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Soil Dynamics and Earthquake Engineering 25 (2005) 795–809
www.elsevier.com/locate/soildyn
Seismic earth pressures on rigid and flexible retaining walls
P.N. Psarropoulosa,*, G. Klonarisb, G. Gazetasa
a
Scho

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Seismic earth pressures on rigid and ﬂexible retaining walls
P.N. Psarropoulos
a,
*
, G. Klonaris
b
, G. Gazetas
a
a
School of Civil Engineering, National Technical University, Athens, Greece
b
Mueser Rutledge Consulting Engineers, New York, USA
Accepted 11 November 2004
Abstract
While limiting-equilibrium Mononobe–Okabe type solutions are still widely used in designing rigid gravity and ﬂexible cantileverretaining walls against earthquakes, elasticity-based solutions have been given a new impetus following the analytical work of Veletsos andYounan [23]. The present paper develops a more general ﬁnite-element method of solution, the results of which are shown to be in agreementwith the available analytical results for the distribution of dynamic earth pressures on rigid and ﬂexible walls. The method is then employedto further investigate parametrically the effects of ﬂexural wall rigidity and the rocking base compliance. Both homogeneous andinhomogeneous retained soil is considered, while a second soil layer is introduced as the foundation of the retaining system. The resultsconﬁrm the approximate convergence between Mononobe–Okabe and elasticity-based solutions for structurally or rotationally ﬂexible walls.At the same time they show the beneﬁcial effect of soil inhomogeneity and that wave propagation in the underlying foundation layer mayhave an effect that cannot be simply accounted for with an appropriate rocking spring at the base.
q
2005 Elsevier Ltd. All rights reserved.
Keywords:
Retaining walls; Dynamic earth pressures; Finite-element analysis; Layered soil; Mononobe–Okabe
1. Introduction
For many decades, the seismic analysis of retaining wallshas been based on the simple extension of Coulomb’s limit-equilibrium analysis, which has become widely known asthe Mononobe–Okabe method ([14,11]). The method,modiﬁed and simpliﬁed by Seed and Whitman [19], hasprevailed mainly due to its simplicity and the familiarity of engineers with the Coulomb method.Experimental studies in the 1960s and 1970s using small-scale shaking table tests proved in many cases that theMononobe–Okabe method was quite realistic, at least if theoutward displacement of the wall (either due to translation,or rotation, or bending deformation) was large enough tocause the formation of a Coulomb-type sliding surface in theretained soil. A signiﬁcant further development on theMononobe–Okabe method has been its use by Richardsand Elms [17] in determining permanent (inelastic)outward displacements using the Newmark sliding block concept [13].However, in many real cases (basement walls, bracedwalls, bridge abutments, etc.) the kinematic constraintsimposed on the retaining system would not lead to thedevelopment of limit-equilibrium conditions, and therebyincreased dynamic earth pressures would be generated [7].Elastic analytical solutions were ﬁrst published by Scott [18],Wood[24],andAriasetal.[3].TheWoodsolutionreferredto
an absolutely rigid wall ﬁxed at its base; the derived elasticdynamic earth pressures are more than two times higher thanthe pressures obtained with the limit-equilibrium methods.This fact, and the scarcity of spectacular failures of retainingwalls during earthquakes, led to the widely-held impressionthattheelasticmethodsareover-conservativeandinappropri-ate for practical use. This was the main reason for the nearlyexclusive use of Mononobe–Okabe (and Seed–Whitman)method in engineering practice.The two groups of methods mentioned above (elastic andlimit-equilibrium) seem to cover the two extreme cases. Theelastic methods regard the soil as a visco-elastic continuum,while limit-equilibrium methods assume rigid plasticbehaviour. Efforts to bridge the gap between the aboveextremes have been reported by Whitman and his
Soil Dynamics and Earthquake Engineering 25 (2005) 795–809www.elsevier.com/locate/soildyn0267-7261/$ - see front matter
q
2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.soildyn.2004.11.020
*
Corresponding author. Tel./fax:
C
1 30 210 894 5275.
E-mail address:
prod@central.ntua.gr (P.N. Psarropoulos).
co-workers ([1,2,12]). Their analyses combine wavepropagation in a visco-elastic continuum with concentratedplastic deformation on a failure surface.Based on the aforementioned categorization, many codesestimate the dynamic earth pressures according to thepotential of the wall to deform. For example, the Greek Regulatory Guide E39/93 [16], referring to the seismicanalysis of bridge abutments, proposes three different casesfor the calculation of the dynamic earth pressures dependingon the ratio between the displacement at the top of the wall
u
to its height
H
(see Fig. 1). As it is possible for the wall–soil system to develop material (or even geometric) non-linearities, it is difﬁcult to distinguish the limits between thethree cases. The main reason is that the displacement
U
cannot be predeﬁned. It is obvious though, that theminimum dynamic earth pressures are predicted in thecase of ﬂexible walls (
u
/
H
O
0.10%), while the dynamicpressures are almost 2.5 times higher in the case of perfectlyrigid immovable walls (
u
/
H
z
0). For intermediate cases, thedynamic earth pressures are somewhere between themaximum and minimum value (see Ref. [15]).It was not until recently that Veletsos and Younan[21–23] proved that the very high dynamic earth pressuresof the elastic methods are attributed to the assumptions of rigid and ﬁxed-based wall, which are an oversimpliﬁcationof reality. To overcome this limitation, they developed ananalytical solution that could account for the structuralﬂexibility of the wall and/or the rotational compliance at itsbase; the latter was achieved through a rotational spring atthe base of the wall. They discovered that the dynamicpressures depend profoundly on both the wall ﬂexibility andthe foundation rotational compliance, and that for realisticvalues of these factors the dynamic pressures are substan-tially lower than the pressures for a rigid, ﬁxed-based wall.In fact, they found out that the dynamic pressures mayreduce to the level of the Mononobe–Okabe solution if either the wall or the base ﬂexibility is substantial.However, these analytical solutions are based on theassumption of homogeneous retained soil, and there arereasons for someone to believe that the potential soilinhomogeneity may lead to signiﬁcant changes in themagnitude and distribution of the dynamic earth pressures.Furthermore, as the presence of the foundation soil layersunder the retained system is only crudely modelled througha rotational spring, these solutions do not account for thepotential horizontal translation at the wall base, which ingeneral may have both an elastic and an inelastic (sliding)component.In the present paper, after a numerical veriﬁcation of theanalytical solution of Veletsos and Younan (utilising theﬁnite-element method), the versatility of the ﬁnite-elementmethod permits the treatment of some more realisticsituations that are not amenable to analytical solution. Sothe modelling was extended to account for: (a) soilinhomogeneity of the retained soil, and (b) translational
Fig. 1. Typical dynamic pressure distributions proposed in seismic bridge codes for seismic analysis of abutments. Situations (a) and (b) correspond to the twoextreme cases: (a) of yielding wall supporting elasto-plastic soil in limit equilibrium, and (b) of undeformable and non-yielding wall supporting purely elasticsoil. For intermediate size of wall displacements:
D
P
Z
0.75(
a
0
g
H
2
).Fig. 2. The systems examined in this study: (a) ﬂexible wall elasticallyrestrained at its base, and retaining a homogeneous soil layer, (b) ﬂexiblewall elastically restrained at its base, and retaining an inhomogeneous soillayer, and (c) rigid gravity wall in a two-layer soil system.
P.N. Psarropoulos et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 795–809
796
ﬂexibility of the wall foundation. Fig. 2 outlines the casesstudied in this paper. Evidently:
†
Case (a) coincides with the single-layer case studied in[23], where the retained soil is characterized byhomogeneity.
†
Case (b) models the same single-layer case, but theretained soil is inhomogeneous, with the shear modulusvanishing at the soil surface.
†
Case (c) refers to a rigid wall founded on a soil stratum.The results show that the inhomogeneity of the retainedsoil leads to reduced earth pressures near the top of the wall,especially in the case of very ﬂexible walls, while thecompliance of the foundation may not easily be modelled bya single rotational spring, due to wave propagationphenomena.
2. Flexible cantilever wall
2.1. Case A: homogeneous soil
Veletsos and Younan in 1997 [23] developed ananalytical approach for evaluating the magnitude anddistribution of the dynamic displacements, pressures, andforces induced by horizontal ground shaking in walls thatare both ﬂexible and elastically constrained against rotationat their base. The simplicity of their analytical toolpermitted the assessment of the effects and relativeimportance of the factors involved.In their model, the soil is considered to act as a uniform,inﬁnitely extended visco-elastic stratum of height
H
. Theproperties of the soil are regarded constant, and deﬁned bythe density
r
, the shear modulus
G
, and Poisson’s ratio
n
.The material damping is presumed to be of the constanthysteretic type and is deﬁned by the critical dampingratio
x
.The layer is free at its upper surface, ﬁxed on a rigid base,and it is retained by a vertical, ﬂexible wall, elasticallyconstrained against rotation at its base. The properties of thewall are described by its thickness
t
w
, mass per unit of surface area
m
w
, modulus of elasticity
E
w
, Poisson’s ratio
n
,and critical damping ratio
x
w
. The stiffness of the rotationalbase constraint is denoted by
R
q
.The bases of the wall and the soil stratum are consideredto be excited by a space-invariant horizontal motion,assuming an equivalent force-excited system.The factors examined are the characteristics of theground motion, the properties of the soil stratum, and theﬂexibilities of the wall and the rotational constraint at itsbase. Emphasis is given on the long-period—effectivelystatic—harmonic excitations. The response for a dynami-cally excited system is then given as the product of thecorresponding static response with an appropriate ampliﬁ-cation (or de-ampliﬁcation) factor.The whole approach is based on the following simplify-ing assumptions:
†
no de-bonding or relative slip is allowed to occur at thewall–soil interface.
†
no vertical normal stresses develop anywhere in themedium, i.e.
s
y
Z
0, under the considered horizontalexcitation.
†
the horizontal variations of the vertical displacements arenegligible.
†
the wall is considered to be massless.While the ﬁrst assumption was made in order to obtain asimpliﬁed model, the other three assumptions were made tosimplify the solution of the resulting equations that describethe behaviour of the model.The main parameters that affect the response of thesystem are the relative ﬂexibility of the wall and retainedsoil, deﬁned by
d
w
Z
GH
3
D
w
(1)and the relative ﬂexibility of the rotational base constraintand retained soil, deﬁned by
d
q
Z
GH
2
R
q
(2)
D
w
in Eq. (1) denotes the ﬂexural rigidity per unit of length of the wall:
D
w
Z
E
w
t
3w
12
ð
1
K
v
2w
Þ
(3)What also affect the response are the characteristics of the input base motion. For a harmonic excitation theresponse is controlled by the frequency ratio
u
/
u
1
, where
u
is the dominant cyclic frequency of the excitation, and
u
1
the fundamental cyclic frequency of the soil stratum.
2.1.1. Numerical modelling
This study focuses primarily on the numerical veriﬁca-tion of the analytical results of Veletsos and Younan, usingexactly the same model for the wall–soil system. Thepurpose was to validate the assumptions of the analyticalsolution and to deﬁne the range of its applicability.Presuming plane-strain conditions, the numerical anal-ysis was two-dimensional, and was performed using thecommercial ﬁnite-element package
ABAQUS
[8]. The ﬁnite-element model of the wall–soil system examined is shownin Fig. 3.By trial analyses it was established that the magnitude of the wall pressures is proportional to the wall height, whichwas also noted in the analytical solution. Taking thatinto account, all the analyses were performed consideringan 8 m-high wall. The wall itself was discretized bybeam elements, of unit longitudinal dimension
P.N. Psarropoulos et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 795–809
797
and thickness
t
w
Z
0.20 m. Given the value of
d
w
, themodulus of elasticity of the wall
E
w
derives from Eqs. (1)and (3), while the Poisson’s ratio is 0.2. The wall mass perunit of surface area
m
w
is presumed to be 2.5 t/m
2
. It isreminded that Veletsos and Younan had regarded the wall asmassless. At the base of the wall a rotational constraint wasplaced, the stiffness of which is denoted by
R
q
. Eq. (2)relates
d
q
with
R
q
.The discretization of the retained soil is made by two-dimensional, four-noded quadrilateral, plane-strainelements. Since, the ﬁnite-element grid cannot extendinﬁnitely, there is a need for using absorbing boundariesin order to simulate the radiation of energy. The latter isachieved using horizontal and vertical viscous dashpots,which absorb the radiated energy from the
P
and
S
waves,respectively. The efﬁciency of the viscous dashpots is ingeneral quite acceptable, but as it depends strongly on theangle of incidence of the impinging wave the dashpots wereplaced 10
H
away from the wall to improve the accuracy of the simulation (see Fig. 3). The soil is presumed to act as avisco-elastic material. Trial analyses indicated that the wallpressures are not directly affected by the shear modulusvalue of the retained soil. The shear modulus value affectsthe wall pressures indirectly via the relative ﬂexibilityfactors (
d
w
and
d
q
), and the eigenfrequency of the soilstratum. Therefore, for the analyses presented in this sectionthe density
r
and the shear wave velocity
V
S
are assumed tobe 1.8 t/m
3
and 100 m/s, respectively. Furthermore, thePoisson’s ratio
n
is presumed to be 1/3, while criticaldamping ratio
x
is 5%.Regarding the wall–soil interface, although the option of de-bonding and relative sliding was available in
ABAQUS
[8],theassumptionofcompletebonding—madebyVeletsosandYounan—was also adopted to permit a comparative study.The excitation was introduced by a prescribed accelera-tion time history on the nodes of the wall and the soil-stratum bases. The case of harmonic excitation wasexamined:
A
ð
t
Þ
Z
A
0
sin
ð
u
t
Þ
;
where
A
0
Z
1 m
=
s
2
(4)The duration chosen for all numerical analyses was suchthat steady state conditions were always reached.
2.1.2. Problem parameters
The dimensionless parameters of the problem examinedare three: the relative ﬂexibility factors
d
w
and
d
q
, and theratio of the cyclic frequency of the excitation to thefundamental cyclic eigenfrequency of the soil layer
u
/
u
1
.The values of parameters were the following:
d
w
Z
0 (rigid wall), 1, 5, and 40
d
q
Z
0 (ﬁxed against rotation), 0.5, 1, and 5
u
/
u
1
Z
1/6 (practically static case), 1 (resonance), and 3(high-frequency motion).
2.1.3. Quasi-static response
Initially, the response of the system under nearly staticexcitation is examined. Practically, that is achieved by aharmonic ground motion with frequency very low comparedto the fundamental eigenfrequency of the soil stratum(
u
/
u
1
Z
1/6). For this value of frequency ratio, all thepossible combinations of the parameters
d
w
and
d
q
areconsidered.The heightwise distributions of the statically inducedwall pressures
s
st
Z
s
st
(
h
) for systems with different valuesof the relative ﬂexibility factors
d
w
and
d
q
are shown inFig. 4. The values of the pressures, plotted on thehorizontal graph axis, are normalized with respect to
a
0
g
H
, where
a
0
is the maximum acceleration at the baseexpressed in g,
g
is the unit weight of the retained soil,and
H
is the height of the wall. On the vertical graph axis,the
y
-coordinate of the corresponding point along theinner side of the wall, normalized with respect to thewall’s height is plotted (
h
Z
y
/
H
). The pressures areconsidered positive when they induce compression on thewall. Note that all the results presented in this paper referto the incremental (dynamic) loads due to horizontalshaking.It is observed that, for relatively high values of
d
w
and
d
q
,negative pressures (i.e. tensile stresses) are developed near
Fig. 3. The ﬁnite-element discretization of the examined single-layer systems. The base of the model is ﬁxed, while absorbing boundaries have been placed onthe right-hand artiﬁcial boundary.
P.N. Psarropoulos et al. / Soil Dynamics and Earthquake Engineering 25 (2005) 795–809
798

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