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**Unformatted text preview: **LIMIT ANALYSIS VERSUS LIMIT EQUILIBRIUM FOR SLOPE STABILITY By H. S. Yu,l Member, ASCE, R. Salgado,2 Associate Member, ASCE, S. W. Sloan,3
and J. M. Kim‘ ABSTRACT: This paper compares conventional limit-equilibrium results with rigorous upper and lower bound
solutions for the stability of simple earth slopes. The bounding solutions presented in this paper are obtained
by using two newly developed numerical procedures that are based on ﬁnite-element formulations of the bound
theorems of limit analysis and linear programming techniques. Although limit-equilibrium analysis is used widely
in practice for estimating the stability of slopes, its use may sometimes lead to signiﬁcant errors as both kinematic
and static admissibility are violated in the method. Because there are no exact solutions available against which
the results of limit-equilibrium analysis can be checked, the present bounding solutions can be used to benchmark
the results of limit-equilibrium analysis. In addition, the lower bound solutions obtained also can be applied
directly in practice because they are inherently conservative. INTRODUCTION
Background and Objectives Because of its practical importance, the analysis of slope
stability has received wide attention in the literature. Limit-
equilibrium analysis has been the most popular method for
slope stability calculations. A major advantage of this approach
is that complex soil proﬁles, seepage, and a variety of loading
conditions can be easily dealt with. Many comparisons of
limit-equilibrium methods [see, for example, Whitman and
Bailey (1967), Fredlund and Krahn (1977), Duncan and
Wright (1980), and Nash (1987)] indicate that techniques that
satisfy all conditions of global equilibrium give similar results.
Regardless of the different assumptions about the interslice
forces, these methods (such as those of Janbu, Spencer, and
Morgenstem and Price) give values of the safety factor that
differ by no more than 5%. Even though it does not satisfy all
conditions of global equilibrium, Bishop’s simpliﬁed method
also gives very similar results. Partly because of this and partly
because of its simplicity, the slice method of limit-equilibrium
analysis proposed by Bishop (1955) has been used widely for
predicting slope stability under both drained and undrained
loading conditions. Because of the approximate and somewhat
arbitrary nature of limit-equilibrium analysis, concern is often
voiced about how accurate these types of solutions really are.
The answer to this question is particularly important in cases
where designs are based on slim margins of safety. The following are objectives of the present paper: (1) To
present rigorous lower and upper bound solutions for the sta-
bility of simple slopes in both homogeneous and inhomoge-
neous soils; and (2) to check the accuracy of the method of
Bishop (1955) by comparing its solutions against those derived
from limit analysis. To overcome the difﬁculties of manually
constructing both statically admissible stress ﬁelds and kine-
matically admissible velocity ﬁelds, two newly developed nu-
merical procedures are used to calculate both upper and lower
bound solutions for the stability of simple earth slopes in both ’Sr. Lect, Dept. of Civ. Engrg., Univ. of Newcastle, NSW 2308, Aus-
tralia. 2Assist. Prof, School of Civ. Engrg., Purdue Univ., IN 47907. 3Assoc. Prof, Dept. of Civ. Engrg., Univ. of Newcastle, NSW 2308,
Australia. ‘Grad. Student, School of Civ. Engrg., Purdue Univ., IN. Note. Discussion open until June 1, 1998. To extend the closing date
one month, a written request must be ﬁled with the ASCE Manager of
Journals. The manuscript for this paper was submitted for review and
possible publication on January 11. 1996. This paper is part of the Jour-
nal of Geotechnical and Geoenvironmental Engineering, Vol. 124, No.
1, January, 1998. ©ASCE, ISSN 1090-0241/98/0001—0001—0011/S4.00
+ $.50 per page. Paper No. 12469. purely cohesive and cohesive-frictional soils. These numerical
procedures are based on ﬁnite-element formulations of the
bound theorems of limit analysis and the best lower and upper
bound solutions are obtained, respectively, by optimizing stat-
ically admissible stress ﬁelds and kinematically admissible ve-
locity ﬁelds using linear programming techniques. Over the years, many limit-equilibrium computer codes
have been developed to locate the most critical failure surface
by using various search strategies. In this study, the limit-equi-
librium results are obtained from the well-known slope stabil-
ity computer code STABL with the option of Bishop’s sim-
pliﬁed method (Kim et al. 1997). Limit Analysis Method The limit analysis method models the soil as a perfectly
plastic material obeying an associated ﬂow rule. With this ide-
alization of the soil behavior, two plastic bounding theorems
(lower and upper bounds) can be proved (Drucker et a1. 1952;
Chen 1975). According to the upper bound theorem, if a set of external
loads acts on a failure mechanism and the work done by the
external loads in an increment of displacement equals the work
done by the internal stresses, the external loads obtained are
not lower than the true collapse loads. It is noted that the
external loads are not necessarily in equilibrium with the in—
ternal stresses and the mechanism of failure is not necessarily
the actual failure mechanism. By examining different mecha-
nisms, the best (least) upper bound value may be found. The
lower bound theorem states if an equilibrium distribution of
stress covering the whole body can be found that balances a
set of external loads on the stress boundary and is nowhere
above the failure criterion of the material, the external loads
are not higher than the true collapse loads. It is noted that in
the lower bound theorem, the strain and displacements are not
considered and that the state of stress is not necessarily the
actual state of stress at collapse. By examining different ad-
missible states of stress, the best (highest) lower bound value
may be found. The bound theorems of limit analysis are particularly useful
if both upper and lower bound solutions can be calculated,
because the true collapse load can then be bracketed from
above and below. This feature is invaluable in cases for which
an exact solution cannot be determined (such as slope stability
problems), because it provides a built-in error check on the
accuracy of the approximate collapse load. Limit-Equilibrium Method The limit-equilibrium method has been used to analyze
slope stability problems in soil mechanics for many years by JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / JANUARY 1998 /1 assuming that soil at failure obeys the perfectly plastic Mohr-
Coulomb criterion [e.g., Fellenius (1926) and Terzaghi
(1943)]. However, over the last 30 years the analysis of slopes
using the limit-equilibrium method has been signiﬁcantly re-
fined by using various methods of vertical slices [e.g., Bishop
(1955), Janbu (1954), Morgenstcrn and Price (1965), and
Spencer (1967)]. An excellent review of popular limit-equilib-
rium techniques for predicting slope stability can be found in
Nash (1987) or Graham (1984). Using a global equilibrium condition, the limit-equilibrium
approach is purely static as it neglects altogether the plastic
ﬂow rule for the soil (i.e., constitutive relation). If the soil at
failure is assumed to be a rigid perfectly plastic material obey-
ing an associated ﬂow rule, then collapse mechanisms selected
by the limit-equilibrium method are usually kinematically in-
admissible. In addition, the static admissibility of the stress
ﬁeld also is not satisﬁed, because some arbitrary assumptions
are made to remove statical indeterminacy and, more impor-
tantly, only a global equilibrium condition (rather than equi-
librium conditions at every point in the soil) is satisﬁed. Mich-
alowski (1989) showed that upper bound solutions based on
kinematically admissible rigid-block velocity ﬁelds (associated
with the linear Mohr-Coulomb criterion) also satisfy global
force equilibrium equations. Hence an upper bound limit anal-
ysis solution also may be regarded as a special limit-equilib-
rium solution but not vice versa. However, by no means. can
these two methods be regarded as equivalent (Collins 1974;
Chen 1975). Based on the bound theorems of limit analysis,
it can be concluded that, in general, the limit-equilibrium
method is of an approximate and arbitrary nature and the re-
sults obtained from this method are neither upper bounds nor
lower bounds on the true collapse loads. Any attempt to val-
idate the limit-equilibrium approach by comparing different
limit-equilibrium solutions, without reference to a more rig-
orous analysis, is considered to be inconclusive. PROBLEM OF STABILITY OF SIMPLE SLOPES The slope geometry analyzed in this paper is shown in Fig.
1. TWO types of analysis are considered: undrained stability
analysis of purely cohesive slopes and drained stability anal-
ysis of cohesive-frictional slopes. The purely cohesive soil un-
der undrained loading conditions is modeled by a rigid per-
fectly plastic Tresca yield criterion with an associated ﬂow
rule. The strength of the cohesive soil is determined by the
undrained shear strength S“, which may increase linearly with
depth as is the case in normally consolidated clays (Gibson
and Morgenstcrn 1962; Hunter and Schuster 1968). Under
drained loading conditions, a perfectly plastic Mohr—Coulomb
model is used to describe the soil behavior. For this case, the
strength parameters are the effective cohesion c’ and the ef-
fective friction angle d)’. Both of these quantities are assumed
to be constant throughout the slope. For simplicity, the effect
of seepage (or pore pressures) on the stability of cohesive-
frictional slopes has not been included in this study. The so-
lutions obtained are therefore only relevant for fully drained
loading conditions where the effect of pore pressures can be
neglected. Recent work by Miller and Hamilton (1989) and
Michalowski (1994) suggests that it is also possible to incor-
porate the effect of pore pressures in limit analysis, but this
extension will not be covered here. . The solutions for the simple slopes considered in this paper
are relevant to excavations and man-made ﬁlls built on soil or
rock (Taylor 1948; Gibson and Morgenstcrn 1962; Hunter and
Schuster 1968; Duncan et al. 1987). Duncan et al. (1987) have
showed that stability charts for simple slopes also can be used
to obtain reasonably accurate answers for more complex prob—
lems if irregular slopes are approximated by simple slopes and Firm Base FIG. 1. Geometry of Slmple Slopes and Parameters for De-
ecrlblng Increasing Strength with Depth In Slope Stablllty Anal-
yele carefully determined averaged values of unit weight, cohesion,
and friction angles are used. FINITE-ELEMENT LIMIT ANALYSIS Because of the difﬁculties of constructing statically admis-
sible stress ﬁelds manually, the application of limit analysis
has in the past almost exclusively concentrated on the upper
bound method (Chen 1975; Chen and Liu 1990). In fact, the
authors are not aware of any rigorous lower bound solutions
for the stability of slopes in cohesive-frictional soils. Although
the upper bound solutions may be used as an estimate for the
true collapse load, it is the lower bound solutions that are
generally more useful in practice, because they are inherently
conservative. Finite-Element Lower Bound Limit Analysis The use of ﬁnite elements and linear programming to com-
pute rigorous lower bounds for soil mechanics problems ap-
pears to have been ﬁrst proposed by Lysmer (1970). Although
Lysmer’s procedure was potentially very powerful, its useful-
ness was limited initially by the slowness of the algorithms
that were available for solving large linear programming prob-
lems. In recent years, signiﬁcant progress has been made in
developing more efﬁcient algorithms for solving large linear
programming problems (Sloan 1988a). Detailed discussions of
the recent developments in ﬁnite-element formulations of the
lower bound theorem may be found in Anderheggen and
Knopfel (1972), Bottero et a1. (1980), Sloan (1988b), and Yu
and Sloan (1991a,b). In this study, the formulation of Sloan
(1988b) has been used because it has proven to be very efﬁ-
cient and robust when applied to large practical problems. The lower bound formulation under conditions of plane
strain uses the three types of elements shown in Fig. 2. The
stress ﬁeld for each of these elements is assumed to vary lin-
early. The extension elements may be used to extend the so-
lution over a semiinﬁnite domain and therefore provide a com-
plete statically admissible stress ﬁeld for inﬁnite half-space
problems. Because this paper is concerned mainly with the
stability of ﬁnite depth slopes resting on a ﬁrm base, extension
elements are needed only at the left and right boundaries of
the problems (shown in Fig. 4). In fact, the extension elements
shown in Fig. 2 can be used readily to extend the stress ﬁelds
into a semiinﬁnite domain if a slope in an inﬁnitely deep layer
needs to be analyzed (i.e., the depth factor D = 00). Examples 2/ JOURNAL OF GEOTECHNICAL AND GEOENVIFIONMENTAL ENGINEERING /JANUARY 1998 (0,2. 0,2. r m) (ax3‘ay3’rxy3) (011,034,101) 3-noded triangular element Direction of Baension —e 4
Direction of Extension 4-noded rectangular extension element 3~noded triangular extension element X FIG. 2. Elements Used for Lower Bound Limit Analyels of how this can be done can be found in Yu and Sloan (1994).
As will be shown later in this paper, the depth factor has a
very small effect on the stability of slopes provided the value
of depth factor D is greater than approximately 4 [in agree-
ment with the slope stability charts presented by Taylor
(1948)]. It may therefore be reasonable to assume that the
stability solutions for a slope in a ﬁnite layer with a large D
value (say D 2 4) would be very close to those of a slope in
a layer of inﬁnite depth. A lower bound solution is obtained by insisting that the
stresses obey equilibrium and satisfy both the stress boundary
conditions and the yield criterion. Each of these requirements
imposes a separate set of constraints on the nodal stresses. In
the lower bound ﬁnite-element analysis, statically admissible
stress discontinuities are permitted at edges shared by adjacent
triangles and also along borders between adjacent rectangular
extension elements. The Mohr-Coulomb and Tresca yield
functions can be shown to plot as a circle in the Zr,y versus
(a, -- 0",) stress space, where 1;, is shear stress and 0’, and cry
are normal stresses. To avoid nonlinear constraints occurring
in the constraint matrix, a key idea behind the numerical lower
bound technique is to use an internal linear approximation of
the Mohr-Coulomb or the Tresca yield surface in the 27,,y ver-
sus (0'; — 0),) stress space. In a typical lower bound analysis, a statically admissible
stress ﬁeld is sought that maximizes either a collapse load over
some part of the boundary or the magnitude of body forces
acting within a region. Both the collapse load and soil unit
weight can be used to deﬁne an objective function for linear
programming calculations. In the simple slope problem con-
sidered in this paper, it is convenient to ﬁnd a static stress ﬁeld
that maximizes the unit weight. As a result, we will treat the
unit weight as the unknown and optimize it directly. A typical
lower bound mesh, together with the applied boundary con-
ditions, is shown in Fig. 4(a). Note that the rectangular exten-
sion elements are arranged to ensure that the computed stress
ﬁeld is statically admissible throughout the slope, so that the
solution is a rigorous lower bound solution. Finite-Element Upper Bound Limit Analysis The ﬁrst formulation of the upper bound theorem, which
used constant-strain triangular ﬁnite elements and linear pro-
gramming, appears to have been developed by Anderheggen
and Knopfel (1972) who analyzed plate problems. This for-
mulation was later generalized by Bottero et a1. (1981) and
Sloan (1989) to include velocity discontinuities in plane strain limit analysis. When these constant-strain ﬁnite-element for-
mulations are used, the grid must be arranged so that four
triangles form a quadrilateral, with the central node lying at
the intersection of the diagonals. If this pattern is not used, the
elements cannot provide a sufﬁcient number of degrees of
freedom to satisfy the incompressibility condition that accom-
panies undrained failure (Nagtegaal et al. 1974). To overcome
this limitation, Yu et al. (1992) developed a six-noded qua-
dratic element for upper bound limit analysis. This formulation
can be used to model an incompressible velocity ﬁeld without
resorting to special grid arrangements and also is more efﬁ-
cient than an equivalent three—noded formulation with the
same number of nodes. However, it does suffer from the same
shortcomings as the formulation of Bottero et a1. (1980) and
Sloan (1989), in that the direction of shearing for each velocity
discontinuity must be speciﬁed a priori. Very recently, a new
upper bound formulation that permits a large number of dis-
continuities in the velocity ﬁeld has been derived by Sloan
and Kleeman (1995). This method is again based on the three—
noded triangle, but a velocity discontinuity may occur at any
edge that is shared by a pair of adjacent triangles, and the sign
of shearing is chosen automatically during the optimization
process to give the least-dissipated energy rate. In this study, the upper bound formulation developed by
Sloan and Kleeman (1995) is used to calculate upper bound
solutions for slope stability. A typical constant-strain triangular
element used in the upper bound analysis is shown in Fig. 3. (”2aV2) (“3, v3) (“I‘vl) (111,112,...,A,,) FIG. 3. Trlangular Element Used for Upper Bound LImlt Anal-
yele a,,=0 1=0
,xsaaaaxmﬁ
,xaaaaaaaan
ma” &%%%%%%X%M Z%§%%%%%%%%%%%%XE %
XX%
XXX QV
XXXXXXQXD
IXXXXXX
XXXXXXX u=v= v=0
0 %
XX XXXXXX X%X%%X XXXXXX
%X%XXX
XXXXXX XX
X 0 (b)
FIG. 4. Typlcal FInlte Element Meshes Used In leIt Analyels JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING /JANUAFIY 1998 / 3 Within each element, the velocities are assumed to vary line-
arly. Each node has two velocity components and each element
has p plastic multiplier rates in, X2, X,, where p is the
number of planes used to linearize the yield criterion. An upper bound solution is obtained by requiring that the
velocity ﬁelds obey an associated plastic ﬂow rule and satisfy
the velocity boundary conditions. Each of these requirements
imposes a separate set of constraints on the nodal velocities.
To remove the stress terms from the ﬂow rule equations, and
thus provide a linear relationship between the unknown nodal
velocities and plastic multiplier rates, an external linear ap-
proximation to the yield surface in the stress space of 213,,
against (a, — 0,) is employed to ensure that the solution ob-
tained is truly an upper bound. In a typical upper bound analysis, a kinematically admis-
sible velocity ﬁeld is sought that minimizes the amount of
dissipated power. The dissipated power can be expressed in
terms of the unknown plastic multiplier rates and the discon-
tinuity parameters used to deﬁne an objective function [see
Sloan and Kleeman (1995)]. For the slope stability problem
considered in this paper, we will minimize the unit weight
directly and this can be achieved by equating the power ex-
pended by the external loads to the internal power dissipation
[see Sloan (1995) for details]. A typical upper bound mesh is
shown in Fig. 4(b). This is very similar to the equivalent lower
bound grid of Fig. 4(a) except that the extension elements are
no longer necessary because a rigid velocity boundary condi-
tion used in the upper bound limit analysis will ensure that the
solution obtained is a rigorous upper...

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