--
~-
-
--
Barbosa, M. R., Morris, D. V. & Sarma, S. V. (1989). Geotechnique, 39, No.3, 471-483
Factor of safety and probability of failure
of rockfill embankments
M. R. BARBOSA.,D. V. MORRIS. and S. K. SARMAt
The limit equilibrium method is applied to aDalyse
the stability of rockfill embaDkmeDts. Stability
charts that utilize CODtOursof zero critical acceleratioD factor are geDerated. These charts are valuable as design aids aDd are preseDted iD a form
ameDable for the direct computatioD of the factor
of safety, which is a more familiar parameter of
stability aDalysis. Reliability theory is theD applied
iD the form of a secoDd momeDt format, to
compute the probability of failure of a rockfill
embaDkmeDt. StreDgth parameters are characterized usiDg three slightly differeDt density fUDctions,
to study the effect of truDcatioD ODthe probability
of failure of the system. The Use of the logistic distributioD is iDtroduced, as a means of computiDg
the failure probability of the structure without the
aid of tables. The parameters Decessary to compute
both the factor of safety aDd its reliability, have
beeD evaluated aDd are preseDted iD graphical
form, iD terms of commODdesigD variables. A wide
spectrum of practical combiDations of geometries
aDd streDgth parameters are covered by the study.
KEYWORDS: dams; failure; limit state design!
analysis; statistical analysis.
La methode de I'equilibre limite est employee pour
aDalyser la stabilite des remblais rocheux. OD preseDte des abaques de stabilite qui utiliseDt des CODtours de facteur zero d'acceleratioD critique. Ces
abaques SODt utilisables pour I'etablissemeDt de
projets et preDDeDtUDeforme qui se prete au calcul
direct du facteur de securite, qui est UDparametre
mieux CODDU
dans l'aDalyse de stabilite. La theorie
de la fiabilite s'applique alors comme un deuxieme
moment de force pour calculer la probabilite de
rupture d'un remblai rocheux. Des parametres de
resistance SODtcaracterises Ii I'aide de trois fonctions de densitie on peu differeDtes, afiD d'etudier
I'effect de la troDcature sur la probabilite de
rupture du systeme. On introduit I'emploi de la distribution logistique comme moyen de calculer la
probabilite de rupture de la construction sans I'aide
de d'abaques. Les parametres Decessaires pour calcoler Ie facteur de securlte et sa fiabilite ont ete
evalues et sont presentes en forme de diagrammes
seloD des variables ordinaires de constrUCtiOD.
L'etude compreDd one gamme eteDdue de combiDaisons pratiques de geometries avec des para- .
metres de resistance.
NOTATION
ao constant, intercept of the core strength
aXIs
at constant, slope of the design curves/
failure functions
a2 constant, equal to - 1.
CO Vt coefficient of variation on the rockfill
strength
COV2 coefficient of variation on the core
strength
Cwt weight of wedge 1 in terms of hi H2
Cw2 weight of wedge 2 in terms of 1YtH2
c' shear strength parameter in terms of
C2' effective cohesion along core section of
slip plane 2
Cu undrained shear strength
d length of interslice failure plane
E normal force acting between wedges I,
and 2
F. factor of safety
H height of the embankment
Kc critical acceleration factor
It length of slip plane 1
12 length of core section of slip plane 2
13 total length of slip plane 2
M function of a vector random variable
mt failure function
Nt normal force acting along the base of
wedge 1
PF water force acting between wedges 1 and
2
effective stress
c' effective cohesion along interslice failure
plane
Ct' effectivecohesion along slip plane 1
Pf probabilityoffailure
S shear force acting between wedges 1 and
2
Tt shear force acting along the base of
wedge 1
Discussion on this Paper closes 5 January 1990; for
further details see p. ii.
· Texas A & M University.
t Imperial College, London.
471
472
BARBOSA, MORRIS AND SARMA
T2', shear force component along core section
of wedge 2
T."
2
shear force component along rockfin
section of wedge 2
U 1 water force acting along the base of
wedge 1
U2' water force component along core section
of wedge 2
U"2 water force component along rockfin
section of wedge 2
WI
weight of wedge 1
W2 weight of wedge 2
Xl
random variable representing the rockfin
strength
X2
random variable representing the core
strength
(Xl angle defining inclination of slip plane 1
(X2 angle defining inclination of slip plane 2
P reliability index
PI angle of inclination of embankment
(upstream side)
P2 core inclination angle (upstream side)
P3 angle of inclination of embankment
(downstream side)
P4 core inclination angle (downstream side)
() inclination of interslice failure plane
')11 unit weight of rockfill material
')12 unit weight of core material
IlM expected value of the failure function
III
mean value of the rockfill strength
112 mean value of the core strength
ell( . ) normal distribution function
,p' effective angle of shearing resistance
,pI' rockfill effective angle of shearing resistance
A,.'
core effective angle of shearing resistance
'f' 22
variance of the failure function
O'M2
0'1
variance of the rockfill strength
0'/
variance of the core strength
INTRODucnON
The use of limit equilibrium analysis is well established in studies of earthwork failures (Skempton
& Hutchinson, 1969; Chandler, 1984),but further
development is by no means at an end. One
variation that offers economy in computation for
slope and embankment stability is that proposed
by Sarma (1973),utilizing the 'critical acceleration
factor' Kc to determine the horizontal acceleration (as a fraction of gravity) just required to
bring the structure into limiting equilibrium. The
method is equivalent in degree of rigour to the
rigorous Bishop method of slices, although it
requires less iteration to arrive at an acceptable
result. Such an analysis is also particularly suitable for probabilistic evaluation of safety, if
adapted in accordance with statistical principles.
This Paper demonstrates how this can be done
efficiently,for the case of rockfill embankments,
which it is hoped will be of interest to designers.
An efficient probabilistic procedure is significant, in view of the trend in engineering analysis
to move away from consideration of a single
factor of safety, with which uncertainties can only
be treated in a superficial fashion, and design criteria cannot really be considered in a systematic
and explicit way (Baecher, 1973a; Ellingwood &
Galambos, 1982). Originally variations of the
Monte Carlo method were first used as a relatively crude and expensive (if powerful) procedure
to produce predictions of the probability of
failure, but the development of reliability theory
has generated more sophisticated procedures. A
number of general analyses of slopes and
embankments that use probability have been
published (Priest & Brown, 1983; Ramachandran
& Hosking, 1985; Vanmarcke, 1977 and 1980;
Veneziano, 1983),in some cases with closed form
solutions.
This Paper applies both limit equilibrium
analysis and reliability theory to generate charts
of the probability of failure of rockfill embankments, using the second moment method for
direct computation. The methodology is simple
enough to be applied in a design situation by an
engineer who may only be acquainted with basic
statistical and probability principles.
Although the technique is of general applicability to any dam cross-section, the present study
covers only rockfill embankments with central
clay cores under static (not seismic) stability (i.e.
zero critical acceleration). Results are presented in
the form of stability charts, which can easily represent different embankment geometries, and are
a convenient way of evaluating preliminary
designs.
METHODOF ANALYSIS
The method of analysis utilized in this study
assumes a failure mechanism composed of two
rigid wedges sliding along planar surfaces, as
shown in Fig. 1. The first experimental evidence
concerning the validity of this mechanism was
published by Sultan & Seed (1967),who, in order
H
Fig. 1. Failure mechanism for a rocldUl embankment
-
---
--
ROCKFILL
EMBANKMENT
to investigate the behaviour of sloping core dams,
carried out some model tests with an embankment constructed of cohesionless material overlying an impervious core. As shown by Sultan &
Seed the mode of failure of all cross-sections
tested was observed to follow a regular pattern,
consisting of two wedgessliding along planar surfaces. The inclination of the interslice plane was
found to be around 35° from the vertical. Subsequent work has not resulted in the determination of any significantly different mechanisms,
and this particular configuration appears to have
established itself as the most appropriate one for
the analysis of this type of embankment.
The force diagram acting on each of the two
wedgesis shown in Fig. 2. An additional horizontal force is included to simulate inertia effects.
This force is usually expressed in terms of the
weight of the material and a factor called the
critical acceleration factor Kc. By definition, Kc is
the acceleration, given as a fraction of the acceleration of gravity, which produces a state of limiting equilibrium (Sarma, 1973). This concept is
valid under either static or dynamic stability.
Using the dimensionless parameters')' = ')'2/Yl'
D = d/H, Ll = 11/H, L2 = 12/H, L3 = 13/H, e =
e'/')'IH, Ct = Cl'/')'IH, C2= C2'/')'IH,~ = W2/Wl,
Cwl = Wl(!')'IH2), Cw2= W2(t'YIH2), it can be
(Xs.Ys)
PF--
~
KcWI
~E
S
-:J
Id
(0,0)
~
\
NI\ UI
r;
'=(N,-U,)tanf;
~'= (E-PF)tanq>
(a)
(XI.YI)
(0.0)
(b)
Fig. 2. Force diagram for sOdingwedges:(a) wedge1;
(b) wedge2
473
STABILITY
shown that in general, for a rockfill embankment,
Kc may be computed from
Kc = (81 + 82 + 83 + 84 + 8s)/86.
where 81
82
83
=
(1)
R1
= R3R4 ~
= (2ed - PF
-
tan c,6')[(R4 1) sin ~
- (R1 - R3 R4) cos 15]/Cwl
84 = (2c1L1 - U tan cPl')(COS 01:1
- Rl sin 0I:1)/CWI
8s = (2c2 L2 - U2' tan cP2'+ 2cl'(L3 - U2" tan cPt')[R4(COS 01:2
1
86
Rt
R2
- R3 sin 0I:2)]/Cw1
= 1 + R4 ~
= tan (cPt' - 01:1)
= 12/13
tan cPt' + R2(2 - R2)
x tan cP2'- tan 01:2
1 + (1 - R2)2 tan cPl' tan 01:2
+ R2(2 - R2) tan cP2' tan 01:2
(1
R3=
R4
L2)
-
= 11 --
R2)2
--
Rt tan (c,6' 15)
R3 tan (cP'-15)
The failure mechanism assumes that normal
stresses along the failure planes are linearly distributed. This hypothesis results in a more conservative approach for the same geometry and
material properties compared with a previous
study carried out by Sarma and Barbosa (1985)
where a constant normal stress was assumed to
act along failure plane 2.
Equation (1) can be used to analyse rockfill
dams with either a central or sloping core section.
It is a rigorous solution in the sense that it
satisfies conditions for static equilibrium. Kc
depends only on the material properties and the
geometrical characteristics of the embankment
and failure mechanism. Equation (1) can also be .
used to compute the factor of safety of the structure. By definition, Kc = 0 represents the value of
unity for the factor of safety. To arrive at this
condition, generally three or four trials are
required, in which the strength of the materials
comprising the cross-sectionis reduced by a
factor representing the factor of safety Fa.
Graphical solutions to evaluate F. become therefore unnecessary, and what is more important,
the most critical inclination of the failure planes is
found as part of the solution.
This analysis assumes no drainage or dissipation of excess pore water pressures in the
impervious core of the dam. Thus, a total stress
approach, based on the undrained shear strength
of the core material, will be adequate. The rockfill, however, will have rates of dissipation of pore
pressures so great that no significant build-up are
.
474
BARBOSA, MORRIS AND SARMA
expected. Therefore, it seems also adequate to
assume that this part of the embankment will
behave as if composed of a fully drained material,
whose strength can be conveniently described by
its angle of shearing resistance (assuming c1' = 0).
Thus the terms S3' S4 and Ss in equation (1) can
be simplified to
S3
= (2cD)[(R4 - 1) sin ~
- (R1 -
cot{3, = 1.50
{32= 55°
Kc= 0.00
0.Q8
~
~
.8-
Failure function for critical
mechanism
controlled by <5< (90° -{32)
r.
i5>
c:
!!!0.04
1ij
.,
is
u
R3 R4) cos ~]/C...1
S4 =0
Ss = (2c2L2)[R4(cos(X2- R3 sin (X2)]/C...1
The factor R3 can now simply be written as tan
(0 0(2)'where 0 is given by tan -1 [(1 - RZ)2 tan
4>1'] and the expression for the dimensionless
parameter C2becomes cJYIH.
-
STABILITYCHARTS
Because the traditional analysis of slopes and
embankments is a costly and time consuming
operation, simplifiedprocedures have always provided a significant aid in the selection Of initial
design alternatives. This has motivated the development of stability charts, which by their very
nature are based on different hypotheses and
account for different field conditions (Taylor,
1937; Bishop & Morgenstern, 1960; Hunter &
Schuster, 1968;Charles & Soares, 1984).
The stability charts developed in the present
study have some advantages over previous ones.
From an analytical standpoint, the charts permit
the evaluation of the' factor of safety of the structure, defined as that factor by which the strength
must be reduced in order to reach a state of limiting equilibrium. In addition, as shall be seen later,
the charts may be used to determine the probability associated with F,. For failure conditions, the
factor of safety to be considered is unity, and the
curves presented herein apply directly. It is possible, however, to treat any other condition in a
similar manner.
In a design situation, an inverse procedure may
be applied. After a decision is reached as to the
level of acceptable risk, several geometries can be
easily investigated to determine the best alternative without having to perform the usual stability
calculations in each case. This eases the entire
selection process and permits the designer to concentrate on evaluating the economic factors for
the assessment of the total risk involved.
As shown in Fig. 3, the selected configuration
consists of a curve, defined in terms of dimensionless strength parameters, that describes the
behaviour of a particular cross-section under a
constant critical acceleration factor. The design
curve is an envelope of two possible valid failure
------o0.2
0.6
1
Rockfill strength
1.4
1.8
tan 4>;
Fig. 3. Envelopeshowing two failure modes of rockfill
embankment
mechanisms-valid in the sense that no tension is
implied in any of the points where the soil
strength is verified (i.e. failure planes). The difference between the two mechanisms lies in the position of the interslice failure plane. The upper part
of the envelope relates to a mechanism which has
this plane included entirely within the rockfill
portion of the embankment (i.e. ~ < (90 - P2»'
whereas in the lower part, the interslice failure
plane has a fixed position located at the boundary between the rockfill and core materials (i.e.
~ = (90 - P2»'
The upper part of the envelope can be
described by a straight line. This behaviour has
been observed regardless of the geometry of the
embankment or level of the critical acceleration.
The lower part of the envelope, however, tends to
have a slight curvature, concave upwards. This
has been observed to be the case when dealing
with high or low values of rockfill strength
parameters. However, a linear approximation for
the lower part of the envelope is not unreason~
able. In the present procedure, no graphical solutions are applied, other than the representation of
the curve, which is generated numerically. Therefore, the analysis of wedge mechanisms by
resolution of forces and the application of iterative processes for the computation of the factor
of safety become unnecessary. More important
perhaps is the fact that no restrictions are
imposed on the configuration of the slip surface.
A set of curves has been generated for those
rockfill embankments most often encountered in
practice, namely
cot Pl = 1.00, 1,25, 1,50, 1,75,2,00,2,25,2,50
P2 = 30 -+ 70° (as applicable)
Figure 4 shows the curves applicable to the
analysis of a rockfill embankment under static
conditions. These curves represent contours of
ROCKFILL
EMBANKMENT
475
STABILITY
oolP, = 1.00
Kc = 0.00
0.08
OOlp, = 1.25
Kc = 0.00
J:
~o
.s::.
'6>
c
!!! 0.04
P2= 55°
u;
!!!
o
60°
()
I
1
I
55°
60°
65°
l8Op'\1
o
0.2
1-4
1
I
0.6
1.8
0.2
1-4
OOIP, = 1.50
Kc = 0.00
oolp, = 1.75
Kc = 0.00
0.08
J:
>:.
-;s
.s::.
'6>
c
!!! 0.04
u;
P2= 40"
~
45°
()
I'
I
I
1
lanp,
o
0.2
1.8
(a)
\! ~
l8Op,
6;--60"
0.6
1.4
~
1.8
(e)
0.2
112= 40"
~
'"
\ !~
!
0.6
I
45°
55°
65°60"
~
1.4
1.8
(d)
0.08
OOlp, = 2.25
Kc = 0.00
oolP, = 2.00
Ke= 0.00
J:
~o
:;C>
~0'04
u;
CD
P2= 35°
I
<3
40"
0' ""P\~55"'5O"
0.2
0.6
1-4
1
(e)
P2= 35°
1.8
1
1-4
Rockfillstrength Ian .pi
(I)
1.8
0.08
ootp, = 2.50
Kc = 0.00
J:
~o
:;
g'0.04
!!!
u;
!!!
P2= 30°
I
8
l8Op,
O'
0.2
~
35°
"""
\ : 550 50°
~
I
Fig. 4. Stability
I
0.6
charts
45°
1-4
1
Rockfillstrength Ian .pi
(9)
1.8
whenKe= 0: (a) cot 'I = 1-(M);
(b) cot 'I = 1'25;(c)cot 'I = 1'50; (d)cot'I = 1'75; (e) cot
'I = 2'00; (0 cot 'I = 2'25; (g) cot 'I = 2'50
476
BARBOSA, MORRIS AND SARMA
Kc = 0 (i.e. factor of safety of unity). Each point
on the curves is associated with a combination of
strength parameters that yield a condition of limiting equilibrium, thus they can also be identified
as failure functions. Furthermore, each point in
the curve is related to the most critical failure
mechanism that would be encountered for the
particular problem under consideration. This
eliminates the need to include several geometrical
parameters-mostly dealing with the configuration of the slip surface-in the presentation of
the results.
Any embankment slope or core inclination that
falls within the range of values considered in this
study, but not explicitly included in the present
set of charts, can be investigated by a simple
interpolation technique. The linear equations
describing the upper and lower portions of the
envelope are, respectively
aou+ atU tan 4>t'+ a2(cJY1H)= 0
(2a)
aot + att tan 4>t'+ a2(cJYtH) = 0
(2b)
where aou, ao1, at u, atl and a2 are constants
determined numerically. Geometrically, ao and al
represent simply the ordinate intercept and the
Core
20
o
incfination
angle
40
{3'2
80
60
0.5
!
,o
-0,2
III
Q;
a.
a.
20.0.3
CD
,
.a
ia. -0.4
a.
2CD
a.
-0.6
~
slope of a straight line (a2 being equal to -1).
Figs 5 and 6 give information concerning the
values of these constants for the two portions that
form the failure envelope. These values and those
relevant to the co-ordinates for the point of intersection of the two failure envelopes (Fig. 7)
provide sufficient information to define practically any cross-section not covered by the present
study. Except for the intercept values for the
lower part of the failure envelope, there seems to
be a consistent relationship with the core inclination angle P2' In the case of the former, only
upper and lower limits are given because the
parameter ao1 does not appear to be sensitive to
the embankment slope cot Pl'
Under static conditions, curves can be prepared for core inclinations up to about 50-70°the higher values relating to steeper embankment
slopes. This can be seen in Fig. 5, which shows
the relation of the slope of the upper and lower
portions of the failure envelopes. For design situations in which a steeper core is employed, the
results of the analysis indicate that the limiting
case of shear sliding along the face of the
embankment may be more appropriate, in which
case the failure function is described by a vertical
~
~
~
~
~<9.
ib'<J'
~
!!
.s
>
~~
a:"'d''<J.
it. '3- ~o '" 1:>
-0.8
0.1
(a)
(a)
o
0.08
-0
III
~_o-
Q;
-0.02
~
C.
0.06
i -0.04
Q;
~
1.50
o8. -0.06
.5
>
..."
8
:
!~\I1'\\8
2
0
0 ___D__O
:J ~....--
2
11<_el\\I1'\\
/-"""0
\,..U..
.....-
o.o~o
Ci5
40
60
80
Core inclination angle P20
(b)
-0.08
Fig. 6. Core streogtb iDten:eptagainst '2: <a)for upper
envelope;(b) for lowerenvelope
-0.1
(b)
Fig. 5. Stability charts: <a) for upper envelope;(b) for
lower envelope
U9~
~
!! 0.04
ROCKFILL
EMBANKMENT
477
STABILITY
1.8
I
:e:
c
!!
~1-4
c
cotp,= 1.00
~
8
='"
c
1
~
1n
= 0.6
~
a:
0.2
(a)
:t:
.;:;
8
s=
c;,
c
~
1n0.02
~
20
~
~:'=1.00
~ 0.04
c;;
c
=e
1.50
:.~~~
:--:::::
40
8
60
Core inclination angle
(b)
80
P20
Fig. 7. Strengtb co-ordinate against 112for point of intersection of two envelopes: <.) for rocldill; (b) for core
tainties were not considered, except for the fact
that most designers would work with averages or
perhaps, values on the lower side of the mean, if
more conservatism were deemed necessary. To
complement the analysis, reliability theory can be
applied, thus providing a more rational basis for
selectingdesign parameters.
In principle, it is required to find a failure
surface,as a function of random and deterministic
variables, that divides the basic variable space
into two regions (Fig. 8): a failure region, which
contains all the combinations of random variables that would result in failure (Le.area where
M < m.), and a safe region, which contains all the
combinations of random variables that would not
result in failure (Le.area where M > m.).
The basic variable space is composed of geometric quantities, material strength parameters
and load configurations. If all these are combined
and expressed in an analytical model that is used
in the stability analysis, it is possible to find the
expression for the failure surface. The level of
complexity of the problem will dictate the procedure to be followed in the definition of. the
failure function. In some cases,it may be an easily
line that intersects the rockfill strength axis at tan
P. (shown by the broken line in Figs 4(a-g). The
stability of the cross-section may then simply be
assessed from the equivalent analysis of an infinite slope composed of granular material. In other
words, the factor of safety is given by
F.
= tan
q,.'/tan P.
M = function of random
variables
M=B2X2+B,X, +ao
(3)
BASICCONCEPTSOF RELIABILITYTHEORY
In the case of stability of slopes, the current
object is to determine the relation between the
factor of safety and the probability that its value
may fall below a certain assigned value (usually
unity).
The stability charts that were presented in the
preceding section can be used to determine the
factor of safety, provided that the geometry of the
embankment and the strength of the materials are
known. As each curve represents a condition of
limiting equilibrium the factor of safety is found
by first locating the point defined by the mean
strength value of the rockfill stength and that of
the core expressed as the ratio cJy.H (P. and P2,
for instance). A segment is then drawn from this
point to the origin and the co-ordinates of the
M>m,
M<m,
Failure region
Safe region
Sx,X S.2
x,
point intersectingthe curveare obtained(P3 and
P4)' The factor of safety is the ratio P./P 3 (or
P2IP4).
Up to this point the analysis is in principle
deterministic because model or parameter uncer-
Sm
m,
.
Fig. 8. Representation of Hnear fallore function
m
478
BARBOSA, MORRIS AND SARMA
determined explicit function whereas in others
(such as in the analysis of slopes and embankments) the failure function may have to be defined
numerically (Parkinson, 1978).
From the analytical formulation of the problem,
an expression can emerge for the joint density
function that describes the failure function, from
which the probability of failure can be calculated.
However, because of the complexity of evaluating
an integral that involves a multiple joint density
function and the frequent lack of knowledge of
the relation between the variables involved, only
the Monte Carlo methods and the level 2
methods can be used to determine the failure
probability. Descriptions of these approaches can
be found in the current literature (e.g. Baker,
1983; Ramachandran, 1984; Ayyub & Haldar,
1984).
In spite of their simplicity, Monte Carlo techniques do not attract much interest because of the
relatively large number of trials that are sometimes necessary to reach a satisfactory result.
Level 2 methods, however, provide a good alternative and when properly used, can be extremely
valuable for comparing different design configurations.
A level 2 method makes use of a given failure
function to compute the probability of failure
from
(4)
Pr=4.>(-P)
where, 4.>(.)is the normal distribution function,
for which tables are readily available. The parameter P is commonly known as the reliability
index, and is useful for characterizing the degree
of safety (Whitman, 1984). There are currently
two basic approaches that can be used to evaluate p: the first-order second moment method and
the advanced second moment method.
The two methods are based on the knowledge
of the first and second moments (i.e. mean and
variance) of the distribution of the random variables. The first-order seCondmoment method utilizes a Taylor series expansion of the failure
function about the mean values of the random
variables, and a truncation point at the linear
terms. Assuming independent variables, the
approximate expressions for the mean JlMand
variance UM2of the failure function, respectively
are
L (a,Jl,)+ ao,
UM2~ L (a,u,)2,
JlM~
i = 1, n
(5)
i = 1, n
(6)
where ao, a, are constants. The argument of <I>in
equation (4) then becomes P = JlMiUMin which
case P represents the distance (given as a number
of standard deviations) that separates the mean
from the origin. For most engineering applica-
tions, the linear approximation will be sufficient
to arrive at reasonable estimates of the probability of failure. This method will yield the correct
result only if the failure function is linear and is
composed of normally distributed random variables. In the particular case of rock slopes, where
a circular failure mechanism may prevail, the
assumption of linearity does not appear to be
valid. Hoek & Bray (1981) showed that for this
case, the failure function is non-linear. Moreover,
the method does not provide for the treatment of
non-normally distributed variables. The method
also yields different results for different mechanically equivalent formulations of the same problem
(Ayyub & Haldar, 19~4; Thoft-Christensen &
Baker, 1982).
Alternatively, in the advanced second moment
method (Hasofer & Lind, 1974; Rackwitz &
Fiessler, 1978) the Taylor series is expanded
about a point on the failure surface, called the
design or checking point, which gives a good
approximation of the probability content of
the failure region in the original variable space
(Ramachandran, 1984).The method also provides
for cases where the failure function is non-linear
and the variables are non-normal, and various
algorithms have been developed to account for
these conditions (Rackwitz & Fiessler, 1978).The
method also has the advantage that the problem
is treated with use of an invariant formulation.
CHARACfERIZA
nON OF STRENGTH
PARAMETERS
The uncertainties that surround soil properties
and give origin to their randomness, come from
sources such as geology, mathematical model,
material properties (e.g. strength, unit weights
and so on) and loads (Baecher, 1983b). Of all
these sources, perhaps the most important is the
strength parameter uncertainty. It has been
shown (Lumb, 1966; 1970), that the normal
density is an adequate function for description of
the strength properties of a soil. Subsequent
studies (Schulze, 1971; Lumb, 1974; Matsuo &
Asaoka, 1977)tend to support the validity of the
normal density function for modelling shear
strength (particularly when only one component
is involved) with the additional advantage of
having a constant coefficient of variation with
depth (Matsuo & Asaoka 1976, 1977).Soils with
two shear strength components are sometimes
better modelled by means of the beta density
function (Harrop-Williams, 1986).
When appropriate, the normal or Gaussian
density function is one of the simplest forms to
describe test data and is defined by two well
known and readily obtainable parameters: the
ROCKFILL
EMBANKMENT
479
STABILITY
0.60
-4.0
2.0
4.0
Fig. 9. Density functions (after Jenkins, 1982)
mean and the standard deviation. The most
common objection, however, is that strength
properties, particularly those pertaining to soils,
cannot, on physical grounds, take negative values.
It is possible therefore, that errors may be introduced by describing the rockfill or core strengths
by an unbounded normal density function. This
becomes more critical when mode values are low
(for materials of low resistance),or when working
with a large scatter of data. As these two situations may arise in the design of an embankment,
it is of interest to provide some evidence that the
normality condition represents a reasonable
assumption.
The evaluation of probabilities associated with
a normal density function requires numerical
integration or the use of tables. Even though
these tables can be found in most textbooks that
deal with basic statistics, the lack of a closed form
solution to determine the cumulative normal distribution function hinders the development of a
complete reliability analysis. One way to overcome this is to assume a similar distribution for
characterizing soil strength parameters and is
capable of providing similar results to those
obtained by using the Gaussian curve without
significantloss of accuracy.
A frequency distribution that fits this category
is the so-called logistic density function. This
function has been used to characterize a variety
of processes but is seldom encountered in engineering applications, due in part perhaps to its
relatively low exposure in this field and the determination of the parameters that define it.
The logistic distribution has been used to
analyse life test data (Plakett, 1959),in the Saturn
S-II programme (Peterson, 1965) and has also
been applied to study system maintainability
modelling (Requlinski, 1970). A comparison
between the standardized normal and the stan-
dardized logistic density functions is shown in
Fig. 9. In terms of one random variable, the logistic density function is defined as
J; (x)
x
=
1t(exp[ -1t(x - J.L)/(f.J3])
(f.J3(1
+ exp [ -
1t(x
(7)
- J.L)/(f.J3])2
and its cumulative distribution function is given
by
1
Fx(x) =
(8)
1 + exp [ - 1t(x- J.L)/(f.J3]
where J.Land (f are the mean and standard deviation of the logistic density function. Standard
procedures are available to compute J.Land (f for
the logistic curve. This is done by least squares
(Pearl &. Reed, 1920; Berkson, 1944), using
maximum-likelihood techniques (Wilson &
Worcester, 1943; Berkson, 1957; Plackett, 1958;
Harter & Moore, 1967), working with the
minimax estimator (Berkson & Hodges, 1961)
and also by using the minimum-chi-square technique (Berkson, 1955). With the appropriate
choice of parameters, the logistic density function
has the interesting property of yielding probability values practically equal to those of the normal
density function. This is especially true when the
logistic function is forced to have the same
maximum ordinate as the normal function, as
shown in Fig. 10. It transpires that a good
approximation of probability can be obtained by
using the parameters of the normal curve for the
logistic density function. A further advantage of
the logistic function is that it provides a means of
evaluating probabilities in a closed form fashion.
The logistic density function can also be modified to represent physical quantities that have
only positive values associated with them, such as
the strength of soils. In these cases, a truncated
logistic distribution may be more appropriate to
480
BARBOSA, MORRIS AND SARMA
0-60
-4-0
-2-0
Fig. 10. Density fUDCtiollSwith p
4.0
= 0-0
and (J
model the data, and a thorough study of this
function has been carried out by Jenkins (1982).It
was demonstrated that truncation of the logistic
density function is unnecessary unless small
values of failure probabilities are being considered for design. The probability of failure must
be of the order of 0,05 or less before significant
differences become apparent between the truncated and untruncated functions. For higher
failure probabilities there is neglible variation in
the final results, and easier use has meant that the
unbounded distribution is usually preferred, as
has been the case in this study.
SAFETYAND RELIABILITYOF
ROCKFILLEMBANKMENTS
At basic levels of reliability analysis, a failure
function (or safety margin) for an embankment
can be defined in terms of the strength parameters
of the materials that comprise the cross-section.
As was shown previously, the curves of Kc = 0
(i.e.F. = 1)serve the purpose of a failure function.
For a rockfill embankment, it is possible to
approximate failure modes by two linear functions obtained from a numerical analysis and
described using linear regression. These linear
functions are defined in terms of the rockfill
strength, tan 4Jt', and the undrained core strength
expressed in the form of a dimensionless parameter, cu/YtH (Fig. 4).
For this type of structure, the assumption of
linearity appears to be well justified and the
normal density function appears to be adequate
to model the strength parameters. As an alternative the logistic density function may also be used.
Thus, the first-order second moment method provides an easy and straightforward procedure to
compute the failure probability, and equations (5)
= 1-1366
(after Jenkins,
1982)
and (6) specifically become
IlM
= a21l2 +
UM2= (a2 (2)2
atilt + ao
(9)
+ (atUt)2
(10)
where Ilt, Ut and 1l2' U2 are the mean and stan-
dard deviation of the rockfill and core strength
parameters respectively. The constants ao and at
are obtained from Figs 5 and 6; a2 is always
equal to -1. Equations (9)and (10)can be evaluated for the upper and lower portions of the
failure envelope of the cross-section under consideration. The probability of failure is then computed from
Pc = P[U] + P[L] - P[U]P[L]
(11)
where P[U] and P[L] are the probabilities of
failure as calculated from the upper and lower
portions of the envelope that defines the failure
function of the rockfill embankment cross-section
(Fig. 3). Equation (11) implicitly assumes statistical independence between modes of failure.
This assumption provides a conservative estimate
of the probability of failure as the two modes are
believedto be positively correlated.
Alternatively, the logistic cumulative distribution function (equation (8» can be rearranged and
expressed in terms of the reliability index, to give
for either the upper or lower portion of the
envelope
P[U or L]
= 1/(1 + exp
[np/..j3])
(12)
which provides an easy and simple way of assessing the probability of failure.
In terms of a single failure region (upper or
lower portion of the envelope), it can be shown
that the factor of safety, as defined in the present
study (i.e. ratio of peak to mobilized strength), is
ROCKFILL
EMBANKMENT
Table 1. Soil properties
given by
Rockfill
')'1
III
COV1
F. = {p2- a1Jl.l)/aO
Core
o
43°
20 kNfm3
0.932
0,05
C'
4>1'
60 kNfm2
Cu
4>/
o
')'2
112
COV2
20 kNfm3
0,0375
0.15
"
"'.::::~,
~~x-
~
x/
CF
x...
~8
TC
x_x
~o
"0
-x
CF
CC)."-o~"-
TC Totalcost
CF
Cost offailure
CC
Costofconstruction
0.01
0
.
'0-0-0
3
0.001
Y
~
x
.~
'0
the face of the embankment
tion (14), is obtained from
at
0.1
Probability
(14)
It has already been mentioned that, in the case
of steep cores (i.e.P2 greater than about 70°),the
slip mechanism tends to betome shallower until it
reaches a limiting state in the form of shear
failure along the face of the embankment. As the
stability of the cross-section in this case depends
only on the rockfill strength and the inclination of
the embankment, the reliability index becomes a
function of only the upper mechanism. From
observing the stability charts, as this condition is
approached, the parameter al decreases. It can be
shown that for the limiting case of shear along
TC
Cotp,= 1.50
~5
P = [alJl.l - Jl.2][1- I/F.]/f1M
x
x
(13)
Equation (13)is particularly useful for estimating the factor of safety when used in connection
with the constants ao and al' especially for those
geometries not explicitly included in the stability
charts. Equation (13)can be substituted back into
equation (9) and thus the following relation for
the reliability index is obtained in terms of F.
7
CoIP'-2~
481
STABILITY
of failure P,
p, as
given in equa-
lim
- coP = C~v.1 [1 - I/FJ
~
(15)
where COVl is the coefficient of variation of the
rockfill strength. Bearing in mind that for this
Fig. 11. Total cost against probabiHty of failure
Table 2. Trial cross-sections
Cot
P1
60
65
50
55
45
50
40
45
1.25
1.50
1.75
2.00
Table 3.
P2
45
48
50
52
55
60
65
F.
1.08
1-16
1-13
1.28
1.20
1.32
1.30
1.37
P2
aU
0
a1u
a0 1
0.31
0,40
0.23
0.27
0.21
0.25
0.18
0.22
-0.32
-0,46
-0,24
-0.33
-0,23
-0,34
-0,21
-0,32
0.055
0.060
0,048
0,053
0.045
0.048
0,040
0.045
Pc
all
- 0.029
0.096
0.018
0.13
0,035
0.052
0,0074
0.025
0.0051
- 0,039
-0,018
- 0.028
-0,018
- 0.028
-0,015
- 0.026
Cost analysis for embankment witb cot Pl = I.SO
F.
0,93
1.07
1-13
1-16
1.28
1.34
1.43
aou
0.19
0.21
0.23
0.25
0.27
0.34
-
aU
1
-0'15
-0,20
-0,24
-0,27
-0,33
-0,45
-
a0 1
a1 1
Costs
Pr
0.043
0.047
0,048
0,050
0,053
0,056
-0,009
-0,014
-0,018
-0,022
0.95
0.31
0,13
0,079
- 0.028
0,035
- 0.040
0.0032
0,060
-0,052
-
CC.
CCr
CC
TC
1.43
1.28
1.20
1-12
1.00
0.82
2.48
3.00
3-32
3,60
4,00
4.60
3.91
4.28
4.52
4.72
5,00
5.42
5,40
4.81
4.76
4,87
5.07
5.43
0,66
5-16
5.82
5.82
482
BARBOSA, MORRIS AND SARMA
condition the factor of safety is given by equation
(3), equation (15) then becomes a useful expression for computing the probability of failure of
surface slips.
NUMERICALEXAMPLE
Consider typical results of laboratory data
from a site investigation, shown in Table 1. It is
desired to evaluate the stability of a 80 m high
dam. Fig. 4 may be used to select a wide variety
of possible cross-sections that satisfy stability
requirements for the above soil strength parameters. Some of these cross-sections are presented
in Table 2. Also included in this table are the constants ao and a1 (Figs 5 and 6) and the probability of failure computed from equation (12).
It is further assumed that an embankment with
cot P1 = 1.50 represents an adequate option. The
total cost of the structure is calculated from the
sum of the cost of construction (CC) and the cost
of failure (CF) times the probability of failure(Pr)
TC = CC + (CF)Pr
For simplicity,the cost of construction is based
on the cost of construction of the rockfill added
to the cost of construction of the core section.
The cost of failure is assumed to be 0,40 times the
cost of construction. All these values have been
normalized to the cost of construction of the core
of a cross-section having a core inclination angle
of pz = 55°. The results are shown in Table 3.
These same results are presented in graphical
form in Fig. 11, which also includes those of a
second alternative having an embankment inclination expressed by cot P1 = 2.00. Based on the
assumptions made, a good design option, minimizing costs, appears to be related to a crosssection having a core inclination of about 50°.
SUMMARYAND CONCLUSIONS
The stability of rockfill embankments has been
considered and stability charts generated that
allow direct computation of the factor of safety.
Because these curves represent conditions of limiting equilibrium, they were subsequently used to
define safety margins or failure functions, so that
reliability theory could be rigorously applied.
The second moment format can be used to
determine the probability of failure. As the failure
modes for rockfill embankments can be approximated by linear functions, it is believed that the
first-order second moment approach provides a
good estimate of this probability, provided that
the strength properties of the materials can be
characterized by the normal distribution.
As an alternative to the Gaussian curve, the
logistic distribution function was considered. Estimates of failure probabilities obtained with this
function are practically the same, even when compared with the truncated normal distribution,
with the major advantage that this function can
be easily evaluated without the aid of tables or
numerical integration techniques.
ACKNOWLEDGEMENTS
The Authors express their appreciation to Dr
K. Ramachandran of Imperial College of Science
and Technology, for his critical review and comments of the manuscript and to the Civil Engineering Department of Texas A & M University
for supporting this work.
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