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/
2
,
Q
3
r
z
1
2
rr
=
(12.21b)
.
+
5
2
R
R
(
R
z
)
Q
1
2
r
z
/
,
=
(12.21c)
.
+
2
2
R
R
z
R
2
3
Qrz
=
(12.21d)
5
2
R
Q
R
z
r
Figure12.7 Boussinesq's problem
Corresponding strains and deformations can be determined by using the elastic
stress-strain relation, according to (5.2). Elaboration shows that the vertical
displacement
w
due to the load
Q
in the centre becomes
/
2
,
Q
(
)
z
2
)
w =
(12.22)
.
+
2
2
ER
R
This expression shows that under the load at the surface the displacement
becomes infinite, which is a practical inconvenience. In practice, (12.21a) is used
in the following form
z
2
with
N =
3/2 (1+
r
2
/z
2
)
5/2
v
= QN/
(12.23)
If one assumes an influence area in the underground limited by an expanding
cone under 45
o
, implying that at depth
z
the load is mainly carried in a zone
restricted to
z
2
, then
N =
1
.
Boussinesq's solution can be used for superposition.
For a uniform load
q
in a circle with radius
r
0
and the vertical displacement in the
centre becomes
2
)
/E =
2
Q/
2
)
w =
2
qr
0
(1
r
0
E
e
with
E
e
= E/
(1
(12.24a)
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