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Slope Stability
updated June 7, 2007
Haryo Dwito Armono, M.Eng, Ph.D
Types of Slope FailuresTypes of Slope Failures
FallsFalls
Topless
Slides
Spreads
Flows
Falls
Slope failures consisting
Topless
Similar to a fall,p gof soil or rock fragments that droprapidly down a l
except that it begins with a mass of rock of stiff l t ti fslope
Most often occur in steep rock slopes
clay rotating away from a vertical joint
steep rock slopes
Usually triggered by water pressure orwater pressure or seismic activity
Slides
Slope failures that
Spreads
Similar to translational pinvolve one ore more blocks of earth that
d l b
slides except that the block separate and
t th lmove downslope by shearing along well defined surfaces or thin
move apart as they also move outward
Can be very destructivedefined surfaces or thin shear zones
Can be very destructive
Flows
Downslope movement of
Types of Slides
Rotational slidespearth where earth resembles a viscous fl id
Most often occur in homogeneous materials such as fills or soft claysfluid
Mudflow can start with a snow avalanche or be in
such as fills or soft clays
Translational slidesMove along planar shearsnow avalanche, or be in
conjuction with flooding
Move along planar shear surfaces
Compound slides
Complex and composite slides
The driving force (gravity) overcomes the g (g y)resistance derived from the shear strength of the soil along the rupture surfaceg p
Slope stability analysisSlope stability analysis
Calculation to check the safety of naturalCalculation to check the safety of natural slopes, slopes of excavations and of
t d b k tcompacted embankments.
The check involves determining and comparing the shear stress de elopedcomparing the shear stress developed along the most likely rupture surface to the shear strength of soil.
Factor of SafetyFactor of Safetyf
sd
Fττ
=
factor of safety with respect to strength
average shear strength of the soil
average shear stress developed along the potential failure surface
d
f
d
Fs
ττ
==
= a e age s ea st ess de e oped a o g t e pote t a a u e su ace
tan
cohesion
d
f c
c
τ
τ σ φ= +
= cohesion
angle of friction
av
c
φσ
== erage normal stress on the potential failure surface
tan
subscript 'd' refer to potential failure surfaced d d
similarly
cτ σ φ= +
s
p p
tanF =
tand d
c
c
σ φσ φ
++
when Fs = 1, the slope is in a state of impending of failured dφ
.
In general Fs > 1.5 is acceptable
Stability of SlopeStability of Slope
Infinite slope without seepageInfinite slope without seepage
Infinite slope with seepage
Finite slope with Plane Failure Surface (Cullman's Method)(Cullman s Method)
Finite slope with Circular Failure Surface(Method of Slices)
Infinite slopeInfinite slope
β
L
β
L tancFs
φ= +β
WF
B
d
a
T
Na
β
WF
B
d
a
T
Na
2cos tan tanFs
Hγ β β β= +
For granular soils c=0 Fs is independent
βR
F
H
A
c
b
Ta
Tr
NrβR
F
H
A
c
b
Ta
Tr
Nr
For granular soils, c 0, Fs is independent of height H, and the slope is stable as long as β < φ
RR
LL
W
Direction of seepage
d
a
N
L
W
Direction of seepage
d
a
N
L
' tancF
γ φ
β
H
B
c
b
Ta
Na
Trβ
H
B
c
b
Ta
Na
Tr
2cos tan tansat sat
FsH
γ φγ β β γ β
= +
β βRA
r
Nr
β βRA
r
Nr
Finite slope with Plane Failure SurfaceFinite slope with Plane Failure Surface
W
B C
β φ+W
Ta
Na
tan
unit weight of soil
f cτ σ φγ
= +=H
2d
cr
β φθ
+=
RTr Nr
Aβ θ
unit weight of soil γ=
4 sin cos
1 cos( )cr
cH
β φγ β φ
⎡ ⎤= ⎢ ⎥− −⎣ ⎦A ( )γ β φ⎣ ⎦
Finite slope with Circular Failure SurfaceFinite slope with Circular Failure Surface
Slope Failure Shallow Slope FailureSlope Failure Shallow Slope FailureO
Toe Circle
Firm Base
OBase Failure
L L
O
L L
Slope Circle
Firm Base
Midpoint circle
Firm BaseFirm Base Firm Base
Slopes in Homogeneous Clay (φ=0)Slopes in Homogeneous Clay (φ=0)For equilibrium, resisting and driving moment about O:
M M OOR d
2d 1 1 2 2
1 1 2 2
M = M
c r W l W l
W l W l
θ = −
−
O
Radius = r
C Dθ
O
Radius = r
C Dθf ucτ =
1 1 2 2d 2
cr
c
W l W l
θ= l2 l1
A B
FW1
cdH
l2 l1
A B
FW1
cdH
u
d d
c
c c
critical when Fs is minimum, trials to find critical plane
fFsτ
= =
→E
W2
cd cd
Nr (normal reaction)E
W2
cd cd
Nr (normal reaction)
solved analitically by Fellenius (1927) and Taylor (1937)
cucr
cH =
m
presented graphically by Terzaghi &Peck, 1967 in Fig 11.9 Braja
γ
m is stability number
Swedish Slip Circle Method
Slopes in Homogeneous Clay (φ>0)Slopes in Homogeneous Clay (φ>0)tanf cτ σ φ= +f φ
( )dc , , ,H fγ α β θ φ⎡ ⎤= ⎣ ⎦
( ), , ,c
f mH
α β θ φ= =( )
where m is stability numbercrHγ
Method of SlicesMethod of Slices
For equilibrium
α= cosr n nN W
( )α φ=
=
Δ +=
∑1
cos tann p
n n nn
c L WFs
α=
=
=
∑1
sinn p
n nn
FsW
αΔ ≈ th where b is the width of n slice
cosn
nn
bL
n
Bishop MethodBishop MethodBishop (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 .
= 1n p
( )α
φ
α
=
=
+=
∑
∑1 ( )
1tan
sin
n p
n nn n
n p
n n
cb Wm
FsW
φ αα
=
= +
∑1
tan sincos
n nn
n
where
mα α= +( ) cosn nmFs
The ordinary method of slices is presented as learning tools It is rarely used because is toolearning tools. It is rarely used because is too conservative
Computer ProgramsComputer Programs
STABLSTABL
GEO-SLOPE
etc
BibliographyBibliographyHoltz, R.D and Kovacs, W.D., “An Introduction to GeotechnicalHoltz, R.D and Kovacs, W.D., An Introduction to Geotechnical
Engineering”
Das, B.M., “Soil Mechanics”
Das, B.M., “Advanced Soil Mechanics”