Seismic Displacement of Retaining Wall with Reinforced Backfill

Saran, S.K. Viladkar, M.N.1

Scientist Professor

e-mail: saranswami@yahoo.co.in e-mail: sumanfce@iitr.ernet.in

Geotechnical Engineering Division, Central Building Research Institute, Roorkee1Department of Civil Engineering, IIT Roorkee, Roorkee

ABSTRACT

In this paper firstly the salient features of the procedure of obtaining the displacement of a retaining wall having

reinforced backfill under seismic condition have been presented. Both the translational and rotational modes of

vibration of reinforced wall have been considered. The mass of retaining wall is lumped at its centre of gravity.

The analysis indicated that the displacement depends on (i) frequency of excitation, (ii) acceleration, (iii) yield

displacement and (iv) amount of reinforcement. Variation of its displacement with respect to these four parameters

have been studied and presented herein. It was found that the displacement becomes very large when excitation

frequency becomes closer to either of the two natural frequencies. Increase in excitation acceleration increases

the displacement. The displacement of the wall decreases with increase in the amount of yield displacement and

amount of reinforcement at all frequencies.

Indian Geotechnical Conference – 2010, GEOtrendz

December 16–18, 2010

IGS Mumbai Chapter & IIT Bombay

1. INTRODUCTION

Rigid retaining walls are susceptible to failure if their

displacements under static and dynamic conditions are not

properly predicted. Therefore, in the design of retaining

walls, a displacement criterion becomes important. The effect

of displacement on the behaviour of retaining wall with

reinforced backfill under static condition was thoroughly

investigated by the authors and complete details are given

Saran (1998). The design of retaining walls based on

allowable displacement under dynamic conditions has gained

importance in recent years. There are very few methods

available in the published literature for computing

displacements of retaining walls during earthquake namely

Richard-Elms Model (1979) based on Newmark’s approach

(1965), Solution inpure translation (Nandakumaran, 1973),

Solution in pure rotation (Prakash et al. 1981), Nadim-

Whitman Model (1983) using finite element technique and

Reddy’s Model (1985) which incorporate both modes of

movement i.e. translation and rotation. However, no work

is available in literature to predict the displacement of rigid

retaining walls with reinforced backfill and subjected to

seismic condition.

In this paper an attempt is made to present a simplified

approach for this problem.

2. MATHEMATICAL MODEL AND ANALYSIS

Fig. 1(a) shows a section of a rigid retaining wall of height

H having reinforced backfill. The backfill properties are

given in terms of density (γ), angle of internal friction (φ)

and soil modulus (ηh). The reinforcement in the backfill is

provided at vertical spacing of ∆H. f* represents the

apparent coefficient of friction between the soil and

reinforcement.

Fig. 1: Proposed Mathematical Model of Wall for

Displacement Analysis Under Seismic Conditions

The backfill soil has been replaced by closely spaced

independent elastic springs by sliding elements (Fig. 1(a)).

Spring constants are computed using the concept of soil

modulus which depends on the type of soil. It varies linearly

with depth in sands and normally consolidated clays. If nh

is presents the soil modulus, spring constants at various

point of subdivision would be as under (Fig. 1):

214 S.K. Saran and M.N. Viladkar

( )2

1 61 hK hs ∆= η 1(a)

( ) ( )21 hiK hsi ∆−= η 1(b)

( ) ( )21361 hnK hsn ∆−= η 1(c)

where, Ksi in general, is spring constant at any point of

subdivision, i.

The stiffness of slidding elements (KR1

, KR2

,…. KRn

) will

depend on the material and configuration of reinforcement

and properties of backfill soil. This can be obtained by

performing pullout tests. The detailed procedure of

obtaining KR is given elsewhere (Saran, 1998). In the case

when backfill soil is sand (γ = 15.8 kN/m3, φ = 38.5o) and

Netlon geogrids (CE-121) as reinforcement the value of

KR for a sliding element at depth z from top of wall is

obtained as

z

zmkNK R

2.18.1

2000)/(

+= (2)

The above equation is valid when z d” 5.0 m. For z > 5.0 m,

the value of KR is obtained by putting z = 5.0 m in Eq. (2).

The proposed methodology has been developed is based on

the following assumptions:

(i) The earthquake motion is considered as an

equivalent sinusoidal motion with uniform peak

acceleration and the total displacement is equal

to the residual displacement per cycle multiplied

by the number of cycles.

(ii) Soil stiffness (or spring constants) for

displacement of wall towards the backfill and away

from the backfill are different.

(iii) Soil participating in vibration, base friction and

damping of soil are neglected.

To study the response characteristics of the system,

two cases are considered, one in which plastic deformations

does not take place (elastic system) and the other in which

plastic deformations takes place (plastic system).

Analysis of an Elastic System

Wall Moving Away from Backfill (Tension Condition)

The equations of motion for the proposed model (Fig. 1) of

the retaining wall can be written in general terms using D’

Alembert’s principle as follows:

Considering horizontal forces:

( ) ( ){ }[ ] h 1-i - h - H x n

1i

θ∆++ ∑=

sikxM

( ) ( ) tHKRi ωθ sinF 1-i - 2

H - h - H x 0

N

1i

=

∆∆

++∑=

(3)

Considering moments:

( ) ( ){ }[ ] ( ) ( ){ }

( ) ( ) ( ) ( ) 0 H 1-i - 2

H - h-H * 1-i -

2

H - h - H x

h 1-i - h-H * h 1-i - h - H x

N

1i

n

1i

=

∆∆

∆∆

++

∆∆++

∑

∑

=

=

θ

θθ

Hk

kJ

Ri

Si

(4)

where

J = M. r2, r = radius of gyration= A/I and a0=F

0M

The above equation of motion can be reduced to

x + ax = bθ + a0 sin ωt (5)

Also,

x 2

=+

r

bcθθ (6)

where, 11

M

kk

a

n

i

Ri

n

i

Si ∑∑==

+

=

( ) ( ){ } ( ) ( )

H 1-i - 2

H - h-H h 1-i - h-H

11

M

kk

b

N

i

Ri

n

i

Si ∑∑==

∆∆

+∆

=

( ) ( ){ } ( ) ( )

J

kk

c

N

i

Ri

n

i

Si ∑∑==

∆∆

+∆

= 1

2

1

2

H 1-i - 2

H - h-H h 1-i - h-H

The solution of equations (5) and (6) becomes:

( )tsin .

)(

b - -a

a

22

22

o ω

ωω

−

=

cr

x (7)

( ) ( )tsin .

b - .-c - -a

a

222

o ω

ωω

θ

=

b

r(8)

The displacement of the top of rigid retaining wall is

given by

( )θhHxxtop −+= (9)

Wall Moving Towards Backfill (Compression

Condition)

The ratio of stiffness on the compression and tension sides

is denoted by η. Hence, in the passive condition, the values

of a, b, and c changes and these are given by :

a= η (a)a

(10a)

b= η (b)a

(10b)

c= η (c)a

(10c)

where

(a)a, (b)

a and (c)

a are the values of a, b and c respectively in

tension condition. The solution for this condition is similar

to the tension condition as described above.

Analysis of a Plastic System (Tension condition)

Assume that Zy and θ

y are the yield displacements occurring

simultaneously in all the springs. Then the equations of

motion can be written as :

x + a Zy= bθ

y + a

0 sinωt (11)

y2 Z

=+r

bc yθθ (12)

Seismic Displacement of Retaining Wall with Reinforced Backfill 215

Integrating the above equations twice, one gets:

( )212

02

yy C .C tsin a

- 2

t Za - ++= tbx

ω

ωθ (13)

43

2

yy2C .C

2

t c - Z ++

= t

r

bθθ (14)

Let ‘te’ be the time after which displacement of the top of

wall (ytop

) is greater than the yield displacement (Yd) and

the system is in plastic state. Let xe, x

e, θ

e and θ

e, be the

values corresponding to time, te and can be calculated by

using the equations developed for the elastic system. The

following boundary conditions can be applied to evaluate

the constants of integration :

(i) t = te, x = x

e(15a)

(ii) t = te, x = x

e(15b)

(iii) t = te, θ =

e(15c)

(iv) t = tc, = θ

e(15d)

Therefore, we have

Zy = x

e and θ

y = θ

e(16)

( )ω

ωθ e0

01

t cos ataZbxC vvo +−+= (17)

( )ω

ω

ω

ωθ e0e

2

0

2

2

t cos..t -

tsin a

2

ataZbtxxC e

vvooo +−+−= (18)

eyy203 tc - Z

= θθ

r

bC (19)

e

2

e

yy204 t - 2

t c - Z e

r

bC θθθ

= (20)

3. DYNAMIC RESPONSE OF WALL

The philosophy of computation of displacement of wall (or

slip) is: (i) when the wall moves towards the backfill the

slip per cycle is very small much below the yield

displacement, soil remains in elastic state, displacement

after half cycle of loading (i.e. after time Tp/2, T

p being the

time period) became zero. (ii) When the wall moves away

from the backfill stiffness of soil being less displacement

may exceed yield displacement (Yd), the system comes in

plastic state giving slip after the next half cycle (i.e. after

time Tp).

For illustration a wall of 6 m height having trapezoidal

section (a’ =1.0 m, b’ = 2.5 m) having backfill soil (γ =

15.8 kN/m3, hη = 600 kN/m4, η= 2.0) is considered. Netlon

geogrids with f* = 0.4 are taken as reinforcements. Amount

of the reinforcement is varied by taking different number

of reinforcement layers (N). In a typical case variation of

slip per cycle with time is given in Fig. 2.

Variations of slip per cycle with excitation frequency

(ω), excitation acceleration (a0), yield displacement (Y

d)

and amount of reinforcement (N) are shown in Fig.s 3 to 6

respectively.

Fig. 2: Variation of Wall Displacement with Time

It can be seen from Fig. 3, that, slip per cycle becomes

very large when ω is close to either of the two natural

frequencies. For the values of operating frequency away

(approximately 1.25 times) from the natural frequencies

the slip per cycle becomes small.

Fig. 3: Variation of Slip Per Cycle of Wall with Frequency

Fig. 4 shows that the magnitude of slip per cycle

increases with the increase in a0 at all frequencies. Further

the increase in slip per cycle is more rapid for the excitation

frequencies closer to natural frequencies. It can be observed

from Fig. 5, that the slip per cycle decreases with the

increase in yield displacement at all frequencies. It is due

to the fact that the backfill soil comes in plastic condition

for less duration at higher displacement.

216 S.K. Saran and M.N. Viladkar

Fig. 4: Variation of Slip Per Cycle of Wall with

Excitation Acceleration

Fig. 5: Variation of Slip Per Cycle of Wall with Yield

Displacement for Different Excitation Frequencies

Fig. 6 shows that the slip per cycle decreases with

the increase in amount of reinforcement i.e., number of

reinforcement layers, the decrease in slip per cycle is at

the faster rate at the beginning.

Fig. 6: Variation of Slip Per Cycle of Wall with Number

of Reinforcement Layers

4. CONCLUSION

The displacement of a rigid retaining wall depends on (i)

acceleration, frequency and number of cycles of the dynamic

force, and (ii) yield displacement, and (iii) amount of

reinforcement. The effect of frequency is largely enhanced

when it coincides with either of the natural frequency. The

natural frequencies of the system can be changed by

providing adequate reinforcement such that the excitation

frequency becomes away from either of the natural

frequency. Thus the slip of wall may be brought within

limit by this technique.

ACKNOWLEDGEMENT

The authors are thankful to the Director, Central Building

Research Institute, Roorkee and Director, Indian Institute

of Technology, Roorkee for their encouragement and kind

permission to publish this paper.

REFERENCES

Nadim, F. and Whitman, R.V. (1983). Seismically Induced

Movement of Retaining Walls, Journal of Geotechnical

and Geoenvironmental Engineering, ASCE, 109(7),

915-931.

Nandakumaran, P. (1973). Behaviour of Retaining Walls

under Dynamic Loads, Doctoral Thesis, Indian Institute

of Technology Roorkee, Roorkee, India.

Newmark, N.M. (1965). Effect of Earthquakes on Dams

and Embankments, Geotechnique, 15(2), 129-160.

Prakash, S., Puri, V.K. and Khandoker, J.U (1981). Rocking

Displacements of Rigid Retaining Walls During

Earthquake, Companion Paper to Conference on Recent

Advances in Geotechnical Earthquake Engineering and

Soil Dynamics, 3, St. Louis, MO, USA, April-May.

Reddy, R.K., Saran, S. and Viladkar, M.N. (1985),

Prediction of Displacement of Retaining Walls under

Dynamic Conditions, Bulletin of Indian Society of

Earthquake Technology, Paper No. 239, 22(3).

Richard, R., Jr. and Elms, D.G. (1979), “Seismic Behaviour

of Gravity Retaining Walls, Journal of Geotechnical

Engineering Division, ASCE, 105(GT 4), 449-464.

Saran, S.K. (1998), Seismic Earth Pressures and

Displacement Analysis of Rigid Retaining Walls having

Reinforced Sand Backfill, Doctoral Thesis, Indian

Institute of Technology Roorkee, Roorkee, India.