Formula of Passive Earth Pressure and Prediction of Tunneling-Induced Settlement in Anisotropic Ground Based on a Simple Method

Acknowledgments

Notation

$\overline{c}$

cohesion in terms of ${\overline{\sigma }}_{ij}$;

c_v

cohesion along the vertical direction;

$D_{ijkl}^e$, $D_{ijkl}^{ep}$

elastic and elastoplastic stiffness matrices, respectively;

e₀

initial void ratio;

F_ij

fabric tensor;

f

yield function;

f_ani

influence coefficient of soil anisotropy;

g

plastic potential function;

K, G

bulk and shear moduli, respectively;

K_p

Rankine’s passive earth pressure coefficient;

M

critical state stress ratio;

$\tilde{M}$

critical state stress ratio in terms of ${\tilde{\sigma }}_{ij}$;

$\overline{p}$, $\overline{q}$

mean and deviatoric stresses of ${\overline{\sigma }}_{ij}$, respectively;

$\tilde{p}$, $\tilde{q}$

mean and deviatoric stresses of ${\tilde{\sigma }}_{ij}$, respectively;

${\tilde{p}}_{x0}$

preconsolidation pressure in terms of ${\tilde{\sigma }}_{ij}$;

${\overline{q}}_c$

deviatoric stress of ${\overline{\sigma }}_{ij}$ in triaxial compression;

S_max

maximum surface settlement;

α

angle between the depositional plane and the horizontal plane;

Δ

parameter that quantifies the degree of soil anisotropy;

δ_ij

Kronecker delta;

${\epsilon }_{ij}^p$

plastic strain tensor;

${\epsilon }_v^p$

plastic volumetric strain;

ɛ_v, ɛ_d

total volumetric and deviatoric strains, respectively;

γ

unit weight of soil;

$\tilde{\eta }$

stress ratio in terms of ${\tilde{\sigma }}_{ij}$;

κ

slope of the swelling line;

Λ

plastic index;

λ

slope of the normal consolidation line;

ν

Poisson’s ratio;

σ_ij

ordinary stress tensor;

${\overline{\sigma }}_{ij}$

modified stress tensor;

${\tilde{\sigma }}_{ij}$

transformed stress tensor;

σ_p

passive earth pressure;

σ_z, σ_x, σ_y

normal stress components of σ_ij along the z-, x-, and y-axes, respectively;

${\overline{\sigma }}_z$, ${\overline{\sigma }}_x$, ${\overline{\sigma }}_y$

normal stress components of ${\overline{\sigma }}_{ij}$ along the z-, x-, and y-axes, respectively;

σ₁, σ₂, σ₃

major, intermediate, and minor principal values of σ_ij, respectively;

${\overline{\sigma }}_1$, ${\overline{\sigma }}_2$, ${\overline{\sigma }}_3$

major, intermediate, and minor principal values of ${\overline{\sigma }}_{ij}$, respectively;

${\overline{\tau }}_{zx}$, ${\overline{\tau }}_{xz}$

shear stress components of ${\overline{\sigma }}_{ij}$ in the z–x plane;

$\overline{\phi }$

internal friction angle in terms of ${\overline{\sigma }}_{ij}$;

φ_h

internal friction angle along the horizontal direction; and

φ_v

internal friction angle along the vertical direction.

References