ARTICLE IN PRESS

Ocean Engineering 34 (2007) 1947–1954

www.elsevier.com/locate/oceaneng

Stability of waterfront retaining wall subjected to pseudo-static

earthquake forces
Deepankar Choudhury, Syed Mohd. Ahmad
Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
Received 30 October 2006; accepted 26 March 2007
Available online 12 April 2007

Abstract

Waterfront retaining walls supporting dry backfill are subjected to hydrostatic pressure on upstream face and earth pressure on the
downstream face. Under seismic conditions, if such a wall retains a submerged backfill, additional hydrodynamic pressures are
generated. This paper pertains to a study in which the effect of earthquakes along with the hydrodynamic pressure including inertial
forces on such a retaining wall is observed. The hydrodynamic pressure is calculated using Westergaard’s approach, while the earth
pressure is calculated using Mononobe-Okabe’s pseudo-static analysis. It is observed that when the horizontal seismic acceleration
coefficient is increased from 0 to 0.2, there is a 57% decrease in the factor of safety of the retaining wall in sliding mode. For investigating
the effect of different parameters, a parametric study is also done. It is observed that if f is increased from 301 to 351, there is an increase
in the factor of safety in the sliding mode by 20.4%. Similar observations were made for other parameters as well. Comparison of results
obtained from the present approach with [Ebeling, R.M., Morrison Jr, E.E., 1992. The seismic design of waterfront retaining structures.
US Army Technical Report ITL-92-11. Washington DC] reveal that the factor of safety for static condition (kh ¼ 0), calculated by both
the approaches, is 1.60 while for an earthquake with kh ¼ 0.2, they differ by 22.5% due to the consideration of wall inertia in the present
study.
r 2007 Elsevier Ltd. All rights reserved.

Keywords: Hydrodynamic pressure; Seismic active earth pressure; Design; Wall inertia; Sliding; Overturning; Factor of safety; Soil and wall friction angle

1. Introduction submerged backfill and is subjected to an earthquake,

because additional hydrodynamic pressure gets generated
Waterfront retaining structures are extensively used along with the seismic lateral earth pressure on the
across the world and the design of these retaining downstream side of the wall. Several researchers in the
structures is an important topic of research for civil recent past had given solutions for the computation of
engineers. Again, the devastating effects of the earthquakes the seismic lateral earth pressure acting on a rigid retaining
make the problem more complicated compared to the static wall. The pioneering work by Okabe (1924) and Mononobe
design procedure for the waterfront retaining wall. Hence, and Matsuo (1929), which is commonly known as
the stability of the waterfront retaining wall under the Mononobe–Okabe method (see Kramer, 1996) by con-
earthquake conditions must be studied carefully. Under the sidering the pseudo-static seismic accelerations, is still
static condition, for a typical waterfront retaining wall, being used worldwide, to compute the seismic lateral earth
supporting a dry backfill, the only disturbing force for the pressure. The work done by Ebeling and Morrison (1992)
stability of the wall is the lateral earth pressure from the considered both the seismic active earth pressure and
downstream side, while on the upstream side, the disturb- hydrodynamic pressure for the design of the waterfront
ing force is the hydrostatic pressure. However, the situation retaining walls. Again, the hydrodynamic pressure, which
changes when such a waterfront retaining wall retains a tends to destabilize the wall, was described by Kim et al.
(2005). A study of such a waterfront retaining wall and its
Corresponding author. Tel.: +91 22 2576 7335; fax: +91 22 2576 7302. behaviour under the action of the above-mentioned forces
E-mail address: dc@civil.iitb.ac.in (D. Choudhury). needs to be carried out to assess its stability.

0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2007.03.005
 ARTICLE IN PRESS
1948 D. Choudhury, S.M. Ahmad / Ocean Engineering 34 (2007) 1947–1954

Nomenclature Pae seismic active earth thrust

Pdyn hydrodynamic pressure
b, H width and height of the wall Pstat, P0 stat hydrostatic pressure on upstream and
h height of the water on upstream side downstream side
F dr ; F df driving force for restrained and free water Pw pressure due to water
conditions ru pore pressure ratio
Fr resisting force W weight of the wall
FSoverturningr factor of safety for overturning mode of y point of application of Pae
failure for restrained water condition b ground inclination with respect to hori-
FSoverturningf factor of safety for overturning mode of zontal
failure for free water condition d wall friction angle
FSslidingr factor of safety for sliding mode of failure gw, gs, gc specific weight of water, soil, and concrete
for restrained water condition ḡ , gwe equivalent specific weight of the soil and
FSslidingf factor of safety for sliding mode of failure water due to submergence
for free water condition gsat, gd saturated and dry specific weight of the soil
k hydraulic conductivity of the soil f soil friction angle
kh, kv horizontal and vertical seismic acceleration m coefficient of base friction
coefficient y wall inclination with respect to vertical
Kae seismic active earth pressure coefficient c seismic inertia angle

However, a very few literature proposed the analysis Steps for the analysis of the rigid retaining wall by
of waterfront retaining wall under the combined action of considering the hydrodynamic pressure generated due to
forces due to water and seismic earth pressure, as most of the submerged backfill along with the seismic active earth
the literature deals with the individual forces acting on the pressures were given only by Ebeling and Morrison (1992).
waterfront retaining wall. For example, the effect of wave However, one of the important aspect of considering the
action on caisson, vertical and sloping walls and other wall inertia, the effect of which on the stability of a
coastal structures had been studied by Kirkgoz and Mengi retaining wall has already been well established, as is
(1987) and by Kirkgoz (1990, 1991 and 1995). Muller and reported by Richards and Elms (1979), Choudhury and
Whittaker (1993) investigated the effect of wave impact on Nimbalkar (2007) and Nimbalkar and Choudhury (2007) is
the sloping walls, while a comparative study for the not addressed properly in the above-mentioned analysis.
evaluation of the design wave impact pressure is again Hence, till today, the complete solution for the combined
reported by Muller and Whittaker (1996). Experimental effect of seismic active earth pressure and hydrodynamic
studies to assess the behaviour of the vertical wall were pressure on the waterfront retaining wall with the
reported by Ramsden (1996) with the details of the consideration of wall inertia is scarce.
development of an empirical expression for calculating The present method completely describes the behaviour
the forces and moments on a vertical wall due to long of a waterfront retaining wall from the stability considera-
waves, bores and surges. For the studies related to the tion in terms of the sliding and overturning modes of
hydrodynamic pressure, Chakrabarti et al. (1978) had failure under earthquake condition. This study is extremely
shown its effect on cellular type cofferdams. New method essential for the design purpose of the waterfront retaining
of analysis for the quay wall including the effect of wall under seismic condition. A generalized case of a
hydrodynamic pressure was described by Nozu et al. waterfront retaining wall, supporting a submerged backfill
(2004). Again, the seismic active earth pressures acting on on one side and water on the other side, under seismic
the rigid retaining wall for dry soil were computed by using conditions including seismic inertial forces is considered.
different methods of analyses like the limit equilibrium
method (Seed and Whitman, 1970; Richards and Elms, 2. Method of analysis
1979; Choudhury and Singh, 2006; Choudhury and
Nimbalkar, 2006, 2007; Nimbalkar and Choudhury, A typical waterfront retaining wall with vertical face
2007), approximate elastic solutions (Matsuo and Ohara, (i.e., y ¼ 01), width ‘b’ and height ‘H’ is shown in Fig. 1. It
1960), two-dimensional wave propagation theory or shear- retains backfill to its full height on one side, referred to as
beam model (Scott, 1973; Veletsos and Younan, 1994; Wu the ‘downstream side’, and water to a height of ‘h’ on the
and Finn, 1999), finite element techniques (Nadim and other side, called as the ‘upstream side’ of the wall. The
Whitman, 1983; Gazetas et al., 2004), numerical simulation ground surface of the backfill is assumed to be horizontal
by using geotechnical software FLAC (Green et al., 2003). (i.e., b ¼ 01) and is submerged to the same level (i.e., ‘h’) up
But none of the above solutions considered the effect of to which the water is standing on the upstream side of the
hydrodynamic pressure. retaining wall. A free body diagram of the wall showing
 ARTICLE IN PRESS
D. Choudhury, S.M. Ahmad / Ocean Engineering 34 (2007) 1947–1954 1949

b gsat kh
c ¼ tan1 , (3)
ḡð1  kv Þ
 2  2 !
h h
ḡ ¼ gsat þ 1 gd . (4)

Gravity Wall
Water Partially submerged
H H
backfill soil H
h Pae h It is to be noted that the pore pressure ratio ‘ru’, which is
Pw Pw defined as the ratio of excess pore pressure to the initial
vertical stress, incorporated in Eq. (1) above is a simplified
way (as per Ebeling and Morrison, 1992) of simulating the
Fig. 1. A typical gravity type waterfront retaining wall. effect of the excess pore pressure generated due to cyclic
shaking of the soil during an earthquake.

direction of the wall 2.2. Seismic inertia forces on the wall

movement

upstream side downstream side

Due to earthquake, additional inertia forces will be
developed in the wall and for vertical and horizontal
b
directions, these forces are given by kv  W and kh  W
respectively. Though different combinations of these
kv.W Pae.sinδ
Pae inertia forces with respect to the direction of vertical and
kh.W δ H horizontal seismic acceleration coefficients kv and kh are
Pae.cosδ
h Pdyn h
Pdyn considered, only the critical combination resulting in
Pstat W P'stat y
0.4h h/3 h/3 0.4h maximum seismic active earth pressure, which needs to
O
sliding resistance
be considered for the design is shown in Fig. 2.
normal reaction
2.3. Forces on the wall due to water
Fig. 2. Free body diagram of the wall subjected to different forces.
The forces acting on the wall due to the presence of
water, both on the upstream and downstream sides are
different forces coming onto it from soil, water, and due to calculated as follows.
seismicity along with their respective points of applications
is shown in Fig. 2. Pseudo-static seismic accelerations with 2.3.1. Hydrostatic force
acceleration coefficients kh and kv in the horizontal and The hydrostatic force (Pstat) due to the standing water on
vertical directions, respectively, are assumed to act. the face of the wall is given by,
Basically, the wall is subjected to three kinds of forces Pstat ¼ 12gw h2 . (5)
viz., the seismic earth pressure force, the inertia force on
the wall and force due to the presence of water (both on the It acts at a height of h/3 from the base of the wall.
upstream and downstream sides) and each of these are However, for calculating the hydrostatic pressure on the
calculated as follows. wall from the downstream side (P0 stat), gw in Eq. (5) is
replaced by gwe (as given by Ebeling and Morrison, 1992),
and can be calculated as
2.1. Seismic earth pressure  
gwe ¼ gw þ ḡ  gw ru . (6)
The seismic active earth pressure on the wall is calculated Thus,
using the pseudo-static Mononobe–Okabe’s approach.
Similar to the analysis of Ebeling and Morrison (1992) P0stat ¼ 12gwe h2 . (7)
and the expression given by Kramer (1996), the basic
expression for the calculation of the total seismic active
earth thrust (Pae) has been modified to consider the effect 2.3.2. Hydrodynamic force
of submergence in the backfill and the existence of excess The hydrodynamic force (Pdyn) acting on the vertical
pore pressure, and is given as, face of the wall is calculated using the Westergaard’s (1933)
Pae ¼ 12K ae H 2 ḡð1  kv Þð1  ru Þ, (1) approach and is given as

where
7
Pdyn ¼ 12 k h g w h2 . (8)

cos2 ðf  y  cÞ
K ae ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 , (2)
cos c cos2 y cosðd þ y þ cÞ 1 þ sinðd þ fÞ sinðf  b  cÞ= cosðd þ y þ cÞ cosðb  yÞ
 ARTICLE IN PRESS
1950 D. Choudhury, S.M. Ahmad / Ocean Engineering 34 (2007) 1947–1954

It acts at a height of 0.4h from the base of the wall. On are then given as
the upstream side, this hydrodynamic force acts in a Fr Pstat þ m½ð1  kv ÞW þ Pae  sin d
direction opposite to the direction of the hydrostatic force FSslidingr ¼ ¼ 0 (12)
F d r Pstat þ Pdyn þ Pae  cos d þ kh  W
(Ebeling and Morrison, 1992), while on the downstream
side the direction of both the hydrostatic and hydrody- and
namic forces would be towards the wall, thus creating a
Fr Pstat þ m½ð1  kv ÞW þ Pae  sin d
worst possible combination with respect to both the sliding FSslidingf ¼ ¼ 0
F d f Pstat þ 2Pdyn þ Pae  cos d þ kh  W
and overturning modes of failure of the wall. Though
Matsuo and Ohara (1965) had suggested the hydrodynamic (13)
pressure on the downstream side to be around 70% of that where
on the upstream side, but to consider the worst possible
combination of forces for the design of the wall, similar to m ¼ coefficient of base friction ¼ tan f, (14)
the consideration of Ebeling and Morrison (1992), here in
the present study, the same amount of the hydrodynamic W ¼ weight of the wall ¼ bHgc . (15)
pressure is considered both on the downstream and For the generalized design purpose, Eqs. (12) and (13)
upstream side. can be rewritten in the non-dimensional form as follows:
1 2
 
2gw ðh=HÞ þ m ð1  k v Þðb=HÞgc þ 12K ae ḡ  sin d
FSslidingr ¼ 1 2 7 2 1
,
3. Stability of the wall 2gwe ðh=HÞ þ 12kh gw ðh=HÞ þ 2K ae ḡ  cos d þ kh ðb=HÞgc
(16)
Under the action of the above-mentioned forces, the
 
stability of the wall is checked for both the sliding and 1 2 1
2gw ðh=HÞ þ m ð1  k v Þðb=HÞgc þ 2K ae ḡ  sin d
overturning modes of failure using limit equilibrium FSslidingf ¼ 1 2 7 2 1
.
2gwe ðh=HÞ þ 6kh gw ðh=HÞ þ 2K ae ḡ  cos d þ kh ðb=HÞgc
method. Depending on the hydraulic conductivity (k) of
(17)
the soil, two different cases viz., restrained water case
(k103 cm/sec) and free water case (kvery high) may
arise for the generation of the hydrodynamic pressure in 3.2. Factor of safety against overturning mode of failure
the backfill soil (Kramer, 1996). For the restrained water
case, the movement of water is assumed to be with the Similarly, by assuming that the seismic active earth
movement of the backfill soil particles and thus it is pressure (Pae) acts at y ¼ 0.5H above the base of the wall
assumed that the hydrodynamic pressure is not present. (Ebeling and Morrison, 1992), the factor of safety against
However, for the free water case, the water is having the overturning mode of failure for both the restrained and
enough space to move freely within the soil, hence, the free water cases, respectively, are given as
additional hydrodynamic pressure is considered. Expres- 1
6gw ðh=HÞ
3
þ 12ðb=HÞ2 ð1  kv Þgc þ 12K ae ḡðb=HÞ sin d
FSoverturningr ¼ 1 .
sions for finding out the factor of safety against the sliding 6gwe ðh=HÞ
3
þ ð2:8=12Þkh gw ðh=HÞ3 þ 14K ae ḡ cos d þ 12kh ðb=HÞgc
and overturning modes of failure are detailed in the (18)
following section.
and
1
gw ðh=HÞ3 þ 12ðb=HÞ2 ð1  kv Þgc þ 12K ae ḡðb=HÞ sin d
3.1. Factor of safety against sliding mode of failure FSoverturningf ¼ 1 6 3 5:6 3 1 1
.
6gwe ðh=HÞ þ 12 kh gw ðh=HÞ þ 4K ae ḡ cos d þ 2k h ðb=HÞgc

Considering the equilibrium of all the forces acting in the (19)

horizontal direction (Fig. 2), one can write Using the above proposed Eqs. (16), (17), (18) and (19),
Total resisting force; F r ¼ Pstat þ m½ð1  kv ÞW þ Pae sin d.
one can easily design the section of the waterfront retaining
wall subjected to the combined earthquake and hydro-
(9) dynamic forces.
And, the total driving force for the restrained water case
is 4. Results and discussions
F dr ¼ P0stat þ Pdyn þ Pae cos d þ kh  W . (10)
Combination of the different parameters involved in
Similarly, for the free water case, due to the additional Eqs. (16)–(19) can predict the stability of the waterfront
hydrodynamic force, there would be an extra component of wall under earthquake. However, the chosen values must
the same and the total driving force will be be of practical significance and the phenomenon of shear
F d f ¼ P0stat þ 2Pdyn þ Pae  cos d þ kh  W . (11) fluidization must be avoided. To avoid the phenomenon of
shear fluidization, as proposed by Richards et al. (1990) for
The respective factor of safety of the wall against sliding the dry soil, and subsequently modified by Ebeling and
mode of failure for both the restrained and free water cases Morrison (1992) for the wet soil, the following expression
 ARTICLE IN PRESS
D. Choudhury, S.M. Ahmad / Ocean Engineering 34 (2007) 1947–1954 1951

Table 1 2.5
Values/range of different parameters chosen for the present study b/H = 0.4; φ = 30° ; δ = φ/2; kv= kh/2; ru= 0.2

2.0 h/H = 0.0 (restrained and free)

Parameter Value/range
h/H = 0.4 (restrained)

FSoverturning
1.5 h/H = 0.8 (restrained)
b/H 0.4 h/H = 0.4 (free)
h/H 0, 0.4, 0.8 h/H = 0.8 (free)
kh 0, 0.1, 0.2, 0.3, 0.4 1.0
kv 0, kh/2, kh
ru 0.2 0.5
gc, gsat, gd, gw 25, 19, 16 and 10 kN/m3 respectively
f (degree) 25, 30, 35, 40
d (degree) f/2, 0, f/2 0.0
0.0 0.1 0.2 0.3 0.4
Horizontal seismic acceleration coefficient, kh

Fig. 4. Factor of safety in overturning mode for different h/H values.

3.5
b/H = 0.4; φ = 300; δ = φ/2; kv= kh/2; ru = 0.2 3.5
3.0 b/H= 0.4; h/H = 0.8; δ = φ/2; kv= kh/2; ru= 0.2
h/H =0.0 (restrained and free) 3.0
2.5 h/H =0.4 (restrained) φ = 35° (restrained)
h/H =0.8 (restrained) 2.5 φ = 30° (restrained)
2.0
FSsliding

h/H =0.4 (free)

h/H =0.8 (free) φ = 25° (restrained)

FSsliding
2.0 φ = 35° (free)
1.5
φ = 30° (free)
1.5
1.0 φ = 25° (free)
1.0
0.5
0.5
0.0
0.0 0.1 0.2 0.3 0.4 0.0
Horizonal seismic acceleration coefficient, kh 0.0 0.1 0.2 0.3 0.4
Horizontal seismic acceleration coefficient, kh
Fig. 3. Factor of safety in sliding mode for different h/H values.
Fig. 5. Effect of f on sliding stability.

need to be satisfied for the present study. 2.5

b/H=0.4; h/H = 0.8; δ = φ/2; kv= kh/2; ru= 0.2
cof (20)
2.0 φ = 35° (restrained)
Different values of the parameters and their ranges φ = 30° (restrained)
FSoverturning

φ = 25° (restrained)
considered in the present analysis are given in Table 1. 1.5
φ = 35° (free)
Effect of the various parameters on the sliding and φ = 30° (free)
overturning stability of the wall with respect to the value 1.0 φ = 25° (free)
of water to wall height ratio (h/H), soil friction angle (f),
wall friction angle (d), and the coefficients of horizontal 0.5
and vertical seismic accelerations (kh and kv) are discussed
0.0
in Figs. 3–10. 0.0 0.1 0.2 0.3 0.4
Horizontal seismic acceleration coefficient, kh
4.1. Effect of the horizontal seismic acceleration coefficient
(kh) Fig. 6. Effect of f on overturning stability.

From Fig. 3 it is observed that for a particular value of

h/H ratio, the factor of safety in sliding mode decreases decrease in the factor of safety by about 57% for an
drastically with an increase in the value of horizontal increase in kh from 0 to 0.2 is observed for both the cases.
seismic acceleration coefficient kh. It shows that a stable Similar trend is noted for the overturning mode also
waterfront retaining wall can fail with the increase in (Fig. 4). Another important observation which can be
horizontal seismic acceleration. As an illustration, for made from Figs. 3 and 4 is that the effect of h/H ratio on
chosen values of b/H ¼ 0.4, h/H ¼ 0.4, f ¼ 301, d ¼ f/2, the factor of safety for lower kh values is significant as can
kv ¼ kh/2 and ru ¼ 0.2, the factor of safety against sliding be observed from a large difference in values of factor of
both for the restrained and free water cases is 2.43 when safety for a particular kh value, say 0 or 0.1. But the effect
there is no earthquake (i.e., kh ¼ 0); while the same reduces of h/H ratio on the factor of safety in both sliding and
to 1.07 for the restrained water case and to 1.04 for the free overturning modes of failure is marginal at higher values of
water case when the value of kh is increased to 0.2. Hence, a kh. However, it may be noted that at higher values of kh,
 ARTICLE IN PRESS
1952 D. Choudhury, S.M. Ahmad / Ocean Engineering 34 (2007) 1947–1954

2.5 3.0
b/H = 0.4; h/H = 0.8; φ = 30°; kv= kh/2; ru= 0.2 b/H = 0.4; h/H = 0.8; φ = 30°; δ = φ/2; ru= 0.2
2.0 2.5 kv= 0.0 (restrained)
δ = -φ/2 (restrained)
δ = 0.0° (restrained) kv= kh/2 (restrained)
2.0

FSoverturning
δ = φ/2 (restrained) kv= kh (restrained)
1.5
FSsliding

δ = -φ/2 (free) kv= 0.0 (free)

δ = 0.0° (free) 1.5
kv= kh/2 (free)
1.0 δ =φ/2 (free)
kv= kh (free)
1.0
0.5 0.5

0.0 0.0
0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4
Horizontal seismic acceleration coefficient, kh Horizontal seismic accleration coefficient, kh

Fig. 7. Effect of d on sliding stability. Fig. 10. Effect of kv on overturning stability.

2.5 the value of soil friction angle f is nearly constant for all
b/H=0.4; h/H = 0.8; φ = 30°; kv= kh/2; ru= 0.2 values of kh and is true for the overturning case also
2.0 δ = -φ/2 (restrained) (Fig. 6). Also, the trend is similar both for the free and
δ = 0.0° (restrained) restrained water cases, except for the fact that the value for
δ = φ/2 (restrained)
FSoverturning

1.5 δ = -φ/2 (free) free water case is slightly lower than the value of restrained
δ = 0.0° (free) water case. Another important observation from these
δ = φ/2 (free)
1.0 figures is that with an increase in f, the shear fluidization
phenomenon can be avoided.
0.5

4.3. Effect of wall friction angle (d)

0.0
0.0 0.1 0.2 0.3 0.4
Horizontal seismic acceleration coefficient, kh From Figs. 7 and 8, the effect of wall friction angle (d) on
the sliding and overturning stability of the wall is observed.
Fig. 8. Effect of d on overturning stability. It is found that as d increases from 0 to f/2, the stability of
the wall increases. For example, for kh ¼ 0.1, b/H ¼ 0.4,
3.0 h/H ¼ 0.8, f ¼ 301, kv ¼ kh/2 and ru ¼ 0.2 in Fig. 7, the
b/H = 0.4; h/H = 0.8; φ = 30°; δ = φ/2; ru = 0.2
factor of safety against the sliding mode is 1.11 when d is 0
2.5 kv= 0.0 (restrained)
kv= kh/2 (restrained)
and increases to 1.20 when d is changed to f/2, i.e., an
2.0 kv= kh (restrained) increase of about 8.1% in the factor of safety value for a
change in d from 0 to f/2 for the restrained water case. For
FSsliding

kv= 0.0 (free)

1.5 kv= kh/2 (free) the same data, in case of free water, the increase in the
kv= kh (free)
1.0
factor of safety against sliding is about 8.0% for a change
in d from 0 to f/2. For the overturning mode (Fig. 8), the
0.5 similar trend is observed for the factor of safety value with
change in d.
0.0
0.0 0.1 0.2 0.3 0.4
Horizontal seismic acceleration coefficient, kh 4.4. Effect of the vertical seismic coefficient (kv)

Fig. 9. Effect of kv on sliding stability. As shown in Fig. 9, with the increase in the vertical
seismic acceleration coefficient, kv from 0 to kh, the factor
the stable wall under static condition has failed in both the of safety against sliding mode reduces (for kh ¼ 0.1,
sliding and overturning modes (FSo1 in Figs. 3 and 4). b/H ¼ 0.4, h/H ¼ 0.8, f ¼ 301 and ru ¼ 0.2) by about
4%, which may be considered as marginal. Overturning
4.2. Effect of soil friction angle (f) mode of failure of the wall shows the similar behaviour as
can be seen from Fig. 10.
Figs. 5 and 6 respectively show the variation of factor of
safety in the sliding and overturning modes of failure for 5. Comparison of results
different f values. With the increase in the value of f from
301 to 351, there is 20.4% increase in the factor of safety For the purpose of verifying the present methodology for
against the sliding mode of failure (Fig. 5) for kh ¼ 0.1, design, the results obtained must be compared with existing
b/H ¼ 0.4, h/H ¼ 0.8, d ¼ f/2, kv ¼ kh/2 and ru ¼ 0.2. The works. However, as already mentioned, for the waterfront
rate of decrease in factor of safety value with decrease in retaining wall subjected to combined hydrodynamic
 ARTICLE IN PRESS
D. Choudhury, S.M. Ahmad / Ocean Engineering 34 (2007) 1947–1954 1953

2.0 approach is stable while the present analysis shows that it

b/H = 0.4; h/H=0.8; φ = 30°; δ = φ/2; kv= kh/2; ru= 0.2 has failed. This difference is due to the presence of
1.5
Ebeling and Morrison (1992) additional inertial force considered in the present analysis
Present
which is more logical. Similar observation can be made for
the overturning case also as shown in Fig. 12.
FSsliding

1.0

6. Conclusions
0.5
The study shows the importance to develop a separate
0.0 design methodology for a waterfront retaining wall under
0.0 0.1 0.2 0.3 0.4 earthquake condition. The stability of the wall decreases
Horizontal seismic acceleration coefficient, kh significantly during an earthquake. An easy methodology
to design the section of the waterfront retaining wall
Fig. 11. Comparison between present analysis and the one adopted by subjected to the combined action of hydrodynamic
Ebeling and Morrison (1992) for sliding stability.
pressure and seismic active earth pressure is described
through the closed-form solutions to obtain the factor of
2.0 safety against sliding and overturning modes of failure.
b/H = 0.4; h/H=0.8; φ = 30°; δ = φ/2; kv= kh/2; ru= 0.2
From the typical results it is observed that for a given wall
Ebeling and Morrison (1992)
1.5 Present
section and other soil and water parameters remaining
constant, the factor of safety in overturning mode is less
FSoverturning

than the factor of safety in sliding mode under earthquake

1.0
condition. Parameters like soil friction angle (f), wall
friction angle (d), and horizontal and vertical seismic
0.5 accelerations (kh and kv), water to wall height ratio (h/H)
have significant effect on the stability of the wall, and out
0.0 of these, the factor of safety value is very much sensitive to
0.0 0.1 0.2 0.3 0.4 the f and d. Comparison of the present results with those
Horizontal seismic acceleration coefficient, kh obtained by Ebeling and Morrison (1992) suggests that
wall inertia needs to be considered as it has a significant
Fig. 12. Comparison between present analysis and the one adopted by effect on the stability of the wall.
Ebeling and Morrison (1992) for overturning stability.

Acknowledgement
pressure and seismic active earth pressure, the only work
The authors gratefully acknowledge the critical reviews
which can found out is the one by Ebeling and Morrison
and helpful suggestions made by the anonymous reviewers.
(1992). Figs. 11 and 12 present a comparison between the
results of the present study and the one obtained by the
approach used by Ebeling and Morrison (1992). A keen References
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