This presentation is all about various approaches used to find stability of slope

1. STABILITY OF
SLOPES
Prepared By:
Arbaz Kazi
Asst. Professor
VCET, Vasai (W)

2. CONTENTS
• Slope – Definition
• Types of Slope
• Causes of failure of slopes
• Patterns of Slope failure
• Aims of Slope stability
• Types of factor of safety
• Types of slope failure
• Stability Analysis of Slope
❑Infinite Slope
❑Finite Slope
• Taylor’s Stability Number
• Method’s of Preventing Slope failure

3. ➢ A surface of which one end or side is at a higher
level than another; a rising or falling surface.
➢ A slope can be defined as an inclined body between
ground and air.
➢ An earth slope is inclined or unsupported part of a
soil mass.
Slope - Definition

4. SLOPES OF EARTH ARE OF TWO TYPES
1. Natural slopes
- slopes exist in hilly areas

5. 2. Man made slopes
❖ The sides of cuttings
❖ The slopes of embankments constructed for roads railway lines, canals etc.
❖ The slopes of earth dams constructed for storing water.

6. THE SLOPES WHETHER NATURAL OR ARTIFICIAL MAY BE
1. Infinite slopes
▪ The term infinite slope is used to designate a constant
slope of infinite extent.
▪ The long slope of the face of a mountain
2. Finite slopes
▪ Finite slopes are limited in extent.
▪ The slopes of embankments and earth dams are
examples of finite slopes.

7. CAUSES OF FAILURE OF SLOPES
✓ Gravitational force
✓ Force due to seepage water
✓ Erosion of the surface of slopes due to flowing water
✓ The sudden lowering of water adjacent to a slope
✓ Forces due to earthquakes

8. PATTERNS OF SLOPE FAILURE

9. Aims of Slope Stability Analysis
❑To assess the stability of slopes under short-term (often
during construction) and long-term conditions.
❑To assess the possibility of landslides involving natural or
existing engineered slopes.
❑To analyze landslides and to understand failure mechanisms
and the influence of environmental factors.
❑To enable the redesign of failed slopes and the planning and
design of preventive and remedial measures, where
necessary.
❑To study the effect of seismic loadings on slopes and
embankments.

10. WHAT CAUSES SLOPE TO FAIL?
• The failure of a soil mass occurs along a plane or a curved surface
when a large mass of soil slides with respect to remaining mass.
• In other words we can say there is downward and outward
movement of soil mass in case of slope failures.
• A slope failure occurs when the forces causing failure are greater
than the shearing resistance (shear strength) developed along
critical surface of failure.
• The factors leading to slope failure is divided in two types:
➢ The factors which causes increase in shear stresses.
➢ The factors which causes decrease in shear strength.

11. • In slope stability analysis we determine the Factor of
Safety as a ratio of resisting forces to driving forces
Fs = Resisting / Driving
• Theoretically, any slope with a Factor of Safety less
than one will fail and any slope with a factor of safety
greater than one will not.
• Design focuses on the soil parameters and geometry
that will provide the maximum factor of safety.
SLOPE STABILITY

12. Sometimes, the analysis of an existing slope will be what
is called a parametric study – that is establishing a factor
of safety and performing an analysis that back calculates
the strength parameters.
• The engineer will then determine his/her confidence
level as to whether or not the soil has that strength
through experience, lab, and/or field data.
SLOPE STABILITY

13. TYPES OF FACTOR OF SAFETY
❑Factor of safety with respect to shearing strength (Fs):
It is defined as the ratio of available shear strength to
mobilised shear strength.
𝐹𝑠 = 𝜏/𝜏 𝑚
❑Factor of safety with respect to cohesion (Fc): It is
defined as the ratio of available cohesion to mobilised
shear cohesion.
𝐹𝑐 = 𝑐/𝑐 𝑚
❑Factor of safety with respect to friction(FØ): It is
defined as the ratio of available friction to mobilised
friction.
𝐹∅ = ∅/∅ 𝑚

14. Types of Slope Failures
Slope Failure Toe Failure
Base Failure

15. ASSUMPTIONS MADE IN ANALYSIS
• Stress system is assumed to be 2-dimensional
• Stresses in 3rd direction are taken as zero
• Coulomb’s equation is used for determining shear
strength. i.e. ( s = c + 𝜎 ∗ tan ∅)
• It is assumed that seepage conditions and water level
are known and pore water pressure can be estimated
• The shearing strength at all points are enough to
mobilize all available shear strength.

16. Stability Analysis of Infinite Slope
Cohesionless Soil
1. Dry State
The factor of safety of infinite slopes for dry state may be
written as
𝐹𝑠 = Τtan ∅ tan 𝑖
2. Submerged State
The factor of safety of infinite slopes for submerged state
may be written as
𝐹𝑠 = Τtan ∅ tan 𝑖
3. Steady Seepage along slope
The factor of safety of infinite slopes for steady seepage
along slope may be written as
𝐹𝑠 = 𝛾′
∗ Τtan ∅ 𝛾𝑠𝑎𝑡 ∗ tan 𝑖

17. Stability Analysis of Infinite Slope
Purely Cohesive Soil
1. Dry State
The factor of safety of infinite slopes for dry state may be
written as
𝐹𝑠 = Τ𝑐 𝛾 ∗ cos 𝑖 ∗ sin 𝑖 ∗ 𝐻
2. Submerged State
The factor of safety of infinite slopes for submerged state
may be written as
𝐹𝑠 = Τ𝑐 𝛾′ ∗ cos 𝑖 ∗ sin 𝑖 ∗ 𝐻
3. Steady Seepage along slope
The factor of safety of infinite slopes for steady seepage
along slope may be written as
𝐹𝑠 = Τ𝑐 𝛾𝑠𝑎𝑡 ∗ cos 𝑖 ∗ sin 𝑖 ∗ 𝐻

18. Stability Analysis of Infinite Slope
Cohesive Friction Soil
1. Dry State
The factor of safety of infinite slopes for dry state may be
written as
𝐹𝑠 = Τ𝑐 + 𝛾 ∗ 𝐻 ∗ 𝑐𝑜𝑠2 𝑖 ∗ tan ∅ 𝛾 ∗ cos 𝑖 ∗ sin 𝑖 ∗ 𝐻
2. Submerged State
The factor of safety of infinite slopes for submerged state
may be written as
𝐹𝑠 = Τ𝑐 + 𝛾′ ∗ 𝐻 ∗ 𝑐𝑜𝑠2 𝑖 ∗ tan ∅ 𝛾′ ∗ cos 𝑖 ∗ sin 𝑖 ∗ 𝐻
3. Steady Seepage along slope
The factor of safety of infinite slopes for steady seepage
along slope may be written as
𝐹𝑠 = Τ𝑐 + 𝛾′
∗ 𝐻 ∗ 𝑐𝑜𝑠2
𝑖 ∗ tan ∅ 𝛾𝑠𝑎𝑡 ∗ cos 𝑖 ∗ sin 𝑖 ∗ 𝐻

19. Stability Analysis of Finite Slope
Following are the methods which are used for
stability analysis of finite slope:
1.Wedge failure method
2.Friction circle method
3.Swedish circle method (method of slices)
• Analysis of purely cohesive soil (Ø = 0)
• Analysis of c-¢ soil

20. Wedge Failure Method
A wedge failure occurs when a soil deposit has a specific plane of
weakness. The stratified deposits generally fail along the interface. Fig
shows a soil mass resting on an inclined layer of impermeable soil. There
is a tendency of the upper mass to slide downwards along the plane of
contact AB

21. • The forces trying to cause sliding is the tangential components of
the weight (W) along the plane of contact
𝑇 = 𝑊 ∗ sin 𝛼
Where α = angle which plane Ab makes with horizontal
• The force tending to resist the sliding depends upon the cohesion
‘c’ and the frictional force and is given by
𝑆 = 𝑐 ∗ 𝐿 + 𝑊 ∗ cos 𝛼 ∗ tan ∅
Where L = length of failure surface.
• FOS is given by:
𝐹 =
𝑐 ∗ 𝐿 + 𝑊 ∗ cos 𝛼 ∗ tan ∅
𝑊 ∗ sin 𝛼

22. The friction circle method is based on the fact that the resultant reaction
between the two portions of the soil mass into which the trial slip circle
divides the slope will be tangential to a concentric smaller circle of
radius r sin Ø, since the obliquity of the resultant at failure is the angle of
internal friction (Ø) (This, of course, implies the assumption that friction
is mobilised in full).
This smaller circle is called the ‘friction circle’ or ‘Ø - circle’.
The forces acting on the sliding wedge are:
i. weight W of the wedge of soil
ii. reaction R due to frictional resistance, and
iii. cohesive force 𝐶 𝑚 mobilised along the slip surface
Friction Circle Method

23. Friction Circle Method

24. Friction Circle Method
Following procedure is adopted for finding FOS with friction circle
method:
1. Consider slope as shown in fig.
A
B
Toe of a Slope
Crest of a Slope
2. Draw trial slip surface (Arc AC) passing through toe with “O” as
the centre and “R” as the radius.

25. 3. Find the centroid of the sliding soil mass ABCA and calculate it’s
weight “W”

26. 4. Let the slip circle be considered to be made up of a number of
elementary arcs each of length “ΔL”

27. 5. Let the cohesive force acting
along this element opposing
the sliding of soil be 𝑪 𝒎*ΔL
6. Total Cohesive force along arc
AC forms a force Polygon,
also the closing side AC of
polygon represents magnitude
and direction of resultant
cohesive force

28. 7. Let the length of chord AC be 𝐿 𝑐and length of Arc AC be ‘L’.
The magnitude of Resultant force C = 𝑪 𝒎* 𝐿 𝑐. The position of
resultant can be obtained using Varigno’s theorem
C*a = ∑ 𝑪 𝒎*ΔL*R
𝑪 𝒎* 𝑳 𝒄*a = 𝑪 𝒎*L*R
a =
𝑳
𝑳 𝒄
*R
L > 𝑳 𝒄
Hence a > R

29. 8. On mobilization of
frictional resistance .
Let P be the soil
reaction opposing the
sliding of the soil mass
as shown. Reaction due
to Frictional resistance
‘P’ is inclined at an
angle ‘Ø’ to the normal
at the point of action as
shown

30. 9. By knowing the magnitude and direction of ‘W’ and direction and
line of action of other forces, the force triangle can be completed.
Measuring the magnitude of ‘C’ the FOS can be computed as
shown.
The mobilised cohesion 𝐶 𝑚 = Τ𝐶 𝐿 𝑐
FOS w.r.t cohesion is given by
10. The minimum FOS can be obtained by locating critical slip
surface.
𝐹𝑐 = 𝑐/𝑐 𝑚

31. • This method was developed by Swedish
engineers assuming failure surface to be an arc of
a circle
• Two cases are considered in this method
1.Analysis of purely cohesive soil (Ø = 0)
2.Analysis of c-Ø soil
Swedish Circle Method

32. ANALYSIS OF PURELY COHESIVE
SOIL

33. • In this method, no. of trial slip circles are considered and factor of safety for each is
worked out.
• The circle corresponding to minimum factor of safety is called as critical slip circle
• Let AD be trial slip circle, with R as radius and O as centre of rotation. Let W be the
weight of wedge ABDA acting through its centroid
• Driving Moment = W x e
• Let Cu be the unit cohesion and L’ be the length of slip arc AD =
2𝜋𝑟𝛿
360
, hence the
shear resistance developed along slip surface will be equal to Cu x L’ which acts at
radial distance R from O.
• Resisting Moment = Cu x L’ x R
• F = Mr/Md = Cu x L’ x R/ W x e
• Let Cm = mobilised shear resistance necessary to keep equilibrium
• W x d = Cm x L’ x R , therefore Cm = W x e / L’ x R
• F = Cu/Cm = Cu x L’ x R/ W x e

34. ANALYSIS OF COHESIVE
FRICTION SOIL
❑In order to test the stability of the slope of a cohesive
frictional soil (c - Ø 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.

36. ❑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 𝑀 𝑑 = T x R

37. ✓ 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 ΔL is the curved length of each slice
then the resisting force from Coulomb's equation is equal to
(c*ΔL + N*tan ∅).
✓ 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.

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

40. Method-2 :
➢ ∑N and ∑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.
➢ 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-curve respectively.

41. The areas under these curves represent ∑N and ∑T, The
areas under these curves are measured by planimeter and
multiplied by the unit weight of the soil to obtain ∑N and
∑T.

42. Method-3 :
❖ A simplified rectangular plot method has been suggested
by Singh (1962) to determine ∑N and ∑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

43. Taylors Stability Number
✓ If the slope angle i, height of embankment H, the effective unit
weight of material , angle of internal friction ', and unit
cohesion c' are known, the factor of safety may be determined.
✓ Taylor (1937) conceived the idea of analyzing the stability of a
large number of slopes through a wide range of slope angles and
angles of internal friction, and then representing the results by an
abstract number which he called the "stability number". This
number is designated as Sn.
𝑆 𝑛 =
𝑐
𝐹𝑐 ∗ 𝛾 ∗ 𝐻

46. Methods of Preventing Slope failure
VIBROFLOTATION PROVISION OF BERM AT TOE
VEGETATION COVER PROVISION OF SHEET PILE