-
Active pressure on gravity walls supporting purelyfrictional
soils
D. Loukidis and R. Salgado
Abstract: The active earth pressure used in the design of
gravity walls is calculated based on the internal friction angle
ofthe retained soil or backfill. However, the friction angle of a
soil changes during the deformation process. For drained load-ing,
the mobilized friction angle varies between the peak and
critical-state friction angles, depending on the level of
shearstrain in the retained soil. Consequently, there is not a
single value of friction angle for the retained soil mass, and the
activeearth pressure coefficient changes as the wall moves away
from the backfill and plastic shear strains in the backfill
increase.In this paper, the finite element method is used to study
the evolution of the active earth pressure behind a gravity
retainingwall, as well as the shear patterns developing in the
backfill and foundation soil. The analyses relied on use of a
two-surfaceplasticity constitutive model for sands, which is based
on critical-state soil mechanics.
Key words: finite elements, plasticity, retaining walls,
sands.
Rsum : La pression active des terres utilise dans la conception
des murs gravitaires est calcule partir de langle defriction
interne du sol ou du remblai retenu. Cependant, langle de friction
dun sol change durant le processus de dforma-tion. Dans le cas dun
chargement drain, langle de friction mobilis varie entre langle de
friction au pic et celui ltatcritique, dpendant du niveau de
dformation en cisaillement dans le sol retenu. En consquence, il ny
a pas de valeurunique dangle de friction pour une masse de sol
retenue, et le coefficient de pression active des terres varie
mesure quele mur se spare du remblai et que les dformations
plastiques en cisaillement augmentent dans le remblai. Dans cet
article,la mthode par lments finis est utilise pour tudier
lvolution de la pression active des terres derrire un mur de
soutne-ment gravitaire, ainsi que les patrons de cisaillement qui
se dveloppent dans le remblai et dans le sol de fondation. Les
ana-lyses sont ralises laide dun modle constitutif de plasticit
deux surfaces pour des sables, qui est bas sur lamcanique de ltat
critique des sols.
Motscls : lments finis, plasticit, murs de soutnement,
sables.
[Traduit par la Rdaction]
IntroductionThe active earth pressure acting on the back of a
retaining
wall controls its design. The active earth pressure is
ex-pressed as the product of the vertical effective stress s 0v
inthe retained soil mass or backfill1 and the active earth
pres-sure coefficient KA. The earliest and simplest methods for
thecalculation of the active earth pressure for purely
frictionalbackfills are those based on the Coulomb and Rankine
theo-ries. For a backfill with horizontal surface, the Rankine
solu-tion is mathematically exact for a vertical and smooth
wallbackface. Coulombs solution assumes a planar slip surfaceand is
equivalent to an upper bound solution. For a horizon-tal backfill
and a vertical wall backface, Coulombs solutionyields
1 KA cos2f
cosdf1 sin f d sinf=cosdp g2
Caquot and Kerisel (1948) produced solutions in tabulatedform,
assuming slip surfaces with logarithmic spiral shape.More recently,
Paik and Salgado (2003) estimated the activeearth pressure behind
rigid walls by improving the formula-tion of Handy (1985), which
considers soil arching concepts.Limit analysis has also been used
to study the active earth
pressure problem. Rigorous upper bound values for KA
estab-lished by Chen (1975) and Soubra and Macuh (2002) usinglimit
analysis are in very close agreement with the values of Ca-quot and
Kerisel (1948). Sokolovski (1965) solved the problemof active and
passive earth pressure using the method of charac-teristics. More
recently, Lancellotta (2002) provided a rigorouslower-bound
solution for active pressures in closed form:
2 KA cosd1 sinfcosd
sin 2f sin 2d
pedarcsin sind=sinf tanf
Received 21 March 2011. Accepted 26 September 2011. Published at
www.nrcresearchpress.com/cgj on 20 December 2011.
D. Loukidis. Department of Civil and Environmental Engineering,
University of Cyprus, Nicosia 1678, Cyprus.R. Salgado. School of
Civil Engineering, Purdue University, W. Lafayette, IN 47907-1284,
USA.
Corresponding author: D. Loukidis (e-mail:
[email protected]).1The paper is not restricted to backfilled
walls. To call attention to applicability of the discussion to
walls supporting natural ground aswell as completely backfilled
walls, the terms retained soil mass and backfill are used
interchangeably throughout.
78
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The upper bound values for KA by Coulombs solution,Chen (1975),
and Soubra and Macuh (2002) are very closeto the corresponding
lower bound values using Lancellottasequation (eq. [2]); the
differences do not exceed 7%. The KAvalues by Sokolovski (1965) lie
between the narrow banddefined by these lower and upper
bounds.Lower and upper bounds produced by limit analysis are
valid for perfectly plastic soils following an associated
flowrule (dilatancy angle j equal to the friction angle f). In
thecase of materials commonly used for backfills (sands
andgravels), j is significantly lower than f. In fact, the
complex-ity of soil behavior goes beyond the difference between
jand f, as discussed in detail later, but this does not appearto
have been studied in connection with the analysis of re-taining
walls.Three stability checks are traditionally done in wall
design,
namely bearing capacity failure, sliding, and toppling. In
es-sence, these checks deal with assuring vertical, horizontal,and
moment equilibrium of the wall. While these separatechecks are easy
for engineers to understand and apply, thehorizontal and moment
resistances that the foundation soil(including any embedment in
front of the wall) can provideto the wall are in fact coupled with
the vertical bearing ca-pacity. For example, toppling failure
occurs in theory whenthe foundation load eccentricity e becomes
greater than one-half the foundation width B. Unless the wall base
is restingon rock or very stiff soil, the wall foundation will
actuallyfail due to the excessively high contact pressure at the
wallbase caused by large load eccentricity before toppling. In
ad-dition, wall sliding on its base is more likely to have the
formof a shallow one-sided bearing capacity mechanism with alarge
horizontal displacement component (Loukidis et al.2008) rather than
pure sliding along the basesoil interface.This paper aims to
investigate the gravity wallsoil interac-tion and the development
of these different failure scenariosby modeling the soil mechanical
behavior in a realistic wayin a series of finite element (FE)
analyses. This allows thedevelopment of displacement and stress
fields within the soilthat are not constrained by the simplifying
assumptions ofperfect plasticity and associativity. These results
are useful ininforming design decisions, the most important of
whichbeing how to calculate the active pressures on the backfaceof
the wall. The FE analyses, which take into account
nonas-sociativity, stress dependence of sand strength and
dilatancy,stress-induced anisotropy, fabric-induced anisotropy, and
pro-gressive failure, focus on the evolution of KA with wall
dis-placement u. This permits establishing the soil friction
anglevalue that is suitable for the estimation of the design
KAvalue, which is the one that corresponds to the wall
displace-ment required to bring the wall to an ultimate limit
state(ULS).
Problem statement
Dependence of active earth pressure on wall movementIn methods
of analysis currently used in design practice,
the main input for the calculation of KA for purely
frictionalbackfills is the internal friction angle f of the soil.
Thesemethods, which include the Rankine, Coulomb, and Lancel-lotta
methods discussed earlier, assume that f is constant, i.e.,its
value is the same at all points inside the backfill and
does not change as the wall moves. This would be validfor a very
loose backfill, where all soil elements reach fail-ure directly at
critical state (CS), with f equal to the CSfriction angle fc.
However, most practical cases involvebackfills consisting of medium
dense and dense sands andgravels, which are strain-softening
materials when shearedunder drained conditions, meaning that the
mobilized fric-tion angle of an element of any of these soils will
firstreach a peak value fp and then decrease towards fc. Cer-tain
regions inside the backfill mass will fail and start tosoften early
in the loading process. The shear strain leveldeveloped in these
regions may be large enough for thefriction angle to drop to its CS
value fc before the wallreaches a ULS, while f is close to fp in
other regions.This phenomenon is commonly referred to as
progressivefailure. In addition, fp depends strongly on the level
ofmean effective stress p, which varies from point to pointinside
the backfill and evolves continuously during wallmovement. It
should also be noted that, given that retainingwalls have a much
larger length than width, the deformationof the backfill and
foundation soil happens under plane-strain conditions (so the CS
friction angle is the plane-strainCS friction angle (Loukidis and
Salgado 2009)). Given thatthe friction angle varies from point to
point in the backfill,the representative f value to be used in KA
calculationmethods assuming perfect plasticity and associated flow
isunknown; it cannot be determined precisely based on intu-ition or
judgment.Let us idealize the gravity wall initial condition as one
in
which there has been no horizontal movement; as a result,the
coefficient of lateral earth pressure K is equal to its at-rest
value (K0). If we allow the wall to move away from thebackfill, K
first decreases to a minimum value KA, min (pointM in Fig. 1) and
then increases to an ultimate (residual)value KA, cr (point C in
Fig. 1). Between points M and C, thesupported soil is in an active
state. Point M is associated withan active state for which the
average mobilized f in the sup-ported soil is closer to fp than fc.
Point C is associated withfull mobilization of CS (f fc) along all
failure surfaces(shear bands) formed in the retained soil.To design
a wall, we are interested in the value of KA at a
limit state (KA, LS), which is not necessarily equal to
eitherKA, cr or KA, min. At present, there are two approaches to
deter-mine KA (Salgado 2008), one based on calculations using
anestimate of fp and the other using an estimate of fc. The for-mer
approach, which is most common in practice, wouldunderpredict the
active earth pressure on the wall at the limitstate, making it
unconservative. On the other hand, using fcmay be overly
conservative, since a well-designed wall wouldnot move as much as
to cause more than 20% shear strain inthe shear bands developing in
the backfill before the wallreaches its limit state (Salgado 2008).
The following sectionexamines in more detail what happens between
points M andC, and what would constitute an appropriately defined
ULSfor a gravity wall.
Wall limit state (WLS)To establish KA, LS, we need first to
establish a way to
identify the ultimate WLS. We must stress that, in establish-ing
a limit state, we are unconcerned with what the value of
Loukidis and Salgado 79
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the factor of safety (FS) would be to account for design
un-certainties and with serviceability limit states, which must
behandled separately. At early stages of wall movement awayfrom the
backfill, the total horizontal resistance Hr and mo-ment resistance
Mr increase at high rates, since the strains inthe foundation soil
are still small and the soil stiffness istherefore large. At the
same time, the driving horizontal forceHd and moment Md either
decrease, as the earth pressure co-efficient decreases from K0 to
KA, min, or increase at smallrates after the earth pressure
coefficient bottoms and starts toincrease from KA, min to KA, cr
(Fig. 1). The variations of thesequantities with wall displacement
can be written mathemati-cally as dHr > dHd and dMr > dMd.
Beyond a certain pointin the process (e.g., a certain amount of
wall crest displace-ment u), the resistance starts increasing at a
lesser rate thanthe driving action. This happens first for one of
the two resis-tances (Hr or Mr), so that this stage of the loading
process ismathematically identified as the state at which either
dHr 1, the wall is stable, meaning that equilibrium(Hd = Hr and Md
= Mr) is reached before the stationary stateor limit state is
reached (Fig. 1). Artificial external forceswould need to be
applied to the wallsoil system to bring itto the limit state, which
we define as identical to the station-ary state first reached by
the wall (i.e., if, by the addition ofexternal force, the Hd Hr
reaches its stationary state beforeMd Mr, then the limit state is
defined by the horizontalforce, not moment). This is analogous to
having a foundationelement (e.g., a footing or pile) supporting a
vertical load Qdless than its limit bearing capacity. To bring the
foundationelement to its bearing capacity ULS, we must apply an
artifi-cial external force Qext to the foundation element until it
col-lapses, which happens when the foundation resistance Qrattains
its maximum value QL. At this stage, both the Qr ver-sus settlement
curve and the Qext = Qr Qd versus settlementcurve reach stationary
(maximum) points (since Qd is con-
Fig. 1. Evolution of coefficient of lateral earth pressure and
total resisting horizontal force, Hr, and moment, Mr, and total
destabilizing hor-izontal force, Hd, and moment, Md, with
displacement of wall crest.
80 Can. Geotech. J. Vol. 49, 2012
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stant). In the case of a retaining wall, the load to be
sup-ported is not constant because of the dependence of KA onwall
displacement. Therefore, the WLS can only be identifiedby the
stationary point of the Hr Hd (or Mr Md) versus ucurve and not of
the Hr (or Mr) versus u curve. If FS < 1,Hd Hr (or Md Mr) is
always greater than zero and thewall is unstable, meaning that
artificial external forces wouldneed to be applied to the wall to
establish equilibrium. Inother words, wall movement never manages
to mobilizeenough resistance to balance driving forces. In this
case, thestationary point corresponds to the minimum value of
theseexternal forces or, equivalently, the maximum value of Hr Hd
(or Mr Md). The WLS is therefore the state at whichthe rate of
increase (mobilization) of wall base resistance (interms of either
force or moment) becomes smaller than therate of increase of the
destabilizing actions (mainly the earththrust).The goal of this
paper is to study the evolution of KA with
wall displacement and establish an indication of the KA
value(and the appropriate f value for its calculation) at WLS
con-ditions. For this purpose, we perform FE analyses of the
re-taining wall stability problem using a two-surface
constitutivemodel for sands, which is based on CS soil mechanics.
TheFE analyses are performed for a rigid gravity wall with arough
vertical backface supporting soil with level surfaceand purely
frictional nature. The analyses apply to the typicalcase of
backfilled walls but also to walls retaining naturalground if sandy
or gravelly in nature. Both the retained soiland foundation soils
are sands. The analyses do not simulatethe several complex stages
involved in the construction ofgravity walls (such as backfill
laying and compaction), whichwould lead to different initial stress
conditions, but such isnot the focus of the analyses, which aim
instead at bringingout the details of the mechanics of wall loading
not ad-dressed in the literature and the implications and
insightsthat they offer.
FE methodology
FE meshThe analyses use unstructured meshes consisting of
eight-
noded, plane-strain quadrilateral elements with 12
quadraturepoints. A typical FE mesh is shown in Fig. 2. It includes
thewall, the backfill soil, and the foundation soil. The wall hasa
rectangular cross section, with width B and height H. Thethickness
of the backfill soil layer is equal to H. The retain-ing wall is
embedded a distance D into the foundation soil.All analyses start
from an ideal state of the retained soil,reached without the wall
having moved or rotated (as if thebackfill soil had been placed in
one lift instantaneously).The wall is modeled as a block of linear
elastic materialwith very large Youngs modulus so that it can be
consid-ered rigid.No interface elements are placed between the soil
and the
wall; i.e., wall and soil share the same nodes along the
corre-sponding contact planes. As a consequence, slippage
betweenthe wall and backfill occurs due to the formation inside
thesoil mass of a shear band parallel to the wall backface.
Thisroughness condition is realistic given the rough
materialscommonly used for gravity walls, such as masonry,
concrete,and cribs containing stone.
It is well known that analyses involving materials thatsoften
and follow a nonassociative flow rule suffer from theproblem of
solution nonuniqueness. This means that, as themesh gets refined,
the FE analysis results change, and con-vergence to a unique
solution does not happen. To tacklethis problem, FE analyses should
either employ a regulariza-tion approach (such as Cosserat or
gradient plasticity) or usemeshes with element sizes consistent
with the known shearband thickness. The thickness of the soil
elements inside theshear bands simulating slippage between a rough
structureand granular soil is an important factor for the accurate
pre-diction of the shear resistance acting on the structure
(Louki-dis and Salgado 2008). Hence, the thickness of the
backfillsoil elements that are in contact with the wall backface is
setequal to 520 times the mean particle diameter of the sand(D50).
This is roughly the thickness of the shear bands thatform in sandy
soils, as observed in a number of experimentalstudies (e.g., Uesugi
et al. 1988; Vardoulakis and Sulem1995; Nemat-Nasser and Okada
2001). Due to restrictions inmemory allocation and analysis
runtime, the element sizes inother locations where shear bands are
expected to develop (i.e.,inside the sliding wedge and in the
foundation soil) werelarger than 520D50. Element size inside the
sliding wedgeforming behind the wall was of the order of 500D50.
Asshown later in the paper, this choice of the element sizehas only
a small impact on the analysis accuracy.
Constitutive modelThe constitutive model used in this study is
the two-surface
plasticity model based on CS soil mechanics developedoriginally
by Manzari and Dafalias (1997). The model wassubsequently modified
by Li and Dafalias (2000), Papadimi-triou and Bouckovalas (2002),
Dafalias et al. (2004), andLoukidis and Salgado (2009). The model
parameters weredetermined by Loukidis and Salgado (2009) for two
sands:air-pluviated or dry-deposited Toyoura sand (Iwasaki et
al.1978; Fukushima and Tatsuoka 1984; Lam and Tatsuoka1988;
Yoshimine et al. 1998) and water-pluviated or slurry-deposited
clean Ottawa sand (Carraro et al. 2003; Carraro2004; Murthy 2006;
Murthy et al. 2007). Toyoura sand isa fine sand (D50 0.2 mm) with
angular to subangular par-ticles, while Ottawa sand is a
medium-sized sand (D50 0.4 mm) with rounded to subrounded
particles. The modelconsiders four distinct surfaces having the
form of opencones in stress space: the bounding surface, dilatancy
sur-face, CS surface, and yield (loading) surface. Bounding
andcritical surfaces represent peak and CS shear strengths,
re-spectively. The dilatancy surface divides the stress spaceinto
two regions: inside the dilatancy surface, the soil plas-tic
behavior is contractive; outside it, it is dilative. Theyield
(loading) surface defines a very narrow conical do-main inside
which the soil develops no plastic strain. Theyield surface hardens
kinematically upon shearing, leadingto the development of plastic
strains prior to failure.Through this feature, the model simulates
accurately the be-havior of the soil at small and large strains.
The constitutivemodel takes into account the inherent anisotropy of
sandsthrough the use of a fabric tensor (Dafalias et al. 2004),and
the assumption that the position of the CS line in thevoid ratio
(e) mean effective stress (p) space depends onthe direction of
loading relative to the axis of sand deposi-
Loukidis and Salgado 81
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tion. It also accounts for the dependence of the soil friction
an-gle on the intermediate principal stress ratio b
(stress-inducedanisotropy). Details of the constitutive model
formulation,the determination of its input parameters, and its use
insimulating element response in laboratory tests can befound in
Loukidis and Salgado (2009). The parameter val-ues for the two
sands considered in this study are shownin Table A1 in the appendix
along with a short summaryof their role in the model.
FE algorithmsThe FE analyses were performed using the
open-source
code SNAC (Abbo and Sloan 2000). The stressstrain rateequations
of the constitutive model were integrated using asemi-implicit
Euler algorithm with subincrementation and er-ror control, details
of which can be found in Loukidis (2006),
and a relative stress error tolerance of 0.01%. The FE analy-ses
were performed using the modified NewtonRaphsonglobal solution
scheme, with the elastic stiffness matrix asthe global stiffness
matrix.All analyses start with an initial stage in which the
geo-
static stress field is established in the FE mesh. The
geostaticstage includes two phases. In the first phase, gravity is
ap-plied to the mesh as a body force loading, and a uniformpressure
equal to g(H D) is applied on the free surface ofthe soil in front
of the wall. These loadings are applied inone increment (i.e.,
instantaneously). In addition, a geostaticstress state is
prescribed at every Gauss-quadrature point inthe mesh. The
kinematic hardening stress (normalized back-stress) tensor of the
constitutive model is initialized so thatthe stress state lies at
the axis of the conical yield surface(the initial stress state is
inside the elastic domain). Becausethe initial vertical stress
values are set to be consistent withapplied gravity loading,
equilibrium is reached instantlythrough the execution of a single
global solution step. In thenext phase, the uniform pressure acting
on the free surface ofthe soil in front of the wall is removed in a
small number ofsolution increments. During creation of the
geostatic stressfield, the wall is not allowed to move horizontally
but is freeto move vertically. The geostatic stage is followed by
themain analysis stage during which the wall is allowed tomove
according to the scheme described next.
Wall loadingTo achieve the goals of this study, we must be able
to im-
pose large wall displacements from the initial position inwhich
the wall is in equilibrium with soil in an at-rest condi-tion. In
the beginning of the analysis, the wall is fully sup-ported at two
points, namely the crest (node C) and the toe(node T), shown in
Fig. 2, where the corresponding horizon-tal reactions are RC,0 and
RT,0, respectively (Fig. 3). These arethe forces required for full
equilibrium, given the tractionsexerted on the wall by the
surrounding soil at rest. Equiva-lently, the wall is prevented to
move horizontally or rotate be-
Fig. 2. Typical mesh and boundary conditions used in the FE
analyses. DOF, degree of freedom.
Fig. 3. Schematic showing the forces acting on the wall,
includingthe reactions, on nodes C and T due to the applied
displacement onthese nodes. EA, x, EA, y, horizontal and vertical
components of theactive earth thrust, respectively; Fx, horizontal
foundation reaction;Fy, vertical foundation reaction.
82 Can. Geotech. J. Vol. 49, 2012
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cause of the external application of a horizontal force Fext,0
=RC,0 + RT,0 and a moment Mext,0 = RC,0/H. The analysis pro-ceeds
by the application of outward horizontal displacementincrements DuC
and DuT (i.e., displacements pointing awayfrom the backfill) at
nodes C and T (Fig. 3), while monitor-ing the values of external
force Fext = RC + RT, where RCand RT are external reactions, and
external moment Mext =RC/H. Applying static equilibrium principles,
the excess ofthe driving forces throughout the process of wall
movementmust be balanced by artificial external forces, which exist
ex-clusively for the purpose of performing the analysis. It can
beshown that Fext = Hd Hr and Mext = Md Mr, given that
theunbalanced forces at the end of each increment of
theNewtonRaphson solution are minimal (less than 1% of theexternal
forces). As the wall moves, Fext and Mext decreaseprogressively.
The wall is allowed to move vertically, sinceno restraints are
imposed on its nodes in the vertical direc-tion. The prescribed
displacements uC and uT are not constantduring the analysis and are
not equal to each other. Theirmagnitude varies in such a way that
Fext and Mext (and conse-quently RC and RT) change in the same
proportion. This isachieved by setting DuT = 0 (a pure rotation
step) after anyanalysis increment, resulting in Mext/Mext,0 >
Fext/Fext,0, andDuT = DuC (a pure translation step) after any
analysis incre-ment, resulting in Mext/Mext,0 Fext/Fext,0. This
scheme relieson the fact that the wall rotation has a stronger
effect on therate of increase of Mext than of Fext, while wall
translation hasa stronger effect on the rate of increase of Fext
than of Mext.The increment DuC is always equal to a specified value
ofthe order of 106H. Hence, the analyses consist of
alternatingphases of pure wall rotation and pure wall translation.
Appli-cation of the loading in this manner, combined with the
veryfine incrementation used in the present analyses, results
inMext/Mext,0 Fext/Fext,0 (both ratios thus denoted by the
singlevariable Y) throughout the analysis. As a result, if the wall
isstable or marginally stable, Mext and Fext become equal tozero
simultaneously at which point the wall is completely un-supported
by external reactions (which means that this be-comes a point of
equilibrium at which the wall comes to arest). In addition,
Mext/Mext,0 and Fext/Fext,0 reach their mini-mum value (Ymin)
simultaneously, which happens whendMd = dMr and dHd = dHr.
Therefore, referring to our pre-vious discussion of the WLS, Ymin
is reached at the WLS.It should be noted that there is an infinite
number of load-
ing path formulations that can bring the wall to a limit
state,and the formulation presented here is just one of them.
Thepresent wall loading formulation makes it possible to
performdisplacement-controlled analyses, instead of
load-controlledanalyses that drive the wall to its limit state by
increasingthe soil unit weight or a surcharge pressure.
Displacementcontrolled analyses allow the wall to move beyond the
limitstate all the way to CS (a requirement of this study) for
allpossible outcomes (stable, marginally stable, and
unstablewalls). In contrast, load-controlled analyses cannot
proceedpast the point of limit state. This is because any increase
ofthe applied load past this point results in unbalanced forcesthat
cannot be mitigated, since they increase with eachNewtonRaphson
iteration. The formulation used here offersalso simplicity,
allowing clear understanding of the mechan-ics involved and
straightforward derivation of conclusions
for a practical problem that is considerably complex once
ex-amined using rigorous mechanics.In most field cases, the active
state will be mobilized grad-
ually, and the wall base will translate and rotate as the
back-fill is constructed before reaching full height. Moreover,
thebackfills placed behind gravity walls in practice are
com-pacted, resulting in initial stress conditions in each
layerlarger than the K0 conditions assumed in this paper due
tolocked-in stresses (which are difficult to simulate,
requiringthree-dimensional FE analysis). These factors would
generatedifferent stress paths in the soil mass than those produced
inour analyses. Problems involving materials that soften andfollow
a nonassociative flow rule exhibit path dependence, i.e.,the
results depend on the stress paths followed at the
stressintegration points of the mesh. Hence, it is expected thatthe
results of these analyses would be somewhat differentif the exact
backfill construction process were simulated.However, discrepancies
due to wall motion during backfillconstruction are believed to be
small because most of thewall displacement will occur when the
backfill height isnear the wall height, since the earth thrust
increases at leastquadratically with the rate of backfill height,
taking alsointo account that the soil friction angle would decrease
dueto the increase in mean effective stress as the backfill
rises.Discrepancies due to non K0 initial conditions would
existmostly during the early stages of the predicted response,
de-creasing as the active state were approached.
Results of FE simulations
Finite element analyses were performed for values of wallwidth
B, ranging from 1.5 to 3 m and wall height rangingfrom 6 to 8 m.
The sand unit weight g was set equal to18 kN/m3. The wall unit
weight was also set equal to 18 kN/m3,which corresponds more
closely to the unit weight of ma-sonry, gabion, or a crib wall
rather than a concrete wall.The coefficient K0 was set equal to 0.5
in all analyses. Nosurcharge is placed on the backfill free
surface. The rangeof the wall dimensions was chosen such that the
wall FS isnot excessively high or excessively low. As will be
shownlater, the FS of the wall configurations analyzed is in
the0.52.0 range. Analyses are performed for Toyoura and Ot-tawa
sands, with relative density DR ranging from 30% to90%. For the
sake of simplicity, the foundation soil is as-sumed to be of the
same type and density as the backfillsoil.
Collapse mechanism patternsMost of the analyses were performed
with the loading
scheme described in the previous section, which subjects thewall
to both rotation and horizontal translation in such a waythat the
stabilizing external reactions RC and RT decrease pro-portionally
to each other. For comparison purposes, analyseswere performed with
both the wall rotating about its heelwithout translating
horizontally (pure rotation case) andtranslating horizontally
without rotating (pure translation).Figure 4 shows contours of the
incremental maximum shearstrain gmax (= 1 3, where 1 and 3 are the
major and mi-nor principal strains, respectively) from analyses
with purerotation, pure translation, and combination of rotation
and
Loukidis and Salgado 83
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translation. The deformation patterns depicted in these
plotscorrespond to states well beyond the WLS.The failure mechanism
in the backfill consists of a wedge-
shaped sliding mass delimited by the wall backface and anoblique
shear band originating from the heel of the wall.This shear band,
which is nearly straight, with a slight curva-ture at its lower
part, will be referred to in the remainder ofthe paper as the main
shear band. A shear band running par-allel to the wall backface
also forms in all analyses, repre-senting sliding between the
backfill sliding mass (wedge)and the wall. In Figs. 4a and 4b, we
see that families of sec-ondary shear bands develop inside the
sliding wedge. One ofthe shear band families runs parallel to the
main shear band.The shear bands of the other family form an angle
with re-spect to the vertical of the same magnitude as the first
familybut with opposite sign. This is consistent with
observationfrom the experiments performed by Milligan (1974) and
thenewly reinterpreted radiographs of those experiments byLeniewska
and Mrz (2001), as well as from the FE analy-ses by Gudehus and
Nbel (2004). The families of the secon-dary shear bands vanish when
the wall movement is a puretranslation (Figs. 4c, 4d), although a
few secondary shearbands that do not propagate fully, fading inside
the slidingmass, still form. Gudehus and Nbel (2004) also show
thatthe web of secondary shear bands inside the sliding
wedgepresent in the problem of a rotating wall is absent in thecase
of a purely translating wall.Below the wall base, a bearing
capacity mechanism forms.
The shape of this mechanism resembles that of mechanisms
presented by Loukidis et al. (2008) for the case of surfacestrip
footings on purely frictional elastic perfectly plasticmaterial
loaded by eccentric and inclined loads. For analyseswith a wall
movement that contains a translational component(Figs. 4b4d), the
base mechanism is largely one-sided, con-sisting of a fan region
and a passive wedge. The same type ofmechanism can be seen in the
examples of Fig. 5. This isconsistent with failure patterns
observed in footings subjectedto inclined loads irrespectively of
the value of the load eccen-tricity (as long as the eccentric load
lies on the side of thefooting base the horizontal component of the
inclined loadpoints to). For a purely rotating wall (Fig. 4a), most
of theshearing in the base failure mechanism is concentrated in
ashear band that has the shape of a roughly circular arc, withits
end points lying on the two edges of the wall base. Louki-dis et
al. (2008) observed a similar pattern for footingsloaded by
vertical eccentric loads.Figure 5 shows contours of the incremental
gmax from
analyses of walls that translate and rotate (the main
loadingscheme used in the present paper) with a retained mass
ofloose and dense sands. It is evident that the inclination angleof
the shear bands in the retained soil mass with respect tothe
horizontal is larger in the case of dense than loose sand.Based on
the contours shown in Fig. 5, the shear band incli-nation angle
with respect to the horizontal is approximately65 for DR = 90% and
55 for DR = 45%. The inclinationfor 90% relative density is
comparable to the values of theshear band inclination observed in
the centrifuge experimentsof Wolf et al. (2005) in very dense sands
simulating the Rankine
Fig. 4. Examples of contours of incremental gmax from analyses
with a wall subjected to different modes of movement: (a) pure
rotation;(b) rotation and translation; (c) pure translation (B = 3
m); (d) pure translation (B = 1.5 m).
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problem (smooth wall, horizontally unrestrained backfill
base)for wall displacements similar to those in the present
problem.Regarding the wall base failure mechanism, it is hard
to
distinguish a separate passive failure mechanism in the soilin
front of the wall above the wall base level. Any potentialpassive
failure mechanism at the wall toe is fully encom-passed by the
bearing capacity mechanism. Thus, the stressdistribution above the
toe of the wall (Fig. 3) contributes tothe wall stability, not as a
passive resistance independentof the resistance at the wall base
but as part of the lateral ca-pacity of an embedded footing. This
observation reinforcesthe notion that the resistance provided by
the soil below thebase of the foundation and that in front of the
wall arecoupled, meaning that the lateral resistance of the wall
shouldbe analyzed as the problem of an embedded strip footing
sub-jected to eccentric, inclined loading. This holds throughoutthe
process of the loading of the wall, even when a limitbearing
capacity mechanism has not yet formed.An analysis is also done for
a backfill consisting of two re-
gions: a triangular region that is in contact with the wall
back-face consisting of Toyoura sand with DR = 75% and theremaining
soil consisting of loose Toyoura sand (DR = 45%),as shown in Fig.
6. Similar backfill cross section is frequentlyencountered in quay
walls, where a granular material withlarge strength is placed in
contact with the wall backface,with the goal of reducing the earth
thrust that would be ex-erted on the wall if the backfill were made
entirely of loose
fill sand. In Fig. 6, we see that the main shear band in
thebackfill delimiting the sliding mass changes inclination at
thepoint it crosses the boundary between the dense and loosesand
layers. The inclination of the main shear band inside thedense sand
and the loose sand is 66 and 57, respectively.
Earth pressure evolution with wall movementFigure 7 shows
examples of the normal (horizontal) stress
distribution along the back of the wall. All analyses startfrom
geostatic stress conditions (K = K0), and thus a triangu-lar stress
distribution with depth. With increasing wall dis-placement, the
horizontal stress decreases progressively untila minimum active
pressure state (MPS) is reached. From thatpoint on, the average
horizontal stress increases, but at amuch lower rate than the rate
at which it decreased earlier.Before the MPS, the stress
distribution is smooth; afterwards,local peaks and valleys develop.
This is a consequence of bi-furcation and the shear banding that
develops inside the slid-ing mass. The local minima in the stress
distributions roughlycoincide with the intersection of secondary
shear bands withthe wall backface.In all three analyses shown in
Fig. 7, the stress distribution
before the minimum active state is reached is intenselycurved at
the lower third of the wall height. In fact, beyonda certain depth,
the horizontal stress decreases with depth, aconsequence of soil
arching, as noted by other authors (e.g.,Handy 1985; Paik and
Salgado 2003). The curvature of the
Fig. 5. Contours of incremental gmax from analyses with dense
and loose (a, c) Ottawa sand and (b, d) Toyoura sand.
Loukidis and Salgado 85
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stress distribution in the case of a nonrotating wall (Fig.
7c)is more pronounced and happens at a shallower depth than inthe
analyses with rotating walls (Figs. 7a, 7b). As a conse-quence, the
point of application of the horizontal earth thrustfor translating
walls is higher than for rotating walls. This isconsistent with
observations from experimental (Fang andIshibashi 1986) and
numerical (Potts and Fourie 1986; Dayand Potts 1998) studies. After
the minimum active state, thecurvature of the stress distribution
decreases, and the averagedistribution resembles again a triangular
distribution, exceptfor purely translating walls.The evolution of
the lateral earth pressure coefficient K
with crest displacement uC is shown in Figs. 8 and 9. The
Kcoefficient is calculated from the lateral earth thrust, which
isin turn calculated by integrating numerically the
horizontalstress across the entire height of the wall. The stresses
aretaken at the centroid of the elements that are in contact
withthe wall. The results shown in Fig. 8 are all for Toyoura
sandwith 60% relative density but different values of H, B, and
D.All analyses shown, except one, are for a wall subjected toboth
translation and rotation. The K drops sharply towards aminimum
value (KA, min) at uC approximately equal to 0.003Hand,
subsequently, rises smoothly, approaching an asymptoticvalue (KA,
cr) related to the development of CS in the mainshear band and
inside the sliding wedge. According toFig. 8, KA, min and KA, cr
are practically independent of thewall dimensions and the
embedment, which seem to only af-fect the rate of increase towards
CS. These differences in therate of increase are due to the
resulting small differences inthe proportion of uT over uC between
these analyses (i.e., dif-ferences in how much of the motion is
translation versus ro-tation). The ratio uT/uC at the KA, min state
(MPS) in the
analyses of Fig. 8 ranges from 0.17 to 0.28 and is
roughlyproportional to the wall safety factor. Figure 8 also
includesthe response from the analysis with a purely rotating
wall,which is in sharp contrast with the other analyses. The KA,
minfor the purely rotating wall is about 38% higher than for
wallsthat both translate and rotate. More importantly, KA, min
isreached at uC equal to 0.016H, a much larger displacementthan for
the other curves in Fig. 8. Moreover, transition fromKA, min to KA,
cr is more gradual, K appearing to be almost con-stant for a large
range of uC values after the attainment of KA,min.Model tests by
Fang and Ishibashi (1986) demonstrate thatMPS is easily attained
for a purely translating wall, withonly 0.0004H of wall
displacement (in our purely translat-ing wall analyses, the
corresponding value is 0.001H). Incontrast, this state is not
reached in a model test for a wallin pure rotation about its base,
even with 0.008H of crestdisplacement. Large displacements for a
purely rotating wall,of the order of 0.015H, were needed in the
model tests byMilligan (1974) to reach the active state, which is
comparableto the value of 0.016H resulting from the present
analysis.Data reported by Fang and Ishibashi (1986) also supportthe
fact that a purely translating wall and a wall that bothtranslates
and rotates develop similar KA, min values, but theKA, min for a
purely rotating wall is distinctively larger.These findings suggest
that the absence of a translation
component in the wall movement has an important effect onthe KA,
min. Observed differences between wall problems in-volving
different movement modes are a consequence of thepath dependence
and progressive failure inherent in problemsinvolving
strain-softening materials. In contrast, analyses withperfectly
plastic materials following the MohrCoulomb fail-ure criterion
produce KA values that dont depend on the wall
Fig. 6. Contours of incremental gmax from analysis with
composite backfill.
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Fig. 7. Examples of distribution of horizontal stress acting on
thewall backface at different stages during analyses, with H = 7 m,
B =1.5 m, and D = 0.5 m and different modes of wall movement:(a)
Ottawa sand; (b, c) Toyoura sand.
Fig. 8. Variation of normalized lateral earth pressure
coefficient withwall crest displacement from analyses with medium
dense Toyourasand (DR = 60%).
Fig. 9. Variation of normalized lateral earth pressure
coefficient withwall crest displacement from analyses of a wall
that is allowed totranslate and rotate, with H = 7 m, B = 1.5 m,
and D = 0.5 m:(a) Toyoura sand; (b) Ottawa sand.
Loukidis and Salgado 87
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movement mode. As to the value of KA for a purely rotatingwall,
Ichihara and Matsuzawa (1973) measured static KA of0.16 for Toyoura
sand with DR = 85% for a smoother wallthan considered here. This
value still compares well with thevalue of 0.172 for our analyses
with purely rotating wall andDR = 60%.Finite element simulations
for soils that strain soften and
follow nonassociated flow rules suffer from mesh depend-ence. To
assess the accuracy of our simulations, an analysiswith element
size equal to 100D50, instead of 500D50, in theregion the sliding
wedge develops was performed, and resultsare also compared in Fig.
8. The differences between thecurves from the fine mesh analysis
and the correspondingcoarse mesh analysis do not exceed 8%.Figure 9
shows the K/K0 evolution resulting from analyses
with the same wall configuration (H = 7 m, B = 1.5 m, D =0.5 m)
for Toyoura and Ottawa sands with different values ofrelative
density. The figure also shows the KA/K0 value re-sulting from FE
analyses for an elastic perfectly plasticsoil following the
MohrCoulomb failure criterion (MCanalyses) with f fc and j = 0
using the same code andloading scheme as the main series of
analyses. The fc forplane-strain conditions (i.e., fc; PS) for
Toyoura and Ottawasand predicted by the two-surface constitutive
model de-scribed earlier is 36.6 and 34.6, respectively. These
valuesare roughly 45 larger than the fc values for triaxial
com-pression conditions (31.6 and 30.2, respectively). To placethe
KA calculated from fc in this manner in context, addi-tional
results are shown in Table 1. These results includethose of MC
analyses for associated flow (j = f) and non-associated flow with j
= 0 as well as the correspondinglimit analysis lower and upper
bounds. The results for an as-sociated flow rule are inside the
lower and upper bounds.The KA values for j = 0 are about 18%
greater than thosefor associated flow and lie above the KA range
delimited bythe limit analysis bounds. As discussed in the
introduction,limit analysis holds for an associated flow rule, and
numeri-cal analysis is currently the only way to arrive at
solutionsvalid for problems with j
-
Table 1. KA/K0 from FE analyses with elastic perfectly plastic
soil and corresponding limit analysis lower and upper bounds.
FEM (elastic perfectly plastic) Lower bound
(Lancellotta 2002)Upper bound(Chen 1975)*
Upper bound (Soubra andMacuh 2002)*f () j = f j = 0
36.6 0.403 0.480 0.404 0.393 0.39634.6 0.437 0.505 0.444 0.432
0.436
Note: FEM, finite element method.*Interpolated values based on
plotted or tabulated data.
Fig. 10. Evolution of key problem variables with increasing wall
crest displacement for (a) loose and (b) dense Toyoura sand and
wall withH = 7 m, B = 1.5 m, and D = 0.5 m. Deax/H, relative height
of active thrust application; FSH, FS based on horizontal
equilibrium; FSM, FSbased on moment equilibrium; m (= tand),
mobilized friction coefficient at wall backface.
Loukidis and Salgado 89
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only at very large wall displacements (Figs. 10a and 11a).The
WLS is reached when the slow base resistance develop-ment can no
longer match the increase in destabilizing ac-tions due to the
increase of KA towards KA, cr. The WLS canbe described
mathematically as the state at which dHr = dHdor dMr = dMd; beyond
the WLS, gains in resistance do notmatch gains in driving forces
(i.e., dHr < dHd or dMr 0 and dMd > 0) with wall
dis-placement past the KA, min state. As a consequence, the
walldisplacement required to reach the WLS is smaller than
thatrequired to reach base failure.Although WLS and base failure
state occur at distinctively
different wall crest displacements, the Hb and Mb values atWLS
are practically identical to the peak Hb and Mb values.This occurs
because, after attainment of the WLS, Hb and Mbincrease at very
small rates towards their peak values due tothe development of
regions of intense plastic straining in thefoundation soil.
Consequently, the peak Hb and Mb values,which can be determined in
practice with relative ease based
Fig. 11. Evolution of key problem variables, with increasing
wall crest displacement for (a) loose and (b) dense Ottawa sand and
wall withH = 7 m, B = 1.5 m, and D = 0.5 m.
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on bearing capacity calculations, can be used as good
ap-proximations of the values of Hb and Mb at the WLS.Table 2
summarizes the values of certain key variables of
interest in design at the state of minimum active pressure andat
the WLS: the normalized coefficient KA/K0, the mobilizedfriction
angle at the wallbackfill interface dmob, normalizedcrest
displacement uC/H, and the relative height Deax/H ofthe point of
application of the active earth thrust from theheel of the wall.
Table 2 also contains the value of the FSmobilized at the WLS. The
reported FS is taken as the small-est value of the FS against
overturning FSM (based on mo-ment equilibrium about the wall toe)
and sliding FSH(based on horizontal equilibrium). The FS values are
calcu-lated using the following equations:
3a FSH X
stabilizing forcesXdestabilizing forces
HbEA; x
1 RC RTEA; x
3b FSM X
stabilizing momentsXoverturning moments
WB=2 MOb
EA; xDeax EA; yB 1RCH
EA; xDeax EA; yBwhere EA, x and EA, y are the horizontal and
vertical compo-nents of the active earth thrust, W is the wall
self-weight,Deax is the height from the base of the point of
applicationof the earth thrust, and MOb is the moment of the base
resis-tance taken about the toe of the wall (not to be confused
withMb). The peak values of the mobilized FS values do not hap-pen
at the WLS (Figs. 10 and 11). This is because the ex-trema of the
FSH and FSM depend on the denominators ineqs. [3a] and [3b]. The
minimum value Ymin would occur si-multaneously with the peak
mobilized FS and the peak totalbase horizontal or moment resistance
only if these denomina-tors were constant during the
analysis.Figure 12 compares KA,min, KA, LS, and the inferred KA, cr
val-
ues for analyses with rotating and translating walls. We see
thatthe KA, LS values lie approximately halfway between the
mini-mum and CS values of KA. The value of KA, LS is 25%80%greater
than KA,min, with the differences increasing with increas-ing
relative density. Figure 12a also plots results from analysesNos.
1317 (Table 2), which have different wall dimensions (H,B) and
embedment D but the same sand relative density (DR =60%). Figure
12a indicates that the wall dimensions and embed-ment have a more
pronounced effect on KA, LS than KA,min.
Wall displacement to reach characteristic statesThe ratio uC/uT,
resulting from the loading scheme adopted
for the analyses of walls moving in both rotation and
transla-tion, is in the 38 range for MPS. Beyond the MPS,
uC/uTincreases a further 20%40% by the time the WLS isreached,
remaining practically constant for the remainder ofthe analysis,
with the exception of the simulation for DR =90% for which uC/uT
can reach values in the 1012 range.In the analyses in which the
wall is allowed to rotate and
translate, the crest displacement required for reaching theMPS
is in the 0.003H0.006H range (Fig. 13), with the ex-Ta
ble2.
Summaryof
FEM
results
with
respectto
stateof
minim
umlateralearthpressure
coefficientandwalllim
itstate.
No.
Loading
mode
H(m
)W
(m)
DR(%
)D(m
)Sand
Minim
umK
Astate
WLS
KA,m
in/K
0d m
ob()
feq
fc;TX()
u C/H
Deax/H
KA,L
S/K
0
d mob
()
feq
fc;TX()
u C/H
Deax/H
FSFailu
remode
1Rot+hor
71.5
600.5
Toyoura
0.249
35.9
15.1
0.004
0.337
0.383
30.8
5.9
0.021
0.260
1.04
Horizontal
2Rotation
71.5
600.5
Toyoura
0.344
30.8
8.6
0.016
0.267
0.344
30.8
8.5
0.016
0.266
1.38
Mom
ent
3Horizontal
71.5
600.5
Toyoura
0.269
38.0
13.0
0.001
0.358
0.396
30.8
5.1
0.007
0.334
1.65
Horizontal
4Rot+hor
71.5
300.5
Toyoura
0.408
31.4
4.2
0.006
0.319
0.461
30.9
1.2
0.163
0.311
0.47
Mom
ent
5Rot+hor
71.5
450.5
Toyoura
0.321
34.1
9.6
0.003
0.325
0.407
30.8
4.4
0.032
0.290
0.62
Mom
ent
6Rot+hor
71.5
750.5
Toyoura
0.192
37.7
20.2
0.004
0.342
0.326
30.9
9.9
0.020
0.254
1.43
Horizontal
7Rot+hor
71.5
900.5
Toyoura
0.136
38.5
26.3
0.004
0.336
0.247
31.0
16.0
0.019
0.257
2.01
Horizontal
8Rot+hor
71.5
300.5
Ottawa
0.456
29.7
3.1
0.010
0.328
0.483
29.6
1.6
0.160
0.319
0.40
Mom
ent
9Rot+hor
71.5
450.5
Ottawa
0.374
31.3
7.8
0.006
0.324
0.472
29.6
2.2
0.054
0.296
0.50
Mom
ent
10Rot+hor
71.5
600.5
Ottawa
0.304
31.8
12.7
0.006
0.327
0.391
29.6
7.1
0.025
0.292
0.82
Mom
ent
11Rot+hor
71.5
750.5
Ottawa
0.239
31.7
18.0
0.006
0.330
0.306
29.6
12.9
0.015
0.292
1.23
Horizontal
12Rot+hor
71.5
900.5
Ottawa
0.178
31.7
23.7
0.006
0.329
0.240
29.8
18.2
0.013
0.300
1.73
Horizontal
13Rot+hor
61.5
600.5
Toyoura
0.250
37.7
14.7
0.003
0.335
0.370
30.8
6.8
0.021
0.269
1.42
Horizontal
14Rot+hor
81.5
600.5
Toyoura
0.257
36.1
14.4
0.003
0.332
0.342
30.8
8.7
0.018
0.286
0.66
Mom
ent
15Rot+hor
72.0
600.5
Toyoura
0.248
37.9
14.9
0.003
0.340
0.405
30.8
4.5
0.023
0.260
1.49
Horizontal
16Rot+hor
72.5
600.5
Toyoura
0.248
36.6
15.1
0.003
0.337
0.399
30.8
4.9
0.020
0.269
1.98
Horizontal
17Rot+hor
71.5
600.2
Toyoura
0.247
38.0
15.0
0.003
0.344
0.349
30.8
8.2
0.021
0.277
0.83
Mom
ent
18Rot+hor
71.5
75+45
0.5
Toyoura
0.195
33.9
20.4
0.005
0.354
0.270
30.8
14.1
0.015
0.306
1.38
Horizontal
Note:
Rot+hor,rotatio
nandhorizontal;fc;TX,C
Sfrictio
nanglecorrespondingto
triaxial
compression
conditions;feq,equivalentsand
internal
frictio
nangle.
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ception of Ottawa sand, with DR = 30%. According to theCanadian
Geotechnical Society (1992), the uC required fordeveloping the
active earth pressure corresponding to peaksoil strength (i.e.,
uCp) is 0.001H for dense and 0.004H forloose sand. Our results are
of the same order of magnitudebut dont exactly match the values
proposed by the CanadianGeotechnical Society (1992). In addition,
our analyses sug-gest that the relative density has no significant
effect on uCpunless the tendency for strain softening is very weak
to prac-tically absent (e.g., DR = 30%). One would expect that
uCpshould decrease with increasing relative density as the
failurestrain decreases with increasing relative density in
laboratorytests, which in turn is due to the fact that the sands
stiffnessincreases with relative density. However, in the present
prob-lem, the stress and deformation conditions vary in both
the
horizontal and vertical directions, in contrast with
laboratorytests. In Fig. 9, we see that, as expected, the initial,
roughlylinear, response prior to MPS is much stiffer for dense
sandthan for loose sand. As K approaches KA, min, the
responsebecomes nonlinear and is smoother for large DR values
thanfor small ones. As a consequence, KA, min for dense sands
isreached at roughly the same wall displacement as for looseand
medium dense sands. This is a consequence of the pro-gressive
failure developing prior to the attainment MPS. Theprogressive
failure is evident by the fact that the mobilizedfriction angle on
the wallsoil interface is very close to theCS value by the time MPS
is reached, as demonstrated laterin the paper. Progressive failure
is more intense in dense thanin loose sands, counterbalancing the
effect of sand stiffnesson the wall displacement required to reach
MPS.The crest displacement uC,LS required to reach WLS is in
the range of 0.013H0.026H, except for DR = 30%45%. Inaddition,
for a given sand and relative density, the ratio uC,LS/uCp
increases with increasing FS. The uC,LS lies in the rangeof
displacements for which the wall foundation has not yetcollapsed
(i.e., the peak base resistance has not yet beenreached) but is
very compliant, yielding considerably foreven small changes in
foundation load.In most of the analyses with a rotating and
translating wall,
the crest displacement required to reach the peak Hb is in
the0.01H0.09H range (corresponding to toe displacement
of0.01B0.065B). The mobilization of the full horizontal
basecapacity requires displacements that exceed those required
toreach the WLS in the retained soil by 10%150%. The peak
Fig. 12. Minimum, limit state, and critical active earth
pressurecoefficient from analyses with (a) Toyoura sand and (b)
Ottawa sand.
Fig. 13. Wall crest displacement required to reach MPS (uCp)
andWLS (uC, LS) from analyses with Toyoura and Ottawa sands.
Analy-sis numbers follow the numbering shown in Table 2.
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moment base resistance requires even larger wall movementin all
analyses performed in this study.
The variation of the earth pressure coefficient K with uCcan be
described mathematically by the following equation:
4 K fc 1K0 KA; cr=K0 KA;min guCp cuCc 1uCp=K0 KA;min uCc=KA; cr
KA;min uCpc1 KA; cr
The input parameters in eq. [4] are the characteristic valuesfor
the earth pressure coefficient, K0, KA, cr, and KA, min, thecrest
displacement uCp at which KA, min is attained, and a fit-ting
parameter c. According to eq. [4], K is equal to K0 foruC = 0 and
tends to KA, cr asymptotically for large values ofuC. The parameter
c is introduced to control the rate at whichK increases towards KA,
cr after the attainment of KA, min. Byfitting eq. [4] to the
results of this study, shown in Figs. 8and 9, we obtain c in the
1.72.1 range, with an averagevalue of two. An estimate of the KA,
LS to use for wall designcalculations could be obtained using eq.
[4], with c = 2.0,uC = 0.025H for medium dense and dense sands, and
0.06Hfor loose sands.
Point of application of lateral earth thrustAn important
parameter for retaining wall stability calcula-
tions is the location along the wall height at which the
lateralearth thrust EA, x acts. Figure 14 shows the values of the
ver-tical distance Deax of the point of application of EA, x from
thewall base obtained from the analyses of a wall that both
ro-tates and translates. We see that, for the MPS, Deax is
roughlyequal to the widely used value of (1/3)H. Beyond that
state,Deax starts decreasing, reaching a minimum value
almostcoincidentally with the WLS. This decrease is negligible
forloose sand, but it can be up to 25% for dense sand. Thevertical
distance Deax subsequently increases but at a verysmall rate.The
same trends of Deax with increasing wall displacement
were observed in the experiments of Fang and Ishibashi(1986).
Specifically, for a purely rotating wall, Deax starts de-creasing
from an initial value of 0.333H towards a minimumvalue of 0.22H and
then rises slowly (but never exceeding0.28H, even at uC = 0.008H).
Similar trends are found inIchihara and Matsuzawa (1973). Fang and
Ishibashi (1986)also present data that supports the fact that Deax
for purelyrotating walls is less than 0.3H, while it is around H/3
if themotion has a translational component.
Mobilized resistance along wallbackfill interfaceThe mobilized
friction coefficient m (= tand) on the wall
face reaches a peak value at very early stages of the
analyses,before the attainment of the MPS, suggesting the
verticalshear band along the wall backface forms well before
themain shear band delimiting the sliding wedge. This peakvalue is
strongly dependent on the relative density of thesand (Figs. 10 and
11). After the peak, m decreases quicklytowards a residual value
that is consistent with the develop-ment of CS (f fc;PS, j = 0)
inside the thin backfill soilelements that are in contact with the
wall. The residual valuesfor the angle d are 30.8 and 29.6 for
Toyoura sand and Ot-tawa sand, respectively. These are consistent
with the theoret-ical values calculated as dc = arctan(sinfc;PS)
(Loukidis andSalgado 2008). Although the vertical shear band along
the
wall backface forms well before the main shear band, the
mo-bilized frictional angle dmob at MPS is clearly larger than
dc(Fig. 15) for both rotating and translating walls and
purelytranslating walls. The ratio dmob/dc ranges from 1.0 to
1.07for Ottawa sand, while for Toyoura sand, it ranges from 1.02to
1.25 (corresponding to dmob from 31.4 to 38.5). The uCpfor a purely
rotating wall is so large that, by the time MPS isreached, dmob has
become equal to dc.
Equivalent value of sand friction angle for calculation ofKA at
limit stateIt is of practical interest to assess what the
appropriate
(equivalent) value of the sand internal friction angle feq is
foruse in the calculation of KA,min and KA, LS using an
analyticalformula widely used in practice, such as Coulombs
solution(eq. [1]), to obtain a value of KA, LS that is in agreement
withthe present numerical simulations. Figure 16 shows the
differ-ence between feq and the CS friction angle fc;TX
correspondingto triaxial compression conditions. The feq values are
back-calculated using eq. [1] from the KA values resulting fromthe
FE analyses. We consider fc;TX instead fc;PS because itis easier to
estimate it through either empirical relationshipsor a few triaxial
compression tests. Even the frequently per-formed shear box tests
would yield fc estimates that arecloser to fc;TX than fc;PS.
According to Fig. 16a, the feqfor MPS is 326 larger than fc;TX,
depending on the valueof the sand relative density. However, for
calculating KA, LS,feq is only 118 larger than fc;TX (Fig. 16b).
Figure 16bshows that existing walls are not necessarily poorly
designed,even if the design is based on the prevailing practice of
as-suming the soil to be perfectly plastic with a peak value
of(triaxial compression) friction angle to calculate KA:
practi-tioners would rarely use friction angles exceeding fc;TX
bymore than 15 for a dense sand or more than 2 for a loosesand. So,
whether by accident or proper intuition and judg-ment by engineers
working on this problem years ago, stand-ard practice uses friction
angles that are roughly consistentwith WLS rather than the state of
mobilization of peakstrength in the backfill. Figure 17 shows the
difference be-tween feq and the CS friction angle fc;PS
corresponding toplane-strain conditions. The values plotted in Fig.
17 areabout 4.55 smaller than those in Fig. 16. In Fig. 17b, wesee
that, for loose sand, feq for KA, LS is smaller than fc;PS.At first
sight, this would seem to be a violation of the basicprinciple of
soil mechanics that the minimum value of thesand friction angle is
that for CS, but all analytical methodsfor calculating KA presented
in the introduction produce re-sults that are valid for an
associated flow rule (f = j) and,thus, underestimate the actual KA
by roughly 20% (see Table 1).On the other hand, the FE simulations
discussed in this paperuse a model that captures the sand dilatancy
realistically. Hadwe had a formula that predicted KA for realistic
j values, allresulting feq fc;PS values would have been
positive.
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ConclusionsThis paper presents the results of a set of FE
analyses of a
gravity wall subjected to the action of a mass of sand that
thewall initially retains at a state of rest. After the wall is
al-lowed to move due to the action of the retained soil mass,the
pressures on the wall evolve towards active pressures.Based on the
FE results and focusing on walls that are freeto rotate and
translate, we arrive at the following conclusions:
1. The attainment of the minimum value of the active
earthpressure coefficient (KA, min) requires wall crest
displace-ments of the order of 0.001H0.010H.
2. The attainment of KA, min corresponds to a mobilized wallsoil
interface friction angle d that is larger than the CSinterface
friction angle dc by a factor ranging from 1.0(loose sand) to 1.3
(dense sand).
3. For dense and medium dense sand, the crest
displacementrequired to bring the wall to its limit state is in
the0.013H0.026H range. At the WLS, the mobilized inter-face
friction angle has already reached the CS value of dc.
4. The limit state KA (KA, LS) lies between KA, min and
thecorresponding CS value KA, cr. The KA, LS values are lar-ger
than KA, min by a factor of 1.11.8, with the differ-ences
increasing with increasing relative density.
5. The WLS does not necessarily coincide with the mobiliza-tion
of the maximum base resistance, which may requiremuch larger wall
displacements.
6. The height of application of the lateral earth thrust at
limitstate conditions is less than one-third, ranging from 0.25to
0.32, suggesting that the current design practice isslightly
conservative.
7. The equivalent friction angle to be used for the calculation
ofKA values consistent with WLS design can be up to 18higher than
the soil CS friction angle under triaxialcompression conditions.The
results regarding the WLS depend on the base stiff-
ness and strength. Hence, our findings regarding the WLSare
strictly applicable to foundation soils that are like the re-tained
soil (i.e., purely frictional soils). It is expected that, forwalls
founded on stiff clay or weak rock, the WLS may coin-cide with or
even precede the MPS and the WLS, given thehigh stiffness and
brittleness of such geomaterials. Althoughin all analyses the
foundation soil had the same relative den-sity as the backfill, the
results are expected to hold even forcases in which the relative
densities are different. This is be-cause the displacement required
to reach KA, min is practicallyindependent of the density of the
backfill (Fig. 9). In addi-tion, we see in Fig. 9 that the shape of
the curves is thesame for all densities. Hence, what matters
regarding the dis-placement required to attain WLS is the density
of the foun-dation soil. Therefore, the displacement needed to
attainWLS for a loose backfill will not be much different fromthat
for a dense backfill as long as the density of the founda-tion soil
is the same.From a practical standpoint, this study suggests that
the
minimum active earth pressure state is of limited relevanceto
ULS design, since it happens for wall crest displacementsof the
order of only 0.5% the wall height; it is possibly repre-sentative
of a serviceability limit state (SLS). Given that theactive earth
pressure coefficient is a function of the wall dis-
Fig. 14. Vertical distance of point of application of active
earth thrustfrom wall base at (a) minimum earth pressure state and
(b) WLS.
Fig. 15. Ratio of d mobilized along wallbackfill interface at
mini-mum earth pressure state to d corresponding to CS
conditions.
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placement, existing wall design methods can benefit in thefuture
from a focus on estimating KA, using as reference thewell-defined
CS. For example, KA could be calculated usingwell-established
formulas (e.g., Coulombs equation) withrepresentative f values
estimated based on DR and fc;TX andplots such as that of Fig. 16b.
The interface friction angle dcan be conservatively set equal to
the corresponding CSvalue dc to calculate KA for all characteristic
states (MPS,WLS, and CS). Alternatively, the KA for ULS could be
calcu-lated using equations such as eq. [4], with KA, cr, KA, min,
uc,and uCp as input. The coefficient KA, min could be
calculatedusing plots such as that of Fig. 16a, while KA, cr, which
is aninvariant, could be directly calculated using fc;PS ( fc;TX
+4). According to Fig. 13, uCp is practically 0.005. The
dis-placement uc can be set equal to the desired crest
displace-ment value compatible with an ULS established according
todesign code provisions.
The present FE analyses demonstrate that the toppling,sliding,
and vertical bearing capacity failure modes arecoupled. Wall
stability can be assessed by considering twoequilibrium checks, one
pertaining to horizontal equilibriumand one to moment equilibrium,
where the KA estimated us-ing such equations or plots will be the
basic input. The useof equilibrium checks with partial load and
resistance orstrength factors accounting for the uncertainties of
the prob-lem variables (as in load and resistance factors design
(LRFD)or partial factors design (PFD)) instead of global FS
checks(as in working stress design (WSD)) constitute the basis
ofmodern codes. In these equilibrium checks, the horizontaland
moment wall base resistances should be establishedbased on the
appropriately factored bearing capacity failureenvelop of an
embedded strip footing subjected to eccentricand inclined (i.e.,
combined) loading. Following this ap-proach, a vertical equilibrium
check is redundant.
Fig. 17. Difference of equivalent value of friction angle to be
usedin calculation of KA from CS friction angle in plane-strain
condi-tions at (a) minimum earth pressure state and (b) WLS.
Fig. 16. Difference of equivalent value of friction angle to be
usedin calculation of KA from CS friction angle in triaxial
compressionconditions at (a) minimum earth pressure state and (b)
WLS.
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Appendix A
Table A1. Values of constitutive model parameters for Toyoura
and clean Ottawa sands.
ParameterParametersymbol
Parameter value
CommentToyoura sandCleanOttawa sand
Small-strain (elastic)parameters
n 0.15 0.15* Poissons ratioCg 900 611 Parameter controlling the
magnitude of the maximum shear
modulus Gmaxng 0.400 0.437 Exponent controlling the rate of
increase of Gmax with effec-
tive confining stressg1 0.0010 0.000 65 Parameters controlling
the decrease of para-elastic shear
modulus G with shear straina1 0.40 0.47CS Gc 0.934 0.780
Intercept of CS line in ep space
l 0.019 0.081 Parameter controlling inclination of CS line in ep
spacex 0.70 0.196 Parameter controlling curvature of CS line in ep
spaceMcc 1.27 1.21 Critical stress ratio in triaxial (TX)
compression conditions
Bounding surface kb 1.5 1.9 Parameter controlling the increase
of friction angle withsand density
Dilatancy Do 0.90 1.31 Inclination of the stressdilatancy
curvekd 2.8 2.2 Parameter controlling the stress ratio at phase
transformation
Plastic modulus h1 1.62 2.20 Parameters controlling the
magnitude of plastic modulush2 0.254 0.240elim 1.00 0.81 Upper
limit for void ratio for which the plastic modulus be-
comes zerom 2.0 1.2 Parameter controlling stress ratio in
undrained instability
stateStress-inducedanisotropy
c1 0.72 0.71 Ratio of the critical stress ratio in TX extension
to that inTX compression
c2 0.78 0.78* Parameter controlling the value of the magnitude
of inter-mediate principal stress relative to the two other
principalstresses under plain-strain conditions
ns 0.35 0.35* Parameter controlling the magnitude of the
friction angle inplane-strain conditions relative to the friction
angle in TXcompression
Inherent anisotropy a 0.29 0.31 Parameter controlling the
intercept of CS line in ep spaceunder conditions other than TX
compression
kh 0.11 0.39 Parameter controlling the variation of plastic
stiffness, withthe direction of loading relative to the axis of
sand de-position
m 0.05 0.05 Radius of conical yield (loading) surface
*Assumed.
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/CropMonoImages true /MonoImageMinResolution 1200
/MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true
/MonoImageDownsampleType /Average /MonoImageResolution 600
/MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00000
/EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode
/MonoImageDict > /AllowPSXObjects true /CheckCompliance [ /None
] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false
/PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000
0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true
/PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ]
/PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier ()
/PDFXOutputCondition () /PDFXRegistryName (http://www.color.org)
/PDFXTrapped /False
/CreateJDFFile false /SyntheticBoldness 1.000000 /Description
>>> setdistillerparams> setpagedevice