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An Analytical Solution to Seismic Response of a Cantilever Retaining Wall

With Generalized Backfilled Soil

Indrajit Chowdhury

Head of the Department of Civil and Structural Engineering Petrofac International Ltd., Sharjah, U.A.E.

Shambhu P. Dasgupta

Professor of Civil Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India

e-mail: dasgupta@civil.iitkgp.ernet.in

ABSTRACT It is apparent that present day retaining walls are far too flexible and the basic assumption deployed by previous researchers that the wall is infinitely stiff- cannot be justified. Most of the available solutions are a variation of M-O method in one form or the other, trying to incorporate the soil parameters like c-φ soil, or using logarithmic spiral curves etc within the M-O frame work. However, the solutions are valid only for cohesionless soils and cannot be used for c-φ soils, partially saturated back fill, effect of overburden to name some of the often faced conditions in reality. It also does not take into cognizance the effect of vertical acceleration that is often considered for analysis of these walls under Coulomb type of failure of the backfill. A compreshensive analytical solution based on modal analysis is proposed herein that takes into account the effect of time period of the wall, a consideration that has been mostly ignored by previous researchers. Present paper is thus an attempt to re-evaluate this long standing problem and seek solution to many of the open issues cited above.

KEYWORDS: Acceleration, active and passive pressure, back fill, cantilever retaining wall, cohesion, dynamic amplitude, earthquake, failure surface, modal analysis.

INTRODUCTION Retaining walls play an important role in a post earthquake scenario to retain the backfilled

soil in industrial and infrastructure projects. A number of researchers have worked on seismic response of retaining walls, like Mononobe (1929), Okabe (1924), Seed & Whitman (1970) and

Vol. 16 [2011], Bund. C 297 Whitman et al. (1990, 1991), to name some of the pioneering few. However all these researches are based on the assumption that the wall is gravity type where it has an extremely high stiffness, and that the seismic excitation is restricted to soil part only. With the advances of reinforced concrete technology, retaining walls have undergone a significant change in character, and it would be most improbable that a gravity wall will be deployed for retaining back fills even for heavy bridge girders.

Figure 1: Gravity retaining wall and reinforced concrete retaining wall used to retain soil.

Shown in Figure 1 are the cross sections of typical gravity retaining walls used earlier, and RCC retaining walls that are used presently. It is apparent that present day walls are far too flexible and the basic assumption employed by previous researchers that the wall is infinitely stiff- cannot be justified for these retaining walls. A pseudo static approach considered till date for determination of dynamic pressure under seismic load [usually based on Mononobe & Okabe’s (M-O) method] may not be justified. It is apparent that present day constructed walls do have a finite stiffness vis-à-vis time period that will influence the dynamic response of walls under earthquake disturbances.

The M-O method that was considered for a cohesionless dry backfill (c = 0) also has been examined by a number of researchers like, Das & Pur i(1996), Ghosh et al. (2010, 2008, 2007), Saran et al. (1968, 2003), Choudhury et al. (2002, 2004, 2006) to name some of the works. However, most of them are a variation of M-O method in one form or the other, trying to incorporate other soil parameters like c-φ soil, or using logarithmic spiral curves etc within the M-O frame work.

In the recent past, Chowdhury & Dasgupta (2002) derived an approximate solution for such flexible retaining wall based on improved Rayleigh-Ritz technique and showed that results are in variation to pressures derived from M-O method. However, the solution is valid only for cohesionless soil (c = 0) and cannot be used for c-φ soils, partially saturated back fill, effect of overburden to name some of the often faced conditions in reality. It also does not take into cognizance the effect of vertical acceleration that is often considered for analysis of these walls under Coulomb type of failure of the backfill.

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Research carried out in USA by Ostadan & White (1997), Ostadan (2004) has also shown that M-O based methods significantly under predict the dynamic pressure under seismic loads, to the extent that Nuclear Regulatory Board of USA has now stopped using any of the M-O based methods for determining earth pressure for any of their structures.

Present paper is thus an attempt to re-evaluate this long standing problem and seek solution to many of the open issues cited above.

PROPOSED METHOD To start with we take the simplest of the case as shown in Figure 2. Shown herein is a

cantilever retaining wall with dry sandy backfill and the ground has no inclination like in Fig. 1.

Figure 2: A Cantilever retaining wall with dry cohesion less backfill (c=0)

It is to be noted that the same can also be derived from a more generalized soil condition but has been considered first for brevity and also to use it as a benchmark for more generalized cases that will be taken up subsequently.

While performing the analysis it is assumed here that 1. The soil profile under active case is at incipient failure when the failure line makes angle α

= tan (450 +φ/2) as shown in the above figure. 2. Since soil profile is already under failed condition under static load, it will not induce any

stiffness to the overall dynamic response but will only contribute to the inertial effect. 3. Since the cantilever wall is relatively thin, mass contribution of the wall itself may be

Ignored compared to that of the soil. The wall thus contributes only to stiffness of the overall soil-structure system

4. The retaining wall is fixed at the base and foundation compliance has been ignored for the present analysis.

It will be observed that the assumptions made above are identical to what Mononobe or Steedman & Zeng (1990) have assumed in their analysis. Based on above assumptions the analysis is carried out as elaborated hereunder.

Vol. 16 [2011], Bund. C 299

Dynamic response of dry cohesionless backfill (c = 0)

Considering φ as the internal angle of friction, the active and passive coefficient of pressures KA and KP may be expressed as

φφ

sin1

sin1

+−=AK and

φφ

sin1

sin1

−+=PK (1)

The static pressures acting on the wall can be expressed as

zKp sAaz ..γ= (2)

zKp sPpz ..γ= (3)

For the wall considered as a cantilever beam fixed at the base slab, the differential equation of static equilibrium under active soil condition can be expressed as

zK

dz

udEI sA ..

4

4

γ= (4)

where u is displacement of the retaining wall, E = Young’s Modulus and I is moment of inertia of

the R.C.C. wall considered [I = ).(12/1 3tB × , here t is the thickness of the wall; can be taken as an average thickness for variation between top and bottom thickness of the wall], B = width of the wall usually considered as 1.0 m as the analysis is usually carried out per meter width of the wall.

On successive integration of equation (4) we have

1

2

3

3

2

.C

zK

dz

udEI sA +=

γ

(5)

21

3

2

2

6

.CzC

zK

dz

udEI sA ++= γ

(6)

32

2

1

4

224

.CzC

zC

zK

dz

duEI sA +++= γ

(7)

43

2

2

3

1

5

26120

.CzC

zC

zC

zKEIu sA ++++= γ (8)

For the given wall (Figure 2) we consider the boundary conditions

1) At z=0 = 03

3

dz

ud C1=0 8(a)

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2) At z=0 00 22

2

== Cdz

ud 8(b)

3) At z=H 24

..0

4

3

HKC

dz

du sA γ−== 8(c)

4) At z=H 30

..0

5

4

HKCu sA γ== 8(d)

Thus equation (8) can be expressed as

3024

.

120

. 545 HKzHKzKEIu sAsAsA γγγ +−= (9)

Equation (9) can be further expressed in generalized form as

+−= 1

4

5

430

.. 55 ξξγEI

HKu sA (10)

where Hz /=ξ a non-dimensional term that varying between 0-1.

Static deflection of wall under the given condition can be expressed as

EI

HKu sA

static 30

.. 5γ= at 0=ξ (11)

Natural period of the wall is

g

uT staticπ2= (12)

Substituting equation (11) in (12) and considering 31/12.(1 t ),× we have

gEt

HKT sA

A 3

5

97.3γ= (13)

For modal analysis, maximum amplitude (Sd) is expressed as (Clough 1984)

2d aS S /= ω (14)

Vol. 16 [2011], Bund. C 301 where Sa= Spectral acceleration and T/2πω = , natural frequency of the wall.

In terms of code equation (14) can be expressed

2d aS S /= κβ ω (15)

where =κ Modal mass participation factor and is expressed as n n 2

i i i ii 1 i 1

m / m , = =

ϕ ϕ β = A code

factor expressed as ZI/2R where Z= Zone factor I = Importance factor and R = Response reduction factor.

Thus based on equation (15) the dynamic amplitude of the wall can be expressed as

)(

42

2ξ

πκβ fT

Su a= (16)

where 14

5

4)(

5

+−= ξξξf

Equation (16) can be finally expressed as

+−

= 1

4

5

430

55 ξξγκβg

S

EI

HKu asA (17)

Considering M=EI d2u/dz2 and V=EI d3u/dz3 we have,

[ ]33

6ξγκβξ

=g

SHKM asA (18)

Similarly

=g

SzKV asA

z 2

2γκβ (20)

Equations (17), (19) and (20) are exact and give the dynamic displacement, moment and shear for a cantilever retaining wall under earthquake force in fundamental mode for cohesion less dry back fill.

The modal participation factor κ can be expressed in this case as

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1 1n n 2 2 2 2i i i i s s

i 1 i 1 0 0m / m H f ( )d / H f ( ) d

= =

κ = ϕ ϕ = γ ξ ξ ξ γ ξ ξ ξ

(21)

25 51 1

0 0

5 51 d / 1 d

4 4 4 4

ξ ξ ξ ξ κ = ξ − + ξ ξ − + ξ

(22)

Equation (22) may look formidable for calculation (especially for more complicated cases derived later) but can be easily solved numerically. This will be further elaborated by an example in Appendix 1.

Effect of vertical acceleration Sv

From equation (4) we have seen that

zK

dz

udEIp sA ..

4

4

γ== (23)

Thus for the present case dynamic pressure on wall is expressed as

444

4

ξd

d

H

EI

dz

udEIpdyn ==

+−

1

4

5

430

55 ξξγκβg

S

EI

HK asA

(24)

z

g

SKp a

sAdyn

= γκβ

(25)

Now if Sv is the vertical acceleration corresponding to time period TA then the dynamic pressure in vertical direction can be expressed as

z

g

Sp V

sdynV

±= κβγ (26)

Vol. 16 [2011], Bund. C 303

In horizontal direction effect of this pressure can be expressed as

z

g

SKp V

sAdynH

±= γκβ (27)

Thus total dynamic pressure considering the vertical component of acceleration can be expressed as

z

g

SKz

g

SKp v

sAa

sAdyn

±

= γκβγκβ (28)

For maximum pressure we must take the positive sign that gives

z

g

SKz

g

SKp v

sAa

sAdyn

+

= γκβγκβ (29)

As per IS-1893(2002) considering Sv =Sa/2, equation (29) can be expressed as

z

g

SKp a

sAdyn

=2

3γκβ (30)

Considering the effect of vertical acceleration, the dynamic displacement, moment and shear can be expressed as

+−

= 1

4

5

420

55 ξξγκβg

S

EI

HKu asA (31)

=

g

SzKM asA

z 4

3γκβ (32)

=g

SzKV asA

z 4

3 2γκβ (33)

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Equations (31) through (33) show that the displacement, moments and shears get amplified by 50% when effect of vertical acceleration is considered and should not be ignored.

Soil inclined at an angle i with the vertical (Figure 3)

In this case expressions presented vide equations (31), (32) and (33) remains valid except that in this case the active and passive earth pressures are expressed as

φφ

22

22

coscoscos

coscoscoscos

−+−−×=

ii

iiiK A (34)

φφ

22

22

coscoscos

coscoscoscos

−−−+×=

ii

iiiKP (35)

i

Figure 3: Retaining wall with inclined backfill at an angle i

Dynamic response of wall with c-φ backfill

For a general c-φ soil the active earth pressure is expressed [Murthy (1984)] as

ααγ cot2cot2 czp sa −= where, α= 45+φ/2

Substituting this in equation (4) we have

ααγ cot2cot.. 2

4

4

czdz

udEI s −=

(36)

Vol. 16 [2011], Bund. C 305

Proceeding in identical fashion as explained earlier and imposing the boundary conditions as stated in equations 8(a) through 8(d) we have C1=C2=0 and

αγα 243

3 cot243

cot HcHC s−= (37)

4

cotcot

30

42

5

4

ααγ cHHC s −=

(38)

This gives

ααγαγαααγcot

4cot

30cot

243

cotcot

12cot

120

42

52

4342

5 cHHzHzcHczzEIu sss −+−+−= (39)

Equation (39) after some simple algebraic manipulation can be finally expressed as

−+−+−= ψξψξψξξαγ

3

4

31

4

5

430

cot 4525

EI

Hu s (40)

where, αψ tan4

15

H

H c= , a dimensionless parameter, and s

c

cH

γ2= the free standing height of

soil.

Thus ustatic at ξ=0 is expressed as

[ ]ψαγ −= 130

cot25

EI

Hu s

static (41)

Substituting above in equation (12) we have

( )ψγα −= 1cot97.33

5

gEt

HT s

A (42)

Thus for modal analysis the dynamic amplitude is expressed as

( )

−+−+−−

= ψξψξψξξψαγβκ

3

4

31

4

5

41

30

cot..

4525

g

S

EI

Hu as (43)

The dynamic moment and shear can be expressed as

- 306 -

( )

−−

=

2

tan

61cot

232 αψακβγ zHz

g

SM ca

sz (44)

( )

−−

= αψακβγ tan

21cot

22 zH

z

g

SV c

asz (45)

Considering the effect of vertical acceleration Sv = Sa/2, we have

( )

−+−+−−

= ψξψξψξξψαγβκ

3

4

31

4

5

41

20

cot..

4525

g

S

EI

Hu as (46)

( )

−−

=

4

tan3

41cot

232 αψακβγ zHz

g

SM ca

sz (47)

and

( )

−−

= αψακβγ tan

2

3

4

31cot

22 zH

z

g

SV c

asz (49)

The modal mass participation may be expressed as

25 4 5 41 1

0 0

5 4 5 41 d / 1 d

4 4 3 3 4 4 3 3

ξ ξ ξ ξ ξ ξ ξ ξ κ = ξ − + − ψ + ψ − ψ ξ ξ − + − ψ + ψ − ψ ξ (50)

The above derivation is for a general soil that has finite value of c and φ.

When the soil is purely cohesion less i.e. c = 0, 0→ψ equations (46) to (50) degenerates to equations (31) to (33) and equation (22). This shows the correctness of the derivation of the above expressions.

In equation (42) it will be observed that for limiting value of 1→ψ , time period tends to zero

and for 1,ψ > the solution collapses. The physical significance of this is as explained hereunder.

The above solution is valid when the value of c is low so that the soil is adhering to the wall. For high of c ( 1>ψ ), the negative pressure will be sufficiently high to develop tension cracks and

loose contact over the wall for a height (2c/γs)tanα. In such case for evaluation of static pressure, it is usual practice to neglect the cracked portion and consider the wall to be partially loaded by the positive pressure to a height H-2c/γstanα from the base of the wall.

This is a special case and requires separate treatment. This has been dealt with in section (2.8) of this paper.

Vol. 16 [2011], Bund. C 307

For passive case we have

ααγ tan2tan.. 24

4

czdz

udEI s += (51)

As before, after successive integration and imposing the boundary conditions as cited in equations 8(a) through 8(d), we have C1=0 ,C2=0.

ααγtan

3tan

24

32

4

3

cHHC s −−= (52)

ααγtan

4tan

30

42

5

4

cHHC s += (53)

Imposing the above integration constants we have

ααγαγαααγtan

4tan

30tan

243

tantan

12tan

120

42

52

4342

5 cHHzHzcHczzEIu sss ++−−+=

(54)

Equation (54) on simplification gives

+−++−= ppp

s

EI

Hu ψξψξψξξαγ

3

4

31

4

5

430

tan 4525

(55)

Here αψ cot4

15

H

Hcp = , a dimensionless a parameter and Hc is as defined earlier the free

standing height of cohesive soil.

Based on the above

[ ]ps

static EI

Hu ψαγ += 1

30

tan25

(56)

Substituting this in equation (11) we finally have the time period for passive case as

( )ps

P gEt

HT ψγα += 1tan97.3

3

5

(57)

Thus based on modal analysis dynamic amplitude, moments and shears are expressed as

- 308 -

( )

+−++−+

= pppp

asp g

S

EI

Hu ψξψξψξξψαγβκ

3

4

31

4

5

41

30

tan..

4525

(58)

( )

++

=

2

cot

61tan

232 αψακβγ zHz

g

SM c

pa

spz (59)

( )

++

= αψακβγ cot

21tan

22 zH

z

g

SV cp

aspz (60)

Considering effect of vertical acceleration we have

( )

+−++−+

= pppp

asp g

S

EI

Hu ψξψξψξξψαγβκ

3

4

31

4

5

41

20

tan..

4525

(61)

( )

++

=

4

cot3

41tan

232 αψακβγ zHz

g

SM c

pa

spz (62)

( )

++

= αψακβγ cot

2

3

4

31tan

22 zH

z

g

SV cp

aspz (63)

The modal mass participation factor is expressed as

+−++−

+−++−

=1

0

245

1

0

45

3

4

31

4

5

4

3

4

31

4

5

4

ξψξψξψξξξ

ξψξψξψξξξκ

d

d

ppp

ppp

(63a)

Dynamic response of wall with pure intact clay (φ=0) as backfill

This case can be easily derived from the previous general case considering cotα and tanα =1 which gives

( )cs

A gEt

HT ψγ

−= 197.33

5

(64)

( )

−+−+−−

= cccc

as

g

S

EI

Hu ψξψξψξξψγβκ

3

4

31

4

5

41

20..

455

(65)

Vol. 16 [2011], Bund. C 309

( )

−−

=

4

3

41

23 zHz

g

SM c

ca

sz ψκβγ (66)

( )

−−

= zH

z

g

SV cc

asz 2

3

4

31

2

ψκβγ (67)

where H

Hcc 4

15=ψ .

Similarly for passive case considering the vertical acceleration effect can be expressed as

( )pcs

P gEt

HT ψγ

+= 197.33

5

where, cpc ψψ = (68)

( )

+−++−+

= pcpcpcpc

asp g

S

EI

Hu ψξψξψξξψγβκ

3

4

31

4

5

41

20..

455

(69)

( )

++

=

4

3

41

23 zHz

g

SM c

pca

spz ψκβγ (70)

( )

++

= zH

z

g

SV cpc

aspz 2

3

4

31

2

ψκβγ (71)

Dynamic response of wall with c-φ backfill and overburden surcharge q

This type of problem is often faced by engineers in dense urban region and remains a serious problem under earthquake. No solution exists till date for this problem and engineers have to often resort to Finite Element Analysis (FEM) to arrive at a workable result.

Shown in Figure 4 is a retaining wall with c-φ soil as the backfill, and also having a surcharge q at the top. While it is possible to derive the combined pressure due to this inclusion of overburden and then perform successive integration as described earlier, makes the analysis tedious. Considering, we are performing modal analysis we can argue that taken the problem is linear, superimposition of displacement is permissible.

- 310 -

q kN/m2

Figure 4: A Cantilever retaining wall with c-φ backfill and surcharge load q.

Thus in this case we derive the static displacement of the wall for the surcharge load q only and finally add it to equation (40) to arrive at the final static displacement.

Hence, we start with the expression

α24

4

cot.qdz

udEI = (72)

On successive integration as explained earlier and imposing the boundary conditions as cited in equation 8(a) through 8(d) we have C1=0 , C2=0

6

cot 32

3

HqC

α−= (73)

8

cot 42

4

HqC

α−=

Substituting these values we finally have

8

cot

6

cot

24

cot 423224 HqzHqqzEIu

ααα +−= (74)

Equation (74) can be finally written as

+−= 1

3

4

38

cot 442 ξξαEI

Hqu (75)

Equation (75) will be now added to equation (40) to arrive at the total displacement of the system.

Thus total static displacement may now be expressed as

Vol. 16 [2011], Bund. C 311

+−+

−+−+−= 1

3

4

38

cot

3

4

31

4

5

430

cot 4424525 ξξαψξψξψξξαγEI

Hq

EI

Hu s (76)

Equation (76) can be further simplified to

+−+−+−+−= ηηξηξψξψξψξξαγ

3

4

33

4

31

4

5

430

cot 44525

EI

Hu s (77)

In equation (77), as mentioned before, αψ tan4

15

H

H c= and H

q

sγη

4

15= are both

dimensionless parameters.

Thus the maximum static deflection is obtained ( 0=ξ ) as

[ ]ηψαγ+−= 1

30

cot 25

EI

Hu s

static (78)

Substituting equation (78) in equation (12) we finally have

( )ηψγα +−= 1cot97.33

5

gEt

HT s

A

Thus for modal analysis the dynamic amplitude, moments and shears can be expressed as

( )

+−+−+−+−+−

= ηηξηξψξψξψξξηψακβγ

3

4

33

4

31

4

5

41

30

cot 44525

g

S

EI

Hu as

(79)

( )

−−+−

=

sc

asz

qH

zz

g

SM

γαηψακβγ tan

261cot

232 (80)

( )

−−+−

=

sc

asz

qHz

z

g

SV

γαηψακβγ tan

21cot

22 (81)

Equations (79) through (81) gives the displacement, moment and shear under seismic loading for the most general condition of soil.

When there is no overburden i.e., 0→η , the formulas converges to equations (46) to (48) and

represents the case of c-φ soil only. When again, 0→ψ , the equations converges to the case of pure cohesion less soil(c=0).

- 312 -

Finally an interesting case is observed, when, 0→ψ , 0→η and even φ=0 (i.e. )1cot 2 →α . The values obtained in equations (79), (80), (81) are not zero but finite i.e. it converges to a hydrostatic pressure. Thus, if we take density of soil in this particular case (γsat-γw) it reflects a case when the soil is in a liquefied state

The modal mass participation factor in this case is expressed as

+−+−+−+−

+−+−+−+−

=1

0

2445

1

0

445

3

4

33

4

31

4

5

4

3

4

33

4

31

4

5

4

ξηηξηξψξψξψξξξ

ξηηξηξψξψξψξξξκ

d

d

(82)

Considering the effect vertical acceleration we have

( )

+−+−+−+−+−

= ηηξηξψξψξψξξηψακβγ

3

4

33

4

31

4

5

41

20

cot 44525

g

S

EI

Hu as

(83)

( )

−−+−

=

sc

asz

qH

zz

g

SM

γαηψακβγ tan

4

3

41cot

232 (84)

( )

−−+−

=

sc

asz

qHz

z

g

SV

γαηψακβγ tan

2

3

4

31cot

22 (85)

Dynamic response of wall with c-φ backfill partially submerged below water

Shown in Figure 5 is a retaining wall having partially submerged c-φ soil as backfill. The soil is partially submerged for a height H2. In this case considering the complex nature of the pressure diagram it would become difficult to perform the integration with appropriate boundaries.

Vol. 16 [2011], Bund. C 313

Figure 5: A Cantilever retaining wall with c-φ backfill partially submerged.

As such we approach the problem differently for this particular case. Considering the first step in the process is to determine the static deflection and this is obtained based on the loading on the retaining wall, the net horizontal loading on the wall can be expressed as [Murthy (1984)].

( )[ ]s

swsatsA

cHHHcHHP

γαγαγγγα

22

212

22

12 2

cot.cot2cot2

1 ++−−+= (86)

where =sγ Dry unit weight of soil of height H1; =satγ Saturated unit weight of soil of height

H2; =wγ Unit weight of water.

Now if we consider an equivalent dry back fill of density γs which imposes the same load on the wall the displacement of the wall will be same as that as would be induced by load as expressed in equation (86).

Thus considering

( )[ ]s

swsatssAE

cHHHcHHHK

γαγαγγγαγ

22

212

22

122 2

cot.cot2cot2

1

2

1 ++−−+=

We have,

( )[ ]( ) ( ) 2

212

2

221

21

212

21

22

212

)(

4

)(

.2

.

cot4cot

HH

c

HH

HH

HH

c

HH

HHK

s

s

ss

wsatsAE +

++

++

−+

−+=

γγ

γα

γγγγα (87)

Here KAE = An equivalent coefficient of active earth pressure when considering a pressure diagram of zKp sAEaz ..γ= over the height H will give same deflection as that produced by PA in

equation (86).

Thus for the present case the problem now gets simplified considerably when we have

- 314 -

gEt

HKT sAE

A 3

5

97.3γ

= (88)

+−

= 1

4

5

430

55 ξξγκβg

S

EI

HKu asAE (89)

=

g

SzKM asAE

z 6

3γκβ (90)

=

g

SzKV asAE

z 2

2γκβ (91)

Considering vertical acceleration we have

+−

= 1

4

5

420

55 ξξγκβg

S

EI

HKu asAE (92)

=

g

SzKM asAE

z 4

3γκβ (93)

=

g

SzKV asAE

z 4

3 2γκβ (94)

Special case of c-φ soil when it loose contact for some portion at top

For this case, the maximum load on the wall may be expressed as Murthy(1984) as

s

sa

ccH

HP

γααγ 2

22 2

cot2cot2

+−= (95)

Based on the above, for a partially loaded beam, static deflection can be expressed based on fundamentals of beam theory as

Vol. 16 [2011], Bund. C 315

−

+−

=α

γγ

αα

γ

tan2

tan2

4

51

15

)tan2

( 3

ss

sa

staticc

H

c

EI

cHP

u (96)

Here Pa is as expressed in equation (95).

Thus time period, based on equation (12) is expressed as

−

+

−

=α

γγ

αα

γπ

tan2

tan2

4

51

15

tan2

2

3

ss

sA

Ac

H

c

EIg

cHP

T (97)

Maximum dynamic amplitude, Moments and Shears along the depth of the wall are expressed as

−

+−

=

αγ

γ

αα

γκβtan

2

tan2

4

51

15

)tan2

( 3

s

sa

a

cHs

c

EI

cHP

g

Su (98)

2

3

tan2

3

−

=

αγ

κβ

s

aa

z

cH

zP

g

SM

(99)

2

2

tan2

−

=

αγ

κβ

z

aa

zc

H

zP

g

SV

(100)

It is to be noted that in this case z =0 where pressure is zero that is, at a depth (H-2c/γs tanα), from the top of the wall. The modal mass participation can be taken as 2.3 ( for justification of this value refer to the section of Results and Discussion).

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Damping of soil and that of the wall

For the wall considering it as RCC we can consider 5% material damping as is the usual practice.

During earthquake the failed soil wedge ABD (vide Figure 1) will move to and fro during vibration. When the soil body moves towards the wall will generate an active earth pressure and a passive pressure when moving in the opposite direction. It is evident that the soil body during its motion will generate a friction force along slope BD with the soil below the failed surface, and this friction force will oppose the motion and generate a damping that will attenuate the excitation.

The damping force can be approximately estimated as follows.

Considering the free body diagram of the failed wedge ABD the friction acting along the slope BD may be expressed as

)cos(sin αμα −=WFR (101)

where FR= Friction force along the surface BD that resists the motion; W= Weight of the soil body ABD (W=m.g); 2/45 φα += for active case and =μ Internal angle of friction of soil @ φtan .

Considering the resistive friction force FR as the damping force equation (101) can be expressed as

)cos(sinv. αμα −= WC (102)

in which, C= damping of the system, velocity v = Sa/ω where Sa is the seismic acceleration and ω

the natural frequency @ mk / .

Equation (102) on simple algebraic manipulation can be expressed as

−=

g

SC

C

ac 2

cossin αμα (103)

Here Cc= Critical damping =2 mk / .

Considering

cC

C=ζ the damping ratio we have

−=

g

Sa

A

2

cossin αμαζ (104)

Vol. 16 [2011], Bund. C 317 The above gives the damping ratio of the soil in active case. For passive case

+

=

g

Sa

ppp

2

cossin αμαζ , where 2/45 φα −=p

(105)

For a conservative estimate we should consider ζ to be minimum- but having a finite rational value. This is valid either when the numerator in equation (104) and (105) is the minimum or the denominator is the maximum.

For numerator to be minimum, it must be zero which gives φ = 900 which is impossible to achieve. Thus the condition is denominator is maximum. In other words Sa/g is to be the maximum.

For instance maximum value of Sa/g as per IS 1893(2002) is 2.5. Applying this value we have

5

cossin αμαζ −=A (106)

For sandy soil φ value usually varies from 15 degree for very loose sand to 40 degree for very dense sand. Considering the above values, variation in damping ratio, for active and passive cases

are shown in Figure 6.

Figure 6: Variation of damping ratio of soil with friction angle of soil

Comparison of damping ratio for active and passive case

00.050.1

0.150.2

0.25

15 20 25 30 35 40

phi

Dam

ping

ratio Damping ratio

active

Damping ratiopassive

- 318 -

It will be observed that variation of damping ratio with respect to friction angle φ is not widely varying thus an estimated value of 15% in active case and 20% for passive case would cater for soil with almost all levels of in-situ compaction. The damping ratio values also looks quite reasonable and matches the data that are usually considered from experience for practical seismic design of such walls based on FEM.

For c-φ soil the damping ratios can be expressed as

5

coscossin

W

eccH

A

ααμαζ

−−= (107)

and

5

coscossin

W

eccH ppp

p

ααμα

ζ++

=(108)

For the general c-φ soil (Fig. 1) W=0.5γsH2cotα and for c−φ soil with overburden W =

0.5γsH2cotα+qcotα. Replace α by αp vide equation (105) for passive case.

The 5% material damping of wall may be added to above arrive at the design damping ratio.

RESULTS AND DISCUSSIONS To evaluate how the procedure works a 6.0 m high retaining wall with the following soil

properties is analyzed under earthquake force.

Height of wall = 6.0 m; Top thickness = 0.25m; Bottom thickness=0.4m;

Grade of concrete =M30;

Foundation width =4.0m; Foundation thickness=0.6m;

Unit weight of backfill = 22 kN/m3; Angle of internal friction = 28o; Cohesive strength (c) = 10 kN/m2; Overburden (q) = 50 kN/m2; Earthquake zone = Zone IV as per IS-1893.

Damping of soil considered (average) =15%.

The results are worked out for following cases: 1. When the soil is cohesion less with no overburden 2. When the soil is pure clay with no overburden 3. When both cohesion(c) and friction(φ) is present with no overburden 4. General c-φ soil with overburden(q) 5. φ soil with overburden 6. c soil with overburden 7. Soil liquefied under earthquake force(c & φ both considered zero) with overburden

Vol. 16 [2011], Bund. C 319

The basic dynamic parameters that affect response of the cantilever retaining wall under earthquake force and values of maximum bending moment and shear force at base of wall for different type of soil condition as mentioned above are shown in Table-1.

Variation of Bending Moment and Shear force along the depth of wall for various type of soil are shown in Figures 7 and 8.

The combine static plus dynamic pressure by the proposed method and that by M-O method for sandy soil is shown in Figure 9.

Table 1: Analytical Results of the retaining wall by proposed method.

Soil Type Time period

Sa/g Modal Mass Participation

factor(κ)

Maximum Moment (kN.m)

Maximum Shear (kN)

Sandy Soil 0.391 1.75 2.28 54.8 27.4 Clayey Soil 0.43 1.75 2.23 34.9 22.33

c-φ soil 0.09 1.645 1.98 0.6 0.607 c-φ soil with q 0.474 1.75 2.29 112.23 50.96 φ soil with q 0.608 1.566 2.31 256.44 105.5 c soil with q 0.885 1.075 2.31 293.15 126.8

Liquefied soil with q 0.91 1.046 2.31 556.7 215.6 Liquefied soil without q 0.48 1.75 2.28 93.73 46.86

Table 1 depicts the values of moments and shears without the effect of vertical acceleration. If vertical acceleration is taken into cognizance, moments an shears shown above are to be multiplied by a factor 1.5.

Figure 7: Variation of Bending Moment along the depth of the wall.

Comparison of Bending Moment

-100.00

0.00

100.00

200.00

300.00

400.00

500.00

600.00

0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6

Depth(m)

Mom

ent(k

N.m

)

c-phi soil

phi soil

c-soil

c-phi+overburdenc+overburden

phi+overburden

Liquified soil+q

- 320 -

Figure 8: Variation of Shear force along the depth of the wall.

Figures 7 and 8 show the variation of moments and shears for various type of soils along the depth of the wall. It is observed surcharge load in proximity of the wall heavily influences the dynamic response and has an amplifying effect.Thus while designing the retaining walls engineer should carefully consider its effect on the overall dynamic response.

The most critical case is when the soil undergoes liquefaction including overburden.Though due to liquefaction the overburden structure impinging any surcharge load might collapse, but could generate an instantaneous case when the impinging overburden shoots up the moment and shear significantly on the wall that could render either a structural failure of the wall or worse induuce an instabilty by either topling or sliding of the wall – when the effect of failure can be far more damaging.

For the case of sandy soil (c=0) with no overburden, static plus dynamic pressure by the proposed method is compared with M-O method ( static plus dynamic) vide Figure 9 and as predicted at the outset, it is ovserved that conventional M-O method significantly under predicts the dynamic response.

Comparison of Shear force

-50

0

50

100

150

200

250

0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6

Depth(m)

She

ar fo

rce(

kN)

c-phi soil

phi soil

c-soil

c-phi +overburden

c+overburden

phi+overburdenLiquified soil+q

Vol. 16 [2011], Bund. C 321

Figure 9: Variation of Dynamic pressure by proposed method and M-O

method for pure sandy soil (c=0) with no overburden.

Finally the modal mass particpation factor (κ), which is indpendent of soil property is found to be almost invariant for all types soil( varying from a value of 2.28 to 2.3 for all cases except c−φ soil adhering to the wall).Thus from practical design point of view a κ= 2.3 would be a most appropriate value for all cases.

CONCLUSION AND REMARKS A comprehensive analytical solution based on modal analysis is proposed herein that takes into

cognizance effect of time period of the wall, a consideration that has been mostly ignored by previous researchers. It also gives solution to almost all types of soil and loading condtions- that may be expected in a real field design, including liquefaction whose effect on the wall surely needs more research.

Finally a retrospective comment on the work.

The solution is fundamental and almost on the brink of being trivial in approach.

- 322 -

REFERENCES 1. Choudhury D & Subba Rao K.S. (2002) "Displacement - Related Active Earth Pressure",

International Conference on Advances in Civil Engineering (ACE-2002), January 3 - 5, 2002, IIT Kharagpur, India, Vol.2., pp. 1038-1046.

2. Choudhury D, Sitharam T.G.& Subba Rao K.S. (2004) "Seismic design of earth retaining

structures and foundations", Current Science, (ISSN: 0011-3891, IF: 0.694/2003) India, Vol. 87, No. 10: pp. 1417-1425.

3. Choudhury D & Chatterjee S (2006) "Displacement - based seismic active earth pressure

on rigid retaining walls", Electronic Journal of Geotechnical Engineering, (ISSN: 1089-3032), USA, Vol. 11, Bundle C, paper No. 0660.

4. Chowdhury I & Dasgupta S.P.(2002) “Dynamic Analysis of RCC Retaining wall under Earthquake Loading”- ; Electronic Journal of Geo-technical Engineering Vol-8C 2003.

5. Clough R.W.(1984) “Dynamics of Structures” M’cgrawhill Publications New York USA.

6. Das B.M. & Puri V.K.(1996) “Static and dynamic active earth pressure”, Geotechnical and Geological engineering Vol-14, pp-353-356.

7. Ghosh S & Saran S(2007) “Pseudo static Analysis of Rigid Retaining wall for Dynamic Active Earth Pressure” Cenem B.E.College Kolkata India

8. Ghosh S, Dey G.N., and Datta B.N.(2010) “Pseudo static Analysis of Rigid Retaining wall for Dynamic Active Earth Pressure” 12th International Conference of International Association for Computer Methods and Advances in Geomechanics.

9. Ghosh S & Pal J(2010) “Extension of Mononobe-Okabe expression for active earth force on retaining wall backfilled with c-φ soil”14th Symposium on Earthquake Engineering, Indian Institute of Technology Roorkee India Vol-1 pp 522-530.

10. IS-1893(2002) – Code for Earthquake resistant design of Structures; Bureau of Indian

Standard Institution, New Delhi, India.

11. Mononobe N & Matsuo H (1929) “On the determination of earth pressure during earthquakes”, Proc. World Engineering Congress, Tokyo, Vol. 9, Paper 388.

12. Murthy V.N.S. (1984) “Soil Mechanics and Foundation Engineering “ Sai Kripa

Publication Bangalore India.

13. Okabe S. (1924) “General theory of earth pressures and seismic stability of retaining wall and dam”, J. Japanese Society of Civil Engineers, Vol. 12, No. 1.

14. Ostadan F & W. H. White (1997) “Lateral seismic soil pressure, An updated approach”, Bechtel Technical Group Report Los Angles USA

Vol. 16 [2011], Bund. C 323

15. Ostadan F (2004) “Seismic soil pressure on building walls-An Updated approach”, 11th International Conference on Soil Dynamics and Earthquake Engineering. University of California, Berkeley, January.

16. Saran S & Prakash S (1968) “Dimensionless Parameters for static and dynamic earth

pressures behind retaining walls”, Indian Geotechnical Journal Vol. (72(3) pp 295-310. 17. Saran S & Gupta R.P. (2003) “Seismic Earth Pressure behind retaining walls” Indian

Geotechnical Journal Vol. 33(3) pp195-213. 18. Seed H.B. & Whitman R.V. (1970) “Design of earth retaining structures for seismic

loads”, ASCE Specialty Conference on Lateral Stress in Ground and design of Earth Retaining Structures, June.

19. Steedman R.S. & Zeng X (1990) “The Seismic response of Waterfront Retaining walls”,

Proceedings on Specialty Conference on design performance of Earth Retaining Structures, Special Technical Publication 25 Cornell University Ithaca New York pp 897-910.

20. Whitman R.V.(1990) “Seismic Design and Behavior of Gravity Retaining walls”,

Proceedings Specialty Conference on design and performance of Earth Retaining Structures, ASCE, Cornell University, June18-21.

21. Whitman R.V. (1991) “Seismic design of Earth Retaining structures”, Proceedings 2nd

International conference on Recent advances in Geotechnical Earthquake Engineering and Soil Dynamics, St Louis USA, March 11-15.

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APPENDIX

Calculation of modal mass participation For c-φ soil with overburden q as cited in the example the modal mass participation is expressed by equation(82) where ξ varies from 0 to 1 thus taking value of ξ in steps of 0.05 we have.

ξ F(ξ) ξ.F(ξ) ξ.F(ξ)2 0 1.474841 0 0

0.05 1.380686 0.069034 0.00345 0.1 1.286547 0.128655 0.01286

0.15 1.192472 0.178871 0.026830 0.2 1.098550 0.219710 0.043942

0.25 1.004923 0.251231 0.062807 0.3 0.911794 0.273538 0.082061

0.35 0.819437 0.286803 0.100381 0.4 0.728205 0.291282 0.116512

0.45 0.638540 0.287343 0.129304 0.5 0.550985 0.275493 0.137746

0.55 0.466190 0.256405 0.141022 0.6 0.384921 0.230953 0.138571

0.65 0.308073 0.200248 0.130161 0.7 0.236677 0.165674 0.115971

0.75 0.171907 0.128930 0.096697 0.8 0.115096 0.092076 0.073661

0.85 0.067738 0.057577 0.048940 0.9 0.031502 0.028352 0.025516

0.95 0.008241 0.007829 0.007437 1 0 0 0

Sum → 3.430003 1.493883

Thus 296.2493.1

43.3 ==κ ; Here 945.0=ψ and 42.1=η .

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