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Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
21
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Module 5:
Lecture -3 on Stability of Slopes
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slope Stability Analysis Methods
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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):
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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’).
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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).
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Morgenstern-Price method (M-PM)The Morgenstern-Price method (M-PM) also satisfiesboth force and moment equilibriums and assumes theinterslice force function.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Figure for Example 1
Example 1
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 1
This is the factor of safety for the trial failure surfaceselected and is not necessarily minimum factor ofsafety.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 1
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.
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 2
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Table giving computations (After Craig 2004)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example 3
2 m 2 m 2 m 2 m 2 m 2 m 1 m
Fellenius method of slices
b =
l = bsecα
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Solution for Example 2
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Example 3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Solution for Example 3
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Comparison of Slope stability analysis methods
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Comparison of LE methods
Grid and radius option used to search for circular CSS
Entry and exit option used to search for circular CSS
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slice 11 - Bishop Method
36.661
14.727
34.614
40.322
32.668
Slope stability analysis (Geo-slope 2012) Slice free body diagram
1.289
Bishop simplified Method (BSM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Slice 19 - Morgenstern-Price Method
22.893
7.9803
18.608
34.425
17.394
31.108
14.466
Slope stability analysis (Geo-slope 2012) Slice free body diagram
Morgenstern-Price method (M-PM)
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
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
Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
Aryal (2003)
PLAXIS