2. Types of Slope Failures
Falls
Topless
Slides
Spreads
Flows
Falls Topless
Slope failures consisting
p g Similar to a fall,
of soil or rock fragments except that it begins with
that droprapidly down a a mass of rock of stiff
slope
l clay rotating away from a
l t ti f
Most often occur in vertical joint
steep rock slopes
Usually triggered by
water pressure or
seismic activity
3. Slides Spreads
Slope failures that
p Similar to translational
involve one ore more slides except that the
blocks of earth that block separate and
move ddownslope b
l by move apart as they also
t th l
shearing along well move outward
defined surfaces or thin Can be very destructive
shear zones
Flows Types of Slides
Downslope movement of
p Rotational slides
earth where earth Most often occur in
resembles a viscous homogeneous materials
fluid
fl id such as fills or soft clays
Mudflow can start with a Translational slides
snow avalanche, or be in
avalanche Move along planar shear
surfaces
conjuction with flooding
Compound slides
Complex and composite
slides
4. The driving force (g
g (gravity) overcomes the
y)
resistance derived from the shear strength of
the soil along the rupture surface
g p
Slope stability analysis
Calculation to check the safety of natural
slopes, slopes of excavations and of
compacted embankments.
t d b k t
The check involves determining and
comparing the shear stress de eloped
developed
along the most likely rupture surface to
the shear strength of soil.
5. Factor of Safety
τf
Fs =
τd
Fs = factor of safety with respect to strength
τ f = average shear strength of the soil
τ d = average s ea st ess de e oped a o g t e potential failure su ace
a e age shear stress developed along the pote t a a u e surface
τ f = c + σ tan φ
c = cohesion
φ = angle of friction
σ = average normal stress on the potential failure surface
similarly
τ d = cd + σ tan φd
subscript 'd' refer to p
p potential failure surface
c + σ tan φ
Fs =
cd + σ tan φd
when Fs = 1, the slope is in a state of impending of failure.
In general Fs > 1.5 is acceptable
Stability of Slope
Infinite slope without seepage
Infinite slope with seepage
Finite slope with Plane Failure Surface
(Cullman s
(Cullman's Method)
Finite slope with Circular Failure Surface
(Method of Slices)
6. Infinite slope
L
c tan φ
β Fs = +
γ H cos2 β tan β tan β
d
a F
W
Na
B For granular soils, c=0 Fs is independent
soils c 0,
F Ta
c of height H, and the slope is stable as
H
long as β < φ
b Tr
β N
A r
R
L
d
Direction of
γ ' tan φ
a seepage
W c
Na
F =
Fs +
γ sat H cos2 β tan β γ sat tan β
B
Ta
c
H
T
β b r β N
A r
R
Finite slope with Plane Failure Surface
B C
W β + φd
Na
θcr =
Ta 2
τ f = c + σ tan φ
H unit weight of soil = γ
4c ⎡ sin β cos φ ⎤
Tr Nr
R
Hcr =
A
β θ γ ⎢ 1 − cos( β − φ ) ⎥
⎣ ( ⎦
7. Finite slope with Circular Failure Surface
Slope Failure Shallow Slope Failure
O
Toe Circle
Firm Base
O
Base Failure
L L
O
Slope Circle
Midpoint circle
Firm Base Firm Base
Slopes in Homogeneous Clay (φ=0)
For equilibrium, resisting and driving moment about O:
τ f = cu
O
MR = Md
θ C D
c dr 2θ = W1l1 − W2 l 2
Radius = r
W l −W l
cd = 11 2 2 2 l2 l1 cd
rθ H
F
W1
A B
τf cu
Fs = = W2
cd cd cd Nr (normal reaction)
E cd
critical when Fs is minimum, → trials to find critical plane
solved analitically by Fellenius (1927) and Taylor (1937)
cu
Hcr =
γm
presented graphically by Terzaghi &Peck, 1967 in Fig 11.9 Braja
m is stability number
8. Swedish Slip Circle Method
Slopes in Homogeneous Clay (φ>0)
τ f = c + σ tan φ
c d = γ H ⎡f (α , β ,θ ,φ ) ⎤
⎣ ⎦
c
= f (α , β ,θ ,φ ) = m
γ Hcr
where m is stability number
9. Method of Slices
For equilibrium
Nr = Wn cos α n
n=p
∑ ( c ΔL n + Wn cos α n tan φ )
Fs = n =1
n=p
∑W
n =1
n sin α n
bn
ΔLn ≈ where b is the width of nth slice
cos α n
Bishop Method
Bishop (1955) refine solution to the previous
method of slices.
In his method, the effect of forces on the sides
of each slice are accounted for some degree .
n=p
1
∑ ( cb
n =1
n + Wn tan φ )
mα ( n )
Fs = n=p
∑Wn =1
n sin α n
where
tan φ sin α n
mα ( n ) = cos α n +
Fs
The ordinary method of slices is presented as
learning tools. It is rarely used because is too
tools
conservative
10. Computer Programs
STABL
GEO-SLOPE
etc
Bibliography
Holtz, R.D and Kovacs, W.D., “An Introduction to Geotechnical
An
Engineering”
Das, B.M., “Soil Mechanics”
Das, B.M., “Advanced Soil Mechanics”
