Kinematical Analysis of 3D Seismic Stability of Slopes in Unsaturated and Inhomogeneous Soils Using a Pseudodynamic Method

Acknowledgments

Notation

a_h

horizontal seismic acceleration;

a_h0

horizontal seismic acceleration at the slope base;

B

maximum width of the rotational failure mechanism;

b

width of “plane-insert” block;

$B_{max}^{\mathrm{\prime }}$

maximum width of the 3D horn failure part;

c_app

apparent cohesion;

$c_0^{\mathrm{\prime }}$, $c_f^{\mathrm{\prime }}$, and $c_s^{\mathrm{\prime }}$

effective cohesion strength of the slope base, slope face, and slip surface, respectively;

c^3D and c^2D

effective cohesion stress in 3D horn failure part and plane-insert failure part of the slope, respectively;

$D_{c^{\mathrm{\prime }}\text{-}\mathrm{horn}}$ and $D_{c^{\mathrm{\prime }}\text{-}\mathrm{insert}}$

rate of the internal energy dissipation of 3D horn failure part and plane-insert failure part of slope done by effective soil cohesion, respectively;

D_s-horn and D_s-insert

rate of the internal energy dissipation of 3D horn failure part and plane-insert failure part of slope done by apparent cohesion, respectively;

d₁, d₂, and d₃

distances from the centerline of the conical volume to slope crest, slope face, and slope base, respectively;

d′

distance from the center of the cross section to the water table;

E

Young’s modulus;

f_a

amplification factor of soil;

F_s

factor of safety of slope;

g

gravitational acceleration;

G_s

specific gravity of soil;

H

slope height;

H′

auxiliary height of slope;

h_f and h_s

distance from the slope face and slip face to the slope crest, respectively;

k_h

horizontal seismic coefficient;

k_s

saturated hydraulic conductivity;

N

distance from the toe of the slope to the lowest point on the failure surface;

n

distribution of the soil’s pore size;

n^w and n^d

pore size distribution parameter of wet path and dry path, respectively;

n₀

inhomogeneous coefficient;

q

vertical steady flow rate;

R

radius of the conical volume;

r and r′

radius of log-spiral and inner log-spiral, respectively;

r₀ and $r_0^{\mathrm{\prime }}$

initial radius of log-spiral OA and OA′, respectively;

r_h

final radius of the log-spiral;

r_m

average radius of the two log-spirals;

r_p and $r_{p^{\mathrm{\prime }}}$

radius of log-spiral OP and OP′, respectively;

r_u

pore-water coefficient;

S

degree of pore-water saturation;

S_e

normalized degree of saturation;

S_r

residual degree of saturation;

S_t

area of the sliding surface of the slope;

T

shaking period;

t

time;

u_a

pore-air pressure;

(u_a − u_w)

matric suction;

u_w

pore-water pressure;

V_s

shear-wave velocity;

V_us and V_sa

volume of unsaturated and saturated parts in the sliding body, respectively;

W_e-horn and W_e-insert

rate of external forces done by seismic force of the 3D horn failure part and plane-insert failure part of slope, respectively;

W_γ-horn and W_γ-insert

rate of external forces done by soil weight of the 3D horn failure part and plane-insert failure part of slope, respectively;

z₀

vertical distance from the water table to the slope toe elevation;

z′

distance above the slope toe;

z_3D

vertical distance from the any one point of failure block to the water table level;

z_2D

vertical distance from the point on the sliding surface of the plane-insert block to the water table level;

α

inverse of air-entry pressure;

α^w and α^d

inverse of air-entry pressure of wet path and dry path, respectively;

β

slope inclination angle;

β′

auxiliary slope inclination angle;

γ′

effective soil unit weight;

γ_sat

unit weight of saturated soil;

γ_w

unit weight of water;

σ

total stress;

σ′

effective stress;

σ^s

suction stress;

ϕ′

effective internal friction angle;

ξ

damping ratio of soils;

μ

Poisson’s ratio;

λ_s

wavelength of the shear wave;

τ_f

shear strength;

θ

rotational angle;

θ₀

initial rotational angle;

θ_B and θ_C

rotational angles from horizontal line to the line passing through point B and C, respectively;

θ_h

final rotational angle of log-spiral; and

${\theta }_p\mathrm{and}{\theta }_{p^{\mathrm{\prime }}}$

rotational angles from horizontal line to the line passing through point P and P′, respectively.

References