Module 5: Lec tur ure e -3 on Stabilit ility y of of Slo lop pes
Slope Stability Analysis Methods
The or ordina inar ry met metho hod d of of slic lice es In
this met eth hod od,, the po potten enttial fai aillure surfac ace e is as ass sumed to be a c irc ular arar arc c with c en enttre O an and d rad adiius r. The
soil mass (ABCD CD)) above a trial surface (AC) is divided by ver erttic al plan anes es into a series of slic es of width b. The The
bas ase e of ofeac each h slic e is as ass sumed to be a strai aig ght line.
fact acto or of saf afe ety (FS) is defined as the ratio of the avail vaila able shea ear r strength f to the shea ear r strength m which must be mobilized to mai ain ntai ain n a c ondition of limiting equilibrium.
The ordinary method of slices The
Ordinary method (OM) satisfies the moment equilibrium for a circular slip surface, but neglects both the interslice normal and shear forces. The advantage of this method is its simplicity in solving the FOS, since the equation does not require an iteration process.
The method of slices LA = length of arc AC
rsin
O
r X2
b
D C
r
X1 E1
h
A
B l
FBD of slice i
E2
The method of slices
FS =
τ f τ m
The FS is taken to be the same for each slice, implying that there must be mutual support between slides. i.e. forces must act between slices.
1. Total weight of slice W = bh 2. Total normal force N = l ( includes N =
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= ml 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 the moments of the shear forces T on the failure arc AC must be equal the moment of the weight of the soil mass ABCD.
∑ Tr = ∑Wr sin α τ f
∑ (FS ) l = ∑ W sin α τ f l ∑ FS = ∑W sin α
Using T = τ ml =
τ f
(FS )
l
The method of slices For an analysis in terms of effective stress:
(c′ + σ ′ tan φ ′)l ∑ FS = ∑ W sin α FS =
c′ La + tan φ ′∑ N ′
∑ W sin α
(1)
Equation (1) is exact but approximations are introduced in determining the forces N .
The Fellenius (or Swedish ) Solution It is assumed that for each slice the resultant of the interslice forces is zero. The solution involves resolving the forces on each slice normal to the base i.e. N =Wcos - ul Rewriting Equation (1):
FS =
c′ La + tan φ ′∑ (W cos α − ul )
∑ W sin α
The Fellenius (or Swedish ) method of slices r
The components of Wcos and Wsin can be determined graphically foreach slice.
D
r
1
Foran analysis in terms of total stress the parameters c and are used and the value of u =0 u
FS =
7
cu La + tan φ u ∑ (W cosα )
For
u
=0
5
6
4
1
3
A
∑W sin α
4
FS =
2
3
u
cu La
∑W sin α
-
5
C
Bishop simplified Method (BSM) In this solution it is assumed that the resultant forces on the sidesof the slices are horizontal. i.e X1 – X2 =0 For equilibrium the shear force on the base of any slice is: T =
1 FS
(c′l + N ′ tan φ ′)
Resolving forces in the vertical direction: W = N ′ cos α + ul cos α +
c′l FS
sin α +
N ′ FS
tan φ ′ sin α
After some rearrangement and using l = b sec : FS =
∑
1 W sin α
sec α ′ ′ ∑ [c b + (W − ub ) tan φ ] 1 + (tan α tan φ ′ / FS )
Bishop simplified Method (BSM) Bishop (1955) also showed how non-zero values of the resultant forces (X1-X2) could be introduced into the analysis but refinement has only a marginal effect on the factor of safety. The pore water pressure can be related to the total fill pressure at any point by means of dimensionless pore pressure ratio ru = u/ h . For any slice, r u = u/ W/b FS =
∑
1 W sin α
By rewriting:
sec α ′ ′ + − [ c b W ( 1 r ) tan ] φ ∑ u 1 + (tan α tan φ ′ / FS )
Bishop simplified Method (BSM) Bishop’s simplified method (BSM) considers the interslice normal forces but neglects the interslice shear forces. It further satisfies vertical force equilibrium to determine the effective base normal force (N’).
J anbu’s simplified method J anbu’s simplified method (J SM) is based on a composite slip surface (i.e. non-circular) and the FOS is determined by horizontal force equilibrium. As in BSM, the method considers interslice normal forces (E) but neglects the shear forces (T).
Morgenstern-Price method (M-PM) The Morgenstern-Price method (M-PM) also satisfies both force and moment equilibriums and assumes the interslice force function.
Spencer’s method Spencer’s method (SM) is the same as M-PM except the assumption made for interslice forces. A constant inclination is assumed for interslice forces and the FOS is computed for both equilibriums (Spencer 1967)
Example 1 A 45 slope is excavated to a depth of 8m in a deep layer of saturated clay of unit weight 19 kN/m3: the relevant shear strength parameters are c u = 65 kN/ m2 and u = 0. Determine the factor of safety for the trial failure surface specified in Figure. The cross-sectional area ABCD is 70m2.
After Craig (2004)
Example 1
Figure for Example 1
Solution for Example 1
This is the factor of safety for the trial failure surface selected and is not necessarily minimum factor of safety.
Solution for Example 1 The minimum factor of safety can be estimated by using FS = c u/Ns H. Using Taylor’s chart for Ns vs Slope inclination , For = 45 and assuming that D is large, the value of Ns is 0.18.
Taylor’s curves For = 0 soils
> 53
Slope inclination
β [°]
Ns
60
0.191
65
0.199
70
0.208
75
0.219
80
0.232
85
0.246
90
0.261
Example 2 Using the Fellenius method of slices, determine the factor of safety, in terms of effective stress, of the slope shown in Figure for the given failure surface using peak strength parameters c =10 kPa and =29 . The unit weight of the soil above and below the water table is 20 kN/m3.
After Craig (2004)
Solution for Example 2
Table giving computations (After Craig 2004)
Example 3
Fellenius method of slices b =
2m
2m
2m
2m
2m
2m
1m
l = bsec
Solution for Example 2 Fellenius method of slices Slice
l (m)
c′l + tan φ ′∑ ( N − ul ) ∑ FS = ∑T
N=
T=
Wcos
Wsin
c l
ul
N-ul
(kN)
(kN)
(kN)
(N-ul) tan
-13
18.3
4.8
19.7
9.2
93.9
-22.5
16.5
14.8
79.1
36.9
2
148
0
16
22.2
125.8
58.6
0.466
2.1
183
29
154
72.8
8
0.466
2.3
176.1
148
15
0.364
2.8 105.5
37
15
0.364
W
c
(kN)
(kPa)
1
27.7
8
0.466
2.3
24.5
2
96.5
8
0.466
2.1
3
148
8
0.466
4 188.7
8
5 199.8 6 7
tan
FS = (158+267.5)/ 239.4 = 1.78
2
16.5
44.4 16.5 94
18.3
34.2
141.9
61.1
103.9
42
31.4
74.1
27
32.6
30.4
11.4
5.1
1.9
T= 239.4
c l =
158
= 267.5
Example 3 Using the Bishop method of slices, determine the factor of safety in terms of effective stress for the slope detailed in Figure for the specified failure surface. The value of ru is 0.20 and the unit weight of the soil is 20 kN/m3 and the shear strength parameters are c = 0 kN/m2 and = 33
Example 3
Solution for Example 3
Solution for 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
Comparison of LE methods Slope material Properties
Value
Unit wt (kN/m3)
19.64
Cohesion (kPa)
4.31
Friction angle (0)
32
Schematic diagram slope cross-section
After Lambe and Whitman, 1969)
Ordinary method of slices Slice 11 - Ordinary Method
