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Dynamic analysis of RCC retaining walls under earthquake loading
Article · January 2003
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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
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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 MO 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.
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.
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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
KA =
1 − sin φ
1 + sin φ
and K P =
1 + sin φ
1 − sin φ
(1)
The static pressures acting on the wall can be expressed as
paz = K A .γ s .z
(2)
p pz = K P .γ s .z
(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
EI
d 4u
= K A .γ s .z
dz 4
(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 = 1 / 12.( B × t 3 ) , 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
d 3u K A .γ s z 2
+ C1
EI 3 =
2
dz
d 2u K .γ z 3
EI 2 = A s + C1 z + C2
dz
6
4
du K A .γ s z
z2
EI
=
+ C1 + C2 z + C3
dz
24
2
5
3
K .γ z
z
z2
EIu = A s + C1 + C2 + C3 z + C4
120
6
2
(5)
(6)
(7)
(8)
For the given wall (Figure 2) we consider the boundary conditions
1)
At z=0
d 3u
= 0  C1=0
dz 3
8(a)
2)
3)
4)
d 2u
= 0  C2 = 0
dz 2
8(b)
du
K A .γ s .H 4
C
=0 3 =−
dz
24
8(c)
At z=0
At z=H
At z=H u = 0  C4 =
K A .γ s .H 5
30
8(d)
Thus equation (8) can be expressed as
K A .γ s z 5 K A .γ s H 4 z K Aγ s H 5
−
+
EIu =
120
24
30
(9)
Equation (9) can be further expressed in generalized form as
u=

K A .γ s .H 5  ξ 5 5ξ
−
+ 1

30 EI  4
4

(10)
where ξ = z / H a non-dimensional term that varying between 0-1.
Static deflection of wall under the given condition can be expressed as
u static =
K A .γ s .H 5
at ξ = 0
30 EI
(11)
Natural period of the wall is
T = 2π
ustatic
g
(12)
Substituting equation (11) in (12) and considering 1/12.(1× t 3 ), we have
TA = 3.97
K Aγ s H 5
Et 3 g
(13)
For modal analysis, maximum amplitude (Sd) is expressed as (Clough 1984)
Sd = Sa / ω2
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(14)
Vol. 16 [2011], Bund. C
301
where Sa= Spectral acceleration and ω = 2π / T , natural frequency of the wall.
In terms of code equation (14) can be expressed
Sd = κβ Sa / ω2
(15)
n
n
i =1
i =1
2
where κ = Modal mass participation factor and is expressed as  mi ϕi /  mi ϕi , 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
u = κβ
where f (ξ ) =
ξ5
4
−
Sa 2
T f (ξ )
4π 2
(16)
5ξ
+1
4
Equation (16) can be finally expressed as
u = κβ

K Aγ s H 5  S a   ξ 5 5ξ
   −
+ 1
30 EI  g   4
4

(17)
Considering M=EI d2u/dz2 and V=EI d3u/dz3 we have,
M ξ = κβ
K Aγ s H 3  S a  3
  ξ
6
 g 
(18)
K Aγ s z 2  S a 
 
2  g 
(20)
[ ]
Similarly
Vz = κβ
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
n
n
1
 1

κ =  mi ϕi /  mi ϕi 2 =   γ s H 2ξf (ξ)dξ  /   γ s H 2ξf (ξ)2 dξ 
i =1
i =1
0
 0

2 
1  ξ5 5ξ   1  ξ5 5ξ 

κ =   ξ  − + 1 dξ  /  ξ  − + 1 dξ 
  0  4


4
4
 0  4
   


(21)
(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
d 4u
p = EI 4 = K A .γ s .z
dz
(23)
Thus for the present case dynamic pressure on wall is expressed as
pdyn = EI
d 4u EI d
=
dz 4 H 4 dξ 4

K γ H 5  S   ξ 5 5ξ
+ 1
κβ A s  a   −
30 EI  g   4
4

S
 p dyn = κβK Aγ s  a
 g

 z

(24)
(25)
Now if Sv is the vertical acceleration corresponding to time period TA then the dynamic
pressure in vertical direction can be expressed as
S
pV dyn = ±κβγ s  V
 g
- 302 -

 z

(26)
Vol. 16 [2011], Bund. C
303
In horizontal direction effect of this pressure can be expressed as
S
p H dyn = ±κβK Aγ s  V
 g

 z

(27)
Thus total dynamic pressure considering the vertical component of acceleration can be
expressed as
S 
S 
pdyn = κβK Aγ s  a  z ± κβK Aγ s  v  z
 g 
g
(28)
For maximum pressure we must take the positive sign that gives
S 
S 
pdyn = κβK Aγ s  a  z + κβK Aγ s  v  z
 g 
g
(29)
As per IS-1893(2002) considering Sv =Sa/2, equation (29) can be expressed as
 3S 
pdyn = κβK Aγ s  a  z
 2g 
(30)
Considering the effect of vertical acceleration, the dynamic displacement, moment and
shear can be expressed as
u = κβ

K Aγ s H 5  S a   ξ 5 5ξ
   −
+ 1
20 EI  g   4
4

(31)
K Aγ s z 3  S a 
 
M z = κβ
4  g 
(32)
3K Aγ s z 2  S a 
 
4
 g 
(33)
Vz = κβ
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
K A = cos i ×
K P = cos i ×
cos i − cos 2 i − cos 2 φ
cos i + cos 2 i − cos 2 φ
cos i + cos 2 i − cos 2 φ
cos i − cos 2 i − cos 2 φ
(34)
(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
pa = γ s z cot 2 α − 2c cot α where, α= 45+φ/2
Substituting this in equation (4) we have
d 4u
EI 4 = γ s .z. cot 2 α − 2c cot α
dz
- 304 -
(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
C3 =
cH 3 cot α γ s H 4
−
cot 2 α
3
24
γ sH 5
cH 4 cot α
C4 =
cot α −
30
4
2
(37)
(38)
This gives
γ s z5
cz 4
cH 3 z cot α γ s H 4 z
γ sH 5 2
cH 4
2
EIu =
cot α −
cot α +
−
cot α +
cot α −
cot α (39)
120
12
3
24
30
4
2
Equation (39) after some simple algebraic manipulation can be finally expressed as
u=
where, ψ =
γ s H 5 cot 2 α  ξ 5
30 EI
4

−

ξ4
5ξ
4ξ
+ 1 −ψ
+ψ
−ψ 
4
3
3

(40)
15 H c
2c
the free standing height of
tan α , a dimensionless parameter, and H c =
γs
4H
soil.
Thus ustatic at ξ=0 is expressed as
ustatic =
γ s H 5 cot 2 α
30 EI
[1 − ψ ]
(41)
Substituting above in equation (12) we have
T A = 3.97 cot α
γ sH 5
Et 3 g
(1 − ψ )
(42)
Thus for modal analysis the dynamic amplitude is expressed as
u = κ .β .
 ξ 5 5ξ

ξ4
4ξ
 (1 − ψ ) −
+ 1 −ψ
+ψ
−ψ 
4
3
3
 g 
4

γ s H 5 cot 2 α  S a 
30 EI
The dynamic moment and shear can be expressed as
(43)
 z 3 H z 2 tan α 
S 
M z = κβγ s cot 2 α  a (1 − ψ ) − c

2
 g 
6

(44)
 z2

S 
Vz = κβγ s cot 2 α  a (1 − ψ ) − H c z tan α 
 g 
2

(45)
Considering the effect of vertical acceleration Sv = Sa/2, we have
u = κ .β .
 ξ 5 5ξ

ξ4
4ξ
 (1 − ψ ) −
+ 1 −ψ
+ψ
−ψ 
4
3
3
 g 
4

γ s H 5 cot 2 α  S a 
20 EI
(46)
 z 3 3H c z 2 tan α 
 Sa 
M z = κβγ s cot α  (1 − ψ ) −

4
 g 
4

(47)
 3z 2  3 

S 
Vz = κβγ s cot 2 α  a (1 −ψ )
−   H c z tan α 
 g 
 4 2

(49)
2
and
The modal mass participation may be expressed as
2 
4
1  ξ5 5ξ
  1  ξ5 5ξ

ξ4
4ξ
ξ
4
ξ
κ =   ξ  − + 1 − ψ + ψ − ψ dξ  /   ξ  − + 1 − ψ + ψ − ψ  dξ 


 

4
3
3
4
3
3
 0  4
   0  4


(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, ψ → 0 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 ofψ → 1 , time period tends to zero
and for ψ > 1, 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 (ψ > 1 ), 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.
- 306 -
Vol. 16 [2011], Bund. C
307
For passive case we have
d 4u
EI 4 = γ s .z. tan 2 α + 2c tan α
dz
(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.
C3 = −
C4 =
γ sH 4
24
γ sH 5
30
tan 2 α −
tan 2 α +
cH 3
tan α
3
cH 4
tan α
4
(52)
(53)
Imposing the above integration constants we have
EIu =
γ s z5
120
tan 2 α +
cz 4
cH 3 z tan α γ s H 4 z
γ H5
cH 4
tan α −
−
tan 2 α + s
tan 2 α +
tan α
12
3
24
30
4
(54)
Equation (54) on simplification gives
u=
Here ψ p =
γ s H 5 tan 2 α  ξ 5
30 EI

ξ4
5ξ
4ξ
 4 − 4 + 1 +ψ p 3 −ψ p 3 +ψ p 


(55)
15 H c
cot α , a dimensionless a parameter and Hc is as defined earlier the free
4H
standing height of cohesive soil.
Based on the above
ustatic =
γ s H 5 tan 2 α
30 EI
[1 + ψ ]
(56)
p
Substituting this in equation (11) we finally have the time period for passive case as
TP = 3.97 tan α
γ sH 5
Et 3 g
(1 + ψ )
p
Thus based on modal analysis dynamic amplitude, moments and shears are expressed as
(57)
u p = κ .β .
 ξ 5 5ξ

ξ4
4ξ
 (1 + ψ p ) −
+ 1 +ψ p
−ψ p
+ψ p 
4
3
3
 g 
4

γ s H 5 tan 2 α  S a 
30 EI
(58)
 z 3 H z 2 cot α 
S 
M pz = κβγ s tan 2 α  a (1 + ψ p ) + c

2
 g 
6

(59)
 z2

S 
V pz = κβγ s tan 2 α  a (1 + ψ p ) + H c z cot α 
 g 
2

(60)
Considering effect of vertical acceleration we have
u p = κ .β .
 ξ 5 5ξ

ξ4
4ξ
 (1 + ψ p ) −
+ 1 +ψ p
−ψ p
+ψ p 
4
3
3
 g 
4

γ s H 5 tan 2 α  S a 
20 EI
(61)
 z 3 3H c z 2 cot α 
S 
M pz = κβγ s tan 2 α  a (1 + ψ p ) +

4
 g 
4

(62)
 3z 2 3

S 
V pz = κβγ s tan 2 α  a (1 + ψ p )
+ H c z cot α 
2
 g 
 4

(63)
The modal mass participation factor is expressed as
1
κ=
 ξ 5 5ξ

4ξ
ξ4
−
+ 1 +ψ p
−ψ p
+ ψ p  dξ
4
3
3
4

ξ 
0
1
2
 ξ 5 5ξ

4ξ
ξ4
−
+
+
−ψ p
+ ψ p  dξ
1
ξ
ψ
p
0  4 4
3
3

(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
T A = 3.97
u = κ .β .
γ sH 5
Et 3 g
(1 −ψ c )
 ξ 5 5ξ

ξ4
4ξ
 (1 − ψ c ) −
+ 1 −ψ c
+ψ c
−ψ c 
20 EI  g 
4
3
3
4

γ s H 5  Sa 
- 308 -
(64)
(65)
Vol. 16 [2011], Bund. C
where ψ c =
309
 z 3 3H c z 2 
 Sa 
M z = κβγ s  (1 − ψ c ) −
4 
 g 
4
(66)
 3z 2  3 

S 
Vz = κβγ s  a (1 − ψ c )
−  Hc z
 g 
 4 2

(67)
15 H c
.
4H
Similarly for passive case considering the vertical acceleration effect can be expressed as
TP = 3.97
u p = κ .β .
γ sH 5
Et 3 g
(1 + ψ ) where, ψ
pc
pc
=ψ c
 ξ 5 5ξ

ξ4
4ξ
 (1 + ψ pc ) −
+ 1 + ψ pc
− ψ pc
+ ψ pc 
20 EI  g 
4
3
3
4

γ s H 5  Sa 
(68)
(69)
 z 3 3H c z 2 
S 
M pz = κβγ s  a (1 + ψ pc ) +
4 
 g 
4
(70)
 3z 2 3

S 
V pz = κβγ s  a (1 + ψ pc )
+ Hc z
2
 g 
 4

(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.
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
EI
d 4u
= q. cot 2 α
dz 4
(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
C3 = −
q cot 2 αH 3
6
(73)
q cot 2 αH 4
C4 = −
8
Substituting these values we finally have
EIu =
qz 4 cot 2 α q cot 2 αH 3 z q cot 2 αH 4
−
+
24
6
8
(74)
Equation (74) can be finally written as
u=

q cot 2 αH 4  ξ 4 4ξ
+ 1
 −
8 EI
3
3

(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
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Vol. 16 [2011], Bund. C
u=
311
γ s H 5 cot 2 α  ξ 5
30 EI
 q cot 2 αH 4  ξ 4 4ξ

ξ4
5ξ
4ξ
+ 1 −ψ
+ψ
−ψ  +
+ 1 (76)
 −
 −
4
3
3
8 EI
3
4

3

Equation (76) can be further simplified to
u=
γ s H 5 cot 2 α  ξ 5
30 EI

4
−

ξ4
ηξ 4 4ηξ
5ξ
4ξ
+ 1 −ψ
+ψ
−ψ +
−
+η
4
3
3
3
3

In equation (77), as mentioned before, ψ =
15 H c
tan α
4H
and η =
15q
4γ s H
(77)
are both
dimensionless parameters.
Thus the maximum static deflection is obtained ( ξ = 0 ) as
u static =
γ s H 5 cot 2 α
30 EI
[1 − ψ + η ]
(78)
Substituting equation (78) in equation (12) we finally have
T A = 3.97 cot α
γ sH 5
Et 3 g
(1 −ψ + η )
Thus for modal analysis the dynamic amplitude, moments and shears can be expressed as
u=
 ξ 5 5ξ

ξ4
ηξ 4 4ηξ
4ξ
 (1 − ψ + η ) −
+ 1 −ψ
+ψ
−ψ +
−
+η
4
3
3
3
3
 g 
4

κβγ s H 5 cot 2 α  S a 
30 EI
(79)
S
M z = κβγ s cot 2 α  a
 g
 z3 z2 

q
(1 − ψ + η ) −  H c tan α −
γs
2 

6
 z2

S 
q 
Vz = κβγ s cot 2 α  a (1 −ψ + η ) − z  H c tan α − 
γ s 
 g 

2



(80)
(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., η → 0 , the formulas converges to equations (46) to (48) and
represents the case of c-φ soil only. When again,ψ → 0 , the equations converges to the case of
pure cohesion less soil(c=0).
Finally an interesting case is observed, when,ψ → 0 ,η → 0 and even φ=0 (i.e. cot 2 α → 1) .
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
 ξ 5 5ξ

4ξ
ξ4
ηξ 4 4ηξ
−
+ 1 −ψ
+ψ
−ψ +
−
+ η  dξ
4
4
3
3
3
3

κ = 10 
2
5
4
4
ξ

5ξ
4ξ
4ηξ
ξ
ηξ
0 ξ  4 − 4 + 1 − ψ 3 + ψ 3 − ψ + 3 − 3 + η  dξ
1
ξ 
(82)
Considering the effect vertical acceleration we have
u=
 ξ 5 5ξ

ξ4
ηξ 4 4ηξ
4ξ
 (1 − ψ + η ) −
+ 1 −ψ
+ψ
−ψ +
−
+η
4
3
3
3
3
 g 
4

κβγ s H 5 cot 2 α  S a 
20 EI
(83)
 z 3 3z 2 
 Sa 
q 
 H c tan α − 
M z = κβγ s cot α  (1 − ψ + η ) −
4 
γ s 
 g 
4
(84)
 3z 2 3 
S 
q 
Vz = κβγ s cot 2 α  a (1 − ψ + η )
− z  H c tan α − 
2 
γ s 
 g 
 4
(85)
2
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.
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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)].
PA =
[
]
1 2
2c 2
2
2
cot α γ s H1 + (γ sat − γ w )H 2 − 2c cot αH + γ s H1.H 2 cot 2 α +
γs
2
(86)
where γ s = Dry unit weight of soil of height H1; γ sat = Saturated unit weight of soil of height
H2; γ w = 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
[
]
1
1
2c 2
2
2
K AE γ s H 2 = cot 2 α γ s H 1 + (γ sat − γ w )H 2 − 2c cot αH + γ s H 1 .H 2 cot 2 α +
2
2
γs
We have,
K AE = cot 2 α
[γ
H 1 + (γ sat − γ w )H 2
2
s
γ s ( H 1 + H 2 )2
2
]−
2γ s H 1 .H 2
4c 2
4c cot α
+
+
γ s .(H 1 + H 2 ) ( H 1 + H 2 ) 2 γ s 2 ( H 1 + H 2 ) 2
(87)
Here KAE = An equivalent coefficient of active earth pressure when considering a pressure
diagram of paz = K AE .γ s .z 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
u = κβ
K AE γ s H 5
T A = 3.97
Et 3 g
(88)

K AE γ s H 5  S a   ξ 5 5ξ
   −
+ 1
30 EI  g   4
4

(89)
M z = κβ
K AE γ s z 3  S a 
 
6
 g 
(90)
Vz = κβ
K AE γ s z 2  S a 
 
2
 g 
(91)

K AEγ s H 5  S a   ξ 5 5ξ
   −
+ 1
20 EI  g   4
4

(92)
Considering vertical acceleration we have
u = κβ
K AE γ s z 3  S a 
 
M z = κβ
4
 g 
(93)
3K AEγ s z 2  S a 
 
Vz = κβ
4
 g 
(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
Pa =
γ sH 2
2
cot 2 α − 2cH cot α +
2c 2
γs
(95)
Based on the above, for a partially loaded beam, static deflection can be expressed based on
fundamentals of beam theory as
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Vol. 16 [2011], Bund. C
315


tan α ) 3 

γs
2c tan α
1 + 5

 4 
15EI

2c

γ s  H − tan α  
γs


 
Pa ( H −
u static =
2c
(96)
Here Pa is as expressed in equation (95).
Thus time period, based on equation (12) is expressed as


2c
PA  H − tan α 
γs


T A = 2π
15 EIg
3





 5
2c tan α

1 +
 4 γ  H − 2c tan α  
s


γs



(97)
Maximum dynamic amplitude, Moments and Shears along the depth of the wall are expressed
as
S
u = κβ  a
 g





tan α ) 3 

γs
2c tan α
1 + 5


4 
15EI

2c

γs  H − tan α  
γs

 

3
S 
z
= κβ  a  Pa
2
 g 


2c
3 H − tan α 
γs


Pa ( H −
Mz
2c
S 
z2
Vz = κβ  a  Pa
2

 g  
2c
 H − tan α 
γz


(98)
(99)
(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).
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
FR = W (sin α − μ cos α )
(101)
where FR= Friction force along the surface BD that resists the motion; W= Weight of the soil body
ABD (W=m.g); α = 45 + φ / 2 for active case and μ = Internal angle of friction of soil @ tan φ .
Considering the resistive friction force FR as the damping force equation (101) can be
expressed as
C.v = W (sin α − μ cos α )
(102)
in which, C= damping of the system, velocity v = Sa/ω where Sa is the seismic acceleration and ω
the natural frequency @ k / m .
Equation (102) on simple algebraic manipulation can be expressed as
C sin α − μ cos α
=
Cc
S 
2 a 
 g 
(103)
Here Cc= Critical damping =2 k / m .
Considering
ζ =
C
the damping ratio we have
Cc
ζA =
sin α − μ cos α
S 
2 a 
 g 
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(104)
Vol. 16 [2011], Bund. C
317
The above gives the damping ratio of the soil in active case. For passive case
ζp =
sin α p + μ cos α p
S 
2 a 
 g 
, where α p = 45 − φ / 2
(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
ζA =
sin α − μ cos α
5
(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.
Damping ratio
Com parison of dam ping ratio for active and passive
case
0.25
0.2
0.15
0.1
0.05
0
Damping ratio
active
Damping ratio
passive
15
20
25
30
35
40
phi
Figure 6: Variation of damping ratio of soil with friction angle of soil
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
ζA =
sin α − μ cos α −
cH cos ecα
W
(107)
5
and
ζp =
sin α p + μ cos α p +
cH cos ecα p
W
(108)
5
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
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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
Sandy Soil
Clayey Soil
c-φ soil
c-φ soil with q
φ soil with q
c soil with q
Liquefied soil with q
Liquefied soil without q
0.391
0.43
0.09
0.474
0.608
0.885
0.91
0.48
1.75
1.75
1.645
1.75
1.566
1.075
1.046
1.75
Modal Mass
Participation
factor(κ)
2.28
2.23
1.98
2.29
2.31
2.31
2.31
2.28
Maximum
Moment (kN.m)
Maximum
Shear (kN)
54.8
34.9
0.6
112.23
256.44
293.15
556.7
93.73
27.4
22.33
0.607
50.96
105.5
126.8
215.6
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.
Comparison of Bending Moment
600.00
500.00
Moment(kN.m)
c-phi s oil
400.00
phi s oil
c-s oil
300.00
200.00
c-phi
+overburden
c+overburden
100.00
phi+overburden
Liquified s oil+q
0.00
0
0.6
1.2
1.8
2.4
3
3.6
4.2
4.8
5.4
6
-100.00
Depth(m)
Figure 7: Variation of Bending Moment along the depth of the wall.
Comparison of Shear force
250
c-phi soil
Shear force(kN)
200
phi soil
150
c-soil
c-phi +overburden
100
c+overburden
50
phi+overburden
Liquified soil+q
0
-50
0
0.6
1.2
1.8
2.4
3
3.6
4.2
4.8
5.4
6
Depth(m)
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.
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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.
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: 10893032), 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
- 322 -
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 897910.
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.
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.
Thus κ =
ξ
F(ξ)
ξ.F(ξ)
ξ.F(ξ)2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.474841
1.380686
1.286547
1.192472
1.098550
1.004923
0.911794
0.819437
0.728205
0.638540
0.550985
0.466190
0.384921
0.308073
0.236677
0.171907
0.115096
0.067738
0.031502
0.008241
0
Sum →
0
0.069034
0.128655
0.178871
0.219710
0.251231
0.273538
0.286803
0.291282
0.287343
0.275493
0.256405
0.230953
0.200248
0.165674
0.128930
0.092076
0.057577
0.028352
0.007829
0
3.430003
0
0.00345
0.01286
0.026830
0.043942
0.062807
0.082061
0.100381
0.116512
0.129304
0.137746
0.141022
0.138571
0.130161
0.115971
0.096697
0.073661
0.048940
0.025516
0.007437
0
1.493883
3.43
= 2.296 ; Here ψ = 0.945 and η = 1.42 .
1.493
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