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)