Environmental Engineering Reference
In-Depth Information
Figure 3.7.
Case of an infinitely long slope with flow parallel to the slope
In defining the safety factor
F
as the relationship between the ultimate shear
stress τ
f
and the shear stress necessary to the equilibrium, τ,
acting on the potential
failure surface at depth
d
:
τ
τ
f
F
=
[3.6]
the analysis of the equilibrium of a vertical slice of soil leads to the following simple
relationship:
2
(
)
c
'
+
⎡
ρ
gh
+
ρ
'
g d
−
h
⎤
cos
α ϕ
tg
'
⎣
⎦
w
w
F
=
[3.7]
(
)
⎡
ρ
gh
+
ρ
g d
−
h
⎤
sin
α α
cos
⎣
⎦
wsat
w
with the dimensional parameters as given in Figure 3.7 and with ρ the apparent
density, ρ
sat
the saturated density, and with ρ
'=
ρ
sat
−ρ
w
The mass is stable if
F >1
.
Expression [3.7] gives various pieces of information, and in particular:
- for a cohesive material, failure (
F = 1
) will occur at a certain depth
d
; above
this depth, the material is rigid and not plastified; this is not true for a purely
frictional material (
c' = 0
), for which the entire mass plastifies at any depth;
- the depth of the groundwater
h
w
, noticeably influences the stability; the higher
the groundwater (
h
w
small), the more the safety factor is reduced; this is a common
cause for triggering instability;
- for a non-cohesive material (
c' = 0
) and groundwater at terrain level (
h
w
= 0
),
the safety factor is simply written:
