Element Failure Probability of Soil Slope under Consideration of Random Groundwater Level

Acknowledgments

Notation

${\mathbf{A}}^b$

coordinate transformation matrix of the finite element b on the boundary;

${\mathbf{A}}_1^e$

matrix of plastic flow constraint conditions of the finite element e;

${\mathbf{A}}_2^e$

matrix of plastic flow constraint conditions of the finite element e;

${\mathbf{A}}_1^d$

matrix of plastic flow constraint conditions of the velocity discontinuity d;

${\mathbf{A}}_2^d$

matrix of plastic flow constraint conditions of the velocity discontinuity d;

c^r

random variables of the cohesion of the soil materials;

c′^r

random quantity of cohesion after the intensity reduction;

c^r(t_m)

t_mth random number on the materials of the soil cohesion;

c′^r(t_m)

t_mth random number of the soil cohesion after strength reduction;

$H_w^r$

random variable of the groundwater level of the soil slope;

$H_w^r(t_w)$

t_wth random number of the groundwater level of the soil slope;

H_lb

lower bound of the groundwater level of the soil slope;

H_ub

upper bound of the groundwater level of the soil slope;

I_z(t_w, t_m)

failure function of the soil slope corresponding to the random number of the t_m the random shear parameter under the action of the t_wth groundwater level;

I_e(t_w, t_m)

failure function of the finite element e corresponding to the random number of the shear t_mth parameter under the action of the t_wth underground level;

n_b

quantity of the finite elements on the boundary of the soil slope;

n_d

quantity of the velocity discontinuities in the soil slope;

n_e

quantity of finite elements in the soil slope;

n_m

quantity of material for the soil cohesion and the friction angle of the Monte Carlo random number;

n_w

quantity of the Monte Carlo random numbers of the groundwater level of the soil slope;

${\mathbf{p}}_e^r$

pore water pressure vector of finite element e;

$p_{ei}^r(t_w)$

pore water pressure at nodes i (i = 1, 2, 3) in finite element e under the action of t_wth groundwater level;

$p_{ei}^r$

random variable of the pore water pressure at nodes i in finite element e;

$P_F^z$

IFP of the slope under the action of all possible groundwater levels;

$P_F^e$

failure probability of the finite element e in the slope under the action of all possible groundwater levels;

$P_f^e(t_w)$

failure probability of the finite element e in the soil slope under the action of the t_wth groundwater level;

$P_f^z(t_w)$

IFP of the slope under the action of the t_wth groundwater level;

${\mathbf{T}}^b$

transformation matrix of the finite element b on the boundary;

${\mathbf{T}}^d$

transformation matrix of the velocity discontinuity d;

${\mathbf{u}}^b$

velocity vector of the boundary finite element b;

${\mathbf{u}}^d$

velocity vector of the velocity discontinuity d;

${\mathbf{u}}^e$

velocity vector of finite element e;

$u_c^e(t_w,t_m)$

resultant velocity at the centroid of the finite element e corresponding to the random number of the shear t_mth parameter under the action of the t_wth underground water level;

$u_{xi}^d$

velocity of the ith [i = (1, …, 4)] node on the velocity discontinuity plane d along the x-direction;

$u_{yi}^d$

velocity of the ith [i = (1, …, 4)] node on the velocity discontinuity plane d along the y-direction;

$u_{xi}^e$

velocity of nodes i (i = 1, …, 3) in the finite element e along the x-direction;

$u_{xi}^e(t_w,t_m)$

velocity of node i [i = (1, 2, 3)] in the finite element e along the x-direction calculated using the random number c^r(t_m), φ^r(t_m) of the t_mth shear parameter under the action of the t_wth groundwater level;

$u_{yi}^e$

velocity of nodes i (i = 1, …, 3) in the finite element e along the y-direction;

$u_{yi}^e(t_w,t_m)$

velocity of node i [i = (1, 2, 3)] in the finite element e along the y-direction calculated by using the random number c^r(t_m), φ^r(t_m) of the t_mth shear parameter under the action of the t_wth groundwater level;

${\mathbf{W}}_{Ex1}$

external work power done by the dead weight on the velocity of the finite element nodes;

${\mathbf{W}}_{Ex2}$

external power done by concentrated force and distributed load at the velocity of the finite element nodes;

${\mathbf{W}}_{Ex3}^p$

external work power of the pore water pressure in the finite element continuous body;

${\mathbf{W}}_{Ex4}^p$

external work power done by pore water pressure on the finite element velocity discontinuities;

${\mathbf{W}}_{In1}$

internal power of finite elements;

${\mathbf{W}}_{In2}$

internal power of the velocity discontinuities;

Z

limit state function of the soil slope reliability;

γ_a

real volume weight of the soil material;

γ_c(c^r, φ^r, H^r)

random variable of the ultimate value of volume weight that relates to c^r, φ^r, and H^r when the soil reaches the limit state;

${\gamma }_c(c^r(t_m),{\phi }^r(t_m),H_w^r(t_w))$

ultimate volume weight of the soil slope in the limit state when it reaches the instability related to the t_mth random shear parameter under the action of t_wth groundwater level;

γ_e

volume weight of finite element e;

θ_d

inclination angle of the velocity discontinuity d;

θ_b

dip angle of the boundary;

${\mathbf{\lambda }}^d$

vector of nonnegative plastic multiplier of the velocity discontinuity d;

${\mathbf{\lambda }}^e$

vector of nonnegative plastic multiplier of finite element e;

${\lambda }_m^r$

random variable of the safety factor that relates to c^r, φ^r, and H^r;

λ_m(t_w, t_m)

safety factor of the random number corresponding to the t_mth random shear parameter under the action of the t_wth groundwater level;

${\lambda }_{\gamma }^r$

random variable of the overload factor of volume weight that relates to c^r, φ^r, and H^r;

λ_γ(t_w, t_m)

volume weight overload factor corresponding to the random number of the random shear parameter t_m under the action of the t_wth groundwater level;

μ_c

mean value of the material cohesion of the soil;

μ_w

mean groundwater level of the soil slope;

μ_φ

mean value of the friction angle of the soil material;

σ_c

standard deviation of the soil cohesion;

σ_w

standard deviation of the groundwater level of the soil slope;

σ_φ

standard deviation of the friction angle of the soil materials;

φ^r

random variables of the internal friction angle of the soil materials;

${\phi }_d^r$

random quantity of the internal friction angle of the velocity discontinuity plane d;

${\phi }_e^r$

random quantity of friction angle of the finite element e of soil slope;

φ^r(t_m)

t_mth random number of the friction angle of the soil material;

φ′^r

random quantity of the internal friction angle after the intensity reduction; and

φ′^r(t_m)

t_mth random number of the internal friction angle of soil after strength reduction.

References