Slope Stability Analysis Procedures y y Presentation

Slope Stability
y Analysis
y
Procedures
Presentation for AEG/GI Short Course
UC Riverside,
e s de, May
ay 12,, 2012
0
William Kitch, Cal Poly Pomona
1
© William A Kitch 2012
Overview






2
Objectives
Obj
ti
off stability
t bilit analysis
l i
Measures of stability
Available computational methods
Limit equilibrium methods
Stability analysis process
Conclusions & questions
© William A Kitch 2012
Presentation scope

S il or continuous
Soil
ti
rock
k
–

Translational & rotational modes only
–

No debris flow or spreading analysis
St ti & pseudo
Static
d static
t ti stability
t bilit
–
3
Does not cover rock behavior governed by jointing (topples, key
wedge, etc)
No earthquake deformation analysis
© William A Kitch 2012
Objectives of Stability analysis




4
Determine
D
t
i adequacy
d
off an existing
i ti slope
l
Evaluate effectiveness of proposed slope remediation
Back calculate average shear strength of a slope know
to be in failure
Design
g an engineered
g
slope
© William A Kitch 2012
Measures of stability

F t off safety
Factor
f t
s
F

–
where
–
note
s  shear strength available
  equilbrium
ilb i
shear
h stress
t
s

–
5
M


M
resisting
driving
Definition
f
based on shear strength and shear stress is the only
consistent definition
© William A Kitch 2012
Recommended factors of safety
Cornforth (2005)
Minimal Study
Landslide size
Normal Study
Borings
Acceptable F
Borings
Acceptable F
1 or none
1.50
1
1.50
Small
1
1.50
2
1.35
Medium
2
1.40
4
1.25
Large
3
1.30
6
1.20
Very Large
4
1.20
8
1.15
Very Small
Duncan and Wright (2005)
Uncertainty of analysis
6
Cost of failure
Repair costs y incremental cost of safer design
Small
Large
1.25
1.5
Repair costs >> incremental cost of safer design
1.5
2.0 or more
© William A Kitch 2012
Agency requirements
US Army Corps of Engineers (1970)
Required Factor of safety for given condition
Type of slope
Dams, levees,
dikes & other
embankments
End of
construction
Long-term steady
state seepage
Rapid Drawdown
1.3
1.5
1.0 – 1.2
Typical Southern California Agency Requirements
7
Static
Static with pseudo static earthquake load
Temporary slopes
1.5
1.1
1.25
© William A Kitch 2012
Limitations of Factor of safety
Does nott contain
D
t i information
i f
ti about
b t the
th variability
i bilit or
uncertainty of shear strength or mobilized shear stress
Probability of
failure
Probab
bility Density




8
s
Stress

s
Same factor of safety can have different reliability
Probabilistic methods are available to estimate reliability
of slopes
© William A Kitch 2012
Available computational methods

Li it equilibrium
Limit
ilib i
methods
th d
–
–
–
–
–

Finite element methods
–
–
–
–
9
Most common approach
Requires
q
only
y simple
p Mohr-Coulomb soil model
Cannot model progressive failure
Cannot compute displacements
Must search for critical surface
Do not need to search for critical surface, analysis automatically
finds it
Must have a complete stress-strain model for soil
Can compute
p
displacements
p
Can model progressive failure
© William A Kitch 2012
Comparison of limit equilibrium and finite element methods
Limit equilibrium analysis
F = 1.75
10
Finite element analysis
F = 1.74
© William A Kitch 2012
L
Locating
ti complex
l failure
f il
surfaces
f
with
ith FE analysis
l i
su1
 1.0
su 2
su1
su2
su1
 0.6
06
su 2
su1
 0.2
su 2
Griffiths & Lane (1999)
11
© William A Kitch 2012
Limit Equilibrium Approach
1.
Generall shape
G
h
off ffailure
il
surface
f
(planar,
( l
circle,
i l non-circular)
i l )
assumed
–
–
2.
3.
Specific failure surfaced chosen
Some or all of static equilibrium conditions used to compute
eq ilibri m shear stress on failure
equilibrium
fail re ssurface,
rface 
1.
2.
3.
4.
12
Driven by geometry and geology of problem
Determines formulation of the analysis
 Fx = 0
 Fy = 0
 M= 0
Available shear strength, s, along failure surface computed using
Mohr-Coulomb failure criteria (c & )
5.
Factor of safety computed, F = s/
6.
Back to step 2, continue until Fmin found
© William A Kitch 2012


1 unknown, 
1 equation,  FA = 0
Simple planar failure example for  = 0 conditions
 H2
W 
 2 tan 




 FA = 0
H 2 cos 

T  W sin  
2
 H 2 cos    sin  
 
 

H
2




H  sin  cos 

2
2 su
F 
 H  sin  cos 
s
13
W
H/tan

H
T
H/sin
N
weak clay seam with
undrained strength, su
critical surface
© William A Kitch 2012
Simple LE methods




Model
M
d l simple
i l b
butt iimportant
t t cases
Statically determinate problems
Can solve directly for F without assumptions about
distribution of stress within failure mass
Most common and useful methods
–
–
–
14
Planar or single wedge
Infinite slope
Swedish slip circle
© William A Kitch 2012


–
Infinite slope analysis

–
 FA = 0
T  W sin 

D


ER D
W

W sin 
l
   D cos  sin 

W cos 
l
   D cos 2 
From Mohr-Coulomb
s  c    tan  '  c   D cos 2  tan  '
s

l
T
N
15
 FA = 0
 FB = 0
c   D cos 2  tan  '
F 

 D cos  sin 
EL
t
l
cos 
 FB = 0
N  W cos 
t
2 unknowns, F & 
2 equations
W  tD

For c = 0 2
 D cos  tan  ' tan  '

F
 D cos  sin 
tan 
For  = 0, s = su
F
su
 D cos  sin 
© William A Kitch 2012


1 unknown, F
1 equation,  MO = 0
Swedish slip circle for  = 0 conditions
O
l1
a
su1 su

r

W
su2
l

Wa
rl
Shear strength
s  su
l2


 MO = 0
 lr  Wa
F
F
s


su rl
Wa
M
M
resisting
driving
r  suili
F
Wa
16
© William A Kitch 2012
Summary of simple LE methods
17
Procedure
ocedu e
Assumptions
ssu pt o s
Equations
quat o s
used
Variables
a ab es so
solved
ed for
o
Infinite Slope
• Infinitely long slope
• slip surface parallel
to surface
•  F = 0
•  F = 0
• Factor of safety
•  on failure surface
Swedish slip
circle
• =0
•  MO = 0
• Circular slip surface
• Factor of safety
© William A Kitch 2012
Methods of slices
O

r
c1, 1


c2, 2
Wh   0
When
s  c '  ' tan  '
Must determine 
Cannot use simply  MO = 0
zi
i
i
Vi
Wi
Ei
zi+1
Ei+1
Vi+1

Ti
Ni
li
18
© William A Kitch 2012
Equation/unknown count
x

Unknowns







Vi
Wi
Ei

1  MO
1,

n,  Mi

n,  Fx

n,  Fz
Total: 3n + 1 equations
zi+1
Ei+1
Vi+1

Ti
Equilibrium equations

19
F, factor of safety
n values of Ni
n1 values of Ei
n1 values of Vi
n1 values of zi
Total: 4n2 unknowns
zi
z
Ni
li


Must make assumptions to
solve problem
Assumptions made affect
accuracy of solution
© William A Kitch 2012


1 unknown, F
1 equation,  MO = 0
Ordinary method of slices

Assumptions
–
–

Hi
Ignore side
forces
F
Ti
 MO = 0
Ni
Solution
c ' l  W cos   ul  tan  '


F
W sin 
W   Hll cos 
u  pore pressure on base of slice
20
Wi
E
Equations
ti
used
d
–

Ignore side
forces
Unknown
–

Circular surface
Ignore all side forces
li




Can directly solve for F
Simple to implement
Generally conservative
Accuracy poor when pore
pressure high
© William A Kitch 2012


1+n unknowns, F, Ni
1 equation,
 MO = 0
n,  Fz

Simplified bishop method

x

Assumptions
–
–

–
–
21
Wi
Ei
1, F
n, Ni
 MO = 0
n,  Fz
Solution
 c ' l cos   W  ul cos   tan  ' 
  cos    sin  tan  ' / F 

F 
W sin 
zi+1
Ei+1

Ti
Equations used
–

z
Unknown
–

Circular surface
Side forces are horizontal
zi
Ni
li



Requires iterative solution
More accurate the OMS
E il iimplemented
Easily
l
d with
ih
spreadsheet
© William A Kitch 2012
Inclusion of external or internal loads
O
zi
r
k Wi
k Wi
Ei
Wi
zi+1
Ei+1
Ri
Ri
i



22

Ti
i
Know forces included in
existing equilibrium equations
Does not increase number of
unknowns
Solution method the same
Ni
li

Allows for inclusion of
–
–
–
Pseudo static earthquake loads
Forces from pile stabilization
External equipment or
structural loads
© William A Kitch 2012
Uses of non-circular surfaces
Surficial Slide
Weak seam
23
Weak layer
© William A Kitch 2012
Non-circular surface methods



Assumption
A
ti off circular
i l surface
f
simplifies
i lifi problem
bl
By using  MO = 0 number of unknowns substantially
reduced
Method of slices works for non-circular surfaces



Two broad groups of solutions available



24
More unknowns
More equilibrium equations required
Force equilibrium:
F
ilib i
uses  Fx = 0 &  Fz = 0
Full equilibrium: satisfies uses  Fx = 0,  Fz = 0 &  M = 0
All still require
q
assumptions
p
about interslice forces
© William A Kitch 2012
Force equilibrium methods

A
Assume
di ti interslice
direction
i t li forces
f
–
Combined with  Fx = 0 &  Fz = 0 allows for solution for F
Method
Interslice force
assumption
Simplified
p
Janbu ((Janbu et al.1956))
Horizontal
Lowe and Karafiath (1959)
Average of slope of top
and bottom of slice
Corps of Engineers’ modified Swedish method Parallel to average
(US Army Corps of Engineers, 1970)
slope angle
25
© William A Kitch 2012
Force equilibrium solutions sensitive to direction of
interslice force
Figure 6.15 Influence of interslice force inclination on the computed factor of safety for
force equilibrium with parallel interslice forces. (Duncan & Wright, 2005)
26
© William A Kitch 2012
Full equilibrium methods



Add momentt equilibrium
ilib i
tto x & y force
f
equilibrium
ilib i
Still requires assumptions
Two most common methods
–
Spencer (1967)


–
Assumes all interslice forces are parallel
Solves for F and 
Morgenstern and Price (1965)


Assumes V =  f (x) E
f (x) is an assumed function
–  is a scaling constant
–

–
–

27
Solves for F and 
f(x)
Morgenstern & Price more general
Spencer
p
easier to implement
p
When using any full equilibrium method F is insensitive to
assumptions about interslice forces
© William A Kitch 2012
Comparison of full equilibrium methods
28
P
Procedure
d
A
Assumptions
ti
Equations
E
ti
used
V i bl solved
Variables
l d for
f
Spencers
• Interslice forces
parallel
•  Fx = 0
•  Fy = 0
• M=0
•
•
•
•
Morgenstern
& Price
• Interslice forces related •  Fx = 0
by V =  f (x) E
•  Fy = 0
• Form
F
off f (x)
( )
• M=0
•
•
•
•
Factor of safety
Interslice angle 
Interslice force
Location of
interslice force
•  on failure surface
Factor of safety
Scaling factor 
I t li force
Interslice
f
Location of
interslice force
•  on failure surface
© William A Kitch 2012
Data available from full equilibrium method
29
© William A Kitch 2012
Summary
y of applicability
pp
y of LE methods
30
Procedure
Application
Infinite Slope
Homogeneous cohesionless slopes and slopes where the
stratigraphy restricts the slip surface to shallow depths and parallel
to the slope face. Very accurate where applicable.
Swedish Circle
=0
Undrained analyses in saturated clays,  = 0. Relatively thick zones
of weaker materials where circular surface is appropriate.
Ordinary Method
of Slices
Nonhomogeneous slopes and c– soils where circular surface is
appropriate. Convenient for hand calculations. Inaccurate for
effective stress analyses with high pore pressures.
Simplified Bishop
procedure
Nonhomogeneous slopes and c–
c  soils where circular surface is
appropriate. Better than OMS. Calculations feasible by spreadsheet
Force Equilibrium
procedures
Applicable to virtually all slopes. Less accurate than
p
equilibrium
q
p
procedures and results sensitive to
complete
assumed interslice force angles.
Spencer
Applicable to virtually all slopes. The simplest full equilibrium
procedure for computing the factor of safety.
Morgenstern
and Price
Applicable to virtually all slopes. Rigorous, well-established
complete equilibrium procedure.
From Duncan & Wright (2005)
© William A Kitch 2012
Critical details of LE analysis

S
Searching
hi ffor critical
iti l surface
f
–
–

Select appropriate shear strength
–
–

Progressive failure
P
Pre-existing
i ti shear
h
surfaces
f
Check for invalid solutions
–
–
–
31
Check for multiple minima
Special
p
attention required
q
when using
g non-circular surfaces
Tensile forces near crest
Steep exit slopes
Non-convergence of solutions
© William A Kitch 2012
Critical surface search: regional minimum
32
© William A Kitch 2012
Critical surface search: local minimum
33
© William A Kitch 2012
Critical surface search: multiple modes
34
From Duncan & Wright (2005)
© William A Kitch 2012
Progressive failure
35
From Duncan & Wright (2005)
© William A Kitch 2012
Validity of solution: Tension crack at crest

36
Al
Always
check
h k liline off th
thrustt
© William A Kitch 2012
Validity of solution: Tension crack at crest

37
I
Insert
t ttension
i crack
k att crestt if needed
d d
© William A Kitch 2012
Steep exit angle

C cause
Can
–
–
–

Solution
–
–
38
Non-convergence of solution
Very
y high
g stresses
Negative (tensile stresss)
Use Si
U
Simplified-Bishop
lifi d Bi h
For exit slope to be more
shallow
From Duncan & Wright (2005)
© William A Kitch 2012
Preparing for stability analysis




Determine
D
t
i required
i d scope off analysis
l i
Assess risk of project and select appropriate F
Build subsurface model
Determine drainage conditions which apply
–
–


Select appropriate soil strength properties
Id tif expectt failure
Identify
f il
surface
f
geometry
t and
d select
l t
analysis procedure
–

39
End-of-construction undrained condition
Long-term drained condition (both?)
Circular non-cirucular
Select appropriate analysis procedure
© William A Kitch 2012
Performing stability analysis

I
Investigate
ti t potential
t ti l failure
f il
modes
d using
i simple
i l models
d l
–

Adjust subsurface model and analysis method as needed
–

–
Check line of thrust
Sanity check results
–
40
Search all area with local minimum
Consider risk of each significant failure mode
Thoroughly examine computations for critical modes
–

Soil properties, geometry, computational method
Thoroughly investigate all potential failure modes with rigorous
search for critical surface
–

Identify areas where F is low
Similar p
project,
j , hand computation,
p
, other method
© William A Kitch 2012
Software (a very short list)

St d l
Standalone
stability
t bilit packages
k
–
–
–
–

Integrated packages
–
–
–
–
41
STABL/STED
Oasys
y
UTEXAS4
LimitState
RocScience
GeoStudio
gINT
SoilVision
© William A Kitch 2012
Recommended texts



42
Abramson, L
Ab
L. W
W. (2002)
(2002). Slope
Sl
stability
t bilit and
d stabilization
t bili ti
methods. Wiley, New York.
Cornforth D
Cornforth,
D. H
H. (2005)
(2005). Landslides in Practice Investigation, Analysis, and Remedial/Preventative
Options in Soils. John Wiley & Sons.
Duncan, J. M., and Wright, S. G. (2005). Soil Strength
and Slope Stability. John Wiley & Sons, Hoboken, N.J.
© William A Kitch 2012
References









43
Abramson, L.
Abramson
L W.
W (2002).
(2002) Slope stability and stabilization methods
methods. Wiley,
Wiley New York
York.
Cornforth, D. H. (2005). Landslides in Practice - Investigation, Analysis, and
Remedial/Preventative Options in Soils. John Wiley & Sons.
Duncan, J. M., and Wright, S. G. (2005). Soil Strength and Slope Stability. John Wiley & Sons,
Hoboken N
Hoboken,
N.J.
J
Griffiths, D. V., and Lane, P. A. (1999). “Slope stability analysis by finite elements.”
Geotechnique, 49(3), 387–403.
Janbu, N., Bjerrum, L., and Kjærnsli, B. (1956). Veiledning ved Løsning av
Fundamenteringsoppgaver (Soil Mechanics Applied to Some Engineering Problems), Publication
16, Norwegian Geotechnical Institute, Oslo.
Lowe, J., and Karafiath, L. (1959). Stability of earth dams upon drawdown, Proceedings of the
First PanAmerican Conference on Soil Mechanics and Foundation Engineering, Mexico City, Vol.
2, pp. 537–552.
Morgenstern, N. R., and Price, V. E. (1965). “The analysis of the stability of general slip
surfaces”, Geotechnique, 15(1), 79–93.
Spencer, E. (1967). “A method of analysis of the stability of embankments assuming parallel
inter-slice forces”, Geotechnique, 17(1), 11–26.
U.S. Army Corps of Engineers (1970). Engineering and Design:Stability of Earth and Rock-Fill
Dams, Engineer Manual EM 1110-2-1902, Washington, DC, April.
© William A Kitch 2012