Stochastic Finite-Element Analysis of Earth–Rockfill Dams Considering the Spatial Variability of Soil Parameters

Acknowledgments

Notation

A_ρ, A_K

n-dimensional independent standard normal distribution random variable for ρ and K;

a, b

fitting parameters of regression functions between the dry density and Duncan E-B model parameters;

C

autocorrelation matrix;

C₀, C₁, t

fitting parameters;

c_ij

autocorrelation coefficient between © and j position;

G(ρ)

cumulative distribution function of the log-normal distribution of ρ;

G′(ρ), G_K′(K)

modified cumulative distribution function of the log-normal distribution of ρ and K;

g′(ρ)

modified log-normal distribution function of soil dry density;

h

distance between any two dry densities in the monitoring positions;

H_ρ, H_K

modified correlated random field of ρ and K;

${\mathit{H}}_{\mathit{\rho }}^{\mathit{D}},{\mathit{H}}_{\mathit{K}}^{\mathit{D}}$

correlated standard normal random field of ρ and K;

K, n, K_b, m

parameters of Duncan E-B model;

L

lower triangular matrix decomposed from C;

L₀

average length;

l_x, l_y

horizontal and vertical autocorrelation distance;

N_h

number of pairs of two soil dry density separated by h;

R_f

failure ratio of Duncan E-B model;

X

coordinates of the soil positions in the dam;

x_i, x_j

horizontal coordinates of positions i and j;

y_i, y_j

vertical coordinates of positions i and j;

δ

scale of fluctuation;

δ_u, δ_v

fluctuation function;

δ_h, δ_v

horizontal and vertical scale of fluctuation;

Δz

distance between the different soil dry-density monitoring positions;

ϕ₀, Δϕ

shear strength parameter;

Φ_ρ, Φ_K

the cumulative distribution function of the standard normal distribution of ρ and K;

γ (L₀)

variance function ranging from 0 to 1;

γ* (h)

experimental semivariogram;

μ_lnρ, σ_lnρ

mean value and SD of the log-normal distribution of soil dry density;

μ_ρ, σ_ρ

mean value and SD of the of soil dry density;

ρ

soil dry densities;

${\rho }_{L_0}$

a family of moving average processes of the soil dry density;

ρ_max, ρ_min

maximum and minimum soil dry densities measured in situ; and

σ₁, σ₃

major principal and minor principal stress.

References