A Comparison Between Limit Equilibrium and Finite Element Methods for Slope Stability Analysis

Abstract

Analyzing the stability of slopes is a critical aspect of geotechnical design. Computer modeling of slope stability problems has been widely accepted throughout both consulting engineering and research communities, and has provided the ability to reliably predict critical slope failure surfaces using a variety of different approaches. This study focuses on comparisons between safety factors calculated utilizing computer modeling for simple slopes using the extensively used limit equilibrium methods and the less common finite element approach. The comparative analysis is based on a multitude of slope geometry and material property combinations.

Full text

A Comparison Between Limit Equilibrium and Finite
Element Methods for Slope Stability Analysis
M. Yousuf Memon
Missouri University of Science & Technology, Rolla, Missouri
ABSTRACT: Analyzing the stability of slopes is a critical aspect of geotechnical design.
Computer modeling of slope stability problems has been widely accepted throughout both
consulting engineering and research communities, and has provided the ability to reliably predict
critical slope failure surfaces using a variety of different approaches. This study focuses on
comparisons between safety factors calculated utilizing computer modeling for simple slopes
using the extensively used limit equilibrium methods and the less common finite element
approach. The comparative analysis is based on a multitude of slope geometry and material
property combinations.
1.0 INTRODUCTION
1.1 Background
Landslides are known phenomenon that pose a serious risk to human life and safety. An average
of between 25 to 50 people are killed by landslides each year in the United States (USGS). With
a growing world population, the demand for land development in areas previously preserved and
undisturbed of human activity is increasing. As population within cities located in natural valleys
such as Reno, Nevada and Wenatchee, Washington grows, future development expands outward
and upslope the surrounding hills. Development on sloping ground poses an inherent risk related
to the local and global stability of structures and warrants an appropriate level of analysis.
Slope stability is a geotechnical problem that deals with the balance of naturally occurring or
manmade slopes, and requires an understanding of the earth materials and its engineering
properties. Slope stability analysis is conducted to evaluate the stability of existing slopes, as
well as to engineer safe cut and fill slopes for temporary or long-term conditions. Analyzing the
stability of a slope provides ways of determining, relatively unambiguously, whether a given
slope is likely to slide or whether it will remain stable. Mechanical analysis of slope stability
provides us with knowledge of what parameters control landsliding, entirely removing the
guesswork (Cruikshank, 2002).
1.2 Purpose of Study
In this study, an idealized simple slope is modeled with variable slope geometries and soil types,
and is analyzed using limit equilibrium (LE) and finite element (FE) methods. The LE methods
are based on force and moment equilibrium, while the FE methods use the stress-strain
relationships to determine the behavior of the model. For this study, slope analyses were
completed using Rocscience SLIDE and COMSOL Multiphysics for LE and FE methods,
respectively. The purpose of this study was to compare LE and FE methods for slope stability
analysis, and comment on the similarities and differences in the results computed using the
different methods. In addition, comparisons were made between three different methods of

2
calculating the FOS using LE method. Furthermore, this study attempts to provide opinions on
optimizing the use of FE methods to better match the equivalent FOS from LE analysis.
2.0 LITERATURE REVIEW
2.1 Introduction
Slope stability analysis can be performed using a wide range of computer programs that allow
input for various parameters of the soil medium, pore water conditions, method of analysis, and
type of slope failure. This leads to thousands of different combinations of all the variables that
would inevitably provide different results (Duncan and Wright, 2005). While simplifying a slope
stability problem using mathematical equations may lead to significant error, the possibility of
human error still exists with computer modeling. It is therefore considered good industry practice
to make independent checks to minimize errors introduced into the model.
2.2 Slope Failures Mechanisms
Depending on the type of soil or rock mass, groundwater conditions and slope geometry,
different types of slope failures may occur. Slope failure along a weak zone of soil is called a
translational slide and is common in coarse-grained soils and fractured/jointed bedrock
formations. An example of a translational slide is the recent ongoing Rattlesnake Hills Landslide
near Union Gap, Washington that is known to be controlled by movement along a weak, inclined
sedimentary interbed within the igneous basalt flow sequence.
A common type of slope failure in homogenous fine-grained soils is a rotational slide that has its
point of rotation on an imaginary axis parallel to the slope (Duncan and Wright, 2005). Types of
rotational slides include base slides, toe slides and slope slides. Base slides occur when a slope
consisting of soft soil is overlying dense soil/rock at basement, and the slope failure occurs
through the entirety of the soft soil layer below the toe of the slope. Toe slides failures pass
through the toe of the slope, while slope slides occur along failure passes through the slope face.
A fundamental requirement for stability of slopes is that the shear strength of the soil must be
greater than the shear stress required for equilibrium. Therefore the fundamental cause for slope
instability is that the shear strength of the soil is less than the shear stress required to equilibrium
which is either achieved through a decrease in the shear strength of the soil, or through an
increase in the shear stress. A decrease in shear strength may result from increased pore water
pressure, slope cracking, swelling, development of slickensides, decomposition of clayey rock
fills, creep under sustained loads, leaching, strain softening, weathering and cyclic loading.
Examples of increase in shear stress include loading at the top of the slope, water pressure in
cracks at top of the slope, increase in soil weight due to increased water content, excavation at
the bottom of the slope, drop in water level at the base of the slope and earthquake shaking
Duncan and Wright, 2005).
It is important to understand that slope failure is typically a result of a combination of different
causes. In most cases, several causes exist simultaneously and therefore, attempting to decide
which one finally produced failure is not only difficult but also technically incorrect (Sowers,
1977). It is therefore beneficial it most cases to consider all potential causes of failure when
designing new slopes and while analyzing existing slope failures.

3
2.3 Shear Strength and Mohr-Coulomb Model
Shear strength is defined as the maximum value of shear stress that the soil can withstand. A
shear failure in which movement caused by shearing stresses in a soil mass of sufficient
magnitude to move a large slope mass or a slope with its foundation relative to the adjacent
stationary mass. The shear strength of soils is controlled by effective stress, regardless of
whether failure occurs under drained or undrained conditions. The relationship between shear
strength and effective stress can be represented by the Mohr–Coulomb strength envelope.
2.4 Limit Equilibrium Method
Limit equilibrium (LE) methods use the Mohr-Coulomb failure criterion to determine the shear
strength along the slip surface. A state of limit equilibrium exists when the mobilized shear stress
is expressed as a fraction of the shear strength. At failure, the shear strength is fully mobilized
along the critical slip surface. The factor of safety (FOS) against slope failure is calculated as the
ratio of the available shear strength to the mobilized shear strength. The available shear strength
depends on the type of soil and the effective normal stress, whereas the mobilized shear stress
depends on the external forces acting on the soil mass.
In LE analysis, the sliding mass is divided into slices, determination of the shear and normal
inter-slice forces is made, and appropriate force and/or moment equilibrium equations are
satisfied for static equilibrium conditions. The first LE method for a round slip surface was
presented by Fellenius (1936). Bishop (1955) later developed a revised method of circular slip
analysis. Meanwhile, Janbu (1954) presented a technique for non-circular failure surfaces that
isolated a potential sliding mass into a few vertical cuts. Later techniques were developed by
Morgenstern-Price (1965), Spencer (1967), Sarma (1973) and a few others to make further
advances with regards to the various assumptions about inter-slice forces.
The LE methods chosen for this study included Bishop’s simplified method (BSM), Janbu’s
simplified method (JSM), Spencer’s method (SM). These methods are commonly used due to
relatively adequate accuracy while calculating the FOS. All LE methods are based on certain
assumptions for the interslice normal and shear forces, and the basic difference among the
methods is how these forces are determined. Some differences may also be noted in the shape of
the critical slip surface for calculation of the FOS.
BSM satisfies moment equilibrium for FOS and vertical force equilibrium for base normal force,
considers interslice normal force, and applies mostly to circular shear surfaces. JSM satisfies
force equilibriums but does not satisfy moment equilibrium, still considers interslice normal
forces, and is commonly used for composite shear surfaces. Both BSM and JSM have limitations
in satisfying both force and moment equilibrium. The SM considers interslice forces, assumes a
constant interslice force function, as well as satisfies and computes FOS for both moment and
force equilibrium.
2.5 Finite Element Method
Numerical modeling is considered a powerful tool for solving many engineering problems and
has become increasingly popular in geotechnical engineering analysis. The two most common
types of numerical methods are finite element (FE) and finite different (FD) method. The finite
element (FE) method, discussed in this paper, uses the soil stress-strain behavior for slope
stability modeling.

4
Unlike LE methods, there is no presumption regarding the shape and location of the failure slip
surface. This is considered an advantage of this method over the traditional LE methods where
FOS are calculated for a pre-determined failure surface. Since there is no concept of slices in FE
analysis, there is no need for assumptions about the lateral inter-slice forces between adjacent
slices. The FE method preserves global equilibrium until failure is reached, and is able to
monitor progressive failure up to and including overall shear failure (Vinod et al., 2017).
The FE approach divides the model into a number of pieces or elements of a mesh. Stresses and
strains are calculated using the constitutive laws for materials comprising of the slope stability
model. Failure occurs naturally through the zones in which the soil shear strength is unable to
sustain the applied shear stresses. Ultimately, a reduction factor (RF) can be calculated for finite
element methods using the ‘c-φ reduction’ procedure. This approach requires incrementally
reducing the soil strength parameters until the failure occurs. The shear strength reduction
technique enables the FE method to calculate FOS (equivalent RF) for slopes.
2.6 Factor of Safety
The most common method for evaluating the stability of slopes is through determination of a
factor of safety (FOS) against failure under a given set of conditions. The FOS is commonly
defined as the ratio of the resisting forces to driving forces along a potential failure surface. As
previously discussed, for FE analyses, the equivalent to a FOS is calculated as a shear strength
reduction factor. The main assumption in LE is that the FOS is the same at all points along the
slope surface and represents an overall average for assumed slip surface.
In theory, a FOS of 1.0 means the driving and resisting forces are at equilibrium. Greater FOS
indicates increased stability, whereas a lower FOS suggests that the slope is unstable. However,
because the inputs involved in computed values of the FOS are not precise, due to uncertainty of
variables, and therefore the FOS should be larger than simply 1.1 or 1.01 to insure the safety of
the slope from failure (Duncan and Wright, 2005).
2.7 Comparison from Other Studies
A number of studies have previously compared the results of slope stability analyses using the
LE and FE methods. Griffiths and Lane (1999) compared results of six simplified slope models
generated using FE analysis to other traditional LE methods. It was concluded that the FE
method was powerful alternative to traditional LE methods and its widespread use should now be
standardized in geotechnical practice. Another study (Hammah et al., 2004) compared FE
analysis of slopes to several LE methods on a wide range of slope cases. Although the author of
this study recommended adopting the FE method using shear strength RF as an additional robust
and powerful tool for design and analysis, conducting further research using this method us also
recommended.
Cheng et al. (2006) performed slope stability analysis of homogenous and nonhomogeneous
simplified slope cases with various material properties in order to compare the LE results with
the FE method results. The study concluded that the results are generally in good agreement for
homogenous slopes. In was further concluded that both the LE and FE methods have their own
merits and limitations, and the use of FE method is not superior to the use of LE methods in
routine analysis and design. While both methods provide an estimation of the FOS and probable
failure mechanism, engineers should appreciate the limitations of each method when assessing

5
the results of the respective analysis. Khabbaz et al. (2012) presented the results of a study
comparing the limitations and advantages of the LE and FE methods. The findings indicated that
when a simple homogeneous slope is considered, the difference in the FOS and locations of the
critical slip surface are minimal and both methods generate indistinguishable results.
Furthermore, this study states that differences between the FE and LE methods are evident when
a heterogeneous slope is analyzed.
Vinod et al. (2017) concluded that FOS differences between LE and FE methods are very small.
A recent study (Zein et al., 2017) compared results of LE and FE slope stability analysis of
simple slope models founded on untrained clay soils. Results of this study indicated that
although LE methods are simple and relatively fast, relatively accurate and reliable results can be
produced, and such methods may be applied with confidence in routine design for slopes with
simple geometry. Lastly, the study concluded that the FE method is an advanced and valuable
analysis technique, especially while modeling design cases with complex slope geometry,
heterogeneous soil behavior and different loading patterns.
3.0 SIMPLE SLOPE MODEL
3.1 Slope Geometry
It is vastly understood that stability of a slope is a direct function of the height and gradient of
the slope, as well as material properties of the soil/rock material. For purposes of this study, a
simple slope section was modeled with combinations of several different slope heights and
gradients with a variety of different soil conditions as shown in Figure 1. Three different slope
gradients (β) of 1H:1V, 1.5H:1V & 2H:1V were modeled at three different heights (H) of 5m,
10m & 20m, resulting in a total of nine different combinations of slope geometry. The base of
the slope model was assumed to be rigid (or bedrock).
Figure 1. Simple slope geometry with homogenous soil
3.2 Soil Properties
Eight different soil types with unique assumptive soil properties were modeled as homogenous
materials for slope stability analyses using both LE and FE methods. The selection of these soil
types was intended to provide a wide array of strength characteristics ranging from cohesionless
sands to purely cohesive soils, modeled as loose/soft and dense/hard. In order to avoid numerical
errors while completing the analyses, the cohesion parameter for the cohesionless sands was
chosen to be small enough to approximately equate zero. Table 1 provides a summary of the
parameters used to create the various soil types for purposes of this study.
H
β
Material Properties
γ, ϕ, c, E, ν

6
Table 1. List of soil types and assumed elastic/plastic parameters
Soil
Soil
ID
Poisson’s
Ratio, ν
Young’s
Modulus, E
(MPa)
Density
(kg/m3)
Cohesion, c
(kPa)
Friction
Angle, φ
(degrees)
Clean Sand (loose)
S1
0.3
15
1850
1x104
31
Clean Sand (dense)
S2
0.3
25
1950
1x104
34
Silty Sand (loose)
S3
0.3
15
1850
10
30
Silty Sand (dense)
S4
0.3
25
1950
20
32
Sandy Clay (soft)
S5
0.4
3
1300
15
20
Sandy Clay (stiff)
S6
0.4
15
1750
40
28
Lean Clay (soft)
S7
0.4
3
1300
15
0
Lean Clay (stiff)
S8
0.4
15
1750
40
0
4.0 SOFTWARE & ANALYSIS APPROACH
4.1 Limit Equilibrium Analysis
SLIDE software, developed by Rocscience, Inc., was used for LE slope stability analysis. SLIDE
is a two-dimensional based computer program, which can be used to evaluate the stability of
circular or non-circular failure surfaces. SLIDE is widely used in the engineering world for
relatively quick and easy computations of a variety of different slope configurations.
Computations of FOS in SLIDE were based on the Mohr-Coulomb failure criterion without
tension cracks. The model requires basic strength parameters such as the angle of internal
friction (φ) and cohesion (c). Searching of the critical slip surface was achieved by using the
“auto-grid” option. In general, a 50x60 grid was defined, as shown in Figure 2, which resulted in
over 3,000 slip centers.
Figure 2. Auto-grid generated by SLIDE for slope model with β = 1.5H:1V & H = 10m

7
Each point in the slip center grid represents the center of rotation of a series of slip circles. The
number of slices was set to 50 and a half-sine function was used to compute the interslice forces
with tolerance error of 0.5%. Additionally, “slope limits” were set at each end of the slope model
to define the external boundaries of potential slip surfaces.
4.2 Finite Element Analysis
FE slope analyses for this study were completed using COMSOL. COMSOL is a Multiphysics
modeling tool that provides solutions to multiple physical problems based on finite element and
partial differential equations. COMSOL provides a user-friendly interface for mesh generation,
equations configuration, and results visualization.
Two-dimensional slope stability analysis in COMSOL was modeled under plane strain
assumptions. After drawing the given slope geometries and creating soil materials, boundary
conditions were defined for the model. The bottom-most boundary was set as ‘fixed’, while
either of the sides was defined as ‘rollers’. As required in the FE analysis, a ‘free triangular’
mesh with ‘extra fine’ mesh size was applied to the model as shown in Figure 3.
Figure 3. Mesh generated by COMSOL for slope model with β = 1.5H:1V & H = 10m
The Solid Mechanics module was applied for modeling the soil mechanical system and
describing various types of elastic and plastic stress-strain relationships in soils. For this study,
the Mohr-Coulomb model was used with plastic behavior described through the Drucker-Prager
failure criterion. The resulting model is based on the elastic-perfectly plastic theory of soil
mechanics, and both elastic parameters (E, ν) and plastic parameters (c, φ, Ψ) are used in the
model. The Young’s Modulus (E) and Poisson’s ratio (ν) are the two most basic properties
required for elastic stress-strain relationship, while the dilatancy angle (Ψ) is related to
describing the volumetric expansion of soils under plastic flow.
As previously discussed, as an equivalent for FOS, a reduction factor (RF) can be calculated for
finite element methods using the ‘c-φ reduction’ procedure. This approach requires
incrementally reducing the soil strength parameters until the failure occurs. In COMSOL, the
strength parameters are automatically reduced using the ‘auxiliary sweep’ function until the final
calculation step results in a fully developed failure plane.
5.0 SUMMARY OF RESULTS
Since the LE and FE analysis methods are based on different principles, the two approaches
result in uniquely different FOS or RF. This section provides a summary of comparison of the
results computed for 400+ analyses completed for this study. Tables 2a, 2b and 2c present FOS
computed using the BSM, JSM and SM for LE analysis, alongside the equivalent RF calculated
using the FE approach for each of the previously discussed slope geometries and soil types.

8
Table 2a. Calculated FOS and RF for slope model with β = 1H:1V
LE = Limit Equilibrium; FE = Finite Element; BSM = Bishop’s Simplified Method; JSM = Janbu’s
Simplified Method; SM = Spencer’s Method
Table 2b. Calculated FOS and RF for slope model with β = 1.5H:1V
LE = Limit Equilibrium; FE = Finite Element; BSM = Bishop’s Simplified Method; JSM = Janbu’s
Simplified Method; SM = Spencer’s Method
Table 2c. Calculated FOS and RF for slope model with β = 2H:1V
LE = Limit Equilibrium; FE = Finite Element; BSM = Bishop’s Simplified Method; JSM = Janbu’s
Simplified Method; SM = Spencer’s Method
Soil
Slope Height = 5m
Slope Height = 10m
Slope Height = 20m
LE
FE
LE
FE
LE
FE
BSM
JSM
SM
BSM
JSM
SM
BSM
JSM
SM
S1
0.603
0.602
0.602
0.470
0.602
0.602
0.602
0.495
0.602
0.601
0.602
0.495
S2
0.681
0.676
0.676
0.520
0.680
0.675
0.676
0.564
0.680
0.675
0.675
0.570
S3
1.683
1.610
1.684
1.281
1.251
1.185
1.247
1.038
0.999
0.946
0.995
0.906
S4
2.435
2.361
2.435
1.600
1.705
1.631
1.700
1.320
1.297
1.226
1.291
1.144
S5
2.143
2.112
2.141
2.152
1.398
1.355
1.400
1.393
0.981
0.938
0.979
0.957
S6
3.879
3.869
3.890
3.895
2.434
2.388
2.453
2.445
1.658
1.596
1.652
1.613
S7
1.386
1.442
1.441
1.232
0.693
0.718
0.722
0.619
0.346
0.357
0.355
0.313
S8
2.747
2.857
2.885
2.413
1.374
1.423
1.432
1.213
0.686
0.709
0.710
0.606
Soil
Slope Height = 5m
Slope Height = 10m
Slope Height = 20m
LE
FE
LE
FE
LE
FE
BSM
JSM
SM
BSM
JSM
SM
BSM
JSM
SM
S1
0.903
0.902
0.903
0.869
0.903
0.902
0.902
0.895
0.902
0.902
0.902
0.863
S2
1.012
1.012
1.013
0.995
1.012
1.013
1.013
1.000
1.011
1.012
1.012
1.000
S3
2.083
1.927
2.082
1.282
1.609
1.499
1.605
1.113
1.321
1.249
1.315
1.013
S4
2.942
2.738
2.934
1.559
2.131
1.993
2.126
1.313
1.663
1.558
1.659
1.164
S5
2.507
2.368
2.504
2.195
1.678
1.571
1.678
1.620
1.221
1.139
1.221
1.138
S6
4.472
4.271
4.469
3.820
2.890
2.713
2.885
2.556
2.030
1.888
2.024
1.919
S7
1.423
1.434
1.436
1.269
0.712
0.716
0.719
0.632
0.356
0.358
0.356
0.319
S8
2.820
2.842
2.857
2.520
1.410
1.419
1.421
1.256
0.705
0.709
0.712
0.632
Soil
Slope Height = 5m
Slope Height = 10m
Slope Height = 20m
LE
FE
LE
FE
LE
FE
BSM
JSM
SM
BSM
JSM
SM
BSM
JSM
SM
S1
1.204
1.203
1.203
1.020
1.203
1.202
1.203
0.994
1.203
1.202
1.202
1.020
S2
1.351
1.350
1.351
1.157
1.350
1.350
1.350
1.113
1.350
1.350
1.350
1.063
S3
2.469
2.257
2.463
1.273
1.945
1.821
1.945
1.095
1.640
1.562
1.634
1.045
S4
3.408
3.123
3.403
1.591
2.524
2.325
2.519
1.313
2.025
1.909
2.024
1.278
S5
2.816
2.595
2.811
2.200
1.941
1.782
1.937
1.713
1.451
1.340
1.452
1.419
S6
4.978
4.595
4.971
3.900
3.311
3.038
3.304
2.538
2.392
2.199
2.388
2.231
S7
1.455
1.430
1.457
1.300
0.727
0.713
0.729
0.664
0.364
0.357
0.364
0.345
S8
2.883
2.833
2.887
2.578
1.441
1.414
1.442
1.294
0.721
0.707
0.721
0.657

9
For further optimization of the RF obtained from FE analyses, two different dilatancy angles
(Ψ=10° and Ψ=15°) were introduced into the soil properties for all slope models. Tables 3a, 3b
and 3c present FOS computed using the SM for LE analysis, with equivalent RF calculated using
positive dilatancy angles with the FE approach for all cases.
Table 3a. Calculated FOS from SM and RF with positive Ψ for slope model with β = 1H:1V
LE = Limit Equilibrium; FE = Finite Element; SM = Spencer’s Method
Table 3b. Calculated FOS from SM and RF with positive Ψ for slope model with β = 1.5H:1V
LE = Limit Equilibrium; FE = Finite Element; SM = Spencer’s Method
Table 3c. Calculated FOS from SM and RF with positive Ψ for slope model with β = 2H:1V
LE = Limit Equilibrium; FE = Finite Element; SM = Spencer’s Method
Soil
Slope Height = 5m!
Slope Height = 10m!
Slope Height = 20m!
LE!
FE!
LE!
FE
LE
FE
SM
Ψ=10°
Ψ=15°
SM
Ψ=10°
Ψ=15°
SM
Ψ=10°
Ψ=15°
S1
0.602
0.620
0.620
0.602
0.620
0.620
0.602
0.620
0.620
S2
0.676
0.689
0.689
0.676
0.689
0.689
0.675
0.689
0.689
S3
1.684
1.394
1.720
1.247
1.120
1.278
0.995
0.988
1.020
S4
2.435
1.770
2.470
1.700
1.413
1.745
1.291
1.174
1.320
S5
2.141
2.170
2.164
1.400
1.420
1.420
0.979
0.995
1.000
S6
3.890
3.900
3.895
2.453
2.470
2.470
1.652
1.695
1.695
S7
1.441
1.200
0.764
0.722
0.620
0.488
0.355
0.320
0.213
S8
2.885
2.395
1.494
1.432
1.200
0.820
0.710
0.600
0.452
Soil
Slope Height = 5m!
Slope Height = 10m!
Slope Height = 20m!
LE!
FE!
LE!
FE
LE
FE
SM
Ψ=10°
Ψ=15°
SM
Ψ=10°
Ψ=15°
SM
Ψ=10°
Ψ=15°
S1
0.903
0.900
0.920
0.902
0.920
0.920
0.902
0.920
0.920
S2
1.013
1.020
1.020
1.013
1.020
1.020
1.012
1.020
1.020
S3
2.082
1.420
2.095
1.605
1.177
1.620
1.315
1.113
1.339
S4
2.934
1.713
2.945
2.126
1.439
2.164
1.659
1.264
1.695
S5
2.504
2.520
2.500
1.678
1.695
1.695
1.221
1.245
1.245
S6
4.469
4.475
4.413
2.885
2.920
2.900
2.024
2.064
2.064
S7
1.436
1.259
0.684
0.719
0.639
0.364
0.356
0.329
0.200
S8
2.857
2.481
1.359
1.421
1.245
0.713
0.712
0.629
0.413
Soil
Slope Height = 5m!
Slope Height = 10m!
Slope Height = 20m!
LE!
FE!
LE!
FE
LE
FE
SM
Ψ=10°
Ψ=15°
SM
Ψ=10°
Ψ=15°
SM
Ψ=10°
Ψ=15°
S1
1.203
1.100
1.220
1.203
1.095
1.220
1.202
1.113
1.220
S2
1.351
1.256
1.345
1.350
1.213
1.370
1.350
1.213
1.370
S3
2.463
1.420
2.485
1.945
1.194
1.970
1.634
1.148
1.670
S4
3.403
1.629
3.395
2.519
1.506
2.545
2.024
1.294
2.045
S5
2.811
2.820
2.800
1.937
1.964
1.945
1.452
1.470
1.470
S6
4.971
4.969
4.913
3.304
3.320
3.300
2.388
2.400
2.395
S7
1.457
1.278
0.594
0.729
0.649
0.313
0.364
0.330
0.220
S8
2.887
2.545
1.169
1.442
1.269
0.620
0.721
0.650
0.345

10
6.0 COMPARISON OF RESULTS
The results of this study were analyzed for purposes of comparing different methods, slope
geometries and soil types. The conclusions presented herein are focused on comparisons of
computed FOS and RF for slope stability analysis, along with a comparison of the shape/location
of critical slip surfaces and effects of slope geometry and soil properties.
In general, the FOS/RF decreases with taller slope height and steeper slope gradients. Between
FOS computed using the three LE methods, the JSM resulted in slightly lower FOS,
approximately 4% lower than the average FOS. Figures 4a, 4b and 4c show a comparison of FOS
and RF calculated for a given slope model with different soil materials. The green bars show
FOS computed using the three LE methods (BSM, JSM & SM), while the blue bars show the
equivalent RF calculated using three different dilatancy angles (0°, 10°, 15°) in the following
figures.
Figure 4a. Bar chart comparison of FOS & RF calculated for slope model with β = 1H:1V & H = 20m
BSM = Bishop’s Simplified Method; JSM = Janbu’s Simplified Method; SM = Spencer’s Method

11
Figure 4b. Bar chart comparison of FOS & RF calculated for slope model with β = 1.5H:1V & H = 10m
BSM = Bishop’s Simplified Method; JSM = Janbu’s Simplified Method; SM = Spencer’s Method
Figure 4c. Bar chart comparison of FOS & RF calculated for slope model with β = 2H:1V & H = 5m
BSM = Bishop’s Simplified Method; JSM = Janbu’s Simplified Method; SM = Spencer’s Method

12
By modeling the slopes with different soil types, comparisons were made based on the variability
in shear strength properties. For comparisons between FOS and RF, the LE-SM was used due to
the more reliable nature of this approach. Differences in the shape and location of critical failure
surfaces have been discussed for each soil type. The following sections summarize the
comparisons of this study for the different soil types.
6.1 Clean Sands
For clean sands (c = 0), the FOS remained unchanged with different H for a fixed slope gradient.
These results are consistent with the equation for infinite slopes in dry cohesionless soils
(Samtani et al., 2006), whereby the FOS is solely dependent upon the soil friction angle (φ) and
the slope gradient (β). Additionally, FOS were computed to be relatively consistent for all three
LE methods when using cohesionless soil parameters. Furthermore, for β = 1H:1V & β =
1.5H:1V, RF computed using Ψ = 10° & Ψ = 15° provided a close match with FOS from LE
methods. For shallower slope gradients (β = 2H:1V), Ψ of 15° provided a better match between
RF and FOS. It is also noteworthy that in the absence of a positive Ψ, the resulting RF are
significantly lower than the equivalent FOS.
Figure 5. Comparison of critical failure surface between SLIDE (left) and COMSOL (right)
using slope model with β = 2H:1V & H = 20m, Ψ = 15° for Dense Clean Sands
In comparing the resulting slope failure surfaces for clean sands, relatively similar shallow
surficial failures can be seen as shown in Figure 5. The LE method choses a localized zone
within this surficial failure as most critical, whereas FE analysis shows the continuous
distribution of plastic strain across the failure. Higher concentrations of plastic strain can be seen
at the toe of the slope using FE analysis.

13
6.2 Cohesive Soils
For purely cohesive clayey soils (φ = 0), change in β did not result in a significant change in the
computed FOS and RF with fixed H. The computed FOS for the purely cohesive soils was noted
to be higher than the equivalent RF. Also noteworthy for these soils is the greatest relative
difference between FOS and RF when compared to other soil types. Between the three LE
methods, SM resulted in slightly higher and BSM resulted in slightly lower FOS than the
average. In contrast to clean sand soils, positive Ψ selected for this study resulted in significantly
lower RF, whereas Ψ= 0° provided a better match with the equivalent FOS.
Figure 6. Comparison of critical failure surface between SLIDE (left) and COMSOL (right)
using slope model with β = 2H:1V & H = 5m, Ψ = 0° for Stiff Lean Clays
As seen in Figure 6, both LE and FE methods produce deep-seated failure surfaces through the
foundation for purely cohesive soils. The LE method computes the most critical path along a
relatively defined zone, whereas the FE method indicates a much wider zone of critical failure.
Another difference between the two methods is noted to be the horizontal extents of the failure
surfaces. This appears to be due to the difference in how the “slope limits” are defined in SLIDE
utilizing LE approach versus how boundary conditions are defined in FE analyses. The highest
concentrations of plastic strain were noted near the toe of the slope using FE analysis.
6.3 Silty Sands & Sandy Clays
In an effort to evaluate the sensitivity of the shear strength parameters when used in combination,
Silty Sand and Sandy Clay soil types were assumed. Silty Sands were modeled with relatively
higher φ and relatively lower c, whereas Sandy Clays were modeled with relatively lower φ and
relatively higher c.
For Silty Sands, Ψ of 15° provided a better match between RF and FOS. The RF were noted to
be significantly lower in the absence of a positive Ψ, as well as for lower Ψ values (Ψ = 10°). For
Sandy Clays, a relatively close comparison was noted for β = 1H:1V under fixed H. With
shallower β (β = 1.5H:1V & 2H:1V), positive Ψ provided a better match between the RF and the
equivalent FOS. A combination of no Ψ, shallower β and smaller H resulted in significantly
lower RF, as well as the greatest difference compared to the equivalent FOS.

14
Figure 7. Comparison of critical failure surface between SLIDE (left) and COMSOL (right)
using slope model with β = 1.5H:1V & H = 10m, Ψ = 15° for Loose Silty Sand
Figure 8. Comparison of critical failure surface between SLIDE (left) and COMSOL (right)
using slope model with β = 2H:1V & H = 10m, Ψ = 15° for Soft Sandy Clay
In comparing the resulting slope failure surfaces for silty sand and sandy clay soils while using a
Ψ of 15° for the FE analysis, fairly identical toe failure surfaces were observed. Unique to the
silty sand and sandy clay soils, the FE analyses produce a better-defined failure path using
positive Ψ. A small difference in the shape of the failure surface was noted; where the critical
failure surfaces resulting from LE analyses were perfectly circular due to the selected search
criterion, the FE method produces a near-circular zone of failure with a slightly more acute angle
of the failure surface near the toe of the slope. Similar to all other soil types analyzed during this
study, higher concentrations of plastic strain were observed near the toe of the slope from FE
analysis.

15
7.0 CONCLUSIONS
Traditionally, slope stability analyses have been completed using LE methods. The LE methods
provide reasonably reliable results for critical slope surfaces with limited required input.
However, Krahn (2003) states: “LE methods are missing the fundamental physics of stress-strain
relationship, and thus they are unable to compute a realistic stress distribution”. Over the years,
FE analyses have gained more attention due to their ability of modeling the stress-strain behavior
of soil. This is looked upon as the biggest advantage of FE methods over LE methods. The LE
analysis computes FOS for each element along the critical failure surface, whereas a weighted
average RF is computed in the FE analyses. Due to the different fundamental approaches,
comparison between FOS and RF reveals the inherent difference between LE and FE methods.
Both LE and FE methods have their own benefits and limitations, and both methods are used to
provide an estimation of the safety factors and slip surfaces. While similar studies have
highlighted the robust nature of FE analyses to carry out more complex problems, the LE
methods have long been established as the industry standard due to its ability for simpler and
faster analyses. This gives LE analyses a slight edge over FE analyses, which requires a tedious
input of all the input parameters. On the other hand, it is difficult in LE analysis to evaluate the
interslice forces, which depend on a number of factors including the stress-strain and
deformation characteristics of the materials in the slope (Chowdhury, 1978). Therefore, it is
recommended that the users choose the method that best fits the nature and intent of slope
analysis being undertaken (Khabbaz, 2012).
With the understanding of the different analysis approaches between LE and FE methods, as well
as their advantages and limitations, conclusions were drawn from a variety of comparisons made
between a number of slope models. The following is a final summary of all the conclusions
drawn from this study:
§ In general, the LE and FE methods used in this study provide fairly consistent FOS and RF.
When comparing relatively similar critical failure surfaces between the LE and FE
methods, FE analyses show higher concentrations of plastic strain near the toe of the slope.
§ In comparing the different LE methods for slope stability analysis, JSM results in
approximately 4% lower FOS compared to the average FOS using other methods.
Moreover, the SM is considered to be a more reliable LE method due its more complete
approach in satisfying the force and moment equilibriums.
§ Although slope geometry plays a significant role in determining safety factors for stability
of slopes, soil shear strength properties dominate as the primary-most factor in computing
safety factors and producing unique failure surfaces.
§ The stability of slopes in cohesionless soil is controlled by shallow surficial failures.
Hence, slope gradient (β) is the primary factor in determining safety factors for
cohesionless soils. Relatively similar critical failure surfaces were noted between the LE
and FE methods for cohesionless sands.
§ Slope stability analyses of purely cohesive soils result in deep-seated critical failure
surfaces. Slope height (H) was observed to be the primary aspect of slope geometry
affecting slope stability of clayey soils. FOS computed for these soils were notably higher
than the equivalent RF. The LE method resulted in a relatively defined critical failure
surface, whereas the FE method showed a much wider zone of critical failure.

16
§ For more commonly occurring soils with shear strength provided through both cohesion
and internal friction, such as silty sands and sandy clays, reasonably identical failure
surfaces were observed between the LE method and the best-matched FE analyses. The FE
analyses for these soils produce fairly sharp failure surfaces using positive Ψ, and also
appear to take into account strain associated deformations.
§ Results indicate that if a Ψ of 15° is used for FE analysis of all soils, except purely
cohesive clays (φ = 0), the difference between the computed RF and equivalent FOS can be
minimized.
8.0 REFERENCES
Bishop, A.W. (1955). The use of slip circles in stability analysis of slopes. Geotechnique, Vol. 5
No. 1, pp. 7-17.
Chowdhury, R.N. (1978). Slope Analysis. Development in Geotechnical Engineering Vol 22, pp.
137-53.
Cruikshank, K.M., and Johnson, A.M. (2002). Theory of Slope Stability: G 483/583 Anatomy of
Landslides, Portland State University.
Duncan, J.M. and Wright, S.G. (2005). Soil Strength and Slope Stability.
Fellenius, W. (1936). Calculation of the Stability of Earth Dams, Trans. 2nd Cong. on Large
Dams, Vol. 4, p. 445.
Griffiths, D.V., and Lane, P.A. (1999). Slope Stability Analysis by Finite Elements,
Geotechnique, Vol. 49, No. 3, pp. 387-403.
Hammah, R., Yacoub, T., Corkum, B., and Curran, J. (2004). A Comparison of Finite Element
slope stability Analysis with Conventional Limit-Equilibrium Investigation, University of
Toronto.
Janbu, N. (1954). Stability analysis of Slopes with Dimensionless Parameters. Thesis for the
Doctor of Science in the Field of Civil Engineering, Harvard University Soil Mechanics
Series, No. 46.
Khabbaz, H., Fatahi, B., and Nucifora, C. (2012). Finite Element Methods against Limit
Equilibrium Approaches for Slope Stability Analysis, Centre for Built Infrastructure
Research, School of Civil and Environmental Engineering, The University of Technology
Sydney.
Krahn, J. (2003). The limits of limit equilibrium analysis, Canadian Geotechnical Journal, Vol.
44, pp. 643-60.
Morgenstern, N., and Price, V.E., (1965). The Analysis of the Stability of General Slip Surfaces,
Geotechnique, Vol. 15, No. 1, pp.79-93.
Samtani, N.C., and Nowatzki, E.A. (2006). Soils and Foundations, Reference Manual - Volume
I, U.S. Department of Transportation, Federal Highway Administration, Publication No.
FHWA NHI-06-088.
Sarma, S.K. (1973). Stability analysis of embankments and slopes, Geotechnique, Vol. 23, No. 3,
pp. 423-433.

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Sowers, G.F. (1979). Introductory Soil Mechanics and Foundations: Geotechnical Engineering,
4th ed.
Spencer, E. (1967). A method of Analysis of the Stability of Embankments, Assuming Parallel
Interslice Forces. Geotechnique, Vol. 17, pp. 11-26.
U.S. Geological Survey (USGS), Natural Hazards, Frequently asked questions, webpage:
https://www.usgs.gov/faqs/, last visited 10/19/2018.
Vinod B.R., P. Shivananda, Swathivarma R., Bhaskar M.B. (2017). Some of Limit Equilibrium
Method and Finite Element Method based Software are used in Slope Stability Analysis,
International Journal of Application or Innovation in Engineering & Management, Vol. 6,
Issue 9.
Zein, A.K., and Karim, W.A. (2017). Stability of slopes on clays of variable strength by limit
equilibrium and finite element analysis methods, International Journal of GEOMATE, Vol
13, Issue 38, pp.157-164.