2146
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
determined using the theory of elasticity, which may affect the
validity of the getting result. Ignoring of this fact can lead to
getting while calculation overstated or, conversely, understated
safety factors that may result in additional costs of material
sources, or emergency initiation.
co
des the minimum width of the
sti
s applied to
al solution, coincided with the maximum degree of
ac
re
2. PROPOSED APPROACH
The following algorithm for calculating the value of the slope
safety factor K of loaded soil slope, based on the joint
application of the finite element method (FEM) and methods of
mplex function theory (Kolosov 1934, Muskhelishvili, 1966):
a) design scheme of FEM is made up provided with
maximum possible degree of homogeneity of the finite element
mesh observance, which provi
ffness matrix of the system;
b) based on an analytical solution of the first fundamental
boundary value problem of elasticity theory for a homogeneous
isotropic half-plane with a curvilinear boundary (Bogomolov
1996) data handling of boundary condition
uniform
FEM scheme so that the stress values
zx z x
;
;
at the
corresponding points of the investigating region, found by FEM
and analytic
curacy;
c) in the case of heterogeneous geological structure of the
slope, therein geological elements endowing with corresponding
physical and mechanical properties (solid weight
, specific
cohesion C; angle of internal friction
, coefficient of elasticity
E; coefficient of lateral pressure
ξ
(Poisson's
о
ratio
)) are
ex
oint of the soil mass
determined by the following expression
tracted;
d) to the study area
n
external loads are applied, which
according to all parameters coincide with loads specified in
problem specification, but the values of their intensities qn
0.
If in the near the slope area plastic deformation regions are
absent, the construction of most likely slip surface is performed
by the method (Tsvetkov 1979), ensuring fulfillment of the
condition of minimum value of K at each point. At the same
time the safety factor value K
i
in the i-th p
is
i
ixz
i
iz
ix
tg
i
ixz
i
ixz
i
iz
ix
tg iz
ix
i
ix
iz
iK
2 cos
sin)
(5,0
]св 2 2 sin
[
2 cos
sin)
(5,0
)]
(5,0 2 cos )
(5,0[
(2)
where:
св
=
C/tg
- cohesion pressure, numerator and
denominator of the formula (2) determine, respectively, the
numerical values of restraining
F
уд
i
and shearing
F
сдi
forces
acting in the i-th point of the slip surface along the most likely
site of shift. Analysis of formula (2) shows that the value of
K
i
depends on the components of the stress and angle
i
most
likely site shift at each point of sliding surface, as well as
physical properties of soil.
The value of
i
is defined by formulas (4), which arise from
the
ary condition (3)
bound
.02
2
;0
K
K
2 1
)
(
2
2 sin
N
L
xi
yi
xyiN i
(4)
here:
of
pproximate solutions of the mixed problem of elasticity theory
and plasticity theory of soil (Bogomolov 1996). In this case, the
stress at the plastic region (marked by a prime) are given by
w
; 2 4 2)
(
xyi
yi
xi
L
1 )св 2
(
xi
yi
N
If the plastic deformation regions have been developed, but their
sizes are small, then the construction of the most probable slip
urface is carried out by a method built on the basis
s
a
xi
,
sin
sin )св
(
;
sin
sin св 2 ) sin (
;
l
b
zi
z
l
l
zi
xi
zi
zi
(5)
lue of the safety factor coefficient
at the point of
the slope, which is in the plastic area (obv l
where
).
/ (
;5,0)2 1( ; 2
i
i
b
l i
tg b
The va
2/
4
iK
ious y
1
iK
) is
given by
b
l
b
calculated. Note that construction of th
ed out with such a step, that the difference
be
nd shearing forces acting at the same
m
points of slip surface.
The formula for calculating the safety factor valu
provided that the plastic deformation is absent, ha e rm
cos )2 2 cos
)(
(
св
) 2 sin
2 (cos
sin )св
(
i
b св
zi
zi
i
i
iz
iK
.(6)
The numerator and denominator of formula (6) determine
values of appropriate restraining and shearing forces. In m
points of sliding surface with the help of FEM total
displacements from the gravity forces and the external load are
e most probable slip
surface is carri
tween the horizontal coordinates of neighboring points on it is
0,01H
1
.........
1
mx
mx
kx
kx
(where
H
- the height
of the slope).
e) the value of the intensity of the external load is increased
to the calculated value. Once again we construct the most
probable slip surface and the calculation of the total
displacements in its
m
points.
f) the principle of virtual displacements is used, at the same
time the role of the possible movements is carried out by
differences
of movements in corresponding
m
points of the
sliding surface, obtained under the use of two previous steps of
e algorithm. The role of active forces is played by restraining
th
a
e of the slope,
s th fo
(3)
,
)
cos )
(
(
сдi
i
i
сдiF
сдiA
K
(7)
)
cos )
(
(
удi
i
i
удiF удiA
