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50th INDIAN GEOTECHNICAL CONFERENCE

17th – 19th DECEMBER 2015, Pune, Maharashtra, India

Venue: College of Engineering (Estd. 1854), Pune, India

OPTIMUM DESIGN OF GRAVITY RETAINING WALLS OF NARROW

BACKFILL WIDTH

B. Munwar Basha1 and Shaik Moin Ahmed2

ABSTRACT

The significant transportation demand has led to widening of existing highways to increase the right

of way. As the available space at the site is limited, most of the times retaining walls are required to be

constructed in front of rock faces. The design methodology is not very clear at present for earth retaining

structures placed adjacent to rock face with narrow backfill width. The analytical methods for evaluating

active earth pressure like Rankine and Coulomb are inappropriate when the backfill behind a retaining wall

is narrow. This is due to the inadequate development of active thrust wedge in the shape and size as

predicted by these methods. Therefore, an analytical method is developed to obtain a solution for the active

earth pressure exerted by the backfill of narrow width on gravity retaining walls, using the limit equilibrium

method with planar slip surfaces. Three different failure mechanisms of the active wedge are considered,

namely Mechanism 1 (active wedge formed by a single block), Mechanism 2 (active wedge is composed

of two rigid blocks, one slipping with respect to the other) and Mechanism 3 (which considers three blocks).

Using three mechanisms, the magnitude of active earth pressure acting on the gravity wall and its point of

application from the bottom of the wall are computed. The revised active earth pressure and its point of

application have been used to optimize the gravity retaining wall proportions. Totally four modes of failure

are considered, viz overturning of the wall about its toe, sliding of the wall on its base, eccentricity of the

resultant force striking the base of the slab and bearing capacity failure below the base slab. The optimum

dimensions of gravity wall are obtained by imposing the following four constraints in the optimization

routine. The constraints are (i) the factor of safety against overturning ( otFS ) 2 , the factor of safety against

sliding ( sliFS ) 1.5 , (iii) eccentricity of the resultant force which strikes the middle of the base of retaining

wall must be less than one sixth of the base width of the wall, ( / 6e B ) and (iv) Factor of safety against

bearing capacity failure ( bFS ) 2.5 . The influence of interface friction angle between the retaining wall

and backfill soil, and the distance of rock face from the heel of the gravity wall on the magnitude of active

1Assistant Professor, Department of Civil Engineering, IIT Hyderabad, India, [email protected]

2Research Scholar, Department of Civil Engineering, IIT Hyderabad, India, [email protected]

mailto:[email protected]


B.Munwar Basha & Shaik Moin Ahmed

earth pressure, point of application of the active trust and optimum dimensions of the retaining wall are

presented in the form of design charts.

Keywords: Optimization, Narrow Retaining wall, Gravity Retaining wall, Narrow backfill width.

Rock

Retaining Wall

b

B

A

D

E

F

Wedge-1

Wedge-2

Wedge

3

ba h

e

r

y

ymax

Y

lh

C

Fig. 1 Mechanism-3 for narrow backfilled width retaining wall shows formation of an active wedge with

three blocks

Fig. 2 The magnitude of earth pressure coefficient along the depth of the narrow backfilled width

retaining wall

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OPTIMUM DESIGN OF GRAVITY RETAINING WALLS OF NARROW

BACKFILL WIDTH

B. Munwar Basha1, Assistant Professor, Department of Civil Engineering, IIT Hyderabad, India, [email protected]

Shaik Moin Ahmed2 Research Scholar, Dept. of Civil Engg., IIT Hyderabad, India, [email protected]

ABSTRACT: An analytical method is presented to obtain a solution for the active earth pressure exerted by the

backfill of narrow width on gravity retaining walls, using the limit equilibrium method with planar slip surfaces.

Using these three mechanisms, the magnitude of active earth pressure acting on the gravity wall and point of

application of active thrust from the bottom of the wall are computed. The optimum dimensions of the gravity wall

are obtained by considering the four modes of failure, viz overturning of the wall about its toe, sliding of the wall on

its base, eccentricity and bearing capacity failure below the base slab. The influence of various parameters on the

optimum dimensions of the gravity retaining wall are presented in the form of design charts.

INTRODUCTION

Gravity retaining wall is one of the most widely

used earth retaining systems amongst various

categories of retaining walls in civil engineering

practice. The influence of the magnitude of active

earth pressure exerted by the backfill has a

significant impact on the proportions of gravity

wall. In general, gravity retaining wall design is

done using a factor safety approach. Factors of

safety have the advantage of being easily

interpreted in terms of their physical or engineering

meaning. The need for the optimum design of

retaining walls is addressed in the literature.

Studies pertaining to Optimum design of

retaining walls

Hoeg and Murarka (1974) [1] reported results on

the optimum and balanced design of retaining

structures for a gravity retaining wall based on

deterministic procedures and probabilistic

approach. Rhomberg and Street (1981) [2] reported

that there may be many combinations of wall base

proportions that satisfy basic design requirements

of minimum factor of safety and obtained the

dimensions by a trial and error procedure

corresponding to minimum cost. Saribas and

Erbatur (1996) [3] given a methodology for

optimum design of reinforced concrete cantilever

retaining walls using cost and weight of the walls

as objective functions and overturning failure,

sliding failure, no tension condition in the

foundation base, shear and moment capacities of

toe slab, heel slab and stem of wall as constraints.

Narrow backfilled width retaining walls

The significant transportation demand has led to

widening of existing highways to increase the right

of way. As the available space at the site is limited,

most of the times retaining walls are required to be

constructed in front of the rock faces. The design

methodology is not evident at present for earth

retaining structures placed adjacent to rock face

with narrow backfill width. The analytical methods

for evaluating active earth pressure like Rankine

and Coulomb are inappropriate when the backfill

behind a retaining wall is narrow. This is due to the

inadequate development of active thrust wedge in

the shape and size as predicted by these methods.

The following studies are available in the literature

which deal with computation of active earth

pressure behind narrow backfilled width retaining

walls.

Studies pertaining to Experimental

Investigations

Frydman and Keissar (1987) [4] and Take and

Valsangkar (2001) [5] presented the theory for




unyielding retaining walls and reported that there

is a poor agreement between the experimental data

and theoretical predictions in conditions of active

thrust. Woodruff (2003) [6] performed series of

centrifuge model tests on reinforced soil walls

adjacent to a stable face and observed that the slip

line is bilinear. Moreover, this study reported that

failure mechanism is governed by two wedges, one

wedge inclined at an angle a less than the

theoretical Rankine failure plane and the other

wedge developing along the interface between

backfill and stabilized wall.

Studies pertaining to Numerical Investigations

Yang and Liu (2007) [7] presented a finite element

analysis of earth pressure for narrow backfilled

width retaining walls. In addition, Fan and Fang

(2010) [8] reported the numerical analysis of active

earth pressure on rigid retaining walls built near

rock faces.

Studies pertaining to Analytical Investigations

Analytical approaches have been reported for

solving narrow backfilled width retaining wall

problem using the limit equilibrium method. In this

direction, Leshchinsky et al. (2004) [9] proposed

an approach for the calculation of the active thrust

using the simplified Bishop method of slices

assuming that the thrust is applied at 1/3 of the total

wall height. Lawson and Yee (2005) [10] assumed

a thrust wedge governed by a bilinear slip surface

with the angle equal to that given by Rankine’s

method for the condition of active case. James et

al. (2004) [11] presented a methodology for the

calculation of active earth pressure by assuming the

angle of Rankine’s method and Coulomb’s method

for the calculation of the reaction of the rock face.

However, Yang and Zornberg (2006) [12] reported

the value maximizing the active earth pressure

obtained by a trial and error procedure for

computing the angle a, and the rock face reaction

is calculated with Rankine’s method. Greco (2013)

[13] presented the three wedge failure mechanism

characterized by a thrust wedge formed by 1 to 3

rigid blocks depending on the backfill width. The

blocks are limited by slip planes inclined at angles

‘α’, ‘ρ’ and ‘λ’, whose values were chosen with the

criterion of maximizing the active thrust acting on

the wall.

Objectives and Scope of the present study

A review of the literature suggests that the

optimum proportions of gravity retaining walls

were obtained for the adequate development of

active thrust wedge (Rankine and Coulomb

theories) in the shape and size behind the gravity

retaining wall. However, it is clear from the above

discussion that the Rankine and Coulomb theories

are not appropriate for narrow backfilled width

gravity retaining walls. Therefore, the three failure

mechanism proposed by the Greco (2013) [13] is

considered in the present paper for the computation

of active earth pressure exerted by the narrow

backfilled gravity wall. Moreover, deterministic

optimization methodology is proposed to obtain an

optimum proportions of narrow backfilled width

gravity retaining walls. Hence in this paper,

detailed studies conducted on the optimum wall

proportions considering several limit states

namely, overturning, sliding, eccentricity and

bearing capacity.

Computation of Active Earth Pressure (Sa)

The analysis is developed assuming the soil is

cohesionless, without pore pressure and obeys the

Mohr–Coulomb failure criterion and plane strain

conditions. It is also assumed that the wall

movement is sufficient to induce failure along

planes inside the backfill in accordance with

Coulomb’s approach. Three wedge mechanism

proposed by Greco (2013) [13] is used to compute

the active earth pressure behind the narrow

backfilled width gravity retaining walls with

recycled waste or byproducts materials as backfill.

It is based on limit equilibrium method.

The retaining wall considered in the present study

has the inclined back face with an angle b is the

retaining wall back face inclined angle with respect

to the horizontal, h is the height of the retaining wall,

eis the backfill with topographic profile inclined

angle, is the friction angle of backfill soil, is the

unit weight of backfill soil, “b” is the horizontal

distance from the wall heel to planar rock face, h is

the Rock face inclination angle, is the friction

angle between backfill soil and wall and is the

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friction angle between the same backfill soil and

rock face. As assumed by Greco (2013) [13], three

different failure mechanisms of the active wedge are

considered. The mechanism 1 considers an active

wedge formed by a single block as shown in Figure.

1. Further, the active wedge is composed of two

rigid blocks, one slipping with respect to the other

in mechanism 2 as shown in Figure. 2. Similarly, the

mechanism 3 considers three blocks as shown in

Figure.3. The blocks are limited by slip planes

inclined at an angles α, ρ and λ, whose values are

chosen by maximizing the active thrust pressure

acting on the retaining wall.

Fig. 1 shows a schematic cross-section of a wall and

backfill lying behind which fails with Mechanism-1

Mechanism-1

The Figure.1 shows the forces (W, Sa and R1) and

its direction acting on the thrust wedge, where ‘W’

is the weight of the wedge BCD acting downwards,

‘Sa’ is the thrust exerted by the wall on plane BC

which is inclined at an angle ‘ ’ and R1 is the thrust

acting on inclined plane CD. Applying the

conditions of force equilibrium on the wedge BDC,

we can obtain the active thrust Sa given by

2

2

1 sin( )sin( ) sin( )

2 sin sin( ) sin( )aS h

b a b e a

b a e b a

(1)

2

1

1

2a aS h K (2)

The minimum value of failure plane (min

a ) is

obtained by the geometry conditions as shown in

Figure.1



forming two wedges.

1

min

cos sin sintan

cos sin cos

H b

H b

e e ha

h h e

(3)

If the obtained value of ca using Eq. 1 is more than

mina calculated using Eq. 3. (i.e. minca a ), active

earth pressure Sa can be calculated using the

Coulomb’s theory. Otherwise, the mechanism 2

should be used to calculate the active earth pressure


Sa due to inadequate development of active thrust

wedges in the shape and size as predicted by the

Coulomb’s methods.

Mechanism-2

The forces which are acting on two wedges as

shown in Figure. 2 are the self-weights, W1 & W2,

active thrust (Sa) on the plane BC, S2 on plane AD

and reaction (R1) on plane DC. Employing the force



forming three wedges.

equilibrium of the two wedges, the active thrust Sa

is given by

1 2sin( ) sin( )

sin( ) sin( )a

W SS

a h a

b a b a

(4)

2

max 2

2

1 sin( )sin( )( )

2 sin sin( )

sin( )

sin( )

y y

S

r h h e

h r e

r

r h

(5)

btany =

1-cot tan

a

h a (6)

max

cos sinsin

sin( )

H by

e eh

h e

(7)

-1

max

b+ycot -hcot= 90 + tan

h y

h br

(8)

The maximum value of Sa as shown in Eq. 4 can be

obtained by equating the first derivative of Sa with

respect to a and r with boundary conditions

min a a and 180oh r to get cr

and ca is shown below:

0aS

a

and 0aS

r

(9)

The optimization is performed using fminsearchcon

function developed by Nelder- Mead simplex

method in MATLAB. If the obtained value of cr is

less than maxr (i.e. maxcr r ), the active earth

pressure Sa is calculated by substituting the value of

cr in Eq. 4. Otherwise, the mechanism-3 should be

used to calculate the active earth pressure Sa due to

inadequate development of shape and size of two

wedge failures as the failure plane is falling in

between B and C points which forms the third

wedge.

Mechanism-3

The Figure. 3 shows mechanism 3 which is formed

by three wedges. The wedges are subject to its own

weights W1, W2, and W3 , active thrust, Sa, S2, S3,

R1, R2 and R3 on the planes BC, AD, EB, DC, ED

and EF respectively. Applying the conditions of

force equilibrium on the three wedges, we can

obtain the active thrust Sa as given by

1 2sin( ) sin( )

sin( ) sin( )a

W SS

a h a

b a b a

(10)

2 32

sin( ) sin( )

sin( )

W SS

r r b

r h

(11)

33

sin( )

sin( )

WS

l

b l

(12)

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maxmin

cot cot

y Y

b Y Yl

b h

(13)

cot - cot

cot - cot

b y yY

h r

b r

(14)

The maximum value of Sa as shown in Eq. 9 is

obtained by equating the first derivative of Sa with

respect to a , r and l with boundary conditions

min a a , maxh r r and

l b to get ca , cr and cl as shown

below:

0aS

a

, 0aS

r

and 0aS

l

(15)

If the obtained value of cl is less than minl (

c minl l ) then active earth pressure Sa is calculated

using the Eq. (10) by substituting the value of cl .

Otherwise the value of minl is substituted in the

Eq. (10) to get the active thrust Sa and this

mechanism refers to mechanism-3 forced. This

attributes to inadequate development of active thrust

wedge in the shape and size as predicted by the

Mechanism-3 i.e the failure plane is falling in

between points A and D as shown in Figure. 3 which

forms the fourth wedge. Therefore, in order to

compute the value of Sa, higher mechanism is

required to solve four wedge failures. The present

investigation considers maximum of three failure

wedges to obtain active earth pressure. To calculate

the depth of the point of application of the active

earth pressure (ys) from bottom of the wall, the wall

height ‘h’ is divided into ‘n’ number of parts of the

same length z (z = h/n) introducing the points, P0,

P1, P2, . . ., Pn equally spaced between them, where

the point Pi is located at depth zi., If z is sufficiently

small, it can be assumed (with a negligible error)

that the thrust Si is applied to the center of the

segment (Pi-1- Pi) as shown in Figure. 4. The depth

ys is given by 1i i i

a a aS S S (16)

1

1 ( 0.5)

ni

s a

ia

y S z iS

(17)

STABILITY MODES OF GRAVITY

RETAINING WALL

The optimum design of gravity retaining wall is

performed considering four modes of failure, viz

overturning of the wall about its toe, sliding of the

wall on its base, eccentricity and bearing capacity

failures below the base slab.

Fig. 7 The point of application active earth pressure

from the bottom of the wall.

The optimum dimensions are obtained by imposing

the following four constraints in the optimization

routine The constraints are (i) the factor of safety

against overturning ( otFS ) 2 , the factor of safety

against sliding ( sliFS ) 1.5 , (iii) eccentricity of

the resultant force which strikes the base of retaining

wall must be less than one sixth of the base width of

the wall, ( / 6e B ) and (iv) factor of safety against

bearing capacity failure ( bFS ) 2.5. Fig. 8 shows

the dimensions of the gravity wall considered. The

properties of backfill materials are taken from Table

1.

Failure Modes of Retaining Wall

Consider a narrow width backfilled Gravity

retaining wall shown in Figure. 8 with a height of

“h” The calculation of forces and moments acting

on a retaining wall as follows. Expressions for

factors of safety against overturning failure, sliding


failure, eccentricity failure and bearing failure of

wall are given in the following sections.

Fig. 8 Narrow backfilled width gravity retaining

wall with geometric parameters.

Overturning Failure Mode

The factor of safety against overturning about the

toe, that is, about point “a” in Figure.8 may be

expressed as

( )

r

overturning

o

MFS

M

(19)

where, rM Sum of the moments of forces

tending to resist overturning about point “a” in

Figure 8, 0M Sum of the moments of forces

tending to overturn about point “a” in Figure 8,

From the Figure 8 the following forces are

calculated

cos( -90) ( )o a s fM S y t b (20)

t s w

a

w1 t s

r 2 t s w

s3 t

t s w4

L +t +tS cos -90

cot cot

tL +t +

2

cotM = L +t +t

3

2tL +

3

L +t +t cot

2

s

h

h y

w

hw

w

L hw

bb b

b

b

(21)

Sliding Failure Mode

The factor of safety against sliding may be

expressed by the equation

( )

r

Sliding

d

FFS

F

(22)

Where

rF Sum of the horizontal resisting forces

dF Sum of the horizontal driving forces

a S cos( -90) V W b (23)

(24)

2tan (2/3)r w pF V B c P (25)

The passive earth pressure (Pp) is assumed to be zero

in the study.

Eccentricity Failure Mode

For stability, the line of action of the resultant force

must lie within the middle third of the foundation

base. The factor of safety against eccentricity failure

is given by

( )e

Eccentricity

eFS

e (26)

where 6

we

Be (27)

2

w netB Me

V

(28)

net r oM M M (29)

where Bw is the base width of the wall and ‘e’ is the

eccentricity of the resultant force.

Bearing Failure Mode

Factor of safety against bearing capacity failure can

be defined as

( )

max

uBearing Cap

qFS

q (30)

The maximum intensity of soil pressure at toe can

be written as

max

61

w w

V eq

B B

(31)

Ultimate bearing capacity of a shallow foundation

below the base slab of the retaining wall is given by

aS cos( -90) dF b

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2

2

12

2

c cd ci q qd qi

u

w rd ri

cN F F DN F F

qB e N F F

(32)

where , Fcd, Fqd, Frd are depth factors; Fci, Fqi, Fri

Load inclination factors; and Nc, Nq, Nr are the

bearing capacity factors. The following geometric

parameter is optimized in the present study:

/ st h = ratio of front batter width to height of

retaining wall. Table 1 Statistics of input parameters considered in the present study.

Parameters Values Parameters Values

wL /h 0.05 1 20o

fL /h 0.15 2 1(2/3)

tL /h 0.12 e 0o- 20o

hL /h 0.10 η 70o- 90o

18

kN/m3 10o- 40o

2

19 kN/m3

0o

c

24 kN/m3

ψ 0o

c/ H 0.25 b 90o- 120o

RESULTS AND DISCUSSION

Location of the Critical Failure Surface by

Optimization

In this section, the state at which the active earth

pressure ( aS ) achieves a maximum value and the

failure surface corresponding to this state i.e. critical

failure surface is determined. The geometry of the

failure surface is governed by a , r , l for

mechanisms 1, 2, and 3 as shown in Figs. 1, 2 and

3. The main intent of the optimization is to locate

the critical failure surface in order that aS achieves

a maximum value. Although many methods of

optimization are available, “improved Nelder-Mead

simplex method” (Luersen and Riche [14]) is used

herein to optimize the aS function value. The

maximization was accomplished using a multi-

dimensional Nelder-Mead simplex routine available

in Matlab [?] optimization toolbox. Nelder-Mead

simplex algorithm is for nonlinear unconstrained

optimization and it belongs to general class of direct

search methods.

Corresponding to given set of values of soil friction

angle ( ), soil-wall interface friction angle ( ),

friction angle between the backfill soil and rock face

( ψ ), wall back face angle with respect to the

horizontal (b ) and interface friction angle between

base soil and foundation of wall (2 ), the geometry

of the mechanisms 1, 2 and 3 must be optimized to

obtain maximum value of active earth pressure. The

maximization of aS is subjected to bound

constraints can be stated as follows:

cri cri cri

min

max

Find α , ρ and λ which

maximizes

subjected to

aS

a a

h r r

l b

(33)

The above optimization gives the angles of critical

failure surface, cria , crir , cril and corresponding

maximum active earth pressure ( (max)aS ). For this

purpose, 'fminsearch' function of Matlab [15]

optimization toolbox is modified to consider the

lower and upper bound constraints by the use of

algorithm reported in Luersen and Riche [14].

Active earth pressure coefficient is given by,

2

(max) 2 /a aK S H (34)

Fig. 9 Computation of cria and (max)aK using

mechanism 1 for z/h = 0.1 to 0.5.


Fig. 10 Computation of cria , crir and (max)aK

using mechanism 2 for z/h = 0.6.







Fig. 14 Computation of cria , crir , cril and

(max)aK using mechanism 3 for z/h = 1.0.

It can be noted from Figs. 9 to 14 that after 15 to

100 iterations, aS converges to its maximum value

for critical angles cria , crir and cril .

Fig. 15 shows Variation of active earth pressure

coefficient, (max)aK along the depth of the wall

using mechanisms, 1, 2 and 3. Figs. 16, 17 and 18

show the effect of / ratio on (max)aK , point

of application ( /sy h ) and base width of gravity

wall ( /wB h ) using mechanisms 1, 2 and 3. It can

be noted from Figs. 16, 17 and 18 that for constant

ratio of / , the magnitude of aK decreasing

significantly as the rock face distance ( /b h )

decreases from 0.7 to 0.1. Consequently, the

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minimum base width of gravity wall ( /wB H )

reduces with the reduction in rock face distance (

/b h ). The optimized width, /wB H ensures the

safety of the wall with respect to overturning,

sliding, eccentricity and bearing capacity stability

modes simultaneously.

CONCLUSIONS

The present study reports three wedge mechanism

with limit equilibrium method for computing the

active earth pressure coefficient behind narrow

backfilled width gravity walls. The new values of

active earth pressure coefficients and its points of

application are used for the optimum design of

Fig. 15 Variation of active earth pressure

coefficient, (max)aK using mechanisms, 1, 2 and 3.

Fig. 16 Effect of / ratio on (max)aK using

mechanisms 1, 2 and 3.

Fig. 17 Effect of / ratio on /sy h using

mechanisms 1, 2 and 3.

Fig. 18 Effect of / ratio on base width of

gravity wall, /wB h using mechanisms 1, 2 and 3.

gravity retaining walls by considering the

satisfactory performance against overturning,

sliding, eccentricity of the resultant force and

bearing capacity failures. As can be seen from the

present study that the Coulomb’s method produces

an overestimate of Sa values. Moreover, the

presence of the rock face behind the gravity wall

significantly affects the magnitudes of active earth

pressure and its point of application. Hence, due

consideration must be given for computing the

optimum proportions of gravity retaining wall for

the satisfactory performance against external failure

modes. The results presented in this paper can play

a supplementary verification and comparison role to

standard conventional methods. It can be concluded


from the study that a considerable amount of savings

in concrete can be achieved due to presence of rock

face behind the wall.

NOTATIONS

The following notations have been used in the

paper: h = high of the retaining wall, wL top width

of the of wall, fL thickness of the wall footing,

tL length of toe, hL length of heel, unit

weight of the backfill soil, 2

= unit weight of the

foundation soil, c = unit weight of concrete, s =

stability number of the foundation soil (= /c H ), e

= angle of the backfill with horizontal, η = rock

face inclination angle, = friction angle of the f

soil, 1 = friction angle of the foundation soil, =

interface friction angle between backfill soil and

wall, ψ = interface friction angle between the

backfill soil and rock face, b = wall back face

inclined angle with the horizontal, 2 = interface

friction angle between the wall base and foundation

soil, b = horizontal distance of the rock face from

heel of the wall, wB = base width of the wall, and st

= front batter width of the wall.

REFERENCES

1. Hoeg, K. and Murarka, R. (1974), Probabilistic

analysis and design of a retaining wall, Jl. Of

Geotech. Engineering Divison, ASCE, 100(3), 349–366.

2. Rhomberg, E.J. and Street, W.M. (1981),

Optimal design of retaining walls, Jl. of Struct.

Division, ASCE, 107(5), 992–1002.

3. Saribas, A. and Erbatur, F. (1996). Optimization

and sensitivity of retaining structures, Jl.

Geotech. Eng. Div., ASCE, 122(8), 649-656.

4. Frydman, S., and Keissar I. (1987), Earth

pressure on retaining walls near rock faces, Jl.

of Geotech. Engineering Divison, ASCE,

113(6), 586–99.

5. Take, W.A. and Valsangkar, A.J. (2001), Earth

pressures on unyielding retaining walls of

narrow backfill width, Can Geotech Jl., 38,

1220–30.

6. Woodruff, R. (2003), Centrifuge modelling of

MSE-shoring composite walls, Master Thesis,

Boulder: Department of Civil Engineering, the

University of Colorado.

7. Yang, K-H. and Liu NC. (2007), Finite element

analysis of earth pressure for narrow retaining

walls, Jl. of Geoengineering, 2(2), 43–52.

8. Fan, C-C. and Fang, Y-S. (2010), Numerical

solution of active earth pressures on rigid

retaining walls built near rock faces, Comput

Geotech, 37, 1023–1029.

9. Leshchinsky, D., Hu, Y. and Han, J. (2003),

Design implications of limited reinforced zone

space in SRW's, Proceedings of the 17th GRI

Conference on hot Topics in Geosynthetics IV,

LasVegas, Nevada.

10. Lawson, C.R. and Yee, T.W. (2005), Reinforced

soil retaining walls with constrained reinforced

fill zones, Slopes and Retaining Structures

under Seismic and Static Conditions (GSP 140),

ASCE, Reston/VA: 2721–2734.

11. James, M, Li, L. and Aubertin, M. (2004),

Evaluation of the earth pressures in backfilled

stopes using limit equilibrium analysis, In:

Proceedings of the 57th Canadian Geotechnical

Conference, Quebec, Canada, 33–40.

12. Yang, K-H. and Zornberg, J.G. (2006), The

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