Slope Stability Analysis

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Slope Stability Analysis for Landfills and
Embankments Geotechnical Design CGN 4801 By
Kamal Tawfiq, Ph.D., P.E.
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FOR ANALYSIS AND DESIGN SLOPE FAILURE ARE DIVIDED
INTO
I. Planar Failures I- Infinite Slopes II-
Finite Slopes II. Circular Failures III.
Wedge Failures IV. Complex Failures
(Determinate Problems)
Small Depth, Long Failure Surface
Simple Wedge
1- Above the Toe
(Determinate Indeterminate Problems)
2- Through the Toe
3- Deep Seated
Multiple Planar Failure Surface
Combination of Planar Circular
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FACTORS AFFECTING SLOPE STABILITY 1- Soil
Type 2- Geometry of the cross section (Height,
slope angle, etc.) 3- Moisture Content 4- Pore
water pressure 5- Additional loads 6- Shear
Strength reduction 7- Vibrations and Earthquake
By Kamal Tawfiq, Ph.D., P.E.
4
METHODS OF ANALYSIS A state of equilibrium is
said to exist when the shear stress along the
failure surface is expressed as
Stress
Shear Strength
S t / Fs
Shear Stress
Strain
Mathematical Representation of Stress vs.
Strain Relationship
Safety Factor
t c sn tan?
t
t c sn tan?
?
c
s
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FACTOR OF SAFETY
1- For Shear Strength tdeveloped t / FS
tdeveloped (c s tan?) / FS 2- For Shear
Parameters cd c / FS tan?d tan? / FS 3-
For Height of the Slope Hdesign Hc / FS
By Kamal Tawfiq, Ph.D., P.E.
6
H
INFINITE SLOPE I. PLANAR FAILURE or Transitional
Failure
b 1 cosß
1
A- Dry Soil (? soil)
Driving Force, FD W sin ß
b 1 cosß
Normal Component, FV W cos ß
ß
W ? H cosß
ß
Failure Surface
H
N Fv W cos ß
t N tan?
Weight W
By Kamal Tawfiq, Ph.D., P.E.
7
A- Dry Soil (? soil)
W ? H cosß
b 1 cosß
Driving Force FD ? H cos ß sinß Resisting
Force FR ? H cos ß cosß tan? FS FR/FD
G.S.
b 1 cosß
? H cos ß cosß tan?
FS
1
Driving Force, FD W sin ß
? H cos ß sinß
Normal Component, FV W cos ß
tan?
ß
FS
tanß
N Fv W cos ß
ß
Failure Surface
t N tan?
H
Weight W
By Kamal Tawfiq, Ph.D., P.E.
8
B- Submarged Soil (? soil)
_
W ? H cosß
_
Driving Force FD ? H cos ß sinß Resisting
Force FR ? H cos ß cosß tan? FS FR/FD
_
W.T
_
G.S.
? H cos ß cosß tan?
FS
_
b 1 cosß
? H cos ß sinß
1
Normal Component, FV W cos ß
tan?
Driving Force, FD W sin ß
FS
ß
tanß
N Fv W cos ß
ß
t N tan?
Failure Surface
H
Weight W
By Kamal Tawfiq, Ph.D., P.E.
9
C- Seepage Parallel to Slope (? soil)
?w Z
tan?
)
(1-
FS
?soilH cos2ß
tanß
G.S.
Seepage
b 1 cosß
1
Driving Force, FD W sin ß
Normal Component, FV W cos ß
ß
N Fv W cos ß
ß
Failure Surface
t N tan?
H
Weight W
By Kamal Tawfiq, Ph.D., P.E.
10
D- Infinite Slope in c - ? soil (with seepage)
c
u
tan?
)
(1-
FS
?soil H cosß sinß
?soil H cos2ß
tanß
G.S.
If no seepage u 0 If Submarged Slope u 0
? ?
b 1 cosß
Driving Force, FD W sin ß
1
Normal Component, FV W cos ß
ß
N Fv W cos ß
ß
Failure Surface
t N tan?
H
Weight W
By Kamal Tawfiq, Ph.D., P.E.
11
Critical Height Hc at FS 1
c
u
tan?
)
(1-
1
FS
?soil H cosß sinß
?soil H cos2ß
tanß
H
ß
c
- u tan?
N Fv W cos ß
Hc
ß
?soil cos2ß (tanß - tan?)
Weight W
Stability Number Ns
u
c
ru
Ns
pore water pressure ratio
?H
?Hc
General Equation
c
Hc
?soil sinß cosß - tan? (cos2ß - ru)
By Kamal Tawfiq, Ph.D., P.E.
12
General Equation
c
Hc
?soil sinß cosß - tan? (cos2ß - ru)
c
sinß cosß - tan? (cos2ß - ru)
Ns
?Hc
cd
sinß cosß - tan? (cos2ß - ru)
Ns
?H
c
Ns c /?Hc
cd
FS
? 15o
tan?
? 25o
tan?d
? 35o
FS
? 45o
ß
By Kamal Tawfiq, Ph.D., P.E.
13
Stability Number

• A variety of charted solutions exist for the
  simple geometry considered above.
• For the undrained (total stress) analysis of
  slopes charts produced by Taylor are often used.
• The charts are based on the analysis of circular
  failure surfaces, and assume that soil strength
  is given by a Mohr-Coulomb analysis
• Tension cracks are not considered

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Taylors Chart
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Taylors Chart - example
8 m
o
30
Use chart with i 30 degrees, f 5 degrees
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Taylors Chart
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Taylors Chart - example
8 m
o
30
Use chart with i 30 degrees, f 5
degrees Hence
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Taylors Chart - example
8 m
o
30
Use chart with i 30 degrees, f 5
degrees Hence
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Taylors Chart - example

• Zones are marked on the chart indicating whether
  the failure mode will be shallow or deep-seated.
• If a deep-seated failure is indicated the soil
  layer must be sufficiently deep to enable this
  mechanism to occur.
• There is a second chart due to Taylor which can
  be used when the depth of soil below the base of
  the slope is limited
• This chart is only valid for f 0

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Taylors Chart - example with finite depth
8 m
o
30
2 m
Rock
Calculate the Depth Factor D
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Taylors Chart - example with finite depth
8 m
DH
o
30
2 m
Rock
Calculate the Depth Factor D DH 10 m
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Taylors Chart - example with finite depth
8 m
DH
o
30
2 m
Rock
Calculate the Depth Factor D DH 10 m, H 8m D
1.25
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Taylors Chart - example with finite depth
8 m
DH
o
30
2 m
D 1.25
Rock
and F 1.075
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Quiz 4 Fall 2009
For the infinite slope shown below, what is the
pore water pressure at point A ?
b
Soil Surface
b/cos a
Water Table
W
Assumed failure surface
d
dw
T
N
dwcos a
A
U
a
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STATICALLY INDETERMINATE PROBLEMS
METHOD OF SLICES
Center
Center
Xi
1
2
?x
3
4
5
6
7
R
8
11
10
9
Si
ai
ai
Wi cos a
Ei
S ?S
Ti c.L Neffi tan ?
Ti c.L Neffi tan ?
E ?E
Wi
Wi
Wi sin a
ai
L
Ni Neffi ui L
?
Ni Neffi ui L
ai
ai
Resultant
By Kamal Tawfiq, Ph.D., P.E
34
Center
Xi
Unknowns Associated with Force Equilibrium n
Resultant normal forces Ni on the base of each
slice or wedge 1 Safety factor, which permits
the shear forces Ti on the base of each slice
to be expressed in terms of Ni n-1 Resultant
normal forces Ei on each interface between slices
or wedges n-1 Angles ai which express the
relationships between the shear force Si and
the normal force Ei on each interface
?xi
R
S ?S
Si
Ei
Ti c.Li Neffi tan ?
?i
Wi
Ei ?E
Li
bi
Ni Neffi ui Li
?
ai
ai
ai
ai
Resultant
ai
By Kamal Tawfiq, Ph.D., P.E
35
FELLENIUS METHOD (ORDINARY METHOD, SWEDISH METHOD)
Wi cos a
Center
Xi
Ti c.L Neffi tan ?
Wi
Wi sin a
ai
?xi
R
Ni Neffi ui L
Si
Ei
S ?S
Ti c.Li Neffi tan ?
Ei ?E
Wi
Li
MR ?CLi (wicos?i -uiLi)tan?
F
bi


MD ?wi sin?i
Ni Neffi ui Li
?
ai
ai
ai
ai
Resultant
ai
By Kamal Tawfiq, Ph.D., P.E