New Analytical Solutions for Cohesive–Frictional Soils above Deep Active Trapdoors

Acknowledgments

Notation

B

width of trapdoor (m);

B_z

width of horizontal soil layer element at depth z (m);

c

soil cohesion (kPa);

dW

gravity of soil-layer element (kPa);

E

Young’s modulus (MPa);

H

height of covered soils (m);

H₁

height of upper vertical zone in external failure mode (m);

H₂

height of lower spiral zone in external failure mode (m);

K₀

coefficient of earth pressure at rest;

K_a

Rankine active earth-pressure coefficient;

K_h

lateral earth-pressure coefficient;

${\overline{K}}_n$

modified lateral earth-pressure coefficient;

K_p

Rankine passive earth-pressure coefficient;

O

pole of polar coordinate;

P

surcharge on the fill surface (kPa);

P_u

multiplier downward load (kPa);

r

rotation radius (m);

r_a, r_b

end and start points of rotation radius (m);

v

Poisson’s ratio;

α

angle between slip surface and horizontal surface (°);

α₀

initial rupture angle (°);

ΔP

fixed upward load (kPa);

γ

unit weight of covered soil (kN/m³);

η

rotation angle of major principal stress (°);

η_w

rotation angle of major principal stress exerted on the slip surfaces (°);

θ

rotation angle of logarithmic spiral (°);

ρ

soil-arching ratio;

σ₁, ${\sigma }_1^{\ast }$

major principal stresses in original and new coordinates (kPa);

σ₃, ${\sigma }_3^{\ast }$

minor principal stresses in original and new coordinates (kPa);

σ_h, σ_v

horizontal and vertical stresses in the soil element (kPa);

σ_nw, ${\sigma }_{nw}^{\ast }$

normal stresses on the slip surface in original and new coordinate (kPa);

${\overline{\sigma }}_v$

average vertical stress (kPa);

τ_w, ${\tau }_w^{\ast }$

shear stresses on the slip surface in original and new coordinates (kPa);

τ_xy

shear stress in soil element (kPa); and

ϕ

soil friction angle (°).

References