Three-Dimensional Active Earth Pressures for Unsaturated Backfills with Cracks Considering Steady Seepage

Data Availability Statement

Notation

A

loaded boundary;

A_r

stressed area imposed by active earth pressure;

B

width of the retaining wall;

B/H

width-to-height ratio;

B_e

equivalent width;

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

maximum width of the rotational mechanism;

b

width of the plane insert portion;

c_app

apparent cohesion;

c′

effective cohesion;

c′/γH

normalized soil cohesion;

D_c

internal dissipated energy due to the soil effective cohesion;

$D_{c_{\mathrm{app}}}$

internal dissipated energy due to the apparent cohesion;

$D_{c^{\mathrm{\prime }}-3\mathrm{D}}$, $D_{c^{\mathrm{\prime }}-3\mathrm{D}}^{AE}$, $D_{c^{\mathrm{\prime }}-3\mathrm{D}}^{AC}$, $D_{c^{\mathrm{\prime }}-\mathrm{insert}}$, $D_{c^{\mathrm{\prime }}-\mathrm{insert}}^{AE}$, $D_{c^{\mathrm{\prime }}-\mathrm{insert}}^{AC}$

D is the internal dissipated energy. The subscript c′ represents the work rates generated by the effective cohesion; 3D and insert represent different portions of the failure mechanism, respectively. The superscripts represent the area of the slip surface;

d₁

distance between the rotation center with the slip surface;

F_i

vector of body force;

$g_1\sim g_5$, $g_1^{\mathrm{\prime }}\sim g_5^{\mathrm{\prime }}$, $f_1\sim f_7$, $f_1^{\mathrm{\prime }}\sim f_7^{\mathrm{\prime }}$

dimensionless functions;

H

wall height;

K_a

active earth pressure coefficient;

k_s

saturated hydraulic conductivity;

L

length of AD;

L′

length of AB;

n

pore-size distribution;

O

rotation center;

P_a

active earth pressure;

q

vertical specific discharge;

R

radius of the curvilinear cone mechanism;

r₀

length of OA;

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

length of OA′;

r_m

distance from the axis of a curvilinear cone to a rotation point;

T_i

vector of the surface force;

u_a

pore-air pressure;

u_w

pore-water pressure;

u_a − u_w

matric suction;

${\dot{u}}_i^{\ast }$

velocity vector;

V

volume;

$W_{P_a}$

external work rates generated by the active earth pressure;

$W_{\gamma }$

external work rates generated by the soil weight;

W_γ−3D, $W_{\gamma -3\mathrm{D}}^{ADE}$, $W_{\gamma -3\mathrm{D}}^{\mathrm{A}BC}$, W_γ−insert, $W_{\gamma -\mathrm{insert}}^{ADE}$, $W_{\gamma -\mathrm{insert}}^{ABC}$

W is the external work rate. The subscript γ represents the work rates generated by the soil unit weight; 3D and insert represent different portions of the failure mechanism, respectively. The superscripts represent the volume of failure block;

z₀

interval between the base and the water table;

z

vertical interval above the water table;

1/α

air-entry value;

β

inclination angle of the wall back;

χ

degree of saturation (0–1);

δ

general angle to specify the slip surface in the rotational plane, as illustrated in Fig. 3;

δ_w

wall interface friction angle;

δ₁, δ₂

boundary of the general angle δ;

${\dot{\epsilon }}_{ij}^{\ast }$

strain rate;

γ

soil unit weight;

γ_w

unit weight of water;

γH/c

normalized critical height;

ω

angular velocity;

φ′

effective internal frication angle;

σ^s

suction stress;

σ′

effective stress;

σ

total stress;

(σ − u_a)

net normal stress;

${\sigma }_{ij}^{\ast }$

stress tensor;

τ

shear strength;

θ

general angle between θ₀ and θ_h; and

θ₀, θ_h, θ_B, θ_C, θ_D

angles to describe the 3D failure mechanism, which are illustrated in Fig. 3.

References