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Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.
Research Article
© Copyright 2014| Centre for Info Bio Technology (CIBTech) 2456
EARTH PRESSURES AGAINST RIGID RETAINING WALLS IN
THE MODE OF ROTATION ABOUT BASE BY APPLYING
ZERO-EXTENSION LINE THEORY
Morshedi S.M.1, Ghahramani A.2, Anvar S.A.2 and Jahanandish M.2 1Department of Geotechnical Engineering, Shiraz Univ., Shiraz, Iran
2Department of Civil Engineering, Shiraz Univ., Shiraz, Iran
*Author for Correspondence
ABSTRACT
This paper illustrates the use of the zero-extension line theory for determining distribution of active and
passive pressures from sand backfill behind a rigid retaining wall rotating about its base. These data
demonstrate a comparatively linear distribution in the active case. However, because of higher remaining
pressure on the lower wall (due to the lack of formation of the active state), the point of application of the
resultant force stands lower than H/3 (where H is the height of the wall). In the passive case, the pressure
distribution is approximately parabola-shaped. It also lacks a passive state formation on the lower wall,
which causes the location of the resultant force to increase in height. A comparison is made among the
obtained results, solutions resulting from Coulomb theory, and recent analytical and numerical research.
These comparisons show that the magnitude of the active force is nearly 20 percent greater than the
Coulomb solutions in the passive case. The resultant force is about 75 percent smaller than the Coulomb
solution, but agrees well with results obtained from Lancellotta and Shiau. Applying the Coulomb theory
is not reliable in either case.
Keywords: Zero-Extension Lines, Earth Pressure, Retaining Wall, Rotational Mode, Finite Difference
INTRODUCTION
Knowledge of the magnitude and distribution of earth pressures against retaining walls is important in the
design of many civil engineering structures. Common methods for stability analysis in geotechnical
problems (including retaining walls) focus more attention on the properties and behavior of soil in the
ultimate state. A number of these methods assume a failure surface and compute the necessary force for
moving the wedge above the assumed surface. In these methods, it is supposed that the soil fails in
accordance with the Mohr-Coulomb criteria. In the Coulomb limit equilibrium, the failure line in the
backfill consists of a straight line. However, Rankine assumed that the failure lines are formed throughout
the soil mass. Boussinesq examined the problem with the same assumption and consideration of the wall
roughness.
Various laboratory studies have assessed the classical theories of the lateral earth pressure as applied to
the design of retaining walls (Caltabiano et al., 1999; Fang and Ishibashi, 1986; Matsuo et al., 1978;
Sherif et al., 1982). These investigations demonstrated that the distribution pattern of lateral pressure
depended on the mode of wall movement. Therefore, the magnitude and point of application of the lateral
force for various wall movements (rotation about top, rotation about base and translation) are not equal,
and these amounts can be different from the Coulomb solutions (Fang and Ishibashi, 1986).
The progressive nature of failure in soil and the necessity of knowing the load-deflection behavior at
loads other than the limit loads have caused researchers to focus on the strain field. Researchers’ studied
the directions where the linear strain is zero. These directions, called zero extension lines (ZEL), have
many applications in understanding soil deformation (James and Bransby, 1971; Roscoe, 1970). A simple
pattern of ZEL has been used for finding the strain field behind a model retaining wall, with good
agreement between the predictions and observations (James and Bransby, 1971). Success in the prediction
of a strain field by the ZEL led the researchers to implement this theory in obtaining the mobilized
strength at different points of the soil mass (Roscoe, 1970). This information was necessary for obtaining
the stress field. It was this idea that led to the development of the method of associated field, which works
Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.
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well in the prediction of load-deflection behavior (Atkinson and Potts, 1975; James et al., 1972; Serrano,
1972). However, this method has a few problems: 1) It requires the use of an iterative process of
computations to achieve convergence and compatibility between the and fields. 2) It requires
elaborate interpolation routines because the velocities and stresses were not computed at the same points.
Attempts have been made to find a way to calculate the stress field by the ZEL alone. Two alternatives
have been presented for this problem. In the first, stresses were calculated by considering the force
equilibrium of the soil elements between the ZEL (Habibagahi and Ghahramani, 1979). In the second,
equilibrium-yield equations written along the stress characteristics were transferred into the ZEL
directions. The second method has more applications and has been shown to lead to the same results as
the first one (Anvar and Ghahramani, 1997).
The present study deals with the determination of the magnitude and distribution of active and passive
pressures behind a rigid retaining wall rotating about its base. For this purpose, equilibrium-yield
equations along the ZEL were transferred into incremental form, then used in writing a computer program
in the MATLAB software environment for drawing ZEL mesh and calculating active and passive
pressures behind a wall. To assess the ZEL method, the obtained results from this method have been
compared with Coulomb-theory solutions.
Zero-Extension Line Theory
If the soil displacements in the x and z directions are represented by U and W respectively, strains are
represented by xzzx ,, , and compressive strain is considered positive, then:
x
W
z
U
z
W
x
Uxzzx ;; (1)
The Mohr circle of strain is shown in Figure1 . If is the angle of dilation of soil, then sin is
related to the volumetric strain, v , and the maximum shear strain, max , by
max
sin v
(2)
Figure 1: Mohr circle and principle directions of strains
It is assumed that the direction of 1 and 1 coincide. If the origin of lines is P on the Mohr strain
circle, then linear strain is zero along the PA and PB directions. These are the zero extension lines. It is
Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.
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clear that both of these lines make the angle 24 with the 1 direction. We can use the
Mohr strain circle and algebraic manipulations to determine the two strain characteristic directions. For
the positive direction,
tandx
dz (3)
For the minus direction,
tandx
dz (4)
Accordingly, the field of zero extension lines and strain characteristics coincide. Through the aid of the
matrix calculations, the following relation between U and W along both characteristic directions is
obtained:
. . 0dU dx dW dz (5)
From Equations (3), (4) and (5), there exists the possibility of calculating U and W at point C if these
parameters and are known at points A and B (Figure 2). Therefore, if the field of ZEL and the
displacements at the boundaries are known, the displacements at the interior points of the field can be
calculated.
Figure 2: Strain characteristics
Equilibrium Equations along Zel
Consider a soil element under stress state xzzx ,, , as shown in Figure 3. It is assumed that the soil is at
the limit state of equilibrium at the same time as is the angle that 1 direction makes with x axis.
Figure 3: Stress field and principle directions Figure 4: Mohr circle of stress
Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.
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The resulting Mohr circle is the same as shown in Figure 4, in which the origin of planes is P . Thus, there
are two directions for yield: PA and PB , both of which connect point P to the tangency points of yield
lines with the Mohr circle. These directions that make angle 24 with 1 axis are yield
lines, which coincide with stress characteristics.
The equilibrium equations of the soil element shown in Figure 3 when subjected to body forces X and
Z are as follows:
;x xz xz zX Zx z x z
(6)
Through the Mohr stress circle and algebraic manipulation, the following relationships between u and
along the stress characteristic lines are gained:
Along the plus characteristics tandxdz :
2 tan tan tan
tan
du u c d X dz dx Z dx dz
c cu c dx dz dx dz
z x z x
(7)
Along the minus characteristics tandxdz :
2 tan tan tan
tan
du u c d X dz dx Z dx dz
c cu c dx dz dx dz
z x z x
(8)
Assume that and
denote the plus and minus stress characteristics that make angle with 1
axis and that and
denote the plus and minus zero extension lines that make angle with 1 axis
(which coincide with 1 axis) (Figure 5).
Figure 5: ZEL and stress characteristics lines
For any function f of the variable x , z then takes its directional derivatives with respect to the plus and
minus stress characteristics and ZEL. Thus, the following equations are obtained:
cos sin.
cos sin
f f
x
f f
z
(9)
Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.
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cos sin.
cos sin
f f
x
f f
z
(10)
From Eqs. (9) And (10), yield-equilibrium equations along stress characteristics and extensive algebraic
and trigonometric manipulations, the following equations containing the differential of u and along and
are be obtained:
For plus direction,
2 tan tan . .
1tan . tan tan .
cos
1tan .
cos
du u c d d X dz dx
Z dx dz u c d d
cdc d
(11)
For minus direction,
2 tan tan .
1tan . tan tan .
cos
1tan .
cos
du u c d d X dz dx
Z dx dz u c d d
cdc d
(12)
where
1 sin .sin sin sin cos; ;
cos .cos cos .cos cos
(13)
These equations are the stress equilibrium equations along the ZEL. It is worth noting that Eqs. (11) And
(12) on ZEL reduce exactly to Eqs. (7) and (8) on stress characteristics if , which holds for the
associative flow of soils. This creates an analytical tool with the known values of the stress state u and ,
displacements U and W and coordinates x and z at the points A and B along minus and plus ZEL
directions. We can find these parameters at the point C (Figure 2).
Practical Examples and Discussions
Equations (3), (4), (5), (11) and (12) were transferred into incremental form, then used in writing a
computer program in the MATLAB software environment for drawing ZEL mesh and calculating of the
active and passive pressures behind a wall. For these purposes, we need a dilation angle and an initial
friction angle for the backfill material. A dense sand, when exposed to the shear, shows a peak and critical
shear strength. As shown by Cole (1967), the angle of dilation remains constant during a large portion of
a shear test; hence, the results presented in Figure 6 were considered as the input for the ZEL program.
This gives a 15 and an initial
25 at a void ratio of 0.534.
The passive and active cases simulate the loading and unloading conditions on an element of soil.
Because of this, the results of shear tests simulating the loading condition cannot be used to predict active
behavior (Ghahramani and Clemence, 1980). The results of several tests indicate that, although the peak
of sin for the active and passive cases is nearly equal, the shear strain for the passive peak is 3 to 8
Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.
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times larger than the shear strain for the active peak of sin . Therefore, it is recommended to use shear
strain results such as in Figure 6 for the passive case and to use the same curve but a reduced on the
order of 1/3 for the active case.
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4
Shear Strain ( )
sin
j
-8.00E-02
-6.00E-02
-4.00E-02
-2.00E-02
0.00E+00
0 0.1 0.2 0.3 0.4
Shear Strain ( )
Vo
lum
etric
Stra
in ( n
)
(a) (b)
Figure 6: a) sin ; b) curves for a dense sand (Cole, 1967)
A general calculation process is as follows: a rotation angle, , is applied to the wall. From the
displacements (U & W) at the points along the wall, we can obtain the displacements throughout the ZEL
mesh. After the shear strain is calculated at the points of mesh from Eq. (1), the angle of internal friction
developed at these points can be obtained from a simple shear test. Finally, from the developed friction
angle and Eqs. (11) and (12), the magnitudes of u and at each point of the mesh and therefore the
pressures behind the wall can be determined.
To assess and compare the results obtained from the ZEL program with the classical theory of earth
pressure, two examples have been examined for the active and passive cases. In these examples, the
height of the wall ism2 , the unit weight of the backfill material is
3
5.18 mKN, the intensity of surcharge is
2
5 mKN and the number of mesh divisions considered is12 .
Active Case: The active earth pressure distribution behind a wall rotating about its base is generally
believed to be hydrostatic, and experimental evidence has proven this for most of the wall depth (Sherif et
al., 1984). However, the stress condition near the bottom of the wall demonstrates a more complicated
situation.
Figure 7a shows typical change of active pressure distribution as a function of wall rotation angle. As is
shown in Figure 7b, the active earth pressure decreases rapidly soon after the wall starts to rotate up to 05.1 . It then increases at a much slower rate as the wall rotation angle increases. When the upper
parts of the wall enter the active state at a rotation angle of about 3.0 in Figure 7a, the remaining higher
pressures at the lower parts of the wall lead the point of application of the resultant force to a lower
position. In these stages, the top of the wall translation is about 0.005 H. This implies that the bottom
point of the wall will never be able to completely enter the active stage because it requires a great
translation, which opposes the adequate serviceability of the retaining wall. This problem was also
pointed out by Sheriff et al., (1984). The obtained magnitude of the total active thrust from the ZEL
analysis for this wall is kN4.10 , which differs from the Coulomb solution of
KN45.8 by about %20 .
Figure 7c shows the changing point of application of the resultant force as a function of wall rotation
angle. It is necessary to mention that the Coulomb theory does not directly determine the lateral pressure
distribution, but the triangular shape of the linear backfill can be shown. From this interpretation of the
Coulomb theory, the point of application of the resultant force is m74.0 from the base. Because of higher
Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.
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pressure at the lower part of the wall, the point of application falls to a lower location m72.0 from the
base of the wall. According to the Coulomb theory, the resultant force is inclined at an angle w from
perpendicular to the back face wall, while the results given from ZEL analysis show that this angle is a
little smaller than w .
If, in the active state, the shear strength is equal through the whole height of the wall (i.e., max ), then
the results obtained from the ZEL analysis will precisely match the Coulomb solution (see Figure 7a).
Such an occurrence, however, is practically impossible.
In addition, a continuous mass of dense sand will degenerate into quasi-rigid blocks separated by rupture
surfaces soon after the shear strain in the mass exceeds that required to cause the sand to reach its peak
stress ratio condition.
Figure 8a shows the initial and deformed ZEL net for the rotation angle 05.1 of a m2 wall. In the
active case, shown in Figure 8b, the ZEL nets for a smooth wall and a rough wall are the approximately
the same.
This implies that the roughness of the wall has very little effect on the active pressures. To demonstrate
this aspect, results are given in Table 1 for ZEL analysis at different magnitudes of w , along with
solutions from Coulomb theory and Lancellotta’s equation (Lancellotta, 2002).
Table 1: Active pressure Coefficient
w ak
Coulomb Solution Lancellotta ZEL Analysis
0 0.1918 0.1918 0.2287
1/3 0.1787 0.1740 0.2210
1/2 0.1772 0.1667 0.2235
2/3 0.1787 0.1601 0.2291
1 0.1861 0.1523 0.2433
Passive Case: Figures 9a and 9b show the changes of passive pressure distribution and passive resultant
force, respectively, as functions of wall rotation angle. It is evident from Figure 9b that the passive
pressure first increases due to shearing and then decreases, similar to the results of the simple shear test.
In this case, the formation of a fully passive state at the wall base is very difficult, if not impossible. At a
rotation angle of about 0.1 , the upper part of the wall reaches the passive state, while the mobilized
friction angle in the lower parts is much less than max .
The strain is high near the top of the wall but becomes much smaller at greater depth. As a consequence,
high stresses will be generated at the top of the wall and cause the resultant force to rise to higher
location.
At later stages, strain in the sand near the top of the wall is large and falls below its peak value. Thus
the stresses at the top of the wall decrease below their early peak value. No rupture surfaces are predicted
near the toe of the wall, even after 5 of wall rotation ( 1.0 ).
As seen in Figure 9a, the passive pressure distribution is not simply triangular. This change of earth
pressure distribution causes the resultant force to rise to a higher location. Figure 9c shows the change of
the point of application of resultant force as a function of wall rotation angle. If according to Figure 9b,
the pressure distribution for 4.2 is chosen as the passive case, and the necessary displacement for
the formation of passive state is about 0.04H, which equals cm0.8 for a
m2 wall.
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0 5 10 15 20 25-2.5
-2
-1.5
-1
-0.5
0
Active Pressure (Kpa)
He
igh
t o
f th
e W
all
(m
)
Coulomb Solution
ZEL Analysis with
ZEL Analysis for
ZEL Analysis for
ZEL Analysis for
ZEL Analysis for
ZEL Analysis for
max
(a)
0 0.5 1 1.5 2 2.5 3 3.50
5
10
15
20
25
To
tal
Ac
tiv
e T
hru
st
(KN
)
Rotation Angle (Deg.)
ZEL Analysis
Coulomb Solution
(b)
0 0.5 1 1.5 2 2.5 3 3.50.71
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
Po
int
of
Ap
pli
ca
tio
n f
rom
th
e B
as
e (
m)
Rotation Angle (Deg.)
ZEL Analysis
Coulomb Solution
(c)
Figure 7: a) Distribution of Active Pressure at Different Wall Rotation Angle; b) Active Resultant
Force versus Wall Rotation Angle; c) Point of Application of Resultant Force versus Wall Rotation
Angle
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-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-2.5
-2
-1.5
-1
-0.5
0
Distance from the Back of the Wall (m)
He
igh
t o
f t
he
Wa
ll (
m)
Mixed Zone
Rankine Zone
Goursat Zone
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-2.5
-2
-1.5
-1
-0.5
0
Distance from the Back of the Wall (m)
He
ig
ht o
f th
e W
all (m
)
(a) (b)
Figure 8: a) Initial and Deformed ZEL Net at Active Condition with 2 w; b) ZEL Net with 0w
The resultant force from this analysis is KN24.435 which, in comparison with the Coulomb theory equal
to KN24.770 , demonstrates the overestimation and lack of safety of the Coulomb solutions. The
Coulomb theory estimated greater passive force of about %75 , which was confirmed in previous
investigations (Fang and Ishibashi, 1986; Ghahramani and Clemence, 1980; Habibagahi and Ghahramani,
1979; Harr, 1966; Jahanandish, 1988; Jahanandish et al., 1989; James and Bransby, 1970; James and
Bransby, 1971; James et al., 1972; Lancellotta, 2002; Matsuo et al., 1978; Roscoe, 1970; Serrano, 1972;
Sherif et al., 1984; Sherif et al., 1982).
The point of application of the resultant force was calculated at m73.0 from the base, whereas the
Coulomb theory was calculated m74.0 from the base.
Figure 10a shows the initial and deformed ZEL net for the rotation angle 2.1 for the
m2 wall. In the
passive case, shown in Figure 10b, the ZEL net for a smooth and a rough wall are different. This implies
that the roughness of the wall has a strong effect on the passive pressures. This condition is for the case
2 w , but as shown in Figure 10c, the error of the Coulomb method increases with the w increase,
and when w becomes greater than 2 , the error amount of this method goes beyond %100 .
Furthermore, the results obtained from ZEL analysis for different magnitudes of w with solutions
resulted from numerical and analytical solutions of Lancellotta and Shiau (Shiau et al., 2008) are given in
Table 2.
Table 2: Passive pressure Coefficient
w
pk
Coulomb
Solution Lancellotta
Shiau ZEL Analysis
LB UB
0 5.2138 5.2138 5.2585 5.2896 5.2135
1/3 10.1502 8.5405 8.3928 9.7448 7.8761
1/2 15.9174 10.3508 11.2902 13.1728 9.6899
2/3 29.005 12.0727 15.5012 17.8164 11.8064
1 122.9841 13.8985 29.37 33.6216 14.5544
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0 100 200 300 400 500 600 700-2.5
-2
-1.5
-1
-0.5
0
Passive Pressure (Kpa)
He
igh
t o
f th
e W
all
(m
)
Coulomb Solution
ZEL Analysis with
ZEL Analysis for
ZEL Analysis for
ZEL analysis for
ZEL Analysis for
ZEL Analysis for
max
(a)
0 1 2 3 4 5 6 7 8100
200
300
400
500
600
700
800
To
tal
Pa
ss
ive
Th
rus
t (K
N)
Rotation Angle (Deg.)
ZEL Analysis
Coulomb Solution
(b)
0 1 2 3 4 5 6 7 80.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
Po
int
of
Ap
plic
atio
n f
rom
th
e B
ase
(m)
Rotation Angle (Deg.)
ZEL Analysis
Coulomb Solution
(c)
Figure 9: a) Distribution of Passive Pressure at Different Wall Rotation Angle; b) Passive Resultant
Force versus Wall Rotation Angle; c) Point of Application of Resultant Force versus Wall Rotation
Angle
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0 0.5 1 1.5 2 2.5 3 3.5-2.5
-2
-1.5
-1
-0.5
0
0.5
Distance from the Back of the Wall (m)
He
igh
t o
f t
he
Wa
ll (
m)
Rankine Zone
Goursat Zone
Mixed Zone
0 0.5 1 1.5 2 2.5 3-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
Distance from the Back of the Wall (m)
He
ig
ht o
f th
e W
all (m
)
(a) (b)
0
200
400
600
800
1000
1200
1400
1600
1800
0 5 10 15 20 25 30
Friction Angle Between Wall and Soil (Deg.)
Pass
ive
Th
rust
(K
pa)
Coulomb Results Calculated From ZEL
(c)
Figure 10: a) Initial and Deformed ZEL Net at Passive Condition with 2 w; b) ZEL Net
with 0w ; c) Variation of Coulomb Solutions and ZEL Results versus Friction Angle between Wall
and Backfill
It is necessary to mention that the numerical analysis of Shiau is related to the translational mode, while
previous investigations demonstrated that the pressure values in this mode are greater than the rotational
one.
CONCLUSIONS
On the basis of the results obtained from the examined examples with the ZEL program, one can conclude
that:
In the mode of rotation about the base for a dense sand backfill, the formation of the active and passive
states occur at the displacement of about H04.0 and H005.0 from the top of the wall, respectively,
which agrees with the presented magnitudes in the geotechnical references.
Because the formation of the active state at the wall base is practically impossible, the presented values
of the active force by the Coulomb theory are less than the real ones, and the results obtained from ZEL
analysis are greater by about %20 .
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The existence of remaining higher pressures at the lower part of the wall causes the point of
application of the resultant force to fall to a lower location, and the pressure distribution is approximately
linear.
The resultant force direction makes a 3.17 angle with the vertical direction crossed to the back
surface of the wall, which is less than the soil-wall friction angle 35.212 .
In the passive case, because of the lower levels of displacement at the lower parts of the wall, the
applied pressures are much less than the anticipated pressures by the Coulomb theory. The results
obtained from the ZEL analysis demonstrate that the Coulomb solutions are about %75 greater than the
real quantities.
The passive pressure distribution has a parabola shape, and the point of application of the resultant
force is about 3H from the base of the wall.
When it is assumed that the max has been mobilized through the whole height of the wall, the results
obtained from the ZEL analysis are equal to the Coulomb solution in the active case. In the passive case,
the Coulomb solutions are greater by about %60 than the ZEL results.
REFERENCES
Anvar SA and Ghahramani A (1997). Equilibrium Equations on Zero Extension Lines and Their
Application to Soil Engineering. Iranian Journal of Science and Technology 21(1) Transaction B 11-34.
Atkinson JH and Potts DM (1975). A Note on Associated Field Solutions for Boundary Value Problems
in a Variable -Variable Soil. Geotechnique 25(2) 379-384.
Behpoor L and Ghahramani A (1987). Zero Extension Line Theory of Static and Dynamic Bearing
Capacity. Proceedings of 8th Asian Regional Conference on Soil Mechanics and Foundation Engineering,
Kyoto, Japan 341-346.
Caltabiano S, Cascone E and Maugeri M (1999). Sliding Response of Rigid Retaining Walls.
Proceedings of 2nd International Conference on Earthquake Geotechnical Engineering Lisboa, Portugal.
Cole ERL (1967). The Behaviour of Soils in Simple Shear Apparatus. Thesis Presented to the University
of Cambridge, Cambridge, England, in Partial Fulfillment of the Requirements for the Degree of Doctor
of Philosophy.
Fang YS and Ishibashi I (1986). Static Earth Pressures with Various Wall Movement. ASCE Journal of
Geotechnical Engineering 112(3) 317-333.
Ghahramani A and Clemence SP (1980). Zero Extension Line Theory of Earth Pressure. Journal of
Geotechnical Engineering Division, ASCE 106(GT6) 631-644.
Habibagahi K and Ghahramani A (1979). Zero Extension Line Theory of Earth Pressure. Journal of
Geotechnical Engineering Division, ASCE 105(GT7) 881-896.
Harr ME (1966). Foundations of Theoretical Soil Mechanics (McGraw-Hill, New York, USA).
Jahanandish M (1988). Zero Extension Line Net and Its Applications in Soil Mechanics. M.Sc Thesis,
Shiraz University, Shiraz, Iran.
Jahanandish M, Behpoor L and Ghahramni A (1989). Load-Displacement Charactristics of Retaining
Walls. Proceedings of 12th International Conference on Soil Mechanics and Foundation Engineering, Rio
de Janiro, Brazil 243-246.
James RG and Bransby PL (1970). Experimental and Theoretical Investigation of Passive Earth
Pressure Problems. Geotechnique, London, England 20(1) 17-37.
James RG and Bransby PL (1971). A Velocity Field for some Passive Earth Pressure Problems.
Geotechnique, London, England, 21(1) 61-83.
James RG, Smith IAA and Bransby PL (1972). The Prediction of Stresses and Deformations in a Sand
Mass Adjacent to a Retaining Wall. Proceedings of 5th European Conference on Soil Mechanics and
Foundation Engineering, Madrid, Spain 1 39-46.
Indian Journal of Fundamental and Applied Life Sciences ISSN: 2231– 6345 (Online) An Open Access, Online International Journal Available at www.cibtech.org/sp.ed/jls/2015/03/jls.htm 2015 Vol. 5 (S3), pp. 2456-2468/Morshedi et al.
Research Article
© Copyright 2014| Centre for Info Bio Technology (CIBTech) 2468
Lancellotta R (2002). Analytical Solution of Passive Earth Pressure. Geotechnique, London, England,
52(8) 617-619.
Matsuo M, Kenmochi S and Yagi H (1978). Experimental Study on Earth Pressure of Retaining Wall
by Field Tests. Soils and Foundations, Japanese Society of Soil Mechanics and Foundation Engineering
18(3) 27-41.
Roscoe KH (1970). The Influence of Strains in Soil Mechanics. Geotechnique, London, England, 20(2)
129-170.
Serrano AA (1972). Generalization of the Associated Field Method. Proceedings of 5th European
Conference on Soil Mechanics and Foundations Engineering, Madrid, Spain 2 355-379.
Sherif MA, Fang YS and Sherif RI (1984). AK and 0K Behined Rotating and Non-Yielding Walls.
ASCE Journal of Geotechnical Engineering Division 110(1) 41-56.
Sherif MA, Ishibashi I and Lee CD (1982). Earth Pressures against Rigid Retaining Walls. ASCE
Journal of Geotechnical Engineering 108(GT5) 679-695.
Shiau JS, Augarde CE, Lyamin AV and Sloan SW (2008). Finite Element Limit Analysis of Passive
Earth Resistance in Cohessionless Soils. Soils and Foundations, Japanese Society of Soil Mechanics and
Foundation Engineering 48(6) 843-850.