Civil Engineering Reference
In-Depth Information
stress and that when considering passive pressure, the vertical pressure due to the soil weight,
γ
h, is a
minor principal stress.
The two major theories to estimate active and passive pressure values are those by Rankine (
1857)
and
by Coulomb (
1776)
. Both theories are very much in use today and both are described below.
7.3 Rankine's theory: granular soils, active earth pressure
7.3.1 Horizontal soil surface
Imagine a smooth (i.e. no friction exists between wall and soil), vertical retaining wall holding back a
cohesionless soil with an angle of shearing resistance
φ
′
. The top of the soil is horizontal and level with
the top of the wall. Consider a point in the soil at a depth h below the top of the wall (Fig.
7.2)
, assuming
that the wall has yielded sufficiently to satisfy active earth pressure conditions.
From the Mohr circle diagram (Fig.
7.3)
it is seen that:
DC
OC
DC
OC
1
−
′
K h
h
γ
OA
OB
OC AC
OC CB
−
+
OC DC
OC DC
−
+
1
1
−
sin
φ
a
K
=
=
=
=
=
=
a
sin
φ
′
γ
+
1
+
It can be shown by trigonometry that
1
1
−
sin
sin
φ
φ
′
φ
′
′
=
tan
2
45
°−
+
2
hence
′
1
1
−
sin
sin
φ
φ
φ
2
K
a
=
′
=
tan
45
°−
+
2
The Mohr circle diagram can be extended to identify the direction of the major principal stress, using
the geometry indicated in Figure
7.4a
. The angle that the failure plane makes with the back of the wall
is clear from the figure and the network of shear planes formed behind the wall is illustrated in Fig.
7.4b
.
The lateral pressure acting on the wall at any depth,
′
σ
3
, is equal to
K
a
′
σ
1
. As the vertical effective stress
′
=
σ
1
h
and K
a
is a constant, the lateral pressure increases linearly with depth (Fig.
7.4c
). This lateral pres-
sure is of course the active pressure. This pressure is given the symbol p
a
, and is defined:
p
=
γ
K h
a
The magnitude of the resultant thrust, P
a
, acting on the back of the wall is the area of the pressure
distribution diagram. This force is a line load which acts through the centroid of the pressure distribution.
In the case of a triangular distribution, the thrust acts at a third of the height of the triangle.
7.3.2 Sloping soil surface
The problem of the ground surface behind the wall sloping at an angle
β
to the horizontal is illustrated
in Fig.
7.5.
The evaluation of K
a
may be carried out in a similar manner to the previous case, but the verti-
cal pressure will no longer be a principal stress. The pressure on the wall is assumed to act parallel to the
surface of the soil, i.e. at angle
β
to the horizontal.
The active pressure, p
a
, is still given by the expression:
p
=
γ
K h
a
