DHARM
STRESS DISTRIBUTION IN SOIL 367
δσz =
3
2
3
2252
(. )
.
()/
qA z
rz
δ
π +
The stress dσz due to the entire ring is given by:
dσz =
3
2
3
2
2
3
2252
3
2252
q
A
z
rz
q
rdr
z
ππrz
(). π
()
().
//()
Σδ
+
=×
+
∴ dσz =
3 3
2252
qz rdr
rz
.
()+ /
qq
z
A
a
O
q/unit area
q. Ad
r dr
Fig. 10.12 Uniform load over circular area
The total vertical stress σz at A due to entire loaded area is obtained by integrating d σz
within the limits r = 0 to r = a.
∴σz =^3 qz^302252
rdr
r rz
ra
= ()+ /
=
z
Setting r^2 + z^2 = R^2 , rdr = R.dR, the limits for R will be z and (a^2 + z^2 )1/2.
∴σz = (^334)
2212
qz
dR
Rz R
Raz
=+
z
()/
= qz
zaz
3
32232
11
−
- L
N
M
M
O
Q
P
()/ P
∴σz = q
az
1 1
1 232
−
L
N
M
M
O
Q
P
{(/)}/ P
...(Eq. 10.36)
This may be written as:
σz = q.KBC ...(Eq. 10.37)
where KBC = Boussinesq influence coefficient for uniform load on circular area,
and, KBC =^1
1
1 232
−
L
N
M
M
O
Q
P
{(/)}az / P
...(Eq. 10.38)