SIGNIFICANCE OF SOIL DILATANCY IN SLOPE STABILITY ANALYSIS
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By Majid T. Manzari,1 Associate Member, ASCE, and
Mohamed A. Nour,2 Student Member, ASCE
ABSTRACT: A finite-element approach is used to analyze the slope stability problem and to examine the effect
of soil dilatancy on the stability of slopes. It is found that soil dilatancy has a significant effect on the stability
of slopes, and that higher values of dilation angle lead to larger stability numbers. Therefore, the stability numbers
obtained from limit analyses (lower/upper bound solutions) are not conservative for granular soils that exhibit
a dilation angle smaller than a soil’s friction angle.
INTRODUCTION
Accurate evaluation of the deformation and stability of soil
structures is the main objective of many geotechnical analyses.
When subjected to monotonic shearing in a drained condition, different soils may exhibit distinctly different patterns
of stress-strain behavior. In dense granular soils and highly
overconsolidated cohesive soils, the stress-strain response is
generally associated with the emergence of peak shear strength
at relatively low shear strain. This peak strength is generally
followed by a critical-state strength as the soil is sheared further. The ductile response is typical of loose granular soils and
normally consolidated cohesive soils whose critical-state
strength is only reached at high strain levels. It is well known
that the volume change of the soil is a major factor contributing to its shear resistance, and that dilatancy, which is the
change of soil volume on shear loading, is the main reason
for the emergence of the peak strength in dense sands and
overconsolidated clays. The peak and critical-state strengths of
a soil are generally expressed in terms of the peak and criticalstate angle of friction. Several empirical relationships are
available to relate the peak angle of friction to soil dilatancy
and the critical-state angle of friction. For example, Bolton
(1986) proposed the following relationship for granular soils:
fpeak = fcrit 1 0.8cmax
(1)
here, fpeak, fcrit, and cmax = peak friction angle, critical-state
friction angle, and the maximum angle of dilation, respectively. It is noted that the peak friction angle is a function of
the maximum dilatancy that soil exhibits in the process of
shearing. The peak shear strength is not widely used in actual
design practice, because it actually represents the maximum
strength of the soil as long as the strains remain less than a
small limit. However, in many actual designs, it may be highly
overconservative to design the structure so that deformation in
the soil remains below the strain necessary to mobilize the
peak shear strength. In most practical cases, the shear strain
on the failure (slip) surface exceeds the small strain necessary
to mobilize the peak shear strength, and the mobilized shear
stress is somewhere between the peak and critical-state shear
strengths. In practice, the stability of soil structures is often
calculated by using the critical-state shear resistance (Powrie
1997). Despite the fact that such a practice is safer than the
use of peak strength, it has led to a common ignorance of the
1
Asst. Prof., Dept. of Civ. and Envir. Engrg., George Washington Univ.,
Washington, D.C. 20052. E-mail: manzari@seas.gwu.edu
2
Doctoral Student, Dept. of Civ. and Envir. Engrg., George Washington
Univ., Washington, D.C.
Note. Discussion open until June 1, 2000. To extend the closing date
one month, a written request must be filed with the ASCE Manager of
Journals. The manuscript for this technical note was submitted for review
and possible publication on January 11, 1999. This technical note is part
of the Journal of Geotechnical and Geoenvironmental Engineering,
Vol. 126, No. 1, January, 2000. qASCE, ISSN 1090-0241/00/00010075–0080/$8.00 1 $.50 per page. Technical Note No. 19995.
importance of dilation on the stability analysis. Current procedures used in earth-pressure calculations and slope stability
analyses generally deal with friction angle only and ignore the
effect of dilation. Rowe (1963) was among the first researchers
who studied the significance of soil dilatancy in the analysis
of the stability of geotechnical structures. He used the stressdilatancy model (Rowe 1962) to demonstrate the significance
of incorporating volume change characteristics of the soil in
the earth-pressure calculations and slope stability analyses. In
his analysis, Rowe used the stress-dilatancy theory to estimate
the shear strength of the soil on the assumed failure surface
used in the stability calculations. Rowe and Peaker (1965)
compared the results of their careful measurements of passive
earth pressure with the results of two different analytical approaches: (1) A traditional Mohr-Coulomb model where the
failure in soil is characterized by two parameters cohesion c
and friction angle f; and (2) a stress-dilatancy model (Rowe
1963), in which a third parameter, angle of interlocking a, is
used to represent soil dilatancy. They found that the results
obtained by traditional Mohr-Coulomb parameters may err on
the unsafe side, whereas the results obtained by the stressdilatancy model provide more accurate results.
It is the objective of this paper to further examine the significance of dilation in an important class of geotechnical analysis, i.e., slope stability. Initially, it may appear that because
soil strength increases with dilation, ignorance of dilation, as
usually performed in limit analysis, will lead to conservative
results in these cases. It is important to note that if a limit
equilibrium approach is used, ignoring the dilation angle (i.e.,
setting the friction angle to be equal to the critical-state friction
angle) will always lead to more conservative results. However,
in cases where a limit plasticity analysis or a finite-element
approach (based on an associative flow rule) is used, ignorance
of dilatancy effects will lead to an implicit consideration of
high dilation angles. It is well known that the collapse loads
for a material with nonassociated flow rule are smaller than
those obtained for the same material when an associated flow
rule is assumed (Chen 1975). Drescher and Detouney (1993)
addressed the issue of nonassociativity in the context of limit
analysis. They proposed including the nonassociativity by using a modified friction angle for materials that follow a nonassociative flow rule. Michalowski and Shi (1995) successfully
used this modified friction angle to calculate the bearing capacity of strip footings on a two-layer soil system consisting
of a sandy soil, which obeys nonassociative flow rule, overlying a clayey soil. The issue of the impact of nonassociative
flow rule on the collapse load of soil structures has also been
investigated by using numerical methods in the works presented by Davis and Booker (1973), Zienkiewicz et al. (1975),
Griffith (1982), Mizuno and Chen (1983), and Vermeer and de
Borst (1984). Most of these works have concentrated on the
analysis of the bearing capacity of foundations. Here the emphasis is on the effect of nonnormality of flow rule (rather
than the general notion of nonassociativity) on the stability of
soil slopes. It will be shown that ignorance of soil dilation that
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FIG. 1.
Typical Deformed Finite-Element Mesh Obtained at End of Slope Stability Analysis
actually amounts to the use of normality and to an implicit
consideration of a high dilation angle leads to nonconservative
solutions in these analyses.
SLOPE STABILITY ANALYSIS AND NORMALITY
CONDITION
Rigorous analysis of soil structures at a limiting collapse
condition is normally conducted by using the lower- and upper-bound theorems of plasticity. The proof of the validity of
these theorems in providing a lower- or upper-bound solution
to a collapse problem is only possible when the material is
(elastic-) perfect plastic and obeys the normality condition, i.e.,
when f = c (Drucker and Prager 1952 and Chen 1975). Yu
et al. (1998) recently applied a finite-element limit analysis to
a slope stability problem and provided a series of very useful
graphs comparing the bounding solutions (upper and lower
limits) with the traditional limit equilibrium solutions, such as
those obtained from the Bishop limit equilibrium method.
In examining the two major assumptions of limit-analysis
methods, it is noted that the critical-state shear resistance is
actually a conservative representation of soil strength; hence,
the perfect plastic approximation of the stress-strain behavior
is acceptable. However, in case of frictional soils, it is clear
that the critical state of a soil is always accompanied by continuous distortion of the soil element at no volume change (c
= 0) and that the assumption of normality is no longer valid.
Therefore, it is reasonable to assume that frictional soils at a
critical-state condition will not follow the normality condition
and that a realistic collapse analysis of soil structures should
be based on a nonassociative flow rule. Unfortunately, the limit
theorems will not be applicable to the materials obeying a
nonassociated flow rule and the solution to the collapse problems for these materials should be sought using a different
approach. Chen (1975) described an upper-bound theorem for
materials with nonassociated flow rule: ‘‘Any set of loads
which produces collapse for the material with associated flow
rule will produce collapse for the same material with nonassociated flow rule.’’ According to this theorem, the collapse
loads for a material with nonassociated flow rule are smaller
than those obtained for the same material when an associated
flow rule is assumed. Here, we will use a nonassociative MohrCoulomb model to demonstrate that the classical solutions for
slope stability problems are not conservative. Upper-bound
theorem for frictional material is the theoretical foundation for
the validity of the results described later in this paper. Standard
nonlinear finite-element analysis, as opposed to finite-element
limit analysis, will be used.
DESCRIPTION OF CONSTITUTIVE MODEL
A classical Mohr-Coulomb model is observed to appropriately model the failure strength of soils. The model is based
on a pyramid shape yield surface with hexagonal section. The
particular shape of the yield surface introduces discontinuity
in the gradient of yield function at the edges of the cone and
at its apex. To remove the singularities associated with the
edges and the apex of the original Mohr-Coulomb model, the
smoothed forms suggested by Zienkiewicz and Pande (1977)
and Sloan and Booker (1986) are used. The constitutive model
also includes a plastic potential with a geometry similar to the
yield surface, but characterized with dilation angle c instead
of friction angle f. By setting f = c, an associative MohrCoulomb model is recovered. A constant dilation angle is used
throughout each analysis.
ANALYSIS PROCEDURE AND DETECTION OF
INSTABILITY
The objective of a slope stability analysis is to find the factor of safety of an existing soil embankment (slope) against
possible sliding on a potential failure surface. For a cohesivefrictional soil (C-f), it is customary to characterize the stability of a slope by using a dimensionless parameter called stability number Ns. The stability number is normally defined as
(gH )/C, where g is the unit weight of the soil, H is the height
of the soil embankment, and C is cohesion of the soil (Chen
1975). For a soil embankment, with a given slope angle that
consists of soil of a given cohesion and friction angle, there
FIG. 2. Vertical Displacements of Three Key Nodes as Function of Body Force Change
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is a critical value for the stability number. If the height of the
embankment exceeds a certain limit, the stability number will
exceed its critical value and the slope will be unstable.
Here, the value of the critical stability number for a specific
embankment is obtained by applying increasing values of a
vertical body force. Instability is signaled by the occurrence
of excessive displacement in a part of the slope that is sliding
with respect to the other part. It was observed that instability
is generally associated with a difficulty to achieve convergence
in the analysis for a relatively small increment of body force
(say, 1024 of the limit value).
To examine the problem, we consider a homogeneous soil
embankment with an angle of 457. The stability of the slope
will be analyzed using the nonassociative plasticity model described earlier in this paper. Different values of a dilation angle
will be assigned to the soil while the friction angle is kept at
a certain value. The analysis will then be repeated for different
values of the friction angle. For a slope with a height of 3 m
and a slope angle of 457, an additional length of 5 m in the
backfill soil was included in the finite-element model. A total
of 6,400 eight-node plane strain finite elements are used in the
analyses. Preliminary analyses with four-node elements
showed that much finer mesh is needed to obtain the same
level of accuracy. A typical deformed shape of the finite-element mesh, at the end of the analysis when the embankment
becomes unstable, is shown in Fig. 1. It is clear that on the
verge of instability a narrow zone of nearly circular shape is
under intense shear strains. This zone plays the role of a failure
zone (or failure surface) in a traditional slope stability analysis
and allows the large displacement of a part of the slope with
respect to the rest of the soil embankment. Fig. 2 shows the
vertical displacements of three key nodes of the finite-element
mesh (T, M, and R, as shown in Fig. 1) as a function of body
force change. This is obtained in a typical analysis for f = c
= 207. All of the nodes are located on the ground surface: One
is at the tip of the slope, one is just to the right of the failure
zone, and one is located in midway between the first two
nodes. From Fig. 2, it is clear that that on the verge of instability, the vertical displacement of the nodes located to the left
of failure zone suddenly increase for a small change of body
force. The vertical displacement of the node located immediately to the right of the failure zone, however, shows very little
change as the body force reaches its final value.
Fig. 3(a) shows the variations of the normalized force (Ns
= gH/C ) versus the normalized displacement [Gd/(HC )] for a
slope of 457 with various values of friction angle. G is the
FIG. 3. (a) Normalized Force-Displacement Curves for Slope of 458 with f 5 c; (b) Comparison between Results of Finite-Element
Analysis and Results Obtained by Limit-Analysis Approach
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / JANUARY 2000 / 77
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elastic shear modulus, and d is the vertical displacement of
the tip of the slope. All of the final values of stability number
Ns closely match [Fig. 3(b)] those obtained by using a limit
analysis approach considering a log-spiral failure surface passing through the toe of the slope (Chen 1975).
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EFFECT OF SOIL DILATANCY ON CRITICAL
STABILITY NUMBER
To examine the effect of dilation on the calculated critical
stability number, a series of analyses were conducted with a
fixed friction angle and with various values of dilation angles.
Typical normalized force-displacement curves obtained in
these analyses are shown in Fig. 4 for two different friction
angle values: i.e., 257 and 307. Here the soil embankment has
a slope of 457 and the dilation angle was chosen to be 257 and
157 in the first case (f = 257), and 307 and 207 in the second
case (f = 307). Again, the normalized displacement Gd/(HC ),
d = vertical displacement of the tip of the slope, and the dimensionless parameter Ns = gH/C are used as the normalized
displacement and force, respectively. It is seen that in both
cases, with a change of dilation angle, the critical stability
number decreases. This stability number reduction from the
values obtained with f = c, is a typical trend that is observed
in all stability calculations in which the dilation angle is not
equal to friction angle. As the difference between dilation angle and friction angle increases, the reduction of stability number becomes more significant.
It is important to note that in cases with a significant difference between f and c, numerical difficulties arise in the
solution of nonlinear finite-element equations. The main
source of these numerical difficulties is the ill-posedness of
the boundary-value problem due to the loss of ellipticity that
is triggered by the emergence of material instability at the level
of constitutive relations. If the material instability due to triggering of localization remains in a contained area within the
structure and the localized area does not spread to the boundaries of the structure, the numerical solution may be continued.
In cases with large differences between friction and dilation
angles, the material instability emerges at relatively early
stages of the solution and the numerical solution cannot be
continued without regularizing the constitutive equations. Ortiz et al. (1987) proposed a relatively simple enhanced finite
element that can be used to overcome such numerical difficulties. The writers have implemented this enhanced finite element, and their experience shows that the additional incompatible modes introduced in this element can recover the
well-posedness placement of the boundary-value problem in
most cases.
Similar analyses for various friction angle values and with
the dilation angle that varies from 0 to f were conducted in
this study. It was generally observed that the stability number
increases with the dilation angle, and therefore, the analyses
based on the assumption of equality of dilation and friction
angles are nonconservative. In some cases the difference between the calculated stability number with a dilation angle of
zero and those obtained from classical methods (f = c) may
be large enough to warrant a direct finite-element analysis.
EFFECT OF SOIL DILATANCY IN SLOPES CREATED
BY EXCAVATION
Similar analyses, such as those described above, are conducted to investigate the effect of soil dilatancy on a soil slope
made by excavation. A level-ground soil deposit, 30-m long
and 10-m high, is divided into 40 horizontal layers. A soil
slope of 457 is then created by sequentially removing the soil
layers from the ground surface (Fig. 5). The initial stress condition in the soil deposit is calculated by applying the selfweight of the soil as a vertical body force. The analysis is
conducted using a friction angle of 207 and several different
dilation angles (2.57, 57, 107, 12.57, 157, and 207). Fig. 6 shows
the deformed finite element after the 33rd layer of the soil is
removed. During this analysis, it was observed that the re-
FIG. 5.
FIG. 4. Normalized Force-Displacement Curves for Slope of
458 with: (a) f 5 258; (b) f 5 308
FIG. 6.
Schematic Representation of Excavation
Finite-Element Mesh after Removal of 33 Layers
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FIG. 7.
Variation of Tip Vertical Displacement in Excavated Soil with Normalized Force (gH/C ), f 5 208
FIG. 8.
Variation of Stability Number with Dilation Angle, for Slope of 458 with f 5 208
moval of the first 32 layers caused very small deformations in
the slope while a sudden increase in tip displacement occurred
when the last layer (33rd) was removed. The sudden increase
of deformation signals the attainment of the critical height for
this slope. The analysis shown in Fig. 6 is conducted with a
f = 207 and c = 12.57.
Fig. 7 shows the variation of the normalized vertical displacement of the tip of the embankment (Gd/H0 C ) as a function of stability number (gH/C ) for three different dilation angles of 2.57, 107, and 207. The value of H (in gH/C ) is the
current height of the slope (or depth of excavation) that increases as more layers are removed. H0 (in Gd/H0 C ) is the
thickness of the soil deposit before the start of the excavation,
d is the vertical displacement of the tip of the embankment,
and G is the elastic shear modulus of the soil. It can be seen
that the stability number decreases as c decreases. From Fig.
7 it is also observed that in the first stages of excavation, the
vertical displacement of the tip of the slope is upward, and as
more soil layers are removed the tip vertical displacement
changes direction and becomes downward, finally becoming
excessive and resulting in slope failure. The heaving of the
tip of the embankment at the initial stages of excavation is
due to the removal of lateral pressure on the top layers. This
pressure relief causes a net volume increase in the soil elements that are still in their elastic range of behavior and is
manifested in an outward horizontal displacement and a vertical heave.
The effect of dilation angle c on the stability number becomes larger as the difference between f and c increases (Fig.
8). Fig. 9 shows the results of the analyses conducted on the
same soil embankment, but with higher friction angles, i.e., f
= 257 and 357. As shown in Fig. 9, the effect of soil dilatancy
on the stability number may become increasingly important as
the friction angle increases. It is noted that the numerical simulations for low dilation angles become exceedingly difficult
as the friction angle becomes larger. In all of the cases shown
in Fig. 9, the end of the analyses correspond to the full development of a localized zone of failure (slip surface) and the
emergence of several negative eigenvalues in the finite-element equations. These analyses further confirm that dilatancy
may have a major effect on the stability of a soil slope. However, the writers believe that additional analyses using more
robust enhanced finite-element techniques, which accommodate localization effect, are necessary to accurately quantify
the dilatancy effects on slope stability.
CONCLUSIONS
Based on a series of finite-element analyses of the slope
stability problem, it is observed that soil dilatancy may have
a significant effect on the stability of slopes. If a soil slope
with a specific friction angle is used, higher values of a dilation angle generally lead to larger stability numbers for the
same friction angle. This observation is very significant in that
the current analysis procedures such as limit analysis are based
on the assumption of equality of dilation and friction angle
(normality). Therefore, the stability numbers obtained from
these analyses (lower/upper-bound solutions) are not conservative for granular soils that exhibit a dilation angle smaller
than a soil’s friction angle.
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FIG. 9.
APPENDIX I.
Variation of Tip Vertical Displacement in Excavated Soil with Normalized Force (gH/C ): (a) f 5 258; (b) f 5 358
REFERENCES
Bolton, M. D. (1986). ‘‘The strength and dilatancy of sands.’’ Géotechnique, London, 36(1), 65–78.
Chen, W. F. (1975). Limit analysis and soil plasticity. Elsevier Science,
Amsterdam.
Davis, E. H., and Booker, J. R. (1973). ‘‘Some adaptations of classical
plasticity theory for soil stability problems.’’ Proc., Symp. Plast. Soil
Mech., A. Palmer, ed., Cambridge University Press, Cambridge, 24–41.
Drescher, A., and Detouney, E. (1993). ‘‘Limit load in translational failure
mechanisms for associative and non-associative materials.’’ Géotechnique, London, 43, 443–456.
Drucker, D. C., and Prager, W. (1952). ‘‘Soil mechanics and plastic analysis
or limit design.’’ Quarterly of Appl. Mathematics, 10, 157–165.
Griffith, D. V. (1982). ‘‘Computation of bearing capacity on layered soils.’’
Proc., 4th Int. Conf. Num. Meth. Geomech., Z. Eisenstein, ed., Balkema,
Rotterdam, The Netherlands.
Michalowski, R. L., and Shi, L. (1995). ‘‘Bearing capacity of footing over
two-layer foundation soils.’’ J. Geotech. Engrg., ASCE, 121(5), 421–
428.
Mizuno, E., and Chen, W. F. (1983). ‘‘Cap models for clay strata to footing
loads.’’ Comp. and Struct., 17, 511–528.
Ortiz, M., Leroy, Y., and Needleman, A. (1987). ‘‘A finite element method
for localized failure analysis.’’ Comput. Meth. Appl. Mech. Engrg., 61,
189–214.
Powrie, W. (1997). Soil mechanics, concepts and applications. E&FN
Spon, London.
Rowe, P. W. (1962). ‘‘The stress-dilatancy relation for static equilibrium of
an assembly of particles in contact.’’ Proc., Royal Soc., London, Series
A, 269, 500–527.
Rowe, P. W. (1963). ‘‘Stress-dilatancy, earth pressure and slopes.’’ J. Soil
Mech. and Found. Div., ASCE, 89(5), 37–61.
Rowe, P. W., and Peaker, K. (1965). ‘‘Passive earth pressure measurements.’’ Géotechnique, London, 15(1), 57–78.
Sloan, S. W., and Booker, J. R. (1986). ‘‘Removal of singularities in Tresca
and Mohr-Coulomb yield functions.’’ Comm. Appl. Numer. Meth., 2,
173–179.
Vermeer, P. A., and de Borst (1984). ‘‘Non-associated plasticity for soils,
concrete, and rock.’’ Heron, 29, 1–64.
Yu, H. S., Selgado, R., Sloan, S. W., and Kim, J. (1998). ‘‘Limit analysis
versus limit equilibrium for slope stability.’’ J. Geotech. and Geoenvir.
Engrg., ASCE, 124(1), 1–11.
Zienkiewicz, O. C., Humpherson, C., and Lewis, R. W. (1975). ‘‘Associated
and non-associated viscoplasticity and plasticity in soil mechanics.’’
Géotechnique, London, 25, 671–689.
Zienkiewicz, O. C., and Pande, G. N. (1977). ‘‘Some useful forms of isotropic yield surfaces for soil and rock mechanics.’’ Finite elements in
geomechanics, Chapter 5, G. Gudehus, ed., Wiley, New York, 179–190.
APPENDIX II.
NOTATION
The following symbols are used in this paper:
C = soil cohesion;
G = shear modulus;
m = inverse of stability number = 1/N;
N = stability number;
f = soil friction angle (7); and
c = soil dilatancy angle.
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