a common point. This equation is the same as [4] since the interslice forces cancel out. Therefore, the factor of safety equation is the same as for the ordinary method ([5]). However, the definition of the normal force is different.

cAN. GEOTECH. J. VOL. 14, 1977

Spencer's method yields two factors of safely for each angle of side forces. However, at some angle of the interslice forces, the two factors of safety are equal (Fig. 3) and both moment and force equilibrium are satisfied. The Corps of Engineers method, sometimes : + : 5 0 t a n to as Taylor's modified Swedish referred l7l ER EL method, is equivalent to the force equilibrium where d = angle of the resultant interslice portion of Spencer'smethod in which the direcforce from the horizontal. iion of the interslice forces is assumed,generally perpendicforces (1967) summed Spencer at an angle equal to the average surface slope' ular to the interslice forces to derive the J anbu's Simplified Method normal force. The same result can be obJanbu's simplified method uses a correction and tained by summing forces in a vertical factor lo to account for the effect of the interhorizontal direction. slice shear forces. The correction factor is related to cohesion, angle of internal friction, t8l lru : o and the shape of the failure surface (Janbu W - (X* - Xt) - Pcoso - S- sina :0 et at. 1956). The normal force is derived from summation of vertical forces ([8])' with the tel IF' : o the intersliceshear forces ignored. -(E* - EL) + P sina - S-cos'z * kW : O ul tan I sin / m" The normal force can be derived from [8] T is obforce and then the horizontal interslice The horizontal force equilibrium equation tained from [9]. is used to derive the factor of safety (i'e' must tl I I ) . The sum of the interslice forces il01 p : I w - ( E * - E r ) t a n o cancel and the factor of safety equation

Spencer's Method Spencer's method assumes there is a constani relationship between the magnitude of the interslice shear and normal forces (Spencer 1967) .

1l

r r 3Pl : l r - + : +

a - c'l sin-+ F

ul tan 6' sin -----7-)rm"

q'l ,

U4l Fo :

Spencer(1967) derived two factor of safety equations.One is based on the summation of moments about a common point and the other on the summation of forces in a direction parallel to the interslice forces. The moment equation is the same as for the ordinary and the simplified Bishop methods (i.e. [4]). The factor of safety equation is the same as [5]. The factor of safety equation based on force equilibrium can also be derived by summing forces in a horizontal direction.

nll

ul) tan {'cos a}

Fn is used to designate the factor of safety uncorrected for the interslice shear forces'

L <

I t.> .lt

[16] P =

The from th llll). I the sim are kept The fac Spence (i.e.lt2 In on tion. ll evaluat

l.,

, l

L

o

h

l

l

t''

I

cos o

F{c'l cos a -t (P - uI) tan $' cos a} -Lcos <,, l f s i na + l k W + A

-

slice to sides ol between a slice z The r is derir forces.

T--

F

s

+LkW+A-I-cos<,.':0 The interslice forces (Et, - -En) must cancel out and the factor of safety equation with respect to force equilibrium reduces to L,LJ at -

f { c ' l c o s o c* ( P

tlsl

lanbu's Janb point a be defi used ar

o r

IFn : o

- ER) + lsIP sin a L(EL

becomes

The cor

L ro

15

20

sr0EF O R C EA N G L EE, ( D E G R E E S ) Frc. 3 Variation of the factor of safety with respect to moment and force equilibrium- vs'- the =^0'02; angle of the side forces. Soil properties-:c'/1h lO"t r" = 0.5. Geometry: slope = 26'5'; V': h e i g h t: 1 0 0 f t ( 3 0 m ) .

Frc. 4. rigorous r

433

FREDLUNDAND KRAHN

The corrected factor of safety is F = loFo

l15l

lanbu's Rigorous Method Janbu's rigorous method assumes that the point at which the interslice forces act can 'line of thrust'. New terms be defined by a used are defined as follows (see Fig. 4) : tr, tn = vertical distance from the base of the slice to the line of thrust on the left and right sides of the slice, respectively; dt = angle between the line of thrust on the right side of a slice and the horizontal. The normal force on the base of the slice is derived from the summation of vertical forces.

are set to zero. For subsequent iterations, the interslice forces are computed from the sum of the moments about the center of the base of each slice.

u7l LM" : 0 XLbl2 + XRbl2 - Erlt,

+ ER [/L + (bl2) tan a - b tan a,f - kwhl2 :0 After rearrangingI I 7], severalterms become negligibleas the width b of the slice is reduced to a width dr. Theseterms are (Xr, - XL) b/2, (E', - Er.) (b/2) Ian a and (Er - EL)b tan a1.Eliminating these terms and dividing by the slicewidth, the shearforce on the right side of a sliceis

t-

il8l

tl6lP:lw-(X^-X,)

Xn : En tana, - (ER - E)tRlb

L

-

c'l sina

+ (bl2) tanul

+ (kw/b)(h/2) ul tan 6' sinal,

F-- +-i---lt^"

The factor of safety equation is derived from the summation of horizontal forces (i.e. [1]). Janbu's rigorous analysis differs from the simplified analysis in that the shear forces are kept in the derivation of the normal force. The factor of safety equation is the same as Spencer'sequation based on force equilibrium (i.e. [12]). In order to solve the factor of safety equation, the interslice shear forces must be evaluated. For the first iteration. the shears

The horizontal interslice forces, required for solving [18], are obtained by combining the summation of vertical and horizontal forces on each slice. t19l

(E* - Er) :

lW - (X* -

Xy)l tan a

-S./cosa*kW The horizontal intersliceforces are obtained by integration from left to right across the slope. The magnitude of the interslice shear forces in [19] lag by one iteration. Each iteration gives a new set of shear forces. The vertical and horizontal components of line loads must also be taken into account when they are encountered. M o r genst ern-P r ic e M ethod The Morgenstern-Price method assumesan arbitrary mathematical function to describe the direction of the intersliceforces.

t20l

xf(x) = Yls

where ). = a constant to be evaluated in solving for the factor of safety, and l(x) = functional variation with respect to x. Figure 5 shows typical functions (i.e. l(x) ). For a constant function, the Morgenstern-Price method is the same as the Spencer method. Figure 6 shows how the half sine function and ), are used to designate the direction of the interslice forces. Frc. 4. Forces acting on each slice for Janbu's rigorous method.

Morgenstern and Price (1965) based their solution on the summation of tangential and

C A N . G E O T E C H J. VOL. 14. t977

f(x) =CONSTANT

f (x)=HALF- SINE

- SINE f (x)"CLIPPED

f(x)= TRAPEZOID

I x

o

x

x

n

I

f ( x )= S P E C I F I E O

x

U

Frc. 5. Functional variation of the direction of the side force with respectto the r direction.

QL

.eL (+)c(

variation of the factor of safety with respect to . The normal force is derived from the vertical force equilibrium equation ( [ 16] ) . Two factor of safety equations are computed, one with respect to moment equilibrium and one with respect to force equilibrium. The moment equilibrium equation is taken with respect to a common point. Even if the sliding surface is composite, a fictitious common center can be used. The equation is the same as that obtained for the ordinary method, the simplified Bishop method, and Spencer's method ([4] and [5]). The factor of safety with respectto force equilibrium is the same as that derived for Spencer's method (1121). The interslice shear forces are computed in a manner similar to that presented for Janbu's rigorous method. On the first iteration, the vertical shear forces are set to zero. On subsequent iterations, the horizontal interslice forces are first computed (t191) and then the vertical shear forces are computed using an assumed ), value and side force function. Xy: El.xf(x) l21l The side forces are recomputed for each

-__,_l I

: ren I i o R Ts

n

iteratio factors values Thesef: similar (Fig. 7

a SeCOI}

point o momen

Compat Atl momen same fo

t.?o r.r5

l22l

f

All n rium ha safetye

t.to F UJ u

a

123) r

L E

for the Morgenstern- o Frc. 6. Sideforcedesignation F Pricemethod. I

The visualiz ponent!

f

normal forces to each slice. The force equilibrium equations were combined and then the Newton-Raphson numerical technique was used to solve the moment and force equations for the factor of safety and ),. In this paper, an alternate derivation for the Morgenstern-Price method is proposed. The solution satisfies the same elements of statics but the derivation is more consistent with that used in the other methods of slices. It also presents a complete description of the

iiiqfrWlffiffiffi

o.95

o.90

o.2

o.4

o.6

o.8

LO

A

Frc. 7. Variation of the factor of safety with respect to moment and force equilibrium vs. tr for the Morgenstern-Price method. Soil properties: c'/7h = O.o2:,o' : 40'; r" : 0.5. Geometry: slope = 26.5'; height : 100 ft (30 m).

Cohesio Friction Weight Normal Earthqua Partial subme Line loa<

F R E D T - U N DA N D K R A H N

Y =tzo pc,r

@'= 2C" c' =600 psf

z---- j

P I E Z O M E T R ILCI N E

C O N D I T I O 2N ( w e o k c'=O, /t 19o 4

80

loo

t20

t40'

FIc. 8. Exampleproblem.

I I I

1

I

iteration. The moment and force equilibrium factors of safety are solved for a range of )" values and a specifled side force function. These factors of safety are plotted in a manner similar to that used for Spencer's method (Fig. 7). The factors of safety vs. ), are fit by a second order polynomial regressionand the point of intersection satisfies both force and moment equilibrium. Comparison ol Methods of Analysis AII methods of slices satisfying overall moment equilibrium can be written in the same form.

122) F_: ^-

From a theoretical standpoint, the derived factor of safety equations differ in (i) the equations of statics satisfied explicitly for the overall slope and (ii) the assumptionto make the problem determinate.The assumptionused changes the evaluation of the interslice forces in the normal force equation (Table l). All methods, with the exception of the ordinary method, have the same form of equation for the normal force.

l 2 4 lr : I w - ( x s . c'l sin- o- + F

- ul)R tan $' fc'lR f ItP

lwx - LPf + lkwe * Aa-r Ld

All methodssatisfyingoverall force equilibrium have the followins form for the factor of safetyequation: - ul)tan{'cosa ^ - --Lr'l cosa * IfP L ' J J 'E t -Lcos<', lrsin a+lkw+ A The factor of safety equations can be visualizedas consistingof the following components: r12-l

Xt)

whete mo = cos cY*

u l t a n 6 '- -s- i- n- - a l ,| l n "'" F )'

( sin a tan 6') / F .

It is possible to view the analytical aspects of slope stability in terms of one factor of safety equation satisfying overall moment equilibrium and another satisfying overall force equilibrium. Then each method becomes 'best-fit regression' solua special case of the tion to the Morgenstern-Price method. Tasrn l. Comparison of factorof safetyequations Factor of safety basedon

Cohesion Friction Weight Normal Earthquake Partial submergence Line loading

Moment equilibrium

Force equilibrium

Zc'lR Z(P-ul)R tan $' 2l4tx LPf Zklte

2c'l cosu 2(P-ul)tan$'cos c

Aa Ld

A I cos
!P sin a

>ktv

Method

Ordinary or Fellenius Simplified Bishop Spencer's Janbu's simplified Janbu's rigorous Morgenstern-Price

Moment equili brium

x x x

Force equilibrium

X X

x x

Normal force equation

t31 t6l t10l t13l t16l I24l

CAN. GEOTECH. J. VOL. 14, 1977

436 Tasrt

Case no. ExampleProblem* Simple2:l slope,40 ft (12m) high, 6' :20",c' :600 psf(29 kPa) Sameas I with a thin, weak layer with

6,:r0.,c,=0

Sameas I excePtwith ru : 0'25 ^ Sameas 2 exceptwith ru : 0.25for both materials 5 Sameas 1 exceptwith a piezometricline 6 Sameas2 exceptwith a piezometricline for both materials

5

2. Comparison of factors of safety for example problem

Ordinary method 1.928

Simplified Bishop method

2.080

Spencer'smethod

2 . O 7 31 4 . 8 1 0 . 2 3 7

Janbu's simplified method

2.041

MorgensternPrice method Janbu's ,f(x) : constant rigorous method**

2.008

2.076 0.254

Casc no.

si4 hr I .288

|.377

1 . 3 7 3 1 0 . 4 90 . 1 8 5

1.448

1 . 4 3 2 1 . 3 7 80 . 1 5 9

1.607

| .766

1. 7 6 1 1 4 . 3 3 0 . 2 5 5

1.735

r .708

1.765 0.244

1.029

1.t24

1.118 7.93 0.139

I .191

1.162

1 . 1 2 40 . 1 1 6

1.693

1. 8 3 4

l .830 13.87 0.247

1.827

| .776

1. 8 3 3 0 . 2 3 4

l 171

1.248

1.245 6.88 0,121

1 333

| .298

1.2s0 0.097

(!

2 $d ht S.{ 4 Sra fr 5 Sr I 6 SI I J

.Co.{ ttToE t65-

*Width of slice is 0.5 ft (0.3 m) and the toleranc€ on the nonlinear solulions is 0 001 *tThe line of thrust is assumedat 0.333

Figure 8 shows an example problem in- of safety based on overall force equilibrium volving both circular and composite failure are far more sensitive to the side force surfaces. The results of six possible combina- assumption. The relationship between the factors of tions of geometry, soil properties, and water conditions are presented in Table 2. This is not meant to be a complete study of 225 the quantitative relationship between various methods but rather a typical example. 2.20 The various methods (with the exception ,/ of the ordinary method), can be compared by The simplified plotting factor of safety vs. L 2.15 STMPLTFTED / Bishop method satisfiesoverall moment equiBISHOP i librium with ,1,= 0. Spencer's method has tr 2.to equal to the tangent of the angle between the l! horizontal and the resultant interslice force. 2.O5 / a Janbu's factors of safety can be placed along t! the force equilibrium line to give an indication o ,tJaNeu's RrGoRous 2.OO of an equivalent ), value. Figures 9 and 10 o show comparative plots for the first two cases F(J ., +- SPENCER t.95 shown in Table 2. u .-MORGENSTERN-PRICE The results in Table 2 along with thope f(x)= CONSTANT r90 from other comparative studies show that the factor of safety with respect to moment equi= 1.928 oRoTNARY librium is relatively insensitive to the interslice force assumption. Therefore, the factors of safety obtained by the Spencer and r.80 0'6 0.4 o.2 o Morgenstern-Price methods are generally A similar to those computed by the simplified Frc. 9. Comparison of factors of safety for case 1' Bishop method. On the other hand, the factors

i

Z|-.*'+-'-'-'-*-=-':

l6C-

|55-

l5OF F

U L

| 43F

U)

! co;t b l U (r F l U

i

35i-

I

t30-

I

| 25F

r20L

o

Frc. lO

safetl't whethe posite. method very sit Morger small il on the

437

AND KRAHN FREDLUND Tnslp 3. Comparison of two solutions to the Morgenstern-Price method** University of Alberta program side force function Constant Case no.

Half sine

University of Saskatchewan SLOPE program side force function Constant

Half sine

Clipped sine*

Example Problem

I Simple2:1 slope,40ft (12m) high, {' :20",c' : 600'sf (29 kPa) 2 Sameas I with a thin, weak layerwith $' : l0', c' : O 3 Sameas I exceptwith r" : 0.25 4 Sameas 2 exceptwith r" : 0.25 for both materials 5 Sameas I exceptwith a Piezometric line 6 Sameas 2 excePtwith a Piezometric line for both materials

2 . 0 8 5 0 . 2 5 7 2 . 0 8 5 0 . 3 1 4 2 . O 7 60 . 2 5 4 2 . 0 7 6 0 . 3 1 8 2 . 0 8 3 0 . 3 9 0 1 . 3 9 40 . 1 8 2 1 . 3 8 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 . 2 0 3 1 . 7 7 20 . 3 5 1 1 . 7 7 00 . 4 3 2 1 . 7 6 5 0 . 2 4 4 1 . 7 6 4 0 . 3 0 4 1 . 7 7 9 0 . 4 1 7 1 . 1 3 70 . 3 3 4 r . l r 7 0 . 4 4 1 r . 1 2 4 0 . 1 1 6 1 . 1 1 80 . 1 3 0 1 . 1 1 30 . 1 3 8 1 . 8 3 8 0 . 2 7 0 1 . 8 3 7 0 . 3 3 1 1 . 8 3 30 . 2 3 4 1 . 8 3 2 0 . 2 9 0 1 . 8 3 2 0 . 3 0 0 1 . 2 6 5 0 . 1 5 9 N o t c o n v e r g i n1g. 2 5 0 0 . 0 9 7 1 . 2 4 5 0 ' l 0 l

1.242 0.104

*coordinates x : 0,-Morgenstern-Price J : 0.5, and t : 1.0' y : o,21. - - , **ioG.ince on both solutions is 0.001.

equation. In the six example cases, the average difference in the factor of safety was a p p r o x i m a t e l y0 . 1 % .

r.65 r.60 r55

t.50 F

I r+s U)

SIMPLIFIE

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\ of factorsof safetyfor case2. Frc. 10. Comparison safety by the various methods remains similar whether the failure surface is circular or composite. For example, the simplified Bishop method gives factors of safety that are always very similar in magnitude to the Spencer and Morgenstern-Price method. This is due to the small influence that the side force function has on the moment equilibrium factor of safety

Comparison of Two Solutions to the Morgenstern-Price Method Morgenstern and Price ( 1965) originally solvedtheir method using the Newton-Raphson numerical technique.This paper has presented an alternate procedure that has been referred 'best-fit regression'method. The two to as the methods of solution were compared using the University of Alberta computer program (Krahn et al. 1971) for the original method and the University of Saskatchewan computer program (Fredlund 1974) for the alternate solution. In addition, it is possibleto compare the above solutions with Spencer'smethod. Table 3 shows a comparison of the two solutions for the example problems (Fig. 8). Figures 1 I and 12 graphically display the comparisons.Although the computer programs use difierent methods for the input of the geometry and side force function and different techniques for solving the equations, the factors of safety are essentially the same. The example casesshow that when the side force function is either a constant or a half sine, the average factor of safety from the University of Alberta computer program difters from the Universitv of Saskatchewan

438

CAN. GEOTECH.J. VOL, 14. 1977

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Frc. 13. Time per stability analysis trial for all methodsof analysis. proximately 9Vo. However, as shown above, this difference does not significantly affect the final factor of safety.

Comparison of Computing Costs

A simple 2: I slope was selectedto compare the computer costs (i.e. CPU time) associated with the various methods of analysis. The F slope was 440 ft long and was divided into u r ? n '.vv L 5-ft slices.The results shown in Fig. 13 were a obtained using the Univeristy of Saskatchewan b t25 SLOPE program run on an IBM 370 model 158 computer. o The simplified Bishop method required I rzo fl 0.012 min for each stability analysis. The ordinary method required approximately 6O% il5 as much time. The factor of safety by Spencer's method was computed using four side force angles. The calculations associated with each side force angle required 0.024 min. The factor o.2 o.6 of safety by the Morgenstern-Price method was computed using six ). values. Each trial required 0.02 1 min. At least three estimatesof the side Frc. 12. Effect of side force function on factor of safety for case4. force angle or ), value are required to obtain the factor of safety. Therefore, the Spencer or computerprogram by less than 0.7%. Using Morgenstern-Price methods are at least six program, the times as costly to run as the simplified Bishop the University of Saskatchewan Spencermethod and the Morgenstern-Price method. The above relative costs are slightly method (for a constantside force function) afiected by the width of slice and the tolerance differ by lessthan 0.2%. The average). values used in solving the nonlinear factor of safety computedby the two programsdiffer by ap- equations.

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Conclusions ( 1 ) The factor of safety equations for all methods of slices considered can be written in the same form if it is recognized whether moment and (or) force equilibrium is explicitly satisfied. The normal force equation is of the same form for all methods with the exception of the ordinary method. The method of handling the interslice forces differentiates the normal force equations. (2) The analytical aspectsof slope stability can be viewed in terms of one factor of safety equation satisfying overall moment equilibrium and another satisfying overall force equilibrium for various tr values. Then each method becomes a special case of the best-fit factor of safety lines. (3 ) The best-fit regression solution and the Newton-Raphson solution give the same factors of safety. They differ only in the manner in which the equations of statics are utilized. (4) The best-fit regressionsolution is readily comprehended.It also gives a complete understanding of the variation of factor of safety with respect to tr. Acknowledgements The Department of Highways, Government of Saskatchewan, provided much of the financial assistancerequired for the development of the University of Saskatchewan SLOPE computer program. This program

439

contains all the methods of analysis used for the comparative study. The authors also wish to thank Mr. R. Johnson, Manager, Ground Engineering Ltd., Saskatoon, Sask., for the comparative computer time study. BrsHop, A. W. 1955. The use of the slip circle in the s t a b i ift y a n a l y s i so f s l o p e s .G e o t e c h n i q u e5, , p p . 7 - 1 7 . F r l r - r r l u s . W . 1 9 3 6 .C a l c u l a t i o n o f t h e s t a b i l i t y o f e a r t h dams. Proceedings of the Second Congress on Large Dams, 4, pp.445-at$. F n r . o r - u N o , D . G . l 9 ' 7 4 .S l o p e s t a b i l i t y a n a l y s i s . U s e r ' s M a n u a l C D - 4 , D e p a r t m e n t o f C i v i l E n g i n e e r i n g ,U n t v e r s i t y o f S a s k a t c h e w a n ,S a s k a t o o n ,S a s k . 1 9 7 5 .A c o m p r e h e n s i v ea n d f l e x i b l e s l o p e s t a b i l i t y program. Presented at the Roads and Transportation A s s o c i a t i o no f C a n a d a M e e t i n g , C a l g a r y , A l t a . J,qNeu. N., Blennuv, L., and KlnenNsrr. B. 1956. S t a b i l i t e t s b e r e g n i n fgo r f y l l i n g e r s k j a e r i n g e ro g n a t u r l i g e skraninger. Norwegian Geotechnical Publication No. 1 6 ,O s l o , N o r w a y . K n , c H N ,J . , P n t c e, V . E . , a n d M o n c e N s r e n N , N . R . 1 9 7 1 . Slope stability computer program for MorgensternP r i c e m e t h o d o f a n a l y s i s .U s e r ' s M a n u a l N o . 1 4 , U n i versity of Albena. Edmonton, Alta. M o n c r r s l r . n N . N . R . a n d P n l c r - .V . E . 1 9 6 - 5T. h e a n a l y s i s o f t h e s t a b i l i t y o f g e n e r a l s l i p s u r f i t c e s .G e o t e c h n i q u e , 15,pp. 70-93. S p e r c E n , E . 1 9 6 7 .A m e t h o d o f a n a l y s i so f t h e s t a b i l i t y o f embankments assuming parallel inter-slice forces. G e o t e c h n i q u e ,1 7 , p p . l l - 2 6 . T e n z e c s t . K . , a n d P E c x , R . B . 1 9 6 7 .S o i l m e c h a n i c si n e n g i n e e r i n gp r a c t i c e . ( 2 n d e t l . \ . J o h n W i l e y a n d S o n s , lnc.. New York. N.Y. W H r r n e r , R . V . a n d B n t l s v , W . A . 1 9 6 7 .U s e o f c o m p u t e r f o r s l o p e s t a b i l i t y a n a l y s i s .A S C E J o u r n a l o f t h e S o i l M e c h a n i c sa n d F o u n d a t i o n D i v i s i o n , 9 3 ( S M 4 ) . W n r c H r , S . 1 9 6 9 .A s t u d y o f s l o p e s t a b i l i t y a n d t h e u n drained shear strengthof clay shales.PhD thesis' University of California,Berkeley, CA.