Civil Engineering Reference
In-Depth Information
2
−
′
cosec
ψ
sin(
ψ φ
)
K
p
=
′
+
′
+
sin(
φ
δ
) sin(
φ
β
)
sin(
ψ δ
− −
)
sin(
ψ β
−
)
the symbols having the same meanings as previously.
The expression reduces to:
′
1
1
+
sin
sin
φ
φ
K
p
=
−
′
when
ψ
=
90°,
δ
=
0° and
β
=
0°.
With passive pressure, unfortunately, the failure surface only approximates to a plane surface when the
angle of wall friction is small.
The situation arises because the behaviour of the soil is not only governed by its weight but also by
the compression forces induced by the wall tending to push into the soil. These forces, unlike the active
case, do not act on only one plane within the soil, resulting in a non-uniform strain pattern and the devel-
opment of a curved failure surface (Fig.
7.20)
.
It is apparent that in most cases the assumption of a Coulomb wedge for a passive failure can lead to
a serious overestimation of the resistance available. Terzaghi (
1943
) first analysed this problem and con-
cluded that, provided the angle of friction developed between the soil and the wall is not more than
φ
′
/3,
where
φ
′
is the operative value of the angle of shearing resistance of the soil, the assumption of a plane
failure surface generally gives reasonable results. For values of
δ
greater than
φ
′
/3, the errors involved can
be very large.
Adjusted values for K
p
that allow for a curved failure surface are given in Table
7.2.
These values apply
to a vertical wall and a horizontal soil surface and include the multiplier cos
δ
as the values in the table
give the components of pressure that will act normally to the wall.
Fig. 7.20
Departure of passive failure surface from a plane.
Table 7.2
Values of K
p
for cohesionless soils (Kerisel
and Absi,
1990)
.
Values of
φ
′
25°
30°
35°
40°
Values of
δ
Values of K
p
0°
2.5
3.0
3.7
4.6
10°
3.1
4.0
4.8
6.5
20°
3.7
4.9
6.0
8.8
30°
-
5.8
7.3
11.4
