Two-dimensional Slope Stability Analysis by limit equilibrium and
strength reduction methods
Wei Wenbing
05900931r
Department of Civil and Structural Engineering
ABSTRACT: The factors of safety and the locations of critical failure surfaces obtained by limit equilibrium
method and strength reduction method are compared for various slopes. For simple homogenous soil slopes, it
is found that the results from these two methods are generally in good agreement. There are however cases
where great differences between the two methods may occur in some special circumstances which should be
considered in analysis.
1 INTRODUCTION
Limit equilibrium method (LEM) is well known to
be a statically indeterminate problem and assumptions on the internal forces distribution are required
for the solution of the factor of safety. Morgenstern
(1992) among others has however pointed that for
normal problems, the factors of safety from different
methods of analyses are similar so that the assumptions on the internal force distribution f(x) are not
major problems except for some particular cases.
LEM requires trial failure surfaces and optimization analysis to locate the critical failure surface
which is a difficult N-P type global optimization
problem. Many different proposals have been suggested in the past and detailed discussions on various methods in locating critical failure surface have
been provided by Cheng (2003). While most of these
methods can work for normal problems, they may be
trapped by the presence of local minimum in the solution and Cheng (2003) has encountered several interesting problems which are trapped by the presence of local minima with the use of classical
gradient type optimization methods. The use of
modified simulated annealing method by Cheng
(2003) is one of the few successful methods which
can escape from local minimum and the method has
been used for many major problems in several countries.
Many researchers have compared the results between strength reduction method (SRM) in finite element analysis and limit equilibrium method and it
is found that generally the two methods will give
similar factors of safety. Most of the studies are
however limited to homogenous soil slopes and the
geometry of the problems is relatively regular with
no special feature (presence of thin layer of soft material or special geometry). Furthermore, there are
only limited studies in the comparisons of critical
failure surface from LEM and SRM as factors of
safety appear to be the primary goal of studies. In
this paper, the two methods are compared under different conditions and both the factors of safety as
well as the locations of the critical failure surfaces
are considered in the analyses. In the present study,
non-associated flow rule (SRM1 and dilation angle=0) and associated flow rule (SRM2 and dilation
angle=friction angle) which are the upper and lower
limit of the flow rule are applied in the SRM analyses.
2 STABILITY ANALYSIS FOR A SIMPLE AND
HOMOGENEOUS SOIL SLOPE
Firstly, a homogeneous soil slope with a slope height
equal to 6m and slope angle equal to 45 degree is
considered, Figure 1.
4m
6m
10m
6m
10m
4m
20m
Figure 1 Discretization of a simple slope model.
In the parametric study, different shear strength
properties are used and LEM, SRM1 and SRM2
analyses are carried out. The cohesion of the soil
varies from 2, 5, 10 to 20 kPa while the friction angle varies from 5, 15, 25, 35 to 45 degree respectively, but the density, elastic modulus and Poisson ratio
of the soil are kept to be 20 kN/m3, 14MPa and 0.3
respectively in all the analysis. Actually, the authors
have found that the results are not sensitive to the
density of soil and this parameter is hence kept constant in the study. The results of analyses are shown
in Table 1. As shown in Figure 1, the size of the
domain for SRM analyses is 20m in width and 10m
in height and there are 3520 zones and 7302 grid
points in the mesh for analysis. Morgenstern-Price’s
method which satisfies both moment and force equilibrium is adopted and the critical failure surface is
evaluated by the modified simulated annealing technique as proposed by Cheng (2003).
a)
9
ground profile
limit equilibrium
SRM1
SRM2
8
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
7
6
factor of
safety
(SRM2,
associated)
0.26
0.52
0.78
1.07
1.44
0.43
0.73
1.03
1.35
1.74
0.69
1.04
1.37
1.71
2.15
1.20
1.59
1.96
2.35
2.83
y(m)
case
Factors of safety by LEM and SRM
factor of
factor
safety
Cohesion Phi
of safe(SRM1,
(kPa)
(degree) ty
non(LEM)
associated)
2
5
0.25
0.25
2
15
0.50
0.51
2
25
0.74
0.77
2
35
1.01
1.07
2
45
1.35
1.42
5
5
0.41
0.43
5
15
0.70
0.73
5
25
0.98
1.03
5
35
1.28
1.34
5
45
1.65
1.68
10
5
0.65
0.69
10
15
0.98
1.04
10
25
1.30
1.36
10
35
1.63
1.69
10
45
2.04
2.05
20
5
1.06
1.20
20
15
1.48
1.59
20
25
1.85
1.95
20
35
2.24
2.28
20
45
2.69
2.67
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
x(m)
10
11
12
13
14
15
16
6
7
8
9
x(m)
10
11
12
13
14
15
16
6
7
8
9
x(m)
10
11
12
13
14
15
16
6
7
8
9
x(m)
10
11
12
13
14
15
16
6
7
8
9
x(m)
10
11
12
13
14
15
16
b)
9
ground profile
limit equilibrium
SRM1
SRM2
8
7
6
y(m)
Table 1
5
4
3
2
1
0
0
1
2
3
4
5
c)
9
ground profile
limit equilibrium
SRM1
SRM2
8
7
5
4
3
2
1
0
0
1
2
3
4
5
d)
9
ground profile
limit equilibrium
SRM1
SRM2
8
7
y(m)
6
5
4
3
2
1
0
0
1
2
3
4
5
e)
9
ground profile
limit equilibrium
SRM1
SRM2
8
7
6
y(m)
From Table 1 and Figure 2 it is found that the factors
of safety and critical failure surfaces as determined
by SRM and LEM are very similar under different
combinations of soil parameters. Most of the factor
of safety values obtained by SRM differ by less than
7 % with respect to LEM solutions except for case
16 (c=20kPa, phi=5degree) where the difference is
up to 12.8 %. The differences between LEM and
SRM by Saeterbo Glamen et al. (2004) are more
than those in the present study and the authors suspect that it is due to the use of manual location of
critical failure surfaces by Saeterbo Glamen et al.
(2004) while the authors adopted global optimization method with a very fine control in obtaining the
minimum factors of safety.
Based on Table 1 and Figure 2, some conclusions
can be made as follows :
1. Most of the FOS values obtained from SRM are
slightly larger than those obtained by LEM with only few exceptions.
2. When the SRM is implemented, the FOS values
with the use of associated flow rule (SRM2) are
slightly above those with the use of non-associated
flow (SRM1), and the differences in the factors of
safety increases with increasing friction angle. These
results are reasonable and are expected.
3. When the cohesion of the soil is small, the differences of FOS between LEM and SRM (SRM1 and
SRM2) are greatest for higher friction angles. When
the cohesion of the soil is large, the differences of
FOS are greatest for lower friction angles. This result is somewhat different from that by Dawson
y(m)
6
5
4
3
2
1
0
0
1
2
3
4
5
Figure 2. Slip surface comparison with increasing friction angle, c=2kPa,  = 15 (a), 25 (b) and 45 (c) and increasing cohesion (phi=5degree), c = 2 (a) , c= 10 (d) and c = 20 (e).
(1999) who pointed out that the differences are
greatest for higher friction angles.
4. The failure surfaces from LEM, SRM1 and SRM2
are similar in most cases. In particular, the critical
failure surfaces obtained by SRM2 appear to be
closer to those by LEM than those based on SRM1.
5. The right end of the failure surface gets closer and
closer to the crest of the slope when the friction angle of the soil is increasing which is also a well
known result. This behavior is more obvious for
those failure surfaces obtained from SRM1.
6. For SRM analyses, when the friction angle of soil
is small, the differences of slip surfaces between
SRM1 and SRM2 are greatest for smaller cohesion.
When the friction angle is large, the differences of
slip surface between SRM1 and SRM2 are greatest
for higher cohesion (Figure 2).
7. It can also be deduced from the results that the potential failure volume of the slope gets smaller with
increasing friction angle but gets greater with increasing cohesion which is also a well known behavior that when the cohesive strength is high, the critical failure surface will be a deeper failure surface.
Although there are some minor differences in the
results between SRM and LEM in this example, in
general the results by these two methods are in good
agreement and the results suggest that the use of
LEM or SRM is satisfactory in general.
3 STABILITY ANALYSIS OF A SLOPE WITH A
SOFT BAND
A special problem with a soft band is constructed by
the authors. The geometry of the slope is shown in
Figure 3 and the soil properties are shown in Table
2. It is noticed that the soil parameters are particularly low for soil layer 2 which has a thickness of 0.5m
only and slope failures in similar conditions have actually occurred in Hong Kong (Fei Tsui Road slope
failure in Hong Kong).
28,15
20,15
Soil1
28,10
8,8
28,9.5
8,7.5
Soil2
0,5
y
5,5
8,7.1
5,4.5
0,0
x
Figure 3
A slope with a thin soft band
Soil3
28,0
Table 2
Soil
name
Soil1
Soil2
Soil3
Soil properties for Fig.3
Friction
Cohesion
Density
angle
(kPa)
(kN/m3)
(degree)
20
35
19
0
25
19
10
35
19
Elastic
modulus
(MPa)
14
14
14
Poisson
ratio
0.3
0.3
0.3
In order to consider the size effect (boundary effect) in SRM, three different numerical models are
developed to perform SRM using Mohr-Coulomb
analysis and the lengths of the domains are 28m,
20m and 12m respectively. In these three SRM
models, various maximum element sizes have been
tried until the results are not sensitive to the number
of elements used for analysis. Since the factors of
safety from this special problem have great differences with those from LEM, the authors have tried
several well known commercial programs and obtained very surprising results from them. For the locations of the critical failure surfaces, the results
from the three SRM models (using different programs) and LEM are generally in good agreement
with minor differences. Majority of the critical failure surfaces lie within layer 2 which has very small
shear strength parameters and this result is as expected.
From Table 3, it is surprising to find that different
programs produce drastic different results for the
factors of safety even though the locations of the
critical failure surface from these programs are very
close. For the cases as shown in Figure1 and other
cases analyzed by the authors, the results are practically insensitive to the domain size (from parametric
study by the authors) while the case as shown in
Figure 3 are very sensitive to the size of domain for
programs A (SRM1 and SRM2) and B (SRM2). Results from program C appear not to be sensitive to
the domain size but is relatively sensitive to the dilation angle which is different from the previous results. SRM1 results from program B is also not sensitive to the domain size but SRM2 results behave
differently. Results from program A appear to be
over-estimated as the soil parameters for the soft
band are low, but the results from this program are
not sensitive to the dilation angle which are similar
to all the other examples in the present study. For
SRM1, the results from program B and C appear to
be reasonable as the results are not sensitive to the
domain sizes while for SRM2, the authors view that
results from program C may be better as the results
are less sensitive to the dilation angle. It is also surprising to find that program D cannot give any result
to this problem after many different trials but the
program work properly for all the other examples in
this study.
Tables 3 : FOS by SRM from different programs. The values in
each cell are based on SRM1 and SRM2 respectively. (min.
FOS=0.927 from Morgenstern-Price’s analysis)
Program/FOS
12m domain 20m domain 28m domain
A
1.03/1.03
1.30/1.28
1.64/1.61
B
0.77/0.85
0.84/1.06
0.87/1.37
C
0.82/0.94
0.85/0.97
0.86/0.97
D
No solution No solution No solution
Figure 4 Locations of critical failure surfaces from LEM and
SRM for soft band problem
Besides the special results as shown above, the
factors of safety from 28m domain analysis appear
to be large for programs A and B as the soil parameters for soil layer 2 are very small. In fact, it is not
easy to define an appropriate factor of safety for this
problem. If the cohesive strength of the top soil is
reduced to zero, the factor of safety can be roughly
estimated as 0.57 from tan/tan, where  is the
slope angle. It can be viewed that for LEM, the cohesive strength 20 kPa for soil 1 help to bring the
factor of safety to 0.927 and a high factor of safety
for this problem is not reasonable. Allowing for tension crack would reduce the factor of safety close to
0.57. Obviously it would be dangerous to adopt factor of safety values of the order 1.3 to 1.64 given by
some of the analysis using SRM.
When the soil properties of soil layer 2 are
changed to the soil properties of soil layer 3, the results are practically independent of the domain size
and all programs can give similar results. The results
from LEM and the results from programs B and C
using unassociated flow rule are basically consistent
and can be considered as the estimations of the factors of safety. This interesting case has illustrated the
limitation of using both SRM and LEM method
when there is a thin layer of soft material and great
care and judgment are required for acceptance of the
results of analysis. The problems as shown in Table
3 may be purely the limitations of some of the commercial programs instead of the limitation of SRM,
but it also illustrate that it is not easy to compute a
good value for this special type of problem for SRM.
The results are highly sensitive to different nonlinear
solution algorithms (which are however not always
clearly explained in the commercial programs). In
this respect, LEM appears to be a better solution for
this special type of problem. Great care, effort and
time are required to achieve a reasonable result from
SRM for this special problem and the result should
also be compared with LEM before the final acceptance.
If the soil properties of soil 2 and 3 are interchanged so that the third layer soil is the weak soil,
the factors of safety from SRM2 are 1.33 (with all
programs) for all the three different domain sizes
while the factor of safety from LEM is 1.29 from
Morgenstern-Price’s analysis. The locations of the
critical failure surface from SRM and LEM for this
case are very similar. It appears that the presence of
a soft band instead of major differences in soil parameters will create the difficulties in SRM analysis.
4 LOCAL MINIMUM IN LEM AND SRM
For LEM, it is well known that many local minima
may exist besides the global minimum and this is also the difficulty in locating critical failure surface by
classical optimization method. The comparisons of
LEM and SRM with respect to local minimum has
not been considered in the past but is actually a very
important issue which is illustrated by the following
example. In SRM, there is no local minimum as the
formation of shear band will attract strain localization in the solution process. To investigate this problem, a relatively simple slope with a total height of
17 m and one soil layer is discussed. The soil parameters are c’=5 kPa and ’=30° while unit weight
is 20 kN/m3. Using LEM, the global minimum factor
of safety is obtained as 1.33 (Figure 5) but several
local minima are found with factors of safety ranging 1.38 to 1.42. It can be viewed that there are several failure surfaces which are potential failure
mechanisms with virtually the same probability of
failure with the concept of LEM. From SRM, only 1
factor of safety is found to be 1.33 which is equal to
the global minimum from LEM. Other possible failure mechanism cannot be determined from SRM
easily. It may thus well be, that the SRM analysis
yields a local failure surface of less importance,
while a more severe global surface remains undetected. If slope stabilization is carried out only for
this failure surface, failure surfaces as shown by
Figure 5 will not be considered in SRM and no stabilization measures will be carried out for these locations. In this respect, LEM is a better tool to the
engineers in slope stability analysis.
30
FOS = 1,40
FOS = 1,41
FOS = 1,38
FOS = 1,38
25
FOS = 1,33
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
50
55
Figure 5. Global and local minima by LEM/SRM analysis.
5 DISCUSSION AND CONCLUSIONS
While most of the researchers concentrate on the
factors of safety between LEM and SRM, the authors have also compared the locations of critical
failure surfaces from these two methods. In simple
and homogenous soil slope, the differences of the
FOS and locations of critical failure surfaces from
SRM and LEM are small and both methods are satisfactory for engineering use. It is found for the analyzed example that when the cohesion of the soil is
small, the difference of FOS is greatest for higher
friction angles. When the cohesion of the soil is
large, the difference of FOS is greatest for lower
friction angles. For the effects of flow rule, the FOS
and locations of critical failure surface are not greatly affected by the choice of the dilation angle which
is important for the adoption of SRM in slope stability analysis. The critical slip surfaces from finite element analyses appeared to be closer to those by
limit equilibrium method when associated flow rule
was applied.
For SRM, the authors have studied the effects of
dilation angle and found it to be small but still noticeable. Drastic different results are determined
from different computer programs for the problem
with soft band which illustrates that SRM is highly
sensitive to the method of modeling and nonlinear
equation solution process (may not be the fault of
SRM). For this special case, the authors have also
found that the factor of safety determined by some
of the programs was very sensitive to the size of elements, tolerance of analysis and number of iteration
allowed and a parametric study on the effects of
these factors is strongly suggested as a routine process. For soft band problems limit equilibrium
method calculations with a reliable global optimization tool is also strongly suggested to be carried out
as a check of the results from the strength reduction
method. Although the problems for SRM in this special problem may not be the fault of SRM, the authors view that SRM has to be used with great care
for a problem with soft band.
Through the present study, the two limitations
of SRM are established : sensitive to nonlinear solution algorithms/flow rule for some special cases and
the inability to determine other critical failure surfaces. For general problem, it is possible that the
use of SRM may miss the location of the next critical failure surface (with a very small difference in
the FOS but a major difference in the location of
critical failure surface) so that the slope stabilization
measures may not be adequate. If SRM is used for
routine analysis and design of slope stabilization
measures, these two major limitations have to be
overcome and the authors suggest that LEM should
be carried out as a reference. If there are great differences between the results from SRM and LEM,
great care and engineering judgment should be exercised in assessing the proper solutions. There is one
practical problem in applying SRM to slope with
soft band. When the soft band is very thin, the number of elements required to achieve a good solution
is extremely large so that very significant computer
memory and time are required. Cheng (2003) has
tried a slope with 1mm soft band and has effectively
obtained the global minimum factor of safety by
simulated annealing method. If SRM is used for a
problem with 1mm thick soft band, it is extremely
difficult to define a mesh with good aspect ratio unless the number of element is tremendous. For SRM,
the authors have used about 1 hour for a small problem (several thousand elements) and several hours
for a large problem (over ten thousand elements) for
program B while program A requires 1-3 days
(small to large mesh). If a problem with 1mm thick
soft band is to be modeled with SRM, the computer
time and memory required will based on the experience from the present study be extremely high and
SRM is practically inapplicable. The authors are also
not aware of any successful application of SRM to
slope with very thin soft band. In this respect, LEM
is more efficient than SRM for this special type of
problem as it can be solved very quickly by the limit
equilibrium method (Cheng 2003) .
It can be concluded that both LEM and SRM
have their own merits and limitations and the use of
SRM is not really superior to the use of LEM in routine analysis and design. Both methods should be
viewed as providing an estimation of the factor of
safety and the probable failure mechanism but engineers should also appreciate the limitations of each
method and solution routines in the programs they
are using in the assessment of the results of analysis.
REFERENCES
Cheng Y.M., Locations of Critical Failure Surface and some Further Studies on Slope Stability Analysis, Computers and Geotechnics, vol.30, p.255-267, 2003.
Dawson, E. M., Roth, W. H. & Drescher, A.. (1999). Slope
stability analysis by strength reduction. Geotechnique 49,
No. 6, 835-840.
Morgenstern N.R., The Evaluation of Slope Stability – A 25
Year Perspective, Stability and Performance of Slopes and
Embankments –II, Geotechnical Special Publication No.
31, ASCE, 1992.
Saeterbo Glamen M.G., Nordal S. and Emdal A (2004). Slope
stability evaluations using the finite element method. NGM
2004, XIV Nordic Geotechnical Meeting. Volume 1. p. A49-A61.