Geotechnical Engineering

(Jeff_L) #1
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)



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