Civil Engineering Reference
In-Depth Information
where
′
cos
β
−
cos
2
β
−
cos
2
φ
K
a
=
cos
β
′
cos
β
+
cos
2
β
−
cos
2
φ
Example 7.1:
Rankine active thrust
Using the Rankine theory, determine the total active thrust on a vertical retaining wall
5 m high if the soil retained has a horizontal surface level with the top of the wall and
has the following properties:
φ
′
=
35°;
γ
=
19 kN/m
3
.
What is the increase in horizontal thrust if the soil slopes up from the top of the wall
at an angle of 35° to the horizontal?
Solution:
1.
Solution A: Soil surface horizontal
1
−
sin
sin
35
°
K
a
=
°
=
0 271
.
1
+
35
Maximum p
a
= × ×
19 5 0 271 25 75
.
=
.
kPa
Thrust
=
area of pressure diagram
25 75 5
2
.
×
=
=
64
kN
2.
Solution B: Soil sloping at 35°
In this case,
β
=
φ
′
. When this happens the formula for K
a
reduces to K
a
=
cos
φ
′
.
Hence
K
a
=
cos
35
° =
0 819
.
K
h
2
5
2
2
Thrust
=
γ
=
0 819 19
.
× × =
194 5
.
kN
a
2
This thrust is assumed to be parallel to the slope, i.e. at 35° to the horizontal.
Horizontal component
=
194 5
.
×
cos
35
° =
159
kN
Increase in horizontal thrust
=
95
kN/m length of wall
7.3.3 Point of application of the total active thrust
We have seen that the total active thrust, P
a
, is given by the expression:
=
1
2
2
P
γ
h K
a
a
where K
a
is the respective value of the coefficient of active earth pressure, h
=
height of wall and
γ
=
unit
weight of retained soil.
The position of the centre of pressure on the back of the wall, i.e. the point of application of P
a
, is
largely indeterminate. Locations suitable for design purposes are shown in Fig.
7.6
and are based on the
Rankine theory (with its assumption of a triangular distribution of pressure).
