INTRODUCTION
Mononobe and Matsuo (1929) and Okabe
(1926) proposed the wellknown MononobeOkabe analysis of seismic lateral
earth pressure. The analysis is a direct modification of the coulomb wedge analysis.
In the analysis, the earthquake effects are replaced by a quasistatic inertia
force whose magnitude is computed on the basis of the seismic coefficient concept.
As in the coulomb analysis, the failure surface is assumed planer in the MononobeOkabe
method, regardless of the fact that the most critical sliding surface may be
curved. Similar to the coulomb’s, the MononobeOkabe analysis may underestimate
the active earth pressure and overestimate the passive earth pressure. This
solution is therefore practically acceptable at least for the active pressure
case, although its applicability to the passive pressure is somewhat in doubt.
Some researches were done after MononobeOkabe Analysis by Lyamin
and Sloan (2002), Chang (2002), Farzaneh
and Askari (2003), Bhasin and Kaynia (2004) and
Hack et al. (2007), to improve this method of
analysis. Their studies were included different method of analyzing such as
numerical solutions using upper bound and lower bound method of limit analysis
theory. As stated in the lowerbound theorem, if an equilibrium state of stress
below yield can be found which satisfies the stress boundary conditions, then
the loads imposed can be carried without collapse by a stable body composed
of elasticperfectly plastic material (Collins, 2005).
Any such field of stress thus gives a safe or lower bound on the collapse or
limit load. The stress field satisfying all these conditions is called statically
admissible stress field.
In this study, the lowerbound method of limit analysis is applied to include the earthquake effect which is investigated in producing some dimensionless charts for computing the seismic active and passive earth pressure.
THEOREMS OF LIMIT ANALYSIS
Figure 1 shows a typical loaddisplacement curve as it might
be measured for a surface footing test. The curve consists of an elastic portion;
a region of transition from mainly elastic to mainly plastic behavior; a plastic
region, in which the load increases very little while the deflection increases
manifold; and finally, a workhardening region. In a case such as this, there
exists no physical collapse load. However, to know the load at which the footing
will deform excessively has obvious practical importance. For this purpose,
idealizing the soil as a perfectly plastic medium and neglecting the changes
in geometry lead to the condition in which displacements can increase without
limit while the load is held constant as shown in Fig. 1.
A load computed on the basis of this ideal situation is called plastic limit
load (Burland et al., 1996). This hypothetical
limit load usually gives a good approximation to the physical plastic collapse
load or the load at which deformations become excessive. The methods of limit
analysis furnish bounding estimates to this hypothetical limit load.

Fig. 1: 
Loaddisplacement curve 
The theorems of limit analysis can be established directly for a general body
if the body possesses the following ideal properties:
• 
The material exhibits perfect or ideal plasticity, i.e. work
hardening or work softening does not occur. This implies that stress point
can not move outside the yield surface 
• 
The yield surface is convex and the plastic strain rates are derivable
from the yield function through the associated flow rule 
• 
Changes in geometry of the body that occur at the limit load are in significant;
hence the equations of virtual work can be applied 
• 
In summary, the limit load is defined as the plastic collapse load of
an ideal body having the ideal properties listed above and replacing the
actual one 
THE LOWER BOUND METHOD
The lowerbound method of limit analysis is different from the upperbound
method in that the equilibrium equation and yield condition instead of the work
equation and failure mechanism are considered (Kramer, 1996).
Moreover whereas the development of the work equation from an assumed collapse
mechanism is always clear, many engineers find the construction of a plastic
equilibrium stress field to be quite unrelated to physical intuition. Without
physical insight there is trouble in finding effective ways to alter the stress
fields when they do not give a close bound on the collapse or limit load. Often
the user employs the existing stress fields from wellknown texts or the more
recent technical literature as a magic handbook and tries to fit his problem
to the particular solutions he finds. Intuition and innovation seem discouraged
by unfamiliarity and apparent complexity (Zhao et al.,
2005). Although the discontinuous fields of stress which will be drawn and
discussed in this Section are simpler to visualize, they too are not often employed
in an original manner by the design engineer (Li et al.,
2009; Merifield et al., 2006). Yet, in fact,
the concepts are familiar to the civil engineer in his terms and can be utilized
by the designer as a working tool.
The conditions required to establish such a lowerbound solution are essentially
as follows:
• 
A complete stress distribution or stress field must be found,
everywhere satisfying the differential equation of equilibrium 
• 
The stress field at the boundary and discontinuities must satisfy the
stress boundary conditions 
• 
The stress field must nowhere violate the yield condition 
ANALYTICAL SOLUTION FOR ACTIVE CASE
The typical 2D wall geometry for the problem of this study is shown in Fig. 2. Assuming a discontinuity surface, Fig. 2 is showing the variation of stresses in the vicinity of the wall (zone A) and beyond the discontinuity surface (zone B). The final target of the calculations is leading to evaluation of P_{ah} and P_{av} which are the stresses subjected to the earthquake affected on the wall. In this solution the following relation is assumed:
where, c and Φ are known as the strength parameters of the material; c represents the cohesion and Φ represents the angle of internal friction. cw is the cohesion and Φw is the internal friction angle between wall and soil. Knowing stresses quantities in element B, the position of S_{b} is drawn in Fig. 3.
The Mohr circle center and radius are considered as S_{a}, S_{b} and r_{a}, r_{b} respectively for zones A and B.
The soil is modeled by MohrCoulomb yield criterion with various quantities
of friction angle and soil cohesion. In a direct application of the MohrCoulomb
criterion for plane strain stability problems, it is implicitly assumed that
the strength of the soil along the failure surface is fully mobilized everywhere
along the surface. This is probably the case in most laboratory tests in which
the tested specimen is assumed representative of a soil element in the soil
mass. This is because the specimen is generally so small that the strain is
practically considered uniform along the failure surface, although boundary
restrains do exist in almost all tests. For simplicity, the effect of seepage
(or pore pressures) on the stability of cohesivefrictional soils has not been
included in this study. It is also possible to incorporate the effect of pore
pressures in limit analysis but this extension is not being covered here.

Fig. 2: 
Stress discontinuity surface, zones A and B 

Fig. 3: 
Assuming S_{b} as (x, 0) 
The position of stresses of zone B is shown in Fig. 3. The
relation of Mohr circle center and radius of zoneB can express by:
Combining Eq. 1 and 2 results in:
Expanding Eq. 3 leads to:
where, x = S_{b}. As S_{b} anr_{b}are calculated, the Mohr circle of zoneB is drawn (Fig. 4).
In Fig. 4 P_{b} is the pole of Mohr circle of zoneB. For computation of the angle between P_{b} and the principle surface (α) in Mohrcircle of zone B, using geometrical relations leads to following equations:
Substituting r_{b} in above equation leads to:

Fig. 4: 
Mohr circle in zone part B 

Fig. 5: 
Assumed Mohr circle in zone part A 
Dismounting the wall specifications, the Mohrcircle of zoneA is drawn (Fig. 5). Using Fig. 5 results in the following equations:
Combining Eq. 4 and 6 results in:
Considering β as an angle through zoneA stresses surface and principle surface, the rotation angle of stresses from zone A to B will become to:
From Fig. 5 and 6, δθ is defined
as following equation (Fig. 7):
Substituting α and β in above equation leads to:
In which δθ is rotation angle of stresses from zone B to A. Using
the relation between two center point of Mohrcircles of zones A and B, reported
by Chen and Liu (1990), the radius Mohr circle of zone
A is derived:
Knowing the quantities of the r_{a} and S_{a}, Mohrcircle
of element A is drawn. According to Fig. 8, depicting with
line throw intersection point of circles (M) and pole B (P_{b}) and
extending, pole of Mohrcircle in zone A is appeared which leads to attaining
the target.

Fig. 8: 
Calculation results 
In summery Calculation algorithm of p_{ah} is defined as follows:
• 
Calculation of S_{b}, r_{b} and α using
Eq. 2, 4 and 6 
• 
Calculation of β using Eq. 12 
• 
Determining the rotation angle between the Mohrcircles (δθ
= βα) then use of Eq. 14 and 16
for calculation of S_{a} and r_{a}, respectively 
• 
Drawing the Mohrcircles, finding pole of element in zone A which leads
to calculation of p_{ah.} 
Next section is discussed about comparison of this mathematical solution and MononobeOkabe Method which results in some tables and graphs.
COMPARISON BETWEEN RESULTS
The MononobeOkabe analysis which is an extension of the coulombs analysis, has been experimentally proved by MononobeMatsuo to be effective in assessing the seismic active earth pressure. It is generally adopted in the current a seismic design of rigid retaining walls. The MononobeOkabe solution is therefore practically acceptable at least for the active pressure case, although its applicability to the passive pressure is somewhat in doubt.
In this section, some results on seismic active pressures as obtained by the
present Analytical method are compared with the method of MononobeOkabe (MO)
which leads to Table 1 to 4. Comparing the
current results with these methods, good agreement is found among them.
In following tables, k_{h} is horizontal seismic coefficient, δ
is friction angle between wall and soil in MononobeOkabe method and K_{AE}
is active seismic lateral pressure coefficient. For Table 1
to 3 the results are coming out for C = 0, Φ_{w}
= Φ_{cs}/2 = 15°, H = 5 m, γ = 17.6 kN m^{3}
and for Table 4 which is compared between Rankin theory and
current study, the results are coming out for Φ = 30°, γ = 18
kN m^{3}.
So it has been found that the application of limit analysis for cohesionless soil stability problems is practically acceptable. The determination of the seismic lateral earth pressure of a fill on a retaining wall when frictional forces act on the back of the wall, is solved conveniently by this analytical method. As it is seen, the results of Analytical solution and MononobeOkabe are practically identical for most cases. By checking out the results of Chung and Chen which are based on upper bound method of limit analysis, it seems that the exact result has a negligible difference with the results of this method.
Table 1: 
Comparison of K_{AE} for C=0, Φ_{w} =
Φ/2, K_{h} = 0.15, K_{v} = .075, H = 5 m, γ =
17.6 kN m^{3} 

Table 2: 
Comparison of K_{AE} for C = 0, Φ_{w}
= Φ/2, H = 5 m, γ = 17.6 kN m^{3} 

Table 3: 
Comparison of K_{AE} for C = 0, Φ_{w}
= Φ_{cs}/2 = 15°, H = 5 m, γ = 17.6 kN m^{3} 


Table 4: 
Comparison of active lateral pressure for Φ = 30°,
γ = 18 kN m^{3} 

NUMERICAL RESULTS FOR ACTIVE CASE
The lower bound solutions obtained can be applied directly in practice and
one of the most usable applications of this study is the possibility of introducing
some practical dimensionless diagrams for calculating the active seismic lateral
force of retaining walls with the considerable accuracy.

Fig. 9: 
Dimensionless diagram for calculating the active seismic lateral
force, K_{h} = 0.1 

Fig. 10: 
Dimensionless diagram for calculating the active seismic lateral
force, K_{h} = 0.2 

Fig. 11: 
Dimensionless diagram for calculating the active seismic lateral
force, K_{h} = 0.3 
Figure 911 illustrate the activeseismic
lateral force in various quantities of friction angle and cohesion of the soil
and soilwall. The dimensionless parameters presented are defined as:
where, γ is soil unit weight, H is the wall height, c is the cohesion
of the soil fill back of the wall and P_{a} is the seismic lateral force
which is affected the wall. For each seismic coefficient, the results for three
different λ of 3, 5 and 10 are given. To account for the effect of c_{w}
and Φ_{w}the results are presented in terms of c_{w} of
0.2, 0.3 and 0.8. As from Fig. 911 are
displayed, by increasing the soil friction angle, the seismic active force is
decreased, as expected. Comparing Fig. 911,
it seems that for a given λ the active seismic force will increase with
increasing c_{w}/c. Also it seems that increasing in λ and k_{h},
leads to increase the seismic lateral pressure.
SOLUTION ALGORITHM FOR PASSIVE CASE
Similar to the solution algorithm of active case, using some changes in calculation
process the passive seismic lateral earth pressure is computed. Also it can
be possible to introduce some practical dimensionless graphs in this case. This
algorithm can be able to cover the lack of certainty which lies with MononobeOkabe
method in passive case.

Fig. 12: 
Mohrcircle of zoneB in passive case 

Fig. 13: 
Assumed Mohrcircle of zoneA 
The Mohrcircle of zoneB in passive case is drawn in Fig. 12.
Using geometrical relations leads to following Equations:
Substituting r_{b} in equation 17 leads to:
Dismounting the wall specifications, the assumed Mohrcircle of zoneA is drawn (Fig. 13). Using Fig. 13 results in similar equation as active case but β is calculated as follows:
From Eq. 19 and 20, δθ is defined
as following equation (Fig. 14):
Substituting α and β in above equation and using the relation between two center point of Mohrcircles of zones A and B, reported by Chen, the Mohrcircle center and radius in zone A are derived.
P_{ph} and P_{pv} in Fig. 14 are the horizontal and vertical passive seismic lateral pressure on the wall, respectively.
NUMERICAL RESULTS IN PASSIVE CASE
In this section, some results on static and seismic passive pressures as obtained
by the present Analytical method are compared with wellknown methods such as
Rankin theory (for static case), Ghahramani and Clemence
(1980), Chen et al. (2003) and modified Dubrova
(1963) which leads to Table 5 and 6.
Comparing the current results with other methods, good agreement is found among
them, so it is concluded that the results of this study is reliable.
As it is seen the result of current solution is close to the other analytical
methods and also it can be conservative, therefore the method obtained can be
applied directly in practice and one of the most usable applications of this
study is the possibility of introducing some practical dimensionless diagrams
for calculating the passive seismic lateral pressure coefficient of retaining
walls with the considerable accuracy. Fig. 15 to 17
illustrate the passive seismic lateral force in various quantities of friction
angle and cohesion of the soil and soilwall. The dimensionless parameters presented
are defined as:
where, P_{p} is the passive seismic lateral force which is affected
on the wall.

Fig. 15: 
Dimensionless diagram for calculating the passive seismic
lateral force, K_{h} = 0.1 

Fig. 16: 
Dimensionless diagram for calculating the passive seismic
lateral force, K_{h} = 0.2 
Table 5: 
Comparison of passive lateral pressure for Φ = 30°,
γ = 18 kN m^{3}, Kh=0 


Table 6: 
Numerical comparison of solution of various analytical methods
for seismic passive earth pressure for vertical wall, ø = 40, ø_{w}
= 2/3ø, Kh = 0.15 


Fig. 17: 
Dimensionless diagram for calculating the passive seismic
lateral force, K_{h} = 0.3 

Fig. 18: 
Results comparison for distinguishing the effect of c_{w}/c 
For each seismic coefficient, the results for three different λ of 3,
5 and 10 are given. To account for the effect of c_{w} and Φ_{w}the
results are presented in terms of c_{w} of 0.2, 0.3 and 0.8. Having
found these parameters for each problem, one can compute the conservative seismic
passive lateral force in a retaining wall.
As from Fig. 15 to 17 are displayed,
by increasing the soil friction angle, the seismic passive force is increased.
Fig. 18 is a comparison between the results for distinguishing
the effect of c_{w}/c; it seems that for a given λ the passive
seismic force will increase with increasing c_{w}/c.

Fig. 19: 
Results comparison for the effect of k_{h} 
Also from Fig. 19 it seems that increasing in λ leads to increase and increasing in k_{h} comes to decrease the seismic lateral force (P_{p}).
EXAMPLE OF APPLICATION
Now it is illustrated how the results in Fig. 9 to 11
and Fig. 15 to 17 can be used to determine
the seismic active and passive lateral force.
Problem. A wall is built back of a soil which has following parameters, the height of the wall H = 5 m, the soil unit weight is γ = 15 kN m^{3}, the soil’s strength parameters c = 10kN m^{2}, Φ = 30 and the soilwall cohesion c_{w} = 5 kN m^{2}. For a seismic coefficient of k_{h} = 0.1 what is the amount of seismic active and passive lateral force?
A procedure for using the results of the presented study to solve the forgoing
problem can be summarized as follows. First active case:
• 
From the value of γ, H and c, the dimensionless parameter
λ = γH/c = 7.5 is calculated 
• 
With k_{h} = 0.1 and c_{w}/c = 0.5, it follows that the
results presented in Fig. 9a should be used to determine
the force 
• 
In Fig. 9a, a straightvertical line passing through
λ = 7.5 is drawn. This straight line will intersect with three curves
which the intersection point of curve with Φ = 30 and c_{w}/c
= 0.5 is selected 
• 
From this intersection point, it can backfigure the following dimensionless
parameter P^{’} from which the lower bound solution of the
seismic active force can be calculated as P= 11.23 kN m^{1} 
• 
Going on the process for passive case, Fig. 15a is
chosen and the lower bound solution of the seismic passive force can be
calculated as P = 1043.15 kN m^{1} 
CONCLUSION
The active and passive seismic lateral pressures on retaining walls are investigated in this paper. An analytical solution is introduced based on lower bound limit analysis method and solution is compared to the wellknown methods such as Mononobe Okabe, Chung and Chen, Ghahramaniclemence, Dubrova and Rankin which good agreement is found among them. Some practical dimensionless diagrams for calculating the active and passive seismic forces on retaining walls with the considerable accuracy are presented. The results show that in active case by increasing the soil friction angle, the seismic active force is decreased, as expected but in passive case inversely, the seismic passive force is increased. Comparing diagrams, it seems that for a given λ = γH/c both active and passive seismic force will increase with increasing c_{w}/c. Also it is found that increasing in λ leads to increase both active and passive seismic lateral force. Comparing diagrams for various quantities of k_{h}, shows that increasing in k_{h} leads to increase the active pressure but the passive pressure decreases.
NOMENCLATURES
c 
= 
Cohesion 
c_{w} 
= 
Cohesion between soil and wall 
Φ 
= 
Internal friction angle 
Φ_{w} 
= 
Internal friction angle between soil and wall 
γ 
= 
Soil unit weight 
z 
= 
Height 
P_{pv} 
= 
Vertical passive seismic lateral pressure 
P_{ph} 
= 
Horizontal passive seismic lateral pressure 
S_{a} 
= 
Center point of Mohrcircle in zone A 
S_{b} 
= 
Center point of Mohrcircle in zone B 
r_{a} 
= 
Radius of Mohrcircle in zone A 
r_{b} 
= 
Radius of Mohrcircle in zone B 
P_{a} 
= 
Pole of zoneA 
P_{b} 
= 
Pole of zoneB 
α 
= 
Angle between P_{b} and the principle surface 
β 
= 
Angle between P_{a} and the principle surface 
δθ 
= 
Rotation angle of stresses from zone B to A 
M 
= 
Intersection of Mohrcircles 
K_{h} 
= 
Seismic coefficient 
λ 
= 
Dimensionless parameter = γH/c 
P_{a} 
= 
Seismic active lateral force 
P_{p} 
= 
Seismic passive lateral force 
p' 
= 
Dimensionless parameter 