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1. Bishop'S method of stability analysis
OR
Effective Stress Analysis by Bishop's Method
In the method of slices, the effect of the horizontal and shearing forces acting the
slices were neglected.
As a consequence, the factor of safety values obtained were conservative and may lead
to uneconomical design, especially in the case of deep slip circles.
Bishop's method (1955) eliminates these errors to a large degree and provides more
accurate results. The analysis is based on the effective stress approach.
The shape of the slip surface is assumed to be circular.
His method also takes into account the pore pressure acting on the slice.

2. Let E, and Ent1
resultant horizontal forces on the
section n and n+1 respectively.
X, and Xt = resultant vertical shear forces
W= weight of the slice
P= total normal force acting on the base of the slice
S= Shear force acting at the base of the slice
Z= height of the slice
l= length of the arc ab of the slice
b= horizontal width of the slice
= angle of the base ab of the slice with horizontal
OR angle between P and the vertical
x = horizontal distance of the slice from the centre of
the rotation
En
P
W
n+1
En+1
FIG. 23.23. BISHOP"S ANALYSIS.
Centre of
rolatkon

3. Total normal stress G on the base of the slice
F= (1)
Therefore,
I
Ifu is the pore pressure, the effective stress is
1
1
1
S=Tl.l
P
T=-c'+ (o-u) tan ¢] (3)
(2)
(Since, Shear strength =c+otan¢)
I-[c + G-u) tan ¢] (4)
Shear force acting on the base of the slice,
P
l.1
En
P
W
n+1
-En+1
FIG. 23.23. BISHOP'S ANALYSIS.
Centre of
rolatlon

4. For equilibrium,
Sliding moment = Restoring or resisting moment
Xw.x-)s.r-Xrlir
Or
Now from equation (4), which is as under:
1
W.x=)T.l. 1.r
T= c +f-u) tan ¢'] (49)
:. F=
Fz+6-u) cano
P
-2[cl + (P -u.l) tano']
F
XWx
-Z[cl+ (P -u. )tao](5)
E,-
n
P
W
n+1
-En+1
FIG. 23.23. BISHOP'S ANALYSIS.
Centre of
rolatlon

5. Let the normal effective force,
P=P-u.l.1 = P-ul
Resolving the forces vertically,
W=Pcos + S sin (6)
(The vertical shear forces, X, and Xm are taken
to be equal and hence to be neutralizing each
other, the error from this assumption being
considered negligible).
Here P =P'+ul
Since, S=T1.1
From equation (4), which is as under:
1
T=c +G-) tan ¢'] (4)
S= [cl+ P-ul) tan ]
1
F
-[cl+P' tan ¢']
E,
W
n+1
-En +1
FIG. 23.23. BISHOP"'S ANALYSIS.
Centre of
rolation

6. Substituting the values of P and S in equation (6),
which is as: W =P cos +S sin 0 (6)
W = (P + ul)cose +
=P cos + ul. cos +
-p' (cose +
Therefore, P =
:. F=
XWx
c l.sin 0
(c l+P tan o )sin9
F
w-ucos 04e
P tan sin0
+1(ucose + ina
cos 9+
We get, FEcl+
F
F
sin 0
tan D sin 8
Substituting the value of P for (P-ul) in equation (5), which is as:
F
cos +
Z[cl+ (P -u. )tano] (5)
F
w-ucos 04
C sin 91
F
tan 0 sin
F
tang' (7)
En
W
0
n+1
En+1
FIG. 23.23. BISHOP'S ANALYSIS.
Centre of
rolatlon

7. Substituting x =rsin., b = l cos0
The pore pressure ratio I,, for any slice is defined as
ub
F=
W
where z = average height of a slice
Substituting the above values in equation (7), we get
1
Since, W (weight of slice) = yzb.1,
Therefore,
W sinb2(c'b + W(1-n)tand')
1+
seco
tane tanp
F
Since this equation contains F on both sides, the
solution should be one by trial and error.
E
W
n+1
-En+1
FIG. 23.23. BISHOP'S ANALYSIS,
Centre of
rolatlon

8. Methods of Improving Stability of Slopes
The slopes which are susceptible to failure by sliding
can be improved and made safe either by reducing
the mass which may cause use sliding or improve the
shear strength of the soil in the failure zone.
1. Geometrical Method
The changing of the slope angle from steep
slope to a gentle slope may increase the
stabilization of slope and the angle is
usually supported by grass bonding
together with soil.
Fig.1 Geometrical method grass bonding
Fig.2 Combination of geometrical method with gabion wall at the toe of the slopes.

9. 2. Drainage Method
Saturation and pore water pressure building up in the subsoil causes slope failure.
Construction of surface drains minimize the building of pore water pressure.
This method is very effective and easy to maintain the surface drains, but it is difficult to
maintain the subsoil drains.
Subsoil drain is mostly found in the retaining structure
as leak holes and cut off drain.
It is capable of discharging more water during heavy
rain to avoid the effects of large amounts of water
absorption by the slope.

10. 3. Retaining Structures Method
This method is generally more costly. However, due to its
flexibility in a constrained site, it is mostly used.
The principle of this method is to use a a retaining
structure to resist the downward forces of the soi mass.
Ground anchors or other tie back system may be used
together with the retaining structures if the driving
4. Grouting and injection of cement or other compounds into
specific zones help in increasing the stability of slopes.
Cement Grouting
Epoxy Injection
RC Retaining Wall
forces are too large to resist.

11. 5. Upstrcam and downstream slope protection
Upstream slope of earth dam is protected against the erosive action of waves by stone
pitching or by stone dumping.
The thickness of the dumped rock should be about 1 m, and should be placed over a
gravel filter of about 0.3 m thickness.
Downstream slope protection
Downstream of slope 1 also
protected in similar manner.
In addition, downstream should be
provided berm at suitable
interval (about 15 m) to intercept
drain water and discharge safely.
Grass, and plants should be
planted soon after the construction
of earth dam.
Dumped stone
called rip rap
Concrete
toe wallI
1 m
0.3 m
Filter in 3 layers
(lowest size i.e.
Badarpur at
bottom and
gravel at top)
Fig. 14.19. Upstream slope protection
of Earthen dams.