STABILITY OF SLOPESSEEPAGE CONTROL MEASURES AND SLOPE PROTECTION a finite slope AB, the stability of which is to be analyzed. The method Consists of assuming a number of trial slip circles, and finding the factor of safety of each. The circle corresponding to the minimum factor of safely is the critical slip circle. Let AD be a trial slip circle, with r as the radius and O as the centre of rotation Let W be the weight of the soil of the wedge ABDA of unit thickness, acting through the centroid G. The driving moment MD will be equal to W x, where x, is the distance of line of action of W from the vertical line passing through the centre of rotation O. if cu is the unit cohesion, and l is the length of the slip arc AD, the shear resistance developed along the slip surface will be equal to cu • l, which act at a radial distance r from centre of rotation O. When slip is imminent in a cohesive soil, a tension crack will always DevelOP by the top surface of the slope along which no shear resistance can develop, The depth of tension crack is given by The effect of tension crack is to shorten the arc length along which shear resistance gets mobilised to AB' and to reduce the angle δ to δ'. The length of the slip arc to be taken in the computation of resisting force is only AB', since tension crack break the continuity at B'. The weight of the sliding wedge is weight of the area bounded by the ground surface, slip circle arc AB' and the tension crack.

1. SWEDISH
SLIP
CIRCLE
METHOD

2. SWEDISH SLIP CIRCLE
The following two cases are
considered :
( 1 ) Analysis of purely cohesive soil
(Ф = 0 soil)
(2) Analysis of a cohesive frictional
soil (c — Ф soil)

3. (1) Ф = 0 Analysis (Purely
cohesive soil) :

4. a finite slope AB, the stability of which is
to be analyzed.
The method Consists of assuming a
number of trial slip circles, and finding
the factor of safety of each.
The circle corresponding to the
minimum factor of safely is the
critical slip circle.
Let AD be a trial slip circle, with r as the
radius and O as the centre of
rotation

5. Let W be the weight of the soil of the wedge
ABDA of unit thickness, acting through the
centroid G.
The driving moment MD will be equal to W x,
where x, is the distance of line of action of W
from the vertical line passing through the
centre of rotation O.
if cu is the unit cohesion, and l is the
length of the slip arc AD, the shear
resistance developed along the slip surface
will be equal to cu • l, which act at a radial
distance r from centre of rotation O.

6. Hence the resisting moment MR will be
equal to cu • I • r.
The length of the slip surface AD is given
by
Driving moment MD = WX
Factor of safety = F= cu • I • r.
WX

7. Effect of Tension Cracks
When slip is imminent in a cohesive soil, a
tension crack will always DevelOP by the
top surface of the slope along which no
shear resistance can develop,
The depth of tension crack is given by

9. The effect of tension crack is to shorten
the arc length along which shear
resistance gets mobilised to AB' and to
reduce the angle δ to δ'.
The length of the slip arc to be taken in the
computation of resisting force is only AB',
since tension crack break the continuity at
B'.
The weight of the sliding wedge is weight
of the area bounded by the ground
surface, slip circle arc AB' and the tension
crack.

10. Further, water will enter in the crack,
exerting a hydrostatic pressure Pw actjng
on the portion DB’ at a height Z0 /3 from
B’ .
Hence an additional driving moment due
to horizontal hydrostatic pressure PW For
depth Z0 Must be taken into account .

12. Effect of Submergence :
Slopes of embankment dams, canal banks,
etc. may be submerged partly or fully
at different times. Fig. shows the cross
section of such a slope. It can be seen from
the figure that the moment about O of the
body of water in the half segment
CEH balances that of the water
contained in the other half segment
FEH.

13. Since the moments of the water pressure
balance each other, the net driving moent
can be obtained by using bulk unit weight of
the soil y above the level of the water
surface and. submerged unit weight y'
below the water surface.
If the slope is fully submerged, the
submerged unit weight (y‘) is used for the
entire wedge section.
In addition the moment due to the water
resting on the slope will be an additional
resisting moment and increase the factor of
safety.

14. Effect of Submergence :

15. (2) c - Ф Analysis (Cohesive
frictionai Soil) :
• In order to test the stability of the slope of
a cohesive frictional soil (c - Ф soil), trial
slip circle is drawn. The material above
the assumed slip circle is divided into a
convenient number of vertical strips or
slices as shown in Fig.
• The forces between the slices are
neglected and each slice is assumed to
act independently as a column of soil of
unit thickness and of width b.

18. • The weight W of each slice is
assumed to act at its centre.
• If the weight of each slice is resolved
into normal (N) and tangential (T)
components, the normal component
will pass through the centre of
rotation O, and hence do hot cause
any driving moment on the slice.
• However, the tangential component
T causes a driving moment MD = T x
r

19. where r is the radius of the slip
circle. The tangential
components of a few slices the
base may cause resisting
moment, in that case T is
considered negative.
 c is unit cohesion and Δ L is the
curved length of each slice then
the resisting force from
Coulomb's equation is equal to
(cΔL + N tanФ).

21. A number of trial slip circles are
chosen and factor of safety of each
is computed.
The slip circle which gives the
minimum factor of safety is the
critical slip circle.

22. Methods of calculating ε N and
ε T:
Method-1 :
• The value of W, εN and εT may be
found by tabulating the values for all
slices as indicated below :

23. Method-2 :
• ε N and ε T may also be
obtained graphically . vertical
line drawn through the centre
of gravity of the slice and
intersecting the top and bottom
surfaces of the slice may be
assumed to represent the
weight of the slice.

24. • This may be resolved graphically into
normal and tangential components.
These components for all the slices are
plotted separately as ordinates on two
horizontal base lines.
• The plotted points are joined by
smooth curves as shown in Fig. They
are called N-curve and T-cnrve
respectively.
• The areas under these curves
represent EN and ET, The areas under
these curves are measured by
planimeter and multiplied by the unit
weight of the soil to obtain E N and ET.

25. Method-3 :
• A simplified rectangular plot
method has been suggested by Singh
(1962) to determine E N and E T.
• In this method the end ordinate of
each slice is assumed to represent As
weight of the slice and it is resolved
into normal and tangential
components as shown Fig. 2.37.

26. • The normal components N, N2 N3 etc. and the
tangential components T1, T2, T3 etc. are
plotted to form the base of N-rectangle and Trectangle as shown in fig. (b) and (c).
• The width of both rectangles being equal to the
width of the slices, in this method all the slice
should be of the same width. However, if the
width of the last slice is not same as that of the
other slice, but is less say mb, where m is
multiplying factor and b is the width of the
each of other slices, then the last N and T is f l
+ m)
• components are reduced by multiplying with
the factor ~ before being plotted z J the
rectangles.
• The area of N and T rectangles multiplied by
the unit weight of the soil gives E N E T
respectively.

30. Method of Locating Centre of
Critical Slip Circle :
• In order to reduce the number of
trials to find the centre of critical
slip circle, Fellinious has given a
method of locating the locus on
which the probable centre may lie.
Felliaious Method of Locating Centre
of Critical Slip Circle

32. • For a homogeneous c - Ф soil, the centre of slip
circle lie on a line PQ, in which [point Q has its
co-ordinates downwards from toe and 4.5H
horizontally away as in Fig. 2.38. The point P is
located at the intersection of the two lines, one
drawn i the toe at an angle a with the slope, and
other drawn from the top end of the slope i
angle p with the horizontal. The angles a and P
are known as the directional angles their values
depend on the slope angle i. The values of a
and p for different values angle i are
given in table below :

33. Values of directional angles a
and P for different values of i

34. • According to Fellinious for purely cohesive
soils (Ф = 0) the centre of critical circle is
located at point P, and for c – Ф soils, the
centre of critical slip circle lies at point
P on the line QP produced. When the line PQ
is obtained, a number of trial centre O1, 0 2 ,
0 3 , etc. are selected above point P on the
line QP produced.
• For each of selected trial centers slip circle is
drawn and factor of safety is computed. These
factors of safety so obtained are plotted as
ordinates on the corresponding centers a
smooth curve is obtained. The centre
corresponding to the lowest factor of safety
the critical circle centre.