Element Failure Risk Assessment of Soil Slope

Acknowledgments

Notation

A^e

area of element e (m²);

a_b

coordinate transformation matrix of finite element b on the boundary;

$a_d^1$

matrix of plastic flow constraint conditions of velocity d;

$a_d^2$

matrix of plastic flow constraint conditions of velocity d;

$a_e^1$

matrix of plastic flow constraint conditions of finite element e;

$a_e^2$

matrix of plastic flow constraint conditions of finite element e;

C_e

failure consequence coefficient of finite element e;

C_f

failure consequence coefficient of the slope;

C_i

failure consequence coefficient of the ith failure mode;

I[t_m]

failure function of the slope that corresponded to the random number of the shear t_mth parameter;

I_e[t_m]

failure function of finite element e that corresponded to the random number of the shear t_mth parameter;

k_m

random variable of safety factor;

$k_{\gamma }$

random variable of overload factor of volume weight;

N_b

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

N_d

number of velocity d in the soil slope;

N_e

number of finite elements in the soil slope;

N_f

number of failure modes;

N_m

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

P[k_m(x)]

failure probability of the failure mode that corresponded to the safety factor of k_m(x);

P_f

failure probability of the slope;

P_f,e

failure probability of finite element e;

P_f,i

failure probability of ith failure mode;

P_f,e∈Part1

failure probability of finite element in Part 1;

P_f,Part1

failure probability in Part 1;

P_f,e∈Part2

failure probability of finite element in Part 2;

P_{f,Part 2}

failure probability in Part 2;

P_f,e∈Part3

failure probability of finite element in Part 3;

P_f,Part3

failure probability in Part 3;

P_f,e∈Part4

failure probability of finite element in Part 4;

P_f,Part4

failure probability in Part 4;

P_f,e∈Partx

failure probability of elements that included all x-failure modes;

$p_i^e$

pore water pressure of triangular element e;

R

failure risk of the slope;

R_e

failure risk of finite element e;

R_i

failure risk of ith failure mode;

${\mathbf{u}}_b$

velocity vector of boundary finite element b;

${\mathbf{u}}_d$

velocity vector of velocity d;

${\mathbf{u}}_e$

velocity vector of finite element e;

$v_c^e$

the centroid velocity of finite element e;

$v_{xk}^d$

velocity of the kth node (k = 1, 2, 3, 4) on velocity d plane in x-direction;

$v_{xj}^e$

velocity of nodes j (j = 1, 2, 3) in finite element e in x-direction;

$v_{yk}^d$

velocity of the kth node (k = 1, 2, 3, 4) on velocity d plane in y-direction;

$v_{yj}^e$

velocity of nodes j (j = 1, 2, 3) in finite element e in y-direction;

${\mathbf{W}}_e$

internal power of finite element;

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

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

${\mathbf{W}}_d$

internal power of velocity d;

${\mathbf{W}}_d^p$

external work power exerted by pore water pressure on velocity d;

${\mathbf{W}}_G$

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

Z

limit state function of slope;

${\mathbf{\lambda }}_d$

vector of nonnegative plastic multiplier of velocity d; and

${\mathbf{\lambda }}_e$

vector of nonnegative plastic multiplier of finite element e.

References