Environmental Engineering Reference
In-Depth Information
lISt oF SYMbolS
Φ(.)
Standard normal distribution function
Φ
−1
(.)
Inverse standard normal distribution function
Φ
θ
(.,.)
Bivariate standard normal distribution function with Pearson's rho θ
φ
θ
(.)
Generator function
ϕ
Friction angle
θ
Copula parameter
θ
Vector of copula parameters
λ
COV scaling factor
λ
U
Coefficient of upper tail dependence
λ
L
Coefficient of lower tail dependence
Γ(.)
γ-Function
μ
Mean value
μ
i
Mean value of
X
i
σ
Standard deviation
σ
2
Variance
σ
i
Standard deviation of
X
i
γ
soil
Unit weight of the backfill
γ
wall
Unit weight of the retaining wall concrete
ρ
Pearson linear correlation coefficient (Pearson's rho)
τ
Kendall rank correlation coefficient (Kendall's tau)
AIC
Akaike information criterion
Arm
a
Vertical lever distance of
P
a
a
,
b
,
H
Geometrical parameters of the retaining wall
Arm
1
and
Arm
2
Horizontal lever distances of
W
1
and
W
2
BIC
Bayesian information criterion
c
Cohesion
CDF
Cumulative distribution function
CQ
Consolidated-quick direct shear test
CD
Consolidated-drained triaxial compression test
CU
Consolidated-undrained triaxial compression test
Cov(.,.)
Covariance
COV
Coefficient of variation
COV
c
Coefficient of variation of
c
COV
ϕ
Coefficient of variation of ϕ
C
(
u
1
,
u
2
, …,
u
n
; θ)
n
-Dimensional copula function
C
(
u
1
,
u
2
; θ)
Bivariate copula function
C
2
(
U
2
|
U
1
)
Conditional distribution of
U
2
given the values of
U
1
D
(
u
1
,
u
2
, …,
u
n
; θ)
n
-Dimensional copula density function
D
(
u
1
,
u
2
; θ)
Bivariate copula density function
FS
Mean factor of safety
FS
n
Nominal factor of safety
F
(
x
1
,
x
2
, …,
x
n
)
Joint CDF of [
X
1
,
X
2
, …,
X
n
]
f
(
x
1
,
x
2
, …,
x
n
)
Joint PDF of [
X
1
,
X
2
, …,
X
n
]
F
(
x
1
,
x
2
)
Bivariate CDF of
X
1
and
X
2
f
(
x
1
,
x
2
)
Bivariate PDF of
X
1
and
X
2
f
(
c
, ϕ)
Bivariate PDF of
c
and ϕ
F
i
(.)
Marginal CDF of
X
i
F
1
(
c
)
Marginal CDF of
c
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