Earth pressure in Earthquake

1. Soil Dynamics and Earthquake Engineering 27 (2007) 957–969
An alternative to the Mononobe–Okabe equations for
seismic earth pressures
George Mylonakis, Panos Kloukinas, Costas Papantonopoulos
Department of Civil Engineering, University of Patras, Rio 26500, Greece
Received 23 July 2006; received in revised form 23 January 2007; accepted 25 January 2007
Abstract
A closed-form stress plasticity solution is presented for gravitational and earthquake-induced earth pressures on retaining walls. The
proposed solution is essentially an approximate yield-line approach, based on the theory of discontinuous stress fields, and takes into
account the following parameters: (1) weight and friction angle of the soil material, (2) wall inclination, (3) backfill inclination, (4) wall
roughness, (5) surcharge at soil surface, and (6) horizontal and vertical seismic acceleration. Both active and passive conditions are
considered by means of different inclinations of the stress characteristics in the backfill. Results are presented in the form of
dimensionless graphs and charts that elucidate the salient features of the problem. Comparisons with established numerical solutions,
such as those of Chen and Sokolovskii, show satisfactory agreement (maximum error for active pressures about 10%). It is shown that
the solution does not perfectly satisfy equilibrium at certain points in the medium, and hence cannot be classified in the context of limit
analysis theorems. Nevertheless, extensive comparisons with rigorous numerical results indicate that the solution consistently
overestimates active pressures and under-predicts the passive. Accordingly, it can be viewed as an approximate lower-bound solution,
than a mere predictor of soil thrust. Compared to the Coulomb and Mononobe–Okabe equations, the proposed solution is simpler, more
accurate (especially for passive pressures) and safe, as it overestimates active pressures and underestimates the passive. Contrary to the
aforementioned solutions, the proposed solution is symmetric, as it can be expressed by a single equation—describing both active and
passive pressures—using appropriate signs for friction angle and wall roughness.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Retaining wall; Seismic earth pressure; Limit analysis; Lower bound; Stress plasticity; Mononobe–Okabe; Numerical analysis
1. Introduction
The classical equations of Coulomb [1–4,10] and
Mononobe–Okabe [5–11] are being widely used for
determining earth pressures due to gravitational and
earthquake loads, respectively. The Mononobe–Okabe
solution treats earthquake loads as pseudo-dynamic,
generated by uniform acceleration in the backfill. The
retained soil is considered a perfectly plastic material,
which fails along a planar surface, thereby exerting a limit
thrust on the wall. The theoretical limitations of such
an approach are well known and need not be repeated
herein [11–13,16–18]. Given their practical nature and
reasonable predictions of actual dynamic pressures (e.g.
Refs. [9,14,16–18]), solutions of this type are expected to
continue being used by engineers for a long time to come.
This expectation does not seem to diminish by the advent
of displacement-based design approaches, as the limit
thrusts provided by the classical methods can be used to
predict the threshold (‘‘yield’’) acceleration beyond which
permanent dynamic displacements start to accumulate
[11,15,19–21,43].
Owing to the translational and statically determined
failure mechanisms employed, the limit-equilibrium Mono-
nobe–Okabe solutions can be interpreted as kinematic
solutions of limit analysis [22]. The latter solutions are
based on kinematically admissible failure mechanisms in
conjunction with a yield criterion and a flow rule for the
soil material, both of which are enforced along pre-
specified failure surfaces [10,19,23,24,40,42]. Stresses out-
side the failure surfaces are not examined and, thereby,
ARTICLE IN PRESS
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doi:10.1016/j.soildyn.2007.01.004
Corresponding author. Tel.: +30 2610 996542; fax: +30 2610 996576.
E-mail address: mylo@upatras.gr (G. Mylonakis).

2. equilibrium in the medium is generally not satisfied. In the
realm of associative and convex materials, solutions of this
type are inherently unsafe that is, they underestimate active
pressures and overestimate the passive [10,24,25,40].
A second group of limit-analysis methods, the stress
solutions, make use of pertinent stress fields that satisfy the
equilibrium equations and the stress boundary conditions,
without violating the failure criterion anywhere in the
medium [25–27]. On the other hand, the kinematics of the
problem is not examined and, therefore, compatibility
of deformations is generally not satisfied. For convex
materials, formulations of this type are inherently safe that
is, they overestimate active pressures and underestimate the
passive [10,25,26]. The best known such solution is that of
Rankine, the applicability of which is severely restricted by
the assumptions of horizontal backfill, vertical wall and
smooth soil–wall interface. In addition, the solution may
be applied only if the surface surcharge is uniform or non-
existing. Owing to difficulties in deriving pertinent stress
fields for general geometries, the vast majority of limit-
analysis solutions in geotechnical design are of the
kinematic type [8–11,26]. To the best of the authors’
knowledge, no simple closed-form solution of the stress
type has been derived for seismic earth pressures.
Notwithstanding the theoretical significance and prac-
tical appeal of the Coulomb and Mononobe–Okabe
solutions, these formulations can be criticized on the
following important aspects: (1) in the context of limit
analysis, their predictions are unsafe; (2) their accuracy
(and safety) diminishes in the case of passive pressures
on rough walls, (3) the mathematical expressions are
complicated and difficult to verify,1
(4) the distribution
of tractions on the wall are not predicted (typically
assumed linear with depth following Rankine’s solution),
(5) optimization of the failure mechanism is required in the
presence of multiple loads, to determine a stationary
(optimum) value of soil thrust, and (6) in the context of
limit-equilibrium analysis, stress boundary conditions are
not satisfied, as the yield surface does not generally emerge
at the soil surface at the required angles of 45
f=2.
In light of the above arguments, it appears that the
development of a closed-form solution of the stress type for
assessing seismically-induced earth pressures would be
desirable. It will be shown that the proposed solution,
although approximate, is mathematically simpler than the
existing kinematic solutions, offers satisfactory accuracy
(maximum deviation for active pressures against rigorous
numerical solutions less than 10%), yields results on the
safe side, satisfies stress boundary conditions, and predicts
the point of application of soil thrust. Last but least, the
solution will be shown to be symmetric with respect to
active and passive conditions, as it can be expressed by a
single equation with opposite signs for friction angle and
wall roughness. Apart from its intrinsic theoretical interest,
the proposed analysis can be used for the assessment and
improvement of other related methods.
2. Problem definition and model development
The problem under investigation is depicted in Fig. 1: a
slope of dry cohesionless soil retained by an inclined
gravity wall, is subjected to plane strain conditions under
the combined action of gravity (gÞ and seismic body
forces ðah gÞ and ðav gÞ in the horizontal and vertical
direction, respectively. The problem parameters are: height
(H) and inclination ðoÞ of the wall, inclination (b) of the
backfill; roughness (d) of the wall–soil interface; friction
angle (f) and unit weight (g) of the soil material, and
surface surcharge (q). Since backfills typically consist of
granular materials, cohesion in the soil and cohesion at the
soil–wall interface are not studied here. In addition, since
the vibrational characteristics of the soil are neglected, the
seismic force is assumed to be uniform in the backfill. Also,
the wall can translate away from, or towards to, the
backfill, under zero rotation. Both assumptions have
important implications in the distribution of earth pres-
sures on the wall, as explained below.
The resultant body force in the soil is acting under an
angle ce from vertical
tan ce ¼
ah
1 av
, (1)
which is independent of the unit weight of the material.
Positive ah (i.e., ce40) denotes inertial action towards the
wall (ground acceleration towards the backfill), which
maximizes active thrust. Conversely, negative ah (i.e.,
ceo0) denotes inertial action towards the backfill, which
minimizes passive resistance. In accordance with the rest of
the literature, positive av is upward (downward ground
acceleration). However, its influence on earth pressures,
although included in the analysis, is not studied numeri-
cally here, as it is usually minor and often neglected in
design [9,21].
ARTICLE IN PRESS
+ ψe
H
z
cohesionless soil
(φ, γ)
+ω
+ah
γ
+av
γ
inclined
backfill
q +β
inclined wall,
roughness (δ)
γ
Fig. 1. The problem under consideration.
1
The story of a typographical error in the Mononobe–Okabe formula
that appeared in a seminal article of the early 1970’s and subsequently
propagated in a large portion of the literature, is indicative of the difficulty
in checking the mathematics of these expressions (Davies et al. [41]).
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969
958

3. In the absence of surcharge, the Mononobe–Okabe
solution to the above problem is given by the well-known
formula [11]:
where PE denotes the limit of seismic thrust on the wall
ðunits ¼ F=LÞ and KE is the corresponding earth pressure
coefficient. In the above representation (and hereafter),
the upper sign refers to active conditions (PE ¼ PAE;
KE ¼ KAE), and the lower sign to passive (PE ¼ PPE;
K ¼ KPE).
A drawback of the above equation lies in the difficulty in
interpreting the physical meaning—especially signs—of the
various terms (Ref. [25, footnote in p. 4]). As will be shown
below, the proposed solution is free of this problem.
To prevent slope failure when inertial action is pointing
towards the wall, the seismic angle ce should not exceed
the difference between the friction angle and the slope
inclination. Therefore, the following constraint applies [9]:
ceof b. (3)
A similar relation can be written for the case where inertial
action is pointing towards the backfill, but it is of limited
practical interest and will not be discussed here.
To analyze the problem, the backfill is divided into two
main regions subjected to different stress fields, as shown in
Fig. 2: the first region (zone A) is located close to the soil
surface, whereas the second (zone B) close to the wall. In
both regions the soil is assumed to be in a condition of
impeding yielding under the combined action of gravity
and earthquake body forces. The same assumption is
adopted for the soil–wall interface, which, however, is
subjected exclusively to contact stresses. A transition zone
between regions A and B is introduced later on.
Fundamental to the proposed analysis is the assumption
that stresses close to the soil surface can be well
approximated by those in an infinite slope, as shown in
Fig. 2. In this region (A), the inclined soil element shown is
subjected to canceling actions along its vertical sides. Thus
equilibrium is achieved solely under body forces and
contact stress acting at its bottom face.
Based on this physically motivated hypothesis, the
stresses sb and tb at the base of the inclined element are
determined from the following expressions [34]:
sb ¼ gz þ
q
cos b
cos2
b, (4a)
tb ¼ gz þ
q
cos b
sin b cos b, (4b)
which are valid for static conditions (ah ¼ av ¼ 0) and
satisfy the stress boundary conditions at the surface.
Eqs. (4) suggest that the ratio of shear to normal stresses
is constant ðtan bÞ at all depths, and that points at the same
depth are subjected to equal stresses. Note that due to
static determinacy and anti-symmetry, the above relations
are independent of material properties and asymptotically
exact at large distances from the wall.
Considering the material to be in a condition of
impeding yielding, the Mohr circle of stresses in region
A is depicted in Fig. 3. The different locations of the stress
point (sb; tb) for active and passive conditions and the
different inclinations of the major principal plane (indi-
cated by heavy lines) are apparent in the graph.
From the geometry of Fig. 3, the normal stress sb is
related to mean stress SA through the proportionality
ARTICLE IN PRESS
soil
surface
z ZONE A
ZONE B
τβ
σβ
γ
unit length
q β
H
z
active
wall length
L = H / cosω
δ
δ
(σw, τw)
(σw, τw)
ω
passive
Fig. 2. Stress fields close to soil surface (zone A) and the wall (zone B).
KE ¼
2PE
gH2
¼
cos2
½f ðce þ oÞ
cos ce cos2o cos½d ðce þ oÞ 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sinðd þ fÞ sin½f ðce þ bÞ
cos½d ðce þ oÞ cosðb oÞ
s
#2
, (2)
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969 959

4. relation
sb ¼ SA½1 sin f cosðD1 bÞ, (5)
where D1 denotes the Caquot angle [23,28] given by
sin D1 ¼
sin b
sin f
. (6)
For points in region B, it is assumed that stresses are
functions exclusively of the vertical coordinate and obey
the strength criterion of the frictional soil–wall interface, as
shown in Fig. 2. Accordingly, at orientations inclined at an
angle o from vertical,
tw ¼ sw tan d, (7)
where sw and tw are the normal and shear tractions
on the wall, at depth z. The above equation is asympto-
tically exact for points in the vicinity of the wall. The
corresponding Mohr circle of stresses is depicted in Fig. 3.
The different signs of shear tractions for active and passive
conditions follow the directions shown in Fig. 2 (passive
wall tractions pointing upward, active tractions pointing
downward), which comply with the kinematics of the
problem. This is in contrast with the widespread view that
solutions based on equilibrium totally ignore the displace-
ment field [29].
From the geometry of Fig. 3, normal traction sw is
related to mean stress SB through the expression
sw ¼ SB½1 sin f cosðD2 dÞ, (8)
where D2 is the corresponding Caquot angle given by
sin D2 ¼
sin d
sin f
. (9)
In light of the foregoing, it becomes evident that the
orientation of stress characteristics in the two regions is
different and varies for active and passive conditions. In
addition, the mean stresses SA and SB generally do not
coincide, which suggests that a Rankine-type solution
based on a single stress field is not possible.
To determine the separation of mean stresses SA and SB
and ensure a smooth transition in the orientation of
principal planes in the two zones, a logarithmic stress fan2
is adopted in this study, centered at the top of the wall. In
the interior of the fan, principal stresses are gradually
rotated by the angle y separating the major principal planes
in the two regions, as shown in Fig. 4. This additional
condition is written as [10]
SB ¼ SA expð2y tan fÞ. (10)
The negative sign in the above equation pertains to the case
where SBoSA (e.g., active case) and vice versa. The above
equation is an exact solution of the governing Kötter
equations for a weightless frictional material. For a
material with weight, the solution is only approximate as
Kotter’s equations are not perfectly satisfied [25–27]. In
other words, the log spiral fan accurately transmits stresses
applied at its boundaries, but handles only approximately
body forces imposed within its volume. The error is
expected to be small for active conditions (which are of
key importance in design), because of the small opening
angle of the fan, and bigger for passive conditions. As a
result, the above solution cannot be interpreted in the
context of limit analysis theorems. Nevertheless, it will be
shown that these violations are of minor importance from a
practical viewpoint.
2.1. Solution without earthquake loading
The total thrust on the wall due to surcharge and gravity
loading is obtained by the well-known expression [10]
P ¼ KqqH þ 1
2
KggH2
, (11)
which is reminiscent (though not equivalent) of the bearing
capacity equation of a strip surface footing on cohesionless
soil. In the above equation, Kq and Kg denote the earth
pressure coefficients due to surcharge and self-weight,
respectively.
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passive
Δ1
φ
β
Δ1−β SA
Δ1+β
Δ1
σ1A
active
soil surface
active
case
passive
case
Δ2+δ
φ
SB
Δ2−δ
δ
δ
Δ2
σ1B
passive
active
wall
plane
active
wall
plane
passive
(σβ,τβ)
(σw, τw)
(σw,τw)
(σβ,τβ)
ZONE A
ZONE B
Fig. 3. Mohr circles of effective stresses and inclination of the major
principal planes in zones A and B.
2
This should not be confused with log-spiral shaped failure surfaces
used in kinematic solutions of related problems.
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969
960

5. Combining Eqs. (5), (8) and (10), and integrating over
the height of the wall, it is straightforward to show that the
earth pressure coefficient Kg is given by [39]
Kg ¼
cosðo bÞ cos b
cos d cos2 o
1 sin f cosðD2 dÞ
1 sin f cosðD1 bÞ
expð2y tan fÞ, ð12Þ
where
2y ¼ D2 ðD1 þ dÞ þ b 2o (13)
is twice the angle separating the major principal planes in
zones A and B (Fig. 4). The convention regarding double
signs in the above equations is as before.
It is also straightforward to show that the surcharge
coefficient Kq is related to Kg through the simple expression
Kq ¼ Kg
cos o
cosðo bÞ
, (14)
which coincides with the kinematic solution of Chen and
Liu [31], established using a Coulomb mechanism. Note
that for a horizontal backfill (b ¼ 0), coefficients Kq and Kg
coincide regardless of wall inclination and material proper-
ties. Eq. (14) represents an exact solution for a weightless
material with surcharge. A simplified version of the above
solutions, restricted to the special case of a vertical wall
with horizontal backfill and no surcharge (o ¼ b ¼ 0;
q ¼ 0), has been derived by Lancelotta [30]. Another
simplified solution, which, however, contains some alge-
braic mistakes (see application example in the Appendix)
and is restricted to active conditions and no surcharge, has
been presented by Powrie [35].
2.2. Solution including earthquake loading
Recognizing that earthquake action imposes a resultant
thrust in the backfill inclined by a constant angle ce from
vertical (Fig. 1), it becomes apparent that the pseudo-
dynamic problem does not differ fundamentally from the
corresponding static problem, as the former can be obtained
from the latter through a rotation of the reference axes by
the seismic angle ce, as shown in Fig. 5. In other words,
considering ce does not add an extra physical parameter to
the problem, but simply alters the values of the other
variables. This property of similarity was apparently first
employed by Briske [32] and later by Arango [8,9] in the
analysis of related problems. Application of the concept to
the present analysis yields the following algebraic transfor-
mations, according to the notation of Fig. 5:
b
¼ b þ ce, (15)
o
¼ o þ ce, (16)
ARTICLE IN PRESS
H
ψe
ω
ψe
ψe
H*
ω*
β
β*
Fig. 5. Similarity transformation for analyzing the pseudo-dynamic
seismic problem as a gravitational problem. Note the modified wall
height ðH
Þ, backfill slope ðb
Þ, and wall inclination ðo
Þ in the
transformed geometry. Also note that the rotation should be performed
in the opposite sense (i.e., clockwise) for passive pressures ðceo0Þ.
ω
z
zone B
2 2
π Δ2−δ
−
zone A
θAB
Δ1 + β
β 2
ACTIVE CONDITIONS
ω
θAB
z
zone A
PASSIVE CONDITIONS
Δ2 + δ
2
2 2
π −
β
zone B
Δ1−β
Fig. 4. Rotation of major principal planes between zones A and B for
active and passive conditions.
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969 961

6. H
¼ H cosðo þ ceÞ= cos o, (17)
g
¼ gð1 avÞ= cos ce, (18)
q
¼ qð1 avÞ= cos ce. (19)
The modification in g and q is due to the change in length of
the corresponding vectors (Fig. 1) as a result of inertial
action. To obtain Eq. (19), it has been tacitly assumed that
the surcharge responds to the earthquake motion in the
same manner as the backfill and, thereby, the transformed
surcharge remains vertical. Note that this is not an essential
hypothesis—just a convenient (reasonable) assumption from
an analysis viewpoint. Understandably, the strength para-
meters f and d are invariant to the transformation.
In the light of the above developments, the soil thrust
including earthquake action can be determined from the
modified expression:
PE ¼ K
qq
H
þ 1
2
K
g g
H2
, (20a)
in which parameters b, o, H, g, and q have been replaced
by their transformed counterparts. The symbols K
q and K
g
denote the surcharge and self-weight coefficients in the
modified geometry, respectively.
Substituting Eqs. (15) through (19) in Eq. (20a) yields the
modified earth pressure expressions
PE ¼ ð1 avÞ½KEqqH þ 1=2 KEggH2
, (20b)
where
KEg ¼
cosðo bÞ cosðb þ ceÞ
cos ce cos d cos2 o
1 sin f cosðD2 dÞ
1 sin f cos½D
1 ðb þ ceÞ
expð2yE tan fÞ,
ð21Þ
which encompasses seismic action and can be used in the
context of Eq. (11). In the above equation,
2yE ¼ D2 ðD
1 þ dÞ þ b 2o ce (22)
is twice the revolution angle of principal stresses in
the two regions under seismic conditions; D
1 equals
Arcsin½sinðb þ ceÞ= sin f, following Eqs. (6) and (15).
The seismic earth pressure coefficient KEq is obtained as
KEq ¼ KEg
cos o
cosðo bÞ
, (23)
which coincides with the static solution in Eq. (14).
The horizontal component of soil thrust is determined
from the actual geometry, as in the gravitational
problem
PEH ¼ PE cosðo dÞ. (24)
2.3. Seismic component of soil thrust
Following Seed and Whitman [8], the seismic component
of soil thrust is defined from the difference:
DPE ¼ PE P, (25)
which is mathematically valid, as the associated vectors PE
and P are coaxial. Nevertheless, the physical meaning of
DPE is limited given that the stress fields (and the
corresponding failure mechanisms) in the gravitational
and seismic problems are different. In addition, DPE
cannot be interpreted in the context of limit analysis
theorems, as the difference of PE and P is neither an upper
nor a lower bound to the true value.
ARTICLE IN PRESS
Table 1
Comparison of results for active and passive earth pressures predicted by various methods
o 0
20
20
f 20
30
40
30
30
d 0
10
0
15
0
20
0
15
0
15
(a) KAg—valuesa
Coulomb 0.490 0.447 0.333 0.301 0.217 0.199 0.498 0.476 0.212 0.180
Kinematic limit analysis [31] 0.490 0.448 0.333 0.303 0.217 0.200 0.498 0.476 0.218 0.189
Zero extension [33] 0.49 0.41 0.33 0.27 0.22 0.17 — — — —
Slip line [28] 0.490 0.450 0.330 0.300 0.220 0.200 0.521 0.487 0.229 0.206
Proposed stress limit analysis 0.490 0.451 0.333 0.305 0.217 0.201 0.531 0.485 0.237 0.217
(b) KPg—valuesb
Coulomb 2.04 2.64 3.00 4.98 4.60 11.77 2.27 3.162 5.34 12.91
Kinematic limit analysis [31] 2.04 2.58 3.00 4.70 4.60 10.07 2.27 3.160 5.09 8.92
Zero extension [33] 2.04 2.55 3.00 4.65 4.60 9.95 — — — —
Slip line [28] 2.04 2.55 3.00 4.62 4.60 9.69 2.16 3.16 5.06 8.45
Proposed stress limit analysis 2.04 2.52 3.00 4.44 4.60 8.92 2.13 3.157 4.78 7.07
The results for d ¼ o ¼ 0 are identical for all methods. Note the decrease in KPg values as we move from top to bottom in each column, and the
corresponding increase in KAg; b ¼ 0
(modified from Chen and Liu [31]).
a
KAg ¼ PA=1
2
gH2
.
b
KPg ¼ PP=1
2
gH2
.
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969
962

7. 3. Model verification and results
Presented in Table 1 are numerical results for gravita-
tional active and passive pressures ðKAg; KPgÞ from the
present solution and established solutions from the
literature. The predictions are in good agreement (largest
discrepancy about 10%), with the exception of Coulomb’s
method which significantly overestimates passive pressures.
Moving from the top to the bottom of each column, an
increase in KAg values and a decrease in KPg values can be
observed. This is easily understood given the non-
conservative nature of the first two solutions (Coulomb,
Chen), and the conservative nature of the last two
(Sokolovskii [28], proposed). This observation does not
hold for the ‘‘zero extension line’’ solution of Habibagahi
and Ghahramani [33], which cannot be classified in the
context of limit analysis theorems.
Results for gravitational active pressures on a rough
inclined wall obtained according to three different methods
as a function of the slope angle b, are shown in Fig. 6. The
performance of the proposed solution is good (maximum
deviation from Chen’s solution about 10%—despite the
high friction angle of 45
) and elucidates the accuracy of
the predictions. The performance of the simplified solution
of Caquot and Kerisel [23] versus that of Chen and Liu [31]
is as expected.
Corresponding predictions for passive pressures are
given in Fig. 7, for a wall with negative backfill slope
inclination, as a function of the wall roughness d. The
agreement of the various solutions, given the sensitivity of
passive pressure analyses, is very satisfactory. Of particular
interest are the predictions of Sokolovskii’s [28] and
Lee and Herington’s [36] methods, which, surprisingly,
exceed those of Chen for rough walls. This trend is
particularly pronounced for horizontal backfill and values
of d above approximately 10
and has been discussed by
Chen and Liu [31].
Results for active seismic earth pressures are given in
Fig. 8, referring to cases examined in the seminal study of
Seed and Whitman [8], for a reference friction angle of 35
.
Naturally, active pressures increase with increasing levels
of seismic acceleration and slope inclination and decrease
with increasing friction angle and wall roughness. The
conservative nature of the proposed analysis versus the
Mononobe–Okabe (M–O) solution is evident in the graphs.
The trend is more pronounced for high levels of horizontal
seismic coefficient ðah40:25Þ, smooth walls, level backfills,
and high friction angles. Conversely, the trend becomes
weaker with steep backfills, rough walls, and low friction
angles.
A similar set of results is shown in Fig. 9, for a reference
friction angle of 40
. The following interesting observations
can be made: First: the predictions of the proposed analysis
are in good agreement with the results from the kinematic
analysis of Chen and Liu [31], over a wide range of material
ARTICLE IN PRESS
H
ω
β
PA
Slope Angle of Backfill, °
0 5 10 15 20 25
Coefficient
of
Active
Earth
Pressure,
K
Aγ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Chen Liu(1990)
Caquot Kerisel (1948)
Proposed Stress Limit Analysis
ω = 0°
ω = −20°
ω = 20°
1
KA=PA/ (
2
H2
)
= 45°, = 2/ 3
δ
Fig. 6. Comparison of results for active earth pressures predicted by
different methods (modified from Chen [10]).
Angle of Wall Friction, °
0 10 20 30
Coefficient
of
Passive
Earth
Pressure,
K
Pγ
0
1
2
3
4
5
Lee Herington (1972)
Chen Liu (1990)
Sokolovskii (1965)
Proposed Stress Limit Analysis
PP H
δ
ω
KP=PP
1
/( H2
)
2
= 0°
= −10°
=−20°
β
= 30°, = 20°
Fig. 7. Comparison of results for passive earth pressures by predicted by
different methods (modified from Chen and Liu [31]).
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969 963

8. and geometric parameters. Second, the present analysis is
conservative in all cases. Third, close to the slope stability
limit (Fig. 9d), or for high accelerations and large wall
inclinations (Fig. 9c), Chen’s predictions are less accurate
than those of the elementary M–O solution. In the same
extreme conditions, the proposed solution becomes ex-
ceedingly conservative, exceeding M–O predictions by
about 35%. Note that whereas the M–O and the proposed
solution break down in the slope stability limit, Chen’s
solution allows for spurious mathematical predictions of
active thrust beyond the limit, as evident in Fig. 9d. Fourth,
with the exception of the aforementioned extreme cases,
Chen’s and M–O predictions remain close over the whole
range of parameters examined. The improvement in the
predictions of the former over the latter is marginal.
Results for seismic passive pressures (resistances) are
shown in Fig. 10 for the common case of a rough vertical
wall with horizontal backfill. Comparisons of the proposed
solution with results from the M–O and Chen’s kinematic
methods are provided on the left graph (Fig. 10a). The
predictions of the stress solutions are, understandably,
lower than those of Chen and Liu, whereas M–O
predictions are very high (i.e., unconservative)—especially
for friction angles above 37
. Given the sensitivity of
passive pressure analyses, the performance of the proposed
method is deemed satisfactory.
An interesting comparison is presented in Fig. 10b:
average predictions from the two closed-form solutions
(M–O solution and proposed stress solution) are plotted
against the rigorous numerical results of Chen and
Liu [31]. Evidently, in the range of most practical interest
ð30
ofo40
Þ, the discrepancies in the results have been
drastically reduced. This suggests that the limit equilibrium
(kinematic) M–O solution and the proposed static solution
overestimate and underestimate, respectively, passive
resistances by the same amount in the specific range of
ARTICLE IN PRESS
Horizontal Seismic Coefficient, ah
0.0 0.1 0.2 0.3 0.4 0.5
Coefficient
of
Seismic
Active
Earth
Pressure,
K
ΑEγ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
M - O Analysis
Proposed Stress Limit Analysis
0.0 0.1 0.2 0.3 0.4 0.5
K
ΑEγ
cos
δ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
M - O Analysis
Proposed Stress Limit Analysis
Horizontal Seismic Coefficient, ah
0.0 0.1 0.2 0.3 0.4 0.5
K
ΑEγ
cos
δ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Horizontal Seismic Coefficient, ah
35°
40°
0.0 0.1 0.2 0.3 0.4 0.5
K
ΑEγ
cos
δ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
M- O Analysis
Proposed Stress Limit Analysis
Horizontal Seismic Coefficient, ah
M- O Analysis
Proposed Stress Limit Analysis
H
PAE
γah
1
KAEγ=PAE/( H2
)
2
H
PAE
δ γah
1
KAEγ=PAE/ ( H2
)
2
H
PAE
δ γah
1
KAEγ=PAE/( H2
)
2
H
PAE
γah
1
KAEγ=PAE/ ( H2
)
2
= 0°
= 20°
= 35° ; = / 2
= 30°
= = 0°
=/ 2
= = 0°
=/ 2
= 0°
= 35°
= = 0° = = 0°
=/ 2
= 0°
= 35°
γ
γ γ
β
γ
δ
Fig. 8. Comparison of active seismic earth pressures predicted by the proposed solution and from conventional M–O analysis, for different geometries,
material properties and acceleration levels; av ¼ 0 (modified from Seed and Whitman [8]).
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969
964

9. properties. Accordingly, this averaging might be warranted
for design applications involving passive pressures.
Results for the earth pressure coefficient due to
surcharge KqE (Eq. (23)) are presented in Fig. 11, for both
active and passive conditions involving seismic action. The
agreement between the stress solution and the numerical
results of Chen and Liu [31] is excellent in the whole range
of parameters examined (except perhaps for active
pressures, where ah ¼ 0:3). As expected, M–O solution
performs well for active pressures, but severely over-
estimates the passive.
3.1. Distribution of earth pressures on the wall: analytical
findings
Mention has already been made that in the realm
of pseudo-dynamic analysis, there is no fundamental
physical difference between gravitational and seismic earth
pressures. Eqs. (4) indicate that stresses in the soil vary
linearly with depth (stress fan does not alter this
dependence), which implies that both gravitational and
seismic earth pressures vary linearly along the back of wall.
In the absence of surcharge, the distribution becomes
ARTICLE IN PRESS
M - O Analysis
Kinematic Limit Analysis (Chen Liu 1990)
Proposed Stress Limit Analysis
M - O Analysis
Kinematic Limit Analysis (Chen Liu 1990)
Proposed Stress Limit Analysis
M - O Analysis
Kinematic Limit Analysis (Chen Liu 1990)
Proposed Stress Limit Analysis
M - O Analysis
Kinematic Limit Analysis(Chen Liu 1990)
Proposed Stress Limit Analysis
H δ
ω
γ
γah
PAE
H γ
δ
H γ
δ
0.0 0.1 0.2 0.3 0.4
K
AEγ
cosδ
0.1
0.2
0.3
0.4
0.5
0.6
Horizontal Seismic Coefficient, ah
0.0 0.1 0.2 0.3 0.4
Coefficient
of
Seismic
Active
Earth
Pressure,
K
AEγ
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
φ / 3
0°
slope
stability
limit
-20 -10 0 10 20
Coefficient
of
Seismic
Active
Earth
Pressure,
K
AEγ
0.0
0.2
0.4
0.6
0.8
1.0
15°
ω = 0°
15°
25 30 35 40 45
Coefficient
of
Seismic
Active
Earth
Pressure,
K
AEγ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.10
ah = 0
0.20
0.30
H δ
β
γ
PAE
PAE
PAE
γah
γah
γah
1
KAEγ= PAE/ ( γH2
)
2
γH2
)
1
KAEγ=PAE/ (
2
γH2
)
1
KAEγ=PAE/ (
2
γH2
)
1
KAEγ=PAE/ (
2
Friction Angle, °
Slope Angle of Backfill, ° Horizontal Seismic Coefficient, ah
= 40°;ah = 0.20 ; = / 2 = 40°; = 0° ; = / 2
=0
δ=/ 2
=
= = 0° ; = 40°
= = 0° ; = 2/3
β = φ / 2
Fig. 9. Comparison of active seismic earth pressures predicted by different methods, for different geometries, material properties, and acceleration levels;
f ¼ 40
, av ¼ 0 (modified from Chen and Liu [31]).
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969 965

10. proportional with depth, as in the Rankine solution.
Accordingly, the point of application of seismic thrust is
located at a height of H=3 above the base of the wall. It is
well known from experimental observations and rigorous
numerical solutions, that this is not generally true. The
source of the difference lies in the distribution of inertial
forces in the soil mass (which is often sinusoidal like—
following the time-varying natural mode shapes of the
deposit), as well as the various kinematic boundary
conditions (wall flexibility, foundation compliance, pre-
sence of supports). Studying the above factors lies beyond
the scope of this article, and like will be the subject of a
future publication. Some recent developments are provided
in the Master thesis of the second author [39] as well as in
Refs. [11,16–18,37,38].
4. Discussion: simplicity and symmetry
It is instructive to show that the proposed solution can
be derived essentially by inspection, without tedious
algebraic manipulations as in the classical equations.
Indeed, basis of Eq. (12) is the familiar Rankine ratio
ð1 sin fÞ=ð1 sin fÞ. The terms cosðD2 dÞ and cosðD1
bÞ in the numerator and denominator of the expression
reflect the fact that stresses sb and sw are not principal.
Both terms involve the same double signs as their multi-
pliers ( sin f and sin f, respectively). Angle b and
associated angle D1 have to be in the denominator, as an
increase in their value must lead to an increase in active
thrust. The exponential term is easy to remember and
involves the same double signðÞ as the other terms in the
ARTICLE IN PRESS
H
γ
PPE
γah γah
δ
25 30 35 40 45
Coefficient
of
Seismic
Passive
Earth
Pressure,
K
PEγ
0
5
10
15
20
25
Kinematic Limit Analysis (Chen Liu1990)
Proposed Stress Limit Analysis
Kinematic Limit Analysis (Chen Liu1990)
Average of M-O Proposed Stress Limit Analysis
ah = 0
Mononobe -Okabe
(ah=0)
25 30 35 40 45
0
5
10
15
20
25
-0.1
-0.2
-0.3
ah = 0
-0.1
-0.2
-0.3
H
γ
PPE δ
1
KPE=PPE/ ( H2
)
2
1
KPE=PPE/ ( H2
)
2
Angle of Internal Friction, ° Angle of Internal Friction, °
= 2 / 3
= 0°, = 0°
a b
Fig. 10. Comparison of results for passive seismic resistance on a rough wall predicted by various methods (modified from Chen and Liu [31]).
H
P
δ
a q
P
H δ
q
a q
Kinematic Limit Analysis (Chen Liu 1990)
Proposed Stress Limit Analysis
Friction Angle, φ o
25 30 35 40 45
0
5
10
15
20
K
PE
q
=
P
PE
/
q
H
Mononobe - Okabe
(ah = 0)
0.1
ah = 0
0.2
0.3
Friction Angle, φ o
25 30 35 40 45
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Kinematic Limit Analysis (Chen Liu 1990)
Proposed Stress Limit Analysis
K
AE
q
=
P
AE
/
q
H
β ω =
= 0o
δ = 2 / 3 φ
0.1
ah = 0
0.2
0.3
ω =
β = 0o
δ = 2 / 3 φ
Fig. 11. Variation of KAEq and KPEq values with f—angle for different acceleration levels.
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969
966

11. numerator. With reference to the factors outside the
brackets, 1= cos dð¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ tan2
d
p
Þ stands for the vectorial
sum of shear and normal tractions at the wall–soil
interface. Factor cos b arises from the equilibrium of the
infinite slope in Eq. (4a). Finally, cosðo bÞ=cos2
o is a
geometric factor arising from the integration of stresses
along the back of the wall, and is associated with the
inclination of the wall and backfill.
In light of the above, the solution for gravitational
pressures can be expressed by the single equation
Kg ¼
cosðo bÞ cos b
cos d cos2o
1 sin f cosðD2 dÞ
1 þ sin f cos½D1 þ b
expð2y tan fÞ, ð26Þ
which is valid for both active conditions (using positive
values for f and dÞ and passive conditions (using negative
values for f and dÞ. It is straightforward to show that this
property is not valid for the Mononobe–Okabe solutions in
Eq. (4). The lack of symmetry in the limit equilibrium
solutions can be attributed to the maximization and
minimization operations involved in deriving the limit
thrusts. An application example elucidating the simplicity
of the solution is provided below.
5. Conclusions
A stress plasticity solution was presented for determining
gravitational and earthquake-induced earth pressures on
gravity walls retaining cohesionless soil. The proposed
solution incorporates idealized, yet realistic wall geometries
and material properties. The following are the main
conclusions of the study:
(1) The proposed solution is simpler than the classical
Coulomb and Mononobe–Okabe equations. The main
features of the mathematical expressions, including
signs, can be deduced by physical reasoning, which is
hardly the case with the classical equations. Also, the
proposed solution is symmetric with respect to active
and passive conditions, as it can be expressed by a
single equation with opposite signs for soil friction
angle and wall roughness.
(2) Extensive comparisons with established numerical
solutions indicate that the proposed solution is safe,
as it overestimates active pressures and under-predicts
the passive. This makes the method appealing for use in
practical applications.
(3) For active pressures, the accuracy of the solution is
excellent (maximum observed deviation from numerical
data is about 10%). The largest deviations occur for
high seismic accelerations, high friction angles, steep
backfills, and negative wall inclinations.
(4) For passive resistances, the predictions are also
satisfactory. However, the error is larger—especially
at high friction angles. Nevertheless, the improvement
over the M–O predictions is dramatic. Taking the
average between the predictions of the M–O solution
and the proposed stress solution (both available in
closed forms) yields results which are comparable to
those obtained from rigorous numerical solutions.
(5) The pseudo-dynamic seismic problem can be deduced
from the corresponding static problem through a
revolution of the reference axes by the seismic angle
ce (Fig. 5). This similarity suggests that the Coulomb
and M–O solutions are essentially equivalent.
(6) Contrary to the overall gravitational-seismic thrust PE,
the purely seismic component DPE ¼ PE P cannot be
put in the context of a lower or an upper bound. This
holds even when PE and P are rigorous upper or lower
bounds.
(7) In the realm of the proposed model, the distribution of
earth pressures on the back of the wall is linear with
depth for both gravitational and seismic conditions.
This is not coincidental given the similarity between the
gravitational and pseudo-dynamic problem.
It should be emphasized that the verification of the
proposed solution was restricted to analytical—not experi-
mental results. Detailed comparisons against experimental
results, including distribution of earth pressures along the
wall, will be the subject of a future publication.
Acknowledgments
The authors are indebted to Professor Dimitrios
Atmatzidis for his constructive criticism of the work.
Thanks are also due to two anonymous reviewers whose
comments significantly improved the original manuscript.
Appendix A. Application example
Active and passive earth pressures will be computed for a
gravity wall of height H ¼ 5 m, inclination o ¼ 5
and
roughness d ¼ 20
, retaining an inclined cohesionless
material with f ¼ 30
, g ¼ 18 kN=m3
and b ¼ 15
, sub-
jected to earthquake accelerations ah ¼ 0:2 and av ¼ 0. The
static counterpart of the problem has been discussed by
Powrie [35].
The inclination of the resultant body force in the backfill
is obtained from Eq. (1):
ce ¼ arctanð0:2Þ ¼ 11:3
. (A.1)
The two Caquot angles are determined from Eqs. (6), (9)
and (15) as
D
1 ¼ sin1
½sinð15 þ 11:3Þ= sin 30 ¼ 62:4
, (A.2)
D2 ¼ sin1
½sinð20Þ= sin 30 ¼ 43:2
. (A.3)
The angle separating the major principal planes in regions
A and B is computed from Eq. (21):
2yE ¼ 43:2 ð62:4 þ 20Þ þ 15 2 5 11:3 ¼ 45:5
.
(A.4)
ARTICLE IN PRESS
G. Mylonakis et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 957–969 967

12. Based on the above values, the earth pressure coefficient is
obtained from Eq. (21):
KAEg ¼
cosð5 15Þ cosð15 þ 11:3Þ
cos 11:3 cos 20 cos2 5
1 sin 30 cosð43:2 20Þ
1 þ sin 30 cos½62:4 þ ð15 þ 11:3Þ
exp þ45:5
p
180
tan 30
¼ 0:82 ðA:5Þ
from which the overall active thrust on the wall is easily
determined (Eq. (11)):
PAE ¼ 1
2
0:82 18 52
¼ 185 kN=m. (A.6)
Both M–O and Chen–Liu solutions yield KAEg ¼ 0:77,
which elucidates the more conservative nature of the
proposed approach.
For the gravitational problem, the corresponding
parameters are D1 ¼ sin1
½sin 15= sin 30 ¼ 31:2
, D2 ¼
sin1
½sinð20Þ= sin 30 ¼ 43:2
,
2y ¼ 43:2 ð31:2 þ 20Þ þ 15 2 5 ¼ 3
, KAg ¼ 0:42.
Thus,
PA ¼ 1
2
0:42 18 52
¼ 94:5 kN=m. (A.7)
The horizontal component of gravitational soil thrust is
determined from Eq. (24)
PAH ¼ 94:5 cosð5 þ 20Þ ¼ 85:6 kN=m. (A.8)
Note that according to Powrie [35], the horizontal
component is (Eq. 9.42, p. 333)
PAH ¼ 1
2
0:395 18 52
ð1 þ tan 5 tan 20Þ ¼ 91:7 kN=m,
(A.9)
which is clearly in error as: (1) Ka, as determined from
Powrie’s equations, should be 0.385—not 0.395; (2) the
sign in front of product ðtan b tan dÞ should be minus
one. (3) Powrie’s equation does not encompass factor
½cosðo bÞ= cos o cos b arising from the integration of
stresses on the back of the wall.
For the passive case, the corresponding parameters are:
ce ¼ Arctanð0:2Þ ¼ 11:3
,
D
1 ¼ sin1
½sinð15 11:3Þ= sin 30 ¼ 7:4
,
2yE ¼ 43:2 þ ð7:4 þ 20Þ þ 15 2 5 þ 11:3 ¼ 86:9
.
The passive earth pressure coefficient and resistance are
obtained from Eqs. (21) and (11):
KPEg ¼
cosð5 15Þ cosð15 11:3Þ
cos 11:3 cos 20 cos2 5
1 þ sin 30 cosð43:2 þ 20Þ
1 sin 30 cos½7:41 ð15 11:3Þ
exp 2yE
p
180
tan 30
¼ 6:31, ðA:10Þ
PPE ¼ 1
2
6:31 18 52
¼ 1420 kN=m. (A.11)
The M–O and Chen–Liu solutions predict KPEg ¼ 10:25
and 8.01, respectively. Note that the average of the two
closed-form solutions, ð10:25 þ 6:31Þ=2 ¼ 8:28, is very
close to the more rigorous result by Chen and Liu.
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