Slope stability analysis by finite elements

Abstract

The majority of slope stability analyses performed in practice still use traditional limit equilibrium approaches involving methods of slices that have remained essentially unchanged for decades, This was not the outcome envisaged when Whitman & Bailey (1967) set criteria for the then emerging methods to become readily accessible to all engineers, The finite element method represents a powerful alternative approach for slope stability analysis which is accurate, versatile and requires fewer a priori assumptions, especially, regarding the failure mechanism, Slope failure in the finite element model occurs 'naturally' through the zones in which the shear strength of the soil is insufficient to resist the shear stresses. The paper describes several examples of finite element slope stability analysis with comparison against other solution methods, including the influence of a free surface on slope and dam stability. Graphical output is included to illustrate deformations and mechanisms of failure. It is argued that the finite element method of slope stability analysis is a more powerful alternative to traditional limit equilibrium methods and its widespread use should now be standard in geotechnical practice.

Full text

Slope stability analysis by ®nite elements
D. V. GRIFFITHSandP.A.LANE{
The majority of slope stability analyses per-
formed in practice still use traditional limit
equilibrium approaches involving methods of
slices that have remained essentially unchanged
for decades. This was not the outcome envisaged
when Whitman & Bailey (1967) set criteria for
the then emerging methods to become readily
accessible to all engineers. The ®nite element
method represents a powerful alternative ap-
proach for slope stability analysis which is accu-
rate, versatile and requires fewer a priori
assumptions, especially, regarding the failure
mechanism. Slope failure in the ®nite element
model occurs `naturally' through the zones in
which the shear strength of the soil is insuf®-
cient to resist the shear stresses. The paper
describes several examples of ®nite element
slope stability analysis with comparison against
other solution methods, including the in¯uence
of a free surface on slope and dam stability.
Graphical output is included to illustrate defor-
mations and mechanisms of failure. It is argued
that the ®nite element method of slope stability
analysis is a more powerful alternative to tradi-
tional limit equilibrium methods and its wide-
spread use should now be standard in
geotechnical practice.
KEYWORDS: dams; limit equilibrium methods; num-
erical modelling; plasticity; slopes.
En grande majorite
Â, les analyses de stabilite
Âde
pente mene
Âes dans la pratique continuent a
Á
utiliser les me
Âthodes traditionnelles d'e
Âquilibre
limite et des syste
Ámes de tranches qui n'ont
pratiquement pas change
Âdepuis des dizaines
d'anne
Âes. Ce n'e
Âtait pas le re
Âsultat envisage
Â
quand Whitman et Bailey (1967) ont e
Âtabli des
crite
Áres pour que ces me
Âthodes alors e
Âmer-
geantes puissent devenir facilement accessibles a
Á
tous les inge
Ânieurs. La me
Âthode d'e
Âle
Âments ®nis
qui repre
Âsente une alternative puissante pour les
analyses de stabilite
Âde pente, est exacte, poly-
valente et demande moins d'hypothe
Áses `a
priori', surtout en ce qui concerne les me
Âca-
nismes de rupture. La rupture de pente dans le
mode
Ále a
Áe
Âle
Âments ®nis se produit `naturelle-
ment' a
Átravers des zones dans lesquelles la
re
Âsistance au cisaillement du sol est insuf®sante
pour re
Âsister aux contraintes tangentielles. Cet
expose
Âde
Âcrit plusieurs exemples d'analyses de
stabilite
Âde pente utilisant les e
Âle
Âments ®nis et
e
Âtablit des comparaisons avec d'autres me
Âth-
odes, comme l'in¯uence d'une surface libre sur
la stabilite
Âd'une pente et d'une digue. Nous
joignons une repre
Âsentation graphique pour il-
lustrer les de
Âformations et me
Âcanismes de rup-
ture. Nous avancËons que la me
Âthode d'e
Âle
Âments
®nis pour analyser la stabilite
Âdes pentes consti-
tue une alternative plus puissante aux me
Âthodes
traditionnelles d'e
Âquilibre limite et que son utili-
sation devrait maintenant devenir une pratique
standard en ge
Âotechnique.
INTRODUCTION
Elasto-plastic analysis of geotechnical problems
using the ®nite element (FE) method has been
widely accepted in the research arena for many
years; however, its routine use in geotechnical
practice for slope stability analysis still remains
limited. The reason for this lack of acceptance is
not entirely clear; however, advocates of FE tech-
niques in academe must take some responsibility.
Practising engineers are often sceptical of the need
for such complexity, especially in view of the poor
quality of soil property data often available from
routine site investigations. Although this scepticism
is often warranted, there are certain types of
geotechnical problem for which the FE approach
offers real bene®ts. The challenge for an experi-
enced engineer is to know which kind of problem
would bene®t from a FE treatment and which
would not.
In general, linear problems such as the predic-
tion of settlements and deformations, the calcula-
tion of ¯ow quantities due to steady seepage or the
study of transient effects due to consolidation are
all highly amenable to solution by ®nite elements.
Traditional approaches involving charts, tables or
Grif®ths, D. V. & Lane, P. A. (1999). Ge
Âotechnique 49, No. 3, 387±403
387
Manuscript received 28 May 1998; revised manuscript
accepted 8 December 1998. Discussion on this paper
closes 3 September 1999; for further details see p. ii.
Colorado School of Mines, Golden.
{UMIST, Manchester.

graphical methods will often be adequate for rou-
tine problems but the FE approach may be valuable
if awkward geometries or material variations are
encountered which are not covered by traditional
chart solutions.
The use of nonlinear analysis in routine geotech-
nical practice is harder to justify, because there is
usually a signi®cant increase in complexity which
is more likely to require the help of a modelling
specialist. Nonlinear analyses are inherently itera-
tive in nature, because the material properties and/
or the geometry of the problem are themselves a
function of the `solution'. Objections to nonlinear
analyses on the grounds that they require excessive
computational power, however, have been largely
overtaken by developments in, and falling costs of,
computer hardware. A desktop computer with a
standard processor is now capable of performing
nonlinear analyses such as those described in this
paper in a reasonable time spanÐminutes rather
than hours or days.
Slope stability represents an area of geotechni-
cal analysis in which a nonlinear FE approach
offers real bene®ts over existing methods. As this
paper will show, slope stability analysis by elasto-
plastic ®nite elements is accurate, robust and
simple enough for routine use by practising en-
gineers. Perception of the FE method as complex
and potentially misleading is unwarranted and
ignores the real possibility that misleading results
can be obtained with conventional `slip circle'
approaches. The graphical capabilities of FE pro-
grams also allow better understanding of the
mechanisms of failure, simplifying the output
from reams of paper to manageable graphs and
plots of displacements.
TRADITIONAL METHODS OF SLOPE STABILITY
ANALYSIS
Most textbooks on soil mechanics or geotech-
nical engineering will include reference to several
alternative methods of slope stability analysis. In a
survey of equilibrium methods of slope stability
analysis reported by Duncan (1996), the character-
istics of a large number of methods were sum-
marized, including the ordinary method of slices
(Fellenius, 1936), Bishop's Modi®ed Method
(Bishop, 1955), force equilibrium methods (e.g.
Lowe & Kara®ath, 1960), Janbu's generalized pro-
cedure of slices (Janbu, 1968), Morgenstern and
Price's method (Morgenstern & Price, 1965) and
Spencer's method (Spencer, 1967).
Although there seems to be some consensus that
Spencer's method is one of the most reliable, text-
books continue to describe the others in some
detail, and the wide selection of available methods
is at best confusing to the potential user. For
example, the controversy was recently revisited by
Lambe & Silva (1995), who maintained that the
ordinary method of slices had an undeservedly bad
reputation.
A dif®culty with all the equilibrium methods is
that they are based on the assumption that the
failing soil mass can be divided into slices. This in
turn necessitates futher assumptions relating to side
force directions between slices, with consequent
implications for equilibrium. The assumption made
about the side forces is one of the main character-
istics that distinguishes one limit equilibrium meth-
od from another, and yet is itself an entirely
arti®cial distinction.
FINITE ELEMENT METHOD FOR SLOPE STABILITY
ANALYSIS
Duncan's review of FE analysis of slopes con-
centrated mainly on deformation rather than stabi-
lity analysis of slopes; however, attention was
drawn to some important early papers in which
elasto-plastic soil models were used to assess stabi-
lity. Smith & Hobbs (1974) reported results of
öu0 slopes and obtained reasonable agreement
with Taylor's (1937) charts. Zienkiewicz et al.
(1975) considered a c9,ö9slope and obtained good
agreement with slip circle solutions. Grif®ths
(1980) extended this work to show reliable slope
stability results over a wide range of soil properties
and geometries as compared with charts of Bishop
& Morgenstern (1960). Subsequent use of the FE
method in slope stability analysis has added further
con®dence in the method (e.g. Grif®ths, 1989;
Potts et al., 1990; Matsui & San, 1992). Duncan
mentions the potential for improved graphical re-
sults and reporting utilizing FE, but cautions
against arti®cial accuracy being assumed when the
input parameters themselves are so variable.
Wong (1984) gives a useful summary of poten-
tial sources of error in the FE modelling of slope
stability, although recent results, including those
presented in this paper, indicate that better accu-
racy is now possible.
Advantages of the ®nite element method
The advantages of a FE approach to slope
stability analysis over traditional limit equilibrium
methods can be summarized as follows:
(a) No assumption needs to be made in advance
about the shape or location of the failure
surface. Failure occurs `naturally' through the
zones within the soil mass in which the soil
shear strength is unable to sustain the applied
shear stresses.
(b) Since there is no concept of slices in the FE
approach, there is no need for assumptions
about slice side forces. The FE method
388 GRIFFITHS AND LANE

preserves global equilibrium until `failure' is
reached.
(c) If realistic soil compressibility data are avail-
able, the FE solutions will give information
about deformations at working stress levels.
(d) The FE method is able to monitor progressive
failure up to and including overall shear
failure.
Brief description of the ®nite element model
The programs used in this paper are based
closely on Program 6.2 in the text by Smith &
Grif®ths (1998), the main difference being the
ability to model more general geometries and soil
property variations, including variable water levels
and pore pressures. Further graphical output cap-
abilities have been added. The programs are for
two-dimensional plane strain analysis of elastic±
perfectly plastic soils with a Mohr± Coulomb failure
criterion utilizing eight-node quadrilateral elements
with reduced integration (four Gauss points per
element) in the gravity loads generation, the stiff-
ness matrix generation and the stress redistribution
phases of the algorithm. The soil is initially as-
sumed to be elastic and the model generates normal
and shear stresses at all Gauss points within the
mesh. These stresses are then compared with the
Mohr± Coulomb failure criterion. If the stresses at a
particular Gauss point lie within the Mohr±Cou-
lomb failure envelope, then that location is assumed
to remain elastic. If the stresses lie on or outside
the failure envelope, then that location is assumed
to be yielding. Yielding stresses are redistributed
throughout the mesh utilizing the visco-plastic algo-
rithm (Perzyna, 1966; Zienkiewicz & Cormeau,
1974). Overall shear failure occurs when a suf®-
cient number of Gauss points have yielded to allow
a mechanism to develop.
The analyses presented in this paper do not
attempt to model tension cracks. Although `no-
tension' criteria can be incorporated into elasto-
plastic FE analyses (e.g. Naylor & Pande, 1981),
this additional constraint on stress levels compli-
cates the algorithm, and, in addition, there is still
some debate as to how `tension' should properly
be de®ned. Further research in this area is war-
ranted.
Soil model
The soil model used in this study consists of six
parameters, as shown in Table 1.
The dilation angle øaffects the volume change
of the soil during yielding. It is well known that
the actual volume change exhibited by a soil dur-
ing yielding is quite variable. For example, a
medium± dense material during shearing might
initially exhibit some volume decrease (ø,0),
followed by a dilative phase (ø.0), leading even-
tually to yield under constant volume conditions
(ø0). Clearly, this type of detailed volumetric
modelling is beyond the scope of the elastic±
perfectly plastic models used in this study, where a
constant dilation angle is implied.
The question then arises as to what value of ø
to use. If øö, then the plasticity ¯ow rule is
`associated' and direct comparisons with theorems
from classical plasticity can be made. It is also the
case that when the ¯ow rule is associated, the
stress and velocity characteristics coincide, so clo-
ser agreement can be expected between failure
mechanisms predicted by ®nite elements and criti-
cal failure surfaces generated by limit equilibrium
methods.
In spite of these potential advantages of using
an associated ¯ow rule, it is also well known that
associated ¯ow rules with frictional soil models
predict far greater dilation than is ever observed in
reality. This in turn leads to increased failure load
prediction, especially in `con®ned' problems such
as bearing capacity (Grif®ths, 1982). This short-
coming has led some of the most successful con-
stitutive soil models to incorporate non-associated
plasticity (e.g. Molenkamp, 1981; Hicks & Bough-
rarou, 1998).
Slope stability analysis is relatively uncon®ned,
so the choice of dilation angle is less important.
As the main objective of the current study is the
accurate prediction of slope factors of safety, a
compromise value of ø0, corresponding to a
non-associated ¯ow rule with zero volume change
during yield, has been used throughout this paper.
It will be shown that this value of øenables the
model to give reliable factors of safety and a
reasonable indication of the location and shape of
the potential failure surfaces.
The parameters c9and ö9refer to the effective
cohesion and friction angle of the soil. Although
a number of failure criteria have been suggested
for modelling the strength of soil (e.g. Grif®ths,
1990), the Mohr±Coulomb criterion remains the
one most widely used in geotechnical practice and
has been used throughout this paper. In terms of
principal stresses and assuming a compression±
negative sign convention, the criterion can be
written as follows:
Table 1. Six-parameter soil model
ö9Friction angle
c9Cohesion
øDilation angle
E9Young's modulus
í9Poisson's ratio
ãUnit weight
SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS 389

Fó9
1ó9
3
2sin ö9ÿó9
1ÿó9
3
2ÿc9cos ö9(1)
where ó9
1and ó9
3are the major and minor princi-
pal effective stresses.
The failure function Fcan be interpreted as
follows:
F,0 stresses inside failure envelope (elastic)
F0 stresses on failure envelope (yielding)
F.0 stresses outside failure envelope (yielding
and must be redistributed)
The elastic parameters E9and í9refer to
Young's modulus and Poissons's ratio of the soil. If
a value of Poisson's ratio is assumed (typical
drained values lie in the range 0:2,í9,0:3), the
value of Young's modulus can be related to the
compressibility of the soil as measured in a one-
dimensional oedometer (e.g. Lambe & Whitman,
1969):
E9(1 í9)(1 ÿ2í9)
mv(1 ÿí9)(2)
where mvis the coef®cient of volume compressi-
bility.
Although the actual values given to the elastic
parameters have a profound in¯uence on the
computed deformations prior to failure, they have
little in¯uence on the predicted factor of safety
in slope stability analysis. Thus, in the absence
of meaningful data for E9and í9, they can be
given nominal values (e.g. E9105kN=m2and
í90:3).
The total unit weight ãassigned to the soil is
proportional to the nodal self-weight loads gener-
ated by gravity.
In summary, the most important parameters in a
FE slope stability analysis are the same as they
would be in a traditional approach, namely, the
total unit weight ã, the shear strength parameters
c9and ö9, and the geometry of the problem.
Gravity loading
The forces generated by the self-weight of the
soil are computed using a standard gravity `turn-
on' procedure involving integrals over each ele-
ment of the form:
p(e)ãVe
NTd(vol) (3)
where Nvalues are the shape functions of the
element and the superscript erefers to the element
number. This integral evaluates the area of each
element, multiplies by the total unit weight of the
soil and distributes the net vertical force consis-
tently to all the nodes. These element forces are
assembled into a global gravity force vector that is
applied to the FE mesh in order to generate the
initial stress state of the problem.
The present work applies gravity in a single
increment to an initially stress-free slope. Others
have shown that under elastic conditions, sequential
loading in the form of incremental gravity applica-
tion or embanking affects deformations but not
stresses (e.g. Clough & Woodward, 1967). In non-
linear analyses, it is recognized that the stress paths
followed due to sequential excavation may be quite
different to those followed under a gravity `turn-
on' procedure; however, the factor of safety ap-
pears unaffected when using simple elasto-plastic
models (e.g. Borja et al. 1989; Smith & Grif®ths,
1998).
In comparing our results with limit equilibrium
solutions which generally take no account of load-
ing sequence, our experience has shown that the
predicted factor of safety is insensitive to the form
of gravity application when using elastic ±perfectly
plastic Mohr± Coulomb models. An example of this
insensitivity is demonstrated later in the paper.
The factor of safety may be sensitive to load-
ing sequence when implementing more complex
constitutive laws, such as those that attempt to
reproduce volumetric changes accurately in an
undrained or partially drained environment. For
example, Hicks & Wong (1988) showed that the
effective stress path could have a big in¯uence
on the factor of safety of an undrained slope.
Determination of the factor of safety
The factor of safety (FOS) of a soil slope is
de®ned here as the number by which the original
shear strength parameters must be divided in order
to bring the slope to the point of failure. (This
de®nition of the factor of safety is exactly the
same as that used in traditional limit equilibrium
methods, namely the ratio of restoring to driving
moments.) The factored shear strength parameters
c9
fand ö9
f, are therefore given by:
c9
fc9=FOS (4)
ö9
farctan tan ö9
FOS
 (5)
This method has been referred to as the `shear
strength reduction technique' (e.g. Matsui & San,
1992) and allows for the interesting option of
applying different factors of safety to the c9and
tan ö9terms. In this paper, however, the same
factor is always applied to both terms. To ®nd the
`true' FOS, it is necessary to initiate a systematic
search for the value of FOS that will just cause the
slope to fail. This is achieved by the program
390 GRIFFITHS AND LANE

solving the problem repeatedly using a sequence of
user-speci®ed FOS values.
De®nition of failure
There are several possible de®nitions of failure,
e.g. some test of bulging of the slope pro®le
(Snitbhan & Chen, 1976), limiting of the shear
stresses on the potential failure surface (Duncan &
Dunlop, 1969) or non-convergence of the solution
(Zienkiewicz & Taylor, 1989). These are discussed
in Abramson et al. (1995) from the original paper
by Wong (1984) but without resolution. In the
examples studied here, the non-convergence option
is taken as being a suitable indicator of failure.
When the algorithm cannot converge within a
user-speci®ed maximum number of iterations, the
implication is that no stress distribution can be
found that is simultaneously able to satisfy both
the Mohr± Coulomb failure criterion and global
equilibrium. If the algorithm is unable to satisfy
these criteria, `failure' is said to have occurred.
Slope failure and numerical non-convergence occur
simultaneously, and are accompanied by a dramatic
increase in the nodal displacements within the
mesh. Most of the results shown in this paper used
an iteration ceiling of 1000 and are presented in
the form of a graph of FOS versus E9ämax =ãH2(a
dimensionless displacement), where ämax is the
maximum nodal displacement at convergence and
His the height of the slope. This graph may be
used alongside the displaced mesh and vector plots
to indicate both the factor of safety and the nature
of the failure mechanism.
SLOPE STABILITY EXAMPLES AND VALIDATION
Several examples of FE slope stability analysis
are now presented with validation against tradi-
tional stability analyses where possible. Initial con-
sideration will be given to slopes containing no
pore pressures in which total and effective stresses
are equal. This is followed by examples of inhomo-
geneous undrained clay slopes. Finally, submerged
and partially submerged slopes are considered in
which pore pressures are taken into account.
Example 1: Homogeneous slope with no foundation
layer ( D 1)
(See Fig. 5 for the de®nition of D.) The homo-
geneous slope shown in Fig. 1 has the following
soil properties:
ö9208
c9=ãH0:05
The slope is inclined at an angle of 26´578(2:1)
to the horizontal, and the boundary conditions are
given as vertical rollers on the left boundary and
full ®xity at the base. Gravity loads were applied
to the mesh and the trial factor of safety (FOS)
gradually increased until convergence could not be
achieved within the iteration limit as shown in
Table 2.
The table indicates that six trial factors of safety
were attempted, ranging from 0´8 to 1´4. Each
factor of safety represented a completely indepen-
dent analysis in which the soil strength parameters
were scaled by FOS as indicated in equations (4)
and (5). Some ef®ciencies are possible in that the
gravity loads and global stiffness matrix are the
same in each analysis and are therefore generated
once only.
The `Iterations' column indicates the number of
iterations for convergence corresponding to each
FOS value. The algorithm has to work harder to
achieve convergence as the `true' FOS is approached.
H
1.2
H
2
H
Rollers
Fixed
Fig. 1. Example 1: Mesh for a homogeneous slope with a slope angle of 26´578(2:1), ö9208,c9=ãH0´05
SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS 391

When FOS 1:4, there is a sudden increase in the
dimensionless displacement E9ämax =ãH2, and the
algorithm is unable to converge within the iteration
limit. Fig. 2 shows a plot of the data from Table 2, and
indicates close agreement between the FE result and
the factor of safety given for the same problem by the
charts of Bishop & Morgenstern (1960).
Figure 3 shows the in¯uence of gravity loading
increment size on displacements in example 1.
With a `failure' factor of safety of FOS 1:4
applied to the soil properties, the four graphs
correspond to the maximum displacement obtained
when gravity was applied in a single increment as
compared with that obtained with two, three or ®ve
equal increments. The ®gure demonstrates that the
displacement obtained with full gravity loading is
barely affected by the increment size.
Figures 4(a) and 4(b) give the nodal displace-
ment vectors and the deformed mesh corresponding
to the unconverged situation with FOS 1:4. The
deformed mesh corresponding to this unconverged
solution gives a rather diffuse indication of the
failure mechanism. This is due to the relatively
crude FE mesh, which must remain continuous
even at `failure'. Conventional FE analysis is un-
able to model gross discontinuities along potential
failure surfaces, although techniques have been
described for enhancing the visualization of the
failure surfaces (e.g. Grif®ths & Kidger, 1995).
More advanced FE methods for modelling shear
bands in conjunction with adaptive mesh re®ne-
ment techniques have been described by Loret &
Prevost (1991) and Zienkiewicz et al. (1995).
Example 2: Homogeneous slope with a foundation
layer ( D 1:5)
Figure 5 shows that a foundation layer of thick-
ness H=2 has now been added to the base of the
slope of example 1, with all other properties and
geometry remaining the same.
The initial mesh and the deformed mesh at
failure are shown in Figs 5(a) and 5(b) respectively.
It is clear from Fig. 5(b) that a mechanism of the
`toe failure' type has been obtained. Fig. 2 indi-
cates that the critical factor of safety is essentially
unchanged from example 1 at FOS 1:4, although
the displacements are increased due to the greater
volume of compressible soil.
This FE result con®rms that the addition of the
foundation layer has not led to any perceptible
change in the factor of safety of the slope. Bishop
& Morgenstern (1960) give FOS 1:752 as one
Table 2. Results from example 1
FOS E9ämax=ãH2Iterations
0´80 0´379 2
1´00 0´381 10
1´20 0´422 20
1´30 0´453 41
1´35 0´544 792
1´40 1´476 1000
Bishop & Morgenstern (1960)
FOS 5 1.380
0.25
0.5
0.75
1
1.3
1.5
1.8
2
2.3
2.5
2.8
3
3.3
3.5
3.8
4
E
′δmax/γ
H
2
0.811
.21
.41
.6
FOS
D 5 1.0 (Example 1)
D 5 1.5 (Example 2)
Fig. 2. Examples 1 and 2: FOS versus dimensionless
displacement. The rapid increase in displacement and
the lack of convergence when FOS 1´4 indicates
slope failure
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.600
.20
.40
.60
.811
.2
E
′δmax/γ
H
2
Gravity factor
1 increment
2 equal increments
3 equal increments
5 equal increments
Fig. 3. In¯uence of gravity increment size on maxi-
mum displacement at failure (FOS 1´4) from exam-
ple 1
392 GRIFFITHS AND LANE

Fig. 4. Example 1: Deformed mesh plots corresponding to the unconverged
solution with FOS 1´4: (a) nodal displacement vectors; (b) deformed
mesh
(a)
(b)
Fig. 5. Example 2: Homogeneous slope with a foundation layer. Slope angle 26´578(2:1), ö9208,c9=ãH0´05,
D1´5: (a) undeformed mesh; (b) mesh corresponding to unconverged solution with FOS 1´4
DH
H
(a)
(b)
SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS 393

possible solution for this example (D1:5, c9=
(ãH)0:05, ö9208, 2:1 slope), although it is
important to check the alternative solution corre-
sponding to D1:0 to verify which gives the
lower FOS. The charts of Cousins (1978) essen-
tially agree with the FE result and indicate that,
with a foundation layer, the critical circular me-
chanism at its lowest point passes fractionally
below the base of the slope and gives a slightly
lower factor of safety than when there is no
foundation layer present.
Solving this example using a proprietary slip
circle program also found the possible `result' of
FOS 1:7 when a failure circle tangent to the
base of the foundation was assumed. It was neces-
sary to force the slip circle to pass through the toe
to obtain the `correct' FOS of 1´4.
This example demonstrates one of the main
advantages of FE slope stability analysis over con-
ventional methods. The FE approach requires no a
priori assumption of the location or shape of the
critical surface. Failure occurs `naturally' within
the zones of the soil mass where the shear strength
of the soil is insuf®cient to resist the shear stresses.
The use of a limit equilibirum method requires, at
the very least, some experience and care on the
part of the user in order to initiate appropriate
search procedures which avoid the possibility of
homing in on the wrong `critical' circle.
Example 3: An undrained clay slope with a thin
weak layer
The next example demonstrates a stability analy-
sis of a slope of undrained clay. Effective stress
analysis of an undrained slope of this type could
be performed with respect to effective stresses by
adding a large apparent ¯uid bulk modulus to the
soil constitutive matrix (Naylor, 1974). In this case,
however, a total stress analysis using a Tresca
failure criterion (öu0) is presented.
Figure 6 shows a slope on a foundation layer
(D2) of undrained clay. The slope includes a
thin layer of weaker material which initially runs
parallel to the slope, then horizontally in the
foundation and ®nally outcrops at an angle of
458beyond the toe. Although this example may
seem contrived, it is not unlike the situation of a
thin, weak liner within a land®ll system. The
factor of safety of the slope was estimated by FE
analysis for a range of values of the undrained
shear strength of the thin layer (cu2) while main-
taining the strength of the surrounding soil at
cu1=ãH0:25.
The FE results shown in Fig. 7 give the com-
puted factor of safety expressed to the nearest
0´05. For a homogeneous slope (cu2=cu1 1), the
computed factor of safety was close to the Taylor
solution (Taylor, 1937) of FOS 1:47 and gave
the expected circular base failure mechanism. As
the strength of the thin layer was gradually re-
duced, a distinct change in the nature of the results
was observed when cu2=cu1 0:6.
Also shown on this ®gure are limit equilibrium
solutions obtained using Janbu's method assuming
both circular (base failure) and three-line wedge
mechanisms following the path of the weak layer.
The discontinuity when cu2 =cu1 0:6 clearly re-
presents the transition between the circular me-
chanism and the non-circular mechanism governed
by the weak layer. For cu2 =cu1 .0:6, the (circular)
base failure mechanism governs the behaviour, and
H
H
2
H
2
H
0.6
H
1.2
H
c
u2
c
u1
c
u1
2
1
2
1
2
H
1.2
H
1
0.6
H
1
0.4
H
0.4
H
c
u2 ,
c
u1
φu 5 0
Fig. 6. Example 3: Undrained clay slope with a foundation layer including a thin weak layer (D2,
cu1=ãH0´25)
394 GRIFFITHS AND LANE

the factor of safety is essentially unaffected by the
strength of the weaker thin layer. For cu2=cu1 ,0:6,
the (non-circular) thin layer mechanism takes over
and the factor of safety falls linearly.
This behaviour is explained more clearly in
Fig. 8, which shows the deformed mesh at failure
for three different values of the ratio cu2=cu1 . Fig.
8(a), corresponding to the homogeneous case
(cu2=cu1 1), indicates an essentially circular fail-
ure mechanism tangent to the ®rm base as pre-
dicted by Taylor. Fig. 8(c), in which the strength
of the thin layer is only 20% of that of the
surrounding soil (cu2 =cu1 0:2), indicates a highly
concentrated non-circular mechanism closely fol-
lowing the path of the thin weak layer. Fig. 8(b),
in which the strength of the thin layer is 60% of
that of the surrounding soil (cu2=cu1 0:6), indi-
cates considerable complexity and ambiguity. At
least two con¯icting mechanisms are apparent.
First, there is a base failure mechanism merging
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
FOS
0.200
.40
.60
.81
c
u2/
c
u1
Finite elements
c
u1/γ
H
5 0.25 Taylor (1937)
FOS 5 1.47
Slope program—wedges
Slope program—circles
Fig. 7. Example 3: Computed factor of safety (FOS)
for different values of cu2=cu1
Fig. 8. Example 3: Deformed meshes at failure corresponding to the un-
converged solution for three different values of cu2=cu1 (a) cu2 =cu1 1´0;
(b) cu2=cu1 0´6; (c) cu2 =cu1 0´2
(a)
(b)
(c)
SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS 395

with the weak layer beyond the toe of the slope,
and second, there is a mechanism running along
the weak layer parallel to the slope and outcrop-
ping at the toe.
Without prior knowledge of the two alternative
mechanisms, a traditional limit equilibrium search
could seriously overestimate the factor of safety.
This is illustrated in Fig. 7 where, for example, a
circular mechanism with cu2 =cu1 0:2 would in-
dicate FOS 1:3, when the correct factor of safety
is closer to 0´6.
Example 4: An undrained clay slope with a weak
foundation layer
In this case, the same slope geometry and ®nite
element mesh as in the previous example has been
used but with a different type of inhomogeneity, as
shown in Fig. 9. The shear strength of the slope
material has been maintained at a constant value of
cu1=ãH0:25, while the shear strength of the
foundation layer has been varied. The relative size
of the two shear strengths has again been expressed
as the ratio cu2 =cu1. Fig. 10 shows the computed
factor of safety for a range of cu2 =cu1 values,
together with classical solutions of Taylor for the
two cases when cu2 cu1 and cu2 cu1. There is
clearly a change of behaviour occurring at
cu2=cu1 1:5, as indicated by the ¯attening out of
the curve. Also shown on this ®gure are limit
equilibrium solutions for both toe and base circle
mechanisms. The discontinuity corresponding to
cu2=cu1 1:5 obviously represents the transition
between these two fundamental mechanisms.
This transition is clearly demonstrated by the FE
failure mechanisms shown in Fig. 11. When
cu2 cu1 (Fig. 11(a)), a deep-seated base mechan-
ism is observed (Fig. 11(a)), whereas a shallow
`toe' mechanism is seen when cu2 cu1 (Fig.
11(c)). The result corresponding to the approximate
transition point at cu2 1:5cu1 (Fig. 11(b)) shows
an ambiguous situation in which both mechanisms
are trying to form at the same time. It is interest-
ing to note that the lower soil must be approxi-
mately 50% stronger than the upper soil before the
toe mechanism becomes the most critical.
The previous two examples have shown that
even in quite simple cases, complex interactions
can occur between con¯icting mechanisms within
heterogeneous slopes which can be detected by
the FE approach. For more complicated stability
problems involving several soil property groups
such as a zoned earth embankment, the FE
approach is arguably the only rational method that
will generate the correct factor of safety and
indicate the location and shape of the critical
mechanism.
2
H
2
H
H
H
c
u1
c
u2
2
H
21
φu 5 0
Fig. 9. Example 4: Undrained clay slope with a weak foundation layer
(D2, cu1=ãH0´25)
Finite element analysis
Base circle analysis
Toe circle anlaysis
Taylor
(
c
u2 ..
c
u1)
FOS 5 2.10
Taylor
(
c
u2 5
c
u1)
FOS 5 1.47
c
u1/γ
H
5 0.25
00
.51
.52
.53
.51234
c
u2/
c
u1
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
FOS
Fig. 10. Example 4: Computed factor of safety (FOS)
for different values of cu2=cu1
396 GRIFFITHS AND LANE

INFLUENCE OF FREE SURFACE AND RESERVOIR
LOADING ON SLOPE STABILITY
We now consider the in¯uence of a free surface
within an earth slope and reservoir loading on the
outside of a slope as shown in Fig. 12.
Regarding the role of the free surface, a rigor-
ous approach would ®rst involve obtaining a
good-quality ¯ow net for free surface ¯ow through
the slope, enabling pore pressures to be accurately
estimated at any point within the ¯ow region. For
the purposes of slope stability analysis, however,
it is usually considered suf®ciently accurate and
conservative to estimate pore pressure at a point
as the product of the unit weight of water (ãw)
and the vertical distance of the point beneath the
free surface. In Fig. 12 the pore pressures at two
locations, A and B, have been calculated using
this assumption.
In the context of FE analysis, the pore pressures
are computed at all submerged (Gauss) points as
described above, and subtracted from the total nor-
mal stresses computed at the same locations follow-
ing the application of surface and gravity loads. The
resulting effective stresses are then used in the
remaining parts of the algorithm relating to the
assessment of Mohr± Coulomb yield and elasto-plas-
tic stress redistribution. Note that the gravity loads
are computed using total unit weights of the soil.
The external loading due to the reservoir is
modelled by applying a normal stress to the face
of the slope equal to the water pressure. Thus, as
shown in Fig. 13, the applied stress increases
linearly with water depth and remains constant
along the horizontal foundation level. These stres-
ses are converted into equivalent nodal loads on
the FE mesh (e.g. Smith & Grif®ths, 1998) and
added to the initial gravity loading.
Example 5: Homogeneous slope with horizontal
free surface
Figure 14 shows a similar slope to that analysed
in example 1, but with a horizontal free surface at a
depth Lbelow the crest. Using the method described
above, the factor of safety of the slope has been
computed for several different values of the draw-
down ratio (L=H), which has been varied from ÿ0:2
(slope completely submerged with the water level
Fig. 11. Example 4: Deformed meshes at failure corresponding to the
unconverged solution for three different values of cu2=cu1 : (a) cu2=cu1 0´6;
(b) cu2=cu1 1´5; (c) cu2=cu1 2´0
(a)
(b)
(c)
SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS 397

0:2Habove the crest), to 1´0 (water level at the base
of the slope). The problem could be interpreted as a
`slow' drawdown problem in which a reservoir, ini-
tially above the crest of the slope, is gradually
lowered to the base, with the water level within the
slope maintaining the same level. A constant total
unit weight has been assigned to the entire slope,
both above and below the water level.
The interesting result shown in Fig. 15 indicates
that the factor of safety reaches a minimum of
FOS 1:3 when L=H0:7. A limit equilibrium
solution shown on the same ®gure indicates a
similar trend (e.g. Cousins, 1978). The special
cases corresponding to L=H0 and L=H1
agree well with chart solutions given, respectively,
by Morgenstern (1963) (F1:85), and Bishop &
Morgenstern (1960) (FOS 1:4). The fully sub-
merged slope (L=H<0) is more stable than the
`dry' slope (L=H>1), as indicated by a higher
factor of safety.
An explanation of the observed minimum is due
to the cohesive strength of the slope (which is
unaffected by buoyancy) and the trade-off between
soil weight and soil shear strength as the drawdown
level is varied. In the initial stages of drawdown
(L=H,0:7), the increased weight of the slope has a
proportionally greater destabilizing effect than the
increased frictional strength, and the factor of safety
falls. At higher drawdown levels (L=H.0:7), how-
Free surface
Embankment
A
h
A
u
A 5
h
AγW
u
B 5
h
BγW
h
B
h
W
B
Reservoir Level
Fig. 12. Slope with free surface and reservoir loading
Free surface
Embankment
Reservoir Level
Linearly increasing
normal stress from
zero to
h
WγW
h
W
Constant normal
stress 5
h
WγW
Fig. 13. Detail of submerged area of slope beneath free-standing reservoir
water showing stresses to be applied to the surface of the mesh as
equivalent nodal loads
H
L
5 0
L
(negative)
L
(positive)
Fig. 14. Example 5: `Slow' drawdown problem. Homo-
geneous slope with a horizontal free surface. Slope
angle 26´578(2:1), ö9208,c9=ãH0´05 (above and
below free surface)
398 GRIFFITHS AND LANE

ever, the increased frictional strength starts to have a
greater in¯uence than the increased weight, and the
factor of safety rises. Other results of this type have
been reported by Lane & Grif®ths (1997) for a slope
which was stable (FOS .1) when `dry' or fully
submerged, but became unstable (FOS ,1) at a
critical value of the drawdown ratio L=H. It should
also be pointed out from the horizontal part of the
graph in Fig. 15, corresponding to L=H<0, that
the factor of safety for a fully submerged slope is
unaffected by the depth of water above the crest.
Excellent agreement with Morgenstern (1963)
for `rapid drawdown' problems has also been
demonstrated for a range of slopes using a similar
approach (Lane & Grif®ths, 1997).
Example 6: Two-sided earth embankment
The example given in Fig. 16 is of an actual
earth dam cross-section including a free surface
which slopes from the reservoir level to foundation
level on the downstream side (Torres & Coffman,
1997). For the purposes of this example, the materi-
al properties have been made homogeneous. Fig. 17
Finite elements
Limit equilibrium
FOS
2
1.8
1.6
1.4
1.2
Morgenstern
(1963)
FOS 5 1.85
Bishop &
Morgenstern
(1960)
FOS 5 1.4
L
/
H
00
.1
20.120.20.20
.30
.40
.50
.60
.70
.80
.911
.1
Fig. 15. Example 5: Factor of safety in `slow' drawdown problem for
different values of the drawdown ratio L=H
Fig. 17. Example 6: Finite element mesh
Reservoir level
17.1
33.5
7.3
18°23°
124.433
.5
H
5 21.3
7.3
Dimensions in metres
Free surface
Fig. 16. Example 6: Two-sided earth embankment with a sloping free surface. Homogeneous dam, ö9378,
c913´8 kN=m2,ã18´2 kN=m3(above and below WT)
SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS 399

shows the FE model used for the slope stability
analysis (Paice, 1997). The boundary conditions
consist of vertical rollers on the faces at the left and
right ends of the foundation layer with full ®xity at
the base. It should be noted that the downstream
slope of the embankment is slightly steeper than the
upstream slope.
A second analysis was also performed with no
free surface corresponding to the embankment be-
fore the reservoir was ®lled. FE slope stability
analysis led to the results shown in Fig. 18. Both
cases were also solved using a conventional limit
equilibrium approach which gave FOS 1:90 with
a free surface and FOS 2:42 without a free sur-
face. The limit equilibrium and FE factors of safety
values were in close agreement in both cases.
Regarding the critical mechanisms of failure,
Figs 19 and 20 show the deformed mesh cor-
responding to the unconverged FE solution as
compared with the slip circle that gave the
lowest factors of safety from the limit equilibrium
approach. As expected, the lowest factor of safety
occurs on the steeper, downstream side of the
embankment in both cases. It should also be noted
that both the FE and limit equilibrium results
indicate a toe failure for the case with no free
surface (Fig. 19), and a deeper mechanism extend-
ing into the foundation layer for the case with a
free surface (Fig. 20). Fig. 21 shows the corre-
sponding displacement vectors from the FE solu-
tions. Reasonably good agreement between the
locations of the failure mechanisms by both types
of analysis is indicated.
Fig. 19. Example 6 with no free surface: (a) deformed mesh corresponding to the unconverged solution by
®nite elements; (b) the critical slip circle by limit equilibrium. Both methods give FOS 2´4
(a)
(
b
)
R
5 62.7m
5.4m
62.4m
5
10
15
20
25
30
35 1.2
E
′δmax/γ
H
2
11
.41
.61
.822
.22
.42
.6
FOS
FOS 5 1.90 FOS 5 2.42
Limit
equilibrium
solutions
No free surface
With a free surface
Fig. 18. Example 6: FOS versus dimensionless dis-
placement
400 GRIFFITHS AND LANE

THE CRITERIA FOR COMPUTER-AIDED ANALYSIS
Whitman & Bailey (1967) looked forward to the
future of computer-aided analysis for engineers and
set criteria by which it could be judged. Their
comments were originally addressed to the automa-
tion of limit equilibrium methods, but they also
commented on the then emerging numerical analy-
sis techniques.
They judged that the system must be suf®ciently
accurate for con®dence in its use and appropriate
for the parameters being input. FE analysis meets
these criteria with a degree of accuracy decided by
the engineer in designing the model.
It should be possible, in a realistic timescale, to
do suf®cient trials to examine all the key modes of
behaviour; to consider different times in the life of
the structure and to vary parameters during design
to test options for cost and ef®ciency. All this is
now possible with FE methods.
Finally, the method of human± machine commu-
Fig. 20. Example 6 with a free surface: (a) deformed mesh corresponding to the unconverged solution by
®nite elements; (b) the critical slip circle by limit equilibrium. Both methods give FOS 1´9
(a)
(b)
R
547.8m
11.5m
43.9m
Fig. 21. Example 6: Displacement vectors corresponding to the unconverged solution by ®nite elements:
(a) no free surface; (b) with a free surface. Only those displacement vectors that have a magnitude .10% of
the maximum are shown
(b)
FOS 5 1.9
(a)
FOS 5 2.4
SLOPE STABILITY ANALYSIS BY FINITE ELEMENTS 401

nication must be user-friendly and readily accessi-
ble. This is partly a matter of program design but
easily achieved. Graphical output greatly enhances
the process of design and analysis over and above
that from the numerical results.
Similarly, Chowdhury (1981), in his discussion
of Sarma (1979), commented on the perceived
reluctance to develop alternatives to limit equili-
brium methods for practice when the tools to do so
were already available. Since then, numerous appli-
cations and experience have veri®ed the possibili-
ties offered by ®nite elements.
The key issues of cost and turnaround time have
been overtaken by the falling cost of powerful
hardware and processor speeds which now make
the FE method available to engineers at less than
the cost of their CAD systems. What remains is
the concern of powerful tools used wrongly. That
is no more true of ®nite elements after years of
application than of limit equilibrium methods,
which can themselves produce seriously misleading
results. Engineering judgement is still essential,
whichever method is being used.
CONCLUDING REMARKS
The FE method in conjunction with an elastic±
perfectly plastic (Mohr± Coulomb) stress±strain
method has been shown to be a reliable and robust
method for assessing the factor of safety of slopes.
One of the main advantages of the FE approach is
that the factor of safety emerges naturally from the
analysis without the user having to commit to any
particular form of the mechanism a priori.
The FE approach for determining the factor of
safety of slopes has satis®ed the criteria for effec-
tive computer-aided analysis. The widespread use
of this method should now be seriously considered
by geotechnical practitioners as a more powerful
alternative to traditional limit equilibrium methods.
ACKNOWLEDGEMENTS
The writers acknowledge the support of NSF
Grant No. CMS 97-13442. Dr Lane was supported
by the Peter Allen Scholarship Fund of UMIST.
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