Limit Analysis for 3D Stability of Unsaturated Inhomogeneous Slopes Reinforced with Piles

Acknowledgments

Notation

a, d

distances from the centerline of the conical volume to the slope crest for toe-failure and base-failure mechanism;

b

width of “plane insert”;

c

cohesion strength;

c_f

cohesion strength of the slope face;

$c_{\mathrm{app}}^{2\mathrm{D}}$

distribution of apparent cohesion along the sliding surface for the 2D portion;

$c_{\mathrm{app}}^{3\mathrm{D}}$

distribution of apparent cohesion along the sliding surface for the 3D portion;

D₁

center-to-center spacing between piles;

D₂

opening between piles;

D_p

rate of the internal energy dissipation by the total resistance power of the anti-slide pile;

D_S

rate of the internal energy dissipation by apparent cohesion;

D_t

rate of the internal energy dissipation by soil cohesion;

e

distances from the centerline of the conical volume to the slope base;

G_S

specific gravity of soil;

h

integration limit along z′;

h_f

distance from the slope face to the slope crest;

k_s

saturated hydraulic conductivity;

L_X

horizontal distance from the top to the toe of the slope;

N_φ

function related to φ;

n₀

inhomogeneous coefficient;

q

vertical steady flow rate;

q/k_s

influence of normalized flow rate;

R

radius of the conical volume;

r

radius of log-spiral;

R_p

reinforcement effect of pile;

r′

radius of inner log-spiral;

r₀

initial radius of log-spiral OA;

$r_0^{\mathrm{\prime }}$

initial radius inner of log-spiral OA′;

r_h

final radius of the log-spiral;

r_m

average radius of the two log-spirals;

r_p

initial radius of log-spiral OP;

S

degree of pore–water saturation;

S_e

normalized degree of saturation;

S_r

residual degree of saturation;

u_a

pore–air pressure;

u_w

pore–water pressure;

u_a − u_w

matric suction;

$W_{\gamma }^{2\mathrm{D}}$

rate of external forces done by the effective soil weight of the 2D part;

$W_{\gamma }^{3\mathrm{D}}$

rate of external forces done by the effective soil weight of the 3D part;

X_F

horizontal distance between the piles position and slope toe;

z′

depth of soil layer from the ground surface;

z₀

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

z_2D

vertical distance from a general point on the sliding surface to the water table level for the 2D part;

z_3D

vertical distance from a general point on the sliding surface to the water table level for the 3D part;

α

inverse of air-entry pressure;

β

slope inclination angle;

β′

auxiliary slope inclination angle;

γ′

effective 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

internal friction angle;

θ₀

initial rotational angle of the failure mechanism;

θ_B

rotational angles from horizontal line to the line passing through point B;

θ_C

rotational angles from horizontal line to the line passing through point C;

θ_h

final rotational angle of log-spiral;

θ_e

rotational angles from horizontal line to the line passing through point E; and

θ_p

rotational angles from horizontal line to the line passing through point P.

References