29-09-2016, 02:43 PM
A Review of Three-Dimensional Slope Stability Analyses based on Limit Equilibrium Method
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ABSTRACT
The stability of slopes is a major concern in the field of geotechnical engineering. Usually
two-dimensional analyses based on limit equilibrium method are implemented in this field
due to their simplicity and effectiveness. Since these methods ignored the features of the third
dimension of slopes, three-dimensional analyses have developed to remove this shortcoming.
However, some limitations still exist in the application of three-dimensional methods. Most of
these methods assume a plane of symmetry for the slope. Moreover, the internal forces of
sliding mass are largely simplified or ignored in the equations of factor of safety. In addition,
the shape of slip is usually simplified or limited by these methods. The mentioned limitations
restrict the applications of three-dimensional methods in practice. The present paper made a
critical review on the applications and limitations of existing three-dimensional slope stability
analyses based on limit equilibrium method.
INTRODUCTION
Slope stability analysis is applied by two-dimensional (2D) and three-dimensional (3D)
analyses. There are several methods to do so, and limit equilibrium method (hereafter, LEM) is
one of the most well-known. Morgenstern and Price (1965) stated that the reasons of this
popularity are ability of this method to consider internal forces of the soil body, pore pressure,
and multi layered slopes. Fredlund and Krahn (1977) and Chen and Chameau (1983a) stated that
the simple theoretical approach, ability to consider major effective factors on the shearing
resistance, and reliable results in practices are among the reasons of popularity of LEM. Yu,
Salgado, Sloan, and Kim (1998) compared the results of LEM with other rigorous methods for
the stability analysis of simple earth slopes. They concluded that LEM could achieve reasonable
results. Askari and Farzaneh (2008) also emphasis on the simplicity as its wide acceptance causes. Moreover, the accuracy of LEM was determined to be satisfactory based on various backanalyses
of failed slopes. During a comparative study with the finite element method (FEM),
Hongjun and Longtan (2011) proved that LEM is a reliable method for assessing the stability of
soil slopes and using in general geotechnical engineering practices. Wright, Kulhawy, and
Duncan (1973) found that the factor of safety (FOS) and the strength parameters of slide surface
are reasonably accurate and the results of limit equilibrium method are reliably comparable with
actual measurements and forecasted behavior of the slope material. The studies of Wright et al.,
Spencer (1967 &1973), and Duncan (1996) indicated that the average value of FOS for those
LEMs that satisfy all conditions of equilibrium are accurately near to the rigorous methods by a
tolerance of ±6 percent. They mentioned that by considering the accuracy of initial data this
accuracy is precise enough for all practical purposes. Consequently, the mentioned advantages
cause the popularity of LEM in geotechnical slope stability.
The most common slope stability methods among engineers were 2D methods due to their
simplicity. These methods simplify the geometry of slope by converting the natural 3D slope to
2D slope. Therefore, the results of analysis may vary in the accuracy. Various 2D stability
analyses have been developed based on LEM in the past decades. This diversity comes from
different assumptions and simplifications in the analysis as well as using different equilibriums in
equation of FOS. These methods can be classified into circular methods (e.g. Swedish circle and
friction circle methods), non-circular methods (e.g. log-spiral procedure), and methods of slices
(e.g. Bishop, 1955; Janbu, 1954; Morgenstern and Price, 1965; and Spencer, 1967). The method
of slices is the most usual method among 2D methods due to its ability to consider different soil
and water conditions, complex geometrics, and the effects of external forces (Fredlund and
Krahn, 1977).
Although 2D slope stability methods are popular, these plain-strain problems idealized slopes
to be symmetric and infinitely long in third dimension. In opposite, the characteristics of natural
and manmade slopes are often varying even along short distances. Therefore, 2D simplification is
sometimes largely intuitive and typically corresponding to the worst-case scenario. The results of
Cavounidis’s (1987) study showed that 3D FOS is usually gently higher than the corresponding
2D factor. Gens, Hutchinson, and Cavounidis (1988) and Mowen, Zengfu, Xiangyu, and Bo
(2011) reported that the mentioned difference might be as large as 30% in some cases. Cheng,
Liu, Wei, and Au (2005) stated that the ability of 3D methods to achieve higher FOS than 2D
analyses is significant, because it offers an economically enhanced design for the slope. Another
major limitation of 2D methods is determining the direction of sliding (DOS). As an initial
assumption in all 2D methods, DOS is assumed to be in parallel with the cross-sectional plane of
the slope. However, the accuracy of this assumption is never guaranteed.
Although 3D analysis removes the mentioned limitations of 2D methods, the existing 3D
analyses still have some limitations and drawbacks in their background theories and applications.
This paper reviews these methods respect to their procedure and application and then discusses
the limitations of existing 3D slope stability analyses based on LEM. It is expected that this
critical review helps the researchers to understand better the existing shortcomings of available
methods in order to make further efforts in developing techniques that are more complete.
PROCEDURE OF LIMIT EQUILIBRIUM METHOD
LEM considers the slope behavior at the verge of failure as static condition, so does not
analyze the stress-strain relationship or the corresponding deformation within the soil body
(Fredlund, 1984). Consequently, it is required to assume the shape of potential failure surface that defines the sliding body. Based on the mentioned conditions FOS is defined as a numerical ratio
to compares the resisting shear strength of the soil with the existing shear stress on the failure
surface (Bishop, 1955). The following principles are mutual in the framework of all LEMs.
a. A kinematically feasible sliding surface is assumed to define the mechanism of failure.
b. Available shearing strength along the assumed slip surface is obtained by using the
application of static principles. Two applied static principles are the assumption of plastic
behavior for soil mass and validity of Mohr-coulomb failure criterion.
c. The comparison of available shear strength and required shear resistance to bring the
equilibrium into limiting condition is made in terms of FOS.
d. The satisfying value of FOS is determined through an iterative process.
By using the mentioned conditions, FOS is defined as the ratio of the available shear strength
of the soil to the required strength to maintain limiting equilibrium. Consequently, the mobilized
shear strength is defined as follows:
S = (1/FOS) {c’ + (σn – uw) tan φ’)} A (1)
where FOS is factor of safety, S is mobilized shear strength force (kN), c’ is cohesion of the soil
in terms of effective stress (kN/m2
), φ’ is angle of internal friction of soil in terms of effective
stress (kN/m2
), and σ’ is normal stress on the sliding surface (kN/m2
), and uw is pore water
pressure on the sliding surface (kN/m2
), and A is the related area of the slip surface (m2
).
An iterative process is usually carried out to find the minimum existing FOS. The iterative
process is commonly started by taking an initial FOS and continues to obtain the lowest
acceptable value. Although all the LEMs have mutual principles, they differ in utilizing static
equilibrium, assumptions and simplifications.
THREE-DIMENSIONAL SLOPE STABILITY METHODS
BASED ON LEM
Three-dimensional slope stability analyses have been presented in the past decades since
1969. Almost all of the existing 3D LEMs were extended from 2D slices methods. In order to
convert 2D methods of slices into 3D slope stability analyses, the slices are needed to be changed
into the columns by adding the third dimension. Consequently, the static conditions of limit
equilibriums of the columns have to be satisfied. The other definitions, simplifications, and
assumptions of 2D methods may or may not bring into the developed 3D method. However, the
assumptions of 3D methods are mostly derived from the related 2D basics; some new definitions
are only available in 3D methods because of the extra dimension. The 3D shape of slip surface,
the asymmetrical shape of slope, the sliding direction, and the inter-column forces are some of
these new meanings. Each 3D method might consider, simplify, or ignore some of the extra
definitions that make different 3D methods even from mutual 2D methods. Figure 1 shows a
typical soil column and its related internal and external forces where for column (i,j), Wi,j is
weight of soil, Lzi,j is external vertical load, Lxi,j and Lyi,j are respectively external horizontal loads
in x- and y-directions, Fevi,j is vertical force induced by earthquake, Fehxi,j and Fehyi,j are
respectively horizontal force induced by earthquake in x- and y-directions, Exi-1,j and Exi,j are intercolumn
normal forces in x-direction, Eyi,j-1 and Exi,j are inter-column normal forces in y-direction,
Xxi-1,j and Xxi,j are vertical inter-column shear forces in x-direction, Xyi,j-1 and Xxi,j are vertical
inter-column shear forces in y-direction, Hxi-1,j and Hxi,j are horizontal inter-column shear forces in
x-direction, Hyi,j-1 and Hxi,j and horizontal inter-column shear forces in y-direction, Si,j is shear strength force at the base of column, and Ni,j and Ui,j are respectively total normal force and pore
water force at the base of column. Some of these forces may be ignored, simplified, or assumed in
different methods. The following section describes the main characteristics and applications of
existing 3D methods based on LEM.
Anagnosti (1969) proposed a 3D method as an extension of Morgenstern and Price’s (1965)
method to calculate FOS of general slip surfaces. A series of limit equilibrium equations were
established on the thin vertical slices of sliding body while the slip surface was not restricted to
any specific shape. An interslice force function was assumed for distribution of interslice shear
forces. This was able to satisfy all equations of equilibrium. Results of Anagnosti’s study
revealed 50% increase in the value of FOS from 2D to 3D analysis. He also showed that 3D FOS
is not sensitive to the distribution function of interslice shear forces.
Baligh and Azzouz (1975) proposed a 3D method for cohesive slopes based on circular arc
method. The shape of slip surface was assumed as a combination of cylindrical center part with
conical or ellipsoidal ends. All shear resistance forces on the slip surface were assumed to act
perpendicular to the axis of rotation. This method used the equation of moment equilibrium about
the axis of rotation to find the value of FOS. A vertical cut in frictionless clay was studied by the
method where the rotation axis of slip surface was placed at the crest of the cut and the cylindrical
slip surface passed through the toe. Both combinations of slip surfaces were utilized and the
effect of shape of slip surface on the ratio of 2D and 3D FOS was studied. The results showed
that 3D FOS is higher than 2D factor, but the corresponding ratio changes based on the shape of
slip surface. Baligh and Azzouz stated that this ratio is always greater than unity, although it
decreases when the ratio of length to height of sliding mass increases.
Hovland (1977) presented a 3D analysis for cohesive and frictional soils as an extension of
ordinary method of slices. Sliding mass over the slip surface was divided into a number of
vertical columns with parallel sides to x and y-axis. Furthermore, the DOS was assumed parallel with vertical plane. Hovland’s method ignored all inter-column forces on the sides of columns as
well as pore-water pressure. Moreover, the sliding mechanism was limited to cone-shaped or
wedge-shaped surface. Hovland found that for cohesive slopes, the 3D FOS is always higher than
corresponding 2D factor, but this result is not limited to this type of soil. His analysis also showed
that the ratio of 3D to 2D FOS is quite sensitive to the magnitude of cohesion, angle of friction
and geometry of slip surface.
Azzouz and Baligh (1978) made an effort to expand the method of Baligh and Azzouz (1975)
to cohesion and frictional slopes. The shape of slip surface was supposed to follow the previous
method. Although the assumptions related to shear resistance forces did not change, two extra
assumptions were introduced for distribution of other forces. The first assumption followed the
ordinary method of slices (Fellenius, 1936) to neglect all interslice forces and calculate normal
stresses from moment equilibrium of each slice. The second one assumed that the vertical
effective stress is the principal major stress equal to the weight of slice, the horizontal stress is the
minor principal stress, and a third principal stress as a coefficient of slice weight acts parallel to
the axis of rotation. Azzouz and Baligh analyzed four embankments and their results showed that
2D analysis underestimates the shear resistance of frictional slopes on steep slip surfaces. They
also stated that their new assumptions provide more reasonable results than the original
assumptions of 2D ordinary method of slices.
Chen and Chameau (1983) presented a 3D method that was able to analyze symmetrical
homogeneous cohesive and frictional slopes with different pore water conditions. The slip surface
was assumed to be a combination of central cylindrical part that was attached to semi-ellipsoidal
ends with an axis of rotation perpendicular to the symmetrical plane. Based on assuming a DOS
parallel with symmetrical plane, all inter-column shear forces in front and back sides of columns
were ignored. As an extra assumption, the remained inter-column forces were assumed to act on
the middle section of each column with arbitrary heights and a constant inclination. Chen and
Chameau stated that this assumption is convenient by considering the small width and length for
columns. They divided inter-column shear forces into cohesion and frictional parts. The elevation
of acting point of these forces were assumed equal to half and one third of the columns height,
respectively and their inclination was assumed equal to the base angle of related side of the
column. In order to determine the FOS and inclination of resultant inter-column forces, both
moment and force equilibriums were considered for each column as well as the whole sliding
body. Chen and Chameau reported up to 10% lower values of 3D FOS than 2D factor for low
cohesion and high friction soils. However, 3D FOS was detected to be higher than 2D factor in
the presence of pore water pressure.
Azzouz and Baligh (1983) extended the 3D method of Baligh and Azzouz (1975) to consider
the effect of applied loads on the stability of slopes. The slope geometry was remained simple and
the slip surface was assumed as a combination of central cylindrical part and attached ellipsoids
(or cones) at the ends. All numerical procedure of finding FOS was similar to Baligh and
Azzouz’s method. In order to study the effects of 3D study, a comparison was made between
plain strain study and 3D slope stability problem due to the distribution of local loads. A
rectangular uniform load was applied on the top of cylindrical part of 3D slope, while the applied
load of plain strain problem was assumed as a strip load with infinite extend. The problem of 2D
case was considered by the circular arc method with an infinite long cylindrical slip surface. The
3D problem was analyzed by Baligh and Azzouz’s method subjected to a rectangular load on top
of the slope. The results showed that 3D analysis has a significance effect on the permitted
quantity of designed load. This effect was realized to be more significant when the FOS, without
considering the extra load, is near to one. Comparative study of Azzouz and Baligh for several practical cases showed that the effect of 3D analysis could increase the capacity of critical load of
2D analysis between five to ten times. This result can significantly increase the economic balance
of slope designs by achieving higher FOS in 3D analysis compare with corresponding 2D study
for the same load.
Dennhardt and Forster (1985) proposed a 3D method to find the FOS of symmetrical slopes
with ellipsoidal slip surface. This method was able to consider a symmetrical external load on the
top of slope. Dennhardt and Forster assumed a distribution of normal stress throughout the slip
surface to overcome the indeterminacy of the problem. The calculated 3D FOS by this method
was reported to be higher than corresponding 2D factor.
Leshchinsky, Baker, and Silver (1985) presented a 3D method for symmetrical slopes based
on limiting equations and variational analysis of Kopacsy (1957). They initialized three functions
as the problem unknowns to achieve the minimum FOS using variational exterimization of these
functions. The unknown functions included equation of slip surface as spherical or cylindrical
shapes, distribution function of normal stress, and direction of shear stress on the failure surface
(DOS). The problem was to determinate mentioned functions in order to achieve the minimum
FOS. Initial analysis of Leshchinsky et al. showed that distribution of normal stress over the slip
surface does not affect FOS. They also found that the direction of shear force on the slip surface
is not related to the distribution of normal forces, but is connected to the shape of slip surface. In
order to simplify the problem, Leshchinsky et al. limited their method to symmetrical slip
surfaces. From the result of two examples in homogeneous soils (one for spherical slip surface
and another for cylindrical slip surface), they found that FOS of spherical slip surfaces is greater
than long cylindrical slip surfaces’.
Ugai (1985) proposed a 3D method for symmetrical vertical cohesive cuts using limit
equilibrium equations and variational calculus. The suitable shape of slip surface was determined
by examining several arbitrary shapes including cone, ellipsoid, cylinder plus plane, combined
cylinder-cone, combined-cone-plane, and combined cylinder-ellipsoid. Finally, the cylindrical
slip surface attached to two curved caps was assumed the possible shape of failure. Ugai defined
a 3D stability factor as a function of FOS to examine the stability of slope. This factor included
FOS, height of the cut, cohesion and unit weight of the soil, and length of the slip surface.
Regarding to the certain values of cohesion and unit weight of the soil, the stability factor was
dependent on the FOS and the ratio of length of the slip surface to height of the cut. The value of
FOS was achieved by using an iterative process and variational calculus to minimize the stability
factor. The results of Ugai’s study revealed higher values of 3D FOS in comparison with
corresponding 2D factors.
Leshchinsky and Baker (1986) developed a 3D method for symmetrical homogeneous slopes
by limiting the method of Leshchinsky et al. (1985) in the third dimension (like a dam in a
narrow valley). A cylindrical shape attached to cap ends was assumed as slip surface. In order to
find the value of 3D FOS, Leshchinsky and Baker applied two force equilibriums along with the
transverse and vertical axis as well as moment equilibrium of half-sliding body about the rotation
axis. Their comparative study revealed that the most noticeable difference between 3D and 2D
FOS occurs in cohesive slopes. They found similar values of FOS for 3D and 2D analyses of
cohesionless slopes when the shallow slip surface tended to be parallel with the slope face.
Baker and Leshchinsky (1987) developed the 3D method of Leshchinsky et al. (1985) to
study symmetrical conical homogeneous slopes. In order to simplify the method, Baker and
Leshchinsky ignored the pore pressure as well as external loads. The shape and location of slip
surface was defined by the equation of surface and its central point. The results of Baker and Leshchinsky showed maximum rate of 1.6 for the ratio of 3D FOS to 2D factor. They concluded
that this ratio decreases consonantly with the cohesiveness; so, for purely frictional soils the
values of 3D and 2D FOS stand equal. In a general conclusion, Baker and Leshchinsky stated that
2D analysis of conical heaps causes conservative result based on material and inclination of the
slope.
Hungr (1987) proposed a 3D method as an extension of Bishop’s (1955) method. In this
symmetrical problem, a rotational surface with circular central cross section was assumed as the
failure surface. Following the assumption of Bishop, Hungr neglected the vertical Inter-column
shear forces on the sides of columns. This method considered vertical force equilibriums of all
columns as well as overall moment equilibrium of sliding mass about the axis of rotation to
establish the equation of FOS. Hungr re-analyzed the examples of Chen and Chameau (1983) and
concluded that the ratio of 3D to 2D FOS is always greater than unity.
Gens, Hutchinson, and Cavounidis (1988) presented a 3D method for homogeneous,
isotropic, and purely cohesive soils. The general shape of slip surface was assumed to be similar
to the study of Azzouz and Baligh (1987), but a specific combined shape of slip surface was
assumed as a cylindrical center part attached to planar or curved ends to establish the analytical
solutions. The equation of 3D FOS was established by considering the moment equilibrium of
sliding mass about the axis of rotation of cylindrical surface. The results of Gens et al. showed
that the ration of 3D FOS to 2D factor is always greater than unity and varies from 1.03 to 1.30.
Leshchinsky and Mullett (1988a & 1988b) proposed two 3D methods for analyzing the
stability of symmetrical homogeneous slopes based on LEM and variational analysis. These
methods considered vertical corners cuts and vertical cuts with longitudinal extension. The
assumed shape of failure surface was an expansion of log-spiral function. This method could be
able to calculate 3D FOS with the presence of pore pressure.
Ugai (1988) proposed a series of 3D methods by developing 2D methods of slices, including
Spencer's (1967), Fellenius’ (1936), simplified Janbu’s (1954), and simplified Bishop’s (1955)
methods. These extended methods were applied on symmetrical slopes with different sliding
surfaces. Taking into account the assumptions of original methods, each 3D method utilized
different assumptions. Ugai achieved greater values of 3D FOS compare with corresponding 2D
factor by using all developed methods, except the extended method of Fellenius.
Xing (1988) proposed a simple 3D method for analyzing the stability of symmetrical concave
slopes with elliptic slip surface. The assumed slip surface required four geometrical properties to
be defined including its central point and three semi radiuses. Following the assumption of
movement in parallel with vertical plane, all inter-column forces perpendicular to the sliding
direction were neglected. The resultant of remained inter-column forces was assumed to act with
a constant inclination over the sliding mass. Xing calculated the value of FOS by using
longitudinal and vertical force equilibrium for each column as well as overall sliding body and
moment equilibrium equation of sliding body about the axis of rotation. An iterative process with
an arbitrary constant inclination of inter-column forces was utilized to find the value of FOS. This
process continued until the assumed inter-column angle satisfied both force and moment
equilibriums at the same values of FOS. Xing concluded that 3D FOS is greater than 2D factor by
a maximum difference of 4.32% for homogeneous slopes that can gain to 10% with the presence
of a weak layer.
Hungr et al. (1989) proposed two 3D methods as extensions of simplified Bishop’s (1955)
and simplified Janbu’s (1954) methods. The assumptions in extended Bishop’s method were
similar to Hungr’s (1987). Vertical Inter-column shear forces were neglected in this method and all other shear forces on the slip surface were assumed parallel with the plane of symmetry. The
shape of slip surface was assumed rotational which also followed the symmetrical assumption of
the problem. For the non-rotational failure surfaces, a reference axis of rotation was assumed
based on the proposed approach of Fredlund and Krahn (1977). In order to find the value of FOS,
the vertical force equilibrium of each column as well as the overall moment equilibrium of sliding
body was satisfied. The 3D FOS obtained by extended Bishop’s method found to be smaller than
the results of 3D rigorous methods. The results of the extended Janbu’s method found to be even
more conservative than extended Bishop’s method in bilinear slip surfaces. Hungr et al. stated
that these conservatisms came from neglecting internal strengths.
Leshchinsky and Huang (1992b) proposed a generalized 3D method by variational analysis
and rigorous limit equilibrium equations as an extension of Leshchinsky and Huang’s (1992a) 2D
method. The shape of slip surface was assumed an extended log-spiral or a general symmetrical
surface. In order to determine the FOS they established a mathematical process. This method was
able to converge on a FOS by solving simultaneous a certain number of linear equations and three
nonlinear equations in an iterative process. However, several combinations of roots were able to
satisfy the convergence criteria of this system, the resultant FOS was found to be the same.
Leshchinsky and Huang stated that the specified convergence criteria could significantly affect
the value of 3D FOS. Their results showed that the ignorance of end effect in back-analysis of
slopes might lead to overestimated FOS as well as strengths parameters of the soil.
Cavounidis and Kalogeropoulos (1992) proposed a 3D method as an extension of Azzouz and
Baligh’s (1978) method to examine vertical cuts in cohesive soils. A combined central cylindrical
slip surface with conical ends was assumed as the slip surface and its axis of rotation was defined
on the crest of cut. The equation of FOS was established based on moment equilibrium as the
ratio of overturning moment of the sliding body to the resistance moment about the axis of
rotation. The results of Cavounidis and Kalogeropoulos showed that 3D FOS is always greater
than corresponding 2D factor. They also found that the difference between 3D and 2D factors of
safety decreases by increasing the length of cylindrical part in a constant length of conical ends.
In opposite, the role of 3D analysis becomes more important by increasing the length of conical
ends in a constant length of slip surface.
Lam and Fredlund (1993) proposed a 3D method based on general LEM of Fredlund and
Krahn (1977). A rotational surface with single direction of movement was assumed as the slip
surface. Moreover, different arbitrary inter-column force functions were used to calculate the
inclination of resultant inter-column forces. The basic definition of these inter-column force
functions was similar to Morgenstern and Price’s (1965) function including five relationships
between normal and shear inter-column forces. Lam and Fredlund decided to ignore three out of
five inter-column forces due to their insignificance role in typical slopes based on their results of
finite element analysis. They also established two different equations of FOS based on moment
and force equilibriums to determinate the condition of problem. The overall value of FOS was
determined to simultaneously satisfy moment and force equilibriums. Lam and Fredlund found
that 3D FOS is relatively insensitive to the form of inter-column force functions and is
significantly greater than 2D factor.
Yamagami and Jiang (1996 & 1997) proposed the first 3D method that was able to find the
DOS instead of using a plane of symmetry or an assumed sliding direction. This method was an
extension of simplified Janbu’s (1954) method that was able to consider general slopes and slip
surface. The inclinations of resultant inter-column forces were assumed to be a related to column
base angle through an unknown factor. The sliding body was divided into several vertical
columns and the direction of sliding was assumed to be in parallel with the vertical plane.
Yamagami and Jiang utilized horizontal and vertical force equilibriums to establish two separate
equations of FOS. They used an interval calculation to minimize the value of FOS by rotating the
main axes of system as changing DOS. The formation of vertical columns and the arrangement of
geometry information of sliding body were recalculated in each step of interval procedure. The
minimum value of FOS in each equation was determined the corresponding DOS. The overall
value of FOS was achieved through simultaneous iterative processes for both equations.
Huang and Tsai (2000) established a 3D method based on two-directional limit equilibriums
to find the 3D FOS for possible sliding directions. The slip surface was assumed as a semi
spherical or partly spherical composite rotational shape with an axis of rotation parallel with
longitudinal axis. A constant global axis was defined over the slope and the sliding body was
discretized by using a grid parallel to x and y-axes. Huang and Tsai ignored all horizontal intercolumn
forces and then, the rest forces were decomposed to corresponding elements of main axis.
In order to find the DOS and its corresponding value of FOS, three different equations were
established. Two equations were extracted from moment equilibriums in x and y directions (twodirectional
equations of FOS) and the third equation was derived from overall moment
equilibrium of sliding body. An interval calculation similar to Yamagami and Jiang (1996 &
1997) was used to find the DOS and directional factors of safety (DFOS). Then, the overall FOS
was calculated from the third equation by using the achieved DFOS.
Huang et al. (2002) proposed an extension of generalized Janbu’s (1957) method as 3D
method that used two-directional force and moment equilibrium to analyze the stability of an
arbitrary-shaped sliding mass. A square grid was used in this method to discretize the sliding
mass into columns. The basic assumptions of this method were similar to Janbu’s method,
however the resultant of inter-column horizontal shear forces of each column was assumed to be
zero as an extra assumption to decrease the degree of indeterminacy of the problem. The
remained vertical inter-column shear forces at the sides of columns were calculated by
introducing extra moment equilibrium for each column. Moreover, the direction of resultant shear
force (DOS) on the base of columns was assumed to be unique and calculated from the
relationship between DFOS by using Secant method. This method was able to calculate the DOS
in the process of finding 3D FOS. The procedure of finding 3D FOS of the sliding mass was
conducted by using two main iterative processes to calculate DFOS and DOS. Then, the
satisfactory values of DFOS and DOS were used to calculate the overall FOS.
Chen, Zhang, and Wang (2003) presented a simplified 3D method as an extension of 2D
Spencer’s (1967) method. The shape of slip surface was assumed rotational and several
assumptions were used to establish the force and moment equilibriums. Moreover, all horizontal
and two out of four vertical inter-column shear forces were ignored and the inclination of
resultant inter-column forces was assumed to be constant. In addition, the direction of shear
strength at the base of column (DOS) was assumed equal to a constant but unknown angle or
equal to a value that was calculated by the product of mentioned constant and a coefficient of
asymmetry. In order to find 3D FOS, overall moment and force equilibriums were utilized. These
equilibriums were used to establish three controlling equations involving three unknowns as FOS,
inclination of resultant inter-column forces, and DOS. Chen et al. determined these unknowns by
using Newton-Raphson method. The results of comparative studies of Chen et al. showed similar
3D FOS to other 3D methods in symmetrical problems, but greater 3D FOS was achieved in
highly asymmetric cases.
Jiang and Yamagami (2004) proposed a 3D method for analyzing the stability of conical
heaps based on limit equilibrium and variational analysis as an extension of Spencer’s (1967)
method. The general assumptions of Spencer’s method were applied on this method and DOS was assumed to be perpendicular to the longitudinal extend of the slope. The resultant of all intercolumn
forces was also assumed to act on the sides of column with assumed direction and
inclination. Jiang and Yamagami established two different equations for FOS with respect to
horizontal force and overall moment equilibriums. Then, FOS was defined by simultaneous
solving of these equations with different values for inclination of inter-column forces. The
intersection point of two resultant plots, achieved from two equations of FOS, resulted overall
FOS. Jiang and Yamagami concluded that using both force and moments equilibrium in finding
FOS increases the reliability of the results for almost all symmetrical slopes.
Cheng and Yip (2007) proposed a series of 3D methods by developing the simplified
Bishop’s (1955), simplified Janbu’s (1954), and Morgenstern and Price’s (1965) methods. They
formulated a 3D asymmetrical problem as an extension of Morgenstern-Price’s method and then
reduced it to simplified Bishop’s and simplified Janbu’s methods. The assumptions of each 3D
method followed the corresponding assumptions of its 2D origin, but the shape of slip surface
was assumed spherical for all the methods. In order to calculate the inter-column shear forces,
Cheng and Yip assumed an arbitrary inter-column force function together with mobilization
factors based on the inter-column force function of Morgenstern and Price. In addition, two
different equations were established to calculate the FOS in x- and y-directions. The solving
process started with assumed DOS. Then, the values of mobilization factors changed in an
iterative process until the resultant FOS satisfied corresponding overall moment equilibrium in
each direction. This iteration continued until achieving a unique value of FOS in both directions.
Zheng (2009) presented a rigorous 3D method that considered the whole failure body instead
of discretizing it into columns. The sliding surface was assumed to have general shape with an
arbitrary DOS. Zheng’s method satisfied six equilibrium conditions for the sliding mass and used
a vector of integration equation for all six equilibrium conditions. The unknown values of these
equations included FOS and total normal stress on the sliding surface that was defined by a
distribution form including five unknowns. Then, he substituted the distribution function into the
mentioned six equations and provided a system of nonlinear equations. This system was solved to
find the corresponding value of FOS, as well as the distribution vector. Zheng stated that a
positive FOS existed in his nonlinear system for a properly selected DOS.
Sun, Zheng, and Jiang (2011) proposed a 3D method as an extension of Morgenstern-Price’s
(1965) method on the system of forces that followed the procedure of Zheng (2009). Therefore,
no assumption was used regarding to internal forces, the sliding body was not divided into
columns, and all equilibrium conditions were satisfied. A triangular mesh was used to cover the
horizontal projection of slip surface for interpolation calculation of soil mass. Another triangular
mesh was used on the slip surface to calculate the related integrals. In addition to the non-linear
system of Zheng, a more sophisticated patch-wise interpolation with a triangular mesh was used
to better approximate the distribution of normal stress on complicated slip surfaces. Finally, an
optimization problem with an objective function including FOS and other five unknowns was
established. The minimum value of FOS and other corresponding unknowns were obtained by
solving the optimization problem. Comparative study of Sun et al. showed greater values of 3D
FOS in comparison with 2D factors, except for a critical section of one case.