Recent developments in landslide studies: probabilistic methods. State-of-the-art-report - Session VII (a)


Chowdhury, R.N.

IV International Symposium on Landslides, Sept. 16-21, 1984, Toronto, Canada (ISL 1984): 209-228

1984


Considering the purpose and place of analysis in the study of slopes and landslides, it is fruitful to examine the recent trend towards probabilistic approaches. In this state-of-the-art paper basic definitions and methodology are reviewed and attention is then given to the modelling of soil variability within a probabilistic framework and its implications. Probabilistic approaches complement conventional deterministic studies and the advantages of these approaches are discussed in this paper. Reference is made to the methodology for predicting the most probable extent of failure of embankments and natural slopes, the modelling of slip surface propagation and progressive failure which is a consequence of the strain-softening nature of earth materials. The potential use of available techniques for predicting the likelihood of successive failure is discussed. Recent work on updating risk and reliability under static and seismic loads and on predicting failure due to rainfall is also reviewed. Different calculation techniques namely Monte Carlo simulation, Taylor series method and Rosenblueth method are compared. Finally attention is given to the role of probabalistic methods in decision making and to the limitations of probabilistic methods.

209.
N.S.W.
Recent
Developments
in
Landslide
Studies:
Probabilistic
Methods
State-of-the-Art-Report—Session
VII
(a)
R.N.
Chowdhury
Department
of
Civil
and
Mining
Engineering,
University
of
Wollongong,
Wollongong,
N.S.W.,
Australia
SYNOPSIS
Considering
the
purpose
and
place
of
analysis
in
the
study
of
slopes
and
landslides,
it
is
fruitful
to
examine
the
recent
trend
towards
probabilistic
approaches.
In
this
state-of-the-art
paper
basic
definitions
and
methodology
are
reviewed
and
attention
is
then
given
to
the
modelling
of
soil
varia-
bility
within
a
probabilistic
framework and
its
implications.
Probabilistic
approaches
complement
conventional
deterministic
studies
and
the
advantages
of
these
approaches
are
discussed
in
this
paper.
Reference
is
made
to
the
methodology
for
predicting
the
most
probable
extent
of
failure
of
embankments
and
natural
slopes,
the
modelling
of
slip
surface
propagation
and
progressive
failure
which
is
a
consequence
of
the
strain-softening
nature
of
earth
materials.
The
potential
use
of
available
techniques
for
predicting
the
likelihood
of
successive
failure
is
discussed.
Recent
work
on
updating
risk
and
reliability
under
static
and
seismic
loads
and
on
predicting
failure
due
to
rainfall
is
also
reviewed.
Different
calculation
techniques
namely
Monte
Carlo
simulation,
Taylor
series
method
and
Rosenblueth method
are
compared.
Finally
attention
is
given
to
the
role
of
probab-
alistic
methods
in
decision
making
and
to
the
limitations
of
probabilistic
methods.
INTRODUCTION
The
Role
of
Analysis
The
use
of
probabilistic
methods
in
the
assess-
ment
of
slope
stability
is
relatively
new
and
the
value
of
such
methods
in
geomechanics
is
continually
being
assessed.
Although
the
appli-
cations
of
statistics
and
probability
has
gained
a
great
deal
of
acceptance
in
recent
years,
some
doubts
about
the
relevance
of
probabilistic
methods
remain.
Deterministic
methods
of
analy-
sis
continue
to
be
used
and
are
unlikely
to
be
replaced
by
probabilistic
methods
in
the
foresee-
able
future.
Widely
accepted
geomechanics
models
can
be
used
as
a
basis
for
either
deter-
ministic
or
probabilistic
analyses.
In
many
respects
the
two
approaches
are
complementary.
Therefore,
it
is
reasonable
to
ask
the
following
questions:
(1)
What
does
a
probabilistic
approach
offer
which
makes
it
attractive
in
the
pract-
ice
of
slope
engineering
and
in
under-
standing
the
performance
of
slopes?
(2)
What
type
of
calculations
or
assessments
can
be
made
with
probabilistic
methods
which
are
outside
the
scope
of
convent-
ional
deterministic
practice?
(3)
Which
techniques
of
analysis
are
convenient
for
use
in
practice?
(4)
What
are
the
main
limitations
in
the
use
of
probabilistic
methods?
In
this
state-of-the-art
paper
an
effort
is
made
to
answer
these
and
similar
questions.
Be-
fore
considering
recent
developments
in
probab-
ilistic
methods
for
slopes
and
landslides,
it
is
necessary
to
give
some
attention
to
the
occurr-
ence
of
failures
and
to
the
role
of
analyses
in
slope
engineering.
For
decades
the
complexity
of
the
behaviour
of
earth
slopes
and
especially
natural
slopes
or
hillsides
has
been
emphasised.
Good
engineering
decisions
are
facilitated
by
careful
observation
in
the
field
and
one
can
hardly
overemphasize
the
value
of
reliable
in-
formation
about
local
geology,
seepage
condit-
ions
and
the
properties
of
earth
materials
at
the
area
that
is
to
be
assessed.
To
justify
the
use
of
a
method
of
analysis,
it
is
necessary
to
understand
the
factors
which
control
stability
and
also
various
mechanisms
of
failure
in
different
geological
settings.
To
adopt
a
valid
method
of
analysis
and
to
use
it
properly
re-
quires
an
appreciation
of
the
assumptions
and
idealisations
which
are
part
of
that
method.
Again
input
data
should
correspond
to
the
corr-
ect
field
situation
which
is
to
be
simulated
or
analysed
e.g.
(a)
short-term
stability
or
long-term
stability
(b)
first-time
slides
or
renewed
movements
along
existing
slip
surfaces
(c)
steady
seepage
condition
or
draw-down
condition
Occurrence
of
Failures
Understanding
the
factors
which
control
the
stability
of
hillsides
and
slopes
requires
a
sound
knowledge
of
the
shearing
resistance
of
earth
materials
forming
the
slopes.
The
influ-
ence
of
changing
loads
and
pore
water
pressures
on
the
shearing
resistance
is
of
particular
relevance
to
slope
stability
whether
one
is
considering
a
highway
cut,
an
embankment,
a
(a)
P
(b)
(c)
F
1
t
c
(e)
210.
mining
excavation,
an
earth
dam
or
a
mining
spoil
pile.
An
appreciation
of
the
principle
of
effective
stress
is
of
paramount
importance.
Often
slides
occur
after
prolonged
or
heavy
rainfall
when
seepage
occurs
throughout
a
slope
(high
positive
pore
water
pressures).
Failures
may
also
be
triggered
by
seismic
activity
i.e.
an
earth
tremor
or
an
earthquake.
Some
failures
occur
without
apparent
provocation
and,
in
such
cases,
it
may
be
difficult
to
identify
the
cause
of
failure.
The
safety
margin
of
natural
slopes
may
fluct-
uate
widely
from
one
season
to
another,
being
high
in
the
dry
season
and
low
after
rainfall.
Over
a
long
period
of
time
the
safety
margin
reduces
gradually
due
to
natural
causes
and
without
any
man-made
disturbance.
A
point
is
reached
when
the
disturbing
forces
just
exceed
the
resisting
forces
and
the
value
of
factor
of
safety
F
falls
below
unity.
Thus
a
slide
may
occur
without
apparent
cause.
This
process
is
illustrated
in
Fig.
1
and
catastrophic
slides
may
sometimes
occur
in
this
way
if
considerable
amount
of
stored
energy
is
released
(Chowdhury,
1980).
Some
failures
which
appear
to
be
unprovoked
are,
in
fact,
the
delayed
result
of
external
dis-
turbance.
The
delay
in
the
occurrence
of
fail-
ures
of
cuttings
in
overconsolidated
London
clay
is
a
well
known
and
well
documented
example
(Skempton,
1977).
In
such
clays
excess
negative
pore
water
pressures
are
developed
due
to
exca-
vation
and
many
years,
even
several
decades,
may
elapse
before
these
pressures
are
fully
dissipated.
As
positive
pore
water
pressures
increase
to
long-term
equilibrium
values,
shear
strength
decreases
in
accordance
with
the
prin-
ciple
of
effective
stress.
Thus
a
slide
may
occur
many
years
after
the
completion
of
an
excavation.
While
this
is
by
no
means
the
only
cause
of
delayed
failure,
there
is
need
for
well-documented
case
histories
and
other
firm
evidence
in
respect
of
other
causes
of
delay
such
as
the
following:
(a)
Rheological
decrease
of
strength
with
time.
(b)
Progressive
failure
including
propagation
of
a
slip
surface,
strain-softening,
creep
and
other
phenomena.
(c)
Delayed
effects
of
seismic
activity.
(d)
Random
changes
in
pore
water
pressure
and
hence
in
shear
strength.
Some
failures
are
preceded
by
visible
or
detect-
ible
signs
of
impending
danger
whereas
others
are
not.
In
general,
it
is
difficult
to
pre-
dict
slides
on
the
basis
of
surface
observation
although
it
may
be
possible
to
distinguish
be-
tween
a
stable
slope
and
one
which
is
only
marginally
stable.
Subsurface
movements
and
pore
water
pressures
are
sometimes
monitored
by
using
appropriate
instrumentation.
Observation-
al
data
obtained
in
this
way
can
prove
to
be
of
considerable
value
in
stability
assessment
and
in
decision-making.
Nevertheless,
it
is
obvious
that
one
needs
to
perform
appropriate
calculat-
ions
to
quantify
stability
and
to
assess
the
influence
of
various
preventive
or
remedial
mea-
sures
on
the
magnitude
of
the
calculated
quantity,
be
it
a
safety
factor
or
a
reliability
index
or
an
estimate
of
deformation.
Within
the
framework
of
such
a
calculation
procedure
or
a
method
of
analysis,
however
simple,
one
may
be
able
to
revise
the
stability
assessments
and
decisions
in
the
light
of
observational
data.
t
c
time
V
tc
(d
)
t,
L
time
slow
creep
phase
Fig.
1.
Idealisation
of
the
gradual
change
in
stability
of
a
natural
slope
;
strain
E,
factor
of
safety
F,
energy
release
W,
velocity
v
and
lateral
pressure
P
are
plotted
against
time;
t
c
is
the
instant
of
failure;after
Chowdhury
(1980)
t
c
211.
Uncertainties
in
Slope
Analysis
The
assessment
of
slopes
and
landslides
is
diffr
icult
because
of
many
uncertainties.
Because
of
the
variability
of
earth
materials
over
a
site,
knowledge
about
shear
strength
is
imperfect
even
after
a
comprehensive
site
investigation.
Pore
water
pressures
may
vary
widely
over
a
site
and,
of
course,
with
time
as
explained
earlier.
Minor
geological
details
may
remain
undetected
and
thus
the
actual
mode
of
failure
may
be
different
from
the
one
assumed
in
an
analysis.
As
a
result
of
observation
and
investigation,
a
great
deal
of
data
may
be
available
for
a
part-
cular
site;
nevertheless
the
geotechnical
engineer
must
select
the
values
of
the
shear
strength
parameter
and
pore
pressures
in
an
analysis.
Depending
on
this
choice,
which
is
always
difficult,
quite
different
values
of
the
safety
factor
'F'
may
be
obtained
within
a
deterministic
framework.
The
engineer
must
rely
very
much
on
subjective
judgement.
Conseq-
uently
the
real
meaning
of
the
calculated
value
of
a
deterministic
index
of
safety
(e.g.
F=1.2
or
F=1.4
or
F=2.1)
is
in
doubt.
It
must
be
emphasised
that
shear
strength
may
vary
consid-
erably
even
within
so-called
homogeneous
soil
deposits.
Decades
before
the
advent
of
probab-
alistic
geomechanics,
Bishop
(1948)
reported
coefficients
of
variation
of
18%-42%
in
the
un-
drained
shear
strength
of
London
clay
as
summ-
arised
in
Table
1.
The
mean
shear
strength
varied
by
a
factor
of
almost
6
from
16
p.s.i.
to
99p.s.i.
strength
in
the
stability
analysis".
Bishop's
comment
indicates,
although
indirectly,
a
con-
cern
that
soil
variability
may
invalidate
con-
ventional
analysis
procedures.
In
the
interven-
ing
decades
statistical
data
of
shear
strength
from
other parts
of
the
world
have
been
collect-
ed
and
one
continues
to
be
surprised
by
the
variability
of
so-called
homogeneous
soil
depos-
its.
Data
from
several
sources
compiled
by
Baecher
et
al
(1980)
is
shown
in
Table
2.
In
connection
with
the
reliability
of
a
long
em-
bankment
Vanmarcke
(1977b)
stated
that
vane
shear
strength
values
differed
by
a
factor
of
3;
the
coefficient
of
variation
being
estimated
at
about
32%.
Some
data
from
coal
mining
areas
in
Australia
are
presented
in
Table
3
which
shows
the
statistical
parameters
of
cohesion
c
and
friction
angle
(I)
obtained
from
field
investigat-
ion
records
at
various
sites
(Nguyen
and
Chowd-
hury,
1984a).
A
summary
of
various
uncertainty
factors
associated
with
the
estimation
of
the
factor
of
safety
of
a
so-called
homogeneous,
clay
slope
is
shown
in
Table
4
(Tang
et
al,
1976).
Several
other
authors
have
also
discuss-
ed
uncertainties
in
slope
analysis
e.g
Alsonso
(1976),
A-Grivas
(1981),
Chowdhury
(1978,
1980,
1981).
Justification
of
a
Probabilistic
Approach
A
major
benefit
of
a
probabilistic
approach
is
that
it
allows
for
a
logical
and
systematic
analysis
of
uncertainty.
Thus
one
may
make
use
of
the
mean
value
of
shear
strength
as
well
as
its
coefficient
of
variation.
The
idealisation
TABLE
I
Coefficients
of
Variation
of
Undrained
Shear
Strength
of
London
Clay
Reported
by
Bishop
(1948)
Depth
Feet
Samples
from
Whole
Site
Samples
from
Pit
218
only
No.
of
Samples
Coefficient
of
Variation
No.
of
Samples
Coefficient
of
Variation
0-1
38
25%
19
20%
1-2
49
18%
17
26%
2-4
77
38%
19
42%
4-8
65
30%
13
35%
8-12
86
36%
-
-
12-16
46
357
-
-
16-20
32
29%
-
-
23-27
12
30%
-
-
30-60
43
427
-
-
75-105
23
33%
Ti
-
-
Bishop
commented
as
follows:
"It
is
important
to
note
that
the
coefficient
of
variation
was
almost
the
same
for
one
pit,
where
the
samples
lay
within
30
feet
of
each
other
horizontally,
as
for
the
whole
site.
This
justifies
the
use
of
average
values
of
shear
of
an
earth
medium
as
a
statistically
homogen-
eous
one
is
preferable
to
the
deterministic
homogeneous
medium
in
which
a
single
value
of
each
parameter
must
be
adopted
for
each
calcul-
ation.
Certainly
even
the
statistically
homo-
geneous
medium
has
its
limitations
and
in
sub-
sequent
sections
reference
will
be
made
to
the
212.
TABLE
II
TABLE
II(Cont'd)
Reported
Coefficients
of
Variation
for
Soil
Properties
of
Different
Soils
-
Various
Locations
;
Compiled
from
Papers
Pre-
sented
at
ICASP,1(Hong
Kong),
2
(Aachen)
and
3(Sydney)
by
Baecher
et
al
(1980)
Material
Property
COV%
Clay
Liquid
limit
5.9
plastic
limit
4
+
clay
content
11.
specific
gravity
0.5+
dry
density
26.4
Clay
Shale
cohesion
(direct
shear,
DS)
94.8
friction
coef-
.ficient
(t),DS
Cohesive
Till
"undisturbed"
compacted
Various
Tills
Silt
Gravelly
Sand
Coarse
Sand
Medium
Sand
Fine
Sand
Marine
Clay
London
Clay
Sandy
Clay
Silty
Sand
Clay
Silt
Ottawa
Sand
(loose)
Ottawa
Sand
(dense)
Clayey
Silt
(unsoaked)
Clayey
Silt
(soaked)
phi
S
Clayey
Silt
c
u
CH
c
triaxial
phi
--
UU
CL
c
UU
phi
--
UU
ML
c
UU
phi
-
UU
CH
c
--
DS
phi
--
DS
CL
c
--
DS
ML
c
--
DS
Road
Subgrade
soil
suction
soil
suction
Material
Property
COV%
Average
over
LL
6.37
16
cohesive
soils
PL
9.55
Road
base
coarse
CBR
17.4
density
3.9
PI
75.0
36.8
Plastic
Clay
cHmpression
ratio
17
to
38
t
Fine
Sands
t
5
to
13
Gravel-Sands
t
5
Coarse
Sand
t
8
to
14
consideration
of
spatial
variability
of
soil
properties
or
probabilistic
modelling
of
soil
profiles
e.g.
Vanmarcke
(1977a,
1982,
Matsuo
and
Asaoka
(1977),
Asaoka
and
Matsuo
(1982),
Asaoka
and
A-Grivas
(1982).
A
probabilistic
approach
recognises
that
any
earth
structure
has
some
probability
of
failure
however
small.
In
contrast,
a
deterministic
approach
leads
the
geotechnical
engineer
to
an
expectation
that
failure
of
a
slope
can
not
occur
under
the
conditions
for
which
a
value
of
F
greater
than
1
was
calculated.
Yet
failures
of
slopes
designed
to
be
'safe'
(i.e.
calculated
F
>
1)
are
not
uncommon.
The
probabilistic
approach
enables
a
study
of
reliability
to
be
made
under
conditions
of
un-
certainty
although
it
does
not
aim
at
a
(yes
or
no)
type
of
answer
in
respect
of
slope
safety.
As
such
it
enhances
engineering
judgement
and
enables
decisions
about
projects
to
be
made
taking
into
account
the
estimated
costs
of
alt-
ernate
designs,
the
estimated
probability
of
failure
for
each
alternative
and
the
estimated
cost
of
failure
e.g.
Kuroda
and
Tang
(1979),
Matsuo
&
Suzuki
(1983).
Quite
apart
from
the
above
advantages
a
probab-
ilistic
approach
may
provide
a
valid
framework
for
modelling
slope
processes
and
for
investi-
gating
aspects
of
the
occurrence
of
failures.
In
this
way
one
can
gain
a
better
insight
into
aspects
of
slope
stability
and
a
keener
appreci-
ation
of
the
risk
associated
with
particular
sites.
In
subsequent
sections
several
examples
are
cited
concerning modelling
of
failure
pro-
pagation,
e.g.
Chowdhury
(1981)
and
Chowdhury
and
A-Grivas
(1982),
simulating
strain-softening
and
progressive
failure,
e.g.
A-Grivas
and
Chowdhury
(1983)
estimating
the
most
probable
extent
of
slope
failure
e.g.
Vanmarcke
(1977b)
and
Chowdhury
(1980a),
predicting
failure
due
to
rainfall
e.g.
Matsuo
and
Ueno
(1979,
1981)
up-
dating
stability
risk
e.g.
Matsuo
and
Asaoka
(1983)
etc.
Design
procedures
based
on
statistical
techniq-
ues
have
also
been
developed
for
rock
slopes
and
open
pit
mines
e.g.
McMahon
(1971,
1975)
and
Baecher
and
Einstein
(1978).
These
procedures
are
based
on
the
modelling
of
structural
discon-
tinuities
and
uncertainties
associated
with
geological
data
and
structural
features
e.g.
angle
of
dip,
direction
of
strike
etc.,
of
dis-
c
-
DS
t
DS
c
triaxial
t
triaxial
c
CU
t
triaxial
c
triaxial
t
triaxial
c
D
t
D
c
CD
t
CD
c
UU
t
--
UU
UU
n
o
e
n
n
e
n
°
e
e
°
c
o
c
log(C
c
)
t
t
c
c
phi
phi
c
phi
c
u
45.6
103.3
17.7
D
13.5
D
1.6
19.9
CU
9.8
UU
18.8
UU
22.3
24.0
2.1
26.9
6.8
25.5
5.4
14.8
14.7
31.0
19.8
29.0
21.6
89.4
29
16
9.8
16
10
17.5
13.3
18.4
16.2
34.2
13.8
14.8
31.6
25.9
14
12
5
51
22
19
55
29
20
64
UU
15
56
22
19
71
12
63
10
4
3
2
5
24
2
23.2
213.
TABLE
III
Statistical
Measurements
of
Strength
Parameters
as
Obtained
from
Field
Investigation
Records
in
Australia
Nguyen
&
Chowdhury
(1984a)
Cohesion,
c
kPa
Angle
of
friction,
(
c)
Mean
Standard
deviation
Coefficient
of
variation
Mean
Standard
deviation
Coefficient
of
variation
Coal
Mining
Location
Source
(E)
(S
c
) (
-
6
-
/S
c
)
0)
(S
)
q)
0/S
)
q)
Within
Australia
480
87
.18
15
5
.33
Ramp
10S
Highwall
Goonyella
335
108
.32
12
1.8
.15
Ramp
11S
Spoil
Goonyella
116
57
.49
9
1.8
.19
Ramp
6N
Spoil
Goonyella
312
81
.26
11
1.7
.15
Terang
Spoil
South
Blackwater
48
19
.39
9
1.3
.14
Terang
Spoil
Interface
123
25
.20
16
2.9
.18
Terang
Spoil
Floor
Joints
TABLE
IV
Summary
of
Various
Uncertainty
Factors
of
the
Conventional
Safety
Factor
Associated
with
a
Slope
Stability
Problem
by
Tang
et
al
(1976)
Factor
Effect
Soil
Type
Range
v
j
o
f
N
1
Change
in
stress
state
Low
sensitivity
(s
t
=
1
-
2)
Medium
sensitivity
(s
t
=
2
-
4)
1.0
-
1.1
1.03
1.05
0.02
0.02
Sensitive
(s
t
=
4
-
8)
1.07
0.02
Unknown
sensitivity
(s
t
=
1
-
8)
1.05
0.03
N
2
Mechanical
disturbance
Shelby
tube
specimen
Bore
hole
specimen
1.0
-
1.6
1.15-
2.25
1.3
1.7
0.13
0.19
N
3
Size
of
specimen
Stiff-fissured
clay
Intact
clay
0.55-
0.85
0.85-
1.00
0.70
0.93
0.12
0.05
N
4
Rate
of
shearing
Slightly
sensitive
to
sensitive
(strength
reduction
3-10%
per
log
increment
of
time)
0.6
-
1.0
0.80
0.14
Very
sensitive
(10
-
14%)
0.45-
0.7
0.58
0.12
unknown
sensitivity
(
3
-
14%)
0.45-
1.0
0.73
0.22
N
5
Sample
orientation
and
anisotropy
Isotropic
C-anisotropy
0.97-
1.08
0.85-
1.20
1.0
1.03
0.03
0.10
M-anisotropy
0.8
-
1.0
0.9
0.06
N
6
Plane
strain
failure
All
soil
types
1.0
-
1.1
1.05
0.03
N
7
Progressive
failure
Stiff
clay
Medium
clay
0.
0
9
-
1.
0.93
0.97
0.03
0.03
(The
range,
mean
and
standard
deviation
of
each
factor
is
shown)
214.
continuities.
In
this
respect
probabilistic
techniques
for
hard
rock
slopes
differ
somewhat
from
those
used
for
soil
slopes
and
for
fract-
ured
and
soft
rock
masses
in
which
uncertainty
concerning
shear
strength
and
pore
water
press-
ure
are
dominant
and
structural
and
geometrical
parameters
may
generally
be
regarded
as
deter-
ministic.
Morriss
and
Stoter
(1983)
discussed
the
use
of
computer
for
sampling
bivariate
structural
data
in
relation
to
open-pit
design.
They
referred
to
Monte
Carlo
sampling
of
structural
and
strength
parameters
to
derive
a
model
for
pit
slope
sections
and
assessed
the
advantages
and
limitations
of
an
optimisation
approach
based
on
probabilistic
techniques.
THE
PROBABILISTIC
APPROACH
Safety
Factor
and
Safety
Margin
as
Random
Vari-
ables
Each
parameter
affecting
slope
stability
may
be
regarded
as
a
random
variable
with
an
associated
probability
distribution
rather
than
as
a
con-
stant.
Variability
of
some
parameters
(e.g.
unit
wt.
and
geometrical
parameters)
has
an
insignificant
influence
on
stability.
Such
parameters
may
be
regarded
as
constants
whereas
it
is
desirable
to
consider
shear
strength
parameters
and
pore
water
pressure
as
random
variables.
Thus
the
factor
of
safety
F
which
may
be
defined
as
the
ratio
of
resisting
forces
or
moments
RF
and
disturbing
forces
or
moments
DF
is
itself
a
random
variable.
Similarly
the
safety
margin
SM
which
may
be
defined
as
the
difference
of
RF
and
DF
is
also
a
random
variable.
Defining
Reliability
and
Failure
Probability
Reliability
is
the
probability
that
a
slope
will
be
safe
i.e.
it
will
survive
and
not
fail
under
given
conditions.
It
is,
therefore,
the
prob-
ability
of
success
p
s
which
is
the
probability
that
the
factor
of
safety
F
will
be
greater
than
one
or
that
the
safety
margin
SM
will
be
greater
than
zero
i.e.
P
s
=
P[F
>
1]
=
P[SM
>
0]
(1)
The
probability
of
failure
pf
is
complementary
to
the
probability
of
success
and
is
defined
p
f
=
P[F
1]
=
P[SM
0]
(2)
It
is
axiomatic
that
Ps
+
Pf
=
1
(3)
Provided
that
the
probability
distribution
of
either
F
or
SM
is
known
the
reliability
p
s
can
be
calculated
as
the
area
under
the
probability
distribution
curve
to
the
right
of
the
ordinate
at
F=1
or
SM=0.
Similarly
the
probability
of
failure
is
the
area
to
the
left
of
the
ordinate
as
shown
in
Fig.2
for
two
well
known
distribut-
ions,
the
normal
or
Gaussian
distribution
and
the
lognormal
distribution.
In
the
absence
of
data,an
assumption
about
the
probability
distribution
is
made
and
the
cal-
culated
value
of
p
s
will
depend
on
the
choice
of
distribution.
For
relatively
high
values
of
pf
the
calculated
magnitude
is
not
sensitive
to
the
assumed
shape
of
the
probability
distribut-
ion.
For
low
values
of
pf
(say
pf
<
10
-
3),
on
the
other
hand,
the
calculated
value
of
pf
is
very
sensitive
to
the
shape
of
the
distribution
because
the
tails
of
the
distribution
become
very
important.
The
distributions
in
common use
are
the
normal
or
Gaussian
distribution,
the
lognormal
distribution and
the
Beta
distribution.
Even
if
the
shape
of
the
distribution
is
assumed,
one
must
calculate
the
basic
statistical
para-
meters
of
the
random
variable
before
the
probab-
ility
of
failure
or
success
can
be
estimated.
For
example
a
normal
distribution
requires
that
the
mean
and
standard
deviation
of
F
or
SM
be
known
i.e.
either
(F
and
SF)
or
(SM
or
SSM)
must
be
calculated
depending
on
whether
calculations
are
being
made
in
terms
of
F
or
SM
and
whether
the
the
distribution
is
known
or
assumed
either
for
F
or
for
SM.
The
distribution
of
a
reliability
or
safety
parameter
such
as
F
may
be
determined
by
repeat-
edly
taking
a
random
set
of
values
of
the
basic
stochastic
parameters
and
using
the
appropriate
formula
for
the
calculations.
The
Monte
Carlo
simulation
technique
discussed
in
a
subsequent
section
of
this
paper
may
be
used
to
obtain
probability
distributions
of
the
safety
factor
of
a
slope.
Distributions
of
soil
properties
must
be
deter-
mined
on
the
basis
of
available
data
and
one
can
check
whether
a
particular
empirical
distribution
follows
any
well
known
mathematical
pdf
(probab-
ility
density
function)
such
as
the
distributions
known
as
normal
or
Gaussian,
lognormal,
exponen-
tial,
beta
etc.
Table
5
shows
empirical
distri-
bution
functions
for
various
random
variables
compiled
by
Baecher
et
al
(1980).
TABLE
V
Probability
Distribution
Functions
for
Various
Soil
Properties
Compiled
by
Baecher
et
al
(1980)
from
Various
Sources
Material
Property
pdf
Compacted
clay
core
Dry
density
Lognormal
Sand
Pile
capacity
Lognormal
Dry
sand
Bearing
capacity
Lognormal
Silt
Void
ratio
Normal
Silt
Uniformity
Co-
efficient
Exponential
Various
sands
Porosity,
void
ratio
Normal
Various
silts
Porosity,
void
ratio
Normal
Various
clays
Porosity,
void
ratio
Normal
Marine
clay
Undrained
shear
strength
Normal
. .
.
215.
TABLE
V
(Cont'd)
Material
Property
pdf
Silty
sand
tan
0'
Normal
Clayey
Silt
tan
0'
Normal
Clayey
Silt
Undrained
shear
strength
Normal
Various
soils
c',
0'
Beta
THE
RELIABILITY
INDEX
Considering
the
two
basic
statistical
parameters
of
the
safety
factor
F
one
can
define
a
measure
of
reliability
which
is
more
sensible
than
the
conventional
safety
factor
itself.
This
useful
measure
may
be
called
the
Reliability
Index
and
denoted
by
R
where
the
safety
margin.
It
is
often
convenient
to
use
a
standardised
variable
U
which
has
zero
mean
and
unit
standard
deviation
i.e.
u
F
-
F
or
SM-
sm
--
g
SM
The
probability
of
failure
may
also
be
defined
as
follows:-
p
f
=
P(F
1)
=
P(U
-R)
=
F
u
(-R)
(7)
in
which
F
u
( )
denotes
the
cumulative
probability
distribution
of
the
standardised
variable.
Assuming
F
or
SM
follows
a
normal
or
Gaussian
distribution
Fu(
)
is
the
widely
tabulated
Stan-
dard
Normal
Distribution.
Tabulated
values
are
generally
not
available
at
relatively
large
positive
values
of
the
standard
normal
variate
U.
The
following
series
approximation
is,
therefore,
useful
when
one
is
dealing
with
relatively
low
failure
probabilities
as
implied
by
large
values
of
U
(6)
F
-
1
R
=
S
F
(4)
,
2
P[U.5-R]
=
F
(-R)=0/2;
R)-'e-R
2
[1-R
-2
+3R
-4
-15R
76
This
equation
reflects
the
understanding
that
reliability
increases
as
the
mean
of
F
increases
but
decreases
as
the
variability
of
F
increases.
The
reliability
index
measures
the
distance
be-
tween
F
and
its
failure
threshold
(which
is
1)
in
units
of
S
F
.
In
terms
of
the
safety
margin,
one
can
write
:
SM
s
-
o
v
sm
sm
in
which
V
SM
is
the
coefficient
of
variation
of
+
105R
-8
-1
(8)
MODELLING
VARIABILITY
-
PROBABILISTIC
SOIL
PRO-
FILE
Soil
Property
as
a
Random
Process
Soil
deposits,
whether
natural
or
man-made
are
generally
non-homogeneous.
Firstly,
one
must
consider
the
major
trends
in
the
variations
of
a
soil
property
laterally
and
with
depth.
For
example
the
undrained
shear
strength
of
a
soft
clay
may
increase
linearly
with
depth.
In
(5)
f
(FS)
Total
area=
1.
Area
Pf
FS
FS
f(FS)
Total
area
=1
Area
Pf
+a)
FS
0
1
FS
Fig.
2.
Two
probability
distributions
of
the
factor
of
safety
FS
(denoted
F
in
the
text),
frequently
used
for
probabilistic
calculations
in
geomechanics;
note
the
definition
of
failure
probability
c
u
(z)
-
u
(z)
S
c
(z)
u
216.
another
case,
the
shear
strength
may
be
genera-
lly
constant
within
a
given
range
of
depth.
Again
there
may
be
two
or
three
layers
with
dis-
tinctly
different
values
of
undrained
shear
strength.
Considerations
like
these
are
part
of
the
procedure
of
subsoil
modelling
on
the
basis
of
conventional
practice.
However,
local
dev-
iations
from
average
values
of
soil
properties
are
not
given
significant
attention.
Consider
a
single
so-called
homogeneous
layer
within
a
soil
medium
(which
has
been
subdivided
into
various
layers
for
convenience
of
analysis).
On
the
basis
of
available
data
one
may
determine
the
mean
of
the
undrained
shear
strength
c
u
and
its
standard
deviation
S
cu
.
The
magnitude
of
the
standard
deviation
(the
intensity
of
fluct-
uation)
is
often
assumed
to
be
constant
but
it
actually
depends
on
the
extent
of
the
area
which
is
investigated.
If
one
considers
a
large
area,
local
variations
tend
to
cancel
each
other
and
the
calculated
standard
deviation
tends
to
be
low.
On
the
other
hand
if
the
area
considered
is
relatively
small,
the
calculated
standard
deviation
tends
to
be
relatively
large.
These
differences
are
a
consequence
of
local
fluctuations
of
shear
strength
from
its
global
mean
value.
Consequently
it
is
necessary
to
represent
a
soil
property
as
a
function
of
the
sampling
distance
especially
when
the
fluct-
uations
of
that
property
may
have
a
significant
influence
on
the
performance
of
the
soil
mass
in
question
or
of
the
associated
geotechnical
structures.
Thus
for
a
natural
clay
deposit
one
may
write
for
c
u
as
a
function
of
depth
z
as
follows:
c
u
=
c
u
(z)
(9)
In
the
case
of
a
long
embankment
c
u
may
be
con-
sidered
as
a
random
function
of
distance
x
along
the
axis
of
the
embankment
i.e.
c
u
=
c
u
(x)
(10)
This
procedure
of
representing
a
soil
property
as
a
random
function
of
distance
or
depth
is
part
of
the
process
known
as
probabilistic
soil
profile
modelling.
This
procedure
is
a
system-
atic
representation
of
local
spatial
variability
and
its
role
is
to
supplement,
rather
than
re-
place,
conventional
subsoil
modelling.
Scale
of
Fluctuation
and
Correlation
Function
To
enable
modelling
of
soil
property
variability,
the
scale
of
fluctuation
of
the
property
denoted
by
6
is
required
in
addition
to
the
mean
and
the
standard
deviation
(the
intensity
of
fluct-
uation
).
Thus
for
undrained
shear
strength
of
a
soil
one
needs
c
u'
s
and
(s
cu
and
for
the
c
u
friction
angle
of
a
soil
one
needs
T,
S
cp
and
6
q)
.
The
scale
of
fluctuation
6
measures
the
distance
within
which
the
soil
property
shows
relatively
strong
correlation
or
persistence
from
point
to
point.
At
two
points
which
are
less
than
a
dis-
tance
6
apart
from
each
other,
the
values
of
the
property
are
likely
to
be
either
both
above
or
both
below
the
mean.
The
distance
6
is
also
closely
related
to
the
average
vertical
distance
between
crossings
or
intersections
of
the
random
function
and
its
mean
(e.g.
the
intersections
of
function
c
u
(z)
and
the
vertical
line
repres-
enting
c
u
in
the
case
of
a
so-called
homogeneous
clay
layer
in
which
shear_strength
is
constant
with
depth;
the
line
for
c
u
will
be
inclined
if
shear
strength
increases
with
depth).
Frequent
fluctuations
about
the
average
imply
a
small
value
of
6
whereas
infrequent
fluctuations
imply
a
large
value
of
6.
Different
scales
of
fluctuation
will,
in
general,
apply
to
different
directions
within
a
soil
mass
as
shown
in
Tables
6
and
7
which
have
been
compiled
for
different
soils
and
different
case
histories.
Techniques
of
modelling
soil
profiles
and,
in
particular,
for
description
of
spatial
variability
have
been
presented
by
Vanmarcke
(1977a)
and
Asaoka
and
A-Grivas
(1982).
The
mean
of
a
soil
property
and
its
intensity
of
fluctuation
(i.e.
its
standard
deviation)
may
depend
significantly
on
the
distance
in
a
particular
direction.
Consequently
it
is
often
useful
to
standardise
a
random
function
such
as
c
u
(z)
by
adopting
the
type
of
approach
used
for
F
in
Eq.(6)
i.e.
U(z)
=
U
c
(z)
-
u
TABLE
VI
Values
of
Correlation
Parameter
6
Obtained
by
Vanmarcke
(1977a
and
b)
Soil
Property
Correlation
parameter
6
C
P
T
Water
Content,
w
Initial
void
ratio,
e
o
Blow
Count
Undrained
shear
strength
1.2m
9ft
10ft
8ft
5m
)
)
)
)
)
In
the
vert-
ical
direct-
ion
Compression
Index,
c
c
Undrained
shear
strength
180ft
46m
)
In
the
hori-
zontal
dire-
ction
By
this
transformation
one
can
obtain
standard-
ised
data
about
a
significant
soil
property
such
as
c
u
.
This
data
has
a
mean
of
zero
and
a
standard
deviation
of
unity
and
the
new
varia-
ble
may,
therefore,
be
regarded
as
statistically
homogeneous.
The
correlation
function
is
the
coefficient
of
correlation
between
values
of
a
soil
property
at
two
points
separated
by
a
given
distance
Az.
For
example,
in
the
case
of
the
parameter
c
u
the
correlation
function
p(Az)
is
the
expected
value
of
the
quantity
in
brackets
below
i.e.
{
c
u
u}{
c
u
(z+Az)-Z
u}
S
cu
u
S
c
The
value
of
p
is
exactly
unity
when
Az=0,
its
value
is
close
to
unity
for
small
Az,and
the
value
decays
as
Az
increases.
p(Az)=p
o
(Az)=E
(12)
217.
TABLE
VII
Typical
Correlation
Distances
for
Various
Soil
Properties
Compiled
by
Bacher
et
al
(1980)
MATERIAL
CPT
r
o
Coastal
sand
CPT
5m
Compacted
clay
(dam
core)
dry
density
(horizontal
layers)
(vertical)
5
5
North
Sea
Clay
CPT
30
Dune
sand
SPT
20
Plastic
clay
dry
density
(vertical)
1.3
Clean
sand
CPT
0.36
Clay
CPT
1.91
Silty
loam
water
content.
w
0.16
The
dimensionless
ratio
between
the
standard
deviation
over
a
given
distance
and
the
point
standard
deviation
is
called
a
reduction
factor
and
is
here
denoted
by
'rf'
i.e.
rf
=
rf
c
(Az)
-
u
Within
the
correlation
distance
the
reduction
factor
may
be
close
to
unity
but
as
Az
increases
the
standard
deviation
decays
relative
to
its
point
value.
The
square
of
the
reduction
factor
is
called
the
variance
function.
At
relatively
large
values
of
Az
the
variance
becomes
inverse-
ly
proportional
to
Az
and
the
variance
function
takes
the
form:
rf2(
z)
=
oz
(14)
This
relationship
defines
the
scale
of
fluctuat-
ion
6
and
enables
its
estimation
from
given
data
on
the
particular
soil
property
considered
i.e.
c
u
in
the
present
discussion.
It
has
been
found
that
Eq.(14)
is
a
good
approx-
imation
for
the
variance
function
at
values
Az
greater
than
6
but
which
are
close
to
6
(rather
than
large
values).
For
values
of
Az
less
than
6
the
value
of
the
variance
function
may
be
taken
close
to
unity,
i.e.
regardless
of
the
underlying
form
of
the
correlation
function
rf
2
(Az)
=
1
for
Az
6
rf
2
(Az)
=
Z-
for
Az
?,
6
(15)
Table
VI
shows
the
value
of
the
parameter
6
for
various
assumed
correlation
functions.
Assuming
that
the
correlation
function
follows
the
squared
exponential
decay
model
the
follow-
ing
approximate
relationship
has
been
derived
between
the
scale
of
fluctuation
6
and
the
average
distance
d
between
the
intersections
of
the
fluctuating
parameter
e.g.
c
u
(z)
and
its
mean
c
u
a
=
2
-
6
1.25
6
(16)
Equivalent
Multi-layer
System
In connection
with
slope
stability
problems
and,
in
particular,
stability
of
soft
clay
slopes
under,undrained
conditions,
Japanese
research
workers
have
proposed
the
transformation
of
a
stochastic
process
or
random
field
into
an
equi-
valent
system
consisting
of
several
homogeneous
soil
layers.
This
approach
has
been
used
for
embankment
stability
problems
over
a
number
of
years
e.g.
Matsuo
and
Asaoka
(1977,
1982).
A
TABLE
VIII
Correlation
Functions
and
the
Values
of
6
after
Vanmarcke
(1977a)
Assumed
Correlation
Function
of
a
Soil
Property
Parameter
6
-
the
scale
of
fluctuation
e
-1AzI/a
2a
e
-(Az/b)2
vr
7
rr
b
e
-1AzI/c
cos(Az/c)
c
[7
1
1. _
(Az)]
e
-1AzI/d
4
d
d
stochastic
process
or
random
function
such
as
c
u
(z)
may
be
replaced
by
an
equivalent
three-
layer
system
with
properties
c
ul
,c
u2
,
c
u3
which
are
regarded
as
mutually
independent
or
uncorre-
lated
random
variables.
The
mean
and
standard
deviation
of
c
u
within
each
layer
is
regarded
as
constant.
For
the
new
imaginary
multi-layer
system
E[c
.1
=
u
.
ui
i
and
cov[c
ui
,
cu
0
.1=
s
cu
i
,i#j
(17)
2
.
=
.
i
j
in
which
i,j
are
any
two
layers
in
the
system.
The
values
of
E[f
L
c
u
(z)dz
dL]
and
cov[f
L
c
u
z(dz),
f
L-
c
u
(z)dL"1
for
the
three
layer
system
are
expected
to
be
the
same
as
those
for
the
original
system
(L
and
L'
are
the
lengths
of
arbitrary
two-dimensional
slip
surfaces
through
the
base
of
a
clay
deposit).
The
approximation
to
a
three-layer
system
was
considered
to
be
quite
satisfactory
at
least
for
the
first
and
second
statistical
moments.
It
is
important
to
note
that
such
a
system
was
derived
for
a
particular
slope
stability
problem
and
for
specific
condition
i.e.
short-term
un-
drained
stability.
A
separate
derivation
would
s
c
(Az)
u
(13)
s
cu
218.
have
to
be
attempted
for
each
individual
geo-
technical
problem
and
an
entirely
different
equivalent
multi-layer
system
would
result
for
each
problem.
Before
considering
some
important
implications
of
spatial
variability
of
soil
deposits,
it
is
of
interest
to
consider
the
common
techniques
used
for
probabilistic
calculations
in
geotech-
nical
engineering
and
especially
with
regard
to
slopes.
METHODS
OF
PROBABILISTIC
CALCULATION
The
factor
of
safety
of
slope
F
is
a
function
of
several
random
variables
such
as
the
soil
cohesion
c,
angle
of
internal
friction
(p,
unit
weight
y,
slope
inclination
a,
slope
height
H,
pore
water
pressure
u
etc.
Some
or
all
of
these
parameters
may
be'regarded
as
stochastic
parameters,
each
with
its
own
probability
dis-
tribution.
The
probability
distribution
of
F
may
be
determined
from
the
assumed
probability
distributions
of
the
basic
stochastic
variables.
One
method
commonly
used
by
geotechnical
engineers
to
do
this
is
the
Monte
Carlo
simul-
ation
technique.
The
use
of
computer
is
almost
essential
because
the
calculations
are
repetit-
ive
and
time-consuming.
Alternatively,
one
may
assume
a
probability
distribution
for
F;
common-
ly
used
assumptions
are
a
normal
or
Gaussian
distribution,
log-normal
distribution
and
a
beta
distribution.
Even
if
the
shape
of
the
distri-
bution
is
assumed
the
statistical
parameters
of
the
distribution
must
be
estimated
or
derived
from
the
statistical
paftmeters
of
the
basic
stochastic
variables.
Two
approximate
methods
are
widely
used
to
calculate
the
statistical
parameters
of
F,
namely
the
Taylor
series
appro-
ximation
and
Rosenblueth's
method.
The
former
method
has
been
in
use
for
a
long
time
and
is
detailed
in
text
books
such
as
Ang
and
Tang
(1975)
and
Harr
(1977).
The
latter
method
was
first
proposed
by
Rosenblueth
(1975)
and
is
very
likely
to
replace
Taylor
series
approach
as
it
gets
widely
known.
The
Monte
Carlo
simulation
technique
enables
the
determination
of
the
distribution
as
well
as
the
estimation
of
the
statistical
moments;
neither
the
Taylor
series
approximation
nor
Rosenblueth's
method
are,
therefore,
required
if
simulation
is
the
approach
adopted.
The
Monte-Carlo
simulat-
ion
technique
involves
the
generation
of
random
numbers
and
a
value
of
F
associated
with
a
set
of
random
values
of
the
basic
stochastic,
varia-
bles
is
simulated.
After
many
values
of
F
have
been
calculated
in
this
way,
the
probability
distribution
of
F
is
generated
and
the
statist-
ical
parameters
may
be
calculated.
The
probab-
ility
of
failure
may
be
estimated
from
the
generated
distribution
as
explained
earlier
or
directly
by
calculating
the
relative
frequency
with
which
F
was
found
to
be
less
than
or
equal
to
1
during
the
simulations.
The
Taylor
series
method
is
a
first
order,
second
moment
approximation.
Given
the
mean
and
the
standard
deviation
of
each
of
the
basic
stochastic
variables,
the
mean
and
standard
deviation
of
F
is
then
calculated;
higher
moments
of
the
basic
variables
could
be
used
for
greater
accuracy
but
this
is
generally
not
done
in
order
to
retain
the
basic
convenience
and
simplicity
of
the
approach.
Moreover,
data
on
higher
statistical
moments
of
soil
parameters
is
lacking.
This
method
has
proved
to
be
successful
in
most
slope
stability
problems.
Derivatives
of
the
function
F
are
required
and
the
method
is
unsuccessful
or
inapplicable
for
those
geotechnical
problems
in
which
one
or
more
derivatives
of
F
can
not
be
found.
Rosenblueth's
method
of
point
estimates
is
an
approximate
numerical
integration
approach.
The
expected
value
of
any
variable
F
is
found
by
adding
several
terms
(four
terms
if
the
basic
variables
are
only
2,
e.g.
c
and
(I);
eight
terms
if
the
basic
variable
are
only
3,
e.g.
c
(p
and
pore
pressure
ratio
r
u
;
sixteen
terms
if
the
basic
variables
are
only
four,
e.g.
cl,
the
shear
strength
parameter
on
the
basal
slip
and
c2,
42
the
shear
strength
parameters
on
the
rear
slip
plane
in
a
slope
stability
problem
involving
a
wedge
type
failure
mechanism).
These
terms
constitute
the
magnitudes
of
F
calculated
at
values
of
the
variables
one
standard
deviat-
ion
on
either
side
of
the
mean;
the
correlation
between
parameters
is
a
multiplying
factor
to
the
terms.
This
method
is
very
convenient
to
use
and
has
proved
to
be
successful.
It
does
not
require
the
calculation
of
the
derivatives
of
F
and
can,
therefore,
be
used
in
any
geo-
technical
and
especially
slope
stability
prob-
lem
regardless
of
how
complex
the
expression
for
F
is.
It
is
noteworthy
that
both
Taylor
series
and
Rosenblueth
methods
include
the
correlation
between
variables;
in
contrast
Monte-Carlo
simulation
technique
appears
to
have
been
used
only
for
independent
(uncorrelat-
ed
variables)
until recently.
Comparisons
between
the
different
techniques
are
essential
for
checking
their
performance
and
for
selecting
the
most
economical
method.
Con-
sidering
several
example
problems
of
slope
stability
and
using
different
assumed
distribu-
tions,
good
comparison
between
Taylor
series
approach
and
Rosenblueth's
method
was
found
by
Derooy
(1981).
In
connection
with
the
study
of
the
stability
of
mining
spoil
piles,
Nguyen
and
Chowdhury
(1984a)
made
a
comparison
between
Monte
Carlo
simulation
technique
and
Rosenblueth's
method
considering
four
independ-
ent
random
variables.
Excellent
agreement
was
found
(See
Table
IX)
and
it
was
concluded
that
Rosenblueth's
method
should
be
used
for
all
practical
applications
in
order
to
avoid
the
time-consuming
and
repetitive
calculations
nec-
essary
for
the
simulation
technique.
Encouraged
by
these
results,
Nguyen
and
Chowd-
hury
(1984b)
developed
a
procedure
for
Monte
Carlo
simulation
considering
the
basic
variables
to
be
correlated.
It
may
be
noted
that
in
all
published
work
concerned
with
geotechnical
stability
Monte
Carlo
simulation
has
been
used
on
the
assumption
that
the
basic
parameters
are
independent
and
uncorrelated.
The
problem
chosen
for
study
was
one
of
water
table
draw
down
near
a
slope
of
a
box-cut
of
an
open
strip
coal
mine.
Comparison
of
results
again
showed
an
excellent
agreement
between
Monte
Carlo
simulation
and
Rosenblueth's
method
for
the
first
two
statistical
moments
(See
Table
X).
The
Taylor
series
method
gives
a
good
approxi-
mation
for
the
first
and
second
moments
but
is
not
successful
for
calculation
of
higher
moments.
The
Rosenblueth
method
gives
reason-
ably
good
results
for
the
higher
moments
as
well
although
not
as
good
as
for
the
first
two
methods.
219.
TABLE
IX
Comparison
of
Statistical
Parameters
of
'F'
estimated
by
Rosenblueth
Method
and
Monte
Carlo
Simulation.'F'
is
the
Factor
of
Safety
of
the
Slope
of
a
Mining
Spoil
Pile
Considering a
Two
Wedge
Failure
Mechanism
Number
of
Simulations
(after
first
line) line)
m
l
p
m
2
S
2
F
m
3
m
4
1
3/2
=m
/m
3
2
a
2
(kurtosis) (kurtosis)
=m
4
/m
2
METHOD
OF
ROSENBLUETH*
1.376
0.0224
-0.0001
U
0.0011
-0.042
2.261
100
1.392
0.0284
0.0017
0.0028
0.367
3.585
200
1.392
0.0285
0.0025
0.0027
0.520
3.417
300
1.393
0.0286
0.0016
0.0026
0.330
3.165
400
1.401
0.0286
0.0018
0.0027
0.369
3.382
500
1.402
0.0310
0.0020
0.0032
0.374
3.334
600
1.392
0.0291
0.0008
0.0026
0.162
3.075
700
1.395
0.0289
0.0011
0.0026
0.226
3.198
800
1.394
0_0290
0.0012
0.0027
0.243
3.236
900
1.403
0.0313
0.0015
0.0032
0.274
3.233
1000
1,401
0.0316
0.0015
0.0033
0.278
3.287
*Rosenblueth's
approximation
using
16
vertices
around
the
mean;
F
is
a
function
of
four
independent
random
variables,
two
shear
strength
parameters
for
each
of
the
two
planar
failure
surfaces.
TABLE
X
Statistical
Parameters
of
Drawdown
Time
Distribution,
Drawdown
Time
t
Being
a
Function
of
Two
Correlated
Stochastic
Variables,
the
Permeability
k
and
the
storage
coefficient
S;
the
Coefficient
of
Variation
of
Both
Variables
was
Assumed
to
be
10%
and
and
Correlation
Coefficient
r
kS
=0.5
(after
Nguyen
and
Chowdhury
1984b)
Method
Mean
of
t
Variance
of
t
Third
Moment Moment
FourthSkewness
Kurtosis
E
s
m
3
m
4
(3
1
(3
2
Monte
Carlo
400
Simulations
46.04
20.24
24.43
1435.0
0.072
3.502
Rosenblueth
46.01
21.54
44.57
1890.81
0.198
4.073
Taylor
Series
45.78
20.96
0
L
2635.8
0
5.999
MOST
PROBABLE
EXTENT
OF
SLOPE
FAILURE
A
Long
Embankment
An
extremely
important
aspect
of
modelling
soil
property
fluctuations
becomes
evident
if
the
slope
stability
problem
is
considered
within
a
three-dimensional
framework.
In
the
case
of
a
long
embankment
the
undrained
shearing
resist-
ance
c
u
may
be
modelled
as
a
random
function
of
distance
along
the
axis
of
the
embankment.
A
cylindrical
slip
surface
is
considered
with
a
dimension
b
in
the
direction
of
the
axis
and
the
resistance
provided
along
the
edges
(assumed
vertical)
is
regarded
as
deterministic
for
con-
venience.
It
is
then
shown
(Vanmarcke,
1977b,
1980)
that
the
reliability
index,
which
is
a
function
of
the
axial
length
b
considered,
is
a
minimum
for
a
value
of
b
=
b
c
such
that
b
c
=
r-1
°
(19)
in
which
d
o
is
a
constant
depending
on
the
characteristics
of
the
cross
section
of
the
em-
bankment
and
the
potential
shear
surface
and
F
is
the
mean
of
the
two
dimensional
safety
factor.
The
probability
of
failure
is
the
maximum
if
the
220.
lateral
extent
of
slope
considered
is
b=b
c
;
the-
refore,
it
may
be
regarded
as
the
most
probable
failure
length.
While
the
critical
extent
of
failure
is
independ-
ent
of
the
standard
deviation
of
F,
the
assoc-
iated
failure
probability
at
any
cross
section
depends
on
both
r
and
S
F
and
hence
on
the
ran-
dom
function
c
u
(x)
and
not
just
its
mean.
The
actual
failure
probability
of
the
slope
increases
as
the
overall
length
B
of
the
embank-
ment
increases
and
for
values
of
B
which
are
significantly
larger
than
b
c
,
Pf(B)
is
a
linear
function
of
B:-
B
-R/2
P
F
(B)
=ceb
in
which
Rb
is
the
reliability
index
considering
the
extent
of
failure
to
be
b
and
in
which
c
=
(i27)
-1
(b/6).
The
maximum
value
of
PF(B)
is
approximately
obtained
by
substituting
b=b
c
.
A
Long
Natural
Slope
The
classical
infinite
slope
equation
predicts
a
factor
of
safety
for
a
long
natural
slope
considering
a
potential
failure
surface
parallel
to
the
slope.
It
does
not
give
any
indication
of
the
extent
of
failure.
However,
it
is
interesting
to
note
that
modelling
the
shear
strength
as
a
random
function
of
distance
L
along
the
slope
leads
to
an
expression
for
the
critical
extent
of
failure
L
c
similar
to
Eq.19.
Again
the
"end
effects"
are
treated
as
determin-
istic
and
instead
of
a
one-dimensional
formula-
tion,
a
two-dimensional
formulation
is
required.
Instead
of
treating
c
u
as
a
random
variable,
the
friction
angle
cp
and
the
pore
pressure
parameter
r
u
should
be
treated
as
random
variables
(Chowdhury
1980b).
Further
developments
require
the
modelling
of
a
natural
slope
as
a
three-
dimensional
problem
and
it
is
found
that
the
critical
length
of
failure
increases
as
the
lateral
extent
b
of
the
potential
failure
mass
is
considered
to
increase.
The
limiting
value
of
the
critical
length
is
that
for
the
2-D
case.
Moreover,
the
value
of
the
parameter
K
(repres-
enting
the
ratio
of
lateral
to
vertical
in
situ
stress)
influences
the
shape
of
the
curve
of
L
c
versus
b
(Chowdhury,
1984a).
This
is
shown
in
Fig.3
in
which
curves
based
on
values
of
K
between
0.5
to
2
have
been
plotted.
PROGRESSIVE
FAILURE
AND
SUCCESSIVE
FAILURES
Failure
Progression
An
interesting
development
in
slope
studies
is
the
consideration
of
the
probability
of
pro-
gressive
failure.
Consider
a
slip
surface
subdivided
into
various
segments
and
let
the
probability
of
local
failure
of
any
segment
i
be
pi
and
of
the
adjacent
segment
be
pi.
Then
the
task
is
to
determine
the
probability
of
failure
of
(i+1)
should
i
fail.
This
may
be
described
as
the
probability
of
progression
and
denoted
by
p..
where
P(SM.
1
0
and
SM.
0)
P--
=
P(SM
i
0)
The
numerator
is
a
joint
probability
of
failure
of
adjacent
slices
and
requires
an
assumption
concerning
the
joint
probability
density
func-
tion
of
the
safety
margins
of
adjacent
slices.
If
such
a
distribution
is
known
or
assumed,
five
independent
statistical
parameters
are
required
for
the
calculations.
These
are
the
mean
and
standard
deviation
of
each
of
the
safety
margins
and
the
correlation
coefficient
between
them.
The
procedure
for
doing
this
and
the
associated
probabilistic
model
are
discussed
in
detail
by
Chowdhury
(1981)
and
Chowdhury
and
A-Grivas
(1982).
In
the
first
of
these
publica-
tions
some
discussion
of
the
time
aspects
of
progressive
failure
and
the
potential
use
of
a
Markovian
model
is
also
included.
While
the
problem
of
failure
progression
in
its
spatial
aspects
has
been
dealt
with
successfully,
the
inclusion
of
soil
variability
would
represent
an
improvement
and
efforts
to
this
end
continue
(Tang
and
Chowdhury,
1983).
The
influence
of
various
stochastic
variables
on
the
probability
of
spatial
progression
is
discussed
elsewhere
(Chowdhury
1984b).
Strain-Softening
An
important
aspect
of
progressive
failure
concerns
the
strain-softening
nature
of
most
earth
materials
and
studies
of
the
variation
of
the
slope
failure
probability
with
an
increase
in
residual
factor
have
been
made,
e.g.,
Derooy
(1980),Chowdhury
(1981),
Chowdhury
and
Derooy
(1984).
The
residual
factor
is
the
proportion
of
the
slip
surface
over
which
the
shear
streng-
th
parameters
have
fallen
from
their
peak
values
to
their
residual
values.
This
type
of
study
can
be
useful
in
design
and
decision-
making
especially
if
field
evidence
of
movements
and
relative
displacements
for
different
parts
of
a
slope
is
available. The
influence
of
the
direction
of
propagations
on
the
safety
factor
and
the
failure
probability
is
also
important.
It
is
interesting
that
the
direction
of
failure
propagation
has
no
influence
on
the
safety
factor
if
"q)=0"
conditions
prevail
(Fig.4)
but
the
influence
may
be
significant
in
the
general
case
(Fig.5).
For
the
"q)=0"
case
only
two
random
variables
are
considered
whereas
four
random
variables
are
to
be
considered
when
c
and
4)
both
change
their
values
with
strain-softening
Inclusion
of
pore
water
pressure
increases
the
number
of
random
variables
to
five.
(20)
Successive
Failures
The
occurrence
of
successive
failures
is
of
trem-
endous
importance
in
slope
engineering
and
land-
slides.
Using
the
same
framework
as
for
failure
progression
one
may
formulate
the
problem
as
one
of
conditional
probability.
However,
the
development
of
a
valid
methodology
requires
consideration
of
a
number
of
factors
which
may
not
be
significant
in
the
case
where
progression
or
development
of
a
single
slip
surface
is
to
be
considered.
Consider
two
of
many
potential
slip
surfaces
within
a
slope
and
denote these
be
A
and
B
and
the
corresponding
safety
margins
by
SMA
and
SMB.
The
probability
of
failure
along
A
may
be
higher
than
that
along
B.
In
fact,
the
slip
surface
A
may
be
critical.
However,
the
probability
of
failure
along
B
(which
is
initia-
lly
estimated
to
be
lower
than
that
along
A)
may
increase
dramatically
given
that
A
has
failed.
One
may
write
for
this
probability
pAB:
(21)
_
P[Failure
B/Failure
A]
PAB
P[Failure
A]
(22)
2
D-
Case
z
=
6m
(31,6448m)
K=0
K-1
K2
For
z=
6m
221.
oe.
z
R
100
200
300
b
(metres)
Fig.
3.
The
critical
length
of
failure
L
c
of
a
natural
slope
in
the
direction
of
slope
as
a
function
of
the
lateral
extent
of
failure
b
assumed
in
the
analysis
after
Chowdhury
(1984)
The
question
is
how
to
define
the
event
of
failure
of
mass
associated
with
A.
It
may
be
the
event
of
some
slip
occurring
along
A
but,
on
the
other
hand,
it
may
be
that
the
whole
of
the
soil
mass
is
considered
to
have
moved
physically
away.
The
methodology
for
determin-
ing
p
AB
should
be
such
that
both
these
possibil-
ities
can
be
handled.
The
actual
value
of
pAB
will
depend
on
the
particular
case
considered
and,
in
fact,
the
expression
for
the
safety
margin
SMB
will
depend
on
how
the
event
of
fail-
ure
of
slope
along
A
is
defined.
A
joint
distribution
of
the
two
safety
margins
must,
of
course,
be
assumed.
To
use
such
a
distribution
it
is
necessary
to
calculate
the
correlation
coefficient
rSMASMB
between
the
two
safety
margins.
The
procedure
for
doing
this
is
being
published
separately.
Provided
the
problem
is
formulated
correctly,
this
approach
to
the
estimation
of
the
probab-
ility
of
successive
failures
is
found
to
be
successful.
The
correlation
between
safety
margins
is
found
to
be
perfect
if
the
soil
mass
is
modelled
as
a
statistically
homogeneous
medium
for
the
"(p=0"
situation.
Therefore,
it
is
fairly
easy
to
explain
extensive
slope
fail-
ures
in
probabilistic
terms
for
which
back-
analyses
within
the
conventional
deterministic
framework
have
proved
to
be
unsatisfactory
e.g.
the
failure
along
the
Kimola
canal
in
Finland
(Leonards,
1982).
Similarly
considering
the
problem
of
long-term
stability
and,
in
particular,
long
natural
slopes,
one
can
predict
the
occurrence
of
successive
failures.
It
is
to
be
expected
that
for
the
case
of
relatively
insensitive
soils,
successive
failure
masses
will
be
of
relatively
similar
size
as
has
been
reported
in
the
litera-
ture
(e.g.
Skempton
&
Hutchinson,
1969).
This
is
because
for
masses
of
similar
geometry
and
size
the
correlation
between
SM
A
and
SM
B
is
likely
to
be
very
high
indeed;
the
higher
this
correlation
the
greater
the
probability
of
a
successive
failure.
Nevertheless,
there
are
222.
19
+
Failure
initiating
at
crest
Failure
initiating
at
toe
Failure
initiating
in
interior
1.8
1.7
1.6
1.5
1.2'
10m
30
.
08
0
Cup
50
kN/m
Cu
r
-,
25
kN/m
c6p=c6r=0
.
0.2
0.4
0.6
Residual
Factor,
Ri
-5
1.0
0.9
0.8
1.0
Fig.
4.
Influence
of
the
direction
of
failure
propagation
on
the
factor
of
safety
of
a
slope
for
various
values
of
the
residual
factor
R
i
-
the
"cp=0"
case
situations
in
which
even
a
small
initial
slip
may
lead
to
a
large
and
catastrophic
slide
e.g.
in
the
case
of
extrasensitive
or
quick
clays.
The
modelling
of
such
problems
in
probabilistic
terms
would
be
very
valuable
indeed.
In
fact,
this
task
may
prove
to
be
one
which
is
lot
very
difficult.
UPDATING
OF
RISK,
BAYESIAN
APPROACHES
AND
SEISMIC
ANALYSIS
The
construction
of
an
embankment
on
a
soft
clay
foundation
is
a
multi-stage
process
because
of
the
low
factor
of
safety
associated
with
such
construction.
The
optimisation
of
this
multi-
stage
construction
process
based
on
observation
combined
with
probabilistic
modelling
has
often
been
discussed
by
Japanese
workers
in
the
last
decade.
In
a
recent
paper
Matsuo
and
Asaoka
(1983)
demonstrated
the
use
of
the
Rosenblueth
method
in
a
situation
where
the
shear
strength
of
the
clay
is
modelled
as
a
random
process.
The
original
system
is
replaced
by
an
equivalent
three-layer
system
and
it
is
shown
that
the
probability
of
failure
may
be
obtained
by
re-
peating
the
conventional
slope
stability
analysis
only
eight
times
for
eight
distinct
soil
profiles.
Further,
the
technique
for
up-
dating
risk
based
on
the
observation
during
construction
is
demonstrated.
Any
stage
of
construction
may
involve
success
or
failure
and
for
each
stage
one
has
a
prior
calculation
of
the
probability
of
failure.
This
can
be
updated
because
if
one
stage
of
construct-
ion
is
successful,
the
risk
of
failure
of
the
next
stage
must
correspondingly
be
less
and
the
calculations
are
made
using
a
Bayesian
formula-
tion.
The
same
sort
of
approach
may
be
used
to
update
the
seismic
stability
of
a
slope.
First
a
pro-
bability
of
failure
is
obtained
for
each
of
two
cases
namely
(1)
without
earthquake
loading
and
(2)
with
earthquake
loading.
Then
if
the
slope
is
constructed
and
survives
the
construction
period
the
risk
under
earthquake
loading
can
be
reduced.
This
approach
is
discussed
by
A-Grivas
and
Asaoka
(1982)
considering
the
bivariate
223.
1-8
1.7
o
Failure
initiating
at
toe.
+
Failure
initiating
at
crest
1.6
1.5
1.4
1.3
in
LL
1.2
FS=1.14
FS=1.08
22
°
\A
C
=30
kN/m
C
r
=10
kN/m
Op=
20
0
Or
=
12
°
0.7
0
0.2
0.4
Residual
Factor
,
Ri
Ci)
1.1
Ln
0
1.0
2
0.9
0.8
5,
5
,9
Ri
=0.65
t
0.6
0.8
1.0
Fig.
5.
The
influence
of
the
direction
of
failure
progression
on
the
factor
of
safety
of
a
slope
-
the
c,(0
case
distributions
for
c
and
tang)
of
soil
and
using
the
simple
method
of
slices
as
well
as
a
modif-
ied
Bishop
method
of
slices.
The
results
for
an
example
problem
are
shown
in
Table
11.
These
approaches
are
based
on
a
Bayesian
formu-
lation
of
the
slope
stability
problem
which
is
in
accord
with
the
modern
geotechnical
philosophy
of
basing
design
on
observational
procedures.
In
a
separate
paper
Matsuo
and
Asaoka
(1982)
discuss
the
Bayesian
calibration
of
embankment
safety
under
earthquake
loading.
The
factor
of
safety
of
an
embankment
is
back-calculated
using
information
on
the
safety
of
the
embankment
after
the
occurrence
of
an
earthquake.
In
addi-
tion
to
soil
strength,
the
seismic
load,
intro-
duced
again
as
a
horizontal
body
force,
is
also
treated
as
a
random
variable.
This
is
a
welcome
development
considering
that
even
after
the
occurrence
of
an
earthquake,
the
loading
will
remain
uncertain.
Six
embankments
with
available
records
after
the
Niigata
earthquake
of
1964
were
analysed.
A
model
for
predicting
the
probability
of
failure
of
a
slope
of
satusrated
cohesionless
soil
due
to
seismically
induced
pore
pressure
has
been
presented
by
Hadj
Hamou
and
Kavzanjian
(1984).
The
failure
probability
is
evaluated
at
the
end
of
each
cycle
of
loading
on
the
basis
of
the
cumulative
distribution
functions
of
pore
press-
224.
TABLE
XI
The
Probability
of
Failure
of
a
Slope
for
Static
and
Seismic
Loading
Conditions
after
A-Grivas
and
Asaoka
(1982)
Based
on
a
Bayesian
Formulation
of
the
Slope
Stability
Problem
Loading
Condition
Bivariate
Normal
Bivariate
Beta
Simple
Slices
Method
Modified
Bishop
Simple
Slices
Method
Modified
Bishop
p
f
%
static
4.65
2.12
5.71
2.64
p
f
%
seismic
i.e.
static
load
and
a
=
0.05g
22.97
10.03
23.89
11.72
pi/
seismics
up-
dated
on
the
basis
that
slope
surviv-
ed
construction
19.21
8.08
19.28
9.33
DATA.
Slope
Height
15.24m,
inclination
2H
to
1V,
water
table
6.1m
below
ground
surface
dipping
sharply
to
the
slope
toe
c
=
14.36
kN/m
2
S
c
=
4.32
kN/m
2
(V
c
=
0.3)
=
tan
=
0.364
,
S
tang)
=
0.073
(V
tamp
0.2)
r
c,tancp
=
-0.50
Soil
unit
weight
18.84
kN/m
3
NOTE:
An
important
conclusion
of
the
paper
referred
above
is
that
p
f
is
smaller
when
the
strength
parameters
are
negatively
correlated
than
when
they
are
independent;
the
assumption
of
independence
is
thus
a
conservative
one.
ures,
earthquake
acceleration
and
soil
para-
meters.
A
simulation
algorithm
is
used
to
solve
for
the
failure
probability
and
an
example
problem
is
solved.
Only
slopes
of
limited
extent
are
considered
in
this
approach.
Results
show
that
the
critical
failure
surface
progresses
away
from
the
face
of
the
slope
during
seismic
loading.
This
progression
is
attributed
to
the
increase
in
excess
pore
water
pressure
with
distance
from
the
face
that
is
predicted
based
on
the
initial
level
of
static
shear.
Although
probabilistic
analysis
of
liquefaction
of
cohesionless
soils
is
outside
the
scope
of
this
paper
one
must
alert
the
reader
to
such
work,
e.g.,
Haldar
and
Tang
(1979)
and
Haldar
(1980),
because
liquefaction
phenomena
even
in
isolated
sand
and
silt
lenses
could
lead
to
landsliding
on
a
large
scale.
Reference
should
also
be
made
here
to
a
3-D
seismic
reliability
analysis
of
earth
slopes
presented
by
Yuceman
and
Vanmarcke
(1983).
A
probabilistic
model
is
presented
for
the
earth-
quake
induced
seismic
coefficient
and
the
risk
of
slope
failure
on
the
basis
of
several
important
assumptions.
PREDICTION
OF
FAILURE
DUE
TO
RAINFALL
Many
slope
failures
occur
as
a
consequence
of
prolonged
or
heavy
rainfall
and
techniques
must
be
developed
to
estimate
the
risk
of
slope
failure
associated
with
particular
duration
and
intensity
of
rainfall.
In
this
connection
an
interesting
concept
introduced
by
Mastsuo
and
Ueno
(1979)
deserves
mention
here.
This
concerns
the
rate
of
increase
of
the
failure
probability
p
f
which
may
be
obtained
by
plotting
the
failure
probability
pf
itself
against
time.
The
shape
of
this
curve
will
depend
on
the
rainfall
inten-
sity-time
plot
(hyetograph).
Examination
of
a
number
of
actual
case
records
of
slopes
in
Japan
was
made
where
sliding
either
occurred
or
did
not
occur.
Analysis
was
made
of
the
vertical
infiltration
of
rainfall
to
determine
the
ground
response
and
the
degree
of
saturation
of
a
slope
at
any
time
during
and
after
a
rainfall.
Using
rainfall
records
and
actual
soil
properties
to-
gether
with
the
infiltration
analysis,
probab-
ility
of
failure
was
plotted
at
different
times
after
the
start
of
rainfall.
The
changing
slope
of
p
f
vs.
time
curve
gives
the
transition
process
i.e.
the
rate
of
increase
of
probability
of
failure.
The
study
found
that
failure
was
always
225.
associated
with
the
maximum
slope
of
the
curve
i.e.
maximum
value
of
pf.
The
values
of
Pf(max)
for
the
non-sliding
cases
were
significantly
smaller
than
the
values
of
p
i
f
at
the
time
of
slide
in
the
sliding
cases.
Authors
reached
the
tenative
conclusion
that
p
.
f
is
closely
related
to
the
velocity
of
deformation
of
a
slope
and
further
experimental
and
theoretical
investigat-
ion
appeared
to
confirm
this
(Matsuo
and
Ueno,
1981).
CONCLUDING
REMARKS
In
this
paper
key
aspects
of
probabilistic
approaches
to
slopes
and
to
landslide
studies
have
been
highlighted.
Reference
has
been
made
to
recent
work
concerning
(a)
the
methodologies
which
are
used
in
practice
to
assess
failure
probability
and
reliability
(b)
the
uncertain-
ties
concerning
slopes
and
the
modelling
of
soil
variability
(c)
the
assessment
of
a
most
pro-
babe
extent
of
failure
-
which
one
is
able
to
do
only
if
soil
variability
is
properly
accounted
for
(d)
the
updating
of
future
risk
based
on
survival
during
a
stage
of
construction
or
during
an
earthquake
(e)
the
modelling
of
failure
pro-
gression
along
a
slip
surface
and
consideration
of
the
strain-softening
character
of
natural
earth
materials
(f)
the
probabilistic
consider-
ation
of
successive
failures,
etc.
Attention
has
been
concentrated
on
risk
assess-
ment
of
individual
slopes.
What
about
assess-
ment
of
large
areas
with
variable
landforms
and
geomorphological
features?
Urbanisation
of
sloping
land
has,
in
recent
years,
created
frequent
stability problems.
Therefore
it
would
be
valuable
to
assess
the
risk
associated
with
the
development
of
sloping
land
on
a
logical
basis.
This
would
require
a
consideration
of
many
slope
angles,
varying
water
table
or
pore
water
pressure,
spatial
variability
of
soils
within
the
region
of
interest
as
well
as
the
relative
size
of
areas
with
different
slope
inclinations
and
soil
properties.
One
approach
may
be
to
calculate
separately
the
probability
of
failure
of
different
zones
within
the
region
taking
into
consideration
all
the
available
statistical
information
on
soil
properties,
pore
water
pressure
and
other
factors.
The
overall
failure
probability
could
then
be
cal-
culated
as
a
weighted
average
of
the
various
local
failure
probabilities.
However,
there
may
be
pitfalls
in
this
approach
unless
consid-
eration
is
given
to
the
consequences
of
failure
of
each
zone
or
each
individual
slope.
For
example
location
may
be
relatively
small
com-
pared
to
other
locations;
yet
failure
of
that
slope
or
zone
may
cause
successive
landsliding
or
progressive
failure.
Again,
the
assessment
of
risk
of
large
areas
should
take
into
account
the
methods
used
for
development
and
the
rate
at
which
urbanisation
proceeds.
The
probab-
ilistic
methodology
for
risk
assessment
will
only
be
successful
if
geotechnical
factors
are
given
due
consideration.
Decision-making
under
uncertainty
can
be
facilitated
by
using
probabilistic
approaches.
For
example,
Haldar
(1980)
demonstrated
a
decision-analysis
framework
for
liquefaction
and
a
logical
approach
for
embankment
design
was
presented
by
Kuroda
and
Tang
(1979).
The
calculation
of
the
probability
of
failure
is
an
important
step
in
finding
the
optimum
solu-
tion
in
terms
of
a
design
variable
such
as
embankment
slope
inclination
or
in
finding
the
best
alternative
among
several
decision
choices.
This
approach
could
be
extended
to
planning
for
prevention
or
remedial
measures
for
landslide
areas
including
areas
where
some
failures
have
already
occurred.
First,
the
geotechnical
engineer
must
identify
the
various
choices
con-
cerning
the
management
of
a
given
area
with
particular
reference
to
its
present
and
potential
use
(e.g.
highway
construction,
residential
development,
recreational
area
etc.).
The
geo-
technical
engineer
must
then
assess
the
relia-
bility
of
each
alternative
solution
and
hence
the
probability
of
failure
as
well.
The
next
step
should
be
to
consider
the
consequences
of
failure
e.g.
damage
to
property,
loss
of
life,
the
costs
incurred
in
reconstruction
etc.
The
total
expected
cost
of
each
alternative
is
simply
a
sum
of
the
initial
cost
(comprising
construction
cost
and
associated
costs
such
as
any
land
that
has
to
be
acquired)
and
the
pro-
duct
of
the
failure
probability
and
the
failure
loss.
The
best
alternative
is
the
one
with
the
least
total
expected
cost.
There
may
be
pitfalls
in
this
approach
as
well.
For
instance,
how
can
one
adequately
reflect
loss
of
life
in
terms
of
dollars.
Nevertheless,
one
has
to
find
an
optimum
strategy
for
design
with
the
knowledge
that,
in
many
slope
problems,
complete
eliminat-
ion
of
failure
is
simply
impossible.
For
example
consider
a
highway
in
a
hilly
region.
There
will
be
choices
which
are
high
in
initial
cost
and
which
lead
to
low
maintenance
and
there
will
be
other
choices
which
are
relatively
low
in
initial
cost
but
lead
to
expensive
maintenance.
A
similar
situation
may
apply
to
other
develop-
ments,
for
instance,
embankment
construction
for
flood
control.
The
geotechnical
engineer
must
make
a
decision
considering
both
types
of
design
and
the
probabilistic
methodology
appears
to
be
a
logical
basis
for
such
decision
making.
Nevertheless,
there
are
situations
in
which
the
avoidance
of
failure
should
be
the
overriding
consideration
e.g.
construction
of
an
earth
dam
(De
Mello,
1977)
the
failure
of
which
could
lead
to
loss
of
life
and
property
on
a
large
scale.
Of
course,
even
if
design
measures
are
intended
to
eliminate
failure
at
all
costs,
there
will
always
be
some
probability
of
failure
due
to
unforeseen
events
or
combination
of
events
or
due
to
unknown
geological
details
or
due
to
unintended
mistakes
in
design
or
construction.
Thus
while
one
may
plan
for
minimum
failure
risk
(rather
than
minimum
total
expected
cost),
100%
probability
of
success
can
not
be
guaran-
teed.
In
areas
susceptible
to
earthquakes,
planning
for
an
important
structure
or
facility
requires
considering
of
failure
due
to
various
causes.
An
interesting
model
for
earthquake-induced
landslides
has
been
presented
by
Keeney(1980)
in
respect
of
a
site
which
is
near
several
faults
(the
number
of
faults
being
N)
each
of
which
may
be
involved
in
the
occurrence
of
an
earthquake.
The
aim
is
to
assess
the
risk
of
an
accident
at
the
given
sloping
site
due
to
landsliding
caused
by
activity
on
any
of
these
faults.
The
probability
that
an
earthquake
on
fault
i
would
lead
to
a
landslide
at
the
site,
denoted
by
P
i
pay
be
defined
as
follows:
P.=.r
M
f.(M)F.(M)
dM
,
i
=
1,
..,N
(23)
226.
in
which,
fi(M)
is
probability
density
function
for
the
largest
magnitude
earthquake
M
to
occur
during
the
life
of
the
project
on
the
ith
fault,
and,
Pi(M)
is
the
probability
that
a
slide
would
occur
at
the
site
due
to
a
magnitude
M
earthquake
on
the
ith
fault.
This
probability
may
be
cal-
culated
on
the
basis
of
data
about
the
shaking
caused
by
various
earthquakes,
slope
stability
and
professional
judgement
about
the
relation-
ship
between
shaking
and
the
occurrence
of
landslides.
The
overall
probability
P
of
a
slide
is
one
minus
the
probability
that
no
slide
will
occur
i.e.
P
=
1
-
[(1-P
1
)(1-P
2
)
....
(1-P
N
)]
(24)
A
case
history
of
a
geotechnical
power
plant
was
presented
by
Keeney
(1980)
to
illustrate
the
above
procedure.
The
functions
fi
and
Pi
were
developed
by
assuming
that
the
most
significant
parameters
affecting
sliding
are
the
earthquake
magnitudes,
the
peak
site
acceleration
and
the
duration
of
site
ground
motions.
The
resulting
probabilities
of
failure
due
to
earthquake-
induced
landsliding
for
the
30
year
project
life
of
the
plant
were
calculated
as
p
l
=
0.007
for
San
Andreas
fault
p
2
=
0.003
for
Maakama
fault
p
3
=
0.002
for
Healdsburg-Roolgers
Creek
The
overall
probability.
of
a
slide
in
the
30
year
period
was
thus
calculated
to
be
0.012.
Finally,
some
comments
on
the
limitations
of
probabilistic
approaches.
The
main
misgiving
concerns
the
choice
of
probability
distribut-
ions.
As
has
already
been
stated
the
calculated
risk
depends
on
the
assumed
probability
dis-
tribution
especially
for
low
risk
projects.
Again
the
absolute
value
of
risk
depends
on
the
methodology
used
for
calculation
and
for
up-
dating
of
risk
based
on
observation.
One
must,
therefore,
be
wary
of
attaching
overwhelming
importance
to
the
absolute
values
of
calculated
magnitude
of
risk.
It
is
the
comparison
of
calculated
risk
for
different
alternatives
that
is
really
important.
Again
the
sensitivity
of
calculated
risk
to
different
parameters
should
be
given
due
consideration.
Comparative
and
sensitivity
studies
based
on
the
probability
of
failure
are
likely
00
be
more
valuable
than
those
based
on
the
convent-
ional
safety
factor.
Whether
the
extra
effort
and
cost
that
may
be
involved
for
imple-
menting
the
probabilistic
approach
is
justified
is
a
metter
to
be
considered
by
the
geotechnical
engineer.
A
detailed
consideration
of
this
aspect
is
outside
the
scope
of
this
paper.
Clearly,
however,
cost
of
analyses
can
be
a
limitation.
The
most
important
limitation
may
be
the
lack
of
statistical
data
about
soil
properties
and
pore
water
pressures
and
about
loads.
Analyses
which
are
not
based
on
reliable
data
are
of
questionable
value
and
could,
in
some
instances,
prove
to
be
misleading.
Therefore,
it
is
of
the
utmost
importance,
that
research
funding
for
soil
exploration
and
statistical
analysis
of
data
should
increase.
Geotechnical
engineers
should
be
encouraged
to
carry
out
extensive
site
investigation
and
to
use
available
techniques
for
determining
distributions
of
soil
properties
and
of
other
relevant
parameters,
and
for
development
of
probabilistic
soil
profiles.
ACKNOWLEDGEMENTS
Geotechnical
studies
at
the
University
of
Wollongong
under
the
direction
of
the
writer
have
been
supported
by
the
Research
Grants
Committee
and
also
by
external
bodies
such
as
the
Australian
Research
Grants
Committee.
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D,
(1981).
How
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present
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slope
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Sweden,
June,
Paper
11/20,
Vol.3,
427-430.
A-Grivas,
D.,
and
Asaoka,
A.
(1982).
Slope
safety
prediction
under
static
and
seismic
loads,
ASCE,
J.Geotech.Eng.Div.,
108,
GT5,
713-729.
Alonso,
E.E.
(1976).
Risk
analysis
of
slopes
and
its
application
to
slopes
in
Canadian
sensitive
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Geotechnique,
Vol.26,
No.3,
453-472.
Asaoka,
A.,
and
A-Grivas,
D.
(1982).
Spatial
variability
of
undrained
strength
of
clays,
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108,
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A.H-S,
and
Tang,
W.H.
(1975).
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Engineering
Planning
and
Design -
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John
Wiley
&
Sons,
New
York.
Baecher,
G.B.,
Chen,
M.,
Ingra,
T.S.,
Lee,
T.,
and
Nucci,
L.R.
(1980).
Geotechnical
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offshore
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MIT
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Sea
Grant
Program,
MIT,
Cambridge
(Mass),
1980.
Baecher,
G.B.,
and
Einstein,
H.H.
(1978).
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models
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pit
optimisation,
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APCOM
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Bishop,
A.W. (1948).
Some
factors
involved
in
the
design
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a
large
earth
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the
Thames
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Proc.
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and
Foundation
Engineering,
Rotterdam,
June,
Paper
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R.N.
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Elsev-
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Amsterdam
and
New
York,
pp.424.
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R.N.
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natural
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