Second-Order Reliability-Based Design of Unsaturated Infinite Soil Slopes

Acknowledgments

Notation

[A]

matrix used to obtain principal curvatures;

a_f

fitting parameter of the SWCC related to AEV (kPa);

a_fω

normalized parameter for a_f (dimensionless);

b_cb

width of the cantilever beam (mm);

b_wb

width of the welded beam (mm);

C_r

axial compression resistance of the concrete beam (kN);

c′

effective cohesion (kPa);

$c_d^{\mathrm{\prime }}$

effective cohesion mobilized along the slip surface (kPa);

F_c

factor of safety with respect to effective cohesion;

$F_{\varphi }$

factor of safety with respect to the angle of friction;

g(x), g(u)

performance function or LSF (dimensionless);

H

depth of slip surface (m);

H_w

suction head (m);

h_cb

height of the cantilever beam (mm);

I_p

plasticity index of the soil;

L_cb

length of the cantilever beam (mm);

L_wb

length of the welded beam (mm);

l

width of soil element (m);

[M]

Hessian matrix having partial derivatives of second order (units of random variable);

m_f

fitting parameter of the SWCC related to residual water content;

N

total number of Monte Carlo simulations;

N_a, T_a

normal and shear forces acting on the slip surface (kN/m);

N_r, T_r

normal and tangential components of R (kN/m);

n

number of random variables;

n_f

fitting parameter of the SWCC related to the slope of the SWCC (dimensionless);

P

forces acting on faces of the slice (kN/m);

$P_{f\mathrm{\_}\mathrm{FORM}}$

probability of failure computed using FORM;

$P_{f\mathrm{\_}\mathrm{MCS}}$

probability of failure computed using MCS;

$P_{f\mathrm{\_}\mathrm{SORM}}$

probability of failure computed using SORM;

P_l

permanent load effect (kN);

P_wb

load acting on welded beam (kN);

P_x

horizontal load acting on cantilever beam (kN);

P_y

vertical load acting on cantilever beam (kN);

[Q]

rotation matrix (dimensionless);

[q₁], [q₂], …, [q_n]

row vectors of matrix [Q];

R

reaction force to W (kN/m);

S

degree of saturation (dimensionless);

S_n

stability number (dimensionless);

S_y

yield strength of the cantilever beam (kPa);

[T]

matrix used for Gram–Schmidt orthogonalization;

[t₁], [t₂], …, [t_n]

row vectors of matrix [T];

t_wb

height of the welded beam (mm);

U

standard normal space (dimensionless);

U_a, U_w

pore-air and pore-water forces (kN/m);

u_a, u_w

pore-air and pore-water pressures (kPa);

(u_a − u_w), ψ

matric suction (kPa);

$u^{*}$

design point on standard normal space;

V_l

variable load effect (kN);

W

weight of soil element (kN/m);

X

physical space;

z_w

depth of water table (m);

α

unit gradient vector (dimensionless);

α₁, α₂, …, α_n

directional cosines of α (dimensionless);

β_FORM, β_SORM

reliability indices computed using FORM and SORM;

β_MCS

reliability index computed using MCS;

γ

unit weight of soil (kN/m³);

γ_w

unit weight of water (kN/m³);

ɛ₁

percentage difference between $P_{f\mathrm{\_}\mathrm{MCS}}$ and $P_{f\mathrm{\_}\mathrm{FORM}}$;

ɛ₂

percentage difference between $P_{f\mathrm{\_}\mathrm{MCS}}$ and $P_{f\mathrm{\_}\mathrm{SORM}}$;

ζ

correction factor (dimensionless);

θ_r

residual water content;

θ_s

saturated water content;

θ_w

volumetric water content;

κ_i

principal curvatures of LSF (dimensionless);

μ_i

mean values of the random variable;

σ, σ′

total and effective stresses (kPa);

σ_i

standard deviation of random variable;

σ_max

design normal stress of the beam material (kPa);

$\varsigma$

fitting parameter of shear strength models (dimensionless);

τ

shear stress (kPa);

τ_d

shear strength mobilized along potential slip surface (kPa);

τ_f

shear strength of soil (kPa);

Φ

ratio of the tangent of the effective friction angle to the tangent of the slope angle (dimensionless);

Φ()

standard normal CDF;

ϕ′

effective angle of friction (°);

${\varphi }_d^{\mathrm{\prime }}$

effective angle of friction mobilized along slip surface (°);

χ

property of effective stress depends on the degree of saturation and varies with respect to the USS model (dimensionless);

ω

slope angle (°);

${\mathrm{\nabla }}^{2}g(u^{*})$

second-order derivatives of LSF at the design point in the U-space; and

$|\mathrm{\nabla }g(u^{*})|$

length of the gradient vector in the U-space.

References