21 - nptel.ac.in · Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay The ordinary method of slices

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

21


Module 5:

Lecture -3 on Stability of Slopes


Slope Stability Analysis Methods


The ordinary method of slicesIn this method, the potential failure surface is assumedto be a circular arc with centre O and radius r.

The soil mass (ABCD) above a trial surface (AC) isdivided by vertical planes into a series of slices of widthb.

The base of each slice is assumed to be a straight line.

The factor of safety (FS) is defined as the ratio of theavailable shear strength τf to the shear strength τm whichmust be mobilized to maintain a condition of limitingequilibrium.


The Ordinary method (OM) satisfies the momentequilibrium for a circular slip surface, but neglects boththe interslice normal and shear forces. The advantageof this method is its simplicity in solving the FOS, sincethe equation does not require an iteration process.

The ordinary method of slices


rsinα

r

r

h

α

b

A B

CD

l

α

X2

X1

E1

E2

α

FBD of slice i

The method of slices

OLA = length of arc AC


The method of slices

m

fFSττ

=The FS is taken to be the same for eachslice, implying that there must be mutualsupport between slides. i.e. forces must actbetween slices.

1. Total weight of slice W = γbh

2. Total normal force N = σl ( includes N′ = σ′l and U = ul)u = PWP at the centre of the base and l is the length of the base.

3. The shear force on the base, T = τml

4. Total normal forces on sides E1 and E2

5. The shear forces on the sides, X1 and X2


The method of slices Considering moments about O, the sum of themoments of the shear forces T on the failure arc ACmust be equal the moment of the weight of the soilmass ABCD.

∑ ∑= αsinWrTr

( )∑ ∑= ατ

sinWlFS

f

Using ( ) lFSlT f

m

ττ ==

∑∑=

ατsinW

lFS f


The method of slicesFor an analysis in terms of effective stress:

∑∑ ′′+′

=αφσ

sin)tan(

Wlc

FS

∑∑ ′′+′

=α

φsin

tanW

NLcFS a

Equation (1) is exact but approximations areintroduced in determining the forces N′.

(1)


The Fellenius (or Swedish ) SolutionIt is assumed that for each slice the resultant of theinterslice forces is zero.

The solution involves resolving the forces on each slicenormal to the base i.e. N′ = Wcosα - ul

∑∑ −′+′

=α

αφsin

)cos(tanW

ulWLcFS a

Rewriting Equation (1):


r

r

α4

A

CD

12

34

567

α1

α3

-α5

The Fellenius (or Swedish ) method of slices

The components of Wcosα andWsinα can be determinedgraphically for each slice.

For an analysis in terms of totalstress the parameters cu and φuare used and the value of u = 0

∑∑+

=α

αφsin

)cos(tanW

WLcFS uau

∑=

αsinWLcFS auFor φu = 0


In this solution it is assumed that the resultant forces onthe sides of the slices are horizontal. i.e X1 – X2 = 0For equilibrium the shear force on the base of any slice is:

( )φ′′+′= tan1 NlcFS

T

Resolving forces in the vertical direction:αφααα sintansincoscos ′

′+

′++′=

FSN

FSlculNW

After some rearrangement and using l = b secα:

∑∑

′+

′−+′=)/tan(tan1

sec]tan)([sin

1FS

ubWbcW

FSφα

αφα

Bishop simplified Method (BSM)


Bishop (1955) also showed how non-zero values of theresultant forces (X1-X2) could be introduced into theanalysis but refinement has only a marginal effect onthe factor of safety.

The pore water pressure can be related to the total fillpressure at any point by means of dimensionless porepressure ratio ru = u/γh .

For any slice, ru = u/W/b

∑∑

′+

′−+′=)/tan(tan1

sec]tan)1([sin

1FS

rWbcW

FS u φααφ

α

By rewriting:



Bishop simplified Method (BSM)Bishop’s simplified method (BSM) considers theinterslice normal forces but neglects the intersliceshear forces. It further satisfies vertical forceequilibrium to determine the effective base normalforce (N’).


Janbu’s simplified methodJanbu’s simplified method (JSM) is based on acomposite slip surface (i.e. non-circular) and the FOSis determined by horizontal force equilibrium. As inBSM, the method considers interslice normalforces (E) but neglects the shear forces (T).


Morgenstern-Price method (M-PM)The Morgenstern-Price method (M-PM) also satisfiesboth force and moment equilibriums and assumes theinterslice force function.


Spencer’s method

Spencer’s method (SM) is the same as M-PMexcept the assumption made for interslice forces. Aconstant inclination is assumed for interslice forcesand the FOS is computed for both equilibriums(Spencer 1967)


A 45 slope is excavated to a depth of 8m in a deeplayer of saturated clay of unit weight 19 kN/m3: therelevant shear strength parameters are cu = 65 kN/m2

and φu = 0. Determine the factor of safety for the trialfailure surface specified in Figure. The cross-sectionalarea ABCD is 70m2.

Example 1

After Craig (2004)


Figure for Example 1

Example 1


Solution for Example 1

This is the factor of safety for the trial failure surfaceselected and is not necessarily minimum factor ofsafety.



The minimum factor of safety can be estimated by using FS = cu/NsγH.

Using Taylor’s chart for Ns vs Slope inclination β, For β= 45° and assuming that D is large, the value of Ns is0.18.


Taylor’s curves

β > 53°

Slope inclination β

β [°] Ns

60 0.191

65 0.199

70 0.208

75 0.219

80 0.232

85 0.246

90 0.261

For φ = 0 soils


Example 2

Using the Fellenius method of slices, determine thefactor of safety, in terms of effective stress, of the slopeshown in Figure for the given failure surface usingpeak strength parameters c′ = 10 kPa and φ′ = 29°. Theunit weight of the soil above and below the watertable is 20 kN/m3.

After Craig (2004)




Table giving computations (After Craig 2004)


Example 3

2 m 2 m 2 m 2 m 2 m 2 m 1 m

Fellenius method of slices

b =

l = bsecα


Slice W(kN)

c′(kPa)

tanφ′ l (m) N=Wcosα

T=Wsinα

c′l(kN)

ul(kN)

N-ul(kN)

(N-ul) tanφ′

1 27.7 8 0.466 2.3 24.5 -13 18.3 4.8 19.7 9.2

2 96.5 8 0.466 2.1 93.9 -22.5 16.5 14.8 79.1 36.9

3 148 8 0.466 2 148 0 16 22.2 125.8 58.6

4 188.7 8 0.466 2.1 183 44.4 16.5 29 154 72.8

5 199.8 8 0.466 2.3 176.1 94 18.3 34.2 141.9 61.1

6 148 15 0.364 2.8 105.5 103.9 42 31.4 74.1 27

7 37 15 0.364 2 16.5 32.6 30.4 11.4 5.1 1.9

Fellenius method of slices ∑∑∑ −′+′

=T

ulNlcFS

)(tanφ

∑T = 239.4

∑c′ l = 158

∑ = 267.5FS = (158+267.5)/239.4

= 1.78



Using the Bishop method of slices, determine thefactor of safety in terms of effective stress for the slopedetailed in Figure for the specified failure surface. Thevalue of ru is 0.20 and the unit weight of the soil is 20kN/m3 and the shear strength parameters are c′ = 0kN/m2 and φ′= 33°

Example 3


Example 3






Comparison of Slope stability analysis methods


Comparison of LE methods

Grid and radius option used to search for circular CSS

Entry and exit option used to search for circular CSS


After Lambe and Whitman, 1969)Schematic diagram slope cross-section

Slope material Properties Value

Unit wt (kN/m3) 19.64

Cohesion (kPa) 4.31

Friction angle (0) 32

Comparison of LE methods


Slice 11 - Ordinary Method

36.661

13.699

29.689

Slope stability analysis (Geo-slope 2012) Slice free body diagram

Ordinary method of slices

1.161


Slice 11 - Bishop Method

36.661

14.727

34.614

40.322

32.668


1.289



Slice 11 - Janbu Method

36.661

15.325

34.2

40.322

32.668

Slope stability analysis (Geo-slope 2012)

Slice free body diagram

Janbu’s simplified method

1.2221.222


Slice 19 - Morgenstern-Price Method

22.893

7.9803

18.608

34.425

17.394

31.108

14.466


Morgenstern-Price method (M-PM)


Ordinary method of slices (OSM)

Bishop’s simplified method (BSM)

Geo-slope 2012 1.161 1.289

Lambe and Whitman (1969)

1.17 1.3

Comparison of factor of safety


Aryal (2003)

PLAXIS

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21 - nptel.ac.in · Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay The ordinary method of slices