Three-Dimensional Seismic Bearing Capacity Assessment of Heterogeneous and Anisotropic Slopes

Acknowledgments

Notation

A_j

element area of the surface $P_{max,j}{P}_{max,j+1}{P}_{max,j+1}^{\mathrm{\prime }}{P}_{max,j}^{\mathrm{\prime }}$;

a_h0 and a_v0

horizontal and vertical seismic accelerations at the slope toe;

B

width of the slope failure mechanism;

b

width of plane insert;

CF and C′F

boundaries of failure mechanism;

C_s, C_sz, S_s, S_sz, y_s1, y_s2, C_p, S_p, C_pz, S_pz, y_p1 and y_p2

dimensionless functions;

c

soil cohesion;

c_h0 and c_ht

horizontal principal cohesions at the slope crest and the slope toe;

c_j

horizontal cohesion at the depth z_j;

c_v

vertical principal cohesion;

c_ɛi,j and $c_{\epsilon i,j}^{\mathrm{\prime }}$

soil cohesions at the points P_i,j and $P_{i,j}^{\mathrm{\prime }}$;

g

gravity acceleration;

H

slope height;

h_i,j and $h_{i,j}^{\mathrm{\prime }}$

vertical distances from points P_i,j and $P_{i,j}^{\mathrm{\prime }}$ to the slope surface;

K_c

anisotropic cohesion coefficient;

k_h and k_v

horizontal and vertical seismic acceleration coefficients at the slope toe;

$\overrightarrow{n_j}$ and $\overrightarrow{{n^{\mathrm{\prime }}}_j}$

unit normal vectors of the vectors $\overrightarrow{P_j{P}_{j+1}}$ and $\overrightarrow{{P^{\mathrm{\prime }}}_j{P^{\mathrm{\prime }}}_{j+1}}$;

P_i,j, P_i+1,j, P_i,j+1, P_i+1,j+1, Q_i,j, Q_i+1,j, Q_i,j+1 and Q_i+1,j+1

points at the failure surface;

R_i,j and $R_{i,j}^{\mathrm{\prime }}$

distances between the centers of the elements P_i,jP_i+1,jP_i+1,j+1 and $P_{i,j}^{\mathrm{\prime }}{P}_{i,j+1}^{\mathrm{\prime }}{P}_{i+1,j+1}^{\mathrm{\prime }}{P}_{i+1,j}^{\mathrm{\prime }}$ to the rotation axis;

P_j and $P_j^{\mathrm{\prime }}$

points at the discretized boundaries of the failure surface;

P_max,j and P_max,j

last points of the failure surface at the radial plane ψ_j and the radial plane ψ_j;

q_cr

ultimate bearing capacity;

q_j

surcharge acting on the surface A_j;

$R_j^{\mathrm{\prime }}$

radius of the radial plane ψ_j;

r_F

rotation radius of point F;

r_h and $r_h^{\mathrm{\prime }}$

rotation radius of points C and C′;

r_j, $r_j^{\mathrm{\prime }}$, r_j+1 and $r_{j+1}^{\mathrm{\prime }}$

rotation radius of points P_j, $P_j^{\mathrm{\prime }}$, P_j+1, and $P_{j+1}^{\mathrm{\prime }}$;

r_u

pore–water pressure coefficient;

r₀ and $r_0^{\mathrm{\prime }}$

rotation radius of points A and A′;

S_h and S_p

areas of surfaces for the horn failure part and plane-strain insert part;

S_i,j and $S_{i,j}^{\mathrm{\prime }}$

areas of the surfaces P_i,jP_i+1,jP_i+1,j+1 and $P_{i,j}^{\mathrm{\prime }}{P}_{i,j+1}^{\mathrm{\prime }}{P}_{i+1,j+1}^{\mathrm{\prime }}{P}_{i+1,j}^{\mathrm{\prime }}$;

S_m

area of the failure surface at the slope crest;

S_r

area of surface at the slope crest and the slope surface;

S_t

area of the velocity discontinuity surface bounding the failure mechanism;

T

period of the harmonic seismic acceleration;

t

time;

u

pore–water pressure;

u_hb and u_vb

horizontal and vertical seismic accelerations at the slope base;

u_hs(z_i, t) and u_vs(z_i, t)

horizontal and vertical displacement at depth z_i and time t;

u_h0 and u_v0

horizontal and vertical seismic accelerations at the slope toe;

V_i,j

volume of the element P_i,jP_i+1,jP_i+1,j+1−Q_i,jQ_i+1,jQ_i+1,j+1 (or P_i,jP_i,j+1P_i+1,j+1−Q_i,jQ_i,j+1Q_i+1,j+1) (Fig. 4);

V_s and V_p

velocity of shear and primary wave propagating;

$\overrightarrow{v_j}$ and $\overrightarrow{{v^{\mathrm{\prime }}}_j}$

unit normal vectors of vectors $\overrightarrow{O{P}_j}$ and $\overrightarrow{O{P^{\mathrm{\prime }}}_j}$;

W_D

internal energy dissipation rate;

W_q

work rate of the static surcharge at the slope crest;

W_qs

work rate of the earthquake-induced inertia forces from the surcharge on the slope crest;

W_u

work rate of pore–water pressure;

W_γ

work rate of the soil weight;

W_γs

work rate of seismic forces;

z_i and z_j

depth from points P_i and P_j to the slope crest;

α_i

angle between line $C_j^{\mathrm{″}}{P}_{i,j}$ with the negative y axis at the local coordinate axis;

β

slope angle;

δθ

angle between lines OP_j and OP_j+1 or lines $O{P}_j^{\mathrm{\prime }}$ and $O{P}_{j+1}^{\mathrm{\prime }}$;

$\overrightarrow{{\delta }_{j+1}}$

unit vector of the vector $\overrightarrow{O{P}_{j+1}}$;

ɛ_i,j

angle between the major principal stress σ₁ with the vertical direction;

γ

soil unit weight;

η

ratio between the vertical seismic acceleration coefficient k_v and the horizontal seismic acceleration coefficient k_h (k_v = ηk_h);

φ

internal friction angle;

φ_i,j and ${\phi }_{i,j}^{\mathrm{\prime }}$

friction angles at points P_i,j and $P_{i,j}^{\mathrm{\prime }}$;

φ_j

friction angle at the depth z_j;

φ₀ and φ_t

friction angles at the slope crest and the slope toe;

λ

angle between the plane perpendicular to major principal stress σ₁ and failure surface;

θ_Gi,j and R_Gi,j

polar coordinates of the barycenter corresponding to the volume V_i,j;

θ_i,j

angle between the point P_i,j and rotation axis;

θ₀, θ_h, θ_j and θ_F

angles between lines OA, OC, OP_j and OF with the horizontal direction;

σ₁

major principal stress;

ω

angular velocity of the failure mechanism;

ϖ

angle velocity of the shear/primary wave;

ξ

soil damping ratio; and

ψ_j

radial plane at the failure surface.

References