A general limit equilibrium model for three-dimensional slope stability analysis
L
L. LAM
Cliftorz Associates Ltd., Calgary, AB T2P 3K7, Canada
AND
D.G.FREDLUND
Departmerzt of Civil Engineering, University of Saskatchervan, Saskaroorz, SK S7N OWO, Canada
Received August 3 1, 1992
Accepted July 21, 1993
A generalized model for three-dimensional analysis, using the method of columns, is presented. The model is
an extension of the two-dimensional general limit equilibrium formulation. Intercolumn force functions of arbitrary shape can be specified to simulate various directions for the intercolumn resultant forces. A unique feature of
the model involves the use of a geostatistical procedure (i.e., the Kriging technique) in modelling the geometry of
the slope, the stratigraphy, the potential slip surface, and the pore-water pressure conditions. The technique simplifies
the data-input procedure and expedites the column discretization and the factor of safety computations. The shape
of the intercolumn force functions was investigated for several slope geometries using a three-dimensional finite element stress analysis. The significance of the intercolumn force functions in three-dimensional stability analyses
was also studied. The model was utilized to study a case history involving an open-pit mining failure. The results
indicate that the model is able to provide a more realistic simulation of the case history than was possible using a
conventional two-dimensional model.
Key words: stability analysis, general limit equilibrium, three-dimensional, method of columns, factor of safety.
Un modble gCnCralisC pour l'analyse tridimensionnelle utilisant la mCthode des colonnes est prCsentCe. Le modkle
est une extension de la formulation bidimensionnelle de 1'Cquilibre limite gCnCrale. Des fonctions de forces entre les
colonnes de forme arbitraire peuvent &re spCcifiCes pour simuler diverses directions des forces resultantes entre
les colonnes. Une particularit6 unique de ce modkle implique l'utilisation d'une procCdure gCostatique (i.e., la technique de Kriging) pour modCliser la gComCtrie de la pente, la stratigraphic, la surface potentielie de glissement,
et les conditions de pression interstitielle. La technique simplifie la procCdure d'entrCe des donnCes et accClkre la discrCtisation des colonnes et les calculs du coefficient de sCcuritC. La forme des fonctions de forces entre les colonnes
a CtC CtudiCe pour plusieurs gComCtries de talus au moyen d'une analyse tridimensionnelle des contraintes par
ClCments finis. La signification des fonctions de forces entre les colonnes dans les analyses de stabilitC tridimentionnelles
a aussi CtC CtudiCe. Le modkle a CtC utilisC pour Ctudier l'histoire d'un cas impliquant la rupture d'une carrikre de
mine ii ciel ouvert. Les rCsultats indiquent que le modkle peut fournir une simulation plus rCaliste de I'histoire du
cas qu'il Ctait possible de le faire en utilisant un modkle conventionnel bidimensionnel.
Mots cle's : analyse de stabilitb, Cquilibre limite gCnCral, tridimensionnel, mCthode de colonnes, coefficient de
sCcuritC.
[Traduit par la rCdaction]
Can. Geotech. J. 30, 905-919 (1993)
Introduction and literature review
All slope failures have a three-dimensional geometry.
However, slope stability analyses have usually been performed using two-dimensional simulations. Since the mid
1970s, increasing attention has been directed toward the
development and application of three-dimensional stability
models. Several three-dimensional methods of analysis have
been proposed in the literature. The limit equilibrium methods
of columns are most popular and are considered most feasible
for practical engineering applications.
Hovland (1977) appears to have been the first to analyze
a three-dimensional slope using the methods of columns.
Hovland's method is an extension of the assumptions associated with the two-dimensional ordinary method. In other
words, all intercolumn forces acting on the sides of the
columns are ignored. The normal and shear forces acting
on the base of each column are derived as components of the
weight of the column. Hovland (1977) determined the threedimensional factor of safety for several example problems.
The solutions indicated that the three-dimensional analysis
of a slope gave a factor of safety that was smaller than the
Prinlcd in Canada I Imprimd au Canndn
two-dimensional f a c t o r of safety f o r s o m e situations.
Cavounidis (1987) showed that the three-dimensional factor
of safety of a slope should always be greater than the twodimensional factor of safety of the same slope provided the
central portion of the sliding mass is the same for the twoand three-dimensional analyses.
The three-dimensional method proposed by Chen and
Chameau (1982) can be considered partly as an extension
of the assumptions associated with the ordinary method,
and partly as an extension of the assumptions associated
with Spencer's (1967) method. In the formulation of Chen
and Chameau (1982), the intercolumn shear forces in the
plane of movement were assumed to be parallel to the base
of the column, and the intercolumn forces perpendicular to
the plane of movement were assumed to have the same inclination throughout the entire sliding mass.
Chen and Chameau's (1982) assumptions and results were
discussed by Hutchinson and Sarma (1985), Cavounidis
(1987), and Hungr (1987). Concern was expressed regarding
Chen and Chameau's finding that the ratio F3/F2 (where F3
is the factor of safety from a three-dimensional analysis,
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CAN.
GEOTECH. J.
VOL. 30, 1993
AXIS OF
ROTATION
FIG.1. Cross-section through a failed mass in the x direction, showing the common axis of rotation and the forces on the column.
d,, moment arm for the weight of a column; d,,,verticai distance from the axis of rotation to the center of the base of a column, rl,, moment
arm for shear resistance on the circular portlon of the slip surface; EL, intercolumn normal force on the left, front plane of a column; H,, horizontal intercolumn shear force on the left, front plane of a column; X,, intercolumn shear force on the left, side plane
of a column; W, weight of a column; X,, intercolumn shear force on the right, side plane of a column; HR, horizontal intercolumn shear
force on the right, front plane of a column; E,, intercolumn normal force on the right, front plane of a column; T, horizontal shear force
at the base of a column in a plane perpendicular to movement; a,, angle between the horizontal and the shear force at the base of a
slice in the direction of movement; O,, angle between the horizontal and the normal force at the base of a column in the plane of
movement; S,, shear force mobilized at the base of the column in the plane of movement; and O,, angle between the vertical and
the normal force at the base of a column in the plane of movement.
and F2 is the factor of safety from a two-dimensional analysis) may be less than 1.0 in certain circumstances. Hutchinson
and S a r m a (1985) pointed out that the ratio F 3 / F 2 can
approach 1.0, but should not fall below 1.0. It was suggested that Chen and Chameau's (1982) findings were erroneous (Hutchinson and Sarma 1985; Cavounidis 1987).
Hungr (1987) proposed a method that was an extension of
the assumptions associated with Bishop's (1954) simplified
method. The vertical intercolumn shear forces acting on
both the longitudinal and the lateral faces of each column
were neglected. The example problem used by Chen and
Chameau (1982) was solved by Hungr (1987) as a comparison between the two methods. Hungr's results indicated
that, for all cases, the ratio F3/F, was greater than 1.0.
Hungr (1987) suggested that Bishop's (1954) simplified
method will produce similar results to more rigorous techniques when using two-dimensional analyses. Therefore, a
direct extension of the methods to three dimensions should
be intuitively expected to exhibit as good a performance as
observed with two-dimensional methods. Cavounidis (1988),
however, suggested that a comparison between the simplified
and the rigorous methods in three dimensions should be
made to prove that the simplifications were justifiable and the
results acceptable.
Hungr et al. (1989) used a three-dimensional method that
was an extension of the assumptions in Bishop's (1954)
simplified and Janbu's simplified two-dimensional models and
they showed comparisons for a number of solutions. Favourable comparisons of a wedge solution, closed-form algorithms, and an actual sliding mass (i.e., the Lodalen cutting) were shown.
All methods of columns proposed in the literature for
three-dimensional slope stability analysis can be considered
as simplified methods. These methods either neglect the
intercolumn forces or make assumptions that have not been
fully verified. The applicability of these simplified methods has been based on the experience gained using twod i m e n s i o n a l a n a l y s e s , a n d t h e accuracy of t h e threedimensional methods has not been independently studied.
This paper presents the theory and implementation of a
more generalized three-dimensional, slope stability model.
The formulation of the proposed method is "general" from
a limit equilibrium standpoint (Fredlund et al. 1981), but i t
should be noted that it is "not general" from a kinematic
standpoint. For example, nothing is stated in the method
regarding the direction of movement or the possibility of a
change in direction within the sliding mass. The shape of
the intercolumn force functions and their significance to
L A M A N D FREDLUND
AXIS OF ROTATION
FIG. 2. Cross-section through a failed mass in the z direction showing a common axis of rotation and the forces acting on a
column. R, distance from the axis of rotation to the slip surface; R,,,, maximum distance from the axis of rotation to the slip surface;
P,, intercolumn normal force on the left, side plane of a column; Q,, horizontal intercolumn shear force on the left side plane of a
column; V,, vertical intercolumn shear force on the left, side plane of a column; Q,, horizontal intercolumn shear force on the
right, side plane of a column; P,, intercolumn normal force on the right side plane of a column; a;,angle between the horizontal and
the shear force, T at the base of a column; and O:, angle between the horizontal and the normal force at the base of a column.
the three-dimensional factor of safety for two example problems are also presented. The application of the proposed
model was also demonstrated using a slope instability case
history.
I
Problem indeterminacy and assumptions
The proposed generalized three-dimensional slope stability model using the method of columns is an extension
of the two-dimensional general limit equilibrium (GLE) formulation (Fredlund and Krahn 1977). This formulation
assumes a slip mechanism where the direction of movement
is in one plane. The factor of safety is defined as that factor
by which the shear strength components must be reduced
to bring the soil mass into a state of limiting equilibrium
along a selected slip surface. Furthermore, it is assumed
that the factor of safety of the cohesive component of shear
strength and the frictional component of shear strength are
equal.
T h e earth mass above the slip surface is divided into
columns and the forces acting on the various faces of each
column must be computed or assumed. Figures 1 and 2 present the cross sections through a failed mass in the x-y and
the z-y planes, respectively. A free-body diagram showing the
various forces acting on a single column is presented in
Fig. 3.
The method of columns is indeterminate in that the number
exceeds the number
as
in Table 1. If there are iz number of columns in the x direction and m number of columns in the z direction. the number of unknowns is 12nin + 2, and the-number of equations
is 4nm + 2. The indeterminacy associated with a limit equilibrium slope stability analysis can be viewed as arising
FIG.3. Free body diagram of a column before using simplifying
assumptions,
from a lack of knowledge regarding the stresses within the
soil mass. To reduce the degree of indeterminacy, the following assumptions can be made.
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CAN.
GEOTECH. J.
VOL. 30, 1993
TABLE1. Summary of knowns and unknowns in solving for three-dimensional
factor of safety
I
Description
Knowns
nm
Unknowns
i zm
rzin
3nm
nrn
11ln
nm
V2M
nrn
nn?
nrn
1
1
C
2
E
2 F,
C F,
C F,
in z direction for each column
in x direction for each column
= 0 in y direction for each column
Mohr-Coulomb failure criterion for each column
C M = 0 about the axis of rotation for the whole sliding mass
C F, = 0 in x direction for the whole sliding mass
=0
=0
N: normal force at base of each column
S,: shear force mobilized at base of each column
a,, a ,, a;: point of application of N
T: shear force in z direction at base of each column
E: intercolumn normal shear force on yz plane
X: intercolumn vertical shear force on yz plane
H: intercolumn horizontal shear force on yz plane
P: intercolumn normal shear force on xy plane
V: intercolumn vertical shear force on xy plane
Q: intercolumn horizontal shear force on xy plane
F,,,: factor of safety by moment equilibrium
F,: factor of safety by force equilibrium
m
g
8 "
Q
-t
%
Q
6
m
column force functions describe the variation of the direction
of the resultants of the normal and intercolumn shear forces.
Mathematically, these intercolumn force functions can be
represented as follows:
111
[21
131
- = A3 f (3)
[41
1, f (4)
P
T
--=A, f (5)
N
[51
FIG.4. Generated slip surface for a uniform slope.
( 1 ) Assume that the point of application of the normal
force N at the base of a column acts through the centre of the
base area. Therefore, the distances from the normal force
to the four bottom corners of the column are defined. As a
result, the number of unknowns is reduced to 9nm + 2.
( 2 ) Assume that all the intercolumn shear forces acting on
the various faces of the column can be related to their respective normal forces by intercolumn force functions. The inter-
x
-E= A , f ( l )
H
2=h2f(2)
v
P
Q
-=
where
f ( 1 ) is a function that describes the manner in which the
EIX force ratio varies in the x direction;
f ( 2 ) is a function that describes the manner in which the
HIE force ratio varies in the x direction;
f ( 3 ) is a function that describes the manner in which the
VlQ force ratio varies in the z direction;
,f(4) is a function that describes the manner in which the
PlQ force ratio varies in the z direction;
f ( 5 ) is a function that describes the manner in which the
TIN force ratio varies in the z direction; and
A,, A?, h3, A3. and A, are the percentages of the intercolumn force functions used when solving for the factor
of safety.
This approach to computing the intercolumn shear forces
is similar to the approach proposed by Morgenstern and
Price (1965) for two-dimensional analyses. With the above
assumptions, the intercolumn shear forces X, H, V , Q , and
T can be calculated once the normal forces (i.e., E, P, and N )
LAM AND FREDLUND
909
FIG.5. Generated slip surface for a nonuniform slope.
are defined. The number of unknowns is therefore reduced
by 5nnz. The five intercolumn force functions also result in
five more unknowns, namely X,, A,, X,, X,, and X,. As a
result, the number of unknowns is now 4nin + 7. To obtain
a solution to the problem, the value of the five h constants
must be defined. In other words, a certain combination of the
five X constants will give a solution to the analysis, and
only the correct combination of the X constants will give
the solution to the analysis. However, n o procedure is
presently available to determine the proper combination of
the X constants.
It is suggested that a three-dimensional stress analysis
can be used to obtain an indication of the shape of the intercolumn force functions. It may then be possible to assume a
relationship between five X constants. As more information
becomes available concerning the X constants, a method of
solution may become available.
Intercolumn force functions
To render the problem determinate, more information
regarding the nature of the intercolumn force functions must
be known. Fan et al. (1986) investigated the nature of the
interslice force function for two-dimensional cases by using
a finite element stress analysis. Homogeneous slopes of different inclinations angles were analyzed to determine the
stress distribution within the earth mass. The internal stresses
were then integrated to determine the interslice force function. A similar approach was used in this study to evaluate
the three-dimensional intercolumn force functions.
The finite element program called ANSYS was used to
define the stress state of two example slopes subjected to
gravitational forces. Example 1 is a simple or uniform slope
(i.e., no spatial variation in the z direction) representing a
failed mass through the central portion of an embankment
(Fig. 4). Example 2 is a nonuniform slope representing a
FIG. 6. A comparison of the five intercolumn force functions
for a uniform slope at z = 100.
failed mass through the corner of an embankment (Fig. 5).
Both s l o p e s a r e a s s u m e d to b e homogeneous, and t h e
stress-strain behaviour of the soil is assumed to be linear.
Young's modulus of elasticity E and Poisson's ratio IJ, of
the soil are taken to be 100 000 kPa and 0.4, respectively.
All five intercolumn force functions XIE, VIP, HIE, QlP,
and TIN are determined by integrating the appropriate stress
over the areas of the sides of a column. Figure 6 illustrates
the five intercolumn force functions through the centre section of the failed mass for example 1. The XIE function of
the sliding mass f o r example 1 i s presented a s a threedimensional surface in Fig. 7. Figure 8 illustrates the five
intercolumn force functions through the centre section of
the failed mass for example 2. The XIE function and the
VIP function of the sliding mass for example 2 are presented in Figs. 9 and 10, respectively.
The intercolumn force functions have the same form as
observed for comparative two-dimensional cases (Fan et al.
1986). The functions are bell-shaped, with the maximum
magnitude located approximately at the centre of the slope.
As the radius of the slip surface changes (i.e., the slip surface changes) in the z directions, the shape of the functions
also change. The intercolumn force functions can be represented by a three-dimensional surface over the entire slip
surface.
Based on the results of the intercolumn force functions
for the above two example slopes, the following conclusions can be made.
(1) For simple or uniform geometries, only the function
of XIE has values of significant magnitude. All the other
functions are zero (Fig. 6). In other words, A,, h2, X3, A,,
and A, can be assumed to be zero.
910
CAN. GEOTECH. J. VOL. 30, 1993
FIG.7. A three-dimensional surface of the intercolumn force
function XIE for a uniform slope.
FIG.9. A three-dimensional surface of the intercolumn force
function XIE for a nonuniform slope.
FIG. 10. A three-dimensional surface of the intercolumn force
function VIP for a nonuniform slope.
EIX and VIP functions rather than all five functions in a
three-dimensional analysis. T h e r e f o r e , the number of
4. When compared with the
unknowns is reduced to 4nm
number of available equations, there are two more unknowns,
X , and X3, that need to be solved by other means. Although
there is no present direct solution for X I and X3, the two
unknowns can be solved using an iterative procedure while
assuming that
(i) the factor of safety with respect to moment equilibrium
F,, must be equal to the factor of safety with respect to
force equilibrium Ff when total equilibrium is satisfied, and
(ii) the most critical factor of safety of the sliding mass
must be the lowest possible factor of safety if all other conditions of the slope remain unchanged.
In other words, the correct combination of X I and X,
should not only give the total equilibrium factor of safety for
+
FIG.8. A comparison of the five intercolumn force functions
for a nonuniform slope at z = 100.
(2) For the nonuniform geometries studied, only the functions of XIE and VIP have values of significant magnitude. All
other functions have magnitudes that are relatively small
(Fig. 8). In other words, X2, X4, and X5 can be assumed to
be zero.
The above findings are important in the formulation of
a generalized three-dimensional stability analysis model.
The findings suggest that it is reasonable to consider only the
LAM AND FREDLUND
PlEZOMRRlC
SURFACE
SOlL SURFACE 1
SOlL SURFACE 2
FIG. 11. Free body diagram of a column after using simplifying assumptions for movement in the x direction.
the sliding mass but also the lowest factor of safety. Therefore, the number of unknowns is reduced to 4nnz 2, which
is the same as the number of equations. The problem is now
determinate, and the factor of safety can be calculated. The
free body diagram of a column, after applying the above
assumptions, is presented in Fig. 11.
+
Normal force N and factor of safety equations
By summing forces on each column in the y direction,
the normal force N acting perpendicular to the base of a
column can be expressed as
where
W is the weight of a column of soil,
X, is the intercolumn shear force on the left, front plane
of a column,
XR is the intercolumn shear force on the right, front plane
of a column,
V , is the intercolumn shear force on the left, side plane of
a column,
V , is the intercolumn shear force on the right, side plane
of a column, and
a, is the angle between the horizontal and the shear force
at the base of a slice, in the direction of movement.
SOlL SURFACE 2
FIG. 12. Modelling a three-dimensional slope using surfaces.
c' is the effective cohesion of the soil,
+' is the effective angle of internal friction,
A is the area of the base of the column, and
U is the pore-water force acting on the base of the column.
The normal force N is a function of the factor of safety F.
The factor of safety is equal to F,, when solving for the
factor of safety with respect to moment equilibrium and F,
when solving for the factor of safety with respect to force
equilibrium.
The factor of safety with respect to moment equilibrium,
F,, can be derived by summing the moment of all the forces
over the entire failed mass about an axis of rotation.
Similarly, the factor of safety with respect to force equilibrium, F , can be derived by summing forces in the x direction over the entire failed mass. The moment and force factors of safety, F,, and F,, can be derived using statical
equilibrium
[71
F, =
C(AC'+
tan+'-~tan+')(cosa.d, + ~ i n a , d , ~ )
C ( N C O S O , ~+, N c o s 0 , d ,
+ Wd,)
where d, is the x moment arm with respect to the axis of
moments, d, is the y moment arm with respect to the axis of
912
7.
CAN. GEOTECI4. J VOL 30, 1993
GRID W N KNOWN X, Z COORDINATES
FIG. 13. Column discretization of a three-dimensional slope
using the Kriging technique.
FIG. 15. Validation example problem 1 : purely cohesive
material with a circular slip surface (from Hungr et al. 1989).
c, cohesion; 6 , friction angle; R, moment arm of the resisting
force, y, unit weight of soil.
FIG. 14. A fully defined column using the Kriging technique.
moments, 0, is the angle between the horizontal and the
normal force at the base of a column, in the plane movement, and 0, is the angle between the vertical and the normal
force at the base of a column in the plane of movement.
Both factor of safety equations, F, and F,, are nonlinear
because the normal force N is also a function of the factor
of safety (eq. [6]). An iterative, back-substitution procedure
is used to solve for the three-dimensional factor of safety.
The sum of the intercolumn normal forces, E, drops out of
[7] and [S], since it must be equal to zero. Since the horizontal shear force Q on the sides of the column is assumed
to be zero, it follows that each row of columns must also
be in longitudinal equilibrium. Therefore, the normal forces
on the end columns, E, and En, must also be zero. This
boundary condition must be satisfied in the solution of [7]
and [S].
There is a smooth transition between the formulations of
the proposed three-dimensional method of columns and the
generalized two-dimensional method of slices. When a
problem approaches the two-dimensional plane strain condition, the angle a, approaches zero, the angle e,, approaches
a,, the equation for in, becomes the same as for the twodimensional case, and a. is the angle between the horizontal
and the base of a column in a
perpendicular to the
plane of movement. The term VL - V, in the equation for the
normal force becomes zero, since there is no spatial variation
in the z direction under plane strain conditions. As a result,
the equation for the normal force is reduced to be the same
LAM AND FREDLUND
Number of Columns
FIG. 16. A comparison of the three-dimensional factors of safety between various models as a function of number of columns
for validation example problem 1.
form as that for the two-dimensional case. Furthermore,
since cos 8, is identical to sin a,, the factor of safety equations are the same as those for the two-dimensional case.
After computing the factor of safety, it is possible to
check and ensure that the location of the line of thrust is
acceptable, that there are no negative normal thrust forces,
and that the vertical sides of the column can develop sufficient shearing resistance (Morgenstern and Price 1965).
Geometric modelling by Kriging
Traditionally, the geometry of a slope has been modelled
through the input of a series of cross sections and the use of
linear interpolation to define the slope between the cross
sections. A unique feature of the present model involves
the use of a geostatistical procedure (i.e., the Kriging technique) for the geometric modelling of the slope.
Kriging is a geostatistical estimation procedure that predicts
the value of a parameter at any location based on ( i ) a
knowledge of the structure of the variability as represented
by a set of known data points, and (ii) the minimization of
the estimation variance over the entire region for which the
prediction is made. Using this technique, weighting coefficients are computed for each of the known data points used
in the interpolation. These coefficients are subsequently
used to compute the magnitude of the parameter under consideration at any other position (McClarty et al. 1991).
Using the Kriging technique, random points in space can
be used to generate a surface containing a series of designated
points. Since a three-dimensional sliding mass consists of
the ground surface, the peizometric surface, the soil interfaces,
and the sliding surface, it is more effective to model a slope
with surfaces rather than cross sections. The Kriging technique can be used to generate all the necessary surfaces.
The generated surfaces are then superimposed to form a
complete description of the geometry (Fig. 12).
A distinct advantage of using the Kriging technique in
the modelling of a slope is its efficiency in the discretization
of the slope into columns. In the discretization process, a
rectangular grid of known x and z coordinates can be superimposed over the entire sliding mass (Fig. 13). The elevations
of all the surfaces (i.e., y coordinates) can be calculated on
the same grid.
The elevations of the ground surface at specified x and
z coordinates of the grid define the top four corners of a
column. The elevations of the slip surface at the same x
and z coordinates define the bottom four corners of a column.
Similarly, the elevations of the piezometric surface and all
the soil surfaces at the same x and z coordinates define the
positions of the piezometric surface and the soil surfaces
within a column. As a result, the geometric characteristic
of each column within the sliding mass is fully defined as
shown in Fig. 14. These coordinates greatly expedite and
simplify the computation process for the factor of safety.
It is possible to input data points along one cross section
and then request that this information be extended in a
lateral direction. T h e result is a simple geometry, threedimensional slope. Horizontal planes can also be placed
through a single designated point. A sloping surface requires
a minim~lmof three designated points. More complex geometries, such as a curved, three-dimensional sliding surface, may require 10-30 designated points. The number of
points required to designate a particular surface depends
upon its complexity.
Implementation and evaluation of the model
The proposed theory has been implemented into a computer
model called 3~-SLOPE.The model has been tested to ensure
that the method is properly implemented and the various
features of the model are functioning properly. Two of the
evaluation example problems are presented in this paper.
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CAN.
GEOTECH. J. VOL. 30,
1993
TABLE
2. A comparison of factors of safety for validation example problem 1 using various
model
li
Model
Three-dimensional factor of safety
% difference
Closed-form solution
CLARA ~01uti0n
3~-SLOPE
solution (1200 columns)
3~-SLOPE
solution (540 columns)
-
ci=29 kN/m2 #'=200
H.12.2m
= 26.5O
------
CASE I . CIRCULAR
SLIDE SURFACE
CASE 2. NON - CIRCULAR
PIEZOMETRICX--WEAK LAYER ( c 1 = o , + ' =
LINE
-Z
-
FIG. 17. Geometry and soil properties for validation example problem 2 (from Fredlund and Krahn 1977). H is the height of
slope; B is the 26.5" that can be removed from the figure since it corresponds to a 2: 1 slope.
Example problem 1 is a homogeneous, purely cohesive
slope with a circular slip surface. A closed-form solution
to this problem, using the ordinary method, has been proposed
by Baligh and Azzouz (1975) and Gens et al. (1988). For
the problem with a spherical slip surface and the material
properties as illustrated in Fig. 15, the three-dimensional
factor of safety was calculated to be 1.402 using the closedform solution. The factor of safety was computed to be
1.422 using the CLARA model (Hungr et al. 1989).
The same problem has been solved using the developed
3~-SLOPE
model. The computed three-dimensional factor of
safety ranged from 1.386 to 1.472 depending on the number
of columns used in discretizing the slope (Fig. 16). The
results indicate that the change in the factor of safety is
quite pronounced when the number of columns is less than
about 400. However, the change is relatively small when
the number of columns is more than about 500. The factor
of safety is equal to 1.402 when the slope is discretized to
about 540 columns. At about 1000 columns, the computed
factor of safety is 1.386. It would seem that the more
columns used to discretize the slope, the more accurate is the
computed factor of safety. No partial columns are taken into
account and as a result there is some oscillation in the factor
of safety versus the number of columns being used in the calculations. It is not known why the 3D-SLOPE factor of safety
is slightly lower (i.e., approximately 1%) than the closed-form
solution when a large number of columns is used.
Table 2 presents a comparison of the three-dimensional
factor of safety of example 1 for various interslice force
function conditions. The limit equilibrium method of columns
computer model provides an approximate solution to the
closed-form solution. However, the computed results from the
3~-SLOPE
program compares well with the closed-form solution of 1.402.
Example problem 2 is a homogeneous slope with a composite slip surface. The example problem was originally
used by Fredlund and Krahn (1977) in a two-dimensional
slope analysis study. Since then, the example problem has
become a benchmark problem used by several researchers in
an attempt to verify their models in both two- and threedimensional cases. The two-dimensional geometry of the
example problem is presented in Fig. 17. Two cases of the
example problem was analyzed. Case 1 is where there is
no water table, and case 2 has an assumed water table.
Three-dimensional analyses of the problem were conducted by Xing (1988). Since the third dimension of the
problem is also considered in a three-dimensional analysis,
the problem was solved as a function of the volume of the
sliding mass. Using Xing's three-dimensional model, the
factor of safety was computed to be 1.553 for the case without water table and a volume of sliding soil of 13 032 m3.
For the case with the water table and a sliding mass volume
of 16 290 m3, the factor of safety was computed to be 1.441.
These problems were also analyzed by Hungr et al. (1989)
using the CLARA model. For the case without the water table
and a sliding mass volume of 13 0 0 0 m3, the threedimensional factor of safety was computed to be 1.62. For
the case with the water table and a sliding mass volume of
LAM A N D FREDLUND
TABLE
3. A comparison of factors of safety for validation example problem 2 between
models
Case 1: without water table
Method
Ordinary
Bishop's simplified
Janbu's simplified
GLE
Xing"
CLARA"
1.553
-
-
1.62
-
-
-
-
Case 2: with water table
3~-SLOPEXingn
1.534
1.607
1.558
1.603
3 ~ - S L O P Eand
CLARA~
1.441
-
-
1.54
-
-
-
-
other
Average
difference
3~-SLOPE
1.447
1.5 11
1.48 1
1.508
%
0.8
1.4
-
"Computed using Xing's (1988) three-dimensional model.
'CLARA is a proprietary product of 0. Hungr Geotechnical Research Inc., West Vancouver, B.C., Canada.
16 000 m', the three-dimensional factor of safety was 1.54.
The same three-dimensional problems were analyzed using
the 3 ~ - S L O P E computer program. Figure 18 is a threedimensional representation of the generated slip surface.
The factor of safety for four assumptions regarding the interslice forces was obtained, and the results are presented in
Table 3. For both cases (i.e., with and without the water
table), good agreement was observed between the results
E those from other three-dimensional
from 3 ~ - S L O Pand
models. As expected, the factor of safety obtained from the
ordinary method compares closely with Xing's (1988) model,
and the factor of safety obtained from Bishop's simplified
method compares closely with the model CLARA of Hungr
et al. (1989). The average percentage difference between
the results is about 1%. The centre of rotation for the circular
portion in case 2 was used as the centre of moments in performing Bishop's simplified type of analysis. This procedure has been shown to be acceptable for noncircular-type
slip surfaces (Fredlund et al. 1992).
Significance of the intercolumn force functions
The two example slopes used to determine the intercolumn
force functions (i.e., Figs. 4 and 5 ) were analyzed using the
developed model. The slopes were a s s ~ ~ m etod be homogeand~ an angle of interneous, with a unit weight of 19 k ~ / m
nal friction of 30". The cohesion of the material was assumed
to be 15 kPa for example problem 1 and 10 kPa for example
Droblem 2.
The significance of the intercolumn force functions in
three-dimensional slope stability analysis was studied by
comparing the factors of safety from the two example slopes
obtained for different forms of the intercolumn force functions. A "reference" factor of safety for the example problems was defined as the factor of safety computed using
the GLE method along with the intercolumn force functions
computed using the finite element stress analysis. Although
there is no single, "true" factor of safety for each example
problem, the "reference" factor of safety would appear to
be an accurate benchmark re~resentation.
To further examine the significance of the intercolumn
force functions, several variations to the example slopes
were studied. These variations include
(i) c$ only analysis where the material was assumed to
be a purely frictional material;
(ii) no water analysis, where the water table in the example problem was neglected; and
analysis, where the shear strength of the
(iii) with
material was assumed to increase with soil suction within
the slope, and c$" the value was assumed to be 10" (Fredlund
et al. 1978).
+"
FIG. 18. The generated slip surface for validation example
problem 2.
The three-dimensional factors of safety for the various
cases were determined using different forms of the intercolumn force functions, which represent the different methods of analysis. Assumptions or force functions considered
were those associated with the ordinary method, Bishop's
simplified method, Janbu's simplified method, and the GLE
method. For the GLE method, factors of safety were obtained
for three types of intercolumn force functions, namely the
constant function, the half-sine function, and the function
computed using the ANSYS stress analyses. The factor of
safety computed using the actual functions computed from
the stress analysis is called the "reference" factor of safety.
The results of the four example problems and the comparison between the various methods are tabulated in Tables 4
and 5. The following conclusions can be drawn.
(1) In all the analyses, the GLE method with the halfsine intercolumn force functions gave the best approximation
to the reference factors of safety. The largest percentage
difference was 1.1%, and the average percentage difference
was 0.3 1%.
(2) The GLE method with the constant intercolumn function gave the second best approximation of the reference
factors of safety. The largest percentage difference was
1.23%, and the average percentage difference was 0.41%.
(3) Bishop's simplified method gave a good approximation
9 16
CAN.
GEOTECH. J .
VOL. 30, 1993
TABLE4. A comparison of the three-dimensional factors of safety of various methods of analysis for example
problem 1 (uniform slope)
GLE
Conditions
Ordinary
Factor of safety, base failure
% difference
Bishop's
simplified
Janbu's
simplified
Constant
function
Half
function
function
ANSYS
1.736
7.76
1.881
0.05
1.732
7.97
1.888
0.1 1
1.881
0.05
1.882
0.576
21.41
0.726
0.95
0.618
15.69
0.734
0.14
0.733
0.00
0.733
Factor of safety, no water analysis
% difference
1.968
7.26
2.123
0.05
1.915
9.75
2.121
0.05
2.122
0.00
2.122
Factor of safety with
% difference
1.806
6.76
1.937
0.00
1.828
5.63
1.937
0.00
1.937
0.00
1.937
0.14
0.08
0.05
0.0 1
Factor of safety for
% difference
+ only
+b
= lo0
Largest % difference
Average % difference
21.41
10.80
0.95
0.26
15.69
9.76
-
-
-
-
NOTE:A uniform slope is a simple geometry with no three-dimensional geometrical aspects.
TABLE5. A comparison of the three-dimensional factors of safety of various methods of analysis for example
problem 2 (nonuniform slope, corner of an embankment)
GLE
Conditions
Ordinary
Bishop's
simplified
Janbu's
simplified
Constant
function
Half
function
ANSYS
function
Factor of safety, base failure
% difference
Factor of safety for
% difference
+ only
Factor of safety, no water analysis
% difference
Factor of safety with
% difference
+b
= 10'
Largest % difference
Average % difference
of the reference factors of safety, particularly in the cases of
uniform slopes. The largest percentage difference was 4.0%,
and the average percentage difference was 1.42%.
(4) Janbu's simplified method (Janbu 1954, 1973) (without the correction factor for intercolumn shear) significantly
underestimated the reference factor of safety. The largest
percentage difference was 17.8%, and the average percentage difference was 9.5%.
(5) The ordinary method gave the poorest approximation of the reference factor of safety. The largest percentage
difference was 33.8%, and the average percentage difference was 14.6%.
The Poplar River coal mine case history
The developed three-dimensional slope stability model
3 ~ - S L O P Ewas used to analyze a case history. The case history
selected is that of a highwall stability problem encountered
at the Poplar River coal mine in southern Saskatchewan
(Clifton et al. 1986). The mine was an open-pit coal mine
developed by Saskatchewan Power Corporation to supply
coal to the Poplar River Power Station near Coronach,
Saskatchewan.
Geometry and stratigraphy
The surface sediments were mainly till with thicknesses
ranging from 0 to 15 m. Over most of the area, a layer of
sand was found underlying the till and a layer of clay was
found between the sand layer and the coal stratum. The clay
layer varied from being silty to highly plastic. T h e clay
stratum appeared to be sheared with slickensided surfaces,
probably resulting from glacial ice thrusting.
A uerched water table was observed within the till laver
over most of the area. The sand layer beneath the till was
found to be saturated, and the clay below the sand layer
was wet and soft. On the other hand, in the area where the
sand was dry, the clay below the sand was dry and stiff.
A failure (TC10) occurred at station 2 4 + 5 5 along the
highwall. The highwall was 15.2 m high with a slope of
about 33". The failure occurred along a slickensided clay
LAM AND FREDLUND
4.3m
7
shoe Load
215 kPa
1
/
/
15.2 m
-----J
Slip Geometry
-
1
I
I
+ 1.22
- 1.06
+ 0.89
+'
I
0
I
I
Factor of
+
1.01
I
-
0 98
<+
-
I
I
1.0 +
10 20 30
Cohesion (kPa)
I
I
-
40
FIG. 19. Cross section showing slip surface and computed two-dimensional factors of safety at TClO (from Clifton et al. 1986).
layer approximately 7.6 m below the crest of the highwall.
0
A layer of wet sand about 1.2 m in thickness was observed
SCALE
above the clay layer. At the time of the failure, the dragline
was walking parallel to the highwall about 20 m away from
the crest of the highwall. The slope failure appeared to be a
result of the high stresses exerted by the shoe of the dragline.
OBSERVED
A cross section showing the geometry of the failed mass,
SLIP PIANE
7.6 m
the inferred position of the slip surface, and the position of
the external load at TClO is shown in Fig. 19. A plan view
showing the geometry and the approximate position of the
SLOUGHING
external load is illustrated in Fig. 20. The external load due
WET SAND
to the shoe of the dragline was approximately 215 kPa
applied on the slope over an area of 21.9 m by 4.3 m. The
distance from the shoe to the crest of the highwall ranged
from 19.5 to 26.0 m.
The unit weight of all the soils was taken to be 19.3 k ~ / m ~ .
Laboratory results showed that the till stratum had an effective cohesion of 6 kPa and an effective angle of internal
friction of 34". An extensive laboratory testing program was
directed towards the evaluation of the residual shear strength
for the clay stratum (Clifton et al. 1986). The results indicated
that for residual conditions, the effective cohesion was zero
and the effective angle of internal friction ranged from 11
to 22.8". The range in residual friction angle reflects the
variation in texture of the clay zone and the degree of shearFIG. 20. Plan showing geometry, crack pattern, and position
ing disturbance in the clay samples.
of external load of the slope at TClO (from Clifton et al. 1986).
Results and discussioiz of slope stability back analyses
The failed slope of TClO was back analyzed using the
surface. The back-analyzed residual angle of friction for
E a two-dimensional model PC-SLOPE'. the clay was determined for both Bishop's simplified method
model 3 ~ - S L O Pand
Figure 21 illustrates the geometry and the slip surface modand the GLE method and the results are shown in Fig. 22.
Since no pore-water measurements were
elled by 3~-SLOPE.
The computed results indicated that the residual angles of
obtained at the time of failure, the slope was analyzed as
friction for the clay were considerably higher when using a
a function of the height of the water table above the slip
two-dimensional analysis than when using a three-dimensional
analysis. Using a two-dimensional analysis gives a residual
angle of friction for the clay between 14 and 15", even for the
'PC-SLOPE
is a proprietory product of Geo-Slope International
condition with no pore-water pressures. This value appeared
Ltd., Calgary, Alta.
I
11~7
-
CAN. GEOTECH. J. VOL. 30, 1993
,0.. 20
-.u
(I)
2
-
16
m
-2
2
m
Y
12
n
.,
-
-
Three-Dimensional
8
4
c
O0
2
3
4
5
6
Water Table Above SLIPSurface (m)
FIG. 22. Computed residual angle of internal friction of clay
from back-analyses as a function of assumed pore-water pressure conditions.
FIG.21. Generated slip surface of the failed slope at location
TClO.
to be relatively high when compared with typical values for
the highly plastic clay.
Since sloughing wet sand was recorded at the time of
failure, it is likely that the slip surface was under the influence of positive pore-water pressures. The pore-water pressure may be a combination of the effect of the water table
and the excess pressures induced by the external loading
from the dragline shoe. The loading of the dragline shoe
may be considered as the loading of a rectangular footing on
the slope. Using design charts published by Newmark (1942),
the vertical stress below the dragline show and the porewater pressure in the clay acting above the slip surface was
estimated to be about 3 m of pore-water pressure head.
For an estimated pore-water pressure head of 3 m, the
residual angle of friction for the clay is about 7.5" when
back analyzed using the three-dimensional model and 17"
when back analyzed using the two-dimensional model
(Fig. 22). Based on the plasticity index PI of the clay at
TClO (i.e., PI = 41.5%), it is reasonable to have a residual
angle of friction of about 7.5" (Sharma et al. 1981). On the
other hand, a residual angle of friction of 17" appears to
be too high.
The nonuniform geometry, the irregular slip surface, and
the external load cannot be adequately handled using a twodimensional stability model. The two-dimensional analysis
will inevitably underestimate the factor of safety of the
slope and overestimate the shear strength of the soils because
of neglecting the "end friction" and the poor simulation of
the external load.
A more realistic simulation of the case history is obtained
using the developed three-dimensional model. Not only is a
more reasonable residual angle of friction determined for
the clay, but the three-dimensional model also provides a
better appreciation of the pore-water pressure conditions at
the time of failure.
Although the three-dimensional model more realistically
models the geometries, it is still necessary to assume a direction of movement for the analysis. The predominant direction of movement was assumed to be the left (Fig. 20).
Conclusions
(1) A generalized three-dimensional stability analysis
, developed and validated. Factors of
model, 3 ~ - S L O P Ewas
E
satisfactorily with
safety obtained from 3 ~ - S L O Pcompared
analytical solutions, and other published example problems.
(2) Results from the stress analysis computer program
ANSYS indicated that the XIE and VIP functions, in x and
z directions, respectively, were the dominant intercolumn
force functions. The other functions are almost equal to
zero and have relatively small values for the types of geometries considered.
(3) When the GLE formulation was used in the 3D-SLOPE
computer program, the computed three-dimensional factors
of safety were relatively insensitive to the form of the intercolumn force filnctions used in the analysis. However, the
most reasonable approximations to the factor of safety were
obtained when a half-sine function or the function from the
stress analysis were assumed for the intercolumn force
functions.
(4) Bishop's simplified method provides a reasonable
approximation of the three-dimensional factor of safety for
the case of a circular slip surface. For noncircular slip surfaces and complex geometries, the factor of safety computed using Bishop's simplified method can differ significantly from those computed using more realistic intercolumn
force functions.
LAM A N D FREDLUND
(5) T h e ordinary method and Janbu's simplified method
(without a correction factor f o r intercolumn shear) considerably underestimated the actual three-dimensional factor
of safety.
(6) A three-dimensional stress analysis should b e performed for complex, nonuniform geometries and the internal
shear and normal stresses used to compute realistic intercolumn force functions. There can then b e used in the threedimensional slope stability analysis.
(7) Results from a case history indicated that the use of
a two-dimensional model considerably underestimated the factor of safety for the problem and, consequently, overestimated
t h e shear strength parameters of t h e soil w h e n used a s a
back analysis.
Acknowledgements
T h e authors would like to acknowledge the support and
cooperation of Clifton Associates Ltd., Regina, Saskatchewan,
a n d t h e S a s k a t c h e w a n P o w e r Corporation, Regina. B o t h
organizations h a v e been of assistance in conducting this
study. T h e authors would also like to acknowledge the technical input of Professor Doug Stead, Department of Geology,
University of Saskatchewan, Saskatoon.
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