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in which K a is the coefficient of active earth pressure; γ soil is the unit weight of the backfill;
and H is the height of the wall. According to Rankine's theory, K a is expressed as
πφ
tan 2
K a
=
(2.33)
42
It is assumed that the back of the retaining wall is fully drained, that is, the water table is
below the base of the wall and it has no effect on the stability of the wall. The active earth
thrust P a will act at a height of H 0 ( Arm a = H 0 ) above the base of the wall with a horizontal
direction. In Figure 2.7 , the following equations can be established, for a wall with a verti-
cal back:
1
2
2
3
Armb a
(2.34)
W
=
γ
bHArm
,
=
bW
,
=
γ
aH
,
=
+
1
wall
1
2
wall
2
2
in which γ wall is the unit weight of the retaining wall concrete.
It is well known that the shear strength parameters c and ϕ of the retained soil have a
significant influence on the probability of overturning failure (Abd Alghaffar and Dymiotis-
Wellington 2007). Therefore, both c and ϕ are treated as random variables. Following
Jimenez-Rodriguez et al. (2006), a Lognormal distribution is adopted to model the distribu-
tions of c and ϕ. The other five parameters, namely H , a , b , γ soil , and γ wall , are assumed to be
constants so that the correlation between c and ϕ can be effectively explored without inter-
ference from the other random variables. The mean and COV of c are assumed to be 12 kPa
and 0.4, respectively. The mean and COV of ϕ are assumed to be 20° and 0.2, respec-
tively. The deterministic parameters are listed as follows: H = 6 m, a = 0.4 m, b = 1.4 m,
γ soil = 18 kN/m 3 , and γ wall = 24 kN/m 3 . In this example, keeping the aforementioned advan-
tages and weaknesses of Pearson's rho in mind, Pearson's rho is adopted to determine the
copula parameters θ to be consistent with the geotechnical engineering practice. On the
basis of ρ in Tables 2.2 through 2.4, a ρ = −0.5 is used to account for the effect of correlation
on the probability of retaining wall overturning failure.
2.5.2 Probability of failure using direct integration
In the current reliability literature, many reliability methods such as the first-order reliabil-
ity method (FORM), second-order reliability method (SORM), and Monte Carlo simulation
(MCS), are available for determining the probability of failure. To remove the errors result-
ing from linearization at the design point underlying the FORM and SORM, and statistical
errors due to sampling sizes underlying the MCS, a direct integration method is adopted to
determine the probability of failure. In this way, the effect of different copulas on the prob-
ability of failure can be identified accurately.
The following performance function is adopted for the retaining wall example:
g ( c , ϕ) = FS ( c , ϕ) − 1
(2.35)
in which FS ( c , ϕ) is determined by Equation 2.31 . It should be noted that the performance
function of the retaining wall example is a cubic equation with respect to cohesion c .
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