Geotechnical Engineering

DHARM

STABILITY OF EARTH SLOPES 321

Effective normal stress σn = (σn – u)

= (γz cos^2 β – u)

(γ is the average unit weight of the slice, which is usually considered saturated.)

Shear stress τ =

W

l

z

sin

sin cos

β

=γββ ...(Eq. 9.3)

Shear strength of soil = σφntan

= (γ z cos^2 β – u) tan φ

Factor of safety against slippage, F =

Shear strength

Shear stress

∴ F =

(cos )tan

sin cos

γβ φ

γββ

zu

z

(^2) −
= [(1/tan β) – (u/γ z sin β cos β)]. tan φ
=^1 − 2
F
HG
I
KJ
u
γβz
φ
cos β
tan
tan
or F =^1 − 2
F
HG
I
KJ
ru
cos
.tan
β tan
φ
β
...(Eq. 9.4)
where ru = u/γ z ...(Eq. 9.5)
ru is called the ‘pore pressure ratio’.
Flow Parallel to the Surface and at the Surface of a Slope in Cohesionless Soil
If there is a flow parallel to the surface and at the surface at a slope in the cohesionless
soil, the flow net is very simple and is depicted in Fig. 9.4.
Equipotentials
b
Flow line
Q
b
P
hw z
Fig. 9.4 Flow parallel to the surface and at the surface
The excess pore water pressure at the centre P of the base of the element, similar to the
previous case, expressed as a head, is represented by the height hw. From the figure, PQ = z cos β,
hw = PQ. cos β
∴ hw = z. cos^2 β
The excess pore water pressure u = γw hw = γw z cos^2 β
∴ ru = u/γ. z =
γβ
γ
w γγw β
z
z
cos
(/).cos
2
=^2 ...(Eq. 9.6)